ST. PETERSBURG STATE UNIVERSITY
THE INTERNATIONAL SOCIETY OF DYNAMIC GAMES
(Russian Chapter)
CONTRIBUTIONS TO GAME THEORY
AND MANAGEMENT
Volume VII
The Seventh International Conference
Game Theory and Management
June 26–28, 2013, St. Petersburg, Russia
Collected papers
Edited by Leon A. Petrosyan and Nikolay A. Zenkevich
St. Petersburg State University
St. Petersburg
2014
УДК 518.9, 517.9, 681.3.07
Contributions to game theory and management, vol. VII. Collected papers
presented on the Seventh International Conference Game Theory and Management /
Editors Leon A. Petrosyan, Nikolay A. Zenkevich. – SPb.: Graduate School of Management SPbSU, 2014. – 438 p.
The collection contains papers accepted for the Seventh International Conference
Game Theory and Management (June 26–28, 2013, St. Petersburg State University,
St. Petersburg, Russia). The presented papers belong to the field of game theory
and its applications to management.
The volume may be recommended for researches and post-graduate students of
management, economic and applied mathematics departments.
Sited and reviewed in: Math-Net.Ru and RSCI. Abstracted and indexed in: Mathematical Reviews, Zentralblatt MATH and VINITI.
c Copyright of the authors, 2014
c St. Petersburg State University, 2014
ISSN 2310-2608
Успехи теории игр и менеджмента. Вып. 7. Сб. статей седьмой международной конференции по теории игр и менеджменту / Под ред. Л.А. Петросяна и
Н.А. Зенкевича. – СПб.: Высшая школа менеджмента СПбГУ, 2014. – 438 с.
Сборник статей содержит работы участников седьмой международной конференции «Теория игр и менеджмент» (26–28 июня 2013 года, Высшая школа менеджмента, Санкт-Петербургский государственный университет, СанктПетербург, Россия). Представленные статьи относятся к теории игр и ее приложениям в менеджменте.
Издание представляет интерес для научных работников, аспирантов и студентов старших курсов университетов, специализирующихся по менеджменту,
экономике и прикладной математике.
Электронные версии серии «Теория игр и менеджмент» размещены в: Math-Net.Ru
и РИНЦ. Аннотации и ссылки на статьи цитируются в следующих базах данных:
Mathematical Reviews, Zentralblatt MATH и ВИНИТИ.
c Коллектив авторов, 2014
c Санкт-Петербургский государственный университет, 2014
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
On Monotonicity of the SM-nucleolus and the α-nucleolus . . . . . . . . . .
Sergei V. Britvin, Svetlana I. Tarashnina
8
Efficient Myerson Value for Union Stable Structures . . . . . . . . . . . . . . . . 17
Hua Dong, Hao Sun, Genjiu Xu
On the Inverse Problem and the Coalitional Rationality
for Binomial Semivalues of Cooperative TU Games . . . . . . . . . . . . . . . . . 24
Irinel Dragan
Stackelberg Oligopoly Games: the Model and the 1-concavity
of its Dual Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Theo Driessen, Aymeric Lardon, Dongshuang Hou
On Uniqueness of Coalitional Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Michael Finus, Pierre von Mouche and Bianca Rundshagen
Quality Level Choice Model under Oligopoly Competition
on a Fitness Service Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich
A Problem of Purpose Resource Use
in Two-Level Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Olga I. Gorbaneva, Guennady A. Ougolnitsky
Multicriteria Coalitional Model of Decision-making over the Set of
Projects with Constant Payoff Matrix in the Noncooperative Game
Xeniya Grigorieva
93
Differential Games with Random Duration:
A Hybrid Systems Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Dmitry Gromov, Ekaterina Gromova
Simulations of Evolutionary Models of a Stock Market . . . . . . . . . . . . . . 120
Gubar Elena
Equilibrium in Secure Strategies in the Bertrand-Edgeworth
Duopoly Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Mikhail Iskakov, Alexey Iskakov
Equilibrium Strategies in Two-Sided Mate Choice Problem
with Age Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Anna A. Ivashko, Elena N. Konovalchikova
Stationary State in a Multistage Auction Model . . . . . . . . . . . . . . . . . . . 151
Aleksei Y. Kondratev
4
Phenomenon of Narrow Throats of Level Sets
of Value Function in Differential Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Sergey S. Kumkov, Valerii S. Patsko
Strictly Strong (n − 1)-equilibrium in n-person
Multicriteria Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Denis V. Kuzyutin, Mariya V. Nikitina, Yaroslavna B. Pankratova
The Nash Equilibrium in Multy-Product Inventory Model . . . . . . . . . . 191
Elena A. Lezhnina, Victor V. Zakharov
Nash Equilibria Conditions for Stochastic Positional Games . . . . . . . . 201
Dmitrii Lozovanu, Stefan Pickl
Pricing in Queueing Systems M/M/m with Delays . . . . . . . . . . . . . . . . . 214
Anna V. Melnik
How to arrange a Singles’ Party: Coalition Formation
in Matching Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Joseph E. Mullat
Evolution of Agents Behavior in the Labor Market . . . . . . . . . . . . . . . . . 239
Maria A. Nastych, Nikolai D. Balashov
An Axiomatization of the Proportional Prenucleolus . . . . . . . . . . . . . . . . 246
Natalia Naumova
Competition Form of Bargaining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Tatyana E. Nosalskaya
Interval Obligation Rules and Related Results . . . . . . . . . . . . . . . . . . . . . . 262
Osman Palancı, Sırma Zeynep Alparslan Gök, Gerhald Wilhelm Weber
Stable Cooperation in Graph-Restricted Games . . . . . . . . . . . . . . . . . . . . 271
Elena Parilina, Artem Sedakov
Power in Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
Rodolfo Coelho Prates
Completions for Space of Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
Victor V. Rozen
Bridging the Gap between the Nash and Kalai-Smorodinsky
Bargaining Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
Shiran Rachmilevitch
Unravelling Conditions for Successful Change Management
Through Evolutionary Games of Deterrence . . . . . . . . . . . . . . . . . . . . . . . . 313
Michel Rudnianski, Cerasela Tanasescu
Applying Game Theory in Procurement. An Approach
for Coping with Dynamic Conditions in Supply Chains . . . . . . . . . . . . 326
Günther Schuh, Simone Runge
5
An Axiomatization of the Myerson Value . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
Özer Selçuk, Takamasa Suzuki
Multi-period Cooperative Vehicle Routing Games . . . . . . . . . . . . . . . . . . 349
Alexander Shchegryaev, Victor V. Zakharov
Mechanisms of Endogenous Allocation of Firms and Workers
in Urban Area: from Monocentric to Polycentric City . . . . . . . . . . . . . . 360
Alexandr P. Sidorov
The Irrational Behavior Proof Condition for Linear-Quadratic
Discrete-time Dynamic Games with Nontransferable Payoffs . . . . . . . . 384
Anna V. Tur
Von Neumann-Morgernstern Modified Generalized Raiffa
Solution and its Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
Radim Valenčík, Ondřej Černík
Subgame Consistent Cooperative Solution of Stochastic
Dynamic Game of Public Goods Provision . . . . . . . . . . . . . . . . . . . . . . . . . . 404
David W.K. Yeung, Leon A. Petrosyan
Joint Venture’s Dynamic Stability with Application
to the Renault-Nissan Alliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Nikolay A. Zenkevich, Anastasia F. Koroleva
Symmetric Core of Cooperative Side Payments Game . . . . . . . . . . . . . 428
Alexandra B. Zinchenko
Preface
This edited volume contains a selection of papers that are an outgrowth of
the Seventh International Conference on Game Theory and Management with a
few additional contributed papers. These papers present an outlook of the current
development of the theory of games and its applications to management and various
domains, in particular, finance, mechanism design, environment and economics.
The International Conference on Game Theory and Management, a three day
conference, was held in St. Petersburg, Russia in June 26-28, 2013. The conference
was organized by Graduate School of Management St. Petersburg State University
in collaboration with The International Society of Dynamic Games (Russian Chapter) and Faculty of Applied Mathematics and Control Processes (SPbSU). More
than 100 participants from 26 countries had an opportunity to hear state-of-the-art
presentations on a wide range of game-theoretic models, both theory and management applications.
Plenary lectures covered different areas of games and management applications.
They had been delivered by Professor Finn Kydland, Nobel Prize in Economic
Sciences, 2004, University of California, Santa Barbara (USA); Professor Burkhard
Monien, Paderborn University (Germany); Professor Bernard De Meyer, Université
Paris 1, Panthéon-Sorbonne (France); Professor Leon Petrosyan, St. Petersburg
State University (Russia).
The importance of strategic behavior in the human and social world is increasingly recognized in theory and practice. As a result, game theory has emerged as a
fundamental instrument in pure and applied research. The discipline of game theory
studies decision making in an interactive environment. It draws on mathematics,
statistics, operations research, engineering, biology, economics, political science and
other subjects. In canonical form, a game takes place when an individual pursues
an objective(s) in a situation in which other individuals concurrently pursue other
(possibly conflicting, possibly overlapping) objectives and in the same time the objectives cannot be reached by individual actions of one decision maker. The problem
is then to determine each individual’s optimal decision, how these decisions interact
to produce equilibrium, and the properties of such outcomes. The foundations of
game theory were laid more than sixty years ago by von Neumann and Morgenstern
(1944).
Theoretical research and applications in games are proceeding apace, in areas
ranging from aircraft and missile control to inventory management, market development, natural resources extraction, competition policy, negotiation techniques,
macroeconomic and environmental planning, capital accumulation and investment.
In all these areas, game theory is perhaps the most sophisticated and fertile
paradigm applied mathematics can offer to study and analyze decision making under
real world conditions. The papers presented at this Seventh International Conference
on Game Theory and Management certainly reflect both the maturity and the
vitality of modern day game theory and management science in general, and of
7
dynamic games, in particular. The maturity can be seen from the sophistication of
the theorems, proofs, methods and numerical algorithms contained in the most of
the papers in these contributions. The vitality is manifested by the range of new
ideas, new applications, the growing number of young researchers and the expanding
world wide coverage of research centers and institutes from whence the contributions
originated.
The contributions demonstrate that GTM2013 offers an interactive program on
wide range of latest developments in game theory and management. It includes
recent advances in topics with high future potential and exiting developments in
classical fields.
We thank Anna Tur from the Faculty of Applied Mathematics (SPbSU) for
displaying extreme patience typesetting the manuscript.
Editors, Leon A. Petrosyan and Nikolay A. Zenkevich
On Monotonicity of the SM-nucleolus and the α-nucleolus
Sergei V. Britvin and Svetlana I. Tarashnina
St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
Bibliotechnaya pl. 2, St.Petersburg, 198504, Russia
E-mail: [email protected]
E-mail: [email protected]
Abstract In this paper two single-valued solution concepts of a TU-game
with a finite set of players, the SM-nucleolus and the α-nucleolus, are considered. Based on the procedure of finding lexicographical minimum, there
was proposed an algorithm allowing to calculate the SM-nucleolus as well as
the prenucleolus. This algorithm is modified to calculate the α-nucleolus for
any fixed α ∈ [0, 1]. Using this algorithm the monotonicity properties of the
SM-nucleolus and the α-nucleolus are studied by means of counterexamples.
Keywords: cooperative TU-game, solution concept, aggregate and coalitional monotonicity, the SM-nucleolus, the α-nucleolus.
1.
Introduction
In this paper we examine two single-valued solution concepts of a transferable utility
game (TU-game) with a finite set of players — the SM-nucleolus (Tarashnina, 2011)
and the α-nucleolus (Smirnova and Tarashnina, 2011). Both of these solution concepts take into account "the blocking power" of a coalition, the amount which the
coalition cannot be prevented from by the complement coalition.
Based on the procedure (Maschler et al., 1979) of finding lexicographical minimum, there was proposed an algorithm (Britvin and Tarashnina, 2013) allowing to
calculate the SM-nucleolus as well as the prenucleolus (Schmeidler, 1969). By introducing the special numbering of coalitions the problem of finding the SM-nucleolus
of a cooperative n-person game is reduced to solving a single linear program with 2n
rows and (n + 1) columns. The initial values of the problem coefficients are 0, 1, −1.
This algorithm is modified to calculate the α-nucleolus for any fixed α ∈ [0, 1].
In this work we consider two properties of single-valued solution concepts of
TU-games: aggregate and coalitional monotonicity. Aggregate monotonicity means
that if the worth of the grand coalition icreases while the worths of all other coalitions remain the same, then the players payoffs should not decrease. Coalitional
monotonicity applies this rule to any coalition S ⊂ N in a game. The Shapley
value (Shapley, 1953) satisfies aggregate and coalitional monotonicity. N. Megiddo
(Megiddo, 1974) presented an example of nine person cooperative TU-game that
shows that another well-known single-valued solution, the nucleolus (Schmeidler,
1969), violates aggregate monotonicity. It is known that the nucleolus does not satisfy coalitional monotonicity too (Young, 1985). In this paper we verify that the
SM-nucleolus does not satisfy aggregate and coalitional monotonicity.
The paper is organized as follows. In section 2 basic definitions and notations
are given. For any fixed α we describe the algorithm of finding the α-nucleolus
(including the SM-nucleolus) in section 3. Using this algorithm we study the monotonicity properties of the considered solution concepts by means of counterexamples
in section 4.
On Monotonicity of the SM-nucleolus and the α-nucleolus
2.
9
Basic definitions and notations
In this paper we consider cooperative games with transferable utilities (TU-games).
A TU-game is a pair (N, v), where N = {1, ..., n} is the set of players and v : 2N → R
is a characteristic function with v(∅) = 0. More information about cooperative game
theory can be found in (Petrosjan et al., 2012) and (Pecherskiy and Yanovskaya,
2004).
The set of all TU-games with the fixed set of players N is denoted by GN .
Consider a game (N, v) from GN . Assume that the players have formed the maximal
coalition N and consider the distribution of v(N ) among all the players. We define
the set of feasible payoff vectors as follows:
X
X(N, v) = {x ∈ Rn |
xi ≤ v(N )}.
i∈N
The set X (N, v) ⊂ X(N, v) such that
0
X 0 (N, v) = {x ∈ Rn |
X
xi = v(N )}
(1)
i∈N
is a set of group rational payoff vectors of the game (N, v). It follows from (1) that
x ∈ X 0 (N, v) if and only if for all S ⊂ N it holds
x(S) + x(N \S) = v(N ),
where
x(S) =
X
xi .
i∈S
Definition 1. A solution of a TU-game on GN is a mapping f that matches for
every game (N, v) ∈ GN the subset f (N, v) of X(N, v).
In the paper we study two single-valued solution concepts: the SM-nucleolus and
the α-nucleolus. To introduce the definitions of these solution concepts, we should
define the excess of a coalition.
Definition 2. The excess e(x, S, v) of a coalition S at x ∈ X 0 (N, v) is calculated
as
e(x, S, v) = v(S) − x(S).
(2)
Let (N, v) be a TU-game. The dual game (N, v ∗ ) of (N, v) is defined by
v ∗ (S) = v(N ) − v(N \S)
for all coalitions S.
Let us clarify the notion of the constructive and the blocking power of S. The
constructive power of S is the worth of the coalition, or exactly what S can reach by
cooperation. By the blocking power of coalition S we understand the amount v ∗ (S)
that this coalition brings to N if the last will be formed — its contribution to the
grand coalition. The difference between v(N ) and v(N \S) is a subject which should
be taken into account in a solution of a game. In our opinion, the blocking power
can be judged as a measure of necessity of S for N — how much S contributes to
N . So, each coalition S is estimated by N in this spirit.
In order to introduce the SM-nucleolus we define the sum-excess of the coalition.
10
Sergei V. Britvin, Svetlana I. Tarashnina
Definition 3. The sum-excess ē(x, S, v) of a coalition S at x ∈ X 0 (N, v) in the
game (N, v) is
1
1
ē(x, S, v) = e(x, S, v) + e(x, S, v ∗ ).
2
2
We define for some ϕ ∈ Rn the mapping θ : Rn → Rn such that ψ = θ(ϕ) ∈ Rn
means that ψ is obtained from ϕ by ordering its components in non-increasing order.
After calculating the sum-excess for each S ⊆ N we obtain the sum-excess vector
ē(x, v) = {ē(x, S, v)}S⊆N of dimension 2n .
Definition 4. The SM-nucleolus of the game (N, v) is the set XSM ⊂ X 0 (N, v)
such that for every x ∈ XSM vector θ({ē(x, S, v)}S⊆N ) is lexicographically the
smallest:
XSM (N, v) = {x ∈ X 0 |θ({ē(x, S, v)}S⊆N ) lex θ({ē(y, S, v)}S⊆N ), ∀y ∈ X 0 (N, v)}.
To introduce the α-nucleolus we define the α-excess of a coalition.
Definition 5. The α-excess eα (x, S, v) of a coalition S at x ∈ X 0 (N, v) is
eα (x, S, v) = αe(x, S, v) + (1 − α)e(x, S, v ∗ ), α ∈ [0, 1].
(3)
After calculating the α-excess for each S ⊆ N we obtain the α-excess vector
eα (x, v) = {eα (x, S, v)}S⊆N of dimension 2n .
Definition 6. The α-nucleolus of the game (N, v) is the set Xα ⊂ X 0 (N, v) such
that for every x ∈ Xα vector θ({eα (x, S, v)}S⊆N ) is lexicographically the smallest:
Xα (N, v) = {x ∈ X 0 (N, v)|θ({eα (x, S, v)}S⊆N ) lex θ({eα (y, S, v)}S⊆N ),
∀y ∈ X 0 (N, v)}.
It is important to note that both solution concepts represent a unique point in
X 0 , so they are single-valued solutions (Smirnova and Tarashnina, 2011).
Obviously, if α = 21 , then
Xα (N, v) = XSM (N, v).
This means that the SM-nucleolus is a special case of the α-nucleolus.
3.
Algorithm
In the literature there was presented an algorithm of finding the SM-nucleolus of
any TU-game (Britvin and Tarashnina, 2013). Here we modify this algorithm for
calculation the α-nucleolus. First, we should replace the excess in the procedure
of finding the lexicographical minimum (Maschler et al., 1979) to the α-excess. We
obtain the following procedure.
On Monotonicity of the SM-nucleolus and the α-nucleolus
11
1. Consider a pair (X 0 , J 0 ), where J 0 consists of all possible coalitions except the
empty one.
2. Recursively find
ut = min max eα (x, S, v),
(4)
x∈X t−1 S⊆J t−1
X t = {x ∈ X t−1 | eα (x, S, v) ≤ ut , ∀S ⊆ J t−1 },
Jt = {S ⊆ J t−1 | eα (x, S, v) = ut , ∀x ∈ X t },
J t = J t−1 \Jt .
3. If J t = ∅, then we stop, otherwise we go to step 2 with t = t + 1.
In the game there may be formed 2n coalitions (including the empty one). We
will not consider the empty coalition. Let us enumerate all the other coalitions in
the following way.
Suppose that n-person game has been built dynamically by adding one player
at each step. In a game with one player the single coalition {1} is number 1.
When the second player enters the game he brings there two additional coalitions.
The sequence of coalitions in ascending order for two-person game is as follows:
{1}, {2}, {1, 2}. Further, for a three-person game we have: {1}, {2}, {1,2}, {3},
{1,3}, {2,3}, {1,2,3}. And so on.
Assume that the sequence of coalitions has been formed in ascending order for
a k-person game. Adding to this game the (k + 1)-th player entails forming 2k
coalitions. We determine the order of the added coalitions. Let coalition {k + 1}
be the first of the added coalitions. Among the remaining coalitions we do not pay
attention to the (k + 1)-th player, then we obtain a set of coalitions for a k-person
game, which is already built in ascending order. Finally, we extend this numbering
to the additional coalitions. As a result, each coalition in a (k + 1)-person game will
be numbered.
For example, the coalitions in 4-person game in ascending order look like
{1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3},
{4}, {1, 4}, {2, 4}, {1, 2, 4}, {3, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}.
Problem (4) for t = 1 is equivalent to the following task

min u1 ,



u1 ≥ eα (x, S, v),

S ⊆ J 0,



x ∈ X 0.
By using formulas (1), (2) and (3) we transform it to the following form

1

min uP,
1
u + i∈S xi ≥ αv(S) + (1 − α)(v(N ) − v(N \S)), S ⊆ J 0 ,

P
i∈N xi = v(N ).
The resulting problem is a linear programming. Given the suggested order of
the coalitions in J 0 we obtain the matrix form
12
Sergei V. Britvin, Svetlana I. Tarashnina

T

min c z,
Az ≥ b,


Aeq z = beq ,
with
z=
1
u
.
x
(5)
The parameters of this linear programming are the following:
 
 

1
1
100
0
1
0 1 0
 
 

 
 

c = 0 , A = I A∗ , I = 1 , A∗ = 1 1 0
 .. 
 .. 
 ..
.
.
.
0
1
011

v(1)
v(2)
v(1, 2)
..
.



v(N ) − v(2, 3, ..., n)

v(N ) − v(1, 3, ..., n)






 + (1 − α) v(N ) − v(3, 4, ..., n) ,



..



.



b = α


v(2, 3, ..., n)
Aeq = 1 1 1 ... 1 ,

... 0 0
... 0 0

... 0 0
,
.. 
..
. .
... 1 1
v(N ) − v(1)
beq = v(N ).
Based on the theorem from (Britvin and Tarashnina, 2013), it is easy to prove
that there exists a unique solution z ∗ of this linear programming. So, the calculation
procedure is stopped and we obtain the α-nucleolus in the form

 ∗
z2 (α)
 z3∗ (α) 


Xα = 
 , α ∈ [0, 1].
..


.
∗
zn+1
(α)
4.
The monotonicity of the SM-nucleolus and the α-nucleolus
In this paper we investigate the monotonicity properties of single-valued solution
concepts of TU-games: aggregate monotonicity and coalitional monotonicity. First,
let us define these properties.
Definition 7. A single-valued solution concept f satisfies aggregate monotonicity
if for every pair of games (N, v) and (N, w) such that
v(N ) < w(N ),
(6)
v(S) = w(S) for all S ⊂ N,
(7)
fi (N, v) ≤ fi (N, w) for all i ∈ N.
(8)
it follows that
On Monotonicity of the SM-nucleolus and the α-nucleolus
13
Definition 8. A single-valued solution concept f satisfies coalitional monotonicity
if for every pair of games (N, v) and (N, w) such that
it follows that
v(T ) < w(T ) for any T ⊂ N,
(9)
v(S) = w(S) for all S ⊂ N, S 6= T,
(10)
fi (N, v) ≤ fi (N, w) for all i ∈ T.
(11)
Let us give the following example (Megiddo, 1974) that contains of two cooperative games and illustrates the absence of aggregate monotonicity of the SMnucleolus.
Example 1. Let N = {1, 2, ..., 9} and ϕ = (1, 1, 1, 2, 2, 2, 1, 1, 1). Consider two groups
of coalitions:
A = {(1, 2, 3), (1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5), (7, 8, 9)},
B = {(1, 2, 3, 6, 7), (1, 2, 3, 6, 8), (1, 2, 3, 6, 9), (4, 5, 6)}.
Define the characteristic function v as

6, if S ∈ A,



9, if S ∈ B,
v=

12, if S = N,


P
i∈S ϕi − 1, otherwise.
The characteristic function w has the following form

6, if S ∈ A,



9, if S ∈ B,
w=

13, if S = N,


P
i∈S ϕi − 1, otherwise.
It is obvious that conditions (6) and (7) hold for characteristic functions v and
w. For some fixed α ∈ [0, 1] we calculate the α-nucleolus for games (N, v) and
(N, w) using the algorithm, presented in section 3. Assuming that the parameter α
is moving along the interval [0, 1] with the step of 0.1, we have the following payoff
vectors presented in Table 1. The special case α = 12 with XSM = Xα is shown in
bold.
Consider the payoffs that player 6 gets according to the α-nucleolus for α ≥ 0.1.
We can see that Xα6 (N, v) > Xα6 (N, w). Therefore, inequality (8) is not satisfied.
So, by means of the counterexample we can verify that aggregate monotonicity does
not hold for the α-nucleolus with 0.1 ≤ α ≤ 1.
By using the dichotomy method for this pair of games we can approximately
calculate the maximum α∗ such that 0 < α∗ < 0.1 for which aggregate monotonicity
is satisfied and for some α > α∗ aggregate monotonicity is not satisfied. In the
current example α∗ ≈ 0.075.
14
Sergei V. Britvin, Svetlana I. Tarashnina
Table 1: The α-nucleolus for (N, v) and (N, w).
α
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Xα (N, v)
Xα (N, w)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.11, 1.11, 1.11, 2.11, 2.11, 2.11, 1.11, 1.11, 1.11)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.11, 1.11, 1.11, 2.22, 2.22, 1.92, 1.11, 1.11, 1.11)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11)
(1,1,1,2,2,2,1,1,1) (1.11,1.11,1.11,2.22,2.22,1.89,1.11,1.11,1.11)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11)
Let us give one more example.
Example 2. Consider the characteristic function v ′ that coincides with v from Example 1:
v ′ (S) = v(S) for all S ⊆ N.
The characteristic function w′ is constructed in the following way
(
7, if S = (4, 7),
w′ (S) =
v(S), otherwise.
It is obvious that conditions (9) and (10) hold for characteristic functions v ′ and
w . For some fixed α ∈ [0, 1] we calculate the α-nucleolus for games (N, v ′ ) and
(N, w′ ) using the algorithm, presented in section 3. Assuming that the parameter
α is moving along the interval [0, 1] with the step of 0.1, we have the payoff vectors
presented in Table 2. The special case α = 21 with XSM = Xα is shown in bold.
′
Table 2: The α-nucleolus for (N, v ′ ) and (N, w′ ).
α
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Xα (N, v ′ )
Xα (N, w′ )
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.01, 1.01, 1.01, 1.98, 1.98, 2.03, 1.11, 0.94, 0.94)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.01, 1.01, 1.01, 1.96, 1.96, 2.05, 1.21, 0.88, 0.88)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.02, 1.02, 1.02, 1.94, 1.94, 2.08, 1.32, 0.82, 0.82)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.03, 1.03, 1.03, 1.93, 1.93, 2.10, 1.43, 0.76, 0.76)
(1,1,1,2,2,2,1,1,1) (1.04,1.04,1.04,1.91,1.91,2.13,1.54,0.70,0.70)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.04, 1.04, 1.04, 1.89, 1.89, 2.16, 1.64, 0.64, 0.64)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.05, 1.05, 1.05, 1.87, 1.87, 2.18, 1.75, 0.59, 0.59)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.06, 1.06, 1.06, 1.85, 1.85, 2.21, 1.86, 0.53, 0.53)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.07, 1.07, 1.07, 1.83, 1.83, 2.23, 1.97, 0.47, 0.47)
(1, 1, 1, 2, 2, 2, 1, 1, 1)
(1.07, 1.07, 1.07, 1.81, 1.81, 2.26, 2.07, 0.41, 0.41)
Consider the payoffs that player 4 gets according to the α-nucleolus for α ≥ 0.1.
We can see that Xα4 (N, v) > Xα4 (N, w). Therefore, inequality (11) is not satisfied.
On Monotonicity of the SM-nucleolus and the α-nucleolus
15
So, by means of counterexample we can verify that coalitional monotonicity does
not hold for the α-nucleolus with 0.1 ≤ α ≤ 1.
By using the dichotomy method for this pair of games we can approximately
calculate the maximum α∗ such that 0 < α∗ < 0.1 for which coalitional monotonicity
is satisfied and for some α > α∗ coalitional monotonicity is not satisfied. In the
current example α∗ ≈ 0.
5.
Conclusion
The result of this work is not surprising. Although aggregate and coalitional monotonicity are considered to be desirable and natural properties of a solution in a
TU-game (Maschler, 1992). There are very few solution concepts satisfying even
aggregate monotonicity, the weakest form of it. In the paper we have investigated
the monotonicity of the SM-nucleolus and come across to some negative conclusion.
In a general game it violates the both aggregate and coalitional monotonicity. At
the same time, the α-nucleolus due to the arbitrary choice of a real parameter α
demonstrates for some α better properties than the SM-nucleolus. The intervals for
which the α-nucleolus satisfies aggregate and coalitional monotonicity are approximately calculated for the given examples. The investigation may be extended and
deepened in the direction of getting analytical formulas for this interval.
In (Tauman and Zapechelnyuk, 2010), the authors argue that monotonicity may
not be a proper requirement for some economic context from which a cooperative
game arises. They provide an example of a simple 4-person game that marks out a
class of economic problems where the monotonicity property of a solution concept
is not as attractive as it may seem at the beginning. So, sometimes there is a
competition between monotonicity and other attractive properties of a solution in
a TU-game.
References
Britvin, S., Tarashnina, S. (2013). Algorithms of finding the prenucleolus and the SMnucleolus in cooperative TU-games. Mathematical Game Theory and its Applications,
5(4), 14–32 (in Russian).
Maschler, M., Peleg, B., Shapley, L. S. (1979). Geometric properties of the kernel, nucleolus
and related solution concepts. Mathematics of Operations Research, 4(4), 303–338.
Maschler, M. (1992). The Bargaining set, Kernel and Nucleolus. Handbook of Game Theory (R. Aumann, S. Hart, eds), Elsevier Science Publishers BV, 591–665.
Megiddo, N. (1974). On nonmonotonicity of the bargaining set, the kernel and the nucleolus
of a game. SIAM Journal on Applied Mathemetics, 27(2), 355–358.
Pecherskiy, S. L., Yanovskaya, E. B. (2004). Cooperative games: soutions and axioms. European University press: St. Petersburg, 443 p. (in Russian).
Petrosjan, L. A., Zenkevich, N. A., Shevkoplyas, E. V. (2012). Game theory. SaintPetersburg: BHV-Petersburg, 432 p. (in Russian).
Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal on
Applied Mathematics, 17(6), 1163-1170.
Shapley, L. S. (1953). A value for n-person games. In: Kuhn and Tucker (eds.) Contributions to the theory of games II. Princeton University press, pp. 307–311.
Smirnova, N., Tarashnina, S. (2011). On generalisation of the nucleolus in cooperative
games. Journal of Applied and Industrial Mathematics, 18(4), 77–93 (in Russian).
Tarashnina, S. (2011). The simplified modified nucleolus of a cooperative TU-game. Operations Research and Decision Theory, 19(1), 150–166.
16
Sergei V. Britvin, Svetlana I. Tarashnina
Tauman, Y., Zapechelnyuk, A. (2010). On (non-) monotonicity of cooperative solutions.
International Journal of Game Theory, 39(1), 171–175.
Young, H. P. (1985). Monotonic solution of cooperative games. International Journal of
Game Theory, 14, 65–72.
Efficient Myerson Value for Union Stable Structures
⋆
Hua Dong, Hao Sun and Genjiu Xu
Department of Applied Mathematics
Northwestern Polytechnical University, Xi’an, 710072,China
E-mail: [email protected],[email protected],[email protected]
Abstract In this work, an axiomatization of a new value for union stable
structures, efficient Myerson value, is shown by average equity, redundant
fairness, superfluous component property and other three properties. And
the independence of the axioms is illustrated. Besides, the difference of three
values, efficient Myerson value, the two-step Shapley value and collective
value, is shown.
Keywords: Union stable structure; average equity; redundant fairness.
1.
Introduction
A situation in which a finite set of players can obtain payoffs by cooperation can
be described by a cooperative game with transferable utility, shortly TU-game,
being a pair consisting of a finite set of players and a characteristic function on
the set of coalitions of players assigning a worth to each coalition of players. In
practice, since the cooperation restrictions exist, only some subgroup of players can
form a coalition. One way to describe the structure of partial cooperation in the
context of cooperative games is to specify sets of feasible coalitions. Algaba, et al
(Algaba, 2000) considered union stable systems as such sets. A union stable system
of two intersecting feasible coalitions is also feasible, which can be interpreted as
follows: players who are common members of two feasible coalitions are able to
act as intermediaries to elicit cooperation among all the players in either of these
coalitions, and so their union should be a feasible coalition. And a TU game with
a union stable system is called a union stable structure. Besides, the union stable
structure is a generalization of games with communication structure and games with
permission structure, which are respectively proposed by Myerson (Myerson, 1977)
and Gilles (Gilles,1992).
Hamiache (Hamiache, 2012) presented a matrix approach to construct extensions of the Shapley value on the games with coalition structures and communication structures. This paper aims to generalize this matrix approach to union stable
structures, a generalized communication structures.
2.
Preliminaries
2.1. Matrix Approach To Shapley Value
A cooperative game with transferable utility, or simply a TU-game, being a pair
(N, v), where N is the finite set of all players, and v : 2N → R is a characteristic
function satisfying v(∅) = 0. The collection of all games with player set N is denoted
by G. A game (N, v) is called an inessential game if for any two disjoint coalitions
⋆
This work was supported by the NSF of China under grants Nos. 71171163, 71271171
and 71311120091.
18
Hua Dong, Hao Sun, Genjiu Xu
S, T ⊆ N , v(S ∪ T ) = v(S) + v(T ). Here the cardinality of any coalition S ⊆ N is
denoted by |S| or the lower case letter s.
A payoff vector for a game is a vector x ∈P
RN assigning a payoff xi to player
i ∈ N . In the sequel, for all S ⊆ N , x(S) = i∈S xi . A single-valued solution is
a function that assigns to any game (N, v) ∈ G a unique payoff vector. The most
well-known single-valued efficient solution is the Shapley value (1953) given by
Shi (N, v) =
X s!(n − s − 1)!
(v(S ∪ i) − v(S)).
n!
i∈N \S
In fact, the explicit expression of Shapley value can be presented as Shi (N, v) =
(M Sh · v)[{i}], where the matrix M Sh = [M Sh ]i∈N,S⊆N \∅ is defined by
[M
Sh
]i,S =
(
(s−1)!(n−s)!
,
n!
s!(n−s−1)!
−
,
n!
if i ∈ S ,
otherwise.
(1)
Next, we recall the axiomatic characterization of Shapley value (Shapley, 1953)
illustrated in Hamiache (Hamiache, 2001)and the matrix approach in Xu, et al
(Xu, 2008) and Hamiache (Hamiache, 2010) for the analysis of the associated consistency.
For all games (N, v) ∈ G, the associated game (N, vλSh ) defined in Hamiache
(Hamiache, 2001)for all parameters λ(0 < λ < n2 ) as follows,
vλSh (S) = v(S) + λ
X
i∈N \S
[v(S ∪ i) − v(S) − v({i})] for all S ⊆ N
Definition 1. A matrix A is called a row (column)-coalition matrix if its rows
(column) are indexed by coalitions S ⊆ N in the lexicographic order. A is called
square-coalitional if it is both row-coalitional and column-coalitional. And a rowcoalition
matrix A = [a]S,T is called row-inessential or inessential, if A = [a]S,T =
P
i∈S ai,T for all S ⊆ N .
Since the associated game is a linear transformation of the original game, the
associated game can be expressed as vλSh = Mλ · v, where Mλ is a square-coalitional
matrix of order 2n−1 , for detailed information, please refer to Xu, et al (Xu, 2008)
and Hamiache (Hamiache, 2010).
The sequence of associated games illustrated in Hamiache (2001) can be expressed by matrix approach in Xu, et al (Xu, 2008) and Hamiache(Hamiache, 2010)
as follows,
k
vkλ = (v(k−1)λ )Sh
λ = Mλ · v(k−1)λ = ... = (Mλ ) · v, for all k ≥ 2 .
And the sequence of games {(N, vkλ )}∞
k=1 converges to an inessential game (N, vL ),
denote the corresponding coefficient matrix as ML , then limk→∞ (Mλ )k = ML , and
ML is inessential.
2.2.
Union Stable Structures
Definition 2. A union stable system is a pair (N, F ) with F ⊆ 2N verifying that
{i} ∈ F for all i ∈ N and for all S, T ∈ F with S ∩ T 6= ∅, S ∪ T ∈ F .
Efficient Myerson Value for Union Stable Structures
19
Given a union stable system (N, F ), B(F ) is called the basis of F , it is denoted
by the set of all feasible coalitions which cannot be expressed as a union of feasible
coalitions with nonempty intersection, the elements of the basis B(F ) are called
supports of F . Especially, the set of non-singleton supports is denoted by C(F ) =
{B ∈ B(F ) : |B| ≥ 2}.
A union stable structure is a triple (N, v, F ), i.e., a TU game (N, v) with union
stable system (N, F ). The set of such union stable structure with player set N is
denoted by U S N .
Definition 3. Let E ⊆ 2N be a set system and S ⊆ N . A set T ⊆ S is called a
′
′
E-component of S if T ∈ E and there exists no T ∈ E such that T ( T ⊆ S.
Especially, the collection of F -component of N is denoted by β = CF (N ) =
{B1 , B2 , ..., Br } with 1 ≤ r ≤ |N | and ∪B∈β B = N , Bi ∩ Bj 6= ∅ for any Bi , Bj ∈ β.
Given (N, v, F ) ∈ U S N , define the intermediate game (β, v βP
) by v β (R) =
F
F
v(∪B∈R B) for all R ⊆ β and the quotient game (N, v ) by v (S) = T ∈CF (S) v(T )
for all S ⊆ N .
For coalition S ⊆ N \ ∅, define coalitions S and S respectively by the following,
S = ∪{K ∈ β|K ⊆ S}, i.e., the maximal union of components of N which belongs
to coalition S. S = ∪{K ∈ β|K ∩ S 6= ∅}, i.e., the minimal union of components of
N covering coalition S.
3.
3.1.
Efficient Myerson Value For Union Stable Structures
Definition
In order to give the formal definition of the efficient Myerson value, two matrices
closely related to union stable structures are constructed.
Let us define a {0, 1}-squared matrix P of order 2n − 1, which is closely related
to union stable structure (N, v, F ). So that for all S, T ⊆ 2N \ ∅,
P [S, T ] =
(
1, if T ∈ CF (S) ,
0, otherwise.
(2)
Note that for all coalitions S ⊆ N , v F (S) = (P ·v)[S], thus ϕi (N, v, F ) = Shi (N, v F ) =
(ML ·P ·v)[i], where ϕ(N, v, F ) is the Myerson value for union stable structure(N, v, F ).
Next, we shall make a modification of the matrix, and define the matrix Q as
follows,


1, if T = S ,
(3)
Q[S, T ] = 1, if T ∈ CF (S \ S),


0, otherwise.
Lemma 1. Given (N, v, F ) ∈ U S N and the intermediate game (β, v β ) defined before, the vector of weights are w = (b1 , b2 , ..., br ), bl = |Bl | for all l ∈ {1, 2, ..., r}.
Then for all Bl ∈ β and all players i ∈ Bl ,
(ML · (Q − P ) · v)[{i}] = Shi (N, (Q − P ) · v)
1
β
= (Shw
Bl (β, v ) − v(Bl )).
bl
20
Hua Dong, Hao Sun, Genjiu Xu
The proof is similar to the computation in Hamiache (Hamiache, 2012),we omit
here. Next we give the definition of the efficient Myerson value and some axioms
that will be used to axiomatize the value.
Definition 4. For all union stable structures (N, v, F ), define the the efficient Myerson value η as
ηi (N, v, F ) = (ML · Q · v)[{i}] = Shi (N, Q · v) for all i ∈ N.
From Lemma 1, it is obvious that for solution η and all i ∈ N ,
ηi (N, v, F ) = ϕi (N, v, F ) +
1
β
(Shw
Bl (β, v ) − v(Bl )).
bl
We can interpret efficient Myerson value in the following sense. In the first step,
every player obtains the payoff of Myerson value. In the second step, since the
allocation rule satisfies efficiency, define a quotient game, and every component
Bl obtains the payoff of weighted Shapley value, following the principle of fairness
β
among the members of component Bl , the surplus Shw
Bl (β, v ) − v(Bl ) is split
equally.
It can be seen that the efficient Myerson value and the collective value, which
was proposed by Kamijo(Kamijo, 2011), is similar, and the difference lies in the
allocation rule of first step. Compared with a prior coalition structure, given a component of union stable system, some subset of the component may be not feasible.
And the formation of the component of union stable system lies on the contribution
of common players, while collective value cannot illustrate the contribution of the
intermediate members make during the cooperation, so the collective value is not
suitable for union stable structures. So they are irreplaceable for each other.
Let Ci (F ) denote the collection given by {C ∈ C(F ) : i ∈ C}, to provide axiomatic characterizations of the efficient Myerson value, the following definitions
and properties are introduced.
3.2.
Axiomatization
Definition 5. A union stable structure (N, v, F ) is called point anonymous if there
exists a function f : {0, 1, ..., |D|} → R with f (0) = 0 such that v F (S) = f (|S ∩ D|)
for all S ⊆ N , where D = {i ∈ N : Ci (F ) 6= ∅}.
Definition 6. For any (N, v, F ) ∈ U S N , a player i ∈ N is called superfluous for
(N, v, F ) if v F (S ∪ i) = v F (S) for all S ⊆ N \ {i}.
Let ψ : U S N → Rn be a solution, then we call
P it satisfies the above properties, if
Efficiency (EFF) For all (N, v, F ) ∈ U S N , i∈N ψi (N, v, F ) = v(N ).
Additivity (ADD) For any (N, u, F ), (N, v, F ) ∈ U S N , ψ(u + v) = ψ(u) + ψ(v),
where (u + v)(S) = u(S) + v(S) for all S ⊆ N .
Average equity (AE) For all unanimity games uT with T ⊆ N \ ∅, if there exists
two components Bl , Bk ∈ β with Bl ∩ T 6= ∅, Bk ∩ T 6= ∅, then
X
X
|Bl |−1
ψi (N, uT , F ) = |Bk |−1
ψj (N, uT , F ).
i∈Bl
j∈Bk
Point anonymity (PA) For all point anonymous union stable structures (N, v, F ),
Efficient Myerson Value for Union Stable Structures
21
there exists b ∈ R such that ψi (N, v, F ) = b for all i ∈ D, ψi (N, v, F ) = 0 otherwise.
Redundant fairness (RF) If there exists two superfluous players i, j ∈ Bk with
Bk ∈ β, then ψi = ψj .
Superfluous component property
(SCP) Given component Bk ∈ β, if v(R ∪ Bk ) =
P
v(R) for all R ⊆ β, then i∈Bk ψi (N, v, F ) = 0.
Theorem 1. The efficient Myerson value is the unique value on U S N that satisfies efficiency, additivity, average fairness, point anonymity, redundant fairness and
superfluous component property.
Proof. It is straightforward to verify that the efficient Myerson value satisfies EFF,
ADD, AE, RF and SCP. In the following, we will only verify the property of point
anonymity.
Let (N, v, F ) ∈ U S N be point anonymous. If D = ∅, then the restricted game
F
v (S) = f (|S ∩ ∅|) = f (0) = 0 for all S ⊆ N . Hence, the efficient Myerson value
ηi (N, v, F ) = 0 for all i ∈ N . Let D 6= ∅, we will show that there exists a unique
component Bk ∈ β such that D ⊆ Bk . Otherwise, assume there are two components
Bi , Bj ∈ β such that D = Bi ∪ Bj , let S = D, we have v F (S) = v(Bi ) + v(Bj ) =
f (|Bi ∩D|)+f (|Bj ∩D|), which contradicts with v F (S) = f (|S∩D|) = f (|D|). Hence,
let us suppose Bk ∈ β is the unique component such that D ⊆ Bk , then for any
β
R ⊆ β, v(|(Bl ∪R)∩D|) = f (|R ∩D|) = v(R) for all Bl 6= Bk , Shw
Bk (β, v ) = v(Bk ).
β
Consequently, for any Bl ∈ β, Shw
Bk (β, v ) − v(Bk ) = 0, the efficient Myerson value
is equal to the Myerson value, i.e., ηi = ϕi = f (|D|)/|D| for all i ∈ D, otherwise,
ηi = 0. Thus the efficient Myerson value verifies point anonymity.
Next, we will show the converse part. Let ψ ∈ Rn be a solution on U S N satisfying
the above six properties. Given T ⊆ N \ ∅, let (N, uT , F ) be a unanimity game with
union stable system. Given c ∈ R, let cuT be a unanimity game uT multiplied
by a scalar c, Then by additivity, it suffices to show that ψ(N, v, F ) is uniquely
determined by the above six properties. For all T ⊆ N \ ∅, , let us consider the
following two cases: T ∈
/ F and T ∈ F .
Case 1 T ∈
/ F , define T ⊆ β by {B ∈ β, B ∩ T 6= ∅}. Then the unanimity game
(β, (cuT )β ) is a T -unanimity game multiplied by c, i.e., (β, cuT ). It is obvious that
any component
Bl ∈ β \ T is superfluous. From superfluous component property,
P
we have i∈Bl ψi (N, cuT , F ) = 0 for all Bl ∈ β \ T . Together with average equity
P
P
together and efficiency, we have that i∈Bl ψi (N, cuT , F ) = c( Bl ∈T |Bl |)−1 |Bl |
P
for all Bl ∈ T , i∈Bl ψi (N, cuT , F ) = 0 otherwise.
Furthermore, we assert that any player i ∈ N is superfluous for (N, cuT , F ) ∈
F
U S N , i.e., given any i ∈ N , uF
T (S) = uT (S ∪ i) for all S ⊆ N . Consequently,
given any Bk ∈ β, due to the redundancy fairness of ψ(N, cuT , F ), then ψi =
ψjPfor all i, j ∈ Bk . From the above arguments, we have that ψi (N, cuT , F ) =
c( Bl ∈T |Bl |)−1 for all i ∈ Bk and Bk ∈ T , ψi (N, cuT , F ) = 0 otherwise. Hence,
for any unanimity game with union stable structure (N, cuT , F ) ∈ U S N with T ∈
/ F,
ψ(N, cuT , F ) is uniquely determined. The remaining task is to show all players are
superfluous.
In the following, we show that any player i ∈ N is superfluous for (N, cuT , F ) ∈
U S N . If there exists a unique component Bk ∈ β such that T ⊆ Bk , then uF
T (S) =
F
uF
T (S ∪ i) = 1, uT (S) = 0 for all S ⊆ N \ Bk . Otherwise, there exists no such
component, then uF
T (S) = 0 for all S ⊆ N . This completes the proof for case 1.
Case 2 T ∈ F , we show that (N, cuT , F ) ∈ U S N is point anonymous. First we show
22
Hua Dong, Hao Sun, Genjiu Xu
that for T ∈ F , (cuT )F (S) = c if and only if T ⊆ S. Due to whether the coalition
S is feasible or not, we distinguish the following two cases:
(1)If S ∈ F , then (cuT )F (S) = cuT (S) = c if and only if T ⊆ S, i.e., T ∩ S = T .
(2)If S ∈
/ F and T ⊆ S, we will show that there exists a unique feasible coalition
K ∈ F and K ⊆ S such that T ⊆ K. If T ∈ CF (S), let K = T , (cuT )F (S) =
cuT (T ) = c. Otherwise, there exists a series of feasible coalitions A1 , A2 , ..., Al ∈ F
with Ai ∩ Aj = ∅ for any i, j = 1, 2, ..., l(l ≥ 2) and i 6= j such that S = ∪lk=1 Ak ,
since S ∈
/ F and |S| ≥ 2. Hence there exists a unique feasible coalition Aj (1 ≤ j ≤ l)
such that T ⊆ Aj , let K = Aj , consequently, (cuT )F (S) = cuT (Aj ) = c. If S ∈
/F
and T * S, it is easy to verify that (cuT )F (S) = 0.
From the arguments above, we have that if T ∈ F , (cuT )F (S) = c if and only
if T ⊆ S. Therefore, there exists a function f : {0, 1, 2, ..., |T |} → R such that
cuF
T (S) = f (|S ∩ T |) for all S ⊆ N where f (0) = f (1) = ... = f (|T | − 1) = 0 and
f (|T |) = c. Hence,(N, cuT , F ) is point anonymous, applying the point anonymity
to the solution ψ, there exists b ∈ R such
P that ψi = b if i ∈ T and ψi = 0 otherwise.
By efficiency, we have that cuT (N ) = i∈T ψi = b|T | = c, let b = c/|T |, thus the
solution ψ(N, v, F ) is uniquely determined by ψi = b for all i ∈ T , ψi = 0 otherwise.
So, ψ(N, v, F ) is unique determined in both cases. Since the efficient Myerson
value verifies the six properties, ψ(N, v, F ) = η(N, v, F ).
Also the axioms of theorem 1 are logically independent as shown by the following alternative solutions.
Example 1. The zero solution given by ψi (N, v, F ) = 0 for all i ∈ N satisfies ADD,
AE, PA, RF and SCP. It does not satisfy efficiency.
Example 2. The equal division given by
( Shw (β,vβ )−v(B )
ψi (N, v, F ) =
k
Bk
0,
|Bk \SU|
+ ψi (N, v, F ), if i ∈ Bk \ SU ,
if i ∈ Bk ∩ SU .
(4)
for all i ∈ Bk , Bk ∈ β, where SU denotes the set of all superfluous players in
(N, v, F ) and the weight system is the same with the definition of Lemma 1. This
solution satisfies all properties except additivity.
(β,v β )−v(B )
Sh
k
Example 3. The solution given by ψi (N, v, F ) = ϕi (N, v, F ) + Bk |Bk |
for
all i ∈ Bk , Bk ∈ β, satisfies EFF, ADD, PA, RF and SCP. It does not satisfy average
equity.
Example 4. The solution given by ψi (N, v, F ) =
all properties except point anonymity.
β
Shw
Bk (β,v )
|Bk |
for all i ∈ Bk satisfies
Example 5. Define the solution ψ(N, v, F ) by ψi (N, v, F )
[ShBk (β,v β )−v(Bk )]wi
P
for
j∈Bk wj
n
w ∈ R with wi 6= wj
=
ϕi (N, v, F )+
all i ∈ Bk ,Bk ∈ β, for some exogenous weight system
for any two players i 6= j in the same component, and
there exists
a
constant
number
a ∈ R such that for any component Bk ∈ β,
P
w(Bk ) =
w
=
|B
|
·
a.
It
is straightforward to verify that this solution
k
i∈Bk i
satisfies EFF, ADD, AE, PA and SCP, except redundant fairness.
Efficient Myerson Value for Union Stable Structures
23
Example 6. The solution given by
 v(N )−α(v)
, if i ∈ D,


|D|
ψi (N, v, F ) = 0,
if i ∈ Bk \ D,

 α(v)
,
if i ∈ N \ Bk .
|N \Bk |
(5)
for all i ∈ Bk , Bk ∈ β, where α : v → R is a linear operator, i.e., satisfying
α(v + w) = α(v) + α(w), and α(v) = 0 when the union stable structure(N, v, F )
is point anonymous, otherwise 0 < α(v) < v(N )/|D|. Since there exists only one
component Bk such that D ⊆ Bk . It is straightforward to verify that this solution
satisfies EFF, ADD, AE, PA and RF. It does not verify superfluous component
property.
4.
Conclusion
This paper mainly focus on the axiomatization of efficient Myerson value for union
stable structures. And three new axioms:average equity, redundant fairness, superfluous component property and other three properties. And the independence of the
axioms is illustrated. Besides, the difference between the value and collective value
is remarked.
References
Algaba, E., Bilbao, J. M., P., R. and J. J. Lopez. (2000). The position value for union
stable systems. Mathematical Methods of Operations Research., 52, 221–236. Princeton
University Press: Princeton, NJ.
Algaba, E., Bilbao, J. M., P., R. and J. J. Lopez. (2001). The Myerson value for union
stable structures. Mathematical Methods of Operations Research., 54, 359–371.
Driessen, T. S. H. (2010). Associated consistency and values for TU games. International
Journal of Game Theory., 39, 467–482.
Gilles RP, Owen G, Brink R. (1992). Games with permission structures: the conjunctive
approach. International Journal of Game Theory., 20, 277–293.
Hamiache, G. (2001). Associated consistency and Shapley value. International Journal of
Game Theory., 30, 279–289.
Hamiache, G. (2010). A matrix approach to Shapley value. International Game Theory
Review., 12 , 1–13
Hamiache, G. (2012). A matrix approach to TU games with coalition and communication
structures. Social Choice and Welfare., 38, 85–100.
Myerson, R. B. (1977). Graphs and cooperation in games. Mathematics of operations research., 2, 225–229.
Shapley, L. S. (1953). A value for n-person games. In:contributions to the Theory of Games
(Tucker, A.W.and Kuhn, H.W.,ends), Princeton University Press.
Xu, G., Driessen T.and Sun, H. (2008). Matrix analysis for associated consistency in
cooperative game theory. Linear Algebra and its Applications., 428, 1571–1586.
van den Brink R, Khmelnitskaya, A and Van der Laan G. (2012). An efficient and fair
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Kamijo, Y. (2011). The collective value: a new solution for games with coalition structures.
TOP., 1–18.
On the Inverse Problem and the Coalitional Rationality
for Binomial Semivalues of Cooperative TU Games
Irinel Dragan
University of Texas, Mathematics, Arlington,
Texas 76019-0408, USA
E-mail: [email protected]
Abstract In an earlier work (Dragan, 1991), we introduced the Inverse
Problem for the Shapley Values and Weighted Shapley Values of cooperative transferable utilities games (TU-games). A more recent work (Dragan,
2004) is solving the Inverse Problem for Semivalues, a more general class of
values of TU games. The Binomial Semivalues have been introduced recently
(Puerte, 2000), and they are particular Semivalues, including among other
values the Banzhaf Value. The Inverse problem for Binomial Semivalues was
considered in another paper (Dragan, 2013). As these are, in general, not
efficient values, the main tools in evaluating the fairness of such solutions
are the Power Game and the coalitional rationality, as introduced in the
earlier joint work (Dragan/Martinez-Legaz, 2001). In the present paper, we
are looking for the existence of games belonging to the Inverse Set, and for
which the a priori given Binomial Semivalue is coalitional rational, that is
belongs to the Core of the Power Game. It is shown that there are games in
the Inverse Set for which the Binomial Semivalue is coalitional rational, and
also games for which it is not coalitional rational. An example is illustrating
the procedure of finding games in the Inverse Set belonging to both classes
of games just mentioned.
Keywords: Inverse Problem, Inverse Set, Semivalues, Binomial Semivalues,
Power Game, Coalitional rationality.
Introduction
In a cooperative transferable utilities game (TU game), (N, v), defined by a
finite set of players N, n = |N | , and the characteristic function v : P (N ) → R,
with v(∅) = 0, where P (N ) is the set of nonempty subsets of N, called coalitions,
the main classical problem is to divide fairly the win of the grand coalition v(N ).
An early solution was the Shapley Value (1953), defined axiomatically, to satisfy
some fairness conditions (the axioms), and proved to be given by the formula
SHi (N, v) =
X
S:i∈S⊆N
(s − 1)!(n − s)!
.[v(S) − v(S − {i})], ∀i ∈ N,
n!
where s = |S| , S ⊆ N. It is easy to prove that SH is always efficient, that is we have
the sum of components equal v(N ). The Shapley Value may belong to the Core of the
game and in this case it is coalitional rational. The Semivalues, introduced by Dubey,
Neyman and Weber (1981), who tried to avoid the efficiency axiom, in general are
not efficient, so that they do not belong to the Core, and coalitional rationality
is a problem in evaluating the fairness. The Binomial Semivalues were introduced
by Puerte (2000), as extensions of the most known Semivalue, the Banzhaf Value
(1965). To evaluate the fairness of such a solution, an algebraic structure is needed,
On the Inverse Problem and the Coalitional Rationality
25
and let us denote by G(N ) the set of all games with the set of players N. Two
operations are defined, addition and scalar multiplication by
v = v1 + v2 ⇔ v(S) = v1 (S) + v2 (S), ∀S ⊆ N,
where (N, v1 ) and (N, v2 ) are any two TU games in G(N ), and
v = γv1 ⇔ v(S) = γv1 (S), ∀S ⊆ N, ∀γ ∈ R,
where (N, v1 ) is any TU game. It is easy to check that G(N ) is a linear vector space
and its dimension is 2n − 1. Now, for every coalition S ⊆ N , the restriction of (N, v)
to S is the game denoted by (S, v). Obviously, this finite set and the operations
shown above define on S again a linear vector space, G(S), and the union of all
spaces G(S), ∀S ⊆ N, is denoted by GN . A value Φ defined on GN is any functional
defined on each G(S) with values in Rs . The Shapley Value is defined on GN by a
formula similar to the first formula above, where N has been changed into S ⊆ N,
and n into s.
In the first section we introduce the Semivalues and the Binomial Semivalues;
the solution of the Inverse Problem for Semivalues is shown in the second section.
The Power Game and the coalitional rationality, together with the main result of
the paper about Binomial Semivalues are discussed in the last section.
1.
Semivalues and Binomial Semivalues
To give the definition of a Semivalue, we need a weight vector pn ∈ Rn , satisfying
a normalization condition
n X
n−1
s=1
s−1
pns = 1,
(1.1)
together with the interpretation: pns is the common weight of all coalitions of size s.
This would be enough for the games in G(N ), but for all the games in GN we need
a sequence of weight vectors defined recursively as follows:
pn−1
= pns + pns+1 ,
s
s = 1, 2, ..., n − 1,
(1.2)
will give the weight vector for the space G(S) with |S| = n − 1. Then, the sequence
of weight vectors pn−2 , pn−3 , ..., p2 , p1 is defined by formulas similar to (1.2), going
up to p11 = 1. From (1.2) it is easy to show that these vectors satisfy a normalization
condition like (1.1). It has been said earlier that (1.2) are the inverse Pascal triangle
conditions, as any sequence shows triangles similar to those present when the Pascal
triangle conditions were defined. Now, we can define the Semivalue associated with
any sequence of weight vectors, p1 , p2 , ..., pn subject to (1.1) and connected by (1.2),
as the value defined on GN by
SEi (T, v, pt ) =
X
S:i∈S⊆T
pts [v(S) − v(S − {i})], ∀i ∈ T, ∀T ⊆ N, T 6= ∅.
(1.3)
Recall that here (T, v) is the TU game, a subgame of (N, v), obtained as a
restriction of the characteristic function to T, so that (T, v) ∈ GT . Notice that for
t 6 n the weight vectors
26
Irinel Dragan
(s − 1)!(t − s)!
, s = 1, 2, ..., t, pts = 21−s , s = 1, 2, ..., t,
t!
give the Shapley Value, and the Banzhaf Value, respectively.
pts =
(1.4)
Example 1. Consider the weight vector p3 = ( 18 , 14 , 83 ), and p2 = ( 38 , 85 ), p1 = (1),
derived via (1.2). Consider the game
v({1}) = v({2}) = v({3}) = 0, v({1, 2}) = v({1, 3}) = v({2, 3}) = v(1, 2, 3}) = 1,
(1.5)
a constant sum game. We may compute the Semivalue of this game, by the formula
(3), to get for the game (1.5) the outcome SE({1, 2, 3}, v, p3) = ( 12 , 12 , 12 ), which is
not efficient, because the sum of components makes a number different of 1. Then,
the Semivalue does not belong to the Core, as the efficiency is missing. As this
is the case in most situations, we have to define the coalitional rationality in some
other way, following the ideas from the earlier work (Dragan/Martinez-Legaz, 2001),
namely to consider the Power Game of the given game, in which the Semivalue is
efficient and may be coalitional rational. Thus we have to define the Power Game.
We may compute also the Semivalues of the subgames
5 5
SE({1, 2}, v, p2) = SE({1, 3}, v, p2) = SE({2, 3}, v, p2) = ( , ).
(1.6)
8 8
They are looking all similar, due to the symmetry in (1.5) of the worth of the
characteristic function of the players. Of course, the Semivalues of singletons are all
zero. Then, we got a new game
w({1}) = w({2}) = w({3}) = 0, w({1, 2}) = w({1, 3}) = w({2, 3}) =
5
,
4
w({1, 2, 3}) =
3
.
2
were we used (6) to satisfy the definition which follows.
Definition 1. For a TU game (N, v), the Power Game, relative to a Semivalue
associated with a weight vector pn , is the game (N, π, pn ) defined by formula
X
π(T, v, pt ) =
SEi (T, v), ∀T ⊆ N,
(1.7)
i∈T
where the components of the Semivalue were given by formula (1.3).
As seen above in example 1, it is not easy to compute the Power Game by means
of (7). However, this may be done by using the following result:
Theorem (Dragan, 2000). Let a Semivalue SE(N, v) be associated with the
weight vector pn , and the Power Game (N, π, pn ), relative to the Semivalue, given
by formula (7). Then, we have
X
π(T, v, pt ) =
[spts − (t − s)pts+1 ]v(S), ∀T ⊆ N,
(1.8)
S⊆T
where
ptt+1
is an arbitrary number.
27
On the Inverse Problem and the Coalitional Rationality
Example 2. Return to the game of Example 1 and recall that the computation of
the Power Game, relative to the Semivalue, by using the definition (7), led to
π({1, 2}, v, p2) = π({1, 3}, v, p2 ) = π({2, 3}, v, p2) =
5
,
4
π({1, 2, 3}, v, p3) =
3
.
2
(1.9)
Now, by using the theorem, as we have the bracket in (8) given by
5
1
9
, and 2p32 − p33 = , 3p33 = ,
(1.10)
4
8
8
with the second equality used three times for coalitions of size two, from (1.10)
and formula (1.8) we get the same worth for the characteristic function as in (1.9).
Beside definition 1, we illustrated the usefulness of (1.8), relative to (1.7).
Now, that we have the Power Game (1.9) for our given game (1.5), and the
Semivalue is efficient in this game, which obviously is always true, we may check
whether or not, the Semivalue is in the Core of the Power Game, and conclude that
this is not true; hence according to the ideas from Dragan/Martinez-Legaz (2001),
the Semivalue of (1.5) is not coalitional rational. Of course, it may be possible that
the Semivalue does belong to the Core of the Power Game, and in this last case
it will be coalitional rational. For example, if we consider the Banzhaf Value, the
most popular Semivalue, defined by p3 = ( 14 , 41 , 14 ), and compute the Power Game,
then we find out that the value is coalitional rational. Looking at our Example 2
we think that it is justified to introduce the following:
2p22 =
Definition 2. The Semivalue of a given game is coalitional rational if it belongs to
the Core of the Power Game relative to the Semivalue (or, the Power Core of the
game).
Now, let us consider the Binomial Semivalues, introduced by Puerte (2000) and
discussed also in the work by Puerte/Freixas (2002), where definition 2 applies.
Definition 3. The Semivalue SE associated with the sequence of normalized weight
vectors p1 , p2 , ..., pn connected by the inverse Pascal triangle relationships, is a Binomial Semivalue, if the weight vectors satisfy also for some number r ∈ (0, 1], the
equalities
pn2
pn
pn
= 3n = ... = nn = r.
n
p1
p2
pn−1
(1.11)
Now, that the concepts of Power Game and coalitional rationality have been
explained and the computation of the Power Game has been given, we notice:
Lemma 1. In GN , the weights of a Binomial Semivalue are given by the equalities
rs−1
, s = 1, 2, ..., t, t 6 n,
(1 + r)t−1
so that the Binomial Semivalue is given by the formula
pts =
SEi (T, v, pt ) =
X
S:i∈S⊆T
(1.12)
rs−1
[v(S) − v(S − {i})], ∀i ∈ T, ∀T ⊆ N, T 6= ∅.
(1 + r)t−1
(1.13)
28
Irinel Dragan
Proof. Follows from (1.11) and the normalization condition (1.1), as well as formula
(1.3).
⊔
⊓
ę
Remarks: (a) From the inverse Pascal triangle relationships it follows that all
weight vectors of the sequence are given by formulas similar to (1.12), obtained for
different values of t = 1, 2, ..., n, and while the game is replaced by the Value, a fact
which justifies the study of the Binomial Semivalues, that should have properties
similar to those of the Banzhaf Value.
Now, the problem to be considered in this paper is: for a given vector L, and
a given TU game (N, v), such that the Binomial Semivalue corresponding
to a parameter r is not coalitional rational, find out in the Inverse Set of
L a TU game (N, w), for which the Binomial Semivalue with parameter r
is the same, but belongs to the Core of the Power Game. Clearly, we have
to explain the procedure in two steps:
• How do we find the Inverse Set of a Binomial Semivalue associated with a
parameter r, and an a priori given value L? From our previous work on general
Semivalues we know that this should be determined by an explicit formula,
hence the Inverse Set will be available. This will be discussed in the second
section.
• In the Inverse Set, how do we get a TU game for which the Binomial Semivalue
of the original game is in the Core of its Power Game? This will be discussed
in the last section. What about games for which the Binomial Semivalue of the
original game does not belong to the Power Core? like the one from example 1.
2.
The Null Space and the Inverse Set
In a recent work (Dragan, 2004), it has been shown that the Semivalue, associated
with a sequence of weight vectors derived from pn by means of formulas of type
(1.2), has a potential function and for a game (N, v), it is given by the formula
X
P (N, v, pn ) =
pns v(S).
(2.1)
S⊆N
Thus, for a Binomial Semivalue (2.1) becomes
P (N, v, pn ) =
X
1
rs−1 v(S).
n−1
(1 + r)
(2.2)
S⊆N
Obviously, to make (2.2) computationally better, the sum may be written as
P (N, v, pn ) =
n
X
1
rs−1 ds (N, v),
(1 + r)n−1 s=1
(2.3)
where ds (N, v) is the sum of worth of the characteristic function for all subcoalitions of size s in the set of players N.
Example 3. Returning to the game (1.5) of Example 1, and the weight vector p3 =
( 18 , 41 , 38 ), we see that d1 (N, v) = 0, d2 (N, v) = 3, d3 (N, v) = 1, so that from (2.3) we
1
2
get P (N, v, p3 ) = (1+r)
2 (3r + r ), where the expressions of weights in terms of the
29
On the Inverse Problem and the Coalitional Rationality
ratio r were used. Now, for any coalition of size two, again from (2.3), we get the
r
potential P (N − {i}, v, p2) = 1+r
, i = 1, 2, 3, Hence, the Binomial Semivalue is
SEi (N, v, p3 ) = P (N, v, p3 ) − P (N − {i}, v, p2 ) =
3r + r2
r
2r
−
=
,
2
(1 + r)
1+r
(1 + r)2
i = 1, 2, 3, (2.4)
which for the Banzhaf Value (r = 1) becomes as above B(N, v) = ( 21 , 12 , 21 ).
In the more recent work (Dragan, 2013), we considered a basis for the space
G(N), consisting of the linearly independent vectors in the set
defined by the formulas
l
wT (T ) =
1
,
ptt
(2.5)
W = {wT ∈ Rn : T ⊆ N, T 6= ∅},
wT (S) =
s−t (−1)
X
l=0
s−t
l
pt+l
t+l
, ∀S ⊃ T,
wT (S) = 0,
otherwise.
(2.6)
For all these vectors were computed the Binomial Semivalues associated with
the weight vectors in the sequence generated by the vector pn , via (1.2), by using
the formulas for the Binomial Semivalues (1.3) obtained in Lemma 1. Here, we try
to compute the same Binomial Semivalues by means of the potential, using
SEi (N, wT , pn ) = P (N, wT , pn ) − P (N − {i}, wT , pn−1 ), ∀i ∈ N,
(2.7)
like in example 3 above. To get the experience needed in this computation let
us consider first an example.
Example 4. From (2.6), the general three person game shows the basis W, that
taking into account (1.2) becomes
1
1
2
1
1 1
1
, 1 − 2 , 0, 1 − 2 + 3 ) = (1, 0, 0, − , − , 0, 2 ),
2
p2
p2
p2
p3
r r
r
1
1
2
1
1
1 1
w{2} = (0, 1, 0, 1 − 2 , 0, 1 − 2 , 1 − 2 + 3 ) = (0, 1, 0, − , 0, − , 2 ),
p2
p2
p2
p3
r
r r
1
1
2
1
1 1 1
w{3} = (0, 0, 1, 0, 1 − 2 , 1 − 2 , 1 − 2 + 3 ) = (0, 0, 1, 0, − , − , 2 ),
p2
p2
p2
p3
r r r
1
1
1
1+r
1+r
w{1,2} = (0, 0, 0, 2 , 0, 0, 2 − 3 ) = (0, 0, 0,
, 0, 0, − 2 ),
p2
p2
p3
r
r
1
1
1
1+r
1+r
w{1,3} = (0, 0, 0, 0, 2 , 0, 2 − 3 ) = (0, 0, 0, 0,
, 0, − 2 ),
p2
p2
p3
r
r
1 1
1
1+r 1+r
w{2,3} = (0, 0, 0, 0, 0, 2 , 2 − 3 ) = (0, 0, 0, 0, 0,
, − 2 ),
p2 p2
p3
r
r
2
1
(1 + r)
w{1,2,3} = (0, 0, 0, 0, 0, 0, 3 ) = (0, 0, 0, 0, 0, 0,
).
p3
r2
w{1} = (1, 0, 0, 1 −
(2.8)
30
Irinel Dragan
Obviously, these are linearly independent vectors and their number equals the
dimension of the space, hence they form a basis. Let us compute the potentials
of all basic vectors, by using formula (2.3). For i, j, k = 1, 2, 3, we have the sums
2
1
d1 (N, w{i} ) = 1, d2 (N, w{i} ) = − , d3 (N, w{i} ) = 2 ,
r
r
1+r
1+r
d1 (N, w{i,j} ) = 0, d2 (N, w{i,j} ) =
, d3 (N, w{i,j} ) = − 2 ,
r
r
(1 + r)2
d1 (N, w{i,j,k} ) = d2 (N, w{i,j,k} ) = 0, d3 (N, w{i,j,k} ) =
,
r2
(2.9)
and from (2.3) we obtain
p3s =
rs−1
,
(1 + r)2
s = 1, 2, 3.
(2.10)
Now, by (2.9) and (2.10), for every basic vector (N, w) shown above in (2.8),
formula (2.1) written as
P (N, w, p3 ) =
will give
P (N, ws , p3 ) =
3
X
1
rs−1 ds (N, w),
(1 + r)2 s=1
1
[1 + (−1)]s−1 = 0, ∀S ⊂ N.
(1 + r)s−1
(2.11)
(2.12)
while the potential of the last game equals 1. Now, the potentials of the subgames
would be computed, by using a formula similar to (2.11), namely
P (N − {i}, wS , p2 ) =
2
1 X s−1
r dS (N, w), ∀S ⊂ N − {i}, i ∈ N.
1 + r s=0
(2.13)
In the same way, we get zero, and the potential of the last game equals 1.
Then, the Semivalues computed by the formula (2.7) are
SE(N, w{i} , p1 ) = (0, 0, 0), ∀i ∈ N,
SEj (N, wN −{i} , p2 ) = −δji ,
i = 1, 2, 3;
(2.14)
SE(N, wN , p3 ) = (1, 1, 1).
A similar approach is hepful in proving, by computing the potentials, the result:
Theorem (Thm.3, Dragan, 2013). Let a Binomial Semivalue be defined by a
parameter r, and let W be the basis of the space provided by formulas (2.5), (2.6).
Then, we have
SE(N, wT , pt ) = 0, ∀T ⊂ N, |T | 6 n − 2, T 6= ∅,
31
On the Inverse Problem and the Coalitional Rationality
SEi (N, wN −{i} pn ) = −1, ∀i ∈ N, SEj (N, wN −{i} , pn ) = 0, j 6= i, ∀i ∈ N, (2.15)
SEi (N, wN , pn ) = 1, ∀i ∈ N.
As a Corollary, by the linearity of the Semivalue, we get the Inverse Set, where
r enters only the basic vectors:
Theorem (Thm.6, Dragan, 2013). Let a Binomial Semivalue for a game (N, w),
defined by a parameter r, be SE(N, v, pn ) = L. Let W given by (2.5), (2.6) be a
basis for the space G(N ). Then, the solution of the Inverse Problem is expressed by
the formula
w=
X
aS wS + aN (wN +
X
i∈N
S⊂N,|S|6n−2
X
wN −{i} ) −
i∈N
Li wN −{i} ,
(2.16)
where the constants multiplying the basic games are arbitrary.
3.
The Power Game and the coalitional rational inverse
Consider in the Inverse Set the family of games, to be called the “almost null games”,
obtained for aS = 0, ∀S ⊂ N, |S| 6 n − 2, S 6= ∅. This family, as seen in the
formula (2.16), is given by
X
X
w = aN (wN +
wN −{i} ) −
Li wN −{i} ,
(3.1)
i∈N
i∈N
where aN is the parameter of the family; of course, the parameter r of the Binomial
Semivalue occurs in the basic vectors. Now, by using the weight vectors (2.6), written
in terms of r, as shown in (Dragan, 2013), we have
wT (T ) =
(1 + r)t−1
, ∀T ⊆ N,
rt−1
s−t
wT (S) =
(−1)
(1 + r)t−1
, ∀S ⊃ T,
rs−1
(3.2)
and wT (S) = 0, otherwise, so that from (3.2) we obtain
wN −{i} (N − {i}) =
(1 + r)n−2
, ∀i ∈ N,
rn−2
wN −{i} (N ) = −
(1 + r)n−2
,
rn−1
(3.3)
(1 + r)n−1
.
(3.4)
rn−1
In this way, from (3.3), (3.4), the components different of zero in (3.1) are:
wN −{i} (N − {j}) = 0, ∀j 6= i,
(1 + r)n−2
, ∀i ∈ N,
rn−2
(3.5)
X
(1 + r)n−2
[aN (r − n + 1) +
Li ].
n−1
r
(3.6)
w(N − (i}) = (aN − Li )
w(N ) =
wN (N ) =
i∈N
32
Irinel Dragan
Now, we compute the Power Game of an arbitrary game in the almost null
family set, given in (3.5), (3.6), where the null values of the characteristic function
are omitted, and we get:
(3.7)
π(N − {i}, v, pn−1 ) = (n − 1)(aN − Li ), ∀i ∈ N,
π(N, v, pn ) =
X
(3.8)
Li .
i∈N
As stated in definition 2 before, a Semivalue of a given game is coalitional rational if it belongs to the Core of the Power Game, or Power Core. If the Binomial
Semivalue is L > 0, and, as seen in (3.8), this is efficient in the Power Game, then
the only Core conditions are those obtained from the coalitions of size n − 1, that
have the worth shown in (3.7), namely
X
Lj > (n − 1)(aN − Li ), ∀i ∈ N,
(3.9)
or
j∈N −{i}
aN 6
X
1
[
n−1
j∈N −{i}
Lj + (n − 1)Li ], ∀i ∈ N.
(3.10)
We proved
Theorem 2. A Binomial Semivalue associated with the parameter r, and given by a
nonnegative vector L ∈ Rn , is coalitional rational, in the Power Game of the game
X
X
w = aN (wN +
wN −{i} ) −
Li wN −{i} ,
(3.11)
i∈N
i∈N
If and only if aN satisfies the inequality
aN 6
X
1
M in{
n−1
j∈N −{i}
Lj + (n − 1)Li },
(3.12)
Notice that there is also an infinite set of games in the almost null Inverse Set
for which the Binomial Semivalue is not coalitional rational.
Example 5. Return to the game considered in Example 1, for which the Banzhaf
Value is B(N, v) = ( 12 , 12 , 21 ), so that the inequality (3.12) is aN 6 1. We compute
the almost null game, relative to the Banzhaf Value by (3.5) and (3.6) , and we
obtain
w({1}) = w({2}) = w({3}) = 1, w({1, 2}) = w({1, 3}) = w({2, 3}) = w({1, 2, 3}) = 1,
(3.13)
that incidentally coincides with the given game (1.5). Obviously, the Banzhaf Value
is the same, that is B(N, w) = ( 12 , 12 , 21 ). Then, we compute the Power Game, of
(3.13) by (3.7) and (3.8) and we get
3
,
2
(3.14)
π({1, 2}, w, p2 ) = π({1, 3}, w, p2 ) = π({2, 3}, w, p2 ) = 1, π({1, 2, 3}, w, p3) =
33
On the Inverse Problem and the Coalitional Rationality
while we have null values for the singletons. Now, in the new game (3.14) the old
Banzhaf Value is efficient and we may see that it is also in the Power Core. This
happened because for our game we have aN = 1, which satisfies (3.12) and we have
to modify only w({1, 2, 3}) = 1, into π({1, 2, 3}, w, p3) = 32 , to get the Banzhaf
Value in the Power Core. Notice that the Power Game does not have the same
Banzhaf Value as the original one, or the almost null game in the Inverse Set; for
example, in our case the Banzhaf Value of (3.14) is B(N, π) = ( 58 , 85 , 58 ). Obviously,
this is not efficient again and the coalitional rationality conditions do not hold.
Consider the same game, but take aN = 23 , and compute again the almost null
game, relative to the Banzhaf Value by (3.5) and (3.6), and we obtain
w({1}) = w({2}) = w({3}) = 0, w({1, 2}) = w({1, 3}) = w({2, 3}) = 2,
w({1, 2, 3}) = 0, (3.15)
which gives the same old Banzhaf Value. Further, we compute the Power Game and
we get
3
,
2
(3.16)
in which the old Banzhaf Value is efficient, but it is not coalitional rational, because
(3.12) does not hold. These two examples illustrate theorem 2 and the technique to
build the game in the almost null inverse family, for which the given Banzhaf Value
is coalitional rational.
π({1, 2}, w, p2 ) = π({1, 3}, w, p2 ) = π({2, 3}, w, p2 ) = 2, π({1, 2, 3}, w, p3) =
References
Banzhaf, J.F. (1965). Weighted voting doesn’t work; a mathematical analysis. Rutgers Law
Review, 19, 317–343.
Dragan, I. (1991). The potential basis and the weighted Shapley value. Libertas Mathematica, 11, 139–150.
Dragan, I. (1996). New mathematical properties of the Banzhaf value. E.J.O.R., 95, 451–
463.
Dragan, I. and Martinez-Legaz, J. E. (2001). On the Semivalues and the Power Core of
cooperative TU games. IGTR, 3, 2&3, 127–139.
Dragan, I. (2004). On the inverse problem for Semivalues of cooperative TU Games. IJPAM, 4, 545–561.
Dragan, I. (2013). On the Inverse Problem for Binomial Semivalue. Proc. GDN2013
Conference, Stockholm, 191–198.
Dubey, P., Neyman, A. and Weber, R. J. (1981). Value theory without efficiency.
Math.O.R.,6, 122–128.
Puente, M. A. (2000). Contributions to the representability of simple games and to the
calculus of solutions for this class of games. Ph.D.Thesis, University of Catalonya,
Barcelona, Spain.
Freixas, J., and Puente, M. A., (2002). Reliability importance measures of the components
in a system based upon semivalues and probabilistic values. Ann.,O.R., 109, 331–342.
Shapley, L. S., (1953). A value for n=person games. Annals of Math.Studies, 28, 307–317.
Stackelberg Oligopoly Games:
the Model and the 1-concavity of its Dual Game
Theo Driessen1 , Aymeric Lardon2 and Dongshuang Hou3
University of Twente,
Department of Applied Mathematics,
P.O. Box 217, 7500 AE Enschede, The Netherlands
E-mail: [email protected]
2
Université Jean Monnet,
Saint-Etienne, France,
E-mail: [email protected]
3
North-Western Politechnical University,
Faculty of Applied Mathematics,
Xi’an, China,
E-mail: [email protected]
1
Abstract This paper highlights the role of a significant property for the
core of Stackelberg Oligopoly cooperative games arising from the noncooperative Stackelberg Oligopoly situation with linearly decreasing demand
functions. Generally speaking, it is shown that the so-called 1-concavity
property for the dual of a cooperative game is a sufficient and necessary
condition for the core of the game to coincide with its imputation set. Particularly, the nucleolus of such dual 1-concave TU-games agree with the
center of the imputation set. Based on the explicit description of the characteristic function for the Stackelberg Oligopoly game, the aim is to establish,
under certain circumstances, the 1-concavity of the dual game of Stackelberg
Oligopoly games. These circumstances require the intercept of the inverse
demand function to be bounded below by a particular critical number arising
from the various cost figures.
Keywords: Stackelberg oligopoly game; imputation set; core; efficiency; 1concavity
1.
Introduction of game theoretic notions
A cooperative savings game (with transferable utility) is given by a pair hN, wi,
where its characteristic function w : P(N ) → R is defined on the power set P(N ) =
{S | S ⊆ N } of the finite set N , of which the elements are called players, while the
elements of the power set are called coalitions. The so-called real-valued worth w(S)
of coalition S ⊆ N in the game hN, wi represents the maximal amount of monetary
benefits due to the mutual cooperation among the members of the coalition, on the
understanding that there are no benefits by absence of players, that is w(∅) = 0. In
the framework of the division problem of the benefits w(N ) of the grand coalition
N among the potential players, any allocation scheme of the
P form x = (xi )i∈N ∈
RN is supposed to meet, besides the efficiency principle
i∈N xi = w(N ), the
so-called individual rationality condition in that each player is allocated at least
the individual worth, i.e., xi > w({i}) for all i ∈ N . Concerning the development
of the solution part, a (multi- or single-valued) solution concept σ assigns to any
cooperative game hN, wi a (possibly empty) subset of its imputation set I(N, w),
that is σ(N, w) ⊆ I(N, w), where
35
Stackelberg Oligopoly Games
I(N, w) = {(xi )i∈N ∈ RN |
X
xi = w(N )
i∈N
and xi > w({i}) for all i ∈ N }.
The best known multi-valued solution concept called core requires the group rationality condition in that the aggregate allocation to the members of any coalition is
at least its coalitional worth, that is
X
→
CORE(N, w) = {−
x ∈ I(N, w) |
xi > w(S)
i∈S
for all S ⊆ N , S 6= N , S 6= ∅} (1.1)
Of significant importance is the upper core bound composed of the marginal contributions mw
i = w(N ) − w(N \{i}), i ∈ N , with respect to the formation of the
grand coalition N in the game hN, wi. Obviously, xi 6 mw
i for all i ∈ N and all
→
−
x ∈ CORE(N, w). In this context, we focus on the following core catcher called
CoreCover
X
CC(N, w) = {(xi )i∈N ∈ RN |
xi = w(N ) and xi 6 mw
i
i∈N
for all i ∈ N } (1.2)
In the framework of the core, a helpful
Ptool appears to be the so-called gap function
g w : P(N ) → R defined by g w (S) = i∈S mw
i − w(S) for all S ⊆ N , S 6= ∅, where
g w (∅) = 0. So, the gap g w (S) of any coalition S measures how much the coalitional
worth w(S) differs from the aggregate allocation based on the individually marginal
contributions. The interrelationship between the gap function and the general inclusion CORE(N, w) ⊆ CC(N, w) is the following equivalence (Driessen, 1988):
CORE(N, w) = CC(N, w)
⇐⇒
0 6 g w (N ) 6 g w (S)
for all S ⊆ N , S 6= ∅ (1.3)
In words, the core catcher CC(N, w) coincides with the core CORE(N, w) only if
the non-negative gap function g w attains its minimum at the grand coalition N . If
the latter property (1.3) holds, the savings game hN, wi is said to be 1-convex.
With every cooperative savings game hN, wi there is associated its dual game
hN, w∗ i defined by w∗ (S) = w(N ) − w(N \S) for all S ⊆ N . That is, the worth of
any coalition in the dual game is given by the coalitionally marginal contribution
with respect to the formation of the grand coalition N in the original game. Partic(w ∗ )
ularly, w∗ (∅) = 0, w∗ (N ) = w(N ), and so, mi
= w∗ (N ) − w∗ (N \{i}) = w({i})
for all i ∈ N . We arrive at the first main result.
Proposition 1.1. Three equivalent statements for any cooperative savings game
hN, wi.
X
∗
I(N, w) 6= ∅ ⇐⇒ w(N ) >
w({i}) ⇐⇒ g (w ) (N ) 6 0
(1.4)
i∈N
36
Theo Driessen, Aymeric Lardon, Dongshuang Hou
In fact, the dual game hN, w∗ i of any cooperative savings game hN, wi is treated
as a cost game such that the core equality CORE(N, w∗ ) = CORE(N, w) holds,
on the understanding that the core of any cost game is defined through the reversed
→
→
inequalities of (1.1). Thus, −
x ∈ CORE(N, w∗ ) iff −
x ∈ CORE(N, w). As the
counterpart to 1-convex savings games (with non-negative gap functions), we deal
with so-called 1-concave cost games (with non-positive gap functions).
Definition 1.2. A cooperative cost game hN, wi is said to be 1-concave if its nonpositive gap function attains its maximum at the grand coalition N , i.e.,
g w (S) 6 g w (N ) 6 0 for all S ⊆ N , S 6= ∅.
(1.5)
Theorem 1.3. Three equivalent statements for any cooperative savings game hN, wi.
(i) The dual game hN, w∗ i is 1-concave, that is (1.5) applied to hN, w∗ i holds
(ii)
w(N ) >
X
w({i})
i∈N
(iii) I(N, w) 6= ∅
and
w(S) 6
X
w({i})
i∈S
and
for all S ⊆ N , S 6= N , S 6= ∅
(1.6)
CORE(N, w) = I(N, w)
Proof. In view of Proposition 1.1, together with CORE(N, w) ⊆ I(N, w), it remains
to prove the implication (iii) =⇒ (ii). By contra-position, suppose
(ii) does not hold
P
in that there exists S ⊆ N , S 6= N , S 6= ∅ with w(S) > i∈S w({i}). Define the
→
1
allocation −
x = (xi )i∈N ∈ RN by xi = w({i}) for all i ∈ S and xi = w({i}) + n−s
·
P
→
w(N ) −
w({j}) for all i ∈ N \S. Obviously, −
x ∈ I(N, w)\CORE(N, w). ✷
j∈N
2.
The Stackelberg oligopoly game
The normal form game of the non-cooperative Stackelberg oligopoly situation1 is
modeled as a cooperative TU-game as follows.
Throughout the paper we fix the set N of firms with (possibly identical) strategy
sets Xi = [0, wi ), i ∈ N , with reference to (possibly unlimited) capacities wi ∈
[0, ∞], i ∈ N , (possibly distinct) marginal costs ci > 0, i ∈ N , and the inverse
demand function p(x) = a − x for all x 6 a and p(x) = 0 for all x > a. In this
framework, the corresponding individual profit functions πi : Πk∈N Xk → R, i ∈ N ,
and coalitional profit functions πT : Πk∈N Xk → R, T ⊆ N , T 6= ∅, are defined by
X
πi ((xk )k∈N ) = (a − X(N ) − ci ) · xi and πT ((xk )k∈N ) =
πj ((xk )k∈N ) (2.7)
j∈T
P
where X(N ) = k∈N xk ∈ R represents the aggregate production and a > 2 · n ·
maxi∈N ci .
The Stackelberg oligopoly model is based on a two-stage procedure. Given that the
1
For a description of the oligopoly situation, we refer to the PhD thesis of Aymeric Lardon
(Lardon, 2011).
37
Stackelberg Oligopoly Games
members of the coalition S are supposed to perform their leadership in the first stage
maximizing its coalitional profit by taking into account the best responses of individual followers j ∈ N \S (i.e., the non-members of S) during the second stage. So, the
second stage is devoted to the maximization problems maxxj ∈Xj πj (xj , (xk )k∈N \{j} )
for all j ∈ N \S.
Theorem 2.1. Let S ⊆ N , S 6= N . Write cT =
P
k∈T
ck for all T ⊆ N , T 6= ∅.
(i) The best response of any individual i ∈ N \S during the second stage is given by
yi =
cN \S + X(N \S)
a − X(S) + cN \S
− ci =
− ci
n−s
n+1−s
(ii) The worth v(S) of coalition S is determined by
(x∗iS )2
∗
1
v(S) =
where
xiS = 2 · a + cN \S − (n + 1 − s) · ciS
n+1−s
is the maximizer of the profit function of player iS ∈ S with the smallest marginal
contribution among members of S, supposing other members of S produce nothing.
Note that x∗i,S > 0 because of a > n · maxi∈N ci .
Proof. Fix coalition S 6= N . For all i ∈ N \S the maximization problem of the
player’s profit function πi ((xk )k∈N ) = (a − X(N ) − ci ) · xi = −(xi )2 + xi · (a − ci −
i
X(N \{i})) is solved through its first order condition ∂π
∂xi = 0, yielding
1
or equivalently,
xi = a − ci − X(N )
xi = 2 · a − ci − X(N \{i})
Summing up the latter equations over all i ∈ N \S yields
X(N \S) = (n − s) · a − cN \S − (n − s) · X(N )
and so, a − X(N ) =
Hence, by substitution, it holds for all i ∈ N \S
yi = (a − X(N )) − ci =
cN \S + X(N \S)
n−s
cN \S + X(N \S)
a − X(S) + cN \S
− ci =
− ci
n−s
n+1−s
This proves part (i). Given these best responses by players in N \S, the maximization problem of the coalitional profit function πS is, due to unlimited capacities,
equivalent to the maximization problem of the profit function of the firm iS ∈ S
with smallest marginal cost among members of S, (i.e., ciS 6 ci for all i ∈ S),
supposing that the other members of S produce nothing. In this framework,
a − xiS + cN \S
− ci
for all i ∈ N \S and thus,
n+1−s
n−s
y(N \S) =
· a − xiS + cN \S − cN \S
n+1−s
1
= a − xis −
· a − xiS + cN \S
n+1−s
yi =
38
Theo Driessen, Aymeric Lardon, Dongshuang Hou
Hence, we focus on the player’s profit function of the form
→
−
πiS ((yi )i∈N \S , xiS , ( 0 )i∈S\{iS } )
= a − X(N ) − ciS · xiS = a − xiS − y(N \S) − ciS · xiS
1
· a − xiS + cN \S − ciS · xiS
n+1−s
1
=
· a + cN \S − (n + 1 − s) · ciS − xiS · xiS
n+1−s
=
The first order condition yields that the maximizer of this quadratic profit function is given by
∗
1
xiS = 2 · a + cN \S − (n + 1 − s) · ciS
and finally,
→
−
v(S) = πiS ((yi )i∈N \S , x∗iS , ( 0 )i∈S\{iS } )
1
=
· a + cN \S − (n + 1 − s) · ciS − x∗iS · x∗iS
n+1−s
(x∗iS )2
1
∗
∗
=
· 2 · xiS − xiS · x∗iS =
n+1−s
n+1−s
It remains to check the non-negativity constraint for the maximizer x∗iS (since
production levels are supposed to be non-negative). Of course, the non-negativity
constraint also applies to any player j ∈ N \S. For that purpose, choose a sufficiently
large in that a > 2 · n · maxi∈N ci (or to be exact, a > 2 · n · maxi∈N ci ). Recall that
production levels of all firms are supposed to be unlimited. This proves part (ii). ✷
In the context of the resulting cooperative TU game, the following significant
notions appear. For any non-trivial coalition T ⊆ N , T 6= ∅, let cT , c̄T , and cT
respectively, denote the aggregate, average, and minimal cost of coalition T , that is
cT =
X
ck
c̄T =
k∈T
c̄c̄T =
1 X
·
(ck )2
|T |
k∈T
cT
|T |
Note that
cT = min{ck |
k ∈ T }.
Moreover, (2.8)
2 X
X
ck − c̄T =
(ck )2 − |T | · (c̄T )2 (2.9)
k∈T
k∈T
Generally speaking, c̄T > cT and moreover, the equality is met only by identical
marginal costs, that is, for any coalition, the average cost equals the minimum cost if
and only if all the marginal costs of its members do not differ. By (2.9), c̄c̄T > (c̄T )2 ,
that is the average of the squares of marginal costs covers the square of the average
cost.
39
Stackelberg Oligopoly Games
Theorem 2.2. Given the normal form game hN, (ck )k∈N , (wk )k∈N , ai of the noncooperative Stackelberg oligopoly situation with unlimited capacities (wi = +∞ for
all i ∈ N ) and possibly distinct marginal costs, then the corresponding cooperative
n-person Stackelberg oligopoly game hN, vi is determined by v(∅) = 0 and for all
S ⊆ N , S 6= ∅,
2
2
a + cN \S − (n + 1 − s) · cS
a + cN \S
n+1−s
=
·
− cS
v(S) =
(2.10)
4 · (n + 1 − s)
4
n+1−s
Here c∅ = 0. In case all marginal costs are identical, say ci = c for all i ∈ N ,
2
1
then (2.10) reduces to v(S) = (a−c)
· n+1−s
for all S ⊆ N , S 6= ∅, and so, the
4
Stackelberg oligopoly game is a multiple of the symmetric n-person cooperative
1
, the imputation set of which degenerates into the single core
game v(s) = n+1−s
1
allocation n · (1, 1, . . . , 1) ∈ Rn .
3.
1-Concavity of the dual game of Stackelberg oligopoly games
Assuming non-emptiness of its core, our goal is to study whether or not the 1convexity property applies to the Stackelberg oligopoly game. For that purpose,
we are interested in the structure of the corresponding non-negative
gap function.
P
Generally speaking, the validity of the
inequality
v(S)
6
v({k})
is equivalent
k∈S
P
to the reversed inequality g v (S) > k∈S g v ({k}) for all S ⊆ N , S 6= N , S 6= ∅.
Together with the non-negativity of the gap function g v , it follows that g v (S) >
g v ({i}) whenever i ∈ S. In words, the gap function g v of the Stackelberg oligopoly
game attains among non-trivial coalitions containing a given player its minimum
either at the one-person coalition or the grand coalition N . According to the next
proposition, the gap of the grand coalition is not minimal and hence, the 1-convexity
property fails to hold for the Stackelberg oligopoly games. However, the solution
concept called τ -value (cf. Tijs, 1981) agrees, concerning its allocation to any player
i, with the efficient compromise between the marginal contribution mvi = v(N ) −
v(N \{i}) and the stand-alone worth v({i}), i ∈ N (treated as upper and lower core
bounds respectively).
Proposition 3.1. Given the non-emptiness of the core of the Stackelberg oligopoly
game hN, vi, the corresponding gap function g v satisfies g v (N ) > g v ({i}) for all
i ∈ N.
Proof. Fix i ∈ N . Recall that g v (N \{k}) = g v (N ) for all k ∈ N . Fix j ∈ N , j 6= i.
Due to the non-emptiness of the core, mvk > v({k}) for all k ∈ N . We conclude that
g v (N ) − g v ({i}) = g v (N \{j}) − mvi + v({i})
X
mvk − v(N \{j}) + v({i})
=
k∈N \{i,j}
>
X
k∈N \{i,j}
=
X
k∈N \{i,j}
mvk −
X
v({k}) + v({i})
k∈N \{j}
(mvk − v({k})) > 0
40
Theo Driessen, Aymeric Lardon, Dongshuang Hou
Here we applied the inequality v(S) 6
P
k∈S
v({k}) to S = N \{j}).
✷
Now we arrive at the main result stating that the core of any Stackelberg oligopoly
game coincides with its imputation set, provided its non-emptiness. By Theorem
1.3, the class of dual games of Stackelberg oligopoly savings games is a significant
class of 1-concave (cost) games. The proof proceeds by checking the validity of
(1.6). Note that the worth of any single player i ∈ N and the grand coalition N
respectively, are given as follows:
v({i}) =
a + cN − (n + 1) · ci
4·n
2
for all i ∈ N , and v(N ) =
(a − cN )2
(3.11)
4
Theorem 3.2. The dual game hN, v ∗ i of the cooperative n-person Stackelberg oligopoly
game hN, vi of the form (2.10) with distinct marginal costs is 1-concave only if the
intercept a > 0 of the inverse demand function is large enough. On the one hand,
∗
g (v ) (N ) 6 0
if and only if
a>
L1
− cN
2
(3.12)
where the critical number L1 represents the lower bound given by
−1 2
2
L1 = c̄N − cN
· (n + 1) · c̄c̄N − [cN + cN ]
∗
∗
g (v ) (S) 6 g (v ) (N )
On the other,
for all S ⊆ N , S 6= ∅
(3.13)
(3.14)
Proof of Theorem 3.2. (The full proof consists of two parts.) Part 1.
(v ∗ )
(N ) 6 0 or equivalently, by Proposition (1.1) 1.1, v(N ) >
P Firstly, we check g
v({i}).
Put
the
substitution
x = a + cN . By using (3.11), it holds that 4 · n ·
i∈N
v({i}) = [x − (n + 1) · ci ]2 for all i ∈ N as well as 4 · n · v(N ) = n · [a − cN ]2 =
n · [x − (cN + cN )]2 . Thus, we obtain the following chain of equalities:
4 · n · v(N ) −
X
i∈N
v({i})
2 X 2
= n · x − (cN + cN ) −
x − (n + 1) · ci
i∈N
2 2
= n · x − 2 · x · (cN + cN ) + cN + cN
−
X
i∈N
2
2
2
x − 2 · x · (n + 1) · ci + (n + 1) · (ci )
= 2 · x · (n + 1) · cN − n · (cN + cN )
41
Stackelberg Oligopoly Games
+ n · cN + cN
2
− (n + 1)2 ·
X
(ci )2
i∈N
2
= 2 · x · cN − n · cN + n · cN + cN − (n + 1)2 · n · c̄c̄N
= n · 2 · x · c̄N − cN
2
2
+ cN + cN − (n + 1) · c̄c̄N
∗
So far, we conclude that g (v ) (N ) 6 0 if and only if
2
2 · x · c̄N − cN > (n + 1) · c̄c̄N − cN + cN
2
where
x = a + cN (3.15)
or equivalently, a > L21 − cN where the critical lower bound L1 is given by (3.13).
Notice that the quadratic term x2 vanishes in the inequality (3.15).
✷
Remark 3.3. For future convenience, we treat an alternative proof of part 1 of
Theorem 3.2 in the appendix. They differ in that this second proof is based on the
variable a itself instead of the variable x. In the new setting, the description of the
critcal lower bound (3.15) has to be replaced by a similar inequality:
2 · a · c̄N − cN > (n2 + 2 · n) · c̄c̄N − (c̄N )2 + c̄c̄N − (cN )2
(3.16)
This second approach yields an alternative description of the same lower bound of
the form
−1 2
2
2
L2 = c̄N − cN
· (n + 2 · n) · c̄c̄N − (c̄N ) + c̄c̄N − (cN )
It is left to the reader to verify the validity of the equality L2 = L1 − 2 · cN .
(3.17)
✷
Proof of Theorem 3.2. (The full proof consists of two parts.) Part 2.
∗
∗
)
Secondly, we check g (v P
(S) 6 g (v ) (N ) for all S ⊆ N , S =
6 ∅, or equivalently,
by Theorem 1.3, v(S) 6
v({i})
for
all
S
⊆
N
,
S
=
6
N , S 6= ∅. Put the
i∈S
fundamental substitutions
x := a + cN
as well as
AS := cS + (n + 1 − s) · cS
From (2.10), we derive the following shortened notation for the worth of any multiperson coalition S as well as the one-person coalitions respectively, in the Stackelberg oligopoly game.
42
Theo Driessen, Aymeric Lardon, Dongshuang Hou
4 · n · v(S) =
2
n · x − AS
n+1−s
=
4 · n · v({i}) = x − (n + 1) · ci
4·n·
X
v({i}) =
i∈S
X
i∈S
n · x2 − 2 · x · AS + (AS )2
2
for all i ∈ N , and next
2
X
i∈S
2
2
x − 2 · x · (n + 1) · ci + (n + 1) · (ci )
= s · x2 − 2 · x · (n + 1) · cS + (n + 1)2 ·
4·n·
for all S ⊆ N ,
n+1−s
X
(ci )2
i∈S
v({i}) − v(S) = α2 · x2 + α1 · x + α0
(3.18)
Our main goal is to describe the coalitional notion of surplus in terms of a
quadratic function of the variable x, say f (x) = α2 · x2 + α1 · x + α0 where α2 > 0.
Definition 3.4. The three real numbers αk , k = 0, 1, 2, are given as follows:
(n − s) · (s − 1)
n
=
n+1−s
n+1−s
1
α1 =
· −2 · (n + 1 − s) · (n + 1) · cS + 2 · n · AS
n+1−s
α2 = s −
=
(3.19)
(3.20)
1
· 2 · n − (n + 1) · (n + 1 − s) · cS + 2 · n · (n + 1 − s) · cS(3.21)
n+1−s
α0 =
X
1
(ci )2 − n · (AS )2
· (n + 1 − s) · (n + 1)2 ·
n+1−s
(3.22)
i∈S
Clearly, α2 > 0 since s 6= n, s 6= 1. Further, it holds that α1 < 0 due to c̄S > cS
as well as
n · (n + 1 − s) < s · (n + 1) · (n + 1 − s) − n
or equivalently, n < s · (n + 1 − s)
So far, we conclude that the quadratic function f (x) = α2 · x2 + α1 · x + α0 attains
−α1
−α1
its minimum at x = 2·α
and the corresponding minimal function values f ( 2·α
)=
2
2
−(α1 )2
4·α2
(α1 )2 .
+ α0 . This minimal function value is non-negative if and only if 4 · α0 · α2 >
For the sake of the forthcoming computational matters, recall that c̄T = ctT
for all T ⊆ N , T 6= ∅, as well as (2.9). In order to apply shortened notation, put the
substitution δs = n − (n + 1) · (n + 1 − s). As one out of two options for a possible
43
Stackelberg Oligopoly Games
representation of α1 , we choose (3.21) to evaluate the square of α1 , as well as the
product 4 · α2 · α0 . Finally, we arrive at a reasonable description of their difference
as stated in the next lemma.
Lemma 3.5. Consider the setting of Definition 3.4. Then the following equality
holds:
(i)
α2 · α0 −
(α1 )2
= s · (s − 1) · (n − s) · (n + 1)2 · c̄c̄S − (c̄S )2
4
2
2
− s · n · (n + 1 − s) · cS − c̄S
(3.23)
(ii) Moreover, a sufficient condition for 4·α2 ·α0 −(α1 )2 > 0 is given by the following
inequality:
2
(s − 1) · (n − s) · (n + 1)2 · c̄c̄S − (c̄S )2 > n · (n + 1 − s)2 · cS − c̄S
(3.24)
(iii) The sufficient condition (3.24) holds.
Proof of Lemma 3.5. The current approach proceeds as follows. Firstly, we
evaluate the square (α1 )2 and secondly, we study the two contributions within the
· n · (AS )2 , while its
product α2 · α0 , particularly the main contribution − (n−s)·(s−1)
n+1−s
second contribution will not be changed at all and kept till the end in the form
(n + 1)2 ·
(n − s) · (s − 1) X
·
(ci )2
n+1−s
i∈S
that is
(n + 1)2 ·
(n − s) · (s − 1)
· s · c̄c̄S
n+1−s
1
In order to apply shortened notation, put ρs = n+1−s
. Firstly, straightforward
2
calculations involving the relevant square (α1 ) and secondly, straightforward calculations involving the remaining part of the product α2 · α0 , yield the following:
2
(α1 )2
2
= (ρs ) · δs · cS + n · (n + 1 − s) · cS
4
2
2
2
2
2
2
= (ρs ) · (δs ) · (cS ) + n · (n + 1 − s) · (cS ) + 2 · δs · cS · n · (n + 1 − s) · cS
2
2
2
2
2
2
2
= (ρs ) · (δs ) · s · (c̄S ) + n · (n + 1 − s) · (cS ) + 2 · δs · s · n · (n + 1 − s) · c̄S · cS
(3.25)
44
Theo Driessen, Aymeric Lardon, Dongshuang Hou
In addition,
2
(n − s) · (s − 1) n · (AS )2
·
= (ρs )2 · (n − s) · (s − 1) · n · cS + (n + 1 − s) · cS
n+1−s
n+1−s
= (ρs )2 ·(n − s)·(s − 1)·n· s2 ·(c̄S )2 + (n + 1 − s)2 ·(cS )2 + 2·s·(n + 1 − s)·c̄S ·cS
+
(3.26)
Summing up the two negative expressions (3.25)–(3.26) to be multiplied by the
square (ρs )2 yields
s2 · (δs )2 + (n − s) · (s − 1) · n · (c̄S )2
(3.27)
+ (n + 1 − s) · n + (n − s) · (s − 1) · n · (cS )2
(3.28)
+ 2 · s · n · (n + 1 − s) · δs + (n − s) · (s − 1) · c̄s · cS
(3.29)
2
2
In order to simplify these calculations, we use the following simple equalities:
(n − s) · (s − 1) + n = s · (n + 1 − s)
(n − s) · (s − 1) · n + n2 = s · n · (n + 1 − s)
δs + (n − s) · (s − 1) = −(n + 1 − s)2
2
2
(3.30)
−s (δs ) +(n−s)(s−1)n +n·s ·(n+1−s)3 = −(n+1−s)(n+1)2(n−s)(s−1)s
The final computations are as follows:
−2
(ρs )
(α1 )2
· α2 · α0 −
= −s · n · (n + 1 − s)3 · (cS )2
4
+ 2 · s · n · (n + 1 − s)3 · c̄S · cS
− s2 · (δs )2 + (n − s) · (s − 1) · n · (c̄S )2
+ (n + 1 − s) · (n + 1)2 · (n − s) · (s − 1) · s · c̄c̄S
2
= −n · s · (n + 1 − s)3 · cS − c̄S
45
Stackelberg Oligopoly Games
− s2 · (δs )2 + (n − s)·(s − 1) · n ·(c̄S )2 + n·s·(n + 1 − s)3 ·(c̄S )2
+ (n + 1 − s) · (n + 1)2 · (n − s) · (s − 1) · s · c̄c̄S
2
= −n · s · (n + 1 − s)3 · cS − c̄S
+ (n + 1 − s) · (n + 1)2 · (n − s) · (s − 1) · s · (c̄c̄S − (c̄S )2 )
The last equality is due to (3.30).
4.
✷
APPENDIX: Alternative Proofs.
Alternative proof of Theorem 3.2. The full proof consists of two parts. Part
1.
(v ∗ )
(N ) 6 0 or equivalently, by Proposition 1.1, v(N ) >
P Firstly, we check g
i∈N v({i}). By using (3.11), we obtain the following chain of equalities:
X
4 · n · v(N ) −
v({i})
i∈N
2 X 2
a + cN − (n + 1) · ci
= n · a − cN −
i∈N
2 X 2 = n · a − cN −
a2 + 2 · a · cN − (n + 1) · ci + cN − (n + 1) · ci
i∈N
= n · a2 − 2 · a · cN + (cN )2 − n · a2 − 2 · a · n · cN − (n + 1) · cN
−
X
i∈N
2
2
2
(cN ) − 2 · (n + 1) · cN · ci + (n + 1) · (ci )
X
= 2 · a · cN − n · cN + n · (cN )2 − n − 2 · (n + 1) · (cN )2 − (n + 1)2 ·
(ci )2
i∈N
= 2 · a · n · c̄N − cN + n · (cN )2 + (n + 2) · (cN )2 − (n + 1)2 · n · c̄c̄N
2
2
2
= n · 2 · a · c̄N − cN + (cN ) + n · (n + 2) · (c̄N ) − (n + 1) · c̄c̄N
∗
So far, we conclude that g (v ) (N ) 6 0 if and only if
2 · a · c̄N − cN > (n + 1)2 · c̄c̄N − (cN )2 − n · (n + 2) · (c̄N )2
(4.31)
46
Theo Driessen, Aymeric Lardon, Dongshuang Hou
or equivalently,
−1 1
L2
2
2
2
a > · c̄N − cN
· (n + 1) · c̄c̄N − (cN ) − n · (n + 2) · (c̄N ) =
2
2
Here the critical number L2 represents the lower bound given by
−1 2
2
2
L2 = c̄N − cN
· (n + 1) · c̄c̄N − (cN ) − n · (n + 2) · (c̄N )
Notice that
−1 2
2
2
· (n + 2 · n) · c̄c̄N − (c̄N ) + c̄c̄N − (cN )
L2 = c̄N − cN
while
−1 2 L1 = c̄N − cN
· (n + 1)2 · c̄c̄N − cN + cN
(4.32)
(4.33)
Recall that c̄c̄N > (c̄N )2 > (cN )2 . Thus, L2 > 0. In fact, it is left to the reader
✷
to verify the validity of the equality L22 = L21 − cN , that is L2 = L1 − 2 · cN .
Alternative proof of Theorem 3.2. The full proof consists of two parts. Part
1.
∗
∗
Secondly, we checkP
g (v ) (S) 6 g (v ) (N ) for all S ⊆ N , S 6= ∅, or equivalently, by
Theorem 1.3, v(S) 6 i∈S v({i}) for all S ⊆ N , S 6= N , S 6= ∅. This second proof
differs from the first one in that it uses different fundamental substitutions
as well as c(S, i) := cS − (n + 1) · ci
yS := a + cN \S
Fix S ⊆ N , S 6= N , S 6= ∅. Note that
(2.10) it holds that
P
i∈S
for all i ∈ S,
instead of x = a + cN
c(S, i) = −(n + 1 − s) · cS . By using
2
n
4 · n · v(S) =
· a + cN \S − (n + 1 − s) · cS
n+1−s
2
n
=
· yS − (n + 1 − s) · cS
Further,
n+1−s
2 2
4 · n · v({i}) = a + cN − (n + 1) · ci = yS + c(S, i)
for all i ∈ N
Recall that
P
i∈S
chain of equalities:
c(S, i) = −(n + 1 − s) · cS for all S ⊆ N . We obtain the following
47
Stackelberg Oligopoly Games
4·n·
X
i∈S
X
2
v({i}) − v(S) =
yS + c(S, i)
i∈S
2
n
· yS − (n + 1 − s) · cS
−
n+1−s
X
=
(yS )2 + 2 · yS · c(S, i) + (c(S, i))2
i∈S
n
2
2
2
−
· (yS ) − 2 · yS · (n + 1 − s) · cS + (n + 1 − s) · (cS )
n+1−s
X
X
= s · (yS )2 + 2 · yS ·
c(S, i) +
(c(S, i))2
i∈S
−
i∈S
n
· (yS )2 + 2 · yS · n · cS − n · (n + 1 − s) · (cS )2
n+1−s
Our main goal is to describe the coalitional notion of surplus in terms of a
quadratic function of the variable y, say g(y) = β2 · y 2 + β1 · y + β0 where β2 > 0. ✷
Definition 4.1. The three real numbers βk , k = 0, 1, 2, are given as follows:
n
(n − s) · (s − 1)
=
n+1−s
n+1−s
β1 = 2 · n · cS − s · (n + 1 − s) · c̄S
β2 = s −
β0 =
X
i∈S
(c(S, i))2 − n · (n + 1 − s) · (cS )2
(4.34)
(4.35)
(4.36)
Clearly, β2 > 0 since s 6= n, s 6= 1. Further, it holds that β1 < 0 due to c̄S > cS
as well as n < s · (n + 1 − s) since s · (s − 1) < n · (s − 1). So far, we conclude
that the quadratic function g(y) = β2 · y 2 + β1 · y + β0 attains its minimum at
−(β1 )2
−β1
1
y = 2·β
and the corresponding minimal function values g( −β
2·β2 ) = 4·β2 + β0 . This
2
minimal function value is non-negative if and only if 4 · β0 · β2 > (β1 )2 . For the sake
cT
of the forthcoming computational matters, recall (3.11) as well as c̄T = |T
| for all
T ⊆ N , T 6= ∅. Recall the fundamental substitution c(S, i) := cS − (n + 1) · ci for
all i ∈ S. Based upon (4.34)–(4.36), we evaluate the square of β1 , as well as the
product 4 · β2 · β0 . Finally, we arrive at a reasonable description of their difference
as stated in the next lemma.
48
Theo Driessen, Aymeric Lardon, Dongshuang Hou
Lemma 4.2. Consider the setting of Definition 4.1. Firstly, we evaluate β0 in
the following form:
β0 =
X
i∈S
=
(c(S, i))2 − n · (n + 1 − s) · (cS )2
X
i∈S
cS − (n + 1) · ci
2
− n · (n + 1 − s) · (cS )2
= s · (cS )2 − 2 · (n + 1) · (cS )2 + (n + 1)2 · s · c̄c̄S − n · (n + 1 − s) · (cS )2
= (−2 · n − 2 + s) · (cS )2 + (n + 1)2 · s · c̄c̄S − n · (n + 1 − s) · (cS )2
Secondly, we add the following chain of computations:
(β1 )2
= β2 · β0 − n · cS − s · (n + 1 − s) · c̄S
4
(n − s) · (s − 1)
=
· (−2 · n − 2 + s) · s2 · (c̄S )2 + (n + 1)2 · s · c̄c̄S
n+1−s
2
− n · (n + 1 − s) · (cS ) − n2 · (cS )2 − 2 · n · s · (n + 1 − s) · cS · c̄S
2
2
2
+ s · (n + 1 − s) · (c̄S )
2
= −n · (n − s) · (s − 1) − n · (cS )2
β2 · β0 −
(n − s) · (s − 1)
2
2
2
+
· (−2 · n − 2 + s) · s − s · (n + 1 − s) · (c̄S )2
n+1−s
+
(n − s) · (s − 1)
· (n + 1)2 · s · c̄c̄S + 2 · n · s · (n + 1 − s) · cS · c̄S
n+1−s
= −n · s · (n + 1 − s) · (cS )2 + 2 · n · s · (n + 1 − s) · cS · c̄S
(n − s) · (s − 1)
+
· (n + 1)2 · s · c̄c̄S
n+1−s
s2
3
−
· (n − s) · (s − 1) · (2 · n + 2 − s) + (n + 1 − s) · (c̄S )2
n+1−s
2
(n − s) · (s − 1)
= −n · s · (n + 1 − s) · cS − c̄S +
· (n + 1)2 · s · c̄c̄S
n+1−s
s2
+ n · s · (n + 1 − s) −
(n − s) · (s − 1) · (2 · n + 2 − s)
n+1−s
+ (n + 1 − s)3 · (c̄S )2
49
Stackelberg Oligopoly Games
Thus,
(β1 )2
(n + 1 − s) · β0 · β2 −
4
2
= −s · n · (n + 1 − s)2 · cS − c̄S + (s − 1) · (n − s) · (n + 1)2 · s · c̄c̄S
+ n · s · (n + 1 − s)2 − s2 · (n + 1 − s)3
− s2 · (n − s) · (s − 1) · (2 · n + 2 − s) · (c̄S )2
2
= −s · n · (n + 1 − s)2 · cS − c̄S + s · (s − 1) · (n − s) · (n + 1)2 · c̄c̄S
− s · (s − 1) · (n − s) · (n + 1)2 · (c̄S )2
2
= −s · n · (n + 1 − s)2 · cS − c̄S
+ s · (s − 1) · (n − s) · (n + 1)2 · c̄c̄S − (c̄S )2
It suffices to prove the next inequality:
2
2
(s − 1) · (n − s) · (n + 1) · c̄c̄S − (c̄S )
2
> n · (n + 1 − s) · cS − c̄S
(4.37)
2
or equivalently, by (2.9)
2
2
X
(s − 1) · (n − s) · (n + 1)2 ·
ci − c̄S > s · n · (n + 1 − s)2 · cS − c̄S
i∈S
By (4.37), we observe the same inequality as in Lemma 3.5 and hence, we may
state the same sufficiency condition.
✷
References
Chander, P., and H. Tulkens, (1997). The Core of an Economy with Multilateral Environmental Externalities. International Journal of Game Theory, 26, 379–401.
Driessen, T. S. H. (1988). Cooperative Games, Solutions, and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands.
Driessen, T. S. H., and H. I. Meinhardt (2005). Convexity of oligopoly games without transferable technologies, Mathematical Social Sciences, 50, 102–126.
Driessen, T. S. H., Hou, Dongshuang, and Aymeric Lardon (2011a). Stackelberg Oligopoly
TU-games: Characterization of the core and 1-concavity of the dual game. Working
Paper, Department of Applied Mathematics, University of Twente, Enschede, The
Netherlands.
Driessen, T. S. H., Hou, Dongshuang, and Aymeric Lardon (2011b). A necessary and sufficient condition for the non-emptiness of the core in Stackelberg Oligopoly TU games.
Working paper, University of Saint-Etienne, France.
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Theo Driessen, Aymeric Lardon, Dongshuang Hou
Lardon, A. (2009). The γ-core of Cournot oligopoly TU games with capacity constraints.
Working paper, University of Saint-Etienne, France.
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of Saint-Etienne, Universite Jean Monnet de Saint-Etienne, France, 13 October 2011.
Norde, H., Do, K. H. P., and S. H. Tijs (2002). OLigopoly Games with and without transferable technologies. Mathematical Social Sciences, 43, 187–207.
Sherali, H. D., Soyster, A. L., and F. H. Murphy (1983). Stackelberg–Nash–Cournot Equilibria, Characterizations and Computations. Operations Research, 31(2), 253–276.
Tijs, S. H. (1981). Bounds for the core and the τ -value. In: Game Theory and Mathematical
Economics, North-Holland Publishing Company, Amsterdam, 123–132.
Zhao, J. (1999). A necessary and sufficient condition for the convexity in oligopoly games.
Mathematical Social Sciences, 37, 187–207.
Zhao, J. (1999). A β-core existence result and its application to oligopoly markets. Games
and Economics Behavior, 27, 153–168.
On Uniqueness of Coalitional Equilibria
Michael Finus,1 Pierre von Mouche2 and Bianca Rundshagen3
University of Bath, Department of Economics,
Bath BA2 7AY, United Kingdom
Email: [email protected]
2
Wageningen Universiteit,
Hollandseweg 1, 6700 EW, Wageningen, The Netherlands
E-mail: [email protected]
3
Universität Hagen, Department of Economics,
Universitätsstrasse 11, 58097 Hagen, Germany
E-mail: [email protected]
1
Abstract For the so-called ‘new approach’ of coalition formation it is important that coalitional equilibria are unique. Uniqueness comes down to
existence and to semi-uniqueness, i.e. there exists at most one equilibrium.
Although conditions for existence are not problematic, conditions for semiuniqueness are. We provide semi-uniqueness conditions by deriving a new
equilibrium semi-uniqueness result for games in strategic form with higher
dimensional action sets. The result applies in particular to Cournot-like
games.
Key words: Coalition formation, Cournot oligopoly, equilibrium (semi-)uniqueness,
game in strategic form, public good.
1.
Introduction
The analysis of coalition formation – in particular in the context of externalities –
has become an important topic in economics. Examples do not only include firms
that coordinate their output or prices in oligopolistic markets (cartels), jointly invest
in research assets (R&D-agreements) or completely merge (joint ventures), but also
countries that coordinate their tariffs (trade agreements and customs unions) or
their environmental policy (international environmental agreements).
Our article contributes to the so-called ‘new approach’ of coalition formation (see
for instance Yi(1997) and Bloch(2003) for an extensive overview). The goal of this
approach is to determine equilibrium coalition structures. As the approach consists
of modelling coalition formation as a 2-stage game with simultaneous actions in each
of both stages, it is important that for each possible coalition structure coalitional
equilibria, i.e. equilibria in the second stage, are unique.1
So for the new approach it is important to have results that guarantee uniqueness
of coalitional equilibria. Conditions should be such that they can be easily checked
for the base games that appeared so far in these models, like Cournot and public
good games. As far as we know, general uniqueness results for coalitional equilibria
1
Roughly speaking, in the first stage, the players choose a membership action which via
a given member-ship rule leads to a coalition structure. In the second stage, the players
are the coalitions in this coalition structure. Each of these coalitions chooses a ‘physical’
action for each of its members in a base game. See e.g. Finus and Rundshagen(2009) for
more. Also see Bartl(2012).
52
Michael Finus, Pierre von Mouche, Bianca Rundshagen
of such games are not present in the literature. There it is just assumed that one
deals with a situation where coalitional equilibria are unique or that one deals with
a simple concrete example where uniqueness explicitly can be shown. Developing
an abstract general uniqueness result is the main objective of this article.
As shown in Section 3., a general equilibrium existence theorem guarantees for
various common cases existence of coalitional equilibria. So existence is not a real
issue, but equilibrium semi-uniqueness is. Especially as for coalitional equilibria one
has to leave the comfortable usual setting of one dimensional action sets. Indeed:
a coalition is formally treated as a meta-player whose action set is the Cartesian
product of the action sets of the players in this coalition.
In order to obtain our semi-uniqueness result for coalitional equilibria we develop
a semi-uniqueness result for Nash equilibria of games in strategic form with higher
dimensional action sets. This result, Theorem 1, can be considered as a variant
of a result in Folmer and von Mouche(2004) to higher dimensions. It can handle
various aggregative2 base-games with one-dimensional action sets. We identify a
class of such games which contains Cournot and public good games and give with
Corollary 3 a result that guarantees that for each possible coalition structure there
exists a unique coalitional equilibrium.
2.
Coalitional equilibria
In this section, we fix the setting and notations and formally define the notion of
coalitional equilibrium.
2.1.
Games in strategic form
A game in strategic form Γ is an ordered 3-tuple
Γ = (I, (Xi )i∈I , (fi )i∈I ),
where I is a non-empty finite set, every Xi is a non-empty set and every fi is a
function
Y
fi :
Xj → IR.
j∈I
The elements of I are called players, Xi is called the action set of player i, the
elements of Xi are called actions
Q of player i, fi is called the payoff function of
player i and the elements of j∈I Xj , being by I indexed families (xj )j∈I with
xj ∈ Xj , are called action profiles.
For i ∈ I, let
ı̂ := I \ {i}.
Q
(z)
For i ∈ I and z = (zj )j∈ı̂ ∈ j∈ı̂ Xj , the conditional payoff function fi : Xi → IR
is defined by
(z)
fi (xi ) := fi (xi ; z);
here (xi ; z) is the by I indexed family with xi for the element with index
Q i and zj
for the element with index j 6= i. An action profile x = (xj )j∈I ∈ j∈I Xj is a
2
I.e. games where the payoff
P function of each player i is a function of his own action xi
and of a weighted sum l γl xl of all actions.
53
On Uniqueness of Coalitional Equilibria
(Nash) equilibrium if, for all i ∈ N , writing again x = (xi ; z), xi is a maximiser of
(z)
fi . We denote by
E
the set of equilibria of Γ .
We need some further notations for the sequel. For C ⊆ I, let
Y
XC :=
Xj .
j∈C
So an element ξ C of XC is a by C indexed family (ξC;i )i∈C with
Q ξC;i ∈ Xi ; for
i ∈ N , we identify X{i} with Xi . And an element of element of C∈C XC is a by C
indexed family
ξ = (ξ C )C∈C = ((ξC;i )i∈C )C∈C .
2.2.
Notion of coalitional equilibrium
Suppose given a game in strategic form Γ = (I, (Xi )i∈I , (fi )i∈I ).
A coalition is a subset of I and a coalition structure of I is a partition of I, i.e.
a set with as elements non-empty disjoint coalitions whose union is I.
Given a coalition structure C, we denote for i ∈ I by Ci the unique element of
C with
i ∈ Ci
Q
Q
and define the mapping J C : C∈C XC → j∈I Xj by
J C ((ξ C )C∈C ) := (ξCj ;j )j∈I .
For a subset D of I the function fD :
fD :=
Q
X
i∈D
C∈C
XC → IR is defined by
fi ◦ J C .
Having these notations, the next definition formalizes the intended notion of
coalitional equilibrium (with base game Γ ) as outlined in section 1..
Definition 1. Given a game in strategic form Γ = (I, (Xi )i∈I , (fi )i∈I ) and a coalition structure C of I, the (with C associated ) game in strategic form ΓC is defined
by
ΓC := (C, (XC )C∈C , (fC )C∈C ). ⋄
A Nash equilibrium of ΓC also is called a coalitional equilibrium of Γ ; more precisely
we speak of a C-equilibrium of Γ . We also will refer to the elements of C as metaplayers. The action sets XC of ΓC are typically more dimensional. Note that if
C = {{1}, {2}, . . . {n}},
then ΓC = Γ and a C-equilibrium of Γ is nothing else than a Nash equilibrium of
Γ . And if
C = {I},
P
a C-equilibrium is nothing else than a maximizer of the total payoff function i∈I fi .
54
3.
Michael Finus, Pierre von Mouche, Bianca Rundshagen
Existence of coalitional equilibria
General equilibrium existence and semi-uniqueness results for games in strategic
form have immediate counterparts regarding coalitional equilibria if they allow for
higher-dimensional action sets.
A powerful standard existence result in Tan et al.(1995)Tan, Yu, and Yuan leads
to the following sufficient conditions for the game ΓC := (C, (XC )C∈C , (fC )C∈C ) to
have a Nash equilibrium:
I. each action set XC is a compact convex subset of a Hausdorff topological linear
space;
II. each payoff function fC is upper-semi-continuous;
III. every fC is lower-semi-continuous in the variable related to XĈ ;
IV. every fC is quasi-concave in ξ C ∈ XC .
It may be useful to note that if each function fi is quasi-concave in (its own
action) xi , this does not necessarily imply that IV holds. Even assuming that each
function fi is concave in each variable is not sufficient.3
A natural question is to ask for simple sufficient conditions such that for each
coalition structure C a C-equilibrium exists. As can be easily verified by the above
existence result, such conditions are for instance: each action set Xi is a segment of
IR, each payoff function fi is continuous and concave.
4.
A higher dimensional equilibrium semi-uniqueness result
In this section we consider a game in Q
strategic form Γ = (I, (Xi )i∈I , (fi )i∈I ) where
each player i ∈ I has action set Xi = j∈Mi Xi;j with Mi a non-empty set and the
Xi;j proper intervals of IR.
Theorem 1. For i ∈ I let Ei := {ei | e ∈ E} and Ei;j := {ei;j | e ∈ E} (j ∈ Mi ).
Suppose the following conditions I-III hold.
I. The partial derivatives
∂fi
∂xi;j
of IR := IR ∪ {−∞, +∞}.
II. There exist functions
(i ∈ I, j ∈ Mi ) exist at every e ∈ E as an element
Φi : Ei → IR (i ∈ I), Θi : {(Φl (el ))l∈I | e ∈ E} → IR (i ∈ I),
and, with Ψi : E → IR (i ∈ I) defined by Ψi (e) := Θi ((Φl (el ))l∈I ), functions
Ti;j : Ei;j × Φi (Ei ) × Ψi (E) → IR (i ∈ I, j ∈ Mi ),
such that for all i ∈ I and j ∈ Mi
∂fi
a. ∂x
(e) = Ti;j (ei;j , Φi (ei ), Ψi (e)) (e ∈ E);
i;j
b. Ti;j is decreasing in each of its three variables, and strictly decreasing in the
first or second.
3
It is worth noting that the sum of quasi-concave functions may fail to be quasi-concave
and a function that is concave in each of its variables may fail to be concave.
On Uniqueness of Coalitional Equilibria
55
III. a. For all i ∈ I: Φi and Θi are increasing.4
b. For all a, b ∈ E: Ψi (a) ≥ Ψi (b) (i ∈ I) or Ψi (b) ≥ Ψi (a) (i ∈ I).
1. For all a, b ∈ E: Ψi (a) = Ψi (b) (i ∈ I) and even Φi (ai ) = Φi (bi ) (i ∈ I).
2. If every Ti;j is strictly decreasing in the first variable, then #E ≤ 1. ⋄
Proof. 1. Suppose a, b ∈ E.
Step 1: Ψi (a) ≥ Ψi (b) (i ∈ I) ⇒ Φi (ai ) ≤ Φi (bi ) (i ∈ I).
Proof: by contradiction assume Ψi (a) ≥ Ψi (b) (i ∈ I) and for some m ∈ I
Φm (am ) > Φm (bm ).
With J the set of elements j ∈ Mm for which am;j is a left boundary point of Xm;j
or bm;j is a right boundary point of Xm;j , we have
am;j ≤ bm;j (j ∈ J).
Now suppose j ∈ Mm \ J. Because a is an equilibrium and am;j is not a left
boundary point of Xm;j , it follows by condition I that Dm;j fm (a) ≥ 0. And, by the
same arguments, Dm;j fm (b) ≤ 0. So by condition IIa we have
Tm;j (am;j , Φm (am ), Ψm (a)) ≥ 0 ≥ Tm;j (bm;j , Φm (bm ), Ψm (b)).
(1)
As Ψm (a) ≥ Ψm (b) and Φm (am ) > Φm (bm ), condition IIb implies
Tm;j (am;j , Φm (am ), Ψm (a)) ≤ Tm;j (am;j , Φm (bm ), Ψm (b)),
(2)
with strict inequality if Tm;j is strictly decreasing in the second variable. (1) and
(2) imply
Tm;j (am;j , Φm (bm ), Ψm (b)) ≥ Tm;j (bm;j , Φm (bm ), Ψm (b)),
with strict inequality if Tm;j is strictly decreasing in the second variable. As Tm;j
is decreasing, and strictly decreasing in the first or second variable, it follows that
am;j ≤ bm;j . Hence, we proved
am;j ≤ bm;j (j ∈ Mm ), i.e. am ≤ bm .
By condition IIIa this implies Φm (am ) ≤ Φm (bm ), a contradiction.
Step 2: Ψi (a) ≥ Ψi (b) (i ∈ I) ⇒ Ψi (a) = Ψi (b) (i ∈ I)
Proof: suppose Ψi (a) ≥ Ψi (b) (i ∈ I). By Step 1: Φi (ai ) ≤ Φi (bi ) (i ∈ I). This
implies, as Θi is increasing, Ψi (a) ≤ Ψi (b). Thus Ψi (a) = Ψi (b).
Step 3: Ψi (a) = Ψi (b) (i ∈ I).
4
Q
Given a finite product r∈J Zr of subsets of IR the relation ≥ (and its dual ≤) on
Q
br (r ∈ J). And a function
r∈J
QZr is defined by: (ar )r∈J ≥ (br )r∈J means ar ≥ Q
f : r∈J Zr → IR is called increasing if for all a, b ∈ r∈J Zr one has a ≥ b ⇒
f (a) ≥ f (b).
56
Michael Finus, Pierre von Mouche, Bianca Rundshagen
Proof: by condition IIb we have Ψi (a) ≥ Ψi (b) (i ∈ I) or Ψi (b) ≥ Ψi (a) (i ∈ I).
Without loss of generality we may assume that Ψi (a) ≥ Ψi (b) (i ∈ I). Step 3 implies
Ψi (a) = Ψi (b) (i ∈ I).
Step 4: Φi (ai ) = Φi (bi ) (i ∈ I).
Proof: by Step 3 we have Ψi (a) = Ψi (b) (i ∈ I). Now apply Step 1.
2. By contradiction suppose #E ≥ 2. Fix a, b ∈ E and i ∈ I and j ∈ Mi such
that ai;j 6= bi;j . We may assume that ai;j > bi;j . By part 1, Ψi (a) = Ψi (b) =: yi and
Φi (ai ) = Φi (bi ) =: wi . As a is an equilibrium and ai;j is not a left boundary point
of Xi;j , it follows that Di;j fi (a) ≥ 0. And, by the same arguments, Di;j fi (b) ≤ 0.
By condition IIa
Ti;j (ai;j , wi , yi ) ≥ 0 ≥ Ti;j (bi;j , wi , yi ).
As Ti;j is strictly decreasing in its first variable, this implies a contradiction.
⊔
⊓
Taking, in Theorem 1, mi = 1 (i ∈ I) and Φi = Id leads to:
Corollary 1. For i ∈ I let Ei := {ei | e ∈ E}. Sufficient for #E ≤ 1 is that the
following conditions I-III hold.
I. The partial derivatives
II. There exist functions
∂fi
∂xi
(i ∈ I) exist at every e ∈ E as an element of IR.
ϕi : E → IR (i ∈ I),
and functions
ti : Ei × ϕi (E) → IR (i ∈ I),
such that for all i ∈ I
∂fi
a. ∂x
(e) = ti (ei , ϕi (e)) (e ∈ E);
i
b. ti is decreasing in each of its two variables, and strictly decreasing in the
first.
III. a. Every ϕi is increasing.
b. For all a, b ∈ E: ϕi (a) ≥ ϕi (b) (i ∈ I) or ϕi (b) ≥ ϕi (a) (i ∈ I). ⋄
And here is a more practical variant of Theorem 1:
5
Corollary
P 2. For i ∈ I, let ti;j ≥ 0 (j ∈ Mi ), ri ≥ 0, si > 0, Wi :=
Yi := si k∈I rk Wk and define Φi : Xi → IR and Ψi : XI → IR by
X
X
Φi (xi ) :=
ti;j xi;j , Ψi (x) := si
rk Φk (xk ).
j∈Mi
P
j∈Mi ti;j Xi;j ,
k∈I
Suppose the following conditions I, IIa and IIb hold.
I. Each player i’s payoff function fi is partially differentiable with respect to each
variable xi;j .
II. There exist functions
Ti;j : Xi;j × Wi × Yi → IR (i ∈ I, j ∈ Mi ),
such that for all i ∈ I and j ∈ Mi
5
The sum in Wi and Yi is a Minkowski-sum.
57
On Uniqueness of Coalitional Equilibria
∂fi
a. ∂x
(x) = Ti;j (xi;j , Φi (xi ), Ψi (x)) (x ∈ XI );
i;j
b. Ti;j is decreasing in each of its three variables, and strictly decreasing in the
first or second.
1. For all a, b ∈ E: Ψi (a) = Ψi (b) (i ∈ I) and even Φi (ai ) = Φi (bi ) (i ∈ I).
2. If every Ti;j is strictly decreasing in the first variable, then #E ≤ 1. ⋄
5.
Uniqueness of coalitional equilibria for Cournot-like games
In the following definition a class of games in strategic form is introduced for which
we provide sufficient conditions for uniqueness of coalitional equilibria.
Definition 2. A Cournot-like game is a game in strategic form
Γ = (N, (Ki )i∈N , (πi )i∈N )
where every Ki is a proper interval of IR with 0 ∈ Ki ⊆ IR+ and
X
πi (x) = ai (xi ) − xβi i bi (
γl xl )
l∈N
where, with Y :=
–
–
–
–
ai : Ki → IR;
βi ∈ {0, 1};
γi > 0;
bi : Y → IR. ⋄
P
l∈N
γl K l ,
In case Ki is bounded, i.e. where Ki = [0, mi ] or Ki = [0, mi [ we say that player i has
a capacity constraint. Note that some players may have a capacity constraint while
others may not have. The class of Cournot-like games contains various heterogeneous
Cournot oligopoly games: take every βi = 1. It contains6 all homogeneous Cournot
oligopoly games: take in addition all bi equal and each γ = 1. It also contains various
public good games: take every βi = 0. We call βl the type of player l.
In the next theorem and proposition we consider a Cournot-like game Γ and
fix a coalition structure C of N . We suppose that all players belonging to a same
coalition C ∈ C are of the same type βC . Also we suppose for every C ∈ C that
γl = γl′ (l, l′ ∈ C) and in case βC = 1 that bl , bl′ (l, l′ ∈ C).
Theorem 2. Suppose that each function ai and bi is differentiable. Consider the
with the coalition structure C associated game
For C ∈ C, let WC :=
IR (j ∈ C) by
P
ΓC = (C, (KC )C∈C , (πC )C∈C ).
l∈C
γl Kl and define the functions TC;j : Kj × WC × Y →
TC;j (xj , w, y) := Daj (xj ) −
X
w β C γj
Dbi (y) − βC bj (y).
#C · βC + (1 − βC )
i∈C
Suppose every TC;j is decreasing in each of its three variables and strictly decreasing
in its first or second variable.
6
Disregarding Cournot oligopoly games with finite action sets.
58
Michael Finus, Pierre von Mouche, Bianca Rundshagen
1. For all Nash equilibria η, µ of ΓC one has
XX
XX
ηC;i =
µC;i
C∈C i∈C
P
C∈C i∈C
P
and even i∈C ηC;i = i∈C µC;i (C ∈ C).
2. If every TC;j is strictly decreasing in its first variable, then ΓC has at most one
Nash equilibrium. ⋄
Proof. Let γl =: γC (l ∈ C) and in case βC = 1, let bl =: bC (l ∈ C).
Consider the game ΓC . The payoff function of player C ∈ C is
X
X
X
β
πC (ξ) =
(πi ◦ J C )(ξ) =
ai (ξC;i ) − (ξC;i ) C bi (
γm ξCm ;m )
i∈C
i∈C
m∈N
X
X X
β
=
ai (ξC;i ) − (ξC;i ) C bi (
γm ξA;m )
i∈C
A∈C m∈A
X
X
X
β
=
ai (ξC;i ) − (ξC;i ) C bi (
γA
ξA;m ) .
i∈C
If βC = 0, then πC (ξ) =
fore for j ∈ C
P
i∈C
ai (ξC;i ) −
P
A∈C
m∈A
P
P
i∈C bi (
A∈C γA
m∈A ξA;m ) and there-
X
X
X
∂πC
(ξ) = Daj (ξC;j ) − γj
Dbi (
γA
ξA;m ).
∂ξC;j
i∈C
A∈C
m∈A
If βC = 1, then
πC (ξ) =
X
i∈C
and therefore for j ∈ C
X
X
X
ai (ξC;i ) − (
ξC;i )bC (
γA
ξA;m )
i∈C
A∈C
m∈A
X
X
X
X
X
∂πC
(ξ) = Daj (ξC;j ) − bC (
γA
ξA;m ) − γj (
ξC;i )DbC (
γA
ξA;m ).
∂ξC;j
A∈C
m∈A
i∈C
A∈C
m∈A
Noting that for the above functions TC;j one has
TC;j (xj , w, y) := Daj (xj ) − γj pC (w, y) − βC bj (y)
where pC : WC × Y → IR is defined by
P
i∈C Dbi (y) if βC = 0,
pC (w, y) =
wDbC (y) if βC = 1,
we obtain
X
X
X
∂πC
(ξ) = TC;j (ξC;j ,
ξC;i ,
γA
ξA;m ).
∂ξC;j
i∈C
A∈C
m∈A
Having the above, we can apply Corollary 2 which implies the desired results. ⊓
⊔
7
7
Q
Taking I = C, MC = C (C ∈ C), XC;j = Kj (C ∈ C, j ∈ C), XC = j∈MC XC;j =
K
tC;j = 1 (C ∈ C, j ∈ MC ), rC = γC (C ∈ C), sC = P
1 (C ∈ C), fC =
PC (C ∈ C),
C
C (C ∈ C), TC;j = TC;j (C ∈ C, j ∈ MC ), ΦC (ξ C ) =
i∈C πi ◦J = πP
j∈MC ξC;j (C ∈
C) and ΨC (ξ) = D∈C γD ΦD (ξD ) (C ∈ C).
59
On Uniqueness of Coalitional Equilibria
Remark: sufficient for every TC;j to be decreasing in each of its three variables
and strictly decreasing in its first variable is that the following practical condition
holds:
every ai is strictly concave and every bi is increasing and convex.
Proposition 1. Consider the with the coalition structure C associated game
ΓC = (C, (KC )C∈C , (πC )C∈C ).
Given C ∈ C, the following condition guarantees strict concavity of all conditional
payoff functions of player C: the function aC : KC → IR given by
X
aC (ξ C ) :=
ai (ξC;i )
i∈C
is concave and
P
a. if βC = 0, then the function i∈C bi is strictly convex or aC is strictly concave;
b. if βC = 1, then the function y 7→ ybC (y) is convex, and this function is strictly
convex or aC is strictly concave. ⋄
(ξ )
Proof. With ξĈ ∈ KĈ , the conditional payoff function πC Ĉ : KC → IR reads
X
X
X
(ξ )
β
ξC;m + z),
πC Ĉ (ξ C ) =
ai (ξC;i ) +
−(ξC;i ) C bi (γC
i∈C
P
m∈C
i∈C
P
where z = A∈C\C γA m∈A ξA;m The first sum in this expression is by assumption
a concave function.
Case βC = 0: the second equals
X
X
−bi (γC
ξC;m + z)
i∈C
m∈C
(ξ )
and also is concave. As the first or second sum is strictly concave, πC Ĉ is strictly
concave. Case βC = 1: the second sum equals
X
X
−bC (γC
ξC;m + z)
ξC;m
m∈C
m∈C
(ξ )
and also is concave. As the first or second sum is strictly concave, πC Ĉ is strictly
concave.
⊔
⊓
The last paragraph in Section 2, the remark after Theorem 2 and Proposition 1
imply:
Corollary 3. Let Γ be a Cournot-like game with compact action sets, βi = β (i ∈
N ), γi = γi′ (i, i′ ∈ N ) and β = 1 ⇒ bi = bi′ (i, i′ ∈ N ). Suppose each function
ai is differentiable and strictly concave and each function bi is differentiable, increasing and convex. Then for every coalition structure C the game Γ has a unique
C-equilibrium. ⋄
60
Michael Finus, Pierre von Mouche, Bianca Rundshagen
References
Bartl, D. (2012). Application of cooperative game solution concepts to a collusive oligopoly
game. In School of Business Administration in Karviná, editor, Proceedings of the 30th
International Conference Mathematical Methods in Economics, pages 14–19.
Bloch, F. (2003). Non–cooperative models of coalition formation in games with spillovers.
In C. Carraro, editor, Endogenous Formation of Economic Coalitions, chapter 2, pages
35–79. Edward Elgar, Cheltenham.
Finus, M. and B. Rundshagen (2009). Membership rules and stability of coalition structures
in positive externality games. Social Choice and Welfare, 32(0), 389–406.
Folmer, H. and P. H. M. von Mouche (2004). On a less known Nash equilibrium uniqueness
result. Journal of Mathematical Sociology, 28(0), 67–80.
Tan, K. J. Yu, and X. Yuan (1995). Existence theorems of Nash equilibria for noncooperative n-person games. International Journal of Game Theory, 24(0), 217–222.
Yi, S. (1997). Stable coalition structures with externalities. Games and Economic Behavior,
20(0), 201–237.
Quality Level Choice Model under Oligopoly
Competition on a Fitness Service Market
Margarita A. Gladkova1 , Maria A. Kazantseva2 and
Nikolay A. Zenkevich1
St.Petersburg State University,
Graduate School of Management,
Volkhovsky per. 2, St.Petersburg, 199004, Russia
2
Vienna University of Economics and Business,
Augasse 2-6, 1090 Vienna, Austria
E-mail: [email protected]
[email protected]
[email protected]
1
Abstract The growth of complexity of business conditions causes the necessity of innovative approaches to strategic decision-making, instruments and
tools that help them to reach the leading position in mid-term and longterm perspective. One of the instruments that allow increasing company’s
competitiveness is the improvement of the service quality. The goal of the
research is to develop theoretical basis (models) and practical methods of
the service quality level evaluation and choice which is made by the service
provider. Research objectives are: analysis of consumer satisfaction with the
service, development of game-theoretical models of service providers’ interaction, definition of the strategy of service quality level choice, development
of practical recommendations for Russian companies to implement the strategy.
Keywords: quality choice, willingness to pay, exponential distribution, twostage game, Nash equilibrium, optimal quality differentiation, fitness industry.
1.
Introduction
The growth of complexity of an external environment and business conditions,
namely, high development of information and communication technologies and competition boost, predefines the identification of new sources of development of companies competitive abilities and ways to increase management effectiveness. This fact
causes the necessity in development of innovative approaches to strategic decisionmaking, instruments and tools that may help companies to reach the leading position in mid-term and long-term perspective. One of the instruments that allow
increasing company’s competitiveness is the improvement of the service quality.
Contemporary approaches to company management are based on the analysis of
the adding value framework. The value of the service for the consumer is highly
defined by its quality. Therefore, the problem of quality level choice under competition is a very important element of strategic management. The appropriate choice
of service quality level and price provides a company with necessary conditions to
maintain high competitiveness and stable development. At the same time service
quality level is defined by consumer satisfaction. The aim of this paper is to propose evaluated theoretical and applied methodology of quality management under
62
Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich
competition which is correlated with the study the consumer satisfaction and companies strategic interaction in the market. In this paper we will define quality from
the point of view of customers of the investigated service. Customer involvement
in the production of the services creates additional argument for the importance
of quality evaluation from the customers point. In the paper quality of service is a
quantitative estimation of the level of consumer satisfaction with this service.
Assume as well that service quality is a complex notion defined through its characteristics. The characteristics of quality should be measurable, accurate and reliable.
They are measured on the basis of customers’ opinion. Thus, in this paper the quality is evaluated as a cumulative value assigned for the service quality characteristics
and is cumulative measure of consumer’s satisfaction with service quality. Therefore
the higher is consumer satisfaction the higher is the service quality level.
After consumer satisfaction with the service is analyzed and service quality is evaluated, the goal of the research is to develop theoretical basis (models) and practical
methods of the service quality level evaluation and choice which is made by the
service provider. Therefore, the theoretical research objectives are:
– development of game-theoretical models of service providers’ interaction,
– definition of the strategy of service quality level choice.
The empirical part of the research to suggest practical recommendations for
Russian companies to implement the "quality price" strategy that may increase
companies payoff under competition.
The survey was conducted in St. Petersburg in fitness industry. Consumer preferences and satisfaction were defined, game-theoretical analysis of St. Petersburg
industrial market allowed finding the current state and equilibrium service quality
levels. the change in market shares for Fitness clubs of St. Petersburg was evaluated
in equilibrium.
2.
Game-Theoretical Model of Duopoly
Suppose that there are 2 firms on the market which produce homogeneous services
differentiated by quality. Let firm i produces goods with the quality si , and let
s1 < s2 . Assume, that the values of si are known to both firms and consumers.
According to the model, firms use Bertrand price competition. In this case pi is a
price of firm i for the goods with quality si .
The game-theoretical model is presented as dynamic game which consists of the
following stages: a) each firm i chooses its service quality levels si ; b) firms compete
in prices pi .
Consumers differ in their willingness to pay for quality level s, which is described
by the parameter θ ∈ [0, ∞) . This parameter is called inclination to quality. The
utility of a consumer with a willingness to pay for quality θ (consumer θ) when
buying a service of quality s at a price p is equal to:
θs − p, p 6 θs
Uθ (p, s) =
(1)
0, p > θs
It is clear that the consumer θ will purchase the product of quality s at price p
if Uθ (p, s) > 0 and won”t buy a product otherwise.
Quality Level Choice Model under Oligopoly Competition
63
The investigated industrial market is considered to be partially covered.
The model suggests that inclination to quality is exponentially distributed. This
means that the majority of consumers have the willingness to buy services with
the critical level of quality. The case when consumers are eager to buy the lowest
level of quality is considered, but it may be extended to the situation with the
highest level of quality. Therefore, in the model it is assumed that the parameter of
inclination to quality θ is a random variable and has exponential distribution with
density function:
f (x) =
0, x 6 0
λe−λx , x > 0
(2)
The payoff function of the firm i which provides a service of quality si , where
si ∈ [s, s], is the following:
Ri (p1 , p2 , s1 , s2 ) = pi (s1 , s2 ) × Di (p1 , p2 , s1 , s2 ), i = 1, 2
where pi (s1 , s2 ) is the price of the service of the firm i, Di (p1 , p2 , s1 , s2 ) – the demand
function for the service of quality si , which is specified.
Introduce the following variables: θ1 and θ2 .
Consumer with inclination to quality θ is indifferent to the purchase of goods
with the quality s1 and price p1 , if
θs1 − p1 = 0
(3)
θ1 = θ1 (p1 , s1 ) = p1 /s1
(4)
Then we can find that:
θ1 characterizes a consumer, who is equally ready to buy a service with the
quality s1 and price p1 or refuse to buy this service.
Consumer with inclination to quality θ is indifferent to the purchase of services
with quality s1 , s2 and prices p1 , p2 respectively, if:
θs1 − p1 = θs2 − p2
(5)
Therefore, θ2 is equal:
θ2 = θ2 (p1 , p2 , s1 , s2 ) =
p2 − p1
s2 − s1
(6)
θ2 characterizes a consumer, who is indifferent to buy a good with the quality
s1 and price p1 and a good with the quality s2 and price p2 .
64
Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich
Then, demand function Di (p1 , p2 , s1 , s2 ) for firms 1 and 2 can be presented as
following:

θ2 (p1 , pR2 , s1 , s2 )




D
(p
,
p
,
s
,
s
)
=
f (θ)dθ = F (θ2 (p1 , p2 , s1 , s2 )) − F (θ1 (p1 , s1 ));

1
1
2
1
2

θ1 (p1 , s1 )
∞

R


D
(p
,
p
,
s
,
s
)
=
f (θ)dθ = 1 − F (θ2 (p1 , p2 , s1 , s2 )).

2
1
2
1
2


θ2 (p1 , p2 , s1 , s2 )
Then the payoffs of each three firms will be evaluated by the sales return function:
R1 (p1 , p2 , s1 , s2 ) = p1 × D1 (p1 , p2 , s1 , s2 )
R2 (p1 , p2 , s1 , s2 ) = p2 × D2 (p1 , p2 , s1 , s2 , )
where pi (s1 , s2 ) is the price of the service of the firm i with quality si
Game theoretical model of quality choice is a two stages model, when the choice
on each stage is made simultaneously.
– On the first stage firms i choose quality levels si ;
– On the second stage firms i compete in prices pi . It is assumed, that after the
first stage all quality levels are known to both companies and consumers.
This game theoretical model should be solved using the backward induction
method. It means that Nash equilibrium is fined in two steps. On the first step
assuming that the quality levels are known we find prices p∗i (s1 , s2 ) for services
offered by each firm. On the second step, when the prices p∗i (s1 , s2 ) are known we
find quality levels s∗1 , s∗2 in Nash equilibrium for firms 1 and 2 correspondingly.
Taking into account the exponential distribution of inclination to quality θ , we
can rewrite payoff functions as following:

p2 − p1 p

 R (p , p , s , s ) = p × (e−λθ1 − e−λθ2 ) = p × e−λ s11 − e−λ s2 − s1
1 1 2 1 2
1
1
p − p1

−λ 2

R2 (p1 , p2 , s1 , s2 ) = p2 × e−λθ2 = p2 × e s2 − s1
To find equilibrium prices, use the first order condition:

p − p1 p1 
−λ 2

 ∂R1 = e−λ s1 1 − λ p1 − e s2 − s1 1 + λ p1
s1
s2 − s1 = 0
∂p1
p
−
p
1

−λ 2

 ∂R2 = e s2 − s1 1 − λ p2
s2 − s1 = 0
∂p2
The obtained system of equations has unique solution, which may be calculated
numerically using MATLAB algorithm, where an optimal strategy of the second
s2 − s1
company p∗2 =
, and optimal strategy of the first company p∗1 are defined
λ
from the first equation of the system and are unique.
Thus, for instance, if we have information that s1 = 100, s2 = 150, λ = 0, 15
then one can obtain the following equilibrium prices: p∗1 = 115, p∗2 = 333 .
On the second stage of the analysis the companies compete in quality. As the
solution on the first stage is obtained numerically, on this stage we will also use
Quality Level Choice Model under Oligopoly Competition
65
MATLAB to get the optimal quality strategies and numerical values for market
shares and payoffs in equilibrium.
First analyze how the payoff functions change with quality. It can be shown
numerically that payoff function of second company increases with its quality for
any fixed service quality level of the first company. Therefore, the optimal service
quality level strategy of second company is the highest possible quality level: s∗2 = s.
Fig. 1: Companies payoff function with respect to second company service quality level.
Figure 1 presents the payoff values for both companies when company 2 service
quality level is increasing.
Fig. 2: Company 1 payoff function with respect to its service quality level.
Next, determine the quality levels, where the maximum value function of the
first company is achieved. For this purpose, knowing the quality level of second
company, with a predetermined pitch changing the quality service level of the first
company. For each pair of service qualities from the first-order conditions the prices
are obtained in equilibrium. Figure 2 represents the change in payoff of the first
company with respect to its service quality level (see Fig. 2). The graph shows that
there exists a unique quality level of the first company where the company payoff
achieves its maximum value.
When quality levels are calculated, equilibrium price , demand and revenue can
be found.
Table 1 presents an example of an optimal numerical solution for a described
game when the input parameters are: λ = 0, 15; s = 770.
3.
Game-Theoretical Model of Oligopoly
Suppose now that there are 4 firms on some industrial market. Similarly to the previous section, let firm i produce services of quality si , and lets s1 < s2 < s3 < s4 .
Assume, that the values of si are known to all firms and consumers. According to
the model, firms use Bertrand price competition. In this case pi is a price of firm i
66
Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich
Table 1: Numerical solution in duopoly and exponential distribution of inclination to quality.
s1
s2 p 1
p2
D1 D2 R1 R2
460 770 626 2067 0,32 0,50 199 1029
for the goods with quality si .
The game-theoretical model is presented as dynamic game which consists of the
following stages: a) each firm i chooses its service quality levels si ; b) firms compete
in prices pi .
Consumers differ in their willingness to pay for quality level s, which is described
by the parameter θ ∈ [0, ∞) . This parameter is called inclination to quality. The
utility of a consumer is defined as in the previous section.
The investigated industrial market is again considered to be partially covered.
Now again the model when inclination to quality is exponentially distributed is
analyzed. This means that the majority of consumers have the willingness to buy
services with the critical level of quality. The case when consumers are eager to buy
the lowest level of quality is considered, but it may be extended to the situation
with the highest level of quality.
Suppose that there are 4 firms on the market which produce homogeneous services differentiated by quality. The payoff function of the firm i which provides a
service of quality si , where si ∈ [s, s], is the following:
Ri (p, s) = pi (s) × Di (p, s), i = 1, 4
where pi (s) = pi (s1 , s2 , s3 , s4 ) is the price of the service of the firm i, Di (p, s) =
Di (p1 , p2 , p3 , p4 , s1 , s2 , s3 , s4 )âĹŠ the demand function for the service of quality si ,
which is specified.
Introduce the following variables: θ1 , θ2 , θ3 and θ4
Consumer with inclination to quality θ is indifferent to the purchase of goods
with the quality s1 and price p1 , if
θs1 − p1 = 0
Then we can find that:
θ1 = θ1 (p1 , s1 ) = p1 /s1
θ1 characterizes a consumer, who is equally ready to buy a service with the
quality s1 and price p1 or refuse to buy this service.
Consumer with inclination to quality θ is indifferent to the purchase of services
with quality s1 , s2 and prices p1 , p2 respectively, if:
θs1 − p1 = θs2 − p2
Quality Level Choice Model under Oligopoly Competition
67
Therefore, θ2 is equal:
θ2 = θ2 (p1 , p2 , s1 , s2 ) =
p2 − p1
s2 − s1
θ2 characterizes a consumer, who is indifferent to buy a good with the quality
s1 and price p1 and a good with the quality s2 and price p2 .
Consumer with inclination to quality θ is indifferent to the purchase of goods
with quality s2 , s3 and prices p2 , p3 respectively, if:
θs2 − p2 = θs3 − p3
Therefore, θ3 is equal:
θ3 = θ3 (p2 , p3 , s2 , s3 ) =
p3 − p2
s3 − s2
θ3 characterizes a consumer, who indifferent to buy a good with the quality s2
and price p2 and a good with the quality s3 and price p3 .
Consumer with inclination to quality θ is indifferent to the purchase of goods
with quality s3 , s4 and prices p3 , p4 respectively, if:
θs3 − p3 = θs4 − p4
Therefore, θ4 is equal:
θ4 = θ4 (p3 , p4 , s3 , s4 ) =
p4 − p3
s4 − s3
and characterizes a consumer, who indifferent to buy a good with the quality s3
and price p3 and a good with the quality s4 and price p4 .
Then, demand function Di (p1 , p2 , p3 , p4 , s1 , s2 , s3 , s4 ) for firms 1, 2, 3 and 4 can
be presented as following:
68
Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich

θ2 (p1 , pR2 , s1 , s2 )




D
(p
,
p
,
s
,
s
)
=
f (θ)dθ = F (θ2 (p1 , p2 , s1 , s2 )) − F (θ1 (p1 , s1 ));

1
1
2
1
2



θ
(p
,
s
)

1
1
1



θ3 (p2 , pR3 , s2 , s3 )




D
(p
,
p
,
s
,
s
)
=
f (θ)dθ = F (θ3 (p2 , p3 , s2 , s3 ))

2
1
2
1
2



θ
(p
,
p
,
s
,
s
)

2
1
2
1
2

−F (θ2 (p1 , p2 , s1 , s2 ));

θ4 (p3 , pR4 , s3 , s4 )




D
(p
,
p
,
p
,
s
,
s
,
s
)
=
f (θ)dθ = F (θ4 (p3 , p4 , s3 , s4 ))

3
1
2
3
1
2
3



θ
(p
,
p
,
s
,
s
)

3
2
3
2
3



−F (θ3 (p2 , p3 , s2 , s3 ));



∞

R


f (θ)dθ = 1 − F (θ4 (p3 , p4 , s3 , s4 )).
D4 (p3 , p4 , s3 , s4 ) =



θ4 (p3 , p4 , s3 , s4 )
Then the payoffs of each firm will be evaluated by the sales return function:

R1 (p1 , p2 , s1 , s2 ) = p1 × D1 (p1 , p2 , s1 , s2 )



R2 (p1 , p2 , p3 , s1 , s2 , s3 ) = p2 × D2 (p1 , p2 , p3 , s1 , s2 , s3 )
R

3 (p2 , p3 , p4 , s2 , s3 , p4 ) = p3 × D3 (p2 , p3 , p4 , s2 , s3 , p4 )


R4 (p3 , p4 , s3 , p4 ) = p4 × D4 (p3 , p4 , s3 , p4 )
where pi (s) is the price of the service of the firm i.
Game theoretical model of quality choice is a two stages model, when the choice
on each stage is done simultaneously.
– On the first stage firms i choose quality levels si ;
– On the second stage firms i compete in prices pi . It is assumed, that after the
first stage all quality levels are known to both companies and consumers.
The choice on the first stage is made subsequently and on the second stage simultaneously.
This game theoretical model should be solved using the backward induction
method. It means that Nash equilibrium is fined in two steps. On the first step
assuming that the quality levels are known we find prices p∗i (s) for services offered
by each firm. On the second step, when the prices p∗i (s) are known we find quality
levels s∗1 , s∗2 , s∗3 , s∗4 in Nash equilibrium for firms 1, 2, 3, 4 correspondingly.
To solve the problem MATLAB algorithm similar to the one described in Section
2 is used. Here the Service quality is chosen in 4 subsequent steps.
4.
Quality Estimation
In this research the quality is managed using game-theoretical approach which leads
to the problem of quality measurement. The quality is observed from customer point
of view. We introduce integrated service quality which means the composite index
of consumer satisfaction with the service. The quality may have any value from the
unit interval [0,1]. In situation when a customer is totally satisfied with received
service, the quality of service is equal to one.
Service is represented as a set of characteristics, which should be measurable,
precise and reliable. If characteristics are measurable, it is possible to predict them,
Quality Level Choice Model under Oligopoly Competition
69
choose, plan, control and therefore manage. Only in this case the total quality can
be objectively calculated and can be used to provide managerial recommendations.
In order to calculate service quality in the current state the results of questionnaire the program ASPID 3W by Hovanov (2004) is used. ASPID 3W is based
on the method of summary measures. This method is universal and can be used
both for product and service quality evaluation. The main idea of this method is
to summarize all assessments of one complicated object into one united estimate,
which will characterize the quality of this object. The method can be applied to any
multivariate object: complicated technical systems, different versions of managerial,
organizational and investment decisions, consumers’ goods and services, etc.
The main steps of quality calculation using ASPID 3W are:
1. All initial characteristics are summarized in vector x = (x1 , . . . , xm ). Each of
these characteristics is essential for quality calculation, but they became useful
only after summarizing in one united indicator.
2. After that, vector q = (q1 , . . . , qm ) is formed form individual indicators, representing the function qi = q(xi ; i), qi = q(xi ; i) corresponding to the initial
characteristics and evaluating the tested object using m different criteria.
3. The type of synthesized function Q(q) is chosen which is corresponded with
vector q = (q1 , . . . , qm ). Function Q(q) is depended on vector w = (w1 , . . . , wm )
of non-negative parameters which determine relevance of independent indicators
for aggregated estimation: Q = Q(q) = Q(q; w).
4. The meaning of parameters w = (w1 , . . . , wm ) is determined. These parameters are interpreted as the weights which show the influence of independent
characteristics q1 , . . . , qm on Q. Assume that w1 + · · · + wm = 1 .
To sum up, the quality of services offered by each mobile operator is calculated as
weighted sum of all characteristics of services (coverage area, speed and quality of
data communication, quality of voice transmission, availability of mobile services
offices and payment points, number of additional services, availability of tariffs
and their diversity, technical support) multiplied on average price for this service.
Weights are calculated using the results of the survey and are based on customers’
satisfaction.
5.
Experimental Section
5.1. Fitness Industry in St. Petersburg
The main aim of the empirical study is to test theoretical models for some industrial
market. To do that first fitness industry in St. Petersburg, Russia is analyzed and
we find out the quality levels of services offered by the companies in this market.
For this purpose the questionnaire is used. The main research tool is questionnaire
and it was conducted in St. Petersburg.
St. Petersburg and Moscow - two capitals of fitness industry (Moscow accounts
more than 53 % Russia’s national turnover, St. Petersburg - 17 %). In the end of
2008 (pre-crisis period) the center "Evolution - Sports Consulting" estimated that
in St. Petersburg, there were 377 fitness clubs, but in the crisis period their number
decreased dramatically (Fitness market in Russia. The results of 2010 and forecast
for 2014. Analysis of price dynamics).
At the moment the market of fitness services in St. Petersburg has about 350
fitness clubs. According to the Fitness Faculty Company, in St. Petersburg since
70
Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich
the crisis the number of fitness units increased by 20 %, while only in 2012 this
figure raised by another 40 %. In general, the potential growth of the market in the
North-West region, according to the forecast of "TOP Fitness" is estimated in to
be approximately 25-35 %.
At the moment, the St. Petersburg fitness industry is represented by chain and
non-chain clubs, among them a leader in the market are chain players. It should
be noted that this market is fairly strong with respect to local chains and clubs,
which consequently reduces the number of representatives of Moscow (and thus the
Russian leaders) presented in St. Petersburg. For example, the leaders of the market
share in Russia - Russian Fitness Group (World Class clubs and FizKult) and Strata
Partners (Orange Fitness and CityFitness) have, respectively, 5 ( 3 and 2 ) and 1
( 0 and 1) clubs on the market. In addition, most of Moscow chains significantly
weakened during the crisis.
Here is the list of the main leaders of the market, i.e. main chain clubs, which
leadership is determined by the number of clubs on the market of St. Petersburg:
–
–
–
–
–
–
FITNESS HOUSE
SPORT LIFE
ALEX Fitness
Fitness Planet
OLYMP
Extra Sport
However, given the territoriality of competition, it must be mentioned that most
chain clubs are widespread may lose to single clubs or chains with fewer clubs.
Among the well-known medium-sized but successful regional players are:
–
–
–
–
–
FIT FASHION
TAURAS FITNESS
WORLD CLASS
The Flying Dutchman
Neptune and others.
5.2.
Service Quality Levels Evaluation
In order to evaluate current fitness club service quality levels the questionnaire was
used as a main source of information. Dr. Harrington highlights in his works the
essential role of questionnaires in quality evaluation (Harrington 1991) as they help
to estimate the level of customers’ satisfaction with the offered quality of services.
The questionnaire was developed by authors and based on the SERVPERF approach
to service quality evaluation.
In the paper we suppose that the fitness service has five characteristics which
influence the satisfaction of consumers and their choice of the fitness club:
–
–
–
–
–
Fitness club image,
Gym,
Rooms for group training,
Timetable and variety of classes,
Administration.
Quality Level Choice Model under Oligopoly Competition
71
Table 2: Quality characteristics: club image component.
Code
M 1.1
M 1.2
M 1.3
M 1.4
C1.1
H1.2
U 1.1
H1.2
M 1.4
Description
Nice and comfortable location
Comfortable parking space
Pool is big and clean
Interior is comfortable and pleasant
Useful and informative web site
High quality (professional) reputation
Atmosphere of trust and understanding between clients and club workers
The promises on service quality were fulfilled
Variety of additional services (fitness-bar, medical service, etc.)
Each component is evaluated according to five quality dimensions of SERVPERF
approach: tangibles, reliability, responsiveness, willingness to help customers and
provide prompt service, assurance, empathy.
1. Club image or general characteristic of the club. This component describes
the provider of the service, its location, reputation, reliability and other.
2. Gym and rooms for group training are described with the characteristics listed
in the Table 3 and evaluated by the consumers only if they attend the gym and
rooms for group training correspondingly.
Table 3: Quality characteristics: gym and rooms for group training component.
Code
Gym Group rooms
M 2.1
M 3.1
H2.1
H3.1
U 2.1
U 3.1
O2.1
O3.1
M 2.2
M 3.2
C2.2
C3.2
Description
Modern equipment is used
Safety level is good enough
Personnel is professional and has high competences
Personnel is attentive to clients’ interests
Personnel is good-looking
Personnel has individual approach to each client during the
training class
3. Timetable and variety of classes component may significantly influence the
decision of the client to attend group training. That’s why this component is considered separately.
4. Administration includes administrators of the club and sales departments
employee.
Thus, the questionnaire comprehensively evaluates the perceived service quality
by identifying and assessing the importance of each component of the service from
the point of view of the client, and the level of satisfaction and importance of each
quality characteristics.
To check the questionnaire on the criteria of clarity and accuracy the pilot test
was conducted on the 15 respondents. Testing was made in the form of personal
72
Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich
Table 4: Quality characteristics: timetable and variety of classes.
Code Description
M 3.1 Group trainings are interesting and different
C3.1 Timetable is appropriate
C3.2 Timetable is prepared with respect to clientsâĂŹ demands and desires
O3.1 Demands on timetable changes are considered quickly
Table 5: Quality characteristics: administration component.
Code
M 4.1
U 4.1
C4.1
O4.1
H4.1
Description
Personnel is good-looking
Personnel is polite
Personnel has individual approach to each client problem
Personnel solves clients’ problems quickly
Clients are informed quickly and on time
interviews, which made it possible to compile a complete set of requiring corrections
or clarifications of the final questionnaire.
Sample description. The clients of the following four clubs participated in the
interview:
1.
2.
3.
4.
OLYMP
FITNES FAMILY
FITNESS HOUSE
SPORT-LIFE
120 customers of the clubs participated in the survey (30 people from each club).
The sample is uniform in the sense that there were almost equal number of men (40
%) and women (60 % ), family (42 %) and single (58 %) customers. This allows us
to understand better the difference among client groups , as well as to carry out a
comparative analysis. It was also found that the biggest age group of customers of
fitness services in St. Petersburg are the customers in the age of 20-29 years (42 %)
and 30-39 years (34 %) with middle -income (54 %) and above average (37 %). It
is interesting to notice that most of the customers of fitness services are occupied
in the top management (24 %).
Based on the structure of the questionnaire during an interview respondents
rated the quality of fitness services as a set of certain service characteristics. Therefore we present an assessment of service quality as a generalized quantitative characteristic, which is the aggregate indicator of quality, and thus determines the quality
of fitness services in general as an integral quality (Gladkova, Zenkevich, Sorokina,
2011).
In order to calculate the service quality we used ASPID-3W (Hovanov, 1996),
and conducted quality evaluation process, as follows:
1. The evaluation of composite indicators of perceived quality for each of the components of the four clubs: the image of the club, the gym, group training classes,
the variety of classes and timetable, and administrative staff;
Quality Level Choice Model under Oligopoly Competition
73
2. The calculation of integral measures of perceived quality of the components for
each of the four clubs
3. The calculation of the composite indicator of the perceived service quality
for each club : α1 -OLYMP, α2 -FITNESS FAMILY, α3 -FITNESS HOUSE, α4 SPORT LIFE, where αi : 0 < αi 6 1 (respectively, αi = 1 characterizes the
quality perceived by the consumer as the maximum and gives him the most
satisfaction).
The weights for processing ASPID-3W are the averages of the importance of each
characteristic (for step 1) and the average of the importance of components (for step
2 ). In the Table 6 the average importance of each characteristic and component of
the service are presented.
As the prices for the fitness services in each club, we take the standard figures
listed in the price page on the websites and calculate the average for each club (see
Table 7).
As a result of the procedures described above were obtained the evaluation of
integrated fitness service in each club (see Table 8).
5.3. Service Quality Level Choice under Competition
It is assumed that each customer is characterized by inclination to quality (Gladkova, Zenkevich, Sorokina, 2011) , which can be calculated from the information
that we got from the questionnaire. For this purpose the respondents were asked
about the maximum prices they are ready to pay for the fitness service.
In order to apply game-theoretic modeling to determine optimal strategies for service quality for competing clubs on the investigated geographic market, it is necessary to formulate and test hypotheses about the distribution of customers of
fitness clubs by the inclination to quality. Using the Kolmogorov-Smirnov test the
hypothesis on the exponential distribution was tested and accepted at the level of
significance 0.05.
Also from the price information the respondents are ready to pay for the services
we found that the distribution parameter equal to the reciprocal of the sample mean
is equal to 0.0666.
As a result, it was found that the characteristic penchant for quality has an
exponential distribution. In the case of the exponential distribution of inclination
to quality the game-theoretic model presented in the Section 2 can be applied to
the of oligopoly competition of fitness clubs on the investigated geographic market.
This model is a development of a game-theoretical model of duopoly in a vertical
differentiation J. Tirole (Tirole, 1988) , which is applicable in the case of a uniform
distribution of inclination to quality. In addition to the basic model of J. Tirole,
there are also various modifications and improvements of the model for different
cases. The solution of the model is realized in MATLAB as follows:
1. Arrange all the players according to their current quality levels (see Table 8)?
Where the first player is the one with highest current integrated quality level.
• OLYMP - current quality = 0,639,
• FITNESS FAMILY - current quality = 0,597,
• FITNESS HOUSE - current quality = 0,583,
• SPORT LIFE - current quality = 0,614.
2. Using the value of the distribution parameter = 0.0666, we calculate the equilibrium quality strategies of the players, while setting the desired quality for
74
Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich
Table 6: Average importance of each characteristic and component of the service.
Characteristic Name
Nice and comfortable location
Comfortable parking space
Pool is big and clean
Interior is comfortable and pleasant
Useful and informative web site
High quality (professional) reputation
Atmosphere of trust and understanding
The promises on service quality were
fulfilled
Variety of additional services (fitnessbar, medical service, etc.)
Modern equipment is used in the gym
Safety level in the gym is good enough
Group training personnel is professional
and has high competences
Group training personnel is attentive to
clients’ interests
Group training personnel is goodlooking
Group training personnel has individual
approach
Group trainings are interesting and different
Timetable is appropriate
Timetable is prepared with respect to
clients’ demands and desires
Demands on timetable changes are considered quickly
Administrative personnel is goodlooking
Administrative personnel is polite
Administrative personnel has individual
approach
Administrative personnel solves clients’
problems quickly
Clients are informed quickly and on time
Importance Group Name
6,53
5,40
6,25
5,75
5,00
5,92
6,36
6,32
Club Image
Group
Importance
2,929
Gym
3,541
Timetable
3,035
Administration
2,518
6,08
5,96
5,58
4,95
4,79
4,48
4,84
5,23
5,40
5,11
4,93
6,24
6,49
6,22
6,38
60,34
the leading player 0.9. Note that according to the model the quality is established (achieved) by the players in the order selected on the first step. Then,
the equilibrium service quality strategies are as follows:
•
•
•
•
OLYMP - equilibrium quality = 0,9,
FITNESS FAMILY - equilibrium quality = 0,77,
FITNESS HOUSE - equilibrium quality = 0,71,
SPORT LIFE - equilibrium quality = 0,66.
Quality Level Choice Model under Oligopoly Competition
75
Table 7: Prices of fitness services in each club, rub.
Club
Service price, RUB
OLYMP
20 638
FITNESS FAMILY
16 453
FITNESS HOUSE
16 830
SPORT LIFE
14 500
Table 8: Integrated fitness service quality.
Club
Integrated quality level αi
OLYMP
0,639
FITNESS FAMILY
0,597
FITNESS HOUSE
0,583
SPORT LIFE
0,614
3. Further, in accordance with the equilibrium quality players simultaneously set
prices for their services. Prices are set by the players in accordance with the
laws of competition: the player with the highest quality has the right to charge
a higher price, then the player with the lowest level of quality in setting a high
price only to lose their customers.
Table 9: Modeling results.
Club
Integrated Average Equilibrium Equilibrium
quality current price
price
quality
OLYMP
0,639
20,64
22,64
0,900
FITNESS FAMILY
0,597
17,09
17,94
0,758
FITNESS HOUSE
0,583
16,83
16,33
0,684
SPORT LIFE
0,614
14,50
15,09
0,764
The equilibrium price and the equilibrium service qualities are the optimal
strategies for the competition on the considered geographic market.
4. The use of optimal price and quality strategies will lead the companies to the
following market shares:
• OLYMP - 43 %,
• FITNESS FAMILY - 34 %,
• FITNESS HOUSE - 18 %,
• SPORT LIFE - 5 %.
It should be noted that in this model we assume that the players are rational
and tend to increase (or maintain) its own market share and increase revenue
(or maintain the level of revenue).
6.
Results and Discussion
First of all, it is interesting to investigate the market shares of the fitness clubs on
the geographic market. Exploring the fitness clubs, it was found that the behavior
76
Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich
Fig. 3: Respondents distribution by inclination to quality.
of these clubs on the regional market can be described by means of four competitive
roles (Kotler et al , 2007, 489–521).
The leader of this market is the territorial club OLYMP, which appeared earlier
than the others in the market, has a large customer base and is often a leader in
the introduction of price changes, the presentation of new programs, sporting goods
and other services in their package.
FITNESS FAMILY is a contender for the leadership without any doubts, who
uses violent methods to fight for market share.
The follower is the club FITNESS HOUSE, who used soft policy to retain its
market share. Finally, SPORT LIFE which is characterized by the limited services
provided for customers (SPORT LIFE has no pool, also has a small area, only one
hall for group training and gym), focuses on a niche segment, attracting students
and low-income clients to the club (a student club cards).
However, despite the various competitive roles, the major players except SPORT
LIFE have equal shares of presence in the market, which is a consequence of undifferentiated quality and exceptional price competition, which is wide spread among
fitness. SPORT LIFE retains share of 10 % due to low capacity.
According to the modeling results, players will need to improve the services
quality following the leader, as well as to amend the prices (in accordance with
the objective maximization of the revenue). This will lead to the change in their
future market shares. It should be noted that the redistribution of shares will occur
only as a result of specific programs aimed to improve the service quality of leader
club, which should allow differentiating the leader from other players without price
differentiation.
Differentiation in quality is a major problem to retain players’ market shares. It
can be seen that despite the decline in the shares, OLYMP’s competitors still have
to improve the quality of services at little change in prices. For example, FITNESS
HOUSE club in order to retain its share of 17 % will have to improve the quality
by 20 % (according to the evaluation of the integrated quality), while reducing the
cost. An important factor in the evaluation of the results is the ability to set the
optimal strategy of the player. Given that this club does not have the information
obtained by us during this simulation, we can assume that the strategy will not
be implemented, or will not be held lower prices, i.e. strategy will be implemented
partially. In connection with this, the FITNESS HOUSE club’s market share and
revenue will be even smaller, and the rested share and capital will be distributed
among the other competitors.
According to the modeling results it was shown that almost without changing
the quality and prices (changes to 7 % and 4 %, respectively) SPORT LIFE club
Quality Level Choice Model under Oligopoly Competition
77
is able to secure a share of 4 % to settle for a niche market. Note that the player
SPORT LIFE in comparison to its competitors is fairly weak player and supports
its existence only on thee niche market (in practice it acts in a democratic segment).
The share of player SPORT LIFE is rather dependent on the policy of FITNESS
HOUSE. If FITNESS HOUSE doesn’t hold its share the customers can switch to
the SPORT LIFE club and this will increase the competitiveness of the latter.
However, if the implementation of the new policy is successful and if the quality
reached the optimal level, the player FITNESS HOUSE can even win the share
of SPORT LIFE (take its niche consumers the students by offering student card
at a comparable price). This may raise a question of the existence of the fourth
player. As a result of strong competition and service quality improvement, the very
existence of this chain on the market is under consideration with the club gets under
question. The market share of SPORT LIFE reduced almost by half, which is quite
possible, and this will make it unattractive for business owners who may decide
to close this club or remodel it entirely. When SPORT LIFE leaves the market its
market share will be distributed among the remaining players.
Player 3 and 4 are similar enough in the sense of their possible ways of development. As a result of OLYMP’s actions, their market shares will be reduced
significantly despite the improvement of the service quality (the fact that the share
will grow with the clients of players 3 and 4 should be considered when developing
a marketing strategy). The vast number of customers of these clubs may not decrease at all or just slightly due to the development of the fitness industry and the
appearance of new customers. Consequently, in the long term these clubs will lose
its positions on the market and either leave the market or make a total renovation.
Finally, the main competitor, according to the staff of OLYMP, is FITNESS
FAMILY club (quite big club with a low fullness level). It has considerable potential
to increase its market share. The market share value of this club in thee equilibrium,
which was obtained in Section 3, restrained only by a successful and increasing
differentiation of the club OLYMP. FITNESS FAMILY will increase its share from
the "mediocre" position of the above players 3 and 4. Its quality should to be
improved by 29 % when the price only by 5 %, but given the "weak position" in
quality, which are the characteristics of the personnel (gym and group training) as
well as the variety of proposed activities, it can be said that these shortcomings in
the service typical enough for a new player in the fitness service market. As practice
shows, the cubs quickly improve these disadvantages and may become a leader of
the investigated market.
The situation for OLYMP club is rather complicated: it is necessary to improve
the service quality by 40 % , raising the price by 10 % only. It is understandable
that this strategy is aimed not only to the increase of market share but to hold the
aggressive competitor FITNESS FAMILY. OLYMP must devote all its efforts to
achieve the optimal level of service quality to avoid the unfavorable scenario.
7.
Conclusion
Therefore as a result of the research we developed the best strategies for each player
of the investigated territorial market. On the basis of competitive analysis we evaluated the reality of these strategies implementation for each player. In addition,
we also confirmed the competitiveness and potential of leadership of the club FITNESS FAMILY and its "danger" as a competitor for the club OLYMP. Therefore,
78
Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich
Table 10: Strategic quality and price changes for equilibrium achievement.
Change of integrated quality
Club
Value
%
OLYMP
0,26
40,85 %
FITNESS FAMILY 0,17
28,64 %
FITNESS HOUSE 0,13
21,44 %
SPORT LIFE
0,04
6,84 %
Change in price strategy
Value
%
1,997
9,68 %
0,8523
4,99 %
-0,4978
-2,96 %
0,5944
4,10 %
we reiterate the need to improve the service quality of the club OLYMP to preserve
leadership on this market.
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A Problem of Purpose Resource Use in Two-Level Control
Systems ⋆
Olga I. Gorbaneva1 and Guennady A. Ougolnitsky2
1
Southern Federal University,
Faculty of Mathematics, Mechanics, and Computer Sciences,
Milchakova St. 8A, Rostov-on-Don, 344090, Russia
E-mail: [email protected]
2
E-mail: [email protected]
Abstract The system including two level players–top and bottom–is considered in the paper. Each of the players have public (purpose) and private
(non-purpose) interests. Both players take part of payoff from purpose resource use. The model of resource allocation among the purpose and nonpurpose using is investigated for different payoff function classes and for
three public gain distribution types. A problem is presented in the form of
hierarchical game where the Stackelberg equilibrium is found.
Keywords: resource allocation, two-level control system, purpose use, nonpurpose resource use, Stackelberg equilibrium.
1.
Introduction
A wide set of social and economic development problems is solved due to budget
financing, which is performed in different forms (grants, subventions, assignments,
credits) and always has a strictly purpose character, i.e., allocated funds should be
spent only on prescribed needs. Article 289 of the Budget Code of the Russian Federation and Article 15.14 RF Code on Administrative Offences make provisions on
responsibility for non-purpose use of budget funds. Nevertheless, non-purpose use of
budget financing is widespread and can be considered as a kind of opportunistic behavior corresponding to the private interests of the active agents (Willamson, 1981).
Non-purpose use of resources is linked to corruption, especially to “kickbacks”, when
budget funds are allocated to an agent in exchange for a bribe and only partially
used appropriately. They are largely spent on private agent-briber interests.
It is naturally for the resource use problem to be treated in terms of the interest
concordance in hierarchical control systems. This allows for a mathematical apparatus of hierarchical game theory (Basar, 1999), of contract theory (Laffont, 2002),
information theory of hierarchical systems (Gorelik, 1991), active system theory
(Novikov, 2013a) and organizational system theory (Novikov, 2013b). Simultaneously, resource allocation models in hierarchical systems with regard to their misuse
are little studied (Germeyer, 1974) and are analyzed in authors’ investigation line
(Gorbaneva and Ougolnitsky, 2009-2013).
This article is focused on the question how resource allocation among purpose and
non-purpose directions is depended on different public and private payoff function
classes of distributor and resource recipients.
⋆
This work was supported by the the Russian Foundation for Basic Research, project #
12-01-00017
82
2.
Olga I. Gorbaneva, Guennady A. Ougolnitsky
Structure of investigation
We consider a two-level control system which consists of one top level element A1
(resource distributor) and one bottom level element A2 (resource recipient). The top
level has some resource amount (which we assume to be a unit). The distributor
assigns a part of resources to the recipient for purpose use, and the rest for his own
interests. The bottom level assigns in his turn a part of obtained resources for his
own interests (non-purpose use), and the rest for the public interests (purpose use).
Both levels take part in purpose activity profit and have their payoff functions (Fig.
1).
Fig. 1: The structure of modeled system.
The model is built as a hierarchical two-person game in which a Stackelberg
equilibrium is sought (Basar, 1999). A payoff function of each player consists of two
summands: non-purpose activity profit and a part of the system purpose activity
profit. The payoff functions are:
g1 (u1 , u2 ) = a1 (1 − u1 , u2 ) + b(u1 , u2 ) · c(u1 , u2 ) → max;
u1
g2 (u1 , u2 ) = a2 (u1 , 1 − u2 ) + b(u1 , u2 ) · c(u1 , u2 ) → max .
u2
subject to
0 ≤ ui ≤ 1, i = 1, 2,
and conditions on functions a, b and c
ai ≥ 0;
∂ai
∂ai
≤ 0,
≥ 0, i = 1, 2,
∂ui
∂uj6=i
∂bi
bi ≥ 0;
≥ 0, i = 1, 2,
∂ui
∂c
≥ 0, i = 1, 2.
∂ui
Here index 1 relates to the top level attributes (a leading player), index 2 relates to
the bottom level attributes (a following player);
- ui is a share of resources assigned by i-th level to the purpose use (correspondingly,
A Problem of Purpose Resource Use in Two-Level Control Systems
83
1 − ui remains on non-purpose resource use in private interests);
- gi is a payoff function of i-th level;
- ai is a payoff function of i-th level private interest;
- bi is a share of purpose activity profit obtained by i-th level;
- c is a payoff function of purpose system activity (society, organization).
Power, linear, exponential and logarithmic functions are considered as functions a
and c. These functions depend on variables u1 , u2 , and they are cumulative ones,
i.e. a1 = a1 (1 − u1 ), a1 = a2 (u1 (1 − u2 )), c = c(u1 u2 ). In this case a share of
resources being is assigned to the public aims.
The relations a1 = a1 (1 − u1 ), a1 = a2 (u1 (1 − u2 )), reflect the hierarchical structure
of the system. The non-purpose activity income of top level does not depend on the
part of the funds the bottom level assigned for the public aims but the non-purpose
activity income of bottom level depends on the part of the funds the top level gives
him.
Three income types of purpose income distribution b are considered:
1) uniform one, in which the shares in purpose activity income are the same for
both players, in particular, if n = 2
bi =
1
, i = 1, 2,
2
2) proportional one, in which the shares in income are proportional to the shares
assigned to the public aims by the corresponding level, i.e.
u1
,
u1 + u2
u2
b2 =
;
u1 + u2
b1 =
3) constant one, in which:
b1 = b,
b2 = 1 − b;
The player strategy is a share ui of available resources assigned to the public aims.
The top-level player u1 defines and informs the bottom level about it. Then the
second player chooses the optimal value u2 knowing the strategy of the first player.
The investigation aim is to study how the relation of functions a1 , a2 , b1 , b1 , c effects
on the game solution (Stackelberg equilibrium).
The next functions are taken as a non-purpose payoff function:
- power with an exponent less than one (a(x) = axα , α < 1, a > 0),
- linear (a(x) = ax, a particular case of power function with an exponent equaled
to one),
- power with an exponent greater than one, (a(x) = axk , k > 1, a > 0);
- exponential (a(x) = a(1 − exp −λx), λ > 0, a > 0);
- logarithmic (a(x) = alog(1 + x), a > 0).
∂a
∂2a
As a rule, functions are chosen with constraints ∂x
≥ 0, ∂x
2 ≤ 0. The first condition
is satisfied by all functions, the second condition is not satisfied only by function
a(x) = axk , k > 1. The first and the second functions are production functions. The
last two functions are not production ones since the property of scaling production
returns does not hold.
84
Olga I. Gorbaneva, Guennady A. Ougolnitsky
Similarly, the next functions are taken as a purpose payoff function:
- power with an exponent less than one (c(x) = cxα , α < 1, c > 0);
- linear (c(x) = cx);
- power with an exponent greater than one, (c(x) = cxk , k > 1, c > 0);
- exponential (c(x) = c(1 − exp −λx), λ > 0, c > 0);
- logarithmic (c(x) = clog(1 + x), c > 0).
Thirteen of twenty five possible combinations are solved analytically:
1) combinations of similar functions, when a and c are either power, or exponential,
or logarithmic ones;
2) combinations of any non-purpose use function and linear purpose use function;
3) combinations of linear non-purpose use function and any purpose use function.
Six of the rest cases are investigated numerically.
3.
Analytical investigation of different model classes
We consider the case when a1 (u1 , u2 ) = 1 − u1 , a2 (u1 , u2 ) = u1 (1 − u2 ) ,c2(u1 , u2 ) =
u1 u2 , b1 = 12 , b2 = 21 .
Then:
u1 u2
→ max,
u1
2
u1 u2
g2 (u1 , u2 ) = u1 −
→ max
u2
2
g1 (u1 , u2 ) = 1 − u1 +
(1)
(2)
This is the game with constant sum. Function g2 decreases in u2 , therefore the
optimal value u2 ∗ = 0, at which g1 (u1 , 0) = 1 − ui . Function g1 decreases on u1 ,
therefore the value u1 ∗ = 0 is optimal.
So, Stackelberg equilibrium in the game is ST1 = {(0; 0)} , while the player gains
are g1 = a1 , g2 = 0 , i.e. both players use strategy of egoism (assign all available
resources for private aims), but the top level gets maximum, while the bottom level
gets zero.
Consider the case when a1 = a1 (1 − u1 ), a2 = a2 u1 (1 − u2 ), c = (u1 u2 )k is the
production power function.
There may be two fundamentally different cases:
1) k = 1 (linear resource use function);
Then, g1 (u1 , u2 ) = a1 (1 − u1) + b1 u1 u2 , g2 (u1 , u2 ) = a2 u1 (1 − u2) + b2 u1 u2 . We find
optimal strategy of the bottom level:
∂g2
= (b2 − a2 )u1 ,
∂u2
u∗2
=
1, b2 > a2 ,
0, b2 < a2 .
The top level optimizes his gain function:
a1 (1 − u1 ) + b1 u1 u2 , b2 > a2 ,
g1 (u1 , u∗2 ) =
a1 (1 − u1 ),
b 2 < a2 .
∂g1
=
∂u1
b 1 − a1 , b 2 > a2 ,
−a1 , b2 < a2 .
A Problem of Purpose Resource Use in Two-Level Control Systems
Thus, (Fig. 2),
u∗1
=
85
1, (b2 > a2 ) ∧ (b1 > a1 ),
0, (b2 < a2 ) ∨ (b1 < a1 ).
If b2 > a2 and b1 > a1 then both players apply altruistic strategy (u1 ∗ = u2 ∗ = 1),
and g1 = b1 , g2 = b2 . In other cases the leading player behaves egoistically (u1 ∗ = 0),
then g1 = a1 , g2 = 0.
Fig. 2: Game outcomes (3.1)-(3.2).
2) 0 < k < 1 (power resource use function).
Then, g1 (u1 , u2 ) = a1 (1 − u1 ) + b1 (u1 u2 )k , g2 (u1 , u2 ) = a1 u1 (1 − u2 ) + b2 (u1 u2 )k .
We find the bottom level optimal strategy:
∂g2
= −a2 u1 + kb2 (u1 u2 )k−1 = 0,
∂u2
1
∗
u2 =
a2 k−1
( kb
)
2
u1
.
The top level optimizes his payoff function:
g1 (u1 , u2 ∗ ) = b1 (
k
a2 k−1
)
+ a1 (1 − u1 ).
kb2
Since function g1 decreases on u1 , then u1 ∗ = 0.
We consider the case when the payoff function from non-purpose activity is linear,
the payoff function from purpose activity is logarithmic, and a share of the purpose
activity profit is constant for both levels:
a1 (u1 , u2 ) = a1 (1 − u1 ), a2 = a2 u1 (1 − u2 ),
c = c log2 (1 + u1 u2 ), b1 = b, b2 = 1 − b.
86
Olga I. Gorbaneva, Guennady A. Ougolnitsky
Then gain functions are
g1 (u1 , u2 ) = a1 (1 − u1 ) + bc log2 (1 + u1 u2 ) → max,
(3)
g2 (u1 , u2 ) = a2 u1 (1 − u2 ) + (1 − b)c log2 (1 + u1 u2 ) → max,
(4)
u1
u2
subject to
0 ≤ ui ≤ 1, i = 1, 2.
Find the Stackelberg equilibrium. We divide this process into two phases and describe in detail now.
1) First, we solve a bottom level optimization problem. Suppose the value u1 is
known. We find the derivative of g2 with respect to u2 and equate it to zero:
∂g2
(1 − b)cu1
(u1 , u2 ) = −a2 u1 +
= 0.
∂u2
(1 + u1 u2 ) ln 2
We solve the equation. The case u1 = 0 has no practical interest,
therefore
we can
−
1
. Finding
divide both parts of equation by u1 and express u2 : u2 ∗ = u11 (1−b)c
a2 ln 2
the second derivative of the function g2 with respect to u2 , we see that the point
u2 ∗ is a maximum point:
∂ 2 g2
(1 − b)cu1 2
(u
,
u
)
=
−
< 0.
1
2
∂u2 2
(1 + u1 u2 )2 ln 2
Taking into account the restriction on u2 , note that the optimal strategy of the
bottom level player is


a2 ln 2 ≥
(1 − b)c,  0,
(1−b)c
(1−b)c
1
1
∗
u2 = u1 a2 ln 2 − 1 , 0 < u1 a2 ln 2 − 1 < 1,


(1−b)c
1,
a2 ln 2 ≥ 1 + u1 ,
2) Solve a top level problem if the bottom level answer is known. Consider three
cases: a) u2 ∗ = 0 .
In this case g1 (u1 , 0) = a1 (1 − u1 ) + bc log2 1 = a1 (1 − u1 ). Since g1 decreases in
u1 , the top level optimal strategy is u1 ∗ = 0, i.e. if top level knows that the bottom
level assigns all available resources for the private aims, then he gives no resources
to the bottom
level andassigns the resources for his private aims.
1
∗
b) u2 = u1 (1−b)c
a2 ln 2 − 1 .
Then, g1 (u1 , u2 ∗ ) = a1 (1 − u1 ) + bc log2 (1−b)c
a2 ln 2 .
Here, similar to the previous case, the function g1 decreases with respect to u1 . Note
that the bottom level chooses his strategy so that the constant value of resources
is assigned for the public aims. Hence, the more resource is given to the bottom
level by the top one, the more may be spent on the bottom level private aims (as
the difference between resources, which were given by the top level, and constant
value u1 u2 = (1−b)c
a2 ln 2 − 1, which were assigned for the public aims by the bottom
level). And conversely, the less resource is given to the bottom level by the top one,
the less may be spent on bottom level private aims. Hence, taking into account
A Problem of Purpose Resource Use in Two-Level Control Systems
87
the decreasing of function in u1 , it is profitable for the top level to assign as little
as possible resource for the public aims, hence the bottom level assigns as little as
possible for the public aims. So, it is profitable for the bottom level to assign for
the public aims as much resources as the bottom level assigns for the public aims,
namely u1 ∗ = (1−b)c
a2 ln 2 − 1, thereby causing the lower level to spend all the resources
on public aims, i.e. u2 = 1.
c) u2 ∗ = 1.
In this case g1 (u1 , 1) = a1 (1 − u1 ) + bc log2 (1 + u1 ). Maximize this function taking
into account the restriction 0 ≤ u1 ≤ 1.
From the first order conditions
∂g1
bc
(u1 , 1) = −a1 +
= 0.
∂u1
(1 + u1 ) ln 2
we obtain:
u1 ∗ =
bc
− 1.
a1 ln 2
Finding the second derivative of g1 with respect to u1 , we can see that the point
u1 ∗ is a point of maximum:
bc
∂ 2 g1
(u1 , u2 ∗ (u1 )) = −
< 0.
∂u1 2
(1 + u1 )2 ln 2
Taking into account the restriction on u1, the optimal strategy of the bottom level
is

0,
a1 ln 2 ≥ bc,

u∗1 = a1bcln 2 − 1, 0 < a1bcln 2 − 1 < 1,

bc
1,
a1 ln 2 − 1 ≥ u1 ,
So, the Stackelberg equilibrium is


(0; 0),
a2






(1; 1),
a2
ū = bc

− 1; 1 , a2



a1 ln 2


 (1−b)c − 1; 1 , a2
a2 ln 2
>
<
<
>
(1−b)c
bc
ln 2 or a1 > ln 2 ,
(1−b)c
bc
2 ln 2 and a1 < 2 ln 2 ,
a1 (1−b)
and 2 bc
< a1 < lnbc2 ,
b
ln
2
a1 (1−b)
(1−b)c
(1−b)c
and
<
a
<
.
2
b
2 ln 2
ln 2
As can be seen from this formula, if assigning of some resource part for the public
aims is profitable for the bottom level then the top level can enforce the bottom
level to assign all the resources for the public aims. I.e., the bottom level assigns all
the resources either only for public aims or only for private aims.
Consider each branch of the Stackelberg equilibrium:
I. u = (0; 0) if a2 > (1−b)c
or a1 > lnbc2 (Fig.2). In this case for one or two of the
ln 2
players the private activity gives much more profit than the public activity. It is
not profitable for this player to assign the resources for the public aims, but then
another player either has no incentive to assign resources to the public aims (for
the top level) or has no resources (for the bottom level). The players’ gains are
g1 = a1 , g2 = 0.
88
Olga I. Gorbaneva, Guennady A. Ougolnitsky
bc
II. u = (1; 1) if a2 < (1−b)c
2 ln 2 and a1 < 2 ln 2 (Fig.3). In this case for both players the
public activity gives much more profit than the private activity, therefore each of
them assigns all the resources for the public aims. The players’ gains are
g1 = bc, g2 = (1 − b)c.
bc
III. u =
− 1; 1) if a2 < a1 (1−b)
and 2 bc
b
ln 2 < a1 < ln 2 (Fig.3). In this case for
the top level it is profitable to assign only a part of resources for the public aims
(since the both activities profits are comparable) while for the bottom level it is
profitable to assign all the resources for the public aims. The players’ gains are
bc
bc
bc
+ bc log2
, g2 = (1 − b)c log2
.
g1 = 2a1 −
ln 2
a1 ln 2
a1 ln 2
( a1bcln 2
a1 (1−b)
(1−b)c
IV. u = ( (1−b)c
and (1−b)c
a2 ln 2 − 1; 1) if a2 >
b
2 ln 2 < a2 < ln 2 (Fig.3). In this case
for both players it is profitable to assign a part of the resources for the public aims,
since the both activities profits are comparable. The bottom level is going to assign
a fixed value of resources for the public aims and to leave the rest for the private
aims. But the top level gives only this fixed value of resources to the bottom level
thereby he enforces the bottom level to assign all the resources for the public aims.
The players’ payoffs are
(1 − b)c
(1 − b)c
a1 (1 − b)c
+ bc log2
, g2 = (1 − b)c log2
.
g1 = 2a1 −
a2 ln 2
a2 ln 2
a2 ln 2
Fig. 3: Game outcomes (3.3)-(3.4)
Finally, we consider the case when purpose and non-purpose activity functions
are power with an exponent less than one:
α
a1 = a1 (1 − u1 )α , a2 = a2 (u1 (1 − u2 )) ,
c = (u1 u2 )α , b1 = b, b2 = 1 − b.
89
A Problem of Purpose Resource Use in Two-Level Control Systems
Then gain functions are
g1 (u1 , u2 ) = a1 (1 − u1 )α + bc(u1 u2 )α → max,
u1
α
g2 (u1 , u2 ) = a2 (u1 (1 − u2 )) + (1 − b)c(u1 u2 )α → max,
u2
The Stackelberg equilibrium is (Fig. 4):

q
ū = 
1−α
a1
b(1−b) 1−α c 1−α
q
√
α
1
√ α 1−α
1−α
(1−b)c+ 1−α a2 +
b(1−b) 1−α c 1−α
√
1−α
√
1−α
(1−b)c
(1−b)c+
√
1−α
(6)
1
α
1−α
r
(5)
a2
;
We omit the players’ payoffs in this case.
All the thirteen considered cases can be grouped together on the number of outcomes of the game:
1) One outcome, when public and private payoff functions are power with an exponent less than one. In this case for both players it is profitable to assign a part of
resources for the public aims, and another part for the private aims.
2) Two outcomes (0; 0) and (1;1) (Fig. 2), when:
a. The private payoff function is power with an exponent less than one and the
public payoff function is linear;
b. The public and private payoff functions are either linear or power with an exponent greater than one in any combinations.
3) Three outcomes, when private activity function is linear and public payoff function is power with an exponent less than one. In this case for one of the player it is
profitable to assign all the resources for the public aims.
4) Four outcomes (Fig. 3), when one of the functions (either private or public payoff) is linear and another function is logarithmic.
5) Five outcomes (Fig. 4), when a. Public and private payoff functions are linear or
exponential in any combinations except the case when both the functions are linear.
b. Public and private payoff functions are logarithmic.
4.
Numerical investigation of different model classes
We use a numerical investigation for a few cases that could not be solved analytically.
At first we consider a case when the purpose activity function is exponential and
the non-purpose activity function is power with an exponent less than one, purpose
activity profit share is constant for the players:
a1 = a1 (1 − u1 )α , a2 = a2 (u1 (1 − u2 ))α ,
c = c(1 − e−λu1 u2 ), b1 = b, b2 = 1 − b.
In this case the payoff functions are
g1 (u1 , u2 ) = a1 (1 − u1 )α + bc(1 − e−λu1 u2 ) → max,
(7)
g2 (u1 , u2 ) = a2 (u1 (1 − u2 ))α + (1 − b)c(1 − e−λu1 u2 ) → max,
(8)
u1
u2
subject to
0 ≤ ui ≤ 1, i = 1, 2.
90
Olga I. Gorbaneva, Guennady A. Ougolnitsky
Fig. 4: One of the possible cases of the considered game with five outcomes
To find the bottom level optimal strategy we calculate the derivative of g2 with
respect to u2 and equate it to zero:
∂g2
a2 αu1 α
(u1 , u2 ) = −
+ λu1 (1 − b)ce−λu1 u2 = 0.
∂u2
(1 − u2 )1−α
(9)
Prove that the bisection method may be applied for solving this equation. Note that
the second derivative of g2 with respect to u2 is negative,
∂ 2 g2
a2 α(1 − α)u1 α
(u
,
u
)
=
− λ2 u21 (1 − b)ce−λu1 u2 < 0,
1
2
∂u2 2
(1 − u2 )2−α
∂g2
(u1 , u2 ) is monotone.
therefore, the function ∂u
2
∂g2
Then find signs of ∂u2 (u1 , u2 ) at the endpoints of [0,1].
∂g2
∂u2 (u1 , u2 )
∂g2
α
∂u2 (u1 , 0) = −a2 αu1 + λu1 (1 − b)c,
α
1
→u2 →1_ − a2 αu
+ λu1 (1 − b)ce−λu1 u2
0+
(10)
→u2 →1_ −∞.
(11)
If (10) is positive, then the equation may be solved by the bisection method, and
the solution obtained is a maximum point since the second derivative is negative. If
(10) is negative, then bisection method is not applied, but the left part of equation
is monotone then it is negative at the segment [0, 1], hence, function g2 decreases,
then the maximum point is u2 = 0.
That is,
0,
−a2 αu1 α + λu1 (1 − b)c < 0,
∗
u2 =
∈ (0; 1), −a2 αu1 α + λu1 (1 − b)c > 0,
The top level can use this information to enforce the bottom level to choose non-zero
strategy. For the bottom level to choose the positive strategy u2 > 0, it is necessary
to satisfy the condition −a2 αu1 α + λu1 (1 − b)c > 0. When the inequality have been
A Problem of Purpose Resource Use in Two-Level Control Systems
solved for the variable u1 , we obtain u1 >
q
1−α
91
a2 α
λ(1−b)c
For the bottom level not to spend all the resources onqprivate aims, it is recoma2 α
mended for the top level to choose the strategy u1 > 1−α λ(1−b)c
. But he can do it
1−α
q
a2 α
a2 α
only if 1−α λ(1−b)c
< 1, which is equivalent to a2 < λ(1−b)c
.
If the top level cannot use this strategy or this strategy is not profitable for him
then the bottom level choose the strategy u2 = 0. Find then the optimal top level
behavior and his payoff
g1 (u1 , 0) = a1 (1 − u1 )α .
As can be seen, the function g1 decreases in u1 , therefore, u1 = 0.
Draw some
conclusions:
1−α
a2 α
I. If a2 > λ(1−b)c
then the top level cannot effect on the bottom one, in this
case u2 = 0, and therefore u1 = 0. This occurs when the capacity of the bottom level
of non-purpose activity is significantly more than production capacity of purpose
activity.
1−α
a2 α
then the top level can enforce the bottom level to spend
II. If a2 < λ(1−b)c
q
a2 α
some part of resources on the public aims assigning u1 > 1−α λ(1−b)c
. This occurs
when the capacity of the bottom level of purpose activity is significantly more than
production capacity of non-purpose activity.
5.
Conclusion
In this paper a problem of non-purpose resource use is treated in terms of analysis of
control mechanism properties providing the concordance of interests in hierarchical
(two-level) control systems. The interests of players are described by their payoff
functions including two summands: purpose and non-purpose resource use profits.
Different classes of these functions are considered. The top level subject (resource
distributor) is treated as a leading player and the bottom level (resource recipient)
subject is treated as a following player. This leads to the Stackelberg equilibrium
concept. Performed analytical and numerical investigation permits to make the next
conclusions.
In the case when the payoff functions for purpose and non-purpose activities are
power with an exponent less than one it is profitable to assign only a part of resources
for the public aims and another part of them for the private aims for both players.
In the case when one of the payoff functions for purpose or non-purpose activities
is power with an exponent greater than one and another of them is either linear or
power with an exponent greater than one it is profitable to assign all the resources for
only public aims (“egoism” strategy) or for only private aims (“altruism” strategy).
In other cases the next situations may occur:
A) if the effect of the private activities of a player is much more than effect of the
public activity then for a player the “egoism” strategy is profitable;
B) if the effect of the private activities of a player is much less than effect of the
public activity then for a player the “altruism” strategy is profitable;
C) if the effects of the private and public activities of a player are comparable then
for any player it is profitable to assign only a part of resources for the public aims
and the other part for the private aims.
92
Olga I. Gorbaneva, Guennady A. Ougolnitsky
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Multicriteria Coalitional Model of Decision-making over the
Set of Projects with Constant Payoff Matrix in the
Noncooperative Game
Xeniya Grigorieva
St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
University pr. 35, St.Petersburg, 198504, Russia
E-mail: [email protected]
WWW home page: http://www.apmath.spbu.ru/ru/staff/grigorieva/
Abstract Let N be the set of players and M the set of projects. The multicriteria coalitional model of decision-making over the set of projects is
formalized as family of games with different fixed coalitional partitions for
each project that required the adoption of a positive or negative decision
by each of the players. The players’ strategies are decisions about each of
the project. The vector-function of payoffs for each player is defined on the
set situations in the initial noncooperative game. We reduce the multicriteria noncooperative game to a noncooperative game with scalar payoffs
by using the minimax method of multicriteria optimization. Players forms
coalitions in order to obtain higher income. Thus, for each project a coalitional game is defined. In each coalitional game it is required to find in
some sense optimal solution. Solving successively each of the coalitional
games, we get the set of optimal n-tuples for all coalitional games. It is
required to find a compromise solution for the choice of a project, i. e. it is
required to find a compromise coalitional partition. As an optimality principles are accepted generalized PMS-vector (Grigorieva and Mamkina, 2009;
Petrosjan and Mamkina, 2006) and its modifications, and compromise solution.
Keywords: coalitional game, PMS-vector, compromise solution, multicriteria model.
1.
Introduction
The set of agents N and the set of projects M are given. Each agent fixed his participation or not participation in the project by one or zero choice. The participation
in the project is connected with incomes or losses by different parametres which the
agents wants to maximize or minimize. This gives us an optimization problem which
can be modeled as multicriteria noncooperative game. We reduce the multicriteria
noncooperative game to a noncooperative game with scalar payoffs by using the
minimax method of multicriteria optimization. Agents may form coalitions. This
problem we will call as multicriteria coalitional model of decision-making.
Denote the players by i ∈ N and the projects by j ∈ M . The family M of different games are considered. In each game Gj , j ∈ M the player i has two strategies
accept or reject the project. The payoff of the player in each game is determined
by the strategies chosen by all players in this game Gj . As it was mentioned before the players can form coalitions to increase the payoffs components. In each
94
Xeniya Grigorieva
game Gj coalitional partition is formed. The problem is to find the optimal strategies for coalitions and the imputation of the coalitional payoff between the members of the coalition. The games G1 , . . . , Gm are solved by using the PMS-vector
(Grigorieva and Mamkina, 2009; Petrosjan and Mamkina, 2006) and its modifications.
Then having the solutions of games Gj , j = 1, m the optimality principle - “the
compromise solution" is proposed to select the best projects j ∗ ∈ M . The problem
is illustrated by example of the interaction of three players.
2.
State of the problem
Consider the following problem. Suppose
– N = {1 , . . . , n} is the set of players;
– Xi = {0 ; 1} is the set of pure strategies xi of player i , i = 1, n. The strategy
xi can take the following values: xi = 0 as a negative decision for the some
project and xi = 1 as a positive decision;
– li = 2 is the number of pure strategies of player i;
– x = (x1Q
, . . . , xn ) is the n-tuple of pure strategies chosen by the players;
– X=
Xi is the set of n-tuples;
i=1 , n
P
– µi = {ξ (xi )}xi ∈Xi , ξ (xi ) > 0 ∀ xi ∈ Xi ,
ξ (xi ) = 1 is the mixed strategy
xi ∈Xi
of player i; will be used denotation too µi = ξi0 , ξi1 , where ξi0 is the probability
of making negative decision by the player i for some project, and ξi1 is the
probability of making positive decision correspondingly;
– Mi is the set of mixed strategies of the i-th player;
– µ is theQn-tuple of mixed strategies chosen by players for some project;
– M=
Mi is the set of n-tuples in mixed strategies for some project;
i=1, n
– Ki : X → Rr is the vector-function of payoff defined on the set X for each
player i , i = 1, n .
Thus, we have multicriteria noncooperative n-person game G̃ ( x):
D
E
G̃ (x) = N, {Xi }i=1 , n , {Ki (x)}i=1 , n , x∈X .
(1)
Using the minimax method of multicriteria optimization, we reduce the noncooperative n-person game G̃(x) to a noncooperative game G(x) with scalar payoffs:
E
D
G (x) = N, {Xi }i=1 , n , {Ki (x)}i=1 , n , x∈X ,
(2)
where
Ki (x) = max Kis (x) , Kis (x) ∈ Ki (x) , x ∈ X ,
(3)
s=1 , r
Ei (µ) =
X
x1 ∈X1
...
X
[Ki (x) ξ (x1 ) . . . ξ (xn )] , i = 1 , n .
(4)
xn ∈Xn
Now suppose M = {1 , . . . , m} is the set of projects, which require making
positive or negative decision by n players.
95
Multicriteria Coalitional Model of Decision-making
A coalitional partitions Σ j of the set N is defined for all j = 1 , m:
l
n
o
[
Σ j = S1j , . . . , Slj , l 6 n , n = |N | , Skj ∩ Sqj = ∅ ∀ k 6= q,
Skj = N .
k=1
Then we have m simultaneous l-person coalitional games Gj (xΣ j ) , j = 1 , m , in
normal form associated with the game G (x):
n
o
n
o
j (xΣ j )
Gj (xΣ j ) = N, X̃S j
,
H̃
, j = 1, m.
S
j
j
j
j
k
k=1 , l , Sk ∈Σ
k=1 , l , Sk ∈Σ
k
(5)
Here for all j = 1 , m:
– x̃S j = {xi }i∈S j is the l-tuple of strategies of players from coalition Skj , k = 1, l;
k
k
Q
– X̃S j =
Xi is the set of strategies x̃S j of coalition Skj , k = 1, l, i. e. Cartek
–
–
–
–
k
i∈Skj
sian product of the sets of players’ strategies, which are included into coalition
Skj ;
xΣ j = x̃S j , . . . , x̃S j ∈ X̃, x̃S j ∈ X̃S j , k = 1, l is the l-tuple of strategies
1
l
k
k
of all coalitions;
Q
X̃S j is the set of l-tuples in the game Gj (xΣ j );
X̃ =
k
l
k=1,
Q
lS j = X̃S j =
li is the number of pure strategies of coalition Skj ;
k
k
j
i∈Sk
Q
lΣ j =
lS j is the number of l-tuples in pure strategies in the game Gj (xΣ j ).
k
k=1,l
– M̃S j is the set of mixed strategies µ̃S j of the coalition Skj , k = 1, l;
k
k
l j
S
l j
k
P
S
– µ̃S j = µ̃1S j , ... , µ̃S jk , µ̃ξS j > 0 , ξ = 1, lS j ,
µ̃ξS j = 1, is the mixed
k
k
k
k
k
ξ=1
k
strategy, that is the set of mixed strategies of players from coalition Skj , k =
1, l; – µΣ j = µ̃S j , . . . , µ̃S j ∈ M̃, µ̃S j ∈ M̃S j , k = 1, l, is the l-tuple of mixed
1
l
k
k
strategies;
Q
– M̃ =
M̃S j is the set of l-tuples in mixed strategies;
k
k=1, l
– ẼSk (µ̃) is the payoff function of coalition Skj in mixed strategies and defined as
i
X
X h
ẼSk (µ̃) =
...
H̃Sk (xΣ j ) ξ̃ x̃S j . . . ξ˜ x̃S j
, i = 1 , n . (6)
1
x̃
j ∈X̃ j
S1
S1
x̃
S
1
j ∈X̃ j
S
l
l
From the definition of strategy x̃S j of coalition Skj it follows that
k
xΣ j = x̃S j , . . . , x̃S j and x = (x1 , . . . , xn ) are the same n-tuples in the games
1
l
G(x) and Gj (xΣ j ). However it does not mean that µ = µΣ j .
Payoff function H̃S j : X̃ → R1 of coalition Skj for the fixed projects j, j =
k
1, m, and for the coalitional partition Σ j is defined under condition that:
X
Ki (x) , k = 1 , l , j = 1 , m , Skj ∈ Σ j ,
H̃S j (xΣ j ) > HS j (xΣ j ) =
k
k
i∈Skj
(7)
96
Xeniya Grigorieva
where Ki (x) , i ∈ Skj , is the payoff function of player i in the n-tuple xΣ j .
Definition 1. A set of m coalitional l-person games defined by (5) and associated with noncooperative games defined by (1)−(2) is called multicriteria coalitional model of decision-making with constant matrix of payoffs in the noncooperative game.
Definition 2. Solution of the multicriteria coalitional model of decision-making
with constant matrix of payoffs in the noncooperative game in pure strategies is x∗Σ j∗ ,
that is Nash equilibrium (NE) in a pure strategies in l-person game Gj ∗ (xΣ j∗ ), with
∗
∗
the coalitional partition Σ j , where coalitional partition Σ j is the compromise
coalitional partition (see 3.2).
Definition 3. Solution of the multicriteria coalitional model of decision-making
with constant matrix of payoffs in the noncooperative game in mixed strategies
is µ∗Σ j∗ , that is Nash equilibrium (NE) in a mixed strategies in l-person game
∗
∗
Gj ∗ (µΣ j∗ ), with the coalitional partition Σ j , where coalitional partition Σ j is
the compromise coalitional partition (see 3.2).
Generalized PMS-vector is used as the coalitional imputation
(Grigorieva and Mamkina, 2009; Petrosjan and Mamkina, 2006).
3.
Algorithm for solving the problem
3.1.
Algorithm of constructing the generalized PMS-vector in a
coalitional game.
Remind the algorithm of constructing the generalized PMS-vector in a coalitional
game (Grigorieva and Mamkina, 2009; Petrosjan and Mamkina, 2006).
1. Calculate the values of payoff H̃S j (xΣ j ) for all coalitions Skj ∈ Σ j , k = 1, l ,
k
for coalitional game Gj (xΣ j ) by using formula (3).
2. Find NE (Nash, 1951) x∗Σ j or µ∗Σ j (one or more) in the game
Gj (xΣ j ). The
n j o
∗
.
payoffs’ vector of coalitions in NE in mixed strategies E µΣ j = v Sk
k=1, l
Denote a payoff of coalition Skj in NE in mixed strategies by
lΣ j
X
v Skj =
pτ, j H̃τ, S j (x∗Σ j ), k = 1, lΣ j ,
τ =1
k
where
– H̃τ, S j x∗Σ j is the payoff of coalition Skj , when coalitions choose their pure
k
strategies x̃∗S j in NE in mixed strategies µ∗Σ j .
Q kξk
– pτ, j =
µ̃S j , ξk = 1, lS j , τ = 1, lΣ j , is probability of the payoff’s realization
H̃τ, S j
k
k=1,l
x∗Σ j
k
k
of coalition Skj .
The value H̃τ, S j x∗Σ j is a random variable. There could be many l-tuple of NE
k
in the game, therefore, v S1j , ...., v Slj , are not uniquely defined.
∗
The payoff of each coalition in NE
according
to Shapley’s
E µΣ j is divided
j
j
value (Shapley, 1953) Sh (Sk ) = Sh Sk : 1 , ... , Sh Sk : s :
97
Multicriteria Coalitional Model of Decision-making
X (s′ −1) ! (s−s′ ) !
[v (S ′ ) − v (S ′ \ {i})] ∀ i = 1, s ,
Sh Skj : i =
s
!
j
′
(8)
S ⊂Sk
S ′ ∋i
where s = Skj (s′ = |S ′ |) is the number of elements of sets Skj (S ′ ), and v (S ′ ) is
the maximal guaranteed payoff of S ′ ⊂ Sk .
Moreover
s
X
v Skj =
Sh Skj : i .
i=1
Then PMS-vector in the NE in mixed strategies µ∗Σ j in the game Gj (xΣ j ) is
defined as
PMSj (µ∗Σ j ) = PMSj1 (µ∗Σ j ) , ..., PMSjn (µ∗Σ j ) ,
where
3.2.
PMSji (µ∗Σ j ) = Sh Skj : i , i ∈ Skj , k = 1, l.
Algorithm for finding a set of compromise solutions.
Remind the algorithm
(Malafeyev, 2001; p.18).
for
finding
a
set
of
compromise
solutions
j
j
CPMS (M ) = arg min max max PMSi − PMSi .
j
i
j
∗
Step 1. Construct the ideal vector R = (R1 , . . . , Rn ) , where Ri = PMSji =
max PMSji is the maximal value of payoff functions of player i in NE on the set M ,
j
and j is the number of project j ∈ M :


PMS11 ... PMS1n
 ... ... ... 
m
PMSm
1 ... PMSn
↓
...
↓
∗
j1∗
PMS1 ... PMSjnn
Step 2. For each j find deviation of payoff function values for other players from
the maximal value, that is ∆ji = Ri − PMSji , i = 1 , n:


R1 − PMS11 ... Rn − PMS1n
.
∆= 
...
...
...
m
R1 − PMSm
...
R
−
PMS
n
1
n
∆ji∗
j
Step 3. From the found deviations ∆ji for each j select the maximal deviation
= max ∆ji among all players i:
i

  1

→ ∆1i∗1
∆1 ... ∆1n
R1 − PMS11 ... Rn − PMS1n

 =  ... ... ...  ... .
...
...
...
m
m
m
∆m
→ ∆m
R1 − PMS1 ... Rn − PMSn
1 ... ∆n
i∗
m
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Xeniya Grigorieva
Step 4. Choose the minimal deviation for all j from all the maximal deviations
∗
among all players i ∆ji∗∗ = min ∆ji∗ = min max ∆ji .
j
j
j
j
i
The project j ∗ ∈ CPMS (M ) , on which the minimum is reached is a compromise
solution of the game Gj (xΣ j ) for all players.
3.3.
Algorithm for solving the multicriteria coalitional model of
decision-making over the set of projects with constant matrix of
payoffs in the noncooperative game.
Thus, we have an algorithm for solving the problem.
1. Reduce the multicriteria noncooperative n-person game G̃(x) (see (1)) to a
noncooperative game G(x) with scalar payoffs (see (2)) using the minimax method
of multicriteria optimization.
2. Fix a j , j = 1 , m.
3. Construct the coalitional Gj (xΣ j ) associated with the noncooperative game
G(x) for the fixed j.
2. Find the NE µ∗Σ j in the coalitional game Gj (xΣ j ) and find imputation in
NE, that is PMSj µ∗Σ j .
3. Repeat iterations 1-2 for all other j , j = 1 , m.
4. Find compromise solution j ∗ , that is j ∗ ∈ CPMS (M ).
4.
Example
Consider the set M = {j}j=1, 5 and the set N = {I1 , I2 , I3 } of three players, each
having 2 strategies in multicriteria noncooperative game G̃ (x): xi = 1 is “yes" and
xi = 0 is “no" for all i = 1 , 3. The payoff functions of players in the game G̃ (x) are
determined by the table 1.
Table 1: The payoffs of players.
The
I1
1
1
1
1
0
0
0
0
strategies
I2
I3
1
1
1
0
0
1
0
0
1
1
1
0
0
1
0
0
The payoffs of players
I1
I2
I3
(4, 3.43, 1.71) (2, 1.71, 0.86) (1, 0.86, 0.43)
(1, 0.86, 0.43) (2, 1.71, 0.86) (2, 1.71, 0.86)
(3, 2.57, 1.29) (1, 0.86, 0.43) (5, 4.29, 2.14)
(5, 4.29, 2.14) (1, 0.86, 0.43) (3, 2.57, 1.29)
(5, 4.29, 2.14) (3, 2.57, 1.29) (1, 0.86, 0.43)
(1, 1, 0.43) (1.14, 2, 0.86) (1.86, 2, 2)
(0, 0, 0)
(4, 3.43, 1.71) (3, 3, 1.29)
(0, 0, 0)
(4, 3.43, 1.71) (2, 2, 0.86)
Reduce the multicriteria noncooperative n-person game G̃(x) to a noncooperative game G(x) with scalar payoffs using the minimax method of multicriteria
optimization (see (3)). The values of payoff functions of players in the game G (x)
are in the table 2.
Multicriteria Coalitional Model of Decision-making
99
Table 2: The payoffs of players.
The
I1
1
1
1
1
0
0
0
0
strategies
I2
I3
1
1
1
0
0
1
0
0
1
1
1
0
0
1
0
0
The
I1
4
1
3
5
5
1
0
0
payoffs
I2
I3
2
1
2
2
1
5
1
3
3
1
2
2
4
3
4
2
The payoffs of coalition
{I1 , I2 } {I2 , I3 } {I1 , I3 } {I1 , I2 , I3 }
6
3
5
7
3
4
3
5
4
6
8
9
6
4
8
9
8
4
6
9
3
4
3
5
4
7
3
7
4
6
2
6
1. Compose and solve the coalitional game G2 (xΣ 2 ) , Σ2 = {{I1 , I2 } , I3 }, i. e.
find NE in mixed strategies in the game:
η = 3/7 1 − η = 4/7
1
0
0
(1, 1) [6, 1] [3, 2]
0
(0, 0) [4, 3] [4, 2]
ξ = 1/3
(1, 0) [4, 5] [6, 3]
1 − ξ = 2/3 (0, 1) [8, 1] [3, 2] .
It’s clear, that first matrix row is dominated by the last one and the second is
dominated by third. One can easily calculate NE and we have
y = 3/7 4/7 , x = 0 0 1/3 2/3 .
Then the probabilities of payoffs’s realization of the coalitions S = {I1 , I2 } and
N \S = {I3 } in mixed strategies (in NE) are as follows:
ξ1
ξ2
ξ3
ξ4
η1 η2
0 0
0 0 .
1/ 4/
7 21
2/ 8/
7 21
The Nash value of the game in mixed strategies is calculated by formula:
1
2
4
8
36 7
1
1
E (x, y) = [4, 5] + [8, 1] +
[6, 3] +
[3, 2] =
,
= 5 , 2 .
7
7
21
21
7 3
7
3
In the table 3 pure strategies of coalition N \S and its mixed strategy y are given
horizontally at the right side. Pure strategies of coalition S and its mixed strategy
x are given vertically. Inside the table players’ payoffs from the coalition S and
players’ payoffs from the coalition N \S are given at the right side.
Divide the game’s Nash value in mixed strategies according to Shapley value
(8):
Sh1 = v (I1 ) +
Sh2 = v (I2 ) +
1
2
1
2
[v (I1 , I2 ) − v (I2 ) − v (I1 )] ,
[v (I1 , I2 ) − v (I2 ) − v (I1 )] .
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Xeniya Grigorieva
Table 3: The maximal guaranteed payoffs of players I1 and I2 .
Math. Expectation
2.286
4.143
2.714
0.000
v (I1 )
min 1 2.286
min 2 0.000
max 2.286
2.000
1.000
2.429
4.000
v (I1 )
2.000
1.000
2.000
The strategies of N \ S,
the payoffs of S and the payoffs of N \ S
η = 0.43
+1
0
− (1 , 1)
(4 , 2)
ξ = 0.33 + (1 , 2) 
 (3 , 1)
1 − ξ = 0.67 + (2, 1)  (5 , 3)
0
− (2 , 2)
(0 , 4)
1 − η = 0.57
+2
(1 , 2)
(5 , 1) 

(1 , 2) 
(0 , 4)
Find the maximal guaranteed payoffs v (I1 ) and v (I2 ) of players I1 and I2 . For
this purpose fix a NE strategy of a third player as
ȳ = 3/7 4/7 .
Denote mathematical expectations of players’ payoff from coalition S when
mixed NE strategies are used by coalition N \S by ES(i, j) (ȳ) , i, j = 1, 2. In the
table 3 the mathematical expectations are located at the left, and values are obtained by using the following formulas:
ES(1, 1) (ȳ) = 37 · 4 + 47 · 1 ; 37 · 2 + 47 · 2 ; 73 · 1 + 47 · 2 = 2 72 ; 2 ; 1 74 ;
ES(1, 2) (ȳ) = 37 · 3 + 47 · 5 ; 37 · 1 + 47 · 1 ; 73 · 5 + 47 · 3 = 4 71 ; 1 ; 3 76 ;
ES(2, 1) (ȳ) = 37 · 5 + 47 · 1; 73 · 3 + 47 · 2; 37 · 1 + 47 · 2 = 2 57 ; 2 37 ; 147 ;
ES(2,2) (ȳ) = 37 · 0 + 47 · 0 ; 37 · 4 + 47 · 4 ; 37 · 3 + 47 · 2 = 0; 4 ; 2 73 .
Third element here is the mathematical expectation of payoff of the player I3 (see
table 2 too).
Then, look at the table 2 or table 3,
min H1 (x1 = 1, x2 , ȳ) = min 2 27 ; 417 = 2 72 ; v (I1 ) = max 2 72 ; 0 = 2 27 ;
5
min H1 (x1 = 0, x2 , ȳ) = min 2 7 ; 0 = 0; min H2 (x1 , x2 = 1, ȳ) = min 2; 2 37 = 2 ; v (I2 ) = max {2; 1} = 2.
min H2 (x1 , x2 = 0, ȳ) = min {1; 4} = 1; Thus, maxmin payoff for player I1 is v (I1 ) = 2 72 and for player I2 is v (I2 ) = 2.
Hence,
Sh1 (ȳ) = v (I1 ) + 12 5 71 − v (I1 ) − v (I2 ) = 2 27 + 12 5 71 − 2 27 − 2 = 2 75 ;
Sh2 (ȳ) = 2 + 73 = 2 37 .
Thus, PMS-vector is equal:
5
3
1
PMS1 = 2 ; PMS2 = 2 ; PMS3 = 2 .
7
7
3
Multicriteria Coalitional Model of Decision-making
101
Table 4: Shapley’s value in the cooperative game.
The strategies
of players
I1
I2
I3
1
1
1
1
1
2
1
2
1
1
2
2
2
1
1
2
1
2
2
2
1
2
2
2
The payoffs
of players
I1
I2
I3
4
2
1
1
2
2
3
1
5
5
1
3
5
3
1
1
2
2
0
4
3
0
4
2
The payoff
Shapley’s
of coalition
value
HN (I1 , I2 , I3 ) λ1 HN λ2 HN λ3 HN
7
5
9
2.5
3.5
3
9
2.5
3.5
3
9
2.5
3.5
3
5
7
6
2. Solve the cooperative game G5 (xΣ 5 ), Σ5 = {N = {I1 , I2 , I3 }}, see table 4.
Find the maximal payoff HN of coalition N and divide him according to Shapley
value (8), (Shapley, 1953):
Sh1 =
1
1
[v (I1 , I2 ) + v (I1 , I3 ) − v (I2 ) − v (I3 )] + [v (N ) − v (I2 , I3 ) + v (I1 )] ;
6
3
1
1
[v (I2 , I1 ) + v (I2 , I3 ) − v (I1 ) − v (I3 )] + [v (N ) − v (I1 , I3 ) + v (I2 )] ;
6
3
1
1
Sh3 = [v (I3 , I1 ) + v (I3 , I2 ) − v (I1 ) − v (I2 )] + [v (N ) − v (I1 , I2 ) + v (I3 )] .
6
3
Find the guaranteed payoffs:
Sh2 =
v (I1 , I2 ) = max {4, 3} = 4; v (I1 , I3 ) = max {3, 2} = 3;
v (I2 , I3 ) = max {3, 4} = 4 ;
v (I1 ) = max {1, 0} = 1 ; v (I2 ) = max {2, 1} = 2; v (I3 ) = max {1, 2} = 2 .
Then
(2, 1, 1)
Sh1
(1, 2, 2)
= Sh1
(1, 2, 1)
= Sh1
=
1 1 1
1
1 1 5 1
1
+ + [9 − 4] + = + + + = 2 ,
3 6 3
3
3 6 3 3
2
1 1 1
2
1 1 6 2
1
+ + [9 − 3] + = + + + = 3 ,
2 3 3
3
2 3 3 3
2
1 1 1
2
1 1 5 2
(2, 1, 1)
(1, 2, 2)
(1, 2, 1)
Sh3
= Sh3
= Sh3
= + + [9 − 4] + = + + + = 3.
3 3 3
3
3 3 3 3
3. Solve noncooperative game G1 (xΣ 1 ), Σ1 = {S1 = {I1 } , S2 = {I2 } ,
S3 = { I3 }}. In pure strategies NE not exist.
From p. 3 it follows that the guaranteed payoffs v (I1 ) = 1 ; v (I2 ) = 2; v (I3 ) =
2 . Find the optimal strategies with Nash arbitration scheme, see table 5. Then
optimal n-tuple are ((1) , (1) , (2)) and ((2) , (1) , (2)), the payoff in NE equals
((1) , (2) , (2)).
A detailed solution of games for various cases of coalitional partition of players is
provided in (Grigorieva, 2009). Present the obtained solution in (Grigorieva, 2009)
in the table 6.
(2, 1, 1)
Sh2
(1, 2, 2)
= Sh2
(1, 2, 1)
= Sh2
=
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Xeniya Grigorieva
Table 5: Solution of noncooperative game.
The strategies
of players
I1
I2
I3
1
1
1
1
1
2
1
2
1
1
2
2
2
1
1
2
1
2
2
2
1
2
2
2
The payoffs
of players
I1
I2
I3
4
2
1
1
2
2
3
1
5
5
1
3
5
3
1
1
2
2
0
4
3
0
4
2
Pareto-optimality (P)
and Nash arbitration scheme
Nash arbitration scheme P
(4 − 1) (2 − 2) (1 − 2) < 0 (1 − 1) (2 − 2) (2 − 2) = 0 +
(3 − 1) (1 − 2) (5 − 2) < 0 (5 − 1) (1 − 2) (3 − 2) < 0 (5 − 1) (3 − 2) (1 − 2) < 0 (1 − 1) (2 − 2) (2 − 2) = 0 +
(0 − 1) (4 − 2) (3 − 2) < 0 (0 − 1) (4 − 2) (2 − 2) < 0 -
Table 6: Payoffs of players in NE for various cases of the coalitional partitions of players.
Project
Coalitional
partitions
The n-tuple of NE
(I1 , I2 , I3 )
1
Σ1 = {{I1 } {I2 } {I3 }}
2
Σ2 = {{I1 , I2 } {I3 }}
3
Σ3 = {{I1 , I3 } {I2 }}
4
Σ4 = {{I2 , I3 } {I1 }}
5
Σ5 = {I1 , I2 , I3 }
((1) , (1) , (0))
((0) , (1) , (0))
((1, 0) , 1)
((1, 0) , 0)
((0, 1) , 1)
((0, 1) , 0)
(1, (1) , 1)
(1, (0) , 1)
(0, (1) , 1)
(0, (0) , 1)
(1, (0, 1))
(1, 0, 1)
(1, 0, 0)
(0, 1, 1)
Probability
of realization
NE
1
1/7
4/21
2/7
8/21
5/12
1/12
5/12
1/12
1
1
1
1
Payoffs
of players
in NE
((1) , (2) , (2))
((2.71, 2.43) , 2.33)
(2.59, (2.5) , 2.91)
(3, (3, 3))
(2.5, 3.5, 3)
Table 7: The set of compromise coalitional partitions.
Σ1 = {{I1 } {I2 } {I3 }}
Σ2 = {{I1 , I2 } {I3 }}
Σ3 = {{I1 , I3 } {I2 }}
Σ4 = {{I2 , I3 } {I1 }}
Σ5 = {I1 , I2 , I3 }
R
I1
1
2.71
2.59
3
2.5
3
I2
2
2.43
2.5
3
3.5
3.5
I3
2
2.33
2.91
3
3
3
I1 I2 I3
∆ {{I1 } {I2 } {I3 }} 2 1.5 1
2
∆ {{I1 , I2 } {I3 }} 0.29 1.07 0.67 1.07
∆ {{I1 , I3 } {I2 }} 0.41 1 0.09 1
∆ {{I2 , I3 } {I1 }} 0 0.5 0 0.5
∆ {I1 , I2 , I3 }
0.5 0
0 0.5
Multicriteria Coalitional Model of Decision-making
103
Applying the algorithm for finding a compromise solution, we get the set of
compromise coalitional partitions (table 7).
Therefore, compromise imputation are PMS-vector in coalitional game with the
coalition partition Σ4 in NE (1 , (0 , 1)) in pure strategies with payoffs (3 , (3 , 3))
and Shapley value in the cooperative game in NE ((1 , 0 , 1) , (1 , 0 , 0) , (0 , 1 , 1)
– cooperative strategies) with the payoffs (2.5 , 3.5 , 3).
Moreover, in situation, for example, (1 , (0 , 1)) the first and third players give
a positive decision for corresponding project. In other words, if the first and third
players give a positive decision for corresponding project, and the second does not,
then payoff of players will be optimal in terms of corresponding coalitional interaction.
5.
Conclusion
A multicriteria coalitional model of decision-making over the set of projects with
constant payoff matrix in the noncooperative game and algorithm for finding optimal solution are constructed in this paper, the numerical example is given.
References
Grigorieva, X., Mamkina, S. (2009). Solutions of Bimatrix Coalitional Games. Contributions to game and management. Collected papers printed on the Second International
Conference “Game Theory and Management" [GTM’2008]/ Edited by Leon A. Petrosjan, Nikolay A. Zenkevich. - SPb.: Graduate School of Management, SpbSU, 2009, pp.
147–153.
Petrosjan, L., Mamkina, S. (2006). Dynamic Games with Coalitional Structures. Intersectional Game Theory Review, 8(2), 295–307.
Nash, J. (1951). Non-cooperative Games. Ann. Mathematics 54, 286–295.
Shapley, L. S. (1953). A Value for n-Person Games. In: Contributions to the Theory of
Games( Kuhn, H. W. and A. W. Tucker, eds.), pp. 307–317. Princeton University Press.
Grigorieva, X. V. (2009). Dynamic approach with elements of local optimization in a class
of stochastic games of coalition. In: Interuniversity thematic collection of works of St.
Petersburg State University of Civil Engineering (Ed. Dr., prof. B. G. Wager). Vol. 16.
Pp. 104–138.
Malafeyev, O. A. (2001). Control system of conflict. SPb.: St. Petersburg State University,
2001.
Differential Games with Random Duration: A Hybrid
Systems Formulation
Dmitry Gromov1 and Ekaterina Gromova2
Faculty of Applied Mathematics,
St. Petersburg State University,
St.Petersburg, Russia
E-mail: [email protected]
2
Faculty of Applied Mathematics,
St. Petersburg State University,
St.Petersburg, Russia
E-mail: [email protected]
1
Abstract The contribution of this paper is two-fold. First, a new class of
differential games with random duration and a composite cumulative distribution function is introduced. Second, it is shown that these games can be
well defined within the hybrid systems framework and that the problem of
finding the optimal strategy can be posed and solved with the methods of
hybrid optimal control theory. An illustrative example is given.
Keywords: games, hybrid, etc.
1.
Introduction
Game theory as a branch of mathematics investigates conflict processes controlled
by many participants (players). These processes are referred to as games. In this
paper we focus on the duration of games. In differential game theory it is common to
consider games with a fixed duration (finite time horizon) or games with an infinite
time horizon. However, in many real-life applications the duration of a game can
not be determined a priori but depends on a number of unknown factors and thus
is not deterministic any longer.
To take account of this phenomenon, a finite-horizon model with random terminal time is considered. For the first time the class of differential games with random
duration was introduced in (Petrosyan and Murzov, 1966) for a particular case of
a zero-sum pursuit game with terminal payoffs at random terminal time. Later,
the general formulation of the differential games with random duration was given
in (Petrosyan and Shevkoplyas, 2003). Section 2. provides a brief overview of these
results.
Apparently, Boukas, Haurie and Michel, in (Boukas et al., 1990), were first to
consider an optimal control problem with a random stopping time. Apart from
that, in the optimal control theory there have also been papers exploring the idea
of random terminal time applied to non-game-theoretical problems. In particular,
the problem of the consumer’s life insurance under condition of the random moment
of death was discussed in (Yaari, 1965, Chang, 2004).
In many cases the probability density function of the terminal time may change
depending on some conditions, which can be expressed as a function of time and
state. Consider, for instance, the example of the development of a mineral deposit.
The probability of a breakdown may depend on the development stage. At the
Differential Games with Random Duration: A Hybrid Systems Formulation
105
initial stage this probability is higher than during the routine mining operation.
Therefore one needs to define a composite distribution function for the terminal
time as described in Sec. 3. To the best of our knowledge, this formulation has
never been considered before despite its obvious practical appeal. In our view, this
is caused by the limitations of the generally adopted technique for the computation
of optimal strategies.
In non-cooperative differential games players solve the optimal control problem
of the payoff maximization. One of the basic techniques for solving the optimal control problem is the Hamilton-Jacobi-Bellman equation (Dockner et al., 2000). However, in the above described case a solution (i.e., a differentiable value function) to
the HJB equation may not exist. In this case a generalized solution is sought for (the
interested reader is referred to Bardi and Capuzzo-Dolcetta, 1997, Vinter, 2000).
An alternative to the HJB equation is the celebrated Pontryagin Maximum Principle (Pontryagin et al., 1963) which was recently generalized to a class of hybrid
optimal control problems (see, e.g., Riedinger et al., 2003, Shaikh and Caines, 2007,
Azhmyakov et al., 2007). In Sec. 3., we show that the optimization problem for a
differential game with random terminal time and composite distribution function
can be formulated and solved within the hybrid control systems framework.
Finally, in the last section an application of our theoretical results is presented.
We investigate one simple model of non-renewable resource extraction, where the
termination time is a random variable with a composite distribution function. Two
different switching rules are studied and a qualitative analysis of the obtained results
is presented.
2.
Differential Game Formulation
Consider an N -person differential game Γ (t0 , x0 ) starting at the time instant t0
from the initial state x0 , and with duration T − t0 . Here the random variable T
with a cumulative distribution function (CDF) F (t), t ∈ [t0 , ∞), is the time instant
at which the game Γ (t0 , x0 ) ends. The CDF F (t) is assumed to be an absolutely
continuous nondecreasing function satisfying the following conditions:
C1. F (t0 ) = 0,
C2. lim F (t) = 1.
t→∞
Furthermore, there exists an a.e. continuous function f (t) = F ′ (t), called the probability density function (PDF), such that (Royden, 1988)
F (t) =
Zt
t0
f (τ )dτ
∀t ∈ [t0 , ∞).
Let the system dynamics be described by the following ODEs:
ẋ = g(x, u1 , . . . , uN ), x ∈ Rm , ui ∈ U ⊆ comp(R),
x(t0 ) = x0 ,
(1)
where g : Rm × RN → Rm is a vector-valued function satisfying the standard
existence and uniqueness requirements (see, e.g., Lee and Markus, 1967, Ch. 4).
The instantaneous payoff of the i-th player at the moment τ , τ ∈ [t0 , ∞) is
defined as hi (x(τ ), ui (τ )). Then the expected integral payoff of the player i, where
106
Dmitry Gromov, Ekaterina Gromova
i = 1, . . . , N is evaluated by the formula
Ki (t0 , x, u) =
Z∞ Zt
hi (x(τ ), ui (τ ))dτ dF (t) =
t0 t0
Z∞ Zt
hi (x(τ ), ui (τ ))dτ f (t)dt.
(2)
t0 t0
The Pareto optimal strategy in the game Γ (t0 , x0 ) is defined as the n-tuple of
controls u∗ (t) = (u∗1 (t), . . . , u∗n (t)) maximizing the joint expected payoff of players:
(u∗1 (t), . . . , u∗n (t))
= argmax
u
n
X
Ki (t0 , x, u).
(3)
i=1
Hence, the Pareto optimal solution of Γ (t0 , x0 ) is (x∗ (t), u∗ (t)) and the total optimal
payoff V (x0 ) is
V (x0 , t0 ) =
n
X
∗
∗
Ki (t0 , x , u ) =
i=1
n Z∞ Zt
X
i=1 t
0
hi (x∗ (τ ), u∗i (τ ))dτ f (t)dt.
(4)
t0
For the set of subgames Γ (ϑ, x∗ (ϑ)), with ϑ > t0 , occurring along the optimal
trajectory x∗ (ϑ) one can similarly define the expected total integral payoff in the
cooperative game Γ (ϑ, x∗ (ϑ)):
∗
V (x (ϑ), ϑ) =
n Z∞ Z t
X
hi (x∗ (τ ), u∗i (τ ))dτ dFϑ (t),
(5)
i=1 ϑ ϑ
where Fϑ (t) is a conditional cumulative distribution function defined as
Fϑ (t) =
F (t) − F (ϑ)
,
1 − F (ϑ)
t ∈ [ϑ, ∞).
(6)
and the conditional probability density function has the following form:
fϑ (t) =
f (t)
,
1 − F (ϑ)
t ∈ [ϑ, ∞).
(7)
2.1. Transformation of the Integral Functional
Below, the transformation procedure of the double integral functional (2) and its reduction to a single integral is described. We obtain this result by changing the order
of integration; alternative approaches were presented in, e.g., (Boukas et al., 1990,
Chang, 2004). In the following, we assume that the expression under the integral
sign is such that the order of integration in (2) is immaterial. Note that in general
this is not true (see, for example, 1). A detailed account on this issue is presented
in (Kostyunin and Shevkoplyas, 2011).
From now on, without loss of generality we set t0 = 0.
Consider the integral functional of the i-th player:
Z∞ Zt
0
0
hi (τ ) dτ f (t)dt,
Differential Games with Random Duration: A Hybrid Systems Formulation
107
where hi (τ ) is a shorthand for hi (x(τ ), ui (τ )).
Define function a(t, τ ) as follows:
a(t, τ ) = f (t)hi (τ ) · χ{τ 6t} =
f (t)hi (τ ), τ 6 t;
0,
τ >t
Taking into account the above mentioned assumption, we interchange the variables of integration in the double integral. Then we get:
Z∞
0
dt
Zt
0
=
f (t)hi (τ )dτ =
Z∞
0
dτ
Z∞
τ
Z∞
0
dt
Z∞
a(t, τ )dτ =
0
Z∞
f (t)hi (τ )dt = (1 − F (τ ))hi (τ )dτ.
0
In the general case, the expected payoff of the player i in the game Γ (t0 , x0 ) can
be rewritten as:
Z∞
Ki (t0 , x, u) = (1 − F (τ ))hi (x(τ ), ui (τ ))dτ.
(8)
t0
In the same way we get the expression for expected payoff of the player in the
subgame Γ (ϑ, x(ϑ)):
1
Ki (ϑ, x, u) =
1 − F (ϑ)
3.
Z∞
(1 − F (τ ))hi (x(τ ), ui (τ ))dτ.
(9)
ϑ
Hybrid Formulation of a Differential Game
In this section we give the definition of a hybrid control problem and the associated hybrid optimal control problem. It is shown that the differential game with a
composite CDF (CCDF) introduced in Subsection 3.2. fits perfectly in the hybrid
framework. Hence, a hybrid differential game as well as a number of particular cases
are considered, the respective optimal control problems are defined, and the solution
strategies are proposed.
3.1. Hybrid Optimal Control Problem
Below, we give the definition of a hybrid system. For more details, the interested
reader is referred to (Riedinger et al., 2003, Shaikh and Caines, 2007), as well as
(Azhmyakov et al., 2007).
Definition 1. The hybrid system HS is defined as a tuple
HS = (Q, X, U, f, γ, Φ, q0 , x0 ),
where
– Q = {1, . . . , N } is the set of discrete states, XRl is the continuous state, Uq ⊂
Rm , q ∈ Q are the admissible control sets, which are compact and convex, and
Uq := {u(·) ∈ Lm
∞ (0, tf ) : u(t) ∈ Uq , a.e. on[0, tf ]}
represent the sets of admissible control signals.
108
Dmitry Gromov, Ekaterina Gromova
– q0 ∈ Q and x0 ∈ X are the initial conditions.
– fq : X × U → X is the function that associates to each discrete state q ∈ Q a
differential equation of the form
ẋ(t) = fq (x(t), u(t)).
(10)
– γq,q′ : X → Rk is the function that triggers the change of discrete state. Let
q ∈ Q be the current discrete state and x(t) be the state trajectory evolving
according to the respective differential equation (10). The transition to the
discrete state q ′ ∈ Q occurs at the moment χ when γq,q′ (x(χ)) = 0. The set
Γq,q′ = {x ∈ X|γq,q′ (x) = 0} is referred to as the switching manifold.
– When the discrete state changes from q to q ′ , the continuous state might change
discontinuously. This change is described by the jump function Φq,q′ : X → X.
Let χ be the time at which the discrete state changes from q to q ′ , then the
continuous state at t = χ is described as x(χ) = Φq,q′ (x(χ− )), where x(χ− ) =
lim x(t).
t→χ−0
Definition 2. A hybrid trajectory of HS is a triple X = (x, {qi }, τ ), where x(·) :
[0, T ] → Rn , {qi }i=1,...,r is a finite sequence of locations and τ is the corresponding
sequence of switching times 0 = t0 < · · · < tr = T such that for each i = 1, . . . , r
there exists ui (·) ∈ Ui such that:
– x(0) = x0 and xi (·) = x(·)|[ti−1 ,ti ) is an absolutely continuous function in
[ti−1 , ti ) continuously extendable to [ti−1 , ti ], i = 1, . . . , r.
– ẋi (t) = fqi (xi (t), ui (t)) for almost all t ∈ [ti−1 , ti ], i = 1, . . . , r.
– The switching condition xi (ti ) ∈ Γqi ,qi+1 along with the jump condition xi+1 (ti ) =
Φqi ,qi+1 (xi (ti )) are satisfied for each i = 1, . . . , r−1.
Using the introduced notation we can state a hybrid optimal control problem
and characterize an optimal solution to this problem. Let the overall performance
of HS be evaluated by the following functional criterion:
J(x0 , q0 , u) =
Zti
r
X
Lqi (xi (t), ui (t), t)dt,
(11)
i=1 t
i−1
where Lqi : X × U × R>0 , qi ∈ Q, are twice continuously differentiable functions.
Assume that the sequence of discrete states q ∗ is given. Then the necessary conditions for a solution (x∗ , q ∗ , τ, u∗ ) to HS to minimize (11) is given by the following
theorem.
Theorem 1 (Riedinger et al., 2003). If u∗ (t) and (x∗ (t), q ∗ (t), τ ) are the optimal control and the corresponding hybrid trajectory for HS, then there exists a
piecewise absolutely continuous curve p∗ (t) and a constant p∗0 > 0, (p∗ , p∗0 ) 6= (0, 0)
such that
– The tuple (x∗ (t), q ∗ (t), p∗ (t), u∗ (t), τ ) satisfies the associated Hamiltonian system
∂H
ẋ(t) = ∂pqi (x∗ (t), p∗ (t), u∗ (t)),
ṗ(t) = −
∂Hqi
∂x
(x∗ (t), p∗ (t), u∗ (t)),
t ∈ [ti−1 , ti ], i = 1, . . . , r
(12)
Differential Games with Random Duration: A Hybrid Systems Formulation
where
109
Hqi (x∗ (t), p∗ (t), u∗ (t)) =
= p∗0 Lqi (xi (t), ui (t), t) + p∗ (t)fqi (xi (t), ui (t)).
– At any time t ∈ [ti−1 , ti ), the following maximization condition holds:
Hqi (x∗ (t), p∗ (t), u∗ (t)) = sup Hqi (x∗ (t), p∗ (t), u(t)).
(13)
u(t)∈U
– At the switching time ti , there exists a vector π ∈ Rn such that the following
transversality conditions are satisfied:
p∗ (t−
i )=
n
P
pk (ti )
k=1
Hqi−1 (t−
i )
∂Φk
q
i ,qi+1
∂xj
= Hqi (ti ) −
−
n
P
k=1
πki
n
P
(t−
i )+
n
P
πki
k
∂γi,i+1
−
∂xj (ti ),
k=1
∂Φk
i,i+1
pk (ti ) ∂t
(t−
i )−
(14)
k=1
−
∂t (ti )
k
∂γi,i+1
3.2. Composite Cumulative Distribution Function
Let t0 be the initial time, Fi (t), i = 1, . . . , N be a set of CDFs characterizing different
modes of operation and satisfying, along with C1 and C2, the following property:
C3. The CDFs Fi (t) are assumed to be absolutely continuous nondecreasing functions such that each CDF converges to 1 asymptotically, i.e., Fi (t) < 1 ∀t < ∞.
Furthermore, let τ = {τi } s.t. t0 = τ0 < τ1 < · · · < τN −1 < τN = ∞ be an ordered
sequence of time instants at which the switches between individual CDFs occur.
The composite CDF Fσ (t) is defined as follows:

t ∈ [τ0 , τ1 ),

 F1 (t),
Fσ (t) = αi (τi )Fi+1 (t) + βi (τi ), t ∈ [τi , τi+1 ),
(15)


1 6 i 6 N − 1,
where αi (τi ) =
Fσ (τi− )−1
Fi+1 (τi )−1 ,
and βi (τi ) = 1 −
Fσ (τi− )−1
Fi+1 (τi )−1 .
the left limit of Fσ (t) at t = τi− , i.e., Fσ (τi− ) =
lim
t→(τi −0)
Fσ′ (t) and
Here, Fσ (τi− ) is defined as
Fσ (t).
The composite PDF is defined as fσ (t) =
has the following form:


t ∈ [τ0 , τ1 ),
 f1 (t),
fσ (t) = αi (τi )fi+1 (t), t ∈ [τi , τi+1 ),
(16)


1 6 i 6 N − 1.
Proposition 1. Given a set of CDFs Fi (t), 1 6 i 6 N , such that C1-C3 hold for
each Fi (t). Then the composite CDF Fσ defined by (15) satisfies C1-C3.
Proof. See Appendix.
From Lemma 1 it follows that fσ (t) has well-defined finite left and right limits
at points τi , 1 6 i 6 N − 1,
fσ (τi− ) =
lim
t→(τi −0)
fσ (t),
fσ (τi+ ) =
lim
t→(τi +0)
fσ (t),
110
Dmitry Gromov, Ekaterina Gromova
which are not necessarily equal, and is continuous otherwise.
The optimization problem (3) for the CCDF (15) can be written taking into
account the transformation (8):
u∗ (t) = argmax
u
= argmax
u
n
X
Ki (x0 , t0 , u1 , . . . , un ) =
i=1
n ZτN
X
i=1 τ
(17)
(1 − Fσ (τ ))hi (x(τ ), ui (τ ))dτ.
0
3.3. Hybrid Differential Game
The optimization problem (1), (17) can hardly be solved in a straightforward way
due to the special structure of the composite CDF Fσ (t). However, this problem
can be readily formulated as a hybrid optimal control problem. We know that Fσ (t)
is defined by a number of elementary CDFs, (15), and the switching instants τi ,
i = 1, . . . , N . There are two types of switching instants τi corresponding to
a) Time-dependent switches;
b) State-dependent switches.
In the first case, the sequence τ is given; the remaining degrees of freedom are
the values of the state at the switching times τi , i.e., x(τi ). In the second case, the
switching times τi are determined as the solutions to the equations γi (xi (τi− )) = 0,
i.e., the regime changes as the state crosses the switching manifold defined by the
map γ : Rn×l → Rk . We assume that the sequence of operation modes (i.e., discrete
states) is fixed a priori. Therefore, there is no need in performing any combinatorial
optimization and the problem of determining the optimal strategy can be completely
formulated within the framework of hybrid optimal control as shown below.
Time-dependent case To apply the results of Theorem 1 the problem (1), (17)
has to be modified. Namely, we extend the system (1) by one differential equation
modelling the CDF Fσ . Thus, on each interval [τi−1 , τi ) the differential equations
the payoff function are written as
ẋ = g(x, u),
ẋσ = f¯i (t),
Ki (t0 , x, u) =
Rτi
τi−1
where h(x(t), u(t)) =
n
P
(18)
(1 − xσ (t))h(x(t), u(t))dt,
hi (xi (t), ui (t)) is the total instantaneous payoff, and
i=1
f¯i = fσ [τi−1 ,τi ) .
Note that since the switching times are fixed a priory, functions f¯i are well defined.
The respective Hamiltonian functions are
Hi (xt , u, p0 , pt ) = p0 (1 − xσ )h(x, u) + hp, g(x, u)i + pσ f¯i (t),
where p0 = −1, xt (t) = [x(t), xσ (t)]′ , and pt (t) = [p(t), pσ (t)]′ . Solving the Hamiltonian equations (12) together with the maximization condition (13) one obtains
Differential Games with Random Duration: A Hybrid Systems Formulation
111
a solution to (18). To solve (12), a number of boundary conditions has to be defined. First, these are initial and end point conditions x(τ0 ) = x0 , x(∞) = 0, and
xσ (τ0 ) = 0, xσ (∞) = 1. Second, there are constraints imposed on the state and
adjoint variables at switching times τi , i = 1, . . . , N − 1:
x(τi− ) = x(τi ), p(τi− ) = p(τi ),
xσ (τi− ) = xσ (τi ), pσ (τi− ) = pσ (τi ),
Hi−1 (xt (τi− ), pt (τi− ))
(19)
= Hi (xt (τi ), pt (τi )).
With these conditions the problem becomes well-defined. We note that the righthand sides of the differential equations in (18) depend on t. Therefore, on each interval [τi−1 , τi ), an additional condition H(x(τi ), u∗ (τi ), p(τi )) = 0 has to be added. For
details on the time-variant Maximum Principle see, e.g., (Pontryagin et al., 1963,
Ch. 1).
State-dependent case This case is slightly more involved compared to the previous one. The problem is that the switching instants τi are defined from the solution
of the switching condition γi,i+1 = x(τi ) − x̃i = 0 and, thus, not defined a priori.
Looking at (16) one can notice that the composite PDF depends on τi which means
that the functions f¯i (t) in (18) are not well-defined. Therefore, the equations (18)
are modified as shown below
ẋ = g(x, u),
ẋσ = xα fi (t),
ẋα = 0,
xα (τ0 ) = 1
Rτi
Ki (t0 , x, u) =
(1 − xσ (t))h(x(t), u(t))dt.
(20)
τi−1
with Hamiltonian functions modified accordingly
Hi (xt , u, pt ) = p0 (1 − xσ )h(x, u) + hp, g(x, u)i + pσ xα fi (t).
The particularity of this model is that along with the mentioned switching condition γi,i+1 = x(t)−x̃i , there is a jump function associated with xα . When a switching
between discrete states occurs, the state changes discontinuously according to the
jump function
[x(τi ), xσ (τi ), xα (τi )] = Φi,i+1 (x, xσ ) =
h
i
xσ (τi− )−1
= x(τi− ), xσ (τi− ), Fi+1
(τi )−1 .
The intermediate conditions (19) have to be rewritten to take into account the
switching and jump functions:
x(τi ) = x(τi− ),
p(τi− ) = p(τi ) + π,
xσ (τi ) = xσ (τi− ),
pσ (τi− ) =
xα (τi ) =
xσ (τi− )−1
Fi+1 (τi )−1 ,
pσ (τi )
Fi+1 (τi )−1 ,
pα (τi− ) = 0.
Furthermore, the Hamiltonian function is not continuous any longer since the jump
function is time-variant. The condition on the Hamiltonian at switching instants τi
112
Dmitry Gromov, Ekaterina Gromova
is hence
Hqi−1 (τi− ) = Hqi (τi ) + pα (τi )
(xσ (τi ) − 1)fi+1 (τi ) −
(ti ).
(Fi+1 (τi ) − 1)2
The end point conditions remain unchanged. With all conditions imposed, the optimization problem (20) becomes well-defined and can be solved using standard
procedures as illustrated in the following section.
4.
Example
To illustrate the presented approach we consider a simple example of finding a
Pareto optimal solution in the game of resource extraction with N players and
two operation modes. Note that despite its obvious simplicity, this example can
demonstrare rather non-trivial behaviour.
The two CDFs are F1 (t) = 1 − exp(−λ1 t) and F2 (t) = 1 − exp(−λ2 t) with
λ1 , λ2 > 0 and the switching time τ . The resulting CCDF Fσ (t) is defined as
Fσ (t) =
(
1 − exp(−λ1 t),
1−
exp(−λ1 τ )
exp(−λ2 τ )
t ∈ [0, τ ),
exp(−λ2 t), t ∈ [τ, ∞).
(21)
We consider two exponential CDF with rate parameters λ = 0.01 and λ = 0.1.
The corresponding CDFs are shown in Fig. 1.
Fig. 1: Two exponential distributions
Differential Games with Random Duration: A Hybrid Systems Formulation
113
The system dynamics is described by a first order DE:
ẋ(t) = −
N
X
ui (t),
i=1
x(0) = x0 , x(∞) = 0, ui (·) ∈ [0, umax],
(22)
where u(t) is the rate of extraction. The initial amount of resource is set to x(0)=100
and x(∞) is routinely defined as x(∞) = lim x(t).
t→∞
The instanteneous payoff function is chosen as hi (x(t), u(t)) = ln(ui (t)). The
optimal control problem is thus defined to be
min
N
X
i=1
Z∞
N
X
Ki (x, u) = − (1 − Fσ (s))
ln(ui (s))ds.
(23)
i=1
0
Before proceeding to the hybrid formulation, we present the solution to the optimal control problem (23) defined over a single interval. This result is of independent
interest, as this class of optimization problems is fairly common for a wide range of
resource extraction applications (see, e.g., Dockner et al., 2000).
4.1. Optimal Solution to a Single Mode Optimal Control Problem
Consider a more general version of the problem (22), (23) on the interval [t0 , tf ] ⊂
[0, ∞) ∪ {∞} with the boundary conditions x(t0 ) = x0 , x(tf ) = xf , x0 > xf .
Moreover, we assume that there is one single (non-composite) CDF F (t) such that
F (tf ) = 1. The Hamiltonian is written as
H = −ψ
N
X
i=1
ui (t) + ψ0 (1 − F (t))
N
X
ln(ui (t)),
ψ0 = 1.
i=1
The differential equation for the adjoint variable ψ is
ψ̇ = −
∂H
= 0,
∂x
whence we conclude that ψ(t) = ψ ∗ = const for all t.
The optimal controls u∗i are found from the first order extremality condition
∂H
∂ui = 0:
1
u∗i (ψ, t) = ∗ (1 − F (t)).
ψ
2
Moreover, u∗i maximize H as follows from ∂∂uH2 = −ψ0 (1 − F (t)) u12 < 0.
i
i
The value of ψ ∗ is determined from the boundary condition x(tf ) = xf . Solving
(22) and taking into account this condition we find ψ ∗ as
N
ψ =
x0 − xf
∗
Ztf
t0
(1 − F (t))dt
and hence, the optimal controls take the following form:
u∗i (t) =
N
Rtf
t0
x0 − xf
(1 − F (τ ))dτ
(1 − F (t)).
(24)
114
Dmitry Gromov, Ekaterina Gromova
The state x(t) of the system (22) with the control (24) is
∗
x (t) = x0 −
Zt
t0
N
Rtf
t0
x0 − xf
(1 − F (τ ))dτ
(1 − F (s))ds.
Note that the optimal control u∗ (t) exists if the integral in the denominator conRt
verges, i.e. t0f (1 − F (τ ))dτ < ∞ (which might not be the case if tf = ∞). Taking
into account the Bellman optimality principle, the optimal controls u∗i (t) can be
expressed as functions of the current state:
u∗i (t, x(t)) =
x(t) − xf
(1 − F (t)).
Rtf
N (1 − F (τ ))dτ
(25)
t
Hence, from (9) and (25), the value function V (t, x(t)) is given by
I(t)
V (t, x(t)) = −
ln
1 − F (t)
where I(t) = N
Rtf
t
x(t) − xf
I(t)
1
−
1 − F (t)
Ztf
(1−F (s)) ln(1−F (s))ds, (26)
t
(1 − F (τ ))dτ .
Finally, in the framework of the resource extraction problem one may need to
compute the expectation of the state x(t) at the end of the exploration process:


E(x(t)) =
Rtf
t0
Rt

f (t) x0 −
t0 N
= xf + (x0 − xf )
tf
R
t0
tf
R
x0 −xf
(1−F (τ ))dτ

(1 − F (s))ds dt =
F (t)(1−F (t))dt
t0
N
tf
R
.
(1−F (τ ))dτ
t0
In the following, we will assume N = 1 to simplify the notation.
4.2. Time-Dependent Case
We assume that the switching time τ is fixed and equal to τs and the state at time
τs is x(τs ) = xs . Hence, the optimal control problem can be decomposed into two
problems, on the intervals I1 = [0, τs ) and I2 = [τs , ∞).
The optimal control on the first interval [0, τ ) is
u∗ (t) =
(x0 − xs )
(x0 − xs )λ1
(1 − Fσ (t)) =
exp(−λ1 t),
Rτs
(1 − exp(−λ1 τs ))
(1 − Fσ (s))ds
t ∈ I1 .
0
In the same way we define the optimal control on the second interval:
xs
xs λ2
u∗ (t) = R∞
(1 − Fσ (t)) =
exp(−λ2 t),
exp(−λ2 τs )
(1 − Fσ (s))ds
τs
t ∈ I2 .
Differential Games with Random Duration: A Hybrid Systems Formulation
115
Both expressions contain the unknown switching state xs . Solving the optimal
control problem, xs is found as a function of the switching time τs :
xs =
λ1 x0
.
λ2 exp(λ1 τs ) − (λ2 − λ1 )
In Fig. 2, the dependence of the switching state xs on the switching time τs is
shown for two sequences of operation modes.
Fig. 2: Dependence of the optimal switching state x∗s = x(τs ) on the switching time τs .
The continuous line corresponds to the case λ1 = 0.01, λ2 = 0.1, the dotted one –
λ1 = 0.1, λ2 = 0.01
Informally, one can describe these two cases as the "safe" mode first (λ1 =
0.01, λ2 = 0.1), and the "dangerous" mode first (λ1 = 0.1, λ2 = 0.01). The second
case is of particular interest. It turns out that for a small τs the optimal strategy is to
preferrably extract during the "safe" mode. However, as τs grows, the risk that the
system breaks down grows and so, the expected gain in the payoff is compensated
by the risk of an abrupt interruption of the game. As τs reaches a certain value
the optimal strategy becomes to extact as much as possible during the "dangerous"
phase as there is only a slight hope that the process will "survive" until the switching
time τs . Interesting to note that the switching time at which the optimal strategy
changes is determined from the equation
λ21 (exp(λ2 τs ) − 1) − λ22 (exp(λ1 τs ) − 1) = 0.
116
Dmitry Gromov, Ekaterina Gromova
4.3. State-Dependent Case
In the second case, we assume that the switching time τ is determined from the
condition x(τ ) = ax0 , a ∈ [0, 1], where the parameter a describes the extent of
exploration at which the regime changes (i.e., a switching occurs).
As in the previous case, the optimal control problem can be decomposed into
two problems, on the intervals [0, τ ) and [τ, ∞).
Now consider the first interval [0, τ ). The optimal control is
u∗ (t) = Rτ
0
x0 − ax0
(1 − Fσ (s))ds
(1 − Fσ (t)) =
x0 (1 − a)λ1
exp(−λ1 t).
(1 − exp(−λ1 τ ))
The optimal control on the second interval is
ax0
ax0 λ2
u∗ (t) = R∞
(1 − Fσ (t)) =
exp(−λ2 t).
exp(−λ2 τ )
(1 − Fσ (s))ds
τ
The remaining step is to determine the value of the optimal switching time τ ∗ ,
which is equal to
λ1 x0 (1−a)
ln 1 + λ2 −λ1 +λ1 ln(aλ2 x0 )
λ2
e
τ∗ =
.
λ1
The limit case (α = 0) looks as follows:
ln λ ex0 + 1
τs =
λ
We compute the optimal switching time for different values of a and for the two
different sequences of modes. We remind that the parameter a determines the state,
and implicitly the time instant, at which the switching between two modes occur.
The resulting dependencies are shown in Fig. 3, 4.
5.
Conclusions
A new class of differential games with random duration and a composite cumulative
distribution function has been introduced. It has been shown that these games
can be well defined within the hybrid systems framework and that the problem of
finding the optimal strategy can be posed and solved with the methods of hybrid
optimal control theory. An illustrative example along with a qualitative analysis of
the results have been presented.
The further work on the topic will be devoted to the analysis of the cooperative
behaviour in this class of differential games. In particular, we will study the impact
which the change of mode may have of the coalition agreement of the players.
Appendix
Proof of Proposition 1
Property C1 is satisfied since it is satisfied for the function F1 (t):
Fσ (t0 ) = F1 (t0 ) = 0
Differential Games with Random Duration: A Hybrid Systems Formulation
117
Fig. 3: Dependence of the switching time τs on the parameter a for λ1 = 0.1; λ2 = 0.01
Property C2 follows from lim FN (t) = 1 and from the definition of αi (τi ) and
t→∞
βi (τi ):
lim Fσ (t) = αN −1 (τN −1 ) lim FN (t) + βN −1 (τN −1 ) =
t→∞
t→∞
=
−
Fσ (τN
−1 )−1
FN (τN −1 )−1
·1+1−
−
Fσ (τN
−1 )−1
FN (τN −1 )−1
= 1,
where τN −1 is a fixed switching time.
To show that Property C3 holds true for Fσ , we first show that Fσ is continuous.
This follows from the equality of left and right limits at t = τi :
Fσ (τi− )−1
Fσ (τi− )−1
lim Fσ = lim Fi+1
(τi )−1 Fi+1 (t) + 1 − Fi+1 (τi )−1 =
t→τi +
=
t→τi +
Fσ (τi− )−1
Fσ (τi− )−1
Fi+1 (τi )−1 Fi+1 (τi )+1− Fi+1 (τi )−1
= Fσ (τi− ) = lim Fσ
t→τi −
Next, to demonstrate that the function Fσ (t) is non-decreasing we consider two
cases:
i) t1 , t2 ∈ [τi , τi+1 ), i = 0, . . . , N − 1. Then, Fσ (t1 ) 6 Fσ (t2 ) as Fσ (t) is proportional to Fi+1 (t) on [τi , τi+1 ) and Fi+1 (t) is non-decreasing.
ii) t1 ∈ [τi , τi+1 ), t2 ∈ [τj , τj+1 ), i, j = 0, . . . , N − 1, i < j. Taking into account the
continuity property, we have
Fσ (t1 ) 6 Fσ (τi+1 ) 6 . . . 6 Fσ (τj ) 6 Fσ (t2 ).
118
Dmitry Gromov, Ekaterina Gromova
Fig. 4: Dependence of the switching time τs on the parameter a for λ1 = 0.01; λ2 = 0.1
Thus, the function Fσ (t) is non-decreasing.
Finally, we show that Fσ (t) is absolutely continuous. This is equivalent to the
following requirement (Royden, 1988): ∀ε > 0, ∃δ > 0 such that for
P any finite set
of non-intersecting
intervals
(x
,
y
)
from
[t
,
∞),
the
inequality
|yk − xk | 6 δ
k k
0
P
implies
|Fσ (yk ) − Fσ (xk )| 6 ε.
We use the fact that the functions Fi (t), i = 1, . . . , N are absolutely continuous.
ε
Then, for any i = 1, . . . , N and for any εi = 2N
> 0, there exists δi > 0 such that
for
any
finite
set
of
non-intersecting
intervals
(x
(i,k) , y(i,k) ) from [τi−1 , τi ], satisfying
P
|y(i,k) − x(i,k) | 6 δi , holds
k
X
k
|Fi (y(i,k) ) − Fi (x(i,k) )| 6 εi .
(27)
Let δ = min(δi , (τj − τj−1 )), i, j = 1, .P
. . , N . For an arbitrary finite set of nonintersecting intervals (xk , yk ), satisfying
|yk − xk | 6 δ there are two possible
k
variants:
i) Intervals (xk , yk ) are proper subsets of the partition intervals [τi , τi+1 ]. Then,
using the absolute continuity property of Fi and summing over all partition
intervals we get
X
k
|Fσ (yk ) − Fσ (xk )| =
N X
X
i=1
k
|Fi (yk ) − Fi (xk )| < N εi = ε,
Differential Games with Random Duration: A Hybrid Systems Formulation
119
whereas the following convention is employed: |Fi (a) − Fi (b)| = 0, if (a, b) ∩
[τi , τi+1 ] = ∅.
ii) Some intervals of the finite set (xk , yk ) may include switching instants τi(k) .
According to the definition of δ, an interval (xk , yk ) can intersect with at most
two partition intervals. Therefore, one can represent (xk , yk ) as a union of two
intervals (xk , τi(k) ) ⊂ (τi(k)−1
) ⊂ (τi(k) , τi(k)+1
P, τi(k) ), and (τi(k) , ykP
P). In this way,
we
can
subdivide
the
sum
|y
−
x
|
into
two:
|y
−
x
|
=
|yk − τi(k) | +
k
k
k
k
P
|xk − τi(k) | < δ. Summing over all partition intervals and using the triangle
inequality we get
P
k
6
|Fσ (yk ) − Fσ (xk )| =
N P
P
i=1 k
N P
P
i=1 k
|Fi (yk ) − Fi (xk )| 6
|Fi (yk )−Fi (τi(k) )| + |Fi (τi(k) )−Fi (xk )| <
< 2N εi = ε,
where we make use of the same convention as in item i).
The condition (27) is met and thus, Fσ is absolutely continuous. This concludes the
proof.
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Simulations of Evolutionary Models of a Stock Market⋆
Gubar Elena
St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
Universitetskiy pr. 35, St.Petersburg, 198504, Russia
E-mail: [email protected]
fax:+7 (812) 428 71 59
http://www.apmath.spbu.ru
Abstract The main idea of this work is to present some simulation in evolutionary models of agents’ interaction on the stock market. We consider
game-theoretical model of agent’s interaction, which evolving during longtime period. We consider three possible situations on the market, which are
characterized by different types of agents’ behavior.
Keyword: Evolutionary game, ESS strategy, stock market, replicative dynamic, imitation models, imitation dynamics.
1.
Introduction
In this work, we construct and analyze evolutionary models of agents’ behavior
on the stock market in various situations. Consider stock market with large but
finite group of agents. We suppose that each agent has own portfolio of different
companies and can interact with randomly matched opponent. Various situations
on the stock market are characterized by the actions that agents perform with their
blocks of shares. Agents can hold their own blocks of shares, invest or sell the blocks
of shares. Assume that in each model agents use only one pair of declared behavior,
we will consider two variants of pairs: "invest" - "hold" and "invest" - "sell". Each
model of agents’ interaction is defined as basic symmetric two players game with
corresponding payoff matrix. Describe these three situations particularly.
The first situation describes case in which one type of agent’s behavior is hold
the block of shares and receive fixed guaranteed profit from it. The second type of
behavior is to invest the block of shares to get control of the company, but the main
suggestion is that each agent can not invest control of the company independently;
he can invest the control only in cooperation with the other agent, who has block
of shares of the same company. Acquisition the control of the target company can
bring some additional possibilities to the agents, for example they can influence on
the company or receive extra profit.
The second situation on the stock market describes interaction between agents,
which desire "invest" and "sell" the block of shares. Structure of agents’ interaction
is more complicated and will be described in details further.
In the third situation we suppose that agents coordinate their behaviors but
they can get different payoffs. The one behavioral type is "invest" block of shares
and to get the control of the company or large block of shares. The second behavior
⋆
This research was supported by research grant 9.38.245.2014 at St.Petersburg State
University
Differential Simulations of Evolutionary Models of a Stock Market
121
type is "hold" their block of shares and get small, but guaranteed profit from the
own holding of shares.
These three base situations are extended by the addition of new behavior type.
The additional behavior type is to detect purposes of the opponent and replay to the
opponent’s strategy rationally. In other words, in the first situation, in case that one
agent meets his opponent, who wants to hold his block of shares, then he holds too,
but if the agent meets the opponent, who wants to invest the control of the company,
then agents cooperate and invest the control. In the second situation, if agent’s
opponent wants to invest controlling blocks of shares, then rational agent sells their
shares, if his opponent prefers to sell his blocks of shares, then rational agent invests
the shares. In the third situation rational agents behave symmetrically to their
opponents. During the meeting of the rational agents in all modelled situations
both players play Nash equilibrium strategies.
Assume that, in each period, an agent plays one basic model against all other
agents and he chooses a best response to the distributions of actions of the other
agents. The following assumptions are made in order to description the bounded
rationality aspect of the agents (Sandholm, 2009; Subramanian, 2008):
– the number of agents on the stock market is large;
– inertia, that agent agent cannot detect the smallest shifts immediately and
consider possibilities to switching strategies occasionally;
– myopia, that agents choices on current behavior and payoffs do not attempt
include beliefs about the future course of play into their decisions. Each agent
takes into account only the current strategy distribution;
– market agents have limited information about opponents’ behavior, because the
number of agents in interaction is large, exact information about their aggregate
behavior typically is difficult to obtain.
This work pursues some purposes: considering some basic models of interaction
between stock market agents, invasion new behavioral type to the models and analysis the agent’s behavior on the stock market during the long-run period, constructing
imitative evolutionary dynamics for all models.
2.
Basic models
In this section we present three situations on the stock market and describe agents’
behavior and payoffs more detailed. Consider stock markets with h types of shares
and population (group) of agents on the market. Assume that shares have different
investment attractiveness.
Let agent i has own portfolio of h types block of shares Πi = (π1 , . . . , πh ), Denote
h
P
as P r(Πi ) =
nj Aπj – portfolio profit of i-th agent. Where nj is number of shares
j=1
type πj , Aπj – cost of shares type πj . Let Pm in in minimal value of ivestment rate.
Describe symmetric games between two randomly matched individuals. All agents
have own portfolios and they have a choice receive guaranteed profit from own
portfolio allocations or invest money into large block of shares In our game we have
two strategies "invest" and "hold". Strategy Invest forces individuals to buy block
of shares with high attractiveness.
Agents’ interaction can be defined by one of the following basic models:
Situation A:
122
Gubar Elena
Each agent has two strategies "Invest" and "Hold". If agent "Hold" then he has
guaranteed profit P r(Πi )
i
If agent "Invest" he has profit P r(Πi ) + BI , where BI = ViC−C
is investment
i
interest. VI is received value of investment, Ci is cost of an investment.
Suppose that Ci is high and agent can invest only in cooperation with his opponent, he can not investment alone.
Describe agents interaction. Suppose that during single trade session different
blocks of shares can be traded. When agents interact both agent have choice invest
their money into large block of shares with high investment attractiveness or receive
guaranteed profit from own portfolio P r(Πi ). If agent decides to invest into into
large block of shares with high investment attractiveness he can receive additional
profit BI . If one agent wants to invest alone then he incurs investment cost P r(Πi )−
CI but no profit in the same time his opponent has P r(Πi ).
In situation A market agents have two types of behavior. The first type is to
hold the block of share and receive guaranteed profit from it. The second type is
to invest the large block of share to get the control of target company. But agents
have to cooperate with another agent if they prefer to receive the control, because
each agent can not invest the control of company separately.
Matrix below illustrates symmetric game between the agents:
H
I
H (P r(Πi ), P r(Πi ))
(P r(Πi ) + δ, Ci )
I (CI , P r(Πi ) + δ) (P r(Πi ) + BI , P r(Πi ) + BI ).
Define as I ≥ 0 the income, that agent can get during the interaction.Denote
players strategies as H and B, strategy B forces agents to invest and strategy H
forces agents to hold his blocks of shares. We can give following interpretation the
agents payoffs, if both agents hold their blocks of shares, then their get little payoff,
which is equal to I. If one agent wants to invest the control, and other doesn’t,
then the first agent gets 0, because he expends money, but doesn’t have the control,
and the second player gets 3I/2, because he has profit from his block of shares. If
both players want to invest the control and cooperate, then they invest it and get
payoff 2I. Situation (B, B) is more risky, agents should to cooperate and to take
into account own purposes and purposes of their opponents.
Obviously the basic game has three equilibriums (H, H), (B, B), (2/3, 1/3). We
verify that strategies H an B are evolutionary stable, in the sequel denote as ∆ESS
the set of evolutionary stable strategies. Evolutionary stability of some strategy x
means that this strategy gives better payoff against any other strategy y and gives
the best payoff against every alternative best reply y.
Situation B:
In this simple situation we consider one case in which agents enter into competition for the large block of shares with high investment attractiveness.
Each agent has two strategies "Invest" and "sell" and aspire to invest money
into large block of shares with high investment attractiveness. If agent invest then
his payoff is P r(Πi ) + BI . If he sells then he gets P r(Πi ).
Describe interaction: In this situation we suppose that during single trade session
agents can invest into large block of shares with high investment attractiveness or
sell own blocks of shares. If two agents with strategy "Invest" meet each other
then they starts competition for the large block of shares with high investment
Differential Simulations of Evolutionary Models of a Stock Market
123
attractiveness but only one agent can receive it. In this competition agents receive
payoff 1/2(B − CI ), C I is competition costs.
Consider another situation on the stock market and describe interaction between
investers and sellers. Hence we will have another behavioral types for the agents.
One part of agents wants to sell the block of share and thus their corresponding
strategy is S. The other part of agents wants to invest the block of shares, then the
strategy B corresponds to this behavior. As a result we get following situations:
– if two agents with "invest" type of behavior meet, then they start to fight for
the blocks of shares and each agent can get the block of shares or lose it with
probability 1/2;
– if two agent with behavior "sell" meet, then each agents sells their assets and
both agents receive guaranteed profit from the selling;
– if one agent invests the shares and the other sell, then seller has guaranteed profit
and the invester has some profit from the shares and additional possibilities from
large block of shares.
Payoff matrix is presented below:
I
S
I ((BI − CI )/2, (BI − CI )/2)
(BI , Ps ell)
S
(Ps ell, BI )
(P r(Πi )/2, P r(Πi )/2),
here I is agent’s income and C is agent’s costs, I < C both variables are nonnegative.
There are three equilibriums (S, B), (B, S), (e
x, x
e), x
e = (I/C, 1 − I/C). Verification
of evolutionary stability shows that in this game there is the unique evolutionary
stable strategy x
e, x ∈ ∆ESS .
Situation C:
In situation C assume that on the stock market agents provide symmetrical
behavior to the opponents. Both agents can hold their blocks of shares or invest
the large blocks of shares. However they cannot invest or hold assets separately
and have to coordinate their actions with the other agents. Strategy S corresponds
to behavior "hold", and strategy B corresponds behavior "invest". If both agents
hold their shares then they get guaranteed profit but if they invest the large block of
shares or the control of the target company then they receive additional possibilities
(i.e. agents can influence on the companies decisions). Following matrix illustrate
agents’ payoffs:
H
B
H ((I1 ,I1 ) (0,0)
B (0,0) (I2 , I2 ).
where I1 > I2 , I1 , I2 ≥ 0, Ij , j = 1, 2 are agents’ incames. There are three equilibI2
riums in the game (H, H), (B, B), (e
x, x
e) x
e=
. Two pure strategies B and
I1 + I2
H are evolutionary stable, H, B ∈ ∆ESS .
124
3.
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Extended games
Consider an extension of basic models. Suppose the small share of agents, which can
recognize opponents’ behavior during their meeting, invades on the stock market.
This type of agents we will call rational agents or rationalist. We add new strategy
E to each basic game, which describes new type of behavior of the market agents.
Rationality of the player means that agents, which use strategy E can recognize
actions of his opponents and adjust their behaviors in compliance with actions of
the opponents. In each basic game we suppose that rational agents use their best
responses as strategies and if rational agent meets another rational agent, then both
play Nash equilibrium strategies. For each situation on the stock market construct
new payoff matrices.
Situation A:
In basic game we have three equilibriums, hence extended game will have three
variants, which describe agents preferences. We present payoff matrix which depends
on the equilibriums profiles:
H
B
E
H (I, I) (3I/2, 0)
(I, I)
B (0, 3I/2) (2I, 2I)
(2I, 2I)
E (I, I) (2I, 2I) (u(e
x, x
e), u(e
x, x
e))
where values (u(e
x, x
e), u(e
x, x
e)) are players payoffs in the Nash equilibrium situation.
In basic game we have three different equilibriums hence we get three variations of
the extended game.
Situation B:
In situation B after invasion of rational agents,we get following payoff matrix.
Basic game in situation B has three equilibriums hence extended game will have
three modifications:
B
S
E
B ((I − C)/2, (I − C)/2) (I, 1)
(1, 1)
S
(1,I)
(I/2, I/2)
(I, I)
E
(1, 1)
(I, I) (u(e
x, x
e), u(e
x, x
e))
where as in previous case values (u(e
x, x
e), u(e
x, x
e)) is agents payoff on the equilibrium strategies. Strategy E is the strategy of rational agents, which can identify
behavioral type of their opponents.
Situation C:
In situation C we also have three modifications of payoff matrix, depending on
the various agent’s payoffs in Nash equilibriums profiles.
B
H
E
B (I1 , I1 ) (0, 0)
(I1 , I1 )
H (0, 0) (I2 , I2 )
(I2 , I2 )
E (I1 , I1 ) (I2 , I2 ) (u(e
x, x
e), u(e
x, x
e))
where u(e
x, x
e) is equal to agent’s payoff on corresponding equilibrium strategies.
Differential Simulations of Evolutionary Models of a Stock Market
4.
125
Replication by Imitation
For all situations A, B, C we analyze, which behavioral type will prevail on the
market during long-rum period with suggestion that in each situation at initial time
moment small share of rational players are invaded. For all these models we will
consider selection dynamics arising from adaptation by imitation. In all situations
we suppose that all agents in the large group are infinitely live and interact forever.
This assumption can be interpreted in following way, if one agent physically exits
from the market, then he is replaced by another one. Each agent has some pure
strategy for some time interval and then reviews his strategy and sometimes changes
the strategy.
There are two basic elements in this model (Weibull, 1995). The first element
is time rate at which agents in the group review their strategy choice. The second
element is choice probability at which agents change their strategies. Both elements
depend on the current group state and on the performance of the agent’s pure
strategy.
Let K = {H, B, S, E} is the set of agents pure strategies. In each of the previously described situations agents match at random in total group and each agent
use one of pure strategy from the set K. Player with pure strategy i will be called
as i-strategist.
Denote as ri (x) an average review rate of the agent, who uses pure strategy i in
the group state x = (xH , xB , xE ). Variable pij (x) is probability at which i-strategist
switches to some pure strategy j, i, j ∈ K. Here pi (x) = (p1i (x), p2i (x), p3i (x)), i =
H, B, E, S is the resulting probability distribution over the set of pure strategies
and depends on the population state. Value pii (x) is probability that a reviewing
i-strategist does not change his strategy.
Consider imitation process generally in finite large group of agents. Suppose that
each reviewing agent samples another agent at random from the group with equal
probability for all agents and observes with some noise the average payoff to his own
and to the sampled agent’s payoff. If payoff of the samples agents is better then his
own he can switch to the sampled agent’s strategy.
In general case the imitation dynamics is described by the formula:
X
ẋi =
xj rj (x)pij (x) − ri (x)xi , i ∈ K.
(1)
j∈K
In this paper we use special variation of the imitation dynamics of successful agents.
5.
Imitation of Pairwise Comparison
Suppose that each agent samples another stock agent from the total group with
equal probability for all agents and observes the average payoff to his own and
the sampled agent’s strategy. When both players show their strategies then player
who uses strategy i gets payoff u(ei , x) + ε and player, who uses strategy j gets
u(ej , x) + ε′ , where ε, ε′ is random variables with continuously probability distribution function φ. The random variables ε and ε′ can be interpreted as individual
preference differences between agents in the market. Each agent can get various
preferences, i.e. agent, which receive the large block of shares or the control of target company, can be more satisfied, because he can influence to the company or have
additional profit. Other agents, which hold own assets and receive only fixed payoff
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Gubar Elena
can be less satisfied of their profit. Use as distribution function φ(z) = exp(αz),
α ∈ R.
Players compare their payoffs: if the payoff of the sampled agent is better than
of the reviewing agent, he switches to the strategy of the sampled agent. In other
words, if this inequality u(ej , x) + ε′ > u(ei , x) + ε is justify for player with pure
strategy i then he switches to the strategy j.
For the general case the following formula describes the imitation dynamics of
pairwise comparison:


X
ẋi = 
xj (φ[u(ei − ej , x)] − φ[u(ej − ei , x)]) xi , i ∈ K.
(2)
j∈K
To simplify calculations use certain numerical values for the models parameters
and construct dynamics for each extended game.
Situation A:
Using following values for incomes, which are I = 2, α = 1 and values of parameter u are u = 4, 2, 8/3 then we get three different systems of differential equations,
corresponding to various cases of the extended games:
ẋH = (xB (e(4xH +xB −2) − e(−4xH −xB +2) )+
xE (e(−xB +(2−u)xE ) − e(xB +(−2+u)xE ) ))xH ;
ẋB = (xH (e(−4xH −xB +2) − e(4xH +xB −2) )+
xE (e(−2xH +(4−u)xE ) − e(2xH +(−4+u)xE ) ))xB ;
ẋE = (xH (e(xB +(−2+u)xE ) − e(−xB +(2−u)xE ) )+
xB (e(2xH +(−4+u)xE ) − e(−2xH +(4−u)xE ) ))xE .
Situation B:
Let values for income and costs are: I = 2, C = 4, α = 1 then we get three
systems of differential equations with values of parameter u: u = 4, 3, 1 in various
cases:
ẋH = (xB (e(xH +5xS −3) − e(−xH −5xS +3) )+
xE (e(−2xH +(1−u)xE ) − e(2xH +(−1+u)(1−xH −xS )) ))xH ;
ẋS = (xH (e(−xH −5xS +3) − e(xH +5xS −3) )+
xE (e(−2xB +(4−u)xE ) − e(2xS +(−4+u)xE ) ))xB ;
ẋE = (xH (e(2xH +(−1+u)xE ) − e(−2xH +(1−u)xE ) )+
xS (e(2xS +(−4+u)xE ) − e(−2xS +(4−u)xE ) ))xE .
Differential Simulations of Evolutionary Models of a Stock Market
127
Situation C:
Let incomes are I1 = 2, I2 = 1, α = 1, then systems of differential equations
that define pairwise comparison dynamics are following:
ẋB = (xB (e(xH −2xB +1) − e(−xH +2xB −1) )+
xE (e(−xB +(2−u)xE ) − e(xB +(−2+u)xE ) ))xB ;
ẋH = (xH (e(−xH +2xB −1) − e( xH − 2xB + 1))+
xE (e(−2xH +(1−u)xE ) − e(2xH +(−1+u)xE ) ))xH ;
ẋE = (xH (e(xB +(−2+u)xE ) − e(−xB +(2−u)xE ) )+
xB (e(2xH +(−1+u)xE ) − e(−2xH +(1−u)xE ) ))xE ;
Values of parameter u: u = 2, 1, 4/3.
For each system we get numerical solution using next initial states: xE =
0.1, xH = 0.01, 0.02, . . . , 0.98, xB = 0.98, 0.97, . . . , 0.01 and xE = 0.1, xB =
0.01, 0.02, . . . , 0.98, xS = 0.98, 0.97, . . . , 0.01 solution trajectories are presented in
Table 1, where rows represent different situations on the stock market and columns
correspond to various modifications of extended games.
We get that in situation A for all cases of extended games behavioral type
"invest" and behavioral type of rationalist are preferable and that strategies will
survive in the long-run period, however in case 2 we can see that behavior "to hold"
also can be preserved.
In situation B in case 1 only rational agents prevail on the market, in case 2
and 3 mixture of agents, who sell their blocks of shares and rationalists will survive
and situation (xB , xS , xE ) = (0, 1/3, 2/3) will be the rest point of the system.
In situation C we get different variants of prevailed behaviors. In case 1 solutions
trajectories aspire to states xB and xE and on the stock market "investors" and
"rationalists" will be survived in long-run period. In case 2 behaviors "invest" and
partly "hold" will be conserved and and in case 3 only state xB is stable.
Table 1: Imitation dynamics of pairwise comparison.
Case 1
Case 2
Case 3
u=4
u=2
u = 8/3
A
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B
u=4
u=3
u=1
u=2
u=1
u = 4/3
C
6.
Imitation of successful agents
Suppose that the choice probabilities pji (x) are proportional to popularity of j’s
strategy xj , and the proportionality factor is described by the currently payoff to
strategy j. It is thus as if agent observes other agents choices with some small noise
and would imitate another agent from the population with a higher probability
for relatively more successful agents. Denote the weight factor that a reviewing
agent with strategy i attaches to pure strategy j as ω[u(ei , x), x] > 0, where ω is a
continuous function. Then
ω[u(ej , x), x]xj
pji = P
ω[u(ej , x), x]xp
p∈K
Selection dynamics for that model is described by following equations:



 X Pω[u(ej , x), x]xj
ẋi = 
− 1 xi .
ω[u(ej , x), x]xp
j∈K
(3)
p∈K
As in earlier case we have some additional assumptions for choice probability
such as these is not that a reviewing agent necessarily knows the current average
payoff to all pure strategies. It is sufficient that some agent have some possibly noisy
empirical information about payoff to some pure strategies in current use and on
average more agents prefer to imitate an agent with higher payoff than one with
lower average payoff.
In this paper accept as weight function ω = exp(α−1 u(ei , x)), where α is small
noise of observation and get following expression (Sandholm, 2008, 2010):
Differential Simulations of Evolutionary Models of a Stock Market
xi exp(α−1 u(ei , x))
ẋi = P
, i, k ∈ K.
xk exp(α−1 u(ek , x))
129
(4)
k∈K
To simplify calculations, as in previous section, we use certain numerical values
for the models’ parameters and construct dynamics for each extended game.
Situation A:
Let agents’ income is I = 2 then we get three different systems of differential
equations, corresponding to various cases of the extended games:
−1
ẋH =
xH e(α
(xB +2))
−1 (2x +4x +x u)) ;
H )) + x e(α
H
B
E
+ xB e
E
−1
(α (4−4xH ))
xB e
;
ẋB =
(4−4xH )) + x e(α−1 (2xH +4xB +xE u))
xH e(α−1 (xB +2)) + xB e(α−1
E
−1
xE e(α (2xH +4xB +xE u))
ẋE =
;
−1
xH e(α (xB +2)) + xB e(α−1 (4−4xH )) + xE e(α−1 (2xH +4xB +xE u))
xH e
(α−1 (x
B +2))
(α−1 (4−4x
Values of parameter u: u = 4, 2, 8/3.
Situation B:
Let agents’ income is I = 2 and costs are C = 4 then we get three systems of
differential equations describe imitation dynamics of successful agents with values
of parameter u: u = 4, 2, 1.
−1
ẋH =
xH e(α (−2xH +3xS +1))
;
xH e(α−1 (−2xH +3xS +1)) + xS e(α−1 (−3xH −2xS +4)) + xE e(α−1 (xH +4xS +xE u))
ẋS =
xS e(α (−3xH −2xS +4))
;
−1 (−2x +3x +1))
(α
H
B
xH e
+ xB e(α−1 (−3xH −2xS +4)) + xE e(α−1 (xH +4xS +xE u))
−1
−1
xE e(α (xH +4xS +xE u))
ẋE =
;
−1
xH e(α (−2xH +3xS +1)) + xS e(α−1 (−3xH −2xS +4)) + xE e(α−1 (xH +4xS +xE u))
Situation C:
Let incomes are I1 = 2 and I2 = 1 then for different values of parameter u:
u = 2, 1, 4/3. we have following systems of differential equations:
−1
ẋH =
xH e
(α−1 (2−2x
xH e(α (2−2xB ))
;
−1 (1−x ))
B )) + x e(α
H
+ xE e(α−1 (2xH +xB +xE u))
B
−1
ẋB =
xB e(α (1−xH ))
;
−1 (2−2x ))
(α
B
xH e
+ xB e(α−1 1−xH ) + xE e(α−1 (2xH +xB +xE u))
ẋE =
xE e(α (2xH +xB +xE u))
;
−1
xH e(α (2−2xB )) + xB e(α−1 (1−xH )) + xE e(α−1 (2xH +xB +xE u))
−1
Table 2 contains pictures with solution trajectories for each extended game, as
in Table 1 rows represent different situations on the stock market and columns
correspond to various modifications of extended games.
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Gubar Elena
We get that in situation A in case 1 all trajectories converge to stationary state
xE and in cases 2 and 3 the state xH is stable and according behavioral type "hold"
the blocks of shares will prevail on the market.
In situation B, in case 1 stable point is (xB , xS , xE ) = (0, 0, 1), in case 2 system
has only one stable rest point (xB , xS , xE ) = (0, 1/3, 2/3), in case 3 stable point is
(xB , xS , xE ) = (0, 0.6, 0.4), hence we can say that during long-run period behaviors
of rationalists and "sellers" will survive.
In situation C in case 1 solutions trajectories converge to border between xE
and xB and in case 2 and 3 the stable point is (xB , xS , xE ) = (1, 0, 0).
Table 2: Imitation dynamics of successful agents.
Case 1
Case 2
Case 3
A
u=4
u=2
u = 8/3
u=4
Case 1
u=3
Case 2
u=1
Case 3
B
XI
C
u=2
7.
XR
XH
u=1
u = 4/3
Conclusion
This paper’s main contribution is in using general results from evolutionary game
theory to simulation of agents’ interaction on the stock market and analysis the
behavior stability over time. Applying numerical simulation, we get that in some
Differential Simulations of Evolutionary Models of a Stock Market
131
situation behavior of agents with bounded rationality, which can not recognize the
actions of the opponent, will survive in long-run period. And the behavior of rational
players will preserve in some other situations. In future research we are planning to
use other probability distributions for agent’s revision profiles.
References
Weibull, J. (1995). Evolutionary Game Theory. — Cambridge, MA: The M.I.T.Press.
Subramanian, N.(2005). En evolutionary model of speculative attacks. Journal of Evolutionary Economics, 15, 225–250. Springer-Verlag.
Gintis, H. (2000). Game theory evolving. Princeton University Press.
Gubar, E. A.(2008). An Evolutionary models of Trades in the Stock Market. Proceeding of
Workshop on Dynamic Games in Management Science. Montreal, Canada, p.20.
Evstigneev, I. V., Hens, T. and Schenk-Hopp?e, K. R. (2006). Evolutionary stable stock
markets. Economic Theory, 27, 449–468
Sandholm, W. H. (2009).Pairwise Comparison Dynamics and Evolutionary Foundations
for Nash Equilibrium. Econometrica, 769(3), 749–764.
Sandholm, W. H., Durlauf, S. N. and L. E. Blume (2008). Deterministic Evolutionary Dynamics The New Palgrave Dictionary of Economics, 2nd edition, eds., Palgrave Macmillan.
Sandholm, W. H., E. Dokumaci, and F. Franchetti (2010). Dynamo: Diagrams for Evolutionary Game Dynamics, version 0.2.5. http://www.ssc.wisc.edu/ whs/dynamo.
Equilibrium in Secure Strategies in the
Bertrand-Edgeworth Duopoly Model⋆
Mikhail Iskakov1 and Alexey Iskakov2
V.A.Trapeznikov Institute of Control Sciences,
Ul.Profsoyuznaya 65, Moscow 117997, Russia
E-mail: [email protected]
2
V.A.Trapeznikov Institute of Control Sciences,
Ul.Profsoyuznaya 65, Moscow 117997, Russia
E-mail: [email protected], [email protected]
1
Abstract We analyze the Bertrand-Edgeworth duopoly model using a solution concept of Equilibrium in Secure Strategies (EinSS), which provides
a model of cautious behavior in non-cooperative games. It is suitable for
studying games, in which threats of other players are an important factor in
the decision-making. We show that in some cases when Nash-Cournot equilibrium does not exist in the price duopoly of Bertrand-Edgeworth there is
an EinSS with equilibrium prices lower than the monopoly price. The corresponding difference in price can be interpreted as an additional cost to
maintain security when duopolists behave cautiously and secure themselves
against mutual threats of undercutting. We formulate and prove a criterion
for the EinSS existence.
Keywords: Bertrand-Edgeworth Duopoly, Equilibrium in Secure Strategies, Capacity Constraints.
1.
Introduction
It is well known that the model of Bertrand-Edgeworth may not posses a Nash
equilibrium (see e.g. d’Aspremont and Gabszewicz (1985)). Let the receipt function pD(p) be strictly concave and reach its maximum at monopoly price pM .
When D(pM ) > S1 + S2 there is a Nash price equilibrium (p∗ , p∗ ) in the BertrandEdgeworth duopoly model such that D(p∗ ) = S1 +S2 . When D(pM ) < S1 +S2 there
is no (pure strategy) Nash equilibrium in this game. There were several attempts
to restore the concept of equilibrium in this model. For example d’Aspremont and
Gabszewicz (1985) proposed the concept of quasi-monopoly, which restores the existence of pseudo equilibrium when one capacity is quite small compared to the
other. Dasgupta and Maskin (1986) and Dixon (1984) demonstrated the existence
of mixed-strategy equilibrium. However it proved not to be easy to characterize
what the equilibrium actually looks like. Allen and Hellwig (1986a) were able to
show that in a large market with many firms, the average price set would tend to
the competitive price.
In this paper we analyze the Bertrand-Edgeworth duopoly model using a solution
concept of Equilibrium in Secure Strategies (EinSS), which provides a model of
cautious behavior in non-cooperative games (M.Iskakov 2005, 2008). It is suitable
for studying games, in which threats of other players are an important factor in
⋆
This work was supported by the research project No.14-01-00131-a of the Russian Foundation for Basic Research.
Equilibrium in Secure Strategies in the Bertrand-Edgeworth Duopoly Model
133
the decision-making (M.Iskakov and A.Iskakov 2012b). This concept proved to be
effective and allowed to find equilibrium positions in some well-known economic
games that fail to have Nash-Cournot equilibrium. In particular this approach has
been successfully applied for the classic Hotelling’s model with the linear transport
costs (Hotelling 1929). There is no price Nash-Cournot equilibrium in this game
when duopolists choose locations too close to each other (d’Aspremont et al., 1979).
However, there is a unique EinSS price solution for all location pairs under the
assumption that duopolists secure themselves against being driven out of the market
by undercutting (M.Iskakov and A.Iskakov 2012a). Equilibria in secure strategies
have been also successfully characterized for the contest described by Tullock (1969,
1980) as a rent-seeking contest. It is well known that a Nash-Cournot equilibrium
does not exist in a two-player contest when the contest success function parameter is
greater than two. However an EinSS always exists in the Tullock contest (M.Iskakov
et al., 2013). Moreover, when the success function parameter is greater than two,
this equilibrium is unique up to a permutation of players, and has a lower rent
dissipation than in a mixed-strategy Nash equilibrium.
Our aim is to analyze the original Bertrand-Edgeworth duopoly model with
capacity constraints, which may not possess a Nash-Cournot equilibrium. We show
that in some cases when Nash-Cournot equilibrium does not exist there is an EinSS
with equilibrium prices lower than the monopoly price. The corresponding difference
in price can be interpreted as an additional cost to maintain security when duopolists
are cautious and avoid mutual threats. We formulate and prove a criterion for the
EinSS existence.
The remaining paper is organized as follows. In Section 2 we remind the BertrandEdgeworth model. In Section 3 the solution concept that we are going to use for
analyzing the price duopoly game is presented. In Section 4 the equilibria in secure
strategies are characterized for the linear demand function. In Section 5 the obtained results are generalized for the strictly concave receipt functions. Finally we
provide an interpretation and summarize our results in the Conclusion.
2.
Bertrand-Edgeworth duopoly model
In this section we consider a model of price setting duopolists with capacity constraints originated in papers of Bertrand (1883) and Edgeworth (1925). We consider
the market for some homogeneous product with a continuous strictly decreasing consumer’s demand function D(p). There are two firms in the industry i = 1, 2, each
with a limited amount of productive capacity Si such that D(0) > S1 + S2 . As in
Edgeworth’s work we assume these limits as physical capacity constraints, which
are the same at all prices. Firms choose prices pi and play non-cooperatively. The
firm quoting the lower price serves the entire market up to its capacity and the
residual demand is met by the other firm.
All consumers are identical and choose the lower available price on a first-comefirst-serve basis. Following Shubik (1959) and Beckmann (1965) we assume in our
analysis that the residual demand to the firm quoting the higher price is a proportion
of total demand at that price. If duopolists set the same prices firms share the market
in proportion to their capacities. Formally we define the payoff functions of players
134
Mikhail Iskakov, Alexey Iskakov
to be:


p1 min{S1 , D(p1 )}, p1 < p2
1
D(p1 )}, p1 = p2
u1 (p1 , p2 ) = p1 min{S1 , S1S+S
2

p min{S , D(p1 ) max{0, D(p ) − S }}, p > p
2
2
1
2
1
1 D(p2 )


p2 min{S2 , D(p2 )}, p2 < p1
2
D(p2 )}, p2 = p1
u2 (p1 , p2 ) = p2 min{S2 , S1S+S
2

p min{S , D(p2 ) max{0, D(p ) − S }}, p > p
2
2 D(p1 )
1
1
2
1
(1)
In the particular case of a linear demand function D(p) = 1 − p, which we
consider below, the payoff functions can be written as:


p1 min{S1 , 1 − p1 }, p1 < p2
1
(1 − p1 )}, p1 = p2
u1 (p1 , p2 ) = p1 min{S1 , S1S+S
2

p min{S , (1−p1 ) max{0, 1 − p − S }}, p > p
1
1 (1−p2 )
2
2
1
2
(2)


p2 min{S2 , 1 − p2 }, p2 < p1
2
(1 − p2 )}, p2 = p1
u2 (p1 , p2 ) = p2 min{S2 , S1S+S
2

p min{S , (1−p2 ) max{0, 1 − p − S }}, p > p
1
1
2
1
2
2 (1−p1 )
The study of this case will allow us relatively easy to prove the main results in
Section 4. However, these results will be generalized to arbitrary strictly concave
receipt functions in Section 5.
3.
Equilibrium in Secure Strategies
We now proceed to define the solution concept that we are going to use to analyze the Bertrand-Edgeworth duopoly model (1). Below we provide definitions of
Equilibrium in Secure Strategies from (M.Iskakov and A.Iskakov 2012b). Consider
n-person non-cooperative game in the normal form G = (i ∈ N, si ∈ Si , ui ∈ R).
The concept of equilibria is based on the notion of threat and on the notion of
secure strategy.
Definition 1. A threat of player j to player i at strategy profile s is a pair of
strategy profiles {s, (s′j , s−j )} such that uj (s′j , s−j ) > uj (s) and ui (s′j , s−j ) < ui (s).
The strategy profile s is said to pose a threat from player j to player i.
Definition 2. A strategy si of player i is a secure strategy for player i at given
strategies s−i of all other players if profile s poses no threats to player i. A strategy
profile s is a secure profile if all strategies are secure.
In other words a threat means that it is profitable for one player to worsen the
situation of another. A secure profile is one where no one gains from worsening the
situation of other players.
Definition 3. A secure deviation of player i with respect to s is a strategy s′i
such that ui (s′i , s−i ) > ui (s) and ui (s′i , s′j , s−ij ) ≥ ui (s) for any threat {(s′i , s−i ),
(s′i , s′j , s−ij )} of player j 6= i to player i.
Equilibrium in Secure Strategies in the Bertrand-Edgeworth Duopoly Model
135
There are two conditions in the definition. In the first place a secure deviation
increases the profit of the player. In the second place his gain at a secure deviation
covers losses which could appear from retaliatory threats of other players. It is
important to note that secure deviation does not necessarily mean deviation into
secure profile. After the deviation the profile (s′i , s−i ) can pose threats to player i.
However these threats can not make his or her profit less than in the initial profile
s. We assume that the player with incentive to maximize his or her profit securely
will look for secure deviations.
Definition 4. A secure strategy profile is an Equilibrium in Secure Strategies
(EinSS) if no player has a secure deviation.
There are two conditions in the definition of EiSS. There are no threats in the
profile and there are no profitable secure deviations. The second condition implicitly
implies maximization over the set of secure strategies.
Any Nash-Cournot equilibrium poses no threats to players so it is a secure
profile. And no player in Nash-Cournot equilibrium can improve his or her profit
using whatever deviation. Both conditions of the EinSS are fulfilled. Therefore we
obtain
Proposition 1. Any Nash-Cournot equilibrium is an Equilibrium in Secure Strategies.
The Nash equilibrium is the profile, in which the strategy of each player is the
best response to strategies of other players. In a similar way, the strategy of each
player in the EinSS turns out to be the best secure response.
Definition 5. A secure strategy si of player i is a best secure response to
strategies s−i of all other players if player i has no more profitable secure strategy at
s−i . A profile s∗ is the Best Secure Response profile (BSR-profile) if strategies
of all players are best secure responses.
The EinSS is a secure profile by definition. And it must be the best secure
response for each player since otherwise there is a player who can increase the
payoff by secure deviation. Therefore we get:
Proposition 2. Any Equilibrium in Secure Strategies is a BSR-profile.
This property provides a practical method for finding EinSS in three steps: (i)
to analyze threats and determine conditions for secure profile, (ii) to find all BSRprofiles as a solution of the corresponding maximization problem in the set of secure
profiles, and (iii) to verify the definition of EinSS for the found BSR-profiles.
4.
Solution in secure prices for the linear demand function
In this Section we illustrate how to find an EinSS in the simplest case of the linear
demand function D(p) = 1 − p. First of all, let us analyze the threats between
players and define secure profiles in the Bertrand-Edgeworth duopoly model.
Proposition 3. The profile (p1 , p2 ) is a secure profile in the duopoly price game
of Bertrand-Edgeworth with the linear demand function D(p) = 1 − p and payoff
functions (2) if and only if it lies in the set M = {(p1 , p2 ) : 0 < pi 6 p∗ , i = 1, 2},
where p∗ = 1 − S1 − S2 .
136
Mikhail Iskakov, Alexey Iskakov
Proof. (1). Consider the case p∗ < p1 < p2 . If 1 − p1 > S1 player 1 always threatens player 2 by slight increasing his price p1 . If 1 − p1 6 S1 then player 2 can
decrease his price till p̃2 < p1 . In this case, according to (2): ũ1 (p1 , p̃2 ) = p1 (1 −
2 −S2 }
p1 ) max{0,1−p
< p1 (1 − p1 ) = u1 (p1 , p2 ) and ũ2 (p1 , p̃2 ) = p̃2 min{S2 , 1 − p2 } >
1−p2
0 = u2 (p1 , p2 ), and therefore there is always a threat of player 2 to player 1. Symmetrically, if p∗ < p2 < p1 either player 2 threatens player 1 or vice versa. If
p∗ < p2 = p1 there is always a bilateral threat of undercutting by slight decreasing
of price.
(2). If p1 6 p∗ < p2 player 1 always threatens player 2 by increasing his price
till p∗ + 0 < p2 which exceeds p∗ by an arbitrarily small amount. Indeed, in this
case 1 − p1 > 1 − (p∗ + 0) > S1 and according to (2) u1 (p1 , p2 ) = p1 S1 < (p∗ +
2
0)S1 = u1 (p∗ + 0, p2 ). On the other hand, u2 (p∗ + 0, p2 ) = p2 S1−p
S2 < p2 S2 and
1 +S2
S1
S1
∗
u2 (p + 0, p2 ) = p2 (1 − p2 ) 1 − 1−(p∗ +0) < p2 (1 − p2 ) 1 − 1−p1 => u2 (p∗ +
0, p2 ) < u2 (p1 , p2 ). Symmetrically, if p2 6 p∗ < p1 player 2 always threatens player
1.
(3). From the above it follows that all secure profiles must lie in the set M =
{(p1 , p2 ) : 0 < pi 6 p∗ , i = 1, 2}. From the other hand if p1 6 p∗ : u1 (p1 , p2 ) = S1 p1
linearly increases in p1 and does not depend on p2 . Hence there are no threats for
player 1. Symmetrically, if p2 6 p∗ there are no threats for player 2. Therefore
(p1 , p2 ) is a secure profile in the game (2) if and only if it lies in the set M =
{(p1 , p2 ) : 0 < pi 6 p∗ , i = 1, 2}. Let us now find all Best Secure Response profiles in the set M of secure profiles.
Proposition 4. In the duopoly price game of Bertrand-Edgeworth with the linear
demand function D(p) = 1−p and payoff functions (2) there is a unique Best Secure
Response profile (p∗ , p∗ ), where p∗ = 1 − S1 − S2 .
Proof. According to Definition 5 and Proposition 3 any Best Secure Response profile
must lie in the set M = {(p1 , p2 ) : 0 < pi 6 p∗ , i = 1, 2} of secure profiles found
in Proposition 3. According to (2) the payoff functions u1 = S1 p1 and u2 = S2 p2
increase in the set M linearly in p1 and in p2 respectively. Therefore there can
be only one BSR-profile (p∗ , p∗ ) in the set M (otherwise at least one player can
securely slightly increase his price and get a more profitable secure strategy in M ).
Let us now prove that it is indeed a BSR-profile. Any decrease in price in the profile
(p∗ , p∗ ) is not profitable for players. On the other hand, as shown in paragraph 2 of
the proof of Proposition 3 any increase in price in the profile (p∗ , p∗ ) is not secure
for players. Therefore no player has a more profitable secure strategy in (p∗ , p∗ ) and
therefore this profile by definition is a Best Secure Response profile in the game. According to Proposition 2 there can not be other EinSS in the BertrandEdgeworth duopoly game except the BSR-profile (p∗ , p∗ ) found in Proposition 4.
Below we formulate and prove a necessary and sufficient condition for the BSRprofile (p∗ , p∗ ) to be an Equilibrium in Secure Strategies.
Proposition 5. In the game of Bertrand-Edgeworth with the linear demand function D(p) = 1 − p and payoff functions (2) there is an Equilibrium in Secure Prices
(p∗ , p∗ ) where p∗ = 1 − S1 − S2 if and only if
1 − S1
1 − S2
6 p∗ and
6 p∗
2
2
(3)
Equilibrium in Secure Strategies in the Bertrand-Edgeworth Duopoly Model
If the equilibrium price is not less than the monopoly price p∗ > pM =
Nash equilibrium. There are no other EinSS in the game.
1
2
137
it is a
Fig. 1: Equilibria in secure prices in the Bertrand-Edgeworth duopoly model with D(p) =
1 − p in the space of capacity parameters (S1 , S2 ). Dark gray area: EinSS which coincide
with Nash equilibria. Light gray area: EinSS which are not Nash equilibria. White area:
there are neither Nash equilibria nor EinSS.
Equilibria in secure prices in the space of capacity parameters (S1 , S2 ) are shown
in Fig.1. The profile (p∗ , p∗ ) is a Nash equilibrium if S1 + S2 6 12 (dark gray area).
Under the weaker conditions (3) it is an EinSS. The area of EinSS which are not
Nash equilibria are shaded by light gray in Fig.1. If conditions (3) do not hold
this profile is no longer an EinSS and corresponds to an unstable BSR-profile. The
found solution can be compared with the price which would maximize the joint
profits in the industry pM = max{1 − S1 − S2 , 21 }. If Nash equilibrium exists (i.e.
if S1 + S2 6 12 ) then both equilibrium prices coincide. However if EinSS exists and
Nash equilibrium does not exist (i.e. if S1 + S2 > 12 and (3) holds) both EinSS prices
p∗ = 1 − S1 − S2 are lower than the monopoly price pM = 12 . One can interpret
the price difference S1 + S2 − 21 as an additional cost to maintain security in the
situation when players take into account mutual threats of undercutting and behave
cautiously.
Proof. (1) Necessity. Let us consider profile (p∗ , p∗ ) and prove the conditions
2
(3). Suppose for example that p∗ < p̂(S2 ) ≡ 1−S
2 . Then player 1 can deviate
p∗1 → p̂. His payoff will increase since p∗ < p̂ 6 pM = 12 and u1 (p1 , p2 ) is strictly
increasing in p1 if p1 6 pM = 12 according to (2). Any retaliatory threat of player
2 according to (2) can not make the payoff of player 1 less than min u1 (p̂, p2 ) =
p2
min u1 (p̂, p2 ) = u1 (p̂, p2 )|p2 =p̂−0 = p̂ min{S1 , 1 − p̂ − S2 }. The payoff of player 1 in
p2 <p̂
138
Mikhail Iskakov, Alexey Iskakov
the initial profile does not exceed this value. Indeed p(1−p−S2 ) is strictly increasing
2
at p < p̂ = 1−S
and we have u1 (p∗ , p∗ ) 6 p∗ (1 − p∗ − S2 ) < p̂(1 − p̂ − S2 ) and
2
∗ ∗
∗
u1 (p , p ) 6 p S1 < p̂S1 . Therefore the deviation of player 1 into p̂(S2 ) is always a
secure deviation according to Definition 3. Hence profile (p∗ , p∗ ) is not an EinSS.
1
Symmetrically if p∗ < p̂(S1 ) ≡ 1−S
then player 2 can make a secure deviation into
2
∗ ∗
p̂(S1 ) and profile (p , p ) is not an EinSS either. The necessity of (3) is proven.
(2) Sufficiency. Let us now assume that (3) holds (i.e. p̂(S1 ) 6 p∗ and p̂(S2 ) 6
∗
p ). Consider an arbitrary deviation p∗ → p1 of player 1. If p1 < p∗ it can not be a
profitable deviation for player 1. Therefore
p1 > p∗ . Player 1 increases ∗the payoff if
∗
−S2
∗ ∗
∗
∗ 1−p −S2
and only if u1 (p , p ) = p S1 = p 1−p∗ (1−p∗ ) < u1 (p1 , p∗ ) = p1 1−p
1−p∗ (1−p1 ),
∗
∗
i.e. there must be p (1 − p ) < p1 (1 − p1 ). Then there is retaliatory threat of player
2 to deviate from profile (p1 , p∗ ) into profile arbitrarily close to (p1 , p1 − 0). From
p∗ S2 < p1 S2 and p∗ (1 − p∗ ) < p1 (1 − p1 ) it follows that player 2 increases the
payoff at this deviation. The payoff of player 1 in this profile is arbitrarily close to
u1 (p1 , p1 − 0) = p1 min{S1 , 1 − p1 − S2 }|p∗ <p1 = p1 (1 − p1 − S2 ). Since p(1 − p − S2 )
2
2
is strictly decreasing at p > p̂(S2 ) = 1−S
and p1 > p∗ > p̂(S2 ) = 1−S
then
2
2
∗ ∗
∗
∗
u1 (p , p ) = p (1 − p − S2 ) > p1 (1 − p1 − S2 ) = u1 (p1 , p1 − 0). Therefore the
deviation of player 1 into profile (p1 , p∗ ) is not a secure deviation. Symmetrically an
arbitrary deviation of player 2 is not a secure deviation either. No player can make
secure deviation in the profile (p∗ , p∗ ). By definition it is an EinSS. The sufficiency
of (3) is proven.
(3) Nash equilibrium condition. One can easily check that pM 6 p∗ (S1 +
S2 6 21 ) is the maximum condition of functions u1 (p1 ) = u1 (p1 , p∗ ) and u2 (p2 ) =
u2 (p∗ , p2 ) in the points p1 = p∗ and p2 = p∗ respectively. In other words it is a
condition of Nash equilibrium for the profile (p∗ , p∗ ).
(4) Uniqueness of EinSS. It follows from Proposition 2 and the uniqueness
of BSR-profile proven in Proposition 4. 5.
Solution in secure prices for the strictly concave receipt function
Obtained in the previous section results can be generalized to the more general case
of the strictly concave receipt function pD(p). Although the proofs become slightly
more involved they follow the similar logic.
Proposition 6. Let the receipt function pD(p) be strictly concave and reach its
maximum at pM . Then in the game of Bertrand-Edgeworth with payoff functions
(1) there is an EinSS (p∗ , p∗ ) where D(p∗ ) = S1 + S2 if and only if

arg max{p(D(p) − S1 )} 6 p∗
p>0
(4)
arg max{p(D(p) − S2 )} 6 p∗
p>0
If p∗ > pM it is a Nash equilibrium. There are no other EinSS in the game.
Remark. Since the receipt function pD(p) is strictly concave then the function
p(D(p) − S) at a given S is also strictly concave in p and reaches the unique maximum at p > 0. Therefore arg max{p(D(p) − S)} can be considered as a function of
p>0
S. The proof is given in (M.Iskakov, A.Iskakov, 2012b).
The condition (4) can be easily formulated in differential form.
Equilibrium in Secure Strategies in the Bertrand-Edgeworth Duopoly Model
139
Proposition 7. If function pD(p) is differentiable the condition (4) is equivalent
to
d
pD(p) 6 min{S1 , S2 }
(4’)
dp
p=p∗
d
Proof. One can easily check that p̂ = arg max{p(D(p)−S)} <=> dp
pD(p) =
p>0
p=p̂
d
S. Besides dp
pD(p) is strictly decreasing. Therefore p̂ 6 p∗
<=>
d
d
6 dp
pD(p) = S. Hence the equivalence of (4) and (4’).
dp pD(p) ∗
p=p
p=p̂
The obtained results are generally similar to the results obtained for the linear
demand function. In the EinSS all firms set equal prices such that market demand
equals total supply. If the equilibrium price exceeds or equals the monopoly price
this solution coincides with the well-known Nash equilibrium solution. However, in
the EinSS which is not Nash equilibrium the prices are lower than the monopoly
price. The corresponding difference in price can be interpreted as an additional cost
to maintain security in the situation when players behave cautiously and secure
themselves against mutual threats of undercutting.
6.
Conclusion
In this paper we considered the Bertrand-Edgeworth duopoly model, with capacity
constraints which may not possess a Nash-Cournot equilibrium. Whilst the existence
of mixed-strategy equilibrium was demonstrated by Dixon (1984), it has not proven
easy to characterize what the equilibrium actually looks like. It has been argued
that non-pure strategies are not plausible in the context of the Bertrand-Edgeworth
model. Therefore several alternative approaches have been proposed as a response to
the non-existence of pure-strategy equilibrium. Allen and Hellwig (1986b) proposed
a modification of the game, in which firms choose the quantity they are willing to sell
up to at each price. Dastigar (1995) proposed that firms have to meet all demand at
the price they set. Benassy (1989) introduced in the Bertrand-Edgeworth model the
product differentiation. Dixon (1993) explored the Bertrand-Edgeworth model with
”integer pricing” when firms cannot undercut each other by an arbitrarily small
amount. All these approaches in some sense or another change the setting of the
original game and introduce specific ad hoc modification to the Bertrand-Edgeworth
model.
As an alternative approach to analyze the Bertrand-Edgeworth duopoly model
we propose in this paper the concept of Equilibrium in Secure Strategies (EinSS).
We present a new intuitive formulation of EinSS concept from (M.Iskakov and
A.Iskakov, 2012b). An existence condition of the EinSS in the Bertrand-Edgeworth
price duopoly is formulated and proved, which allowed us to extend the set of NashCournot price equilibria in the game. If the equilibrium price exceeds or equals the
monopoly price this solution coincides with the Nash-Cournot equilibrium solution.
However, in the EinSS, which is not Nash-Cournot equilibrium the prices are lower
than the monopoly price. The corresponding difference in price can be interpreted
as an additional cost to maintain security when duopolists secure themselves against
the threats of being undercut.
140
Mikhail Iskakov, Alexey Iskakov
Although the proposed approach does not completely solve the problem of the
non-existence of Nash-Cournot equilibria in the Bertrand-Edgeworth model, it nevertheless provides some advantages. In contrast to the above mentioned ad hoc
equilibrium concepts developed in the context of the Bertrand-Edgeworth model,
the EinSS is a general equilibrium concept that proved to be effective and allowed
to find equilibrium positions in several well-known economic games that fail to have
Nash-Cournot equilibrium (M.Iskakov and A.Iskakov, 2012b). On the other hand
unlike equilibria in mixed strategies it offers a solution in an explicit form and can
be easily interpreted in terms of cautious behavior.
Acknowlegments. We are deeply indebted to C.d’Aspremont who helped us to
find an elegant and intuitively clear formulation of the EinSS concept and inspired
us to apply it to the Bertrand-Edgeworth duopoly model. We are thankful to
F.Aleskerov and D.Novikov for regular discussions on the subject of EinSS concept during seminars at Moscow High School of Economics and at V.A.Trapeznikov
Institute of Control Sciences. This work was supported by the research project
No.14-01-00131-a of the Russian Foundation for Basic Research.
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Equilibrium Strategies in Two-Sided Mate Choice Problem
with Age Preferences⋆
Anna A. Ivashko1 and Elena N. Konovalchikova2
Institute of Applied Mathematical Research,
Karelian Research Center of RAS,
Pushkinskaya str. 11, Petrozavodsk, 185910, Russia
E-mail: [email protected]
URL: http://mathem.krc.karelia.ru
2
Transbaikal State University,
Alekzavodskaya str. 30, Chita, 672039, Russia
E-mail: [email protected]
1
Abstract In the paper the two-sided mate choice model of Alpern, Katrantzi and Ramsey (2010) is considered. In the model the individuals from
two groups (males and females) want to form a couple. It is assumed that the
total number of unmated males is greater than the total number of unmated
females and the maximum age of males (m) is greater than the maximum
age of females (n). There is steady state distribution for the age of individuals. The aim of each individual is to form a couple with individual of
minimum age. We derive analytically the equilibrium threshold strategies
and investigate players’ payoffs for the case n = 3 and large m.
Keywords: mutual mate choice, equilibrium, threshold strategy.
1.
Introduction
In the paper the two-sided mate choice model of Alpern, Katrantzi and Ramsey
(2010) (Alpern et al., 2010) is considered. The problem is following. The individuals
from two groups (males and females) want to form a long-term relationship with
a member of the other group, i.e. to form a couple. Each group has steady state
distribution for the age of individuals. In the model males and females can form a
couple during m and n periods respectively. It is assumed that the total number of
unmated males is greater than the total number of unmated females and m ≥ n. The
discrete time game is considered. In the game unmated individuals from different
groups randomly meet each other in each period. If they accept each other, they
form a couple and leave the game, otherwise they go into the next period unmated
and older. It is assumed that individuals of both sexes enter the game at age 1 and
stay until they are mated or males (females) pass the age m (n). The initial ratio of
age 1 males to age 1 females is given. The payoff of mated player is the number of
future joint periods with selected partner. Payoff of a male age i and a female age
j if they accept each other is equal to min{m − i + 1, n − j + 1}. The aim of each
player is to maximize his/her expected payoff. In each period players use threshold
strategies: to accept exactly those partners who give them at least the same payoff
as the expected payoff from the next period.
⋆
This research is supported by Russian Fund for Basic Research (project 13-01-91158Γ ΦEH_ a, project 13-01-00033-a) and the Division of Mathematical Sciences of RAS
(the program "Algebraic and Combinatorial Methods of Mathematical Cybernetics and
New Information System").
Equilibrium Strategies in Two-Sided Mate Choice Problem with Age Preferences 143
In the literature such problems are called also marriage problems or job search
problems. We use here the terminology of ”mate choice problem”. In papers
(Alpern and Reyniers, 1999; Alpern and Reyniers, 2005; Mazalov and Falko, 2008)
the mutual mate choice problems with homotypic and common preferences are
investigated. In (Alpern et al., 2013) a continuous time model with age preferences is considered. Other two-sided mate choice models were considered in papers
(Gale and Shapley, 1962; Kalick and Hamilton, 1986; Roth and Sotomayor, 1992).
Alpern, Katrantzi and Ramsey (Alpern et al., 2010) derive properties of equilibrium threshold strategies and analyse the model for small m and n. The case n = 2
was considered in paper (Konovalchikova, 2012). In this paper using dynamic programming method we derive analytically the equilibrium threshold strategies and
investigate players’ payoffs for the case n = 3 and large m.
2.
Two-Sided Mate Choice Model with Age Preferences
Denote ai — the number of unmated males of age i relative to the number of females
of age 1 and bj — the number of unmated females of age j relative to the number
of females of age 1 (b1 = 1). The vectors of the relative numbers of unmated males
and females of each age a = (a1 , ..., am ), b = (b1 , ..., bn ) remain constant over time.
Denote the ratio of the rates at which males and females enter the adult popua1
lation by R, R =
= a1 .
b1
n
m
P
P
The total groups of unmated males and females are A =
ai , B =
bj .
A
Denote the total ratio
by r and assume that r > 1.
B
Consider the following probabilities:
i=1
i=1
ai
— the probability a female is matched with a male of age i,
A
B
–
— the probability a male is matched.
A
bj
–
— the probability a male is matched with a female of age j, given that a
B
male is mated.
bj
bj B
–
=
·
— the probability a male is matched with a female of age j.
A
B A
–
Denote Ui , i = 1, ..., m — the expected payoff of male of age i and Vj , j = 1, ..., n
— the expected payoff of female of age j.
Players use the threshold strategies F = [f1 , ..., fm ] for males and G = [g1 , ..., gn ]
for females, where fi = k, k = 1, ..., n — to accept a female of age 1, ..., k, gj =
l, l = 1, ..., m — to accept a male of age 1, ..., l.
3.
Model for n = 3 and m ≥ 3
Consider the two-sided mate choice model with age preferences for the case n = 3
and m ≥ 3.
The expected payoffs of male have the following form
144
Anna A. Ivashko, Elena N. Konovalchikova
V3 =
m−1
P
i=1
V2 =
m−2
P
i=1
V1 =
m−2
P
i=1
ai
am
I{fi = 3} +
≤ 1,
A
A
ai
am−1
am
2I{fi ≥ 2} +
2+
1 ≤ 2,
A
A
A
ai
am−1
am
3+
2+
max{1, V2 },
A
A
A
where I{C} is indicator of event C.
From this it follows that g3 = g2 = m. Also g1 = m, if V2 < 1 or g1 = m − 1, if
V2 ≥ 1.
There are three forms of strategies which are presented in the table:
G1 = [m − 1, m, m]
G2 = [m, m, m]
I. F1 = [1, ..., 1, 2, ..., 2, 3, ..., 3]
| {z } | {z } | {z }
k
l
II. F3 = [2, ..., 2, 3, ..., 3]
| {z } | {z }
m−k−l
k = 1, ..., m − 3, l = 1, ..., m − 3
k
m−k
k = 1, ..., m − 2
III. F2 = [1, ..., 1, 2, ..., 2, 3, ..., 3]
| {z } | {z } | {z }
k
l
m−k−l
k = 1, ..., m − 3, l = 1, ..., m − 3
Note that for female strategy G2 = [m, m, m] in the equilibrium male strategy
it should be f1∗ = 1.
Consider these strategies consequently.
I. Players use strategy profile (F1 , G2 ), where G2 = [m, m, m] (to accept any
partner), F1 = [f1 , ..., fm ] = [1, ..., 1, 2, ..., 2, 3, ..., 3], k = 1, ..., m− 3, l = 1, ..., m− 3.
| {z } | {z } | {z }
k
l
m−k−l
For equilibrium strategies the male’s payoff V2 =
am
1 must be less than 1. It is equivalent to
A
m−1
X
i=1
1−
1
r
i−1
+
m−2
P
i=1
ai
am−1
2I{fi ≥ 2} +
2+
A
A
i−1
k
X
1
2 1−
I{fi ≥ 2} < 0.
r
i=1
Consider the expected payoffs of males
Equilibrium Strategies in Two-Sided Mate Choice Problem with Age Preferences 145
Um =
b1
b2
b3
B
1
1+ 1+ 1+ 1−
0 = < 1,
A
A
A
A
r
Um−1 =
Um−2 =
b2
b3
1
2 b3
1
b1
2+ 2+ 1+ 1−
Um = −
+ 1−
Um < 2,
A
A
A
r
r
A
r
b1
b2
b3
1
3+ 2+
max{1, Um−1 } + 1 −
Um−1 < 3,
A
A
A
r
(1)
b1
b2
b3
1
3+ max{2, Um−i+1 }+ max{1, Um−i+1 }+ 1− Um−i+1 < 3,
A
A
A
r
i = 3, ..., m − 1.
Um−i =
From these expressions it follows that equilibrium strategies are equal to
∗
∗
= 3,
1) fm−1
= fm
∗
∗
2)fm−2 = 3, if Um−1 < 1; fm−2
= 2, if Um−1 ≥ 1,
∗
∗
∗
3) fm−i = 3, if Um−i+1 < 1; fm−i
= 2, if 1 ≤ Um−i+1 < 2; fm−i
= 1, if
2 ≤ Um−i+1 , i = 3, ..., m − 1.
The equilibrium strategies and the optimal payoffs are presented in the theorem.
Theorem 1. If players use the equilibrium strategy profile (F1∗ , G∗2 ), where G∗2 =
[m, m, m], F1∗ = [1, ..., 1, 2, ..., 2, 3, ..., 3], for certain values of k and l (k = 1, ..., m −
| {z } | {z } | {z }
k
l
m−k−l
3, l = 1, ..., m − 3) then the males’ optimal payoffs are equal to

Um = 1 − z,





Um−1 = 2 − z 2 − z,





Um−i = 3 − z i+1 − z i − z i−1 , i = 2, ..., m − 1,
(2)
the equilibrium age distributions are equal to
where z = 1 − 1/r.
a = (R, Rz, Rz 2, ..., Rz m−1 ); b = (1, 0, 0),
1
R=
,
(1 − z)(1 + z + z 2 + ... + z m−1 )
A = r = 1/(1 − z),
Proof. Let the players use strategy profile (F1 , G2 ), where G2 = [m, m, m], F1 =
[1, ..., 1, 2, ..., 2, 3, ..., 3], k = 1, ..., m − 3, l = 1, ..., m − 3.
| {z } | {z } | {z }
k
l
m−k−l
Then the age distributions are equal to
b = (1, 0, 0);
a = (R, a1 (1− 1r ), ..., am−1 (1− r1 )) or a = (R, R(1− r1 ), R(1− 1r )2 , ..., R(1− r1 )m−1 ).
m
n
P
P
Taking into account that Br = A, where A =
ai , B =
bj we get
i=1
R=
i=1
r
.
1 + (1 − 1/r) + (1 − 1/r)2 + ... + (1 − 1/r)m−1
146
Anna A. Ivashko, Elena N. Konovalchikova
Then we can rewrite the expected male’s payoffs (1) in the following recurrent form
(z = 1 − 1/r):
Um =
Um−1
1
= 1 − z,
r
2
1
= + 1−
Um = 2(1 − z) + zUm ,
r
r
Um−i =
3
1
+ 1−
Um−i+1 = 3(1 − z) + zUm−i+1 , i = 2, ..., m − 1.
r
r
Substituting each expression into the next one we get
Um = 1 − z,
Um−1 = 2(1 − z) + zUm = (1 − z)(2 + z),
Um−2 = 3(1 − z) + zUi+1 = (1 − z) 3 + z 2 + 2z ,
!
i−1
P j
Um−i = 3(1 − z) + zUm−i+1 = (1 − z) 3
z − z i−1 + z i , i = 3, ..., m − 1.
j=0
Simplifying the payoffs we obtain (2).
For the equilibrium females’ strategy G∗2 = [m, m, m] the equilibrium males’
strategy F1∗ can be obtained from the system

V2 < 1,




Um−1 < 1,




...




Uk+l+2 < 1,



1 ≤ Uk+l+1 < 2,
...




1 ≤ Uk+2 < 2,




Uk+1 ≥ 2,




...



U2 ≥ 2
for different value of r.
Example 1. For m = 4 and r = 2, we obtain a =
F1∗ = [1, 2, 3, 3], G∗2 = [4, 4, 4].
16 8 4 2
,
, ,
15 15 15 15
, b = (1, 0, 0),
Example 2. F1∗ = [1, ..., 1, 2, 3, 3] for r ∈ (1; 2.191] and m ≥ 4, where r∗ = 2.191 is
the solution of the equation 2r3 − 6r2 + 4r − 1 = 0
F1∗ = [1, ..., 1, 2, 2, 3, 3] for r ∈ [2.191; 2.618] and m ≥ 6, where ãäå r1∗ = 2.191 is
the solution of the equation 2r3 − 6r2 + 4r − 1 = 0, and r2∗ = 2.618 is the solution
of the equation r2 − 3r + 1 = 0,
F1∗ = [1, ..., 1, 2, 3, 3, 3] for r ∈ [2.618; 3.14] and m ≥ 6,
F1∗ = [1, ..., 1, 2, 2, 3, 3, 3] for r ∈ [3.14; 4.079] and m ≥ 7.
Equilibrium Strategies in Two-Sided Mate Choice Problem with Age Preferences 147
II. Consider the case when the female’s strategy is G1 = [m − 1, m, m] (V2 ≥ 1).
The expected males’ payoffs are equal to
b1
b2
b3
B
1 b1
0= −
< 1,
Um = 0 + 1 + 1 + 1 −
A
A
A
A
r
A
b1
b2
b3
1
2 b3
1
Um−1 = 2 + 2 + 1 + 1 −
Um = −
+ 1−
Um < 2
A
A
A
r
r
A
r
b1
b2
b3
1
Um−2 = 3 + 2 +
max{1, Um−1} + 1 −
Um−1 < 3,
A
A
A
r
b2
b3
1
b1
max{2, Um−i+1 } +
max{1, Um−i+1 } + 1 −
Um−i+1 < 3,
Um−i = 3 +
A
A
A
r
i = 3, ..., m − 1.
∗
∗
It follows that fm−1
= fm
= 3, and fi∗ = {1, 2, 3}, i = 1, ..., m − 2 depending on
the values of r.
Consider the case when the male’s strategy is F3 = [2, ..., 2, 3, ..., 3], k = 1, ..., m−
| {z } | {z }
k
m−k
2.
Theorem 2. If players use the equilibrium strategy profile (F3∗ , G∗1 ), where G∗1 =
[m − 1, m, m], F3∗ = [2, ..., 2, 3, ..., 3], for certain values of k (k = 1, ..., m − 2) then
| {z } | {z }
k
m−k
the males’ optimal payoffs are equal to
Um = 1 − z −
1
,
A
Um−1 = 2(1 − z) + zUm ,
am
am
1
i−1
Um−i = 3 − 2
− 1− 2
z
− 1+
z i − z i+1 ,
A (1 − z)
A (1 − z)
A
i = 2, ..., m − 2,
the equilibrium age distributions are equal to


 z m−1 
a = R, Rz, Rz 2, ..., Rz m−1 , b = 
, 0 .
1, m−1
P i 
z
i=0
1 + z + z 2 + ... + z m−2 + 2z m−1
R=
,
(1 − z)(1 + z + z 2 + ... + z m−1 )2
m−1
P i
A=R
z,
i=0
where z = 1 − 1/r.
Proof. The distributions for the age of males and females have form
148
Anna A. Ivashko, Elena N. Konovalchikova
a
m
a = R, Rz, Rz 2, ..., Rz m−1 , b = 1,
, 0 , where z = 1 − 1/r.
A
The ratio of the rates at which males and females enter the adult population has
form
1 + z + z 2 + ... + z m−2 + 2z m−1
, where z = 1 − 1/r.
(1 − z)(1 + z + z 2 + ... + z m−1 )2
am
We have that V2 = 2 −
≥ 1.
A
The expected payoffs have form
R=
Um =
Um−1
1
1
− ,
r
A
2
1
= + 1−
Um ,
r
r
Um−i =
or
Um = 1 − z −
1
am
3
+ 1−
Um−i+1 − 2 , i = 2, ..., m − 2
r
r
A
1
,
A
Um−1 = 2(1 − z) + zUm ,
am
Um−i = 3(1 − z) + zUm−i+1 − 2 =
A am
1
am
i−1
=3− 2
− 1− 2
z
− 1+
z i − z i+1 ,
A (1 − z)
A (1 − z)
A
i = 2, ..., m − 2, where z = 1 − 1/r
For the equilibrium female’s strategy G∗1 = [m − 1, m, m] the equilibrium males’
strategies F3∗ can be obtained from the system

Um−1 < 1,




...



Uk+1 < 1,
1 < Uk < 2,




...



1 < U2 < 2
for different value of r.
III. Finally, consider the case when the female’s strategy is G1 = [m − 1, m, m]
(V2 ≥ 1) and the male’s strategy is F2 = [1, ..., 1, 2, ..., 2, 3, ..., 3], k = 1, ..., m − 3,
| {z } | {z } | {z }
k
l
m−k−l
l = 1, ..., m − 3.
Then the expected payoff of female of age 2 is equal to
k a
P
am
i
V2 = 2 −
−2
and it must be greater then or equal to 1.
A
i=1 A
Then the distributions for the age of males and females have forms
Equilibrium Strategies in Two-Sided Mate Choice Problem with Age Preferences 149
k a
am am P
i
a = (a1 , ..., am ); b = 1,
,
,
A A i=1 A
where
a1 = R, ai= ai−1 (1 − 1/A),
i = 2, ..., k + 1,
1
b3
+1−
, j = k + 2, ..., k + l + 1,
aj = aj−1
A r
1
as = as−1 1 −
, s = k + l + 2, ..., m.
r
The expected payoffs of males are equal to
1
1
− ,
r
A
2 b3
1
+ 1−
Um ,
Um−1 = −
r
A
r
3 b2
b3
1
Ui = −
−2 + 1−
Ui+1 , i = k + l + 1, ..., m − 2,
r
A
A
r
3 b2
b3
1 b3
Uj = −
−3 + 1− +
Uj+1 , j = k + 1, ..., k + l,
r
A
A
A
r
3
b2
b3
1 b2
b3
Us = − 3 − 3 + 1 − +
+
Us+1 , s = 2, ..., k + 1.
r
A
A
r
A
A
Um =
For the equilibrium females’ strategy G∗1 = [m − 1, m, m] the equilibrium males’
strategy F2∗ can be obtained from the system

V2 ≥ 1,




Um−1 < 1,




...




Uk+l+2 < 1,



1 ≤ Uk+l+1 < 2,
...




1 ≤ Uk+2 < 2,




Uk+1 ≥ 2,




...



U2 ≥ 2.
In Table 1. the numerical results for the optimal threshold strategies are given
for different values of r.
Table 1: Equilibrium strategy for m = 5 for different values of r.
Equilibrium
([1, 1, 2, 3, 3], [5, 5, 5])
([1, 2, 3, 3, 3], [4, 5, 5])
([2, 2, 3, 3, 3], [4, 5, 5])
([2, 3, 3, 3, 3], [4, 5, 5])
([3, 3, 3, 3, 3], [4, 5, 5])
A
B
(1, 2.191]
[2.016, 2.79]
[2.85, 4.517]
[4.517, 6.87]
[6.87, +∞)
r=
R
(1, 1.049]
[1.081, 1.191]
[1.209, 1.560]
[1.560, 2.097]
[2.097, +∞)
Acknowlegments. The authors express their gratitude to Prof. V. V. Mazalov for
useful discussions on the subjects.
150
Anna A. Ivashko, Elena N. Konovalchikova
References
Alpern, S. and Reyniers, D. (1999). Strategic mating with homotypic preferences. Journal
of Theoretical Biology, 198, 71–88.
Alpern, S. and Reyniers, D. (2005). Strategic mating with common preferences. Journal of
Theoretical Biology, 237, 337–354.
Alpern, S. Katrantzi, I. and Ramsey, D. (2010). Strategic mating with age dependent preferences. The London School of Economics and Political Science.
Alpern, S. Katrantzi, I. and Ramsey, D. (2013). Partnership formation with age-dependent
preferences. European Journal of Operational Research, 225, 91–99.
Gale, D. and Shapley, L. S. (1962). College Admissions and the Stability of Marriage. The
American Mathematical Monthly, 69(1), 9–15.
Kalick, S.M. and Hamilton, T. E. (1986). The matching hypothesis reexamined. J. Personality Soc. Psychol., 51, 673–682.
Mazalov, V. and Falko, A. (2008). Nash equilibrium in two-sided mate choice problem.
International Game Theory Review. 10(4), 421–435.
Roth, A. and Sotomayor, M. (1992). Two-sided matching: A study in game-theoretic modeling and analysis. Cambridge University Press.
Konovalchikova, E. (2012). Model of mutual choice with age preferences. Mathematical
Analysis and Applications. Transbaikal State University, 10–25 (in Russian).
Stationary State in a Multistage Auction Model⋆
Aleksei Y. Kondratev
Institute of Applied Mathematical Research,
Karelian Research Center of RAS,
Pushkinskaya str. 11, Petrozavodsk, 185910, Russia
E-mail: [email protected]
Abstract We consider a game-theoretic multistage bargaining model with
incomplete information related with deals between buyers and sellers. A
player (buyer or seller) has private information about his reserved price.
Reserved prices are random variables with known probability distributions.
Each player declares a price which depends on his reserved price. If the
bid price is above the ask price, the good is sold for the average of two
prices. Otherwise, there is no deal. We investigate model with infinite time
horizon and permanent distribution of reserved prices on each stage. Two
types of Nash-Bayes equilibrium are derived. One of them is a threshold
form, another one is a solution of a system of integro-differential equations.
Keywords: multistage auction model, Nash equilibrium, integro-differential
equations for equilibrium, threshold strategies.
1.
Introduction
In (Mazalov and Kondratyev, 2012; Mazalov and Kondratyev, 2013) there was considered bargaining model with incomplete information, where a buyer and a seller
have an opportunity to make a deal at only one stage. In (Mazalov et al., 2012)
there was proposed auction model with finite number of steps. We fix time horizon n. A seller and a buyer meet each other at random. Reserved prices s and b
are independent random variables on interval [0, 1] with density functions f (s) and
g(b) accordingly. Seller asks Sk = Sk (s) ≥ s, buyer bids Bk = Bk (b) ≤ b on step
k = 1, 2, . . . n We have a deal on the average of the two prices (Sk (s) + Bk (b))/2
if Bk ≥ Sk . If there is no deal then agents go to the next step k + 1. Differential
equations for equilibrium strategies were found. In this paper we generalize and
research this auction model for case of infinite time horizon.
Let δ be discount factor. Consider a game with infinite time horizon. Let reserved
prices of sellers and buyers s and b at the stage i = 1, 2, . . . have density functions
fi (s), s ∈ [0, 1] and gi (b), b ∈ [0, 1] accordingly. At the stage i players use strategies
Si (s) and Bi (b). If there was a deal then the buyer b and the seller s get outcome
δ i−1 (b − B(b)+S(s)
) and δ i−1 ( B(b)+S(s)
− s) accordingly and in this case they do
2
2
not move to the next stage. Additionally let fix count of new agents appears in the
market at the each stage. We will study this model assuming that when i → ∞
and if agents act optimal then fi (s) and gi (b) tend to the limit density distribution
f (s) and g(b). Hence we research stationary state on the market, when distributions
⋆
This research is supported by Russian Fund for Basic Research (project 13-01-91158Γ ΦEH_ a, project 13-01-00033-a), the Division of Mathematical Sciences of RAS (the
program "Algebraic and Combinatorial Methods of Mathematical Cybernetics and New
Information System") and Strategic Development Program of PetrSU.
152
Aleksei Y. Kondratev
f (s) and g(b) are the same at each stage, i.e. new agents replace making a bargain
players.
2.
Integro-differential equations for Nash equilibrium
To find optimal strategies we count them as functions of reserved prices accordingly
S = S(s) and B = B(b). Let they are differentiable and strictly increasing. Then
inverse functions (differentiable and strictly increasing too) are U = B −1 and V =
S −1 , i.e. accordingly s = V (S) and b = U (B). There is a deal, if B > S. If there
is a deal then we have a deal price (S(s) + B(b))/2. Pay functions are (1) and (2).
Fix buyer’s strategy B(b) and derive the best response of the seller s.
Condition B(b) > S is equivalent to b > U (S). Outcome of the seller equals
Z1 Hs (S) =
U(S)
B(b) + S
− s g(b)db + δG(U (S))Hs (S),
2
1
Hs (S) =
1 − δG(U (S))
Z1 U(S)
B(b) + S
− s g(b)db.
2
(1)
Differentiating (1) with respect to S, we have first order condition
h 1 − G(U (S))
∂Hs (S)
1
′
=
−
(S
−
s)g(U
(S))U
(S)
·
∂S
(1 − δG(U (S)))2
2
Z1 i
B(b) + S
· (1 − δG(U (S))) +
− s g(b)db · δ · g(U (S))U ′ (S) ,
2
U(S)
and so we get integro-differential equation for equilibrium strategies (inverse functions) U (B), V (S)
1 − G(U (S))
2
− (S − V (S))g(U (S))U ′ (S) (1 − δG(U (S)))+
Z1
S
1
δ · g(U (S))U (S)
− V (S) (1 − G(U (S))) +
B(b)g(b)db = 0.
2
2
′
U(S)
The same way let S(s) be seller’s strategy. We find the best response of the
buyer b. His outcome is
Hb (B) =
VZ(B)
0
b−
S(s) + B
2
1
Hb (B) =
1 − δ + δF (V (B))
f (s)ds + δ(1 − F (V (B)))Hb (B),
VZ(B)
0
S(s) + B
b−
2
f (s)ds.
(2)
153
Stationary State in a Multistage Auction Model
Differentiating (2) with respect to B, we have first order condition
h
∂Hb (B)
1
F (V (B)) ′
=
(b−B)f
(V
(B))V
(B)−
·
∂B
(1−δ+δF (V (B)))2
2
· (1−δ+δF (V (B)))−
VZ(B)
b−
0
S(s)+B
2
i
f (s)ds · δ · f (V (B))V ′ (B) ,
and so we get the second integro-differential equation for equilibrium strategies
(inverse functions) U (B), V (S)
F (V (B)) · (1 − δ + δF (V (B)))−
2
VZ(B)
B
1
′
δ · f (V (B))V (B) · (U (B) − )F (V (B)) −
S(s)f (s)ds = 0.
2
2
(U (B) − B)f (V (B))V ′ (B) −
0
Now we have the system of equations for Nash equilibrium
∂U
=
∂t
(1−G(U ))(1−δG(U ))
,
R1
2g(U ) (t−V )(1−δG(U ))−δ( 2t −V )(1−G(U ))− δ2 B(b)g(b)db
(3)
F (V )(1−δ+δF (V ))
(4)
U
∂V
=
∂t
"
2f (V ) (U −t)(1−δ+δF (V
))−δ(U − 2t )F (V
)+ 2δ
RV
S(s)f (s)ds
0
#.
Functions U and V must satisfy U (a) = a, U (c) = 1, V (a) = 0, V (c) = c. From (3)
and (4) it is easy to find
U ′ (a) =
(1 − G(a))(1 − δG(a))
,
R1
δ
δ
2g(a) a(1 − δG(a)) − 2 a(1 − G(a) − 2 B(b)g(b)db
(5)
a
V ′ (c)=
F (c)(1−δ+δF (c))
.
Rc
2f (c) (1−c)(1−δ+δF (c))−δ(1− 2c )F (c)+ 2δ S(s)f (s)ds
(6)
0
To figure out marginal prices a and c assume that there exist finite derivative
V ′ (a) > 0 and density f (0) > 0. Using L’Hopital’s rule we derive
f (V )V ′ (1 − δ + δF (V )) + δF (V )f (V )V ′
t→a 2f (V )[(U ′ −1)(1−δ+δF (V ))+(U −t)δf (V )V ′ −δ(U ′ − 1 )F (V )
2
f (0)V ′ (a)(1 − δ)
V ′ (a)
=
=
,
2f (0)(U ′ (a) − 1)(1 − δ)
2(U ′ (a) − 1)
−δ(U − 2t )f (V )V ′ + 21 δtf (V )V ′ ]
V ′ (a)= lim
and so U ′ (a) = 1.5.
154
Aleksei Y. Kondratev
The same way assume that there exist finite derivative U ′ (c) > 0 and density
g(1) > 0. Using L’Hopital’s rule we derive
−g(U )U ′ (1 − δG(U )) − δ(1 − G(U ))g(U )U ′
t→c 2g(U )[(1−V ′ )(1−δG(U ))−(t−V )δg(U )U ′ −δ( 1 −V ′ )(1−G(U ))
2
U ′ (c)= lim
+δ( 2t
−V
)g(U )U ′
+
1
′
2 δtg(U )U ]
=
−g(1)U ′ (c)(1 − δ)
−U ′ (c)
=
,
′
2g(1)(1 − V (c))(1 − δ)
2(1 − V ′ (c))
and so V ′ (c) = 1.5.
So we find necessary condition for differentiable strictly increasing strategies
to be Nash equilibrium. The case of δ = 0 leads to single-stage auction model
researched in (Mazalov and Kondratyev, 2012).
Fig. 1: Equilibrium strategies
Fig. 2: Deal area (for theorem 1)
Theorem 1. If density functions g(b) and f (s) are continuous on [0, 1], 0 < f (0) <
+∞, 0 < g(1) < +∞. Derivatives 0 < S ′ (0), B ′ (1) < +∞ exist. Then differentiable
and strictly increasing strategies S(s) on [0, c] and B(b) on [a, 1] are Nash equilibrium, if they satisfy (3),(4) on interval (a, c), with respect to boundary conditions
U (a) = a, U (c) = 1, V (a) = 0, V (c) = c. Marginal prices a and c must be derived
from equations U ′ (a) = 1.5, V ′ (c) = 1.5, using (5), (6).
3.
Nash equilibrium with threshold strategies
We derive necessary and sufficient condition for threshold strategies to be Nash
equilibrium in the underlying
155
Stationary State in a Multistage Auction Model
Fig. 3: Threshold strategies
Fig. 4: Deal area (for theorem 2)
Theorem 2. If strategies S(s), B(b) are threshold with price of a deal a ∈ [0, 1],
i.e. S(s) = max{a, s}, B(b) = min{a, b}. Then it is Nash equilibrium if and only if
(∗) Hs=0 (S) on [0, a] has a maximum for S = a,
(∗∗) Hb=1 (B) on [a, 1] has a maximum for B = a.
Proof. The deal is made if seller’s reserved price s ∈ [0, a] and ask price S ∈ [s, a],
S ≤ B(b). Outcome of the seller (1) equals
 a

Z Z1 1
b
+
S
a
+
S

Hs (S) =
− s dG(b) +
− s dG(b) =
1 − δG(S)
2
2
a
S


Za
1
S
1
a

=
− s (1 − G(S)) +
bdG(b) + (1 − G(a)) . (7)
1 − δG(S)
2
2
2
S
It is easy to check that
Hs (S) = Hs=0 (S) +
(1 − δ)s
s
− ,
δ(1 − δG(S)) δ
and in respect that G(S) is increasing, from (∗) we have a result that for any
s ∈ [0, a] seller’s outcome has maximum point S = a.
156
Aleksei Y. Kondratev
We can hold the same reasoning for buyers. The deal is made if buyer’s reserved
price b ∈ [a, 1], and a bid price B ∈ [a, b], B > S(s). From (2) we find his outcome
Hb (B)


ZB
1
 b − B F (B) − 1 sdF (s) − a F (a) .
1 − δ + δF (B)
2
2
2
=
Note that
Hb (B) = Hb=1 (B) −
(8)
a
(1 − b)
(1 − b)(1 − δ)
+
,
δ
δ(1 − δ + δF (B))
and in respect that F (B) is increasing, from (∗∗) we get that for any b ∈ [a, 1]
buyer’s outcome has a maximal value for B = a.
Theorem 3. If distribution functions F (s), G(b) have piecewise-continuous and
limited density functions f (s) ≤ L on [a, 1] and g(b) ≤ M on [0, a], then in theorem
2 for (∗) it is sufficient that
2
δ ≥1−
(1 − G(a))
,
2aM
(9)
F 2 (a)
.
2(1 − a)L
(10)
and condition (∗∗) is true if
δ ≥1−
Proof. At the points of continuity for g(b), differentiating the (7), we derive that
′
Hs=0
(S) =
1
2(1 − δG(S))
2
h
1 − G(S) − 2Sg(S) − δG(S) + δG2 (S)+
+ δSg(S)G(S) + δg(S)
Za
S
i
bg(b)db + δag(S) − δaG(a)g(S) + δSg(S) . (11)
Using that
g(S)
Za
S
bg(b)db ≥ g(S)
Za
Sg(b)db = Sg(S)(G(a) − G(S)),
S
substituting δ = 1 − (1 − δ), it is easy to check that in (11) expression in square
brackets is not less than
2
(1 − G(S)) + g(S)(a − S)(1 − G(a)) − (1 − δ)(−G(S) + G2 (S)+
+ (S − a)g(S)G(a) + (a + S)g(S)) ≥
further as S ≤ a and (9) it results that
2
2
≥ (1 − G(S)) − (1 − δ)(a + S)g(S)) ≥ (1 − G(a)) − (1 − δ)2aM ≥ 0.
′
Hence we prove that derivative Hs=0
(S) is nonnegative on the interval [0, a], so it
leads to (∗).
Stationary State in a Multistage Auction Model
157
At the points of continuity for f (s), differentiating the (8), we find that
′
Hb=1
(B) =
1
2(1 − δ + δF (B))
− δBf (B)F (B) + δf (B)
2
h
ZB
a
− (1 − δ)F (B) + 2(1 − δ)f (B) − δF 2 (B)+
i
sf (s)ds + δaF (a)f (B) − 2(1 − δ)Bf (B) . (12)
Noting that
f (B)
ZB
a
sf (s)ds ≤ f (B)
ZB
a
Bf (s)ds = Bf (B)(F (B) − F (a)),
substituting δ = 1 − (1 − δ), we calculate that in (12) expression in square brackets
is not less than
2(1 − δ)(1 − B)f (B) − (1 − δ)(F (B) − F 2 (B)) − F 2 (B) − δ(B − a)f (B)F (a) ≤
further as B ≥ a and (10) we get that
≤ 2(1 − δ)(1 − B)f (B) − F 2 (B) ≤ 2(1 − δ)(1 − a)L − F 2 (a) ≤ 0.
′
We prove that derivative Hb=1
(B) not positive on [a, 1], and this fact implies (∗∗).
Threshold strategies and deal area are on pic. 3 and pic. 4. Theorem 3 shows
that for any price a ∈ (0, 1), with limited density functions f (s), g(b) and discount
factor δ close to 1 then Nash equilibrium with threshold strategies exists.
Example 1. Let consider a situation of uniform distribution for reserved prices on
the interval [0, 1], i. e. F (s) = s, G(b) = b. As f (s) = 1, g(b) = 1, so in the theorem
2
a2
3 we can set L = M = 1. By (9), (10) we find that if δ ≥ max{1 − (1−a)
2a , 1 − 2(1−a) }
then threshold strategies with price a ∈ (0, 1) are Nash equilibrium. For a = 21 we
get sufficient condition (by using theorem 3) δ ≥ 34 .
Now we calculate explicit minimal value for discount factor δ when it is Nash
equilibrium with threshold at price a = 12 . Derivative of outcome (11) is
1
3 2 3
1 1 2 1
′
Hs=0 (S) =
δS − S + − δa + δa ,
2
2 4
2
(1 − δS)2 4
solving appropriate inequality we derive that for
√
3 − 9 − 6δ + 3δ 2 a2 − 6δ 2 a
S≤
3δ
derivative of seller’s outcome is nonnegative. Hence, we have necessary and sufficient
condition
√
3 − 9 − 6δ + 3δ 2 a2 − 6δ 2 a
.
a≤
3δ
From where we find
"
#
√
3 − δ − δ 2 − 10δ + 9
a ∈ 0,
.
2δ
158
Aleksei Y. Kondratev
The same reasoning for buyers leads to condition
"
#
√
3δ − 3 + δ 2 − 10δ + 9
,1 ,
a∈
2δ
and finally we find that for δ ≥ 23 threshold strategies with price a =
equilibrium in multistage auction model.
4.
1
2
are Nash
Conclusion
We offer multistage double closed auction model. Distribution of reserved prices are
common knowledge. On every stage pairs of agents with different reserved prices
are randomly selected. After then they decide to make a deal or no deal. In classical
version (Chatterjee and Samuelson, 1983) it is single-stage process. In our model if
there is no deal then agents move to the next stage. Outcome is discounted.
We find Nash equilibrium in the model. Strategies are functions of reserved
prices. Assuming the existence of stationary state for distribution of reserved prices
from stage to stage, we research criteria for strategies to be Nash equilibrium. In
theorem 1 we prove criteria for equilibrium in class of strictly increasing strategies,
and in theorem 2 in class of threshold strategies.
References
Chatterjee, K. and Samuelson, W. (1983). Bargaining under incomplete informationg. Operations Research, 31(5), 835–851.
Mazalov, V. V. and Kondratyev, A. Y. (2012). Bargaining model with incomplete information. Vestnik St. Petersburg University. Ser. 10, 1, 33-Ű40.
Mazalov, V. V. and Kondratev, A. Y. (2013). Threshold strategies equilibrium in bargaining
model. Game theory and applications, 5(2), 46–63.
Mazalov, V. V., Mentcher, A. E. and Tokareva, J. S. (2012). Negotiations. Mathematical
theory. Lan. Saint-Petersburg. 304 P.
Mazalov, V. V. and Tokareva, J. S. (2011). Equilibrium in bargaining model with nonuniform distribution for reservation prices. Game theory and applications. 3(2), 37–49.
Myerson, R. and Satterthwait, M. A. (1983) Efficient mechanisms for Bilateral Trading.
Journal of Economic Theory, 29, 265–281.
Myerson, R. (1984) Two-Person Bargaining Problems with Incomplete Information. Econometrica, 52, 461–487.
Phenomenon of Narrow Throats of Level Sets
of Value Function in Differential Games⋆
Sergey S. Kumkov and Valerii S. Patsko
Institute of Mathematics and Mechanics,
S.Kovalevskaya str., 16, Ekaterinburg, 620990, Russia;
Institute of Mathemaics and Computer Sciences, Ural Federal University
Turgenev str., 4, 620083, Ekaterinburg, Russia
E-mail: [email protected] [email protected]
Abstract A number of zero-sum differential games with fixed termination
instant are given, in which a level set of the value function has one or more
time sections that are almost degenerated (have no interior). Presence of
such a peculiarity make very high demands on the accuracy of computational algorithms for constructing value function. Analysis and causes of
these degeneration situations are important during study of applied pursuit
problems.
Keywords: linear differential games, fixed termination instant, level sets of
value function, geometric methods, narrow throats
1.
Introduction
During investigating zero-sum differential games, the main topic is constructing and
studying the value function of the game. One of the traditional approaches to value
function construction is to solve the corresponding Hamilton–Jacobi–Bellman–Isaacs partial differential equation. Another approach is based on the representation
of the value function as a collection of its level sets (Lebesgue sets). These sets are
built by means of a geometric method.
This representation is the most intuitive when the phase vector of the game
is two-dimensional or when the game can be reduced to such a situation. In this
case, any level set is located in a three-dimensional space time × two-dimensional
phase space and can be effectively constructed and visualized to graphic study of
its structure and peculiarities. The result of constructions is often a collection of
polygons that approximate its time sections (t-sections) on some time grid.
A very important thing both from theoretic and numerical points of view is loss
of interior by t-sections of a level set at some instant. Further its evolution (in the
backward time) can lead to complete degeneration of the set (its t-sections become
empty), or can bring back the interior. The last case corresponds to the situation
when we say that the level set has a narrow throat.
Earlier, the authors have investigated appearance of narrow throats in linear differential game with fixed termination instant and terminal convex payoff function
(Kumkov et al., 2005). That game appears during study an interception problem
of one weak-maneuvering object by another one. This paper contains a number of
⋆
This work was supported by the Russian Foundation for Basic Research (projects nos.
12-01-00537 and 13-01-96055), by the Program “Dynamic systems and control theory”
of the Presidium of the RAS (project no.12-Π-1-1002), and by the Act 211 Government
of the Russian Federation 02.A03.21.0006
160
Sergey S. Kumkov, Valerii S. Patsko
examples of another games with convex payoff function, in which there are narrow
throats. Also, we consider games having non-convex payoff. They arise from a pursuit game with two pursuers and one evader. The study is made numerically by
algorithms and programs worked out by the authors.
2.
Games with Convex Payoff Function
2.1. Problem Formulation
Let us consider a zero-sum linear differential game (Krasovskii and Subbotin, 1974;
Krasovskii and Subbotin, 1988):
ż = A(t)z + B(t)u + C(t)v,
t ∈ [t0 ; T ], z ∈Rn , u ∈ P ⊂ Rp , v ∈ Q ⊂ Rq ,
ϕ zi (T ), zj (T ) → min max .
u
(1)
v
The first player governs the control u and minimizes the payoff ϕ; the second player
choosing its control v maximizes the payoff. The sets P and Q that constrain the
players’ controls are convex compacta in their spaces. The payoff function ϕ depends
on values of two components of the phase vector at the termination instant and is
convex.
It is necessary to construct level sets of the value function and study them from
the point of view of narrow throat presence.
2.2.
Equivalent Differential Game
A standard approach to study linear differential games with fixed termination instants and payoff function depending on a part of phase coordinates at the termination instant assumes a passage to a new phase vector; see, for example, (Krasovskii
and Subbotin, 1974; Krasovskii and Subbotin, 1988). These new variables are regarded as the values of the target components forecasted to the termination instant under zero players’ controls. Often they are called zero effort miss coordinates
(Shima and Shinar, 2002; Shinar and Shima, 2002). In our case, we pass to new coordinates x1 and x2 , where x1 (t) is the value of the component zi forecasted from
the current instant t to the termination instant T , and x2 is the forecasted value of
the component zj .
To obtain constructively the forecasted values, one uses a matrix combined of
two rows of the fundamental Cauchy matrix X(T, t) for the system ż = A(t)z. These
rows correspond to the target components of the phase vector. In our case, we use
the ith and jth rows of the Cauchy matrix. The change of variables is described
by the formula x(t) = Xi,j (T, t)z(t). (The subindices i, j of the matrix X denote
taking the corresponding rows of the fundamental Cauchy matrix.)
The equivalent game has the following form:
ẋ = D(t)u + E(t)v,
t ∈ [t0 ; T ], x ∈ R2 , u ∈ P, v ∈ Q, ϕ x1 (T ), x2 (T ) ,
D(t) = Xi,j (T, t)B(t), E(t) = Xi,j (T, t)E(t).
(2)
Further to analyze the evolution in time of the time sections of the level sets of
the value function, it is useful to involve the sets P(t) = D(t)P , Q(t) = E(t)Q, which
are called vectograms of the players at the instant t. The sense of the vectograms
is the collection of velocities that can be given to the system by the players at the
Phenomenon of Narrow Throats of Level Sets
161
corresponding time instant. If one has that Q(t) ⊂ P(t), then it can be said that
at the instant t the first player has (dynamic) advantage. In the case of opposite
inclusion, we say about advantage of the second player.
2.3.
Numerical Construction of Level Sets
Fix a value c and describe construction of an approximation of the level set Wc of
the value function V of game (2). The set will correspond to the chosen constant c.
For a numerical construction, at first, let fix a time grid {tj }, t0 < t1 < . . . <
tN = T . The constructions are made in the backward time from the termination instant T . Let at some instant tj+1 we have an approximation Wc (tj+1 ) of the t-section
Wc (tj+1 ) of the level set Wc . Then the approximation Wc (tj ) of the t-section Wc (tj )
is described by the following formula (Pschenichnyi and Sagaidak, 1970):
∗
Wc (tj ) = Wc (tj+1 ) + (−∆j )D(tj )P −
∆j E(t)Q.
(3)
Here, ∆j = tj+1 − tj ; D(tj ) and E(tj ) are the matrices from dynamics (2) computed
at the instant tj ; P and Q are the sets constraining the controls of the first and
second players. The sign “+” denotes the operation of algebraic sum (Minkowski
∗ denotes the geometric difference (Minkowski difference).
sum), and “ −”
The initial set Wc (T ) for the procedure is taken asa convex polygon Mc close
in the Hausdorff metrics to the convex level set Mc = (x1 , x2 ) : ϕ(x1 , x2 ) ≤ c of
the payoff function. Convexity of the set Mc is due to the convexity of the payoff
function.
It is known that in linear differential games with fixed termination instant,
convexity of the target set provides convexity of all t-sections Wc (tj ) of the corresponding solvability set (the maximal stable bridge). Therefore, in procedure (3) we
can apply algorithms for processing convex sets. In iteration procedures suggested
by one of the authors (Isakova et al., 1984; Kumkov et al., 2005), convex sets in
the plane are described by their support functions. (There is a one-to-one correspondence between
a convex
compact non-empty set S and its support function
ρ(l; S) = max hl, si : s ∈ S , which is positively-homogeneous; here, h·, ·i denotes a
dot product.) With that, to construct the support function of Minkowski sum of two
sets we should just construct the sum of the support functions of the summands. To
obtain the support function of Minkowski difference of two sets, it is necessary to
build convex hull of difference of support functions of the initial sets. Also there is
a very helpful fact that the support function of a convex polygon is piecewise-linear
with areas of linearity in the cones between outer normals to its neighbor edges.
Due to all these properties, it is possible to suggest effective procedures for addition
of sets, subtraction, and convex hull construction.
During the backward constructions, the current section Wc (tj+1 ) is summed
with the dynamic capabilities (−∆j )D(tj )P = (−∆j )P(tj ) of the first player and
further subtracted by dynamic capabilities ∆j E(tj )Q = ∆j Q(tj ) of the second
player. Thus, the change of the t-section is connected to the correlation of the
players’ vectograms. If the first player’s vectogram is “greater” than the vectogram
of the second one (that is, if the first player has advantage), then the t-section
grows in the backward time. In the opposite situation when the second player has
advantage, vice versa, the section contracts in the backward time. If neither P(t) ⊂
Q(t), nor Q(t) ⊂ P(t), then the first player has advantage in some directions and
disadvantage in others. Studying the situation of advantage of one or other player
162
Sergey S. Kumkov, Valerii S. Patsko
allows to estimate qualitatively the evolution of the level set in time without its
exact construction.
2.4.
Examples
One-to-One Interception Problem. In the works (Shinar et al., 1984; Shinar
and Zarkh, 1996; Melikyan and Shinar, 2000), a three-dimensional problem of interception in near space or upper atmosphere is considered. The pursuer P is an
intercept-missile; the evader E is a weak maneuverable target (for example, another
missile or a large aircraft). The geometry of the interception is drawn in Fig. 1. The
three-dimensional problem reduces naturally to a two-dimensional one. The longitudinal velocities of the objects are rather large, and the approach time is small.
Thus, the control accelerations aP and aE that are orthogonal to the current velocity of the corresponding object cannot turn significantly the velocity vectors. Due
to this, the longitudinal motion of the objects can be considered as uniform. Also,
the minimal approach distance, which is the natural payoff in this game, can be
changed by the lateral distance at the instant of nominal longitudinal passage of
the objects. This instant is fixed as the termination one.
The three-dimensional geometric coordinates can be introduced as it is shown in
Fig. 1. The origin O is put at the position of the pursuer P . The axis OX coincides
with the nominal line-of-sight. The axis OY is orthogonal to OX and is located
in the plane defined by the vectors of the nominal velocities of the objects. The
axis OZ is orthogonal to OX and OY .
After excluding the longitudinal motion along the axis OX from consideration,
we pass to a two-dimensional problem of lateral motion in the plane OY Z. The
control of the evader defines its acceleration directly; the pursuer has a more complicated dynamics. Its control affects the acceleration through a link of the first
order:
r̈P = F,
t ∈ [0; T ], rP , rE ∈ R2 , u ∈ P, v ∈ Q,
(4)
Ḟ = (u − F )/lP ,
rP (T ) − rE (T ).
ϕ
x(T
),
y(T
)
=
r̈E = v,
Here, rP and rE are the radius-vectors of the positions of the pursuer and evader in
the plane OY Z; lP is the time constant that describes the inertiality of servomechanisms transferring the control command signal u to the acceleration F ; v is the
Fig. 1: The geometry of the three-dimensional interception. The actual realizations of the
velocity vectors VP (t) and VE (t) are close to the nominal values (VP )col and (VE )col
163
Phenomenon of Narrow Throats of Level Sets
evader’s control; T is the termination instant coinciding with the instant of longitudinal passage of objects along the nominal motions. The sets P and Q constraining
the controls of the players are ellipses. These ellipses are obtained after projection of
the original round vectograms on accelerations (that are orthogonal to the nominal
velocities (VP )col and (VE )col ) into the plane OY Z. The parameters of the ellipse
(the semiaxes) are defined by the maximal acceleration of the corresponding object
(aP or aE ) and by the angle between the vector of its velocity and the line-of-sight
((χP )col or (χE )col ).
To pass to a standard game with the payoff depending on two components of
the phase vector, we use the following change of variables:
z1
z3
z5
z7
= (rP )Y − (rE )Y ,
= (ṙP )Y ,
= (ṙP )Z ,
= (r̈P )Y ,
z2
z4
z6
z8
= (rE )Z − (rE )Z ,
= (ṙE )Y ,
= (ṙE )Z ,
= (r̈P )Z .
(5)
In this case, the payoff function (which is the lateral miss) depends on the values
of z1 and z2 at the instant T :
q
ϕ z1 (T ), z2 (T ) = z12 (T ) + z22 (T ).
Proceeding to a two-dimensional equivalent game, we obtain the dynamics
ẋ = D(t)u + E(t)v,
t ∈ [0; T], x∈ R2 , u ∈pP, v ∈ Q,
ϕ x(T ) = x(T ) = x21 (T ) + x22 (T ),
where
D(t) = ζ(t) · I2 ,
E(t) = η(t) · I2 ,
ζ(t) = (T − t) + lP e−(T −t)/lP − lP ,
(6)
η(t) = −(T − t),
and I2 is a unit 2 × 2 matrix. The sets P and Q are
1/ cos2 (χP )col 0
u ∈ P = u : u′
u ≤ a2P ,
0
1
1/ cos2 (χE )col 0
v ∈ Q = v : v′
v ≤ a2E .
0
1
Example 1. Below, we give the results (Kumkov et al., 2005) of numerical study of
problem (4). The following parameters have been used: lP = 1.0,
u21
u22
v12
v22
2
2
2
+
≤ 1.30 , Q = v ∈ R :
+
≤1 .
P = u∈R :
0.672 1.002
0.712 1.002
Using notations of the original formulation, we have
|VE |
= 1.054,
|VP |
aP
= 1.3,
aE
cos χP = 0.67,
cos χE = 0.71,
lP = 1.
164
Sergey S. Kumkov, Valerii S. Patsko
Fig. 2: Example 1. A general view of the level set of the value function with a narrow throat
Fig. 3: A large view of the narrow throat
In Fig. 2, one can see a general view of the level set Wc computed for c = 2.391,
which is a bit greater than the critical one (that is, to the one corresponding to
the level set, which t-section has no interior at some instant). The main interesting
properties of this tube is that it has the narrow throat and that the direction of
elongation of t-sections changes near the throat. A large view of the narrow throat
is given in Fig. 3. Such a complicated shape of the throat is conditioned by the
process of passage of the advantage from the second player to the first one in this
time interval.
165
Phenomenon of Narrow Throats of Level Sets
This example was computed in the time interval τ ∈ [0; 7]. Here and below,
τ = T − t denotes the backward time. The time step ∆ equals 0.01. The level sets
of the payoff function (that are rounds) and the ellipses of the constraints for the
players’ controls are approximated by 100-gons.
Generalized L.S.Pontryagin’s Test Example. In work (Pontryagin, 1964), the
following differential game
ẍ + αẋ = u,
(7)
ÿ + β ẏ = v.
was taken as an illustration to the theoretic results. Here, α and β are some positive
constants; x, y ∈ Rn ; kuk ≤ µ, kvk ≤ ν. The termination of the game happens when
the coordinates x, y of the objects coincide. The first player tries to minimize the
duration of the game, the second one hinders this. Later, differential games with dynamics (7) and termination conditions depending only on the geometric coordinates
of the objects were called in Russian mathematical literature as “L.S.Pontryagin’s
test example”.
Another well-known example with the dynamics
ẍ = u,
(8)
ẏ = v
and constraints for the player’s controls kuk ≤ µ, kvk ≤ ν was called by L.S.Pontryagin (Pontryagin and Mischenko, 1969) as game “boy and crocodile”. The first
player (the “crocodile”) controls its acceleration and tries to catch the second one
(the “boy”) to some neighborhood. The second player is more maneuverable because
it controls its velocity.
Game (8) is a particular case of the game “isotropic rockets” (Isaacs, 1965),
which dynamics is
ẍ + k ẋ = u,
ẏ = v.
(9)
In works (Pontryagin, 1972; Mezentsev, 1972; ?; Nikol’skii, 1984; Grigorenko, 1990;
Chikrii, 1997), problems with dynamics more complicated than (7), (8), (9) were
studied:
x(k) + ak−1 x(k−1) + · · · + a1 ẋ + a0 x = u,
y
(s)
+ bs−1 y
(s−1)
+ · · · + b1 ẏ + b0 y = v,
u ∈ P,
v ∈ Q.
(10)
(11)
Games having dynamics (10), (11) and termination conditions depending only
on the geometric coordinates x, y, are often called “generalized L.S.Pontryagin’s
test example”. In this paper,
let us assumethat the payoff function is defined by
the formula ϕ x(T ), y(T ) = x(T ) − y(T ). Also, let us count that x, y ∈ R2 .
A variable change similar to (5)
z1 = x1 − y1 ,
z3 = ẋ1 ,
............
(k−1)
z2k−1 = x1
,
z2k+1 = ẏ1 ,
............
(s−1)
z2(k+s)−3 = y1
,
z2 = x2 − y2 ,
z4 = ẋ2 ,
............
(k−1)
z2k = x2
,
z2k+2 = ẏ2 ,
............
(s−1)
z2(k+s)−2 = y2
,
166
Sergey S. Kumkov, Valerii S. Patsko
transforms system (10), (11) to standard form (1):
ż = Az + Bu + Cv,
z ∈ R2(k+s)−2 , u ∈ P, v ∈ Q,
with the matrices A, B, and C that do not p
depend on the time. The payoff function
is terminal and convex: ϕ z1 (T ), z2 (T ) = z12 (T ) + z22 (T ).
There can be other variants of the change, which are more convenient in particular situations, but all of them assume introduction of relative geometric coordinates
(z1 , z2 in our case).
When some experience had been accumulated in numerical study of level sets
with narrow throats in the case of problem from works (Shinar et al., 1984; Shinar
and Zarkh, 1996; Melikyan and Shinar, 2000), the author decided to construct another examples with narrow throats in the framework of games with the dynamics
of the generalized L.S.Pontryagin’s test example.
The most interesting results of constructing level sets of the value function for the
generalized L.S.Pontryagin’s test example are when at least one of the sets P and Q
is not a round (since the level sets of the payoff, which is distance between objects
at the termination instant, are rounds, we need something that destroys uniformity
of the sets). So, let us take the sets P and Q as ellipses with center at the origin
and main axes parallel to the coordinate axes. Then the players’ vectograms P(t)
and Q(t) for all instants are ellipses homothetic to the ellipses P and Q respectively.
As it becomes clear from the previous example, a narrow throat appears when
there is a change of advantage of players. Namely, at the initial period of the backward time the second player should be stronger to contract t-section of level sets.
Then the advantage should pass to the first player to allow him to expand the sections. The easiest way to obtain such a change of advantage is to assign an oscillating
dynamics to one or both players.
The most illustrative way to study the passages of the advantage
is to investigate
tubes of vectograms, that is the sets P = (t, u) : u ∈ P(t) , Q = (t, v) : v ∈ Q(t) .
If one of the tubes includes the other in some period of time, then in this period
the corresponding player has complete advantage.
Example 2. The dynamics is the following:
ẍ + 2 ẋ = u,
ÿ + 0.2 ẏ + y = v,
x, y ∈ R2 ,
u ∈ P,
v ∈ Q.
Here, the first player controls an inertial point in the plane. The second object
is a two-dimensional oscillator. Both objects have a friction proportional to their
velocities. The controls are constrained by the ellipses
u21
u22
v12
v22
2
2
P = u∈R :
+
≤1 ,
Q= v∈R :
+
≤1 .
0.82
0.42
1.52 1.052
The tubes of vectograms appearing in this example are shown in Fig. 4a. Since
the dynamics of the second player describes an oscillating system, the advantage
passes from one player to another several times. At the beginning of the backward
time, the second player has the advantage, but later after a number of passes, the
advantage comes to the first player. An enlarged fragment of the tubes can be seen
in Fig. 4 b.
Phenomenon of Narrow Throats of Level Sets
167
a)
b)
Fig. 4: Example 2. Two views of the vectogram tubes. Number 1 denotes the tube of the
first player (the set P), number 2 corresponds to the second player’s vectogram tube (the
set Q)
Fig. 5 shows a level set Wc for c = 2.45098. This level set breaks (that is, is finite
in time and has empty t-sections from some instant of the backward time). Before
the break, orientation of elongation of the t-sections of Wc (t) changes. Namely,
before the last contraction of the tube, the sections are elongated vertically, and
after it the elongation is horizontal. As in the example in the previous subsection,
this change is due to delicate interaction of the vectogram tubes P(t) and Q(t) in
the time interval of the narrow throat.
If to increase the value of c, the length in time of the level sets grows jump-like.
The level set for c = 2.45100 can be seen in Fig. 6. In Fig. 7, its enlarged fragment is
given, which is near the narrow throat at τ = 11.95. This value of c can be regarded
as critical: for c < 2.45100 level sets break, for c ≥ 2.45100 they are infinite in time.
More exact reconstruction of the level sets corresponding to values c close to the
critical one needs a very accurate computations.
This example was computed in the time interval τ ∈ [0; 20]. The time step
is ∆ = 0.05. The round level sets Mc of the payoff function and the ellipses P
and Q were approximated by 100-gons.
168
Sergey S. Kumkov, Valerii S. Patsko
Fig. 5: Example 2. A broken level set close to the critical one, c = 2.45098
Fig. 6: Example 2. A general view of the level set with a narrow throat, c = 2.45100
Fig. 7: A large view of the narrow throat
169
Phenomenon of Narrow Throats of Level Sets
Example 3. To get an example with a level set of the value function with more than
one narrow throat, we should choose players’ dynamics to provide multiple passage
of advantage in such a way that each of players has it for a quite long time (to allow
the second player contract t-section almost to nothing). The most reasonable way
to get such a situation is to put an oscillating dynamics to both players.
Let the dynamics be the following:
ẍ − 0.025 ẋ + 1.3 x = u,
ÿ + y = v,
x, y ∈ R2 ,
u ∈ P,
v ∈ Q.
Constraints for the players’ controls are equal ellipses:
v12
v22
2
P =Q= v∈R :
+
≤1 .
1.52
1.052
Since the sets P and Q constraining the players’ controls are equal, then at any
instant the players’ vectograms P(t) and Q(t) are homothetic.
In Figs. 8a and 8b, the tubes of players’ vectograms are shown. The difference
of these figures is that in Fig. 8b the second player’s tube is transparent.
Fig. 9 contains a general view of the level set Wc for c = 1.2. In Fig. 10, there is
an enlarged fragment of the set near the first (in the backward time) narrow throat.
The instants of the backward time of the most thin parts of the set are τ1 = 5.65
and τ2 = 8.50.
The level set has been computed in the time interval τ ∈ [0; 16]. The time
step is ∆ = 0.05. Near the narrow throats, the time step was ten times smaller:
∆′ = 0.005. Again, the approximating polygons for the constraints for the controls
and for the payoff level set have 100 vertices.
Note again that despite the players’ vectograms are homothetic, the t-sections of
the level set and the vectograms are not. Absence of this homothety leads to complicated shape of the t-sections of the level set of the value function and, therefore,
complicated shape of narrow throats.
Example 4. The dynamics of this example is described by the relations
ẍ + 0.025 ẋ + 1.2 x = u,
ÿ + 0.01 ẏ + 0.85 y = v,
The constraints are taken as follows:
u2
u2
P = u ∈ R2 : 12 + 22 ≤ 1 ,
2.0
1.3
x, y ∈ R2 ,
Q=
u ∈ P,
v ∈ R2 :
v ∈ Q.
v12
v22
+
≤
1
.
1.52
1.052
The vectograms appearing in this game are given in Fig. 11. The level set Wc
corresponding to c = 0.397 is shown in Fig. 12a. One can see three narrow throats.
An enlarged view of the middle one (which is the narrowest among them) can be
seen in Fig. 12b.
3.
Games with Non-Convex Payoff Function
In the previous section, we demonstrate examples where number of narrow throats is
more than one. One can think that examples of this kind are artificial and, therefore,
170
Sergey S. Kumkov, Valerii S. Patsko
a)
b)
Fig. 8: Example 3. A general view of the vectogram tubes. Number 1 denotes the tube of
the first player (the set P), number 2 corresponds to the second player’s vectogram tube
(the set Q). In subfigure b), the tube of the second player is transparent
Phenomenon of Narrow Throats of Level Sets
171
Fig. 9: Example 3. A general view of the level set with two narrow throats, c = 1.2
Fig. 10: An enlarged view of the first (in the backward time) narrow throat
172
Sergey S. Kumkov, Valerii S. Patsko
rare. During last few years, the authors investigate differential games arising from
consideration of pursuit problems in near space or in upper atmosphere. Descriptions
of dynamics of the objects involved in the pursuit were taken from works by J. Shinar
and his pupils. Games of this type also bring examples having level sets with, at
least, two narrow throats.
3.1.
Problem Formulation
Consider a game
żP1 = AP1 (t)zP1 + BP1 (t)u1 ,
żP2 = AP2 (t)zP2 + BP2 (t)u2 ,
żE = AE (t)zE + BE (t)v,
zP1 ∈ Rn1 , zP2 ∈ Rn2 , zE ∈ Rm , |ui | ≤ µi , |v| ≤ ν
(12)
with three objects moving in a straight line. The objects P1 and P2 described by the
phase vectors zP1 and zP2 are the pursuers. The object E with the phase vector zE
is the evader. The first components zP1 , zP2 , and zE of the vectors zP1 , zP2 , and zE
respectively are the one-dimensional geometric coordinates of the objects.
Two instants T1 and T2 are prescribed. At the instant T1 , the pursuer P1 terminates its pursuit, and the
distance between him and the evader E is measured:
r1 (T1 ) = |zP1 (T1 ) − zE (T1 ). Similarly, the second pursuerP2 stops to pursue at the
instant T2 , when the distance r2 (T2 ) = |zP2 (T2 ) − zE (T2 ) is measured.
The payoff is the minimum of these distances: ϕ = min r1 (T1 ), r2 (T2 ) . The
first player that consists of the pursuers and governs the controls u1 , u2 minimizes
the value of payoff ϕ. The second player, which is identified with the evader E,
maximizes the payoff. All controls are scalar and have bounded absolute value.
Fig. 11: Example 4. A large view of the vectogram tubes. Number 1 denotes the tube of
the first player (the set P), number 2 corresponds to the second player’s vectogram tube
(the set Q)
Phenomenon of Narrow Throats of Level Sets
173
a)
b)
Fig. 12: Example 4. a) A general view of a level set with three narrow throats, c = 0.397;
b) An enlarged view of the narrowest of the throats (the middle one)
3.2. Equivalent Differential Game
Let us pass from system (12) with separated objects to two relative dynamics. To
do this, introduce new phase vectors y (1) ∈ Rn1 +nE −1 and y (2) ∈ Rn2 +nE −1 such
that
(1)
(2)
y1 = zP1 − zE , y1 = zP2 − zE .
(1)
The rest components yi , i = 2, . . . , n1 + nE − 1, of the vector y (1) equal components of the vectors zP1 and zE other than zP1 and zE . In the same way, the rest
(2)
components yi , i = 2, . . . , n2 + nE − 1, of the vector y (2) are the components of
the vectors zP2 and zE other than zP2 and zE . Due to linearity of dynamics (12),
the new dynamics consisting of the two relative ones, is linear too:
ẏ (1) = A1 (t)y (1) + B1 (t)u1 + C1 (t)v, t ∈ [t0 ; T1 ],
ẏ (2) = A2 (t)y (2) + B2 (t)u2 + C2 (t)v, t ∈ [t0 ; T2 ],
y (1) ∈ Rn1 +nE −1 , y (2) ∈ Rn2 +nE −1 ,
(1)
(2)
|ui | ≤ µi , |v| ≤ ν, ϕ = min y1 (T1 ), y1 (T2 ) .
(13)
The payoff function depends now on the first components of the phase vectors of the
individual games. An individual game of the pursuer Pi against the evader E is the
174
Sergey S. Kumkov, Valerii S. Patsko
(i)
game with the dynamics of the vector y (i) and the payoff y1 (Ti ). The dynamics
of the individual games are linked only through the control of the evader.
In each individual game, let us pass to the forecasted geometric coordinates in
the same way as it is done from game (1) to (2). In the game of the pursuer Pi ,
(i)
i = 1, 2, against the evader E, the passage is provided by the matrix X1 (Ti , t)
constructed from the first row of the fundamental Cauchy matrix X (i) (Ti , t) that
corresponds to the matrix Ai . The variable changes are defined by the formulas
(1)
(2)
(1)
x1 (t) = X1 (T1 , t)y (1) , x2 (t) = X1 (T2 , t)y (2) . Note that x1 (T1 ) = y1 (T1 ) and
(2)
x2 (T2 ) = y1 (T2 ).
Dynamics of the individual games is the following:
ẋ1 = d1 (t)u1 + e1 (t)v, t ∈ [t0 ; T1 ],
ẋ2 = d2 (t)u2 + e2 (t)v, t ∈ [t0 ; T2 ],
x1 , x2 ∈ R, |ui | ≤ µi , |v| ≤ ν.
(14)
Here, di (t) and ei (t) are scalar functions:
(i)
(i)
di (t) = X1 (Ti , t)Bi (t), ei (t) = X1 (Ti , t)Ci (t),
i = 1, 2.
In the joint game of the pursuers against the evader, the payoff function is
ϕ = min x1 (T1 ), x2 (T2 ) .
3.3. Numerical Construction of Level Sets
Numerical constructions for the taken formulation are more complicated due to the
following circumstances.
At first, for problem (2), any level set of the payoff function is plunged into the
phase space at the instant T . But for the new formulation, level sets of the payoff
can consist of two parts at two different (generally speaking) instants T1 and T2 . At
second, level sets of the payoff in problem (2) compact. But now the components of
(1)
level sets corresponding to a constant c are infinite strips Mc = {x : |x1 | ≤ c} at
(2)
the instant T1 (that is an infinite strip along the axis x2 ) and Mc = {x : |x2 | ≤ c}
at the instant T2 (an infinite strip along the axis x1 ). Presentation of infinite objects
in a computational program is a quite difficult problem. At third, a realization of
procedure (3) for problem (2) is oriented on work with convex sets. In problem (14),
we need to proceed non-convex time sections.
An algorithm taking into account these considerations and also based on procedure (3) can be formulated as follows.
For definiteness, let us assume that T2 ≤ T1 . The opposite case is considered in
the same way.
For numerical constructions, fix a time grid in the interval [t0 ; T1 ]. It should
(1)
include the instant T2 . At the instant T1 , the set Mc is taken as the start for
the procedure (3). In the case of numerical constructions, the infinite strip is cut
becoming a rectangle with a quite large size along the axis x2 . Then, in the interval (T2 , T1 ], the procedure (3) is applied with the set D(t)P taken as a segment
[−|d1 (t)|µ1 ; |d1 (t)|µ1 ] × {0} (by this, we ignore the action of the second pursuer).
When the construction are made up to the instant T2 , we unite the obtained t(2)
section Wc (T2 + 0) with the set Mc (also cut to a finite size in the case of
numerical constructions). Thus, we get the set Wc (T2 ), which is the start value
175
Phenomenon of Narrow Throats of Level Sets
for the further iterations. In the time interval [t0 ; T2 ], the rectangle vectogram
[−|d1 (t)|µ1 ; |d1 (t)|µ1 ]×[−|d2 (t)|µ2 ; |d2 (t)|µ2 ] of the first player is taken; now, actions
of both players are involved.
The vectogram of
the second player (of the evader)
equals −|e1 (t)|ν, −|e2 (t)|ν ; |e1 (t)|ν, |e2 (t)|ν for all time instants from the grid.
(1)
If T2 = T1 = T , then the start set Mc at the instant T is union of the strips Mc
(2)
and Mc (possibly, cut).
As it was mentioned above, the necessary realization of procedure (3) should be
able to process non-convex sets. Namely, we need operations of Minkowski sum and
difference, which first operand is not convex (the second one is convex because it
is computed as a convex vectogram multiplied by the time step). A helpful fact is
that both operations, sum and difference, can be fulfilled by one operation, namely,
sum. Indeed, it is true that
∗ B = (A′ − B)′ .
A−
Here, the prime denotes set complement. The authors worked out an algorithm
for construction Minkowski sum when the first operand is a union of a number of
simple-connected closed polygonal sets (possibly, non-convex), or is a complement to
such a polygon (in other words, is an infinite closed set with a number of polygonal
holes).
3.4.
Variants of Servomechanism Dynamics
1. A First Order Link. In work (Le Ménec, 2011), a pursuit problem is formulated that includes two pursuers and one evader. Each object has a three-dimensional
phase variable: one-dimensional coordinate, velocity, and acceleration. The acceleration is affected by the control through a link of the first order:
ż1 = z2 ,
ż2 = z3 ,
ż3 = (u − z3 )/l.
(15)
Dynamics of the pursuer in problem (4) is similar, but now the geometric coordinate
is one-dimensional. As above, l is the time constant describing the inertiality of the
servomechanisms.
2. Damped Oscillating Control Contour. In work (Shinar et al., 2013), a game
is considered, in which one of the objects has a damped oscillating control contour:
ż1 = z2 ,
ż2 = z3 ,
ż3 = z4 ,
ż4 = −ω 2 z3 − ζz4 + u.
(16)
Here, ω is the natural frequency of the contour, ζ is the damping coefficient.
3. Tail/Canard Air Rudders. When considering an objects moving in the atmosphere, it is important to take into account position of its rudders with respect to
the center of mass. A corresponding model is set forth in (Shima, 2005):
z̈ = a + du, ȧ = (1 − d)u − a /l.
(17)
The parameter d is defined by the position of the rudder. A positive (negative)
value corresponds to the situation when the rudder is located in front of (behind)
the center of mass. The first situation is called canard control scheme, the second
one is called tail control scheme. The absolute value of d describes now far from the
center of mass the rudder is. The parameter l again is the time constant.
176
Sergey S. Kumkov, Valerii S. Patsko
4. Dual Tail/Canard Scheme. As a development of model (17), work (Shima
and Golan, 2006) suggests a dynamics of an objects, which has both canard and
tail rudders:
ż1 = z2 , ż2 = z3 + dc uc + dt ut , ż3 = (1 − dc )uc + (1 − dt )ut − z3 /l. (18)
Here, the constant dc > 0 describes the capabilities of the canard rudder (then
index c means “canard” here), the constant dt < 0 corresponds to the tail rudder
(the index t means “tail”). The time constant l regarded to be common for inertiality
of both rudders.
In this model, one can see two scalar controls uc and ut (or one vector control u =
(uc , ut )⊤ taken from a rectangle). Therefore, formally this model does not belong
to class (12). But the procedures for construction level sets of the value function
suggested by the authors can be applied to such a dynamics with double scalar
control. The difference is that formula (3) includes now two summands connected
to two controls of the first player.
3.5.
Examples
In this subsection, we assume that the evader has dynamics of type (15).
Example 5. Let both pursuers have the same dynamics of type (15). The parameters
of the game are
µ1 = µ2 = 1.5, ν = 1.0, lP1 = lP2 = 1/0.25, lE = 1/1.0, T1 = T2 = 15.
The level set of the value function corresponding to c = 1.32 is shown in Fig. 13.
In similar problems studied in detail by the authors (Ganebny et al., 2012;
Kumkov et al., 2013), the advantage of a player in an individual game, can be
detected analytically by analyzing the parameters ηi = µi /ν and εi = lE /lPi . When
ηi > 1, ηi εi > 1, the ith pursuer has advantage over the evader. Vice versa, if ηi < 1,
ηi εi < 1, the advantage belongs to the evader. If the parameters do not obey one of
these conditions, then there is a situation of changing advantage of the ith pursuer
over the evader in time.
For the example, the data are such that both pursuers are weaker then the
evader at the beginning of the backward time (near the target set, which is located
in the plane t = T1 = T2 ). Due to this, at the beginning of the backward time, the
Fig. 13: Example 5. Narrow throats in a problem with three objects having dynamics (15)
Phenomenon of Narrow Throats of Level Sets
177
Fig. 14: Example 6. A level set for example 2 with two narrow throats
t-sections start to contract. In Fig. 13, an instant can be distinguished when the
infinite strips (the rectangles elongated along the corresponding axes) degenerate
to a line due to this contraction and disappear. After this instant, the t-sections
consist of two finite disconnected parts that correspond to zones of joint capture.
If the position of the system is in such a zone, then the evader escaping from one
pursuer is captured (with the given miss) by the another one. These parts continue
to contract until an instant when the pursuers get the advantage. Further, the
contraction turns to expansion, and at some instant growing parts joins into one
simple-connected set that continue to grow infinitely.
Since the parameters of both pursuers coincide and the time lengths of both
individual games are equal, the dynamics of the coordinates x1 and x2 are the
same. Therefore, the evolution of t-sections is the same along both coordinate axes.
As it will be seen from the following examples, this is not true if the pursuers’
parameters or game lengths are different.
Example 6. Let both pursuers be equal again, but now they have dynamics (16)
with oscillating control contour. The parameters are the following:
µ1 = µ2 = 0.3, ν = 1.3, ωP1 = ωP2 = 0.5,
ζP1 = ζP2 = 0.0025, lE = 1.0, T = T1 = T2 = 30.
The level set of the value function corresponding to c = 1.6 can be seen in Fig. 14.
In this problem, due to fundamental difference of the pursuers’ and evader’s
dynamics, it is difficult to get analytically the conditions of advantage of one or
other player. Thus, the example is constructed on the base of the evolution of
the players’ vectograms obtained numerically. Presence of two narrow throats is
connected to repeat of a period such that at the beginning the advantage belongs
to the evader and at the end it comes to pursuers. The repeat is provided by the
oscillating type of the pursuers’ dynamics. More throats can be obtained by putting
to the evader an oscillating dynamics too.
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Sergey S. Kumkov, Valerii S. Patsko
Fig. 15: Example 7. The level set W0.525 for the pursuers’ dynamics of type (18); the
pursuers have different parameters of the dynamics
Example 7. Consider now a pair of pursuers both having dynamics (18). Let some
dynamics parameters be different:
aP1 ,max = 1.05, aP2 ,max = 1.15, lP1 = lP2 = 1/0.18807,
dc,1 = dc,2 = 0.605, dt,1 = dt,2 = −0.5, α1 = 0.9, α2 = 0.8,
aE,max = 0.95, dE = 0.157980, lE = 1.0, T1 = 32, T2 = 29.
The value αi defines distribution of the control resource aPi ,max of the ith pursuer
over the rudders:
|uc | ≤ α · aP,max , |ut | ≤ β · aP,max ;
α, β ≥ 0, α + β = 1.
The level set Wc that correspond to c = 0.525 is given in Fig. 15. One can
see that due to difference of the pursuers’ dynamics the contraction of the set is
different along the axes x1 and x2 : degeneration of the infinite strips happens at
different instants. Moreover, the finite parts remaining after degeneration of infinite
strips have sufficiently different sizes along the two coordinate axes.
4.
Conclusion
A level set (Lebesgue set) of the value function corresponding to some value c can
be regarded as a solvability set of a game problem with the payoff equal to c. For
differential games with fixed termination instants, a level set of the value function
is a tube in the space time × phase space along the time axis. It is very important
to establish the law of evolution of time sections of the tubes in time. For example,
if a tube corresponding to some c has a small length in time, then it means that
the zone of guaranteed capture with the miss not greater than c is small too. If a
solvability set has a narrow throat, that is, a period of time where its t-sections are
close to degeneration (to loss of interior), then one should analyze accurately the
Phenomenon of Narrow Throats of Level Sets
179
possibility of practical application of the control law based on such a tube. In the
paper, it is shown that the presence of narrow throats is not rare both in model
differential games and practical pursuit problems.
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line with two pursuers and one evader. Dyn. Games Appl., 2, 228–257.
Grigorenko, N. L. (1990). Mathematical Methods for Control of a Number of Dynamic
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Strictly Strong (n − 1)-equilibrium in n-person Multicriteria
Games
Denis V. Kuzyutin1 , Mariya V. Nikitina2 and Yaroslavna B.
Pankratova3
1
St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
Universitetskii pr, 35, St.Petersburg, 198504, Russia
E-mail: [email protected]
2
International Banking Institute,
Nevski pr, 60, St.Petersburg, 191023, Russia
E-mail:[email protected]
3
St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
Universitetskii pr, 35, St.Petersburg, 198504, Russia
E-mail: [email protected]
Abstract Using some specific approach to the coalition-consistency analysis in n-person multicriteria games we introduce two refinements of (weak
Pareto) equilibria: the strong and strictly strong (n − 1)-equilibriums. Axiomatization of the strictly strong (n − 1)-equilibria (on closed families of
multicriteria games) is provided in terms of consistency, strong one-person
rationality, suitable variants of Pareto optimality and converse consistency
axiom and others.
Keywords: multicriteria games; Pareto equilibria; strong equilibrium;
consistency; axiomatizations.
1.
Introduction
The concept of strictly strong (n − 1)-equilibria (in n-person strategic games and in
multicriteria games) is based on some specific approach to the coalition-consistency
analysis, offered in (Kuzyutin, 1995; Kuzyutin, 2000). Namely, we suppose that trying to investigate the coalition-consistency of some acceptable Nash
equilibrium x, every player i does not consider the deviations of coalitions S, i ∈ S
with her participance (since player i may be sure in her own strategic choice
xi ). This approach allows to make the strong Nash equilibria (Aumann, 1959)
requirements slightly weaker.
We show (in section 2) that the strong and strictly strong (n−1)-equilibrium differs from other closely related solution concepts: coalition-proof equilibrium
(Bernheim et al., 1987) and semi-strong Nash equilibrium (Kaplan, 1992). The axiomatization of strong and strictly strong (n − 1)-equilibria in n-person strategic
games was given in (Kuzyutin, 2000).
In section 3 we explore the same approach to coalition-consistency analysis in
n-person multicriteria games (or the games with vector payoffs) and offer two refinements of the weak Pareto equilibria (Shapley, 1959; Voorneveld et al., 1999).
The axiomatic characterization of strictly strong (n − 1)- equilibria (on closed
families of multicriteria games) is provided in section 4 using the technique offered in
182
Denis V. Kuzyutin, Mariya V. Nikitina, Yaroslavna B. Pankratova
(Peleg and Tijs, 1996; Norde et al., 1996; Voorneveld et al., 1999). In this axiomatization the suitable variants of Pareto-optimality and converse consistency axioms
play a role to distinguish between the strictly strong (n − 1)-equilibria and other
equilibrium solutions in multicriteria games.
2.
Strong and strictly strong (n − 1)-equilibrium in strategic games
Consider a game in strategic form G = (N, (Ai )i∈N , (ui )i∈N , where N is a finite
set
| = n Ai 6= ∅ is the set of player’s i strategies; and ui : A =
Q of players |N
1
A
→
R
is the payoff function of player i ∈ N . A solution (optimality
j
j∈N
principle) ϕ, defined on a class of strategic games Γ , is a function that assigns to
each game G = (N, (Ai )i∈N , (ui )i∈N ∈ Γ a subset ϕ(G) of A. We’ll call a strategy
profile x the optimal situation, if x ∈ ϕ(G).
Q Let S ⊂ N , S 6= ∅, be a coalition;
S ⊂ N , S 6= ∅, N Ů proper coalition; AS = j∈S Aj — a set of all possible players’
i ∈ S strategy profiles.
The concept of strong (Nash) equilibria was offered by Aumann, 1959.
Definition 1. x ∈ A is a strong Nash equilibrium (SNE), if ∀S ⊂ N , S 6= ∅,
∀yS ∈ AS , ∃i ∈ S:
ui (x) ≥ ui (yS , xN \S ),
where yS = (yj )j∈ S, xN \S = (xj )j∈N \S .
Definition 2. x ∈ A is weakly Pareto-optimal (W P O), if ∀y ∈ A, ∃i ∈ N :
ui (x) ≥ ui (y).
Definition 3. x ∈ A is a strictly strong Nash equilibrium (SSN E), if there do not
exist coalition S ⊂ N and yS ∈ AS such that:
ui (yS , xN \S ) ≥ ui (x) ∀i ∈ S,
∃j ∈ S : uj (yS , xN \S ) > uj (x).
Notice that the concept of SN E (as well as SSN E) deals with a r b i t r a r y
deviations of a l l p o s s i b l e coalitions S ⊂ N . We denote by N E(G), SN E(G),
SSN E(G) the set of Nash equilibriums (Nash, 1950), strong Nash equilibriums and
strictly strong Nash equilibriums of G respectively. The following inclusions hold:
N E(G) ⊃ SN E(G) ⊃ SSN E(G).
Unfortunately, the sets SN E(G) and SSN E(G) are often empty (see, for instance, Petrosjan and Kuzyutin, 2008) by the reason of "too strong" requirements
to the solution used in def. 1, 3. We’ll consider an opportunity to make these requirements slightly weaker that leads to new concept of coalition-stable equilibrium.
We guess a game G = (N, (Ai )i∈N , (ui )i∈N ) is "of common knowledge", when
every player knows all players’ strategy sets and payoff functions. Moreover, suppose
that trying to investigate the coalition stability of some acceptable strategy profile
x, every player i does not consider the deviations of coalitions S ∈ i with her
participance since player i may be sure in her own strategic choice xi . The related
motivation was used early for other purposes in Kuzyutin, 1995 to define the istability property in n-person extensive game.
Strictly Strong (n − 1)-equilibrium in n-person Multicriteria Games
183
Definition 4. Let G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ , |N | = n.
1. n ≥ 2: x ∈ A is a strong (n−1)-equilibrium (SN E n−1 ), if for every player i ∈ N
the following condition holds:
∀S ⊂ N \ {i}, ∀ yS ∈ AS , ∃ j ∈ S : uj (x) ≥ uj (yS , xN \S );
2. n = 1: xi ∈ Ai is a strong (n − 1)-equilibrium in one-player game G =
({i}, Ai , ui ), if
ui (xi ) ≥ ui (yi ) ∀yi ∈ Ai
Definition 5. 1. n ≥ 2: x ∈ A is a strictly strong (n−1)-equilibrium (SSN E n−1 ),
if for every player i ∈ N there do not exist a coalition S ⊂ N \ {i} and yS ∈ AS
such that:
uj (yS , xN \S ) ≥ uj (x) ∀ j ∈ S,
2. n = 1: SSN E
n−1
∃ k ∈ S : uk (yS , xN \S ) > uk (x).
coincides with SN E n−1 .
Remark 1. Another possible definition of SN E n−1 (n ≥ 2) is as follows: x is a
SN E n−1 if ∀ S ⊂ N, |S| ≤ n − 1, ∀ yS ∈ AS ∃ j ∈ S:
uj (x) ≥ uj (yS , xN \S ).
However, we guess the def. 4 is more useful to clarify the offered approach every
player (independently of others) holds on to check the coalition stability of x. The
optimally principles SN E n−1 and SSN E n−1 deal with a r b i t r a r y deviations
of c e r t a i n (c r e d i b 1 e) coalitions.
It is clear that
Further we have:
N E(G) ⊃ SN E n−1 (G) ⊃ SSN E n−1 (G).
N E(G) ⊃ SN E n−1 (G) ⊃ SN E(G),
N E(G) ⊃ SSN E n−1 (G) ⊃ SSN E(G).
The example shows that these inclusions may be strict.
Example 1. Let the three-person game
G = (N = {1, 2, 3}, A1 = {x1 , y1 }, A2 = {x2 , y2 }, A3 = {x3 , y3 }, (ui )i∈N ), be
given by the following normal form:
x3
y3
x2
y2
x2
y2
x1 (7, 7, 0) (0, 0, 5) x1 (4, 4, 9) (0, 0, 5)
y1 (0, 0, 5) (5, 5, 15) y1 (0, 0, 5) (0, 0, 0)
For the convenience we’ll restrict ourselves to the players’ pure strategies (player
1 chooses a row, player 2 a column and player 3 a block of the table). Here:
N E(G) = {(y1 , y2 , x3 ), (x1 , x2 , y3 )}; SN E(G) = SSN E(G) = ∅; SN E n−1 (G) =
SSN E n−1 (G) = {x1 , x2 , y3 )}; W P O(G) = {(y1 , y2 , x3 ), (x1 , x2 , x3 )} is the set of
weak Pareto-optimal strategy profiles in G.
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Denis V. Kuzyutin, Mariya V. Nikitina, Yaroslavna B. Pankratova
Certainly, player 3 can not accept the situation (y1 , y2 , x3 ) ∈ N E(G)∩W P O(G)
since the best player 1 and player 2 joint response to the player 3 strategy x3 is
(x1 , x2 ) that leads to the least possible payoff of player 3. At the same time the
strategy profile (x1 , x2 , y3 ) is free from such danger, and satisfies the requirements
of coalition stability from def. 4, 5 although does not satisfy weak Pareto-optimality.
Thus one can use the strong (n − 1)-equilibrium concept to obtain a unique optimal
outcome in strategic game G.
Notice, that the extended analysis of closely related example is offered in
Bernheim et al., 1987 in connection with the coalition-proof Nash equilibrium concept.
Definition 6. Let G = (N, (Ai )i∈N , (ui )i∈N ) be a game, let x ∈ A and let ∅ 6=
S ⊂ N . An internally consistent improvement (ICI) of S upon x is defined by
induction on |S|. If |S| = 1, that is S = {i} for some i ∈ N , then yi ∈ Ai is an ICI
of i upon x if it is an improvement upon x, that is, ui (yi , xN \{i} ) > ui (x).
If |S| > 1 then yS ∈ AS is an ICI of S upon x if:
1. ui (ys , xN \S ) > ui (x) for all i ∈ S,
2. no T ⊂ S, T 6= ∅, S has an ICI upon (yS , xN \S ).
x is a coalition-proof Nash equilibrium (CP N E) if no T ⊂ N , T 6= ∅, has an ICI
upon x.
The reader is refereed to Bernheim et al., 1987 for discussion and motivation.
Definition 7 (Kaplan, 1992). Let G = (N, (Ai )i∈N , (ui )i∈N ) be a game. x ∈ A
is a semi-strong Nash equilibrium (SM SN E), if for every ∅ 6= S ⊂ N and every
yS ∈ N E(GS,x ) there exists i ∈ S such that ui (x) ≥ ui (yS , xN \S ).
Notice that the concept of CP N E (as well as SM SN E) deals only with c e r t
a i n deviations of a l l p o s s i b l e coalitions S ⊂ N .
To clarify the difference between CP N E and SM SN E from the one hand and
the strong (n − 1)-equilibrium concept from the other we consider the following
example.
Example 2. G = (N = {1, 2, 3}, A1 = {x1 , y1 }, A2 = {x2 , y2 }, A3 = {x3 , y3 }, (ui )i∈N ),
is the three-person strategic game:
x3
y3
x2
y2
x2
y2
x1 (9, 9, 0) (4, 10, 0) x1 (4, 4, 9) (0, 0, 5)
y1 (0, 0, 5) (5, 5, 10) y1 (0, 0, 5) (0, 0, 0)
Here: N E(G) = {(y1 , y2 , x3 ), (x1 , x2 , y3 )}; W P O(G) = {(x1 , x2 , x3 ),
(x1 , y2 , x3 ), (y1 , y2 , x3 )}; SN E(G) = SSN E(G) = ∅; CP N E(G) = SM SN E(G) =
N E(G) ∩ W P O(G) = {(y1 , y2 , x3 )}, but SSN E n−1 (G) = {(x1 , x2 , y3 )}.
Notice that (as in example 1) player 3 can reject the strategy profile (y1 , y2 , x3 )
by the reason of other players have the profitable joint deviation (x1 , x2 ) from
(y1 , y2 , x3 ) that is still possible (although (x1 , x2 ) is not ICI of S = {1, 2} upon
(y1 , y2 , x3 )). If such deviation takes place player 3 will receive the least feasible
payoff (independently of further possible deviation y2 of player 2).
Strictly Strong (n − 1)-equilibrium in n-person Multicriteria Games
185
Remark 2. SSN E n−l does not coincide with CP N E (as soon as with SM SN E)
in general case.
Remark 3. Let ϕ be one of the optimality principles: CP N E or SM SN E. SSN E n−1
is not a refinement of ϕ, and ϕ is not a refinement of SSN E n−1 .
3.
Coalition-stable equilibriums in multicriteria games
Now let us turn to so-called multicriteria games (or the games with vector payoffs)
when every player may take several criteria into account. Formally,
let G = (N, (Ai )i∈N , (ui )i∈N ) be a finite multicriteria game, were N is a finite
set of players, |N | = n, Ai 6= ∅ is the finite set
of player i ∈ N ,
Q of pure strategies
r(i)
and for each player i ∈ N the function ui :
A
→
R
maps
each strategy
j
j∈N
profile to a point in r(i)-dimensional Euclidean space. Note that player i in multicriteria game G tries to maximaize r(i) scalar criteria (i.e. all the components of
her vector valued payoff function ui (xi , x−i )).
The concept of equilibrium point for multicriteria games was proposed
by Aumann, 1959 as a natural generalization of the Nash equilibrium concept for
unicriterium games.
Let a, b ∈ Rt , and a > b means that ai > bi for all i = 1, . . . , t; a ≥ b means
that ai ≥ bi for all i = 1, . . . , t, and a 6= b.
The vector a ∈ M ⊆ Rt is weak Pareto efficient (or undominated) in M iff {b ∈
t
R : b > a} ∩ M = ∅. In this case we’ll use the following notation: a ∈ W P O(M ).
Given strategy profile x = (xi , x−i ) in the finite multicriteria game G denote by
Mi (G, x−i ) = {ui (yi , x−i ), yi ∈ Ai }
the set of all player’s i attainable vector payoffs (due to arbitrary choice of his
strategy yi ∈ Ai ).
Q
Definition 8. The strategy profile x = (x1 , . . . , xn ) ∈ j∈N Aj is called (weak
Pareto) equilibrium in multicriteria game G iff for each player i ∈ N there does not
exist a strategy yi ∈ Ai such that:
ui (yi , x−i ) > ui (xi , x−i )
(1)
Note that (1) is equivalent to the following condition:
ui (xi , x−i ) ∈ W P O(Mi (G, x−i )) ∀i ∈ N.
(2)
Let E(G) be the set off all (weak Pareto) equilibriums in multicriteria game G.
Definition 9. The strategy profile x = (x1 , . . . , xn ) is called strong equilibrium (in
a sense of Aumann, 1959 ) in multicriteria game G iff
Y
ui (yS , x−S ) > ui (xS , x−S )
∀S ⊂ N, S 6= ∅ ∄ yS ∈
Aj :
i∈S
.
j∈S
Definition 10. The strategy profile x = (x1 , . . . , xn ) is called strictly strong equilibrium in multicriteria game G iff
Y
ui (yS , x−S ) ≥ ui (xS , x−S )
∀S ⊂ N, S 6= ∅ ∄ yS ∈
Aj :
i∈S
.
j∈S
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Denis V. Kuzyutin, Mariya V. Nikitina, Yaroslavna B. Pankratova
Definition 11. Let G = (N, (Ai )i∈N , (ui )i∈N ) be a finite multicriteria game with
n players, |N | = n.
Q
1. n ≥ 2: x = (x1 , . . . , xn ) ∈ j∈N Aj is a strong (n − 1)-equilibrium if for each
player i ∈ N the following condition holds:
uj (yS , x−S ) > ui (xS , x−S )
∀S ⊂ N \{i}, ∄ yS ∈ AS :
(3)
j∈S
.
2. n = 1: xi ∈ Ai is a strong (n − 1)-equilibrium in one player multicriteria game
G = ({i}, Ai , ui ) if ∄ yi ∈ Ai : ui (yi ) > ui (xi ).
Let SE(G), SSE(G) and SE n−1 (G) be the sets S of all strong equilibriums, strictly
strong equilibriums and strong (n − 1)-equilibriums in multicriteria game G correspondly.
Definition 12. 1. n ≥ 2: x = (x1 , . . . , xn ) ∈ A is a strictly strong (n − 1)equilibrium if for every player i ∈ N the following condition holds:
uj (yS , x−S ) ≥ uj (xS , x−S )
∀S ⊂ N \{i}, ∄ yS ∈ AS :
(4)
j∈S
.
2. n = 1: xi ∈ Ai is a strictly strong (n − 1)-equilibrium in one-person multicriteria
game G = ({i}, Ai , ui ), if ∄ yi ∈ Ai : ui (yi ) ≥ ui (xi ).
The set of all strictly strong (n − 1)-equilibriums in G denote by SSE n−1 (G).
Q
Definition 13. The strategy profile x = (x1 , . . . , xn ) ∈ j∈N Aj is called Pareto
efficient in a multicriteria game G iff
Y
ui (y) ≥ ui (x)
∄y∈
Aj :
(5)
i∈N
.
j∈N
The set of all Pareto efficient strategy profiles in G denote by P OMG (G).
It is clear that
E(G) ⊃ SE n−1 (G) ⊃ SSE n−1 (G),
SSE n−1 (G) ⊃ SSE(G),
P OMG (G) ⊃ SSE(G).
4.
Axiomatization of strictly strong (n − 1)- equilibria in multicriteria
games
In this section we give axiomatization of SSE n−1 correspondence on closed classes of
multicriteria games in terms on consistency, strong one-person rationality, suitable
variants of converse consistency and Pareto-optimality axiom and others.
Let Γ be a set of muliticriteria games G and let ϕ be a solution on Γ .
Definition 14. ϕ satisfies strong one-person rationality (SOP R) if for every oneperson game G = ({i}, Ai , ui ) ∈ Γ
ϕ(G) = {xi ∈ Ai | ∄ yi ∈ Ai : ui (yi ) ≥ ui (xi )}
Strictly Strong (n − 1)-equilibrium in n-person Multicriteria Games
187
Let G = (N, (Ai )i∈N , (ui )i∈N ) be a game, n =| N |≥ 2, let S ⊂ N be a proper
coalition, i.e. S 6= ∅, N .
Definition 15. The proper reduced game GS,x of G (with respect to S and x) is
the multicriteria game GS,x = (S, (Ai )i∈S , (uxi )i∈S ), where
uxi (yS ) = ui (yS , xN \S ) ∀ yS ∈ AS , ∀ i ∈ S.
A family Γ of multicriteria games is r-closed, if G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ ,
S ⊂ N , S 6= ∅, N and x ∈ A imply that GS,x ∈ Γ .
Definition 16. Let Γ be a r-closed family of strategic games. A solution ϕ on Γ
satisfies consistency (CON S), if for every G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ , ∀ S ⊂ N ,
S 6= ∅, N , ∀ x ∈ ϕ(G) the following condition holds:
xS ∈ ϕ(GS,x ).
The CON S property means the restriction xS of the optimal strategy profile
x ∈ ϕ(G) still satisfies the optimality principle ϕ in every reduced game GS,x . If
G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ , and n ≥ 2, then we denote:
ϕ(G)
e
= {x ∈ A | ∀ S ⊂ N, S 6= ∅, N, xS ∈ ϕ(GS,x )}
(6)
Taking (6) into account one can notice that CON S property means ϕ(G) ⊂ ϕ(G)
e
for every G ∈ Γ .
Definition 17. A solution ϕ on Γ satisfies (n − 1)-Pareto optimality for multicrin−1
teria games (P OMG
), if for every G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ with at least two
players (n ≥ 2), for every x ∈ ϕ(G) the following conditions holds:
uj (y−i , xi ) ≥ uj (x−i , xi )
∀ i ∈ N ∄ y−i ∈ A−i :
(7)
j ∈ N \ {i}.
.
n−1
Notice that ϕ satisfies P OMG
iff ∀ x ∈ ϕ(G), ∀i ∈ N
xN \{i} ∈ P OMG (GN \{i},x ).
n−1
Let P OMG
(G) be the set of all strategy profiles x ∈ Πi∈N Ai in G,
satisfying (7).
Definition 18. Let Γ be a r-closed family of strategic games. A solution ϕ satisfies
COCON S∗n−1 (the appropriate version of converse for SSE n−1 ), if for every G =
(N, (Ai )i∈N , (ui )i∈N ) ∈ Γ , n ≥ 2, it is true that:
n−1
x ∈ ϕ(G)
e
and x ∈ P OMG
(G) ⇒ x ∈ ϕ(G)
(8)
In accordance with COCON S∗n−1 property if the restrictions xS of some strategy
profile x ∈ A satisfy optimality principle ϕ in every reduced game GS,x , and x ∈
n−1
P OMG
(G) then x is the optimal strategy profile in the original game G.
Theorem 1. A solution ϕ on a r-closed family of multicriteria games Γ satisn−1
fies CON S, SOP R, P OMG
and COCON S∗n−1 , if and only if ϕ = SSE n−1 (i.e.
ϕ(G) = SSE n−1 (G) for every G ∈ Γ ).
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Denis V. Kuzyutin, Mariya V. Nikitina, Yaroslavna B. Pankratova
n−1
Proof. 1. It is not difficult to verify that SSE n−1 satisfies CON S, SOP R, P OMG
n−1
n−1
and COCON S∗ . Let us check here that SSE
satisfies CON S (for instance). If x ∈ SSE n−1 then ∀ S ⊂ N \{i} ∄ yS ∈ AS :
uj (yS , x−S ) ≥ uj (xS , x−S ) = uj (x)
(9)
j ∈ S.
.
Consider an arbitrary coalition S1 ⊂ N , S1 6= ∅, N and reduced game GS1 ,x .
Let S ⊂ S1 \{i} ⊂ S1 ⊂ N . Using (9) we have that
uj (yS , xS1 \S , xN \S1 ) = uj (yS , xN \S ) ≥ uj (x)
∀ S ⊂ S1 \{i} ∄ yS ∈ AS :
j ∈ S.
.
This means that xS1 ∈ SSE n−1 (GS1 ,x ), i.e. SSE n−1 satisfies CON S.
2. Now let ϕ be a solution on Γ that satisfies the foregoing four axioms. We prove
by induction (on the number of players n) that ϕ(G) = SSE n−1 (G) for every
G ∈ Γ.
By SOP R ϕ(G) = SSE n−1 (G) for every one-person multicriteria game G ∈ Γ .
Now assume that
ϕ(G) = SSE n−1 (G) ∀ G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ,
(10)
where 1 ≤ |N | ≤ k, k ≥ 1, and consider an arbitrary (k + 1)-person multicriteria
game G ∈ Γ . Let x ∈ ϕ(G). From the CON S of ϕ it follows that
(11)
x ∈ ϕ(G).
e
Using the induction hypothesis and the notation (6) we obtain:
n−1
Moreover, by P OMG
of ϕ,
]
ϕ(G)
e
= SSE
n−1
(G).
n−1
x ∈ P OMG
(G).
(12)
COCON S∗n−1 property of
n−1
Taking into account (10), (11), and (12), and
SSE n−1 ,
n−1
we obtain that x ∈ SSE
(G), and, hence, ϕ(G) ⊂ SSE
(G).
Similarly, we may prove that SSE n−1 (G) ⊂ ϕ(G) for every (k + 1)-person
multicriteria game G ∈ Γ . The inductive conclusion completes the proof.
Corollary 1. Let ϕ be a solution on r-closed family of games Γ , that satisfies
n−1
CON S and P OMG
. Then
ϕ(G) ⊂ SSE n−1 (G) ∀ G ∈ Γ, | N |= n ≥ 2.
Proof. Let x ∈ ϕ(G), n ≥ 2. To prove that x ∈ SSE n−1 (G) we need to verify that
for every possible proper coalition
uj (yS , x−S ) ≥ uj (x),
S ⊂ N, S 6= N, ∅, ∄ yS ∈ AS :
j ∈ S.
i.e.
xS ∈ P OMG (GS,x ) ∀ S ⊂ N : s =| S |= 1, 2, . . . , n − 1
(13)
Strictly Strong (n − 1)-equilibrium in n-person Multicriteria Games
n−1
By P OMG
of ϕ
189
xS ∈ P OMG (GS,x ) ∀ S ⊂ N : s = n − 1.
If n = 2 we have already established (13) for all possible proper coalitions. Otherwise
(if n ≥ 3), consider a proper reduced game GS,x , where s = n − 1. By CON S of ϕ
n−1
s−1
xS ∈ ϕ(GS,x ), and by P OMG
xS ∈ P OMG
(GS,x ), i.e.
xT ∈ P OMG (GT,x ) ∀ T ⊂ S, t = |T | = s − 1 = n − 2.
Using the same approach we can establish (13) for every proper coalition S ⊂ N ,
s =| S |= n − 1, n − 2, . . . , 1.
Another axiomatic characterization of the SSE n−1 correspondence involves the
following axioms (Peleg and Tijs, 1996).
Definition 19. Let Γ be a set of multicriteria games, and ϕ be a solution Γ . ϕ
satisfies independence of irrelevant strategies (IIS) if the following condition holds:
if G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ , x ∈ ϕ(G), xi ∈ Bi ⊂ Ai for all i ∈ N , and
G∗ = (N, (Bi )i∈N , (ui )i∈N ) ∈ Γ , then x ∈ ϕ(G∗ ).
A family of games Γ is called s-closed, if for every game G = (N, (Ai )i∈N ,
(ui )i∈N ) ∈ Γ , and Bi ⊂ Ai , Bi 6= ∅, i ∈ N , the game G∗ = (N, (Bi )i∈N , (ui )i∈N ) ∈
Γ . Further, Γ is called closed, if it is both r-closed and s-closed. For example, the
set of all finite multicriteria games is closed.
Definition 20. A solution ϕ on r-closed family of games Γ satisfies the dummy
axiom (DU M ), if for every game G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ and every "dummy
player" d in G (i.e. player d ∈ N such that | Ad |= 1), the following condition holds:
ϕ(G) = Ad × ϕ(GN \{d} , x), where x is an arbitrary strategy profile from A.
Note, that SSE n−1 satisfies IIS and DU M .
Proposition 1. (Peleg B., Tijs S. [1996]) If a solution ϕ on closed family of games
Γ satisfies IIS and DU M , then ϕ also satisfies CON S.
The next axiomatic characterization of SSE n−1 correspondence follows from
the theorem 1 and proposition 1.
Theorem 2. Let Γ be a closed family of multicriteria games. The SSE n−1
n−1
correspondence is the unique solution on Γ that satisfies SOP R, P OMG
,
n−1
COCON S∗ , IIS and DU M .
References
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Journal of Economics Theory, 42, 1–12.
Borm, P., Van Megen, F. and Tijs, S. (1999). A perfectness concept for multicriteria games.
Mathematical Methods of Operation Research, 49, 401–412.
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Kuzyutin, D. (1995). On the problem of solutions’ stability in extensive games. Vestnik of
St.Petersburg Univ., Ser. 1, Iss. 4, 18–23 (in Russian).
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Petrosjan, N. A. Zenkevich. St.Petersburg Univ. Press, 168-177.
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consistency. Games and Economic Behavior, 12, 219–225.
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Peleg, B. and Tijs, S. (1996). The consistency principle for games in strategic form. International Journal of Game Theory, 25, 13–34.
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Voorneveld, M., Vermeulen, D. and Borm, P. (1999). Axiomatization of Pareto equilibria
in multicriteria games. Games and Economic Behavior, 28, 146–154.
The Nash Equilibrium in Multy-Product
Inventory Model
Elena A. Lezhnina, Victor V. Zakharov
St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
Universitetskii pr. 35, St.Petersburg, 198504, Russia
E-mail: [email protected]
Abstract In this paper game theory model of inventory control of a set of
products is treated. We consider model of price competition. We assume that
each retailer can use single-product and multi-product ordering . Demand for
goods which are in stock is constant and uniformly distributed for the period
of planning. Retailers are considered as players in a game with two-level
decision making process. At the higher level optimal solutions of retailers
about selling prices for the non-substituted goods forming Nash equilibrium
are based on optimal inventory solution (order quantity or cycle duration) as
a reaction to chosen prices of the players. We describe the price competition
in context of modified model of Bertrand. Thus at the lower level of the game
each player chooses internal strategy as an optimal reaction to competitive
player’s strategies which are called external. Optimal internal strategies are
represented in analytical form. Theorems about conditions for existences of
the Nash equilibrium in the game of price competition are proved.
Keywords:game theory, non-coalition game, Bertrand oligopoly, Nash equilibrium, logistics.
1.
Introduction
Inventory management of physical goods and other products or elements is an integral part of logistic systems common to all sectors of the economy including industry,
agriculture, and defense. Since the logistic costs account for up to 20% of the costs
of Russian companies under the modern conditions, the issue of reducing costs for
optimization of logistics systems is particularly relevant. The first paper on mathematical modeling in inventory management was written by Harris (Harris ,1915) in
1915. We may also note famous book by Hadley and Whitin (Hadley and Whitin,
1963), as well as books by Hax and Candea (Hax and Candea, 1963) and Tersine
(Tersine, 1994). Inventory control systems with single decision maker capture many
important aspects of inventory management. On the other hand they usually don’t
take into account decisions of other competitors on the market. Game theory is a
mathematical theory of decision making by participants in conflicting or cooperating situations. Its goal is to explain, or to provide a normative guide for, rational
behavior of individuals confronted with strategic decision or involved in social interaction. The theory is concerned with optimal strategic behavior, equilibrium situations, stable outcomes, bargaining, coalition formation, equitable allocations, and
similar concepts related to resolving group differences. The field of game theory may
be divided roughly in two parts, namely non-cooperative game theory and cooperative game theory. Models in non-cooperative game theory assume that each player
192
Elena A. Lezhnina, Victor V. Zakharov
in the game optimizes its own objective and does not care about the effect of its decisions on others. The focus is on finding optimal strategies for each player. Binding
agreements among the players are not allowed. Up to date, many researchers use
non-cooperative game theory to analyse supply chain problems. Non-cooperative
game theory uses the notion of a strategic equilibrium or simply equilibrium to
determine rational outcomes of a game. Numerous equilibrium concepts have been
proposed in the literature (van Damme, 1991). A lot of researches are devoted to
analytical design of contracting arrangements to eliminate inefficiency of decision
making in supply chain with several players like echelon inventory game and local
inventory game (see Cachon review, 2003). Two-level strategic decision making in
Bertrand type competitive inventory model for the first time was treated in (Mansur
Gastratov, Victor Zakharov, 2011). Some widely used concepts are dominant strategy, Nash equilibrium, and sub-game perfect equilibrium. Nash Equilibrium says
that strategies chosen by all players are said to be in Nash equilibrium if no player
can benefit by unilaterally changing their strategy. Nash (Nash, 1951) proved that
every finite game has at least one Nash equilibrium. Historically, most researchers
establish the existence of an equilibrium based on the study of the concavity or
quasi-concavity of profit function. Dasgupta and Maskin (Dasgupta, Maskin, 1986),
Parlar (Parlar, 1988), Mahajan and van Ryzin (Mahajan, van Ryzin, 2001), Netessine et al. (Mahajan, van Ryzin, 2001), among others establish the existence of a
Nash equilibrium based on the two above-mentioned properties of the profit function A starting paper on mathematical models of inventory management was Harris
(1915). Parlar (Parlar, 1988)was the first to analyse the inventory problem in game
theory frameworks. There are two main classes of models depending on whether
price or quantity is regarded as the decision variable. The static models of Cournot
(Cournot, 1838) and Bertrand (Bertrand, 1883) were developed long before modern game theoretic methods. Hence to control the prices is faster and easier than
to good’s quantities, we use the Bertrand oligopoly model of price competition
(Friedman, 1983).
The Harris – Wilson formula is a traditional method for determining the order or
production quantity if you know the total uniform consumption during a period of
time. The formula tries to find an optimal balance between the two costs to minimize
the total cost , which is known as the economic order quantity (EOQ). The classical
EOQ formula is essentially a trade-off between the ordering cost, assumed to be
a flat fee per order, and inventory holding cost. A lager order quantity reduces
ordering frequency, and, hence ordering cost. On the other hend, a smaller order
quantity reduces average inventory but requires more frequent ordering and higher
ordering cost. This formula dating for 1913 is extremely well-known (Harris, 1915).
2.
Preliminaries
2.1. Non-cooperative games
Let’s consider a system
n
n
Γ = hN, {Ωi }i=1 , {Πi }i=1 i .
This system is called a non-cooperative game, where
N = {1, 2, . . . , n} – set of players,
Ωi – set of strategies of player i,
Πi – payoff function of player i.
(1)
The Nash Equilibrium in Multy-Product Inventory Model
193
Players make an interactive decisions simultaneously choosing their strategies
xi from strategy sets Ωi . The agreements and coalition formations are forbidden.
Vector x = (x1 , . . . , xn ) is called situation in the game. As a result players are paid
payoff Πi = Πi (x). A game is a formal representation of a situation in which a number of decision makers (players) interact in a setting of strategic interdependence.
By that, we mean that the welfare of each decision maker depends not only on his
own actions, but also on the actions of the other players. Moreover, the actions that
are best for him to take may depend on what he expects the other players to do.
We say that game theory analyzes interactions between rational, decision-making
individuals who may not be able to predict fully the outcomes of their actions. We
call x⋆ = (x⋆1 , . . . , x⋆n ) a Nash equilibrium if for all admissible strategies xi ∈ Ωi ,
i = 1, . . . , n the following inequalities hold
Πi (x⋆ ) ≥ Πi x⋆1 , x⋆2 . . . , x⋆i−1 , xi , x⋆i+1 , . . . , x⋆n .
Theorem 1. (Kukushkin, Morozov, 1984) In game (1) there exists Nash equilibrium in pure strategies if for each i ∈ N strategy set Ωi is compact and convex, and payoff function Πi (x) is concave with respect to xi and continuous on
Ω = Ω1 × Ω2 × . . . × Ωn .
Assume for any i ∈ N the function Πi (x) is continuously differentiable with
respect to xi . From ( Tirol, 2000) we can see that first-order necessary condition
for Nash equilibrium is the following
∂Πi (x⋆ )
= 0, i ∈ N.
(2)
∂xi
Suppose the payoff function Πi (x), i = 1, . . . , n is concave for all xi ∈ Ωi . In
this case solution of system (2) appears to be a Nash equilibrium in pure strategies
in non-cooperative game
n
n
Γ = hN, {Ωi }i=1 , {Πi }i=1 i .
2.2. Oligopoly
There are two most notable models in oligopoly theory: Cournot oligopoly, and
Bertrand oligopoly. In the Cournot model, firms control their production level,
which influences the market price. In the Bertrand model, firms choose the price to
charge for a unit of product, which affects the market demand.
Definition 1. Non-cooperative oligopoly is a market where a small number of firms
act independently but are aware of each other’s actions.
In the oligopoly model we suppose that:
1. firms are rational;
2. firms reason strategically.
Firms or players meet only once in a single period model. The market then clears
one and for all. There is no repetition of the interaction and hence, no opportunity
for the firms to learn about each other over time. Such models are appropriate for
markets that last only a brief period of time. Cournot and Bertrand oligopolies are
modeled as strategic games, with continuous action sets (either production levels
or prices). We study competitive markets in which firms use price as their strategic
variable.
194
3.
Elena A. Lezhnina, Victor V. Zakharov
The Price Competition
Let’s consider a market with n retailers sell m products: i = 1, . . . , n, j = 1, . . . , m.
In this model each supplier forms for the planning period T single-product orders
to supplier.
Let qij be the quantity of product j in the order; qi = (qi1 , . . . , qim ) – the order
vector of supplier i. After the order was received retailer assigns the prices for every
good for selling: pi = (pi1 , . . . , pim ), i = 1, . . . , n – price vector of player i, where
pij – the price assigned by retailer i for product j.
Assume the demand for items of the product is known and uniform during
a period of planning. Dij (p1j , . . . , pnj ) – demand function for good j with price
appointed by player i and other players. This is the inverse function of pi . In the
price competition the demand on the good depends on prices appointed by other
players. Due to the single-product orders the total inventory cost function of retailer
i could be expressed as
T Ci (p1 , . . . , pn , qi1 , . . . , qim ) =
m X
O Dij (p1j , . . . , pnj )
H qij
=
cj Dij (p1j , . . . , pnj ) + cij
+ cij
,
qij
2
j=1
where
pi = (pi1 , . . . , pim ),
cO
ij – order cost per unit of good j for player i,
cH
ij – holding cost per unit of product j for retailer i during period T ,
cj – procurement price of product j fixed by supplier.
We assume that prices satisfy the conditions:
pij > cj ,
i = 1, . . . , n,
j = 1, . . . , m.
The payoff function is expressed as
Πi (p1 , . . . , pn , qi ) =
=
m
X
j=1
(3)
pij Dij (p1j , . . . , pnj , qi ) − T Ci (p1 , . . . , pn , qi ).
Following single-product inventory game theory model (Mansur Gastratov, Victor
Zakharov, 2011) we consider expanding this model for multi-product one under the
condition of single-product ordering strategies of the players (retailers). As in singleproduct model retailer has to calculate inventory decision on order quantity of a
product as an optimal reaction for the products prices assigned by all players. That
is we also can introduce internal and external strategies of the players as follows.
Definition 2. We define qi = (qi1 , . . . , qim ) as internal strategy, and pi = (pi1 , . . . ,
pim ) as external strategy of player i.
To take into account influence of external strategy of player to internal one we
could find optimal reaction of the retailer to prices assigned by all players. We
would realize two-stage procedure.
195
The Nash Equilibrium in Multy-Product Inventory Model
To find optimal internal strategy retailer has to solve the following problem
min
(qi1 ,...,qim )
T Ci (pi1 , . . . , pim , qi ) =
m X
Dij (p1j , . . . , pnj )
H qij
cj Dij (p1j , . . . , pnj ) + cO
+
c
.
ij
ij
(qi1 ,...,qim )
qij
2
j=1
=
min
The value of the economic order quantity is defined as internal player strategy
qi = (qi1 , . . . , qim ). In this case it is possible to use the Harris-Wilson formula
because of demand function has additive form. Now we have
s
2cO
ij Dij (p1j , . . . , pnj )
∗
qij =
.
(4)
cH
ij
∗
Now it is possible to substitute the optimal qij
in (3). We get the new payoff
function:
m
X
e i (pi1 , . . . , pim ) =
Π
pij Dij (p1j , . . . , pnj , qi ) − T Ci∗ (pi1 , . . . , pim ),
j=1
where
=
m
X
j=1

T Ci∗ (pi1 , . . . , pim ) =

Dij (p1j , . . . , pnj )
cj Dij (p1j , . . . , pnj ) + cO r
+ cH
ij
ij

2cO D (p ,...,p )
ij
1j
ij
r
nj
2cO
ij Dij (p1j ,...,pnj )
cH
ij
2
cH
ij


.

On the next stage the modified price competition of Bertrand oligopoly is considered. Retailers choose the goods’ prices according to the price competition with
other players in non-cooperative game:
E
D n on
n
Γ = N, Π̃i
, {Ωi }i=1 ,
(5)
i=1
where N = 1, . . . , n – set of players,
Ωi – strategy set of player i,
where Ωi = Ωi1 × Ωi2 × . . . , Ωim ,
Ωij = {pij | pij > cj }, i = 1, . . . , n, j = 1, . . . , m.
fi (pi1 , . . . , pim ) – payoff function of player i.
Π
This function depends on external player strategies (pi1 , . . . , pim ) ∈ Ω1 × Ω2 × . . . ×
Ωn .
Every player i chooses external strategy pi ∈ Ωi , which gives the decision of
problem

m
X
max Πi (pi1 , . . . , pim ) = max 
pij Dij (p1j , . . . , pnj )−
(p1 ,...,pn )
−
(p1 ,...,pn )

j=1
m
X

Dij (p1j , . . . , pnj )
cj Dij (p1j , . . . , pnj ) + cO r
+ cH
ij
ij

2cO Dij (p1j ,...,pnj )
j=1
ij
cH
ij
r
2cO
ij Dij (p1j ,...,pnj )
cH
ij
2


 .

196
Elena A. Lezhnina, Victor V. Zakharov
The optimal strategy is achieved by finding Nash equilibrium (Nash strategies),
which is the most commonly used solution concept in game theory. The convexity,
fi (pi1 , . . . , pim ) is necessary and
continuity and differentiability for payoff function Π
sufficient for existence of Nash equilibrium. In terms of convenience we denote
q
q cH
ij
√
Cf
=
+
1
+
cO
ij
ij .
2
Now we can rewrite the cost function as
T Ci∗ (pi1 , . . . , pim ) =
m h
i
X
eij Dij (p1j , . . . , pnj )1/2
cj Dij (p1j , . . . , pnj ) + C
j=1
and payoff function is expressed as
fi (pi1 , . . . , pim ) =
Π
m
m h
i
X
X
fij Dij (p1j , . . . , pnj )1/2 .
=
pij Dij (p1j , . . . , pnj ) −
cj Dij (p1j , . . . , pnj ) + C
j=1
j=1
The existence of Nash equilibrium depends on demand function. There exist two
cases for demand function form.
First case: the case when demand function is in linear form.
Proposition 1. If for every i ∈ N strategy set Ωi is compact and convex and
demand function has a linear form then there exists the Nash equilibrium in the
game (5).
Proof. The demand function is linear, continuous and differentiable. The square
1
root function (Dij (p1j , . . . , pnj )) 2 is concave, continuous and differentiable with
fi (p1 , . . . , pn ) is represented as difference
respect to pij on Ωi . The payoff function Π
of linear function and square root function. From the properties of this function we
get that the payoff function is concave, continua, and differentiable with respect to
pij on Ωi . Which leads to existence of unique Nash equilibrium.
⊔
⊓
Second case: the case of non-linear demand function. In general case for existence
of Nash equilibrium we need the special conditions.
Proposition 2. Let the following conditions be satisfied:
1. for every i ∈ N strategy set Ωi is compact and convex;
2. the demand function Dij (p1j , . . . , pnj ) is continuous and differentiable with respect to pij on Ωi ;
fi (pi1 , . . . , pim ) is concave with respect to pij on Ωi , i = 1, . . . , n,
3. the function Π
j = 1, . . . , m.
D n on
E
fi
Under these conditions the Nash equilibrium in game Γ = N, Π
, {Ωi }ni=1
i=1
exists.
197
The Nash Equilibrium in Multy-Product Inventory Model
Proof. If function Dij (p1j , . . . , pnj ) is continuous and differentiable function on Ωi
fi (pi1 , . . . , pim ) is continuous and differentiable
with respect to pij , then function Π
fi (pi1 , . . . , pim )
with respect to pij on Ωi , i = 1, . . . , n, j = 1, . . . , m. And if function Π
is concave, then all conditions of Theorem 1 are satisfied and there exists the Nash
equilibrium.
⊔
⊓
According to the Teorem 1, there exist the Nash equilibrium in this game. And,
from (Tirol, 2000), the equilibrium point (p⋆1 , . . . , p⋆n ) is found by solving set of
equations:
=
m X
j=1
fi (p1 , . . . , pn )
∂Π
=
∂pij
Dij (pi1 , . . . , pim ) + (pij − cj )
#
eij
C
∂Dij (p1j , . . . , pnj )
− p
,
∂pij
2 Dij (p1j , . . . , pnj )
∂Dij (p1j , . . . , pnj )
−
∂pij
i = 1, . . . , n,
j = 1, . . . , m.
The issue appears when Dij (p1j , . . . , pnj ) = 0. In this case prices set by player are
too high and demand is zero. The player does not participate in price competition.
Thus, we suppose that Dij (p1j , . . . , pnj ) > 0. Substituting the solutions of this
∗
– the optimal goods
equation set (p⋆1 , . . . , p⋆n ) to (4) we will finally get the qij
quantity for player’s order.
4.
The Multi-Product Orders.
As in the first case, we have the market with m products and n retailers: i = 1, . . . , n,
j = 1, . . . , m.
Let τi T be duration of planning period.
Suppose that all retailers set multi-product orders assigning equal duration of
cycles between deliveries as a part of period T . Let τi T be duration of planning
period. Due to uniform demand the following equality takes place for quantity qij
in multi-product order:
qij = τi Dij (p1j , . . . , pnj ).
Assume that ordering cost for multi-product order for player i is equal to cMO
.
i
Then total inventory cost function for player i can be expressed as follows:
m X
cMO
H τi Dij (p1j , . . . , pnj )
i
T Ci (pi1 , . . . , pim , τi ) =
+
cj Dij (p1j , . . . , pnj ) + cij
.
τi
2
j=1
Payoff function is described as
Πi (pi1 , . . . , pim , τi ) =
m
X
j=1
pij Dij (p1j , . . . , pnj )−
m X
cMO
i
H τi Dij (p1j , . . . , pnj )
−
−
cj Dij (p1j , . . . , pnj ) + cij
.
τi
2
j=1
198
Elena A. Lezhnina, Victor V. Zakharov
As in the case of single-product ordering we use the two stage decision making
procedure
To find optimal iternal strategy of player i we have to solve the following problem:
min T Ci (pi1 , . . . , pim , τi ) =
= min
τi
cMO
i
τi
+
m X
τi
cj Dij (p1j , . . . , pnj ) +
τi Dij
cH
ij
i=1
(p1j , . . . , pnj )
2
!
.
This is the problem EQO (optimal economic order). Now it is possible to use
Harris – Wilson formula because of cost function additive form. Using the dependence form
qij = τi Dij (p1j , . . . , pnj ),
it’s possible to find the optimal τi⋆ :
s
τi∗ =
2cMO
i
.
H
j=1 cij Dij (p1j , ..., pnj )
Pm
(6)
Substituting this optimal τi⋆ into the formula of payoff function we obtain
fi (pi1 , . . . , pim ) =
Π

1/2
m
m
X
X
2

=
(pij − cj )Dij (p1j , . . . , pnj ) − √ (cMO
)1/2 
cH
i
ij Dij (p1j , . . . , pnj )
2
j=1
j=1
As a result we get that the payoff function depends on external strategies only.
On the next stage Bertrand oligopoly with price competition is considered. On
the second stage player finds optimal prices according to the competition with other
players. Let’s consider non-cooperative game
D n on
E
n
Γ = N, Π̃i
, {Ωi }i=1 ,
i=1
Ωi – strategy set of player i,
where Ωi = Ωi1 × Ωi2 × . . . , Ωim ,
Ωij = {pij | pij > cj }, i = 1, . . . , n, j = 1, . . . , m.
fi (pi1 , . . . , pim ) – payoff function of player i. This function depends on external
Π
player strategies (pi1 , . . . , pim ) ∈ Ω1 × Ω2 × . . . × Ωn .
Every player i chooses external strategy pi ∈ Ωi , which gives the decision of problem.
The aim of each player is to maximize their payoff in the price competition:
fi (pi1 , . . . , pim )
Π
−→
pi1 ,...,pim
max .
According to the Teorem 1, there exist the Nash equilibrium in this game.
The existence of Nash equilibrium depends on the form of payoff function.
Proposition 3. Suppose the following conditions for i = 1, . . . , n are satisfied:
1. for every i ∈ N strategy set Ωi is compact and convex;
The Nash Equilibrium in Multy-Product Inventory Model
199
2. demand function has a linear form.
Under these conditions Nash equilibrium in pure strategies exists.
Proof. If function Dij (p1j , . . . , pnj ) has a linear form, then it is continuous and
fi (pi1 , . . . , pim ) is
differentiable function on Ωi with respect to pij , and function Π
continuous and differentiable with respect to pij on Ωi , i = 1, . . . , n, j = 1, . . . , m.
fi (pi1 , . . . , pim ) is represented as difference of linear function and square
Function Π
root function. From the properties of this functions follows that the payoff function
is concave, continua and differentiable with respect to pij on Ωi , i = 1, . . . , n,
j = 1, . . . , m. All conditions of Theorem 1 are satisfied and there exists the Nash
equilibrium.
⊔
⊓
Proposition 4. Suppose the following conditions for i = 1, . . . , n are satisfied:
1. for every i ∈ N strategy set Ωi is compact and convex;
2. demand function Dij (p1j , . . . , pnj ) is continuous and differentiable with respect
to pij on Ωi ;
fi (pi1 , . . . , pim ) is concave with respect to pij on Ωi .
3. payoff function Π
Under these conditions Nash equilibrium in pure strategies exists.
Proof. If function Dij (p1j , . . . , pnj ) is continuous and differentiable function on Ωi
fi (pi1 , . . . , pim ) is continuous and differentiable
with respect to pij , then function Π
fi (pi1 , . . . , pim )
with respect to pij on Ωi , i = 1, . . . , n, j = 1, . . . , m. And if function Π
is concave, then all conditions of Theorem 2 are satisfied and there exists the Nash
equilibrium.
⊔
⊓
From (Tirol, 2000), the equilibrium point (p⋆1 , . . . , p⋆n ) is found by solving set of
equations:
fi (p1 , . . . , pn )
∂Π
= 0,
∂pij
i = 1, . . . , n.
After finding optimal strategy value of p⋆ = (p⋆1 , . . . , p⋆n ) it is possible to substitute its to (6) and to calculate the optimal value of period τi⋆ .
5.
Conclusion
In this paper game theory models for multi-product inventory control are treated in
case of competition among retailers. The model of price competition in context of
modied model of Bertran is considered. Each retailer can use two order types: singleproduct and multi-product ordering. Demand for each product is supposed to be
uniform for the period of planning. In game theory model retailers are considered
as players using two-level strategies. At the lower level of the game each player
chooses internal strategy as an optimal reaction to competitive players strategies
which are called external. Optimal internal strategies are represented in analytical
form. Necessary and sufficient conditions for existence of Nash equilibrium in pure
strategies for the cases of linear and non-linear demand functions are proposed.
200
Elena A. Lezhnina, Victor V. Zakharov
References
Nash, J. F. (1951). Non-Cooperative games. Annals of Mathematics, 54, 286–295.
Harris, F. (1915). Operation and Cost. Factory Management Series. A.W. Shaw, Chicago,
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van Damm, E. (1991). Stability and Perfection of Nash Equilibria. Springer-Verlag, Berlin.
Nash, J. F. (1950). Equilibrium Points in n-Person Games. Proceedings of the National
Academy of Sciences, 36, 48–59.
Dasgupta, P., Maskin, E. (1986). The Existence of Equilibrium in Discontinuous Economic
Games. Review of Economics Studies 53, 1–26.
Mahajan, S., van Ryzin, G. J. (2001). Inventory Competition Under Dynamic Consumer
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Netessine, S., Rudi, N., Wang, Y (2003). Dynamic Inventory Competition and Customer
Retention. Working paper.
Parlae, M. (1988). Game Theoretic Analysis of the Substitutable Product Inventory Problem
with Random Demands. Naval Research Logistics, 35, 397–409.
Tirol, Jean (2000). The markets and the market power: The organization and industry
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Galperina and N. A. Zenkevich. SPb: Institute Economic school, in 2 Volumes. V.1.
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State University, 103 p.
Friedman, James (1983).Oligopoly Theory. Cambridge University Press, 260 p.
Bertrand, J. (1883).Theorie mathematique de la richesse sociale. Journal des Savants., 67,
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Paris, L. Hachette, 198 p.
Hax, A. C. and D. Candea (1984). Production and inventory management. Prentice-Hall.
Englewood Cliffs, N.J., 135 p.
Haldey, G. and T. M. Whitin (1963). Analysis of inventory. Prentice-Hall, Englewood
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Amsterdam, pp. 229–340.
Mansur Gasratov, Victor Zakharov (2011). Games and Inventory Management. In: Dynamic and Sustainability in International Logistics and Supply Chain Management.
Cuvillier Verlag, Gottingen.
Nash Equilibria Conditions for Stochastic Positional Games
Dmitrii Lozovanu1 and Stefan Pickl2
Institute of Mathematics and Computer Science,
Academy of Sciences of Moldova,
Academy str., 5, Chisinau, MD–2028, Moldova
E-mail: [email protected]
http://www.math.md/structure/applied-mathematics/math-modeling-optimization/
2
Institute for Theoretical Computer Science, Mathematics and
Operations Research, Universität der Bundeswehr, München
85577 Neubiberg-München, Germany
E-mail: [email protected]
1
Abstract We formulate and study a class of stochastic positional games
using a game-theoretical concept to finite state space Markov decision processes with an average and expected total discounted costs optimization criteria. Nash equilibria conditions for the considered class of games are proven
and some approaches for determining the optimal strategies of the players
are analyzed. The obtained results extend Nash equilibria conditions for deterministic positional games and can be used for studying Shapley stochastic
games with average payoffs.
Keywords: Markov decision processes, stochastic positional games, Nash
equilibria, Shapley stochastic games, optimal stationary strategies.
1.
Introduction
In this paper we consider a class of stochastic positional games that extends deterministic positional games studied by Moulin,1976, Ehrenfeucht and Mycielski, 1979,
Gurvich at al., 1988, Condon, 1992, Lozovanu and Pick, 2006, 2009. The considered
class of games we formulate and study applying the concept of positional games to
finite state space Markov decision processes with average and expected total discounted costs optimization criteria. We assume that the Markov process is controlled
by several actors (players) as follows: The set of states of the system is divided into
several disjoint subsets which represent the corresponding position sets of the players. Additionally the cost of system’s transition from one state to another is given
for each player separately. Each player has to determine which action should be
taken in each state of his position set of the Markov process in order to minimize
his own average cost per transition or the expected total discounted cost. In these
games we are seeking for a Nash equilibrium.
The main results of the paper are concerned with the existence of Nash equilibria for the considered class of games and determining the optimal strategies of the
players. Necessary and sufficient conditions for the existence of Nash equilibria in
stochastic positional games that extend Nash equilibria conditions for deterministic
positional games are proven. Based on the constructive proof of these results we
propose some approaches for determining the optimal strategies of the players. Additionally we show that the stochastic positional games are tightly connected with
Shapley stochastic games (Shapley, 1953) and the obtained results can be used for
studying a special class of Shapley stochastic games with average payoffs.
202
2.
Dmitrii Lozovanu, Stefan Pickl
Formulation of the Basic Game Models and Some Preliminary
Results
We consider two game-theoretic models. We formulate the first game model for
Markov decision processes with average cost optimization criterion and call it the
stochastic positional game with average payoffs. We formulate the second one for
Markov decision processes with discounted cost optimization criterion and call it
stochastic positional game with discounted payoffs. Then we show the relationship
of these games with Shapley stochastic games.
2.1. Stochastic Positional Games with Average Payoffs
To formulate the stochastic positional game with average payoffs we shall use the
framework of a Markov decision process (X, A, p, c) with a finite set of states X,
a finite set of actions A, a transition probability function p : X × X × A → [0, 1]
that satisfies the condition
X
pax,y = 1, ∀x ∈ X, ∀a ∈ A
y∈X
and a transition cost function c : X × X → R which gives the costs cx,y of states
transitions of the dynamical system from an arbitrary x ∈ X to another state
y ∈ X (see Howard, 1960; Puterman, 2005). For the noncooperative game model
with m players we assume that m transition cost functions
ci : X × X → R, i = 1, 2, . . . , m
are given, where cix,y expresses the cost of the system’s transition from the state
x ∈ X to the state y ∈ X for the player i ∈ {1, 2, . . . , m}. In addition we assume
that the set of states X is divided into m disjoint subsets X1 , X2 , . . . , Xm
X = X1 ∪ X2 ∪ · · · ∪ Xm (Xi ∩ Xj = ∅, ∀i 6= j),
where Xi represents the positions set of the player i ∈ {1, 2, . . . , m}. So, the Markov
process is controlled by m players, where each player i ∈ {1, 2, . . . , m} fixes actions
in his positions x ∈ Xi . We assume that each player fixes actions in the states from
his positions set using stationary strategies, i.e. we define the stationary strategies
of the players as m maps:
si : x → a ∈ Ai (x) for x ∈ Xi , i = 1, 2, . . . , m,
where Ai (x) is the set of actions of the player i in the state x ∈ Xi . Without loss of
generality we may consider |Ai (x)| = |Ai | = |A|, ∀x ∈ Xi , i = 1, 2, . . . , m. In order
to simplify the notation we denote the set of possible actions in a state x ∈ X for
an arbitrary player by A(x). A stationary strategy si , i ∈ {1, 2, . . . , m} in the state
x ∈ Xi means that at every discrete moment of time t = 0, 1, 2, . . . the player i uses
the action a = si (x). Players fix their strategy independently and do not inform
each other which strategies they use in the decision process.
If the players 1, 2, . . . , m fix their stationary strategies s1 , s2 , . . . , sm , respectively, then we obtain a situation s = (s1 , s2 , . . . , sm ). This situation corresponds
si (x)
to a simple Markov process determined by the probability distributions px,y in
the states x ∈ Xi for i = 1, 2, . . . , m. We denote by P s = (psx,y ) the matrix of
Nash Equilibria Conditions for Stochastic Positional Games
203
probability transitions of this Markov process. If the starting state x0 is given, then
for the Markov process with the matrix of probability transitions P s we can determine the average cost per transition ωxi 0 (s1 , s2 , . . . , sm ) with respect to each player
i ∈ {1, 2, . . . , m} taking into account the corresponding matrix of transition costs
C i = (cix,y ). So, on the set of situations we can define the payoff functions of the
players as follows:
Fxi 0 (s1 , s2 , . . . , sm ) = ωxi 0 (s1 , s2 , . . . , sm ),
i = 1, 2, . . . , m.
In such a way we obtain a discrete noncooperative game in normal form which is
determined by a finite set of strategies S 1 , S 2 , . . . , S m of m players and the payoff
functions defined above. In this game we are seeking for a Nash equilibrium (see
Nash, 1951), i.e., we consider the problem of determining the stationary strategies
∗
∗
∗
∗
∗
∗
∗
s1 , s2 , . . . , si−1 , si , si+1 , . . . , sm∗
such that
∗
∗
∗
Fxi 0 (s1 , s2 , . . . , si−1 , si , si+1 , . . . , sm ∗ )
∗
∗
∗
∗
≤ Fxi 0 (s1 , s2 , . . . , si−1 , si , si+1 , . . . , sm ∗ ), ∀si ∈ S i , i = 1, 2, . . . , m.
The game defined above is determined uniquely by the set of states X, the
position sets X1 , X2 , . . . , Xm , the set of actions A, the cost functions ci : X × X →
R, i = 1, 2, . . . , m, the probability function p : X × X × A → [0, 1] and the starting
position x0 . Therefore, we denote this game by (X, A, {Xi }i=1,m , {ci }i=1,m , p, x0 ).
In the case m = 2 and c2 = −c1 we obtain an antagonistic stochastic positional game. If pax,y = 0 ∨ 1, ∀x, y ∈ X, ∀a ∈ A the stochastic positional
game (X, A, {Xi }i=1,m , {ci }i=1,m , p, x0 ) is transformed into the cyclic
game (Ehrenfeucht and Mycielski, 1979, Gurvich at al., 1988,
Condon, 1992,
Lozovanu and Pick, 2006). Some results concerned with the existence of Nash equilibria for stochastic positional games with average payoffs have been derived by
Lozovanu at al., 2011. In particular the following theorem has been proven.
Theorem 1. If for an arbitrary situation s = (s1 , s2 , . . . , sm ) of the stochastic
positional game with average payoffs the matrix of probability transitions P s = (psx,y )
induces an ergodic Markov chain then for the game there exists a Nash equilibrium.
If the matrix P s for some situations do not correspond to an ergodic Markov
chain then for the stochastic positional game with average payoffs a Nash equilibrium may not exist. This follow from the constructive proof of this theorem
(see Lozovanu at al., 2011). An example of a deterministic positional game with
average payoffs for which Nash equilibrium does not exist has been constructed
by Gurvich at al., 1988. However, in the case of antagonistic stochastic positional
games saddle points always exist (Lozovanu and Pickl, 2014), i.e. in this case the
following theorem holds.
Theorem 2. For an arbitrary antagonistic positional game there exists a saddle
point.
The existence of saddle points for deterministic positional games with average
payoffs have been proven by Ehrenfeucht and Mycielski, 1979, Gurvich at al., 1988.
204
Dmitrii Lozovanu, Stefan Pickl
2.2. Stochastic Positional Games with Discounted Payoffs
We formulate the stochastic positional game with discounted payoffs in a similar way
as the game from Section 2.. We assume that for the Markov process m transition
cost functions ci : X × X → R, i = 1, 2, . . . , m, are given and the set of states X is
divided into m disjoint subsets X1 , X2 , . . . , Xm , where Xi represents the positions
set of the player i ∈ {1, 2, . . . , m}. The Markov process is controlled by m players,
where each player i ∈ {1, 2, . . . , m} fixes actions in his positions x ∈ Xi using
stationary strategies, i.e. the stationary strategies of the players in this game are
defined as m maps:
si : x → a ∈ A(x)
for
x ∈ Xi ; i = 1, 2, . . . , m.
Let s1 , s2 , . . . , sm be a set of stationary strategies of the players that determine the
situation s = (s1 , s2 , . . . , sm ). Consider the matrix of probability transitions P s =
(psx,y ) which is induced by the situation s, i.e., each row of this matrix corresponds to
si (x)
a probability distribution px,y in the state x where x ∈ Xi . If the starting state x0 is
given, then for the Markov process with the matrix of probability transitions P s we
can determine the discounted expected total cost σxi 0 (s1 , s2 , . . . , sm ) with respect
to each player i ∈ {1, 2, . . . , m} taking into account the corresponding matrix of
transition costs C i = (cix,y ). So, on the set of situations we can define the payoff
functions of the players as follows:
Fbxi 0 (s1 , s2 , . . . , sm ) = σxi 0 (s1 , s2 , . . . , sm ),
i = 1, 2, . . . , m.
In such a way we obtain a new discrete noncooperative game in normal form which
is determined by the sets of strategies S 1 , S 1 , . . . , S m of m players and the payoff
functions defined above. In this game we are seeking for a Nash equilibrium. We denote the stochastic positional game with discounted payoffs by (X, A, {Xi }i=1,m ,
{ci }i=1,m , p, γ, x0 ).
For this game the following result has been proven (Lozovanu, 2011).
Theorem 3. For an arbitrary stochastic positional game (X, A, {Xi }i=1,m ,
{ci }i=1,m , p, γ, x0 ) with given discount factor 0 < γ < 1 there exists a Nash
equilibrium.
Based on a constructive proof of Theorems 1,3 some iterative procedures for
determining Nash equilibria in the considered positional games have been proposed
(see Lozovanu at al., 2011).
2.3.
The Relationship of Stochastic Positional Games with Shapley
Stochastic Games
A stochastic game in the sense of Shapley (see Shapley, 1953) is a dynamic game
with probabilistic transitions played by several players in a sequence of stages, where
the beginning of each stage corresponds to a state of the dynamical system. The
game starts at a given state from the set of states of the system. At each stage
players select actions from their feasible sets of actions and each player receives a
stage payoff that depends on the current state and the chosen actions. The game
then moves to a new random state the distribution of which depends on the previous
state and the actions chosen by the players. The procedure is repeated at a new
Nash Equilibria Conditions for Stochastic Positional Games
205
state and the play continues for a finite or infinite number of stages. The total
payoff of a player is either the limit inferior of the average of the stage payoffs or
the discounted sum of the stage payoffs.
So, an average Shapley stochastic game with m players consists of the following
elements:
1. A state space X (which we assume to be finite);
2. A finite set Ai (x) of actions with respect to each player i ∈ {1, 2, . . . , m}
for an arbitrary state x ∈ X;
3. A stage payoff f i (x, a) with respect to each player i ∈ {1, 2, . . .Q
, m}
for each state x ∈ X and for an arbitrary action vector a ∈ i Ai (x);
Q
Q
4. A transition probability function p : X × x∈X i Ai (x) × X → [0, 1]
that gives the probability transitions pax,y from an arbitrary x ∈ X
Q
to an arbitrary y ∈ Y for a fixed
action vector a ∈ i Ai (x), where
P
Q
a
i
y∈X px,y = 1, ∀x ∈ X, a ∈
i A (x);
5. A starting state x0 ∈ X.
The stochastic game starts in state x0 . At stage t players observe state xt and
simultaneously choose actions ait ∈ Ai (xt ), i = 1, 2, . . . , m. Then nature selects
a state xt+1 according to probability transitions paxtt ,y for fixed action vector at =
(a1t , a2t , . . . , am
t ). A play of the stochastic game x0 , a0 , x1 , a1 , . . . , xt , at , . . . defines
a stream of payoffs f0i , f1i , f2i , . . . , where fti = f i (xt , at ), t = 0, 1, 2, . . . . The t-stage
average stochastic game is the game where the payoff of player i ∈ {1, 2, . . . , m} is
t−1
Fti
1X i
f .
=
t τ =1 τ
The infinite average stochastic game is the game where the payoff of player i ∈
{1, 2, . . . , m} is
i
F = lim Fti .
t→∞
In a similar a Shapley stochastic game with expected discounted payoffs of the
players is defined. In such a game along to the elements described above also a
discount factor λ (0 < λ < 1) is given and the total payoff of a player represents
the expected discounted sum of the stage payoffs.
By comparison for Shapley stochastic games with stochastic positional games
we can observe the following. The probability transitions from a state to another
state as well as the stage payoffs of the players in a Shapley stochastic game depend
on the actions chosen by all players, while the probability transitions from a state
to another state as well as the stage payoffs (the immediate costs of the players) in
a stochastic positional game depend only on the action of the player that controls
the state in his position set. This means that a stochastic positional game can be
regarded as a special case of the Shapley stochastic game. Nevertheless we can see
that stochastic positional games can be used for studying some classes of Shapley
stochastic games.
The main results concerned with determining Nash equilibria in Shapley stochastic games have been obtained by Gillette, 1957, Mertens and Neyman, 1981,
Filar and Vrieze, 1997, Lal and Sinha, 1992, Neyman and Sorin, 2003. Existence
206
Dmitrii Lozovanu, Stefan Pickl
of Nash equilibria for such games are proven in the case of stochastic games with
a finite set of stages and in the case of the games with infinite stages if the total payoff of each player is the discounted sum of stage payoffs. If the total payoff
of a player represents the limit inferior of the average of the stage payoffs then
the existence of a Nash equilibrium in Shapley stochastic games is an open question. Based on the results mentioned in previous sections we can show that in the
case of the average non-antagonistic stochastic games a Nash equilibrium may not
exist. In order to prove this we can use the average stochastic positional game
(X, A, {Xi }i=1,m , {ci }i=1,m , p, x0 ) from section 2. It is easy to observe that this
game can be regarded as a Shapley stochastic game with average payoff functions
of the players, where for a fixed situation s = (s1 , s2 , . . . , sm ) the probability transition psx,y from a state x = x(t) ∈ Xi to a state y = x(t + 1) ∈ X depends only
on a strategy si of player i and the P
corresponding stage payoff in the state x of
player i ∈ {1, 2, . . . , m} is equal to y∈X psx,y cix,y . Taking into account that the
cyclic game represents a particular case of the average stochastic positional game
and for the cyclic game Nash equilibrium may not exist (see Gurvich at al., 1988)
we obtain that for the average non-antagonistic Shapley stochastic game a Nash
equilibrium may not exist. However in the case of average payoffs Theorem 1 can
be extended for Shapley stochastic games.
3.
Nash Equilibria Conditions for Stochastic Positional Games with
Average Payoffs
In this section we formulate Nash equilibria conditions for stochastic positional
games in terms of bias equations for Markov decision processes. We can see that
Nash equilibria conditions in such terms may be more useful for determining the
optimal strategies of the players.
Theorem 4. Let (X, A, {Xi }i=1,m , {ci }i=1,m , p, x) be a stochastic positional game
with a given starting position x ∈ X and average payoff functions
Fx1 (s1 , s2 , . . . , sm ), Fx2 (s1 , s2 , . . . , sm ), . . . , Fxm (s1 , s2 , . . . , sm )
of the players 1, 2, . . . , m, respectively. Assume that for an arbitrary situation s =
(s1 , s2 , . . . , sm ) of the game the transition probability matrix P s = (psx,y ) corresponds
to an ergodic Markov chain. Then there exist the functions
εi : X → R,
i = 1, 2, . . . , m
and the values ω 1 , ω 2 , . . . , ω m that satisfy the following conditions:
1) µix,a +
where
P
y∈X
µix,a
pax,y εiy − εix − ω i ≥ 0, ∀x ∈ Xi ,
P a i
=
px,y cx,y ;
2) min {µix,a +
a∈A(x)
∀a ∈ A(x), i = 1, 2, . . . , m,
y∈X
P
y∈X
pax,y εiy − εix − ω i } = 0,
∀x ∈ Xi , i = 1, 2, . . . , m;
∗
3) on each position set Xi , i ∈ {1, 2, . . . , m} there exists a map si : Xi → A such
that
o
n
X
∗
si (x) = a∗ ∈ Arg min µix,a +
pax,y εiy − εix − ω i
a∈A(x)
y∈X
Nash Equilibria Conditions for Stochastic Positional Games
and
X
µjx,a∗ +
∗
y∈X
∗
207
pax,y εjy − εjx − ω j = 0, ∀x ∈ Xi , j = 1, 2, . . . , m.
∗
The set of maps s1 , s2 , . . . , sm∗ determines a Nash equilibrium situation s∗ =
∗
∗
(s1 , s2 , . . . , sm∗ ) for the stochastic positional game (X, A, {Xi }i=1,m , {ci }i=1,m ,
p, x and
∗
∗
Fxi (s1 , s2 , . . . , sm ∗ ) = ω i , ∀x ∈ X, i = 1, 2, . . . , m.
∗
∗
Moreover, the situation s∗ = (s1 , s2 , . . . , sm∗ ) is a Nash equilibrium for an arbitrary starting position x ∈ X.
Proof. Let a stochastic positional game with average payoffs be given and assume
that for an arbitrary situation s of the game the transition probability matrix
P s = (psx,y ) corresponds to an ergodic Markov chain. Then according to Theorem 1
∗
∗
for this game there exists a Nash equilibrium s∗ = (s1 , s2 , . . . , sm∗ ) and we can
set
∗
∗
ω i = Fxi (s1 , s2 , . . . , sm∗ ), ∀x ∈ X, i = 1, 2, . . . , m.
∗
∗
∗
∗
Let us fix the strategies s1 , s2 , . . . , si−1 , si+1 , . . . , sm ∗ of the players 1, 2, . . . , i −
1, i+1, . . . , m and consider the problem of determining the minimal average cost per
transition with respect to player i. Obviously, if we solve this decision problem then
∗
we obtain the strategy si . We can determine the optimal strategy of this decision
problem with an average cost optimization criterion using the bias equations with
respect to player i. This means that there exist the functions ǫi : X → R and the
values ω i , i = 1, 2, . . . , m that satisfy the conditions:
1) µix,a +
2) min
a∈A(x)
P
y∈X
n
pax,y εiy − εix − ω i ≥ 0,
µix,a +
P
y∈X
∀x ∈ Xi , ∀a ∈ A(x);
o
pax,y εiy − εix − ω i = 0,
∗
∗
∀x ∈ Xi .
∗
∗
Moreover, for fixed strategies s1 , s2 , . . . , si−1 , si+1 , . . . , sm∗ of the corresponding
∗
players 1, 2, . . . , i − 1, i + 1, . . . , m we can select the strategy si of player i where
o
n
X
∗
si (x) ∈ Arg min µix,a +
pax,y εiy − εix − ω i
a∈A(x)
∗
y∈X
∗
and ω i = Fxi (s1 , s2 , . . . , sm ∗ ), ∀x ∈ X, i = 1, 2, . . . , m. This means that conditions
1)–3) of the theorem hold.
Corollary 1. If for a stochastic positional game (X, A, {Xi }i=1,m , {ci }i=1,m ,
∗
∗
p, x) with average payoffs there exist a Nash equilibrium s∗ = (s1 , s2 , . . . , sm∗ )
which is a Nash equilibrium for an arbitrary starting position of the game x ∈ X and
∗
∗
for arbitrary two different starting positions x, y ∈ X holds Fxi (s1 , s2 , . . . , sm∗ ) =
i 1∗ 2∗
m∗
Fy (s , s , . . . , s ) then there exists the functions
εi : X → R,
i = 1, 2, . . . , m
208
Dmitrii Lozovanu, Stefan Pickl
and the values ω 1 , ω 2 , . . . , ω m that satisfy the conditions 1) − 3) from Theorem 4.
∗
∗
So, ω i = Fxi (s1 , s2 , . . . , sm∗ ), ∀x ∈ X, i = 1, 2, . . . , m and an arbitrary Nash
equilibrium can be found by fixing
n
o
X
∗
si (x) = a∗ ∈ Arg min µix,a +
pax,y εiy − εix − ω i .
a∈A(x)
y∈X
Using the elementary properties of non ergodic Markov decision processes with
average cost optimization criterion the following lemma can be gained.
Lemma 1. Let (X, A, {Xi }i=1,m , {ci }i=1,m , p, x) be an average stochastic
∗
∗
positional game for which there exists a Nash equilibrium s∗ = (s1 , s2 , . . . , sm ∗ ),
which is a Nash equilibrium for an arbitrary starting position of the game with
∗
∗
∗
∗
ωxi = Fxi (s1 , s2 , . . . , sm∗ ). Then s∗ = (s1 , s2 , . . . , sm∗ ) is a Nash equilibrium
for the average stochastic positional game (X, A, {Xi }i=1,m , {ci }i=1,m , p, x),
where
cix,y = cix,y − ωxi , ∀x, y ∈ X, i = 1, 2, . . . , m
and
i
∗
∗
F x (s1 , s2 , . . . , sm ∗ ) = 0, ∀x ∈ X, i = 1, 2, . . . , m.
Now using Corollary 1 and Lemma 1 we can prove the following results.
Theorem 5. Let (X, A, {Xi }i=1,m , {ci }i=1,m , p, x) be an average stochastic positional game. Then in this game there exists a Nash equilibrium for an arbitrary
starting position x ∈ X if and only if there exist the functions
εi : X → R, i = 1, 2, . . . , m
and the values ωx1 , ωx2 , . . . , ωxm for x ∈ X that satisfy the following conditions:
P a i
1) µix,a +
px,y εy − εix − ωxi ≥ 0, ∀x ∈ Xi , ∀a ∈ A(x), i = 1, 2, . . . , m,
y∈X
P a i
where µix,a =
px,y cx,y ;
y∈X
2)
min {µix,a
a∈A(x)
+
P
y∈X
pax,y εiy − εix − ωxi } = 0,
∀x ∈ Xi , i = 1, 2, . . . , m;
∗
3) on each position set Xi , i ∈ {1, 2, . . . , m} there exists a map si : Xi → A such
that
o
n
X
∗
si (x) = a∗ ∈ Arg min µix,a +
pax,y εiy − εix − ω i
a∈A(x)
and
µjx,a∗ +
X
y∈X
∗
pax,y εjy − εjx − ω j = 0,
y∈X
∀x ∈ Xi , j = 1, 2, . . . , m.
∗
∗
If such conditions hold then the set of maps s1 , s2 , . . . , sm∗ determines a Nash
equilibrium of the game for an arbitrary starting position x ∈ X and
∗
∗
Fxi (s1 , s2 , . . . , sm ∗ ) = ωxi , i = 1, 2, . . . , m.
209
Nash Equilibria Conditions for Stochastic Positional Games
Proof. The sufficiency condition of the theorem is evident. Let us prove the necessity
one. Assume that for the considered average stochastic positional game there exists
∗
∗
a Nash equilibrium s∗ = (s1 , s2 , . . . , sm ∗ ) which is a Nash equilibrium for an
arbitrary starting position of the game. Denote
∗
∗
σxi = Fbxi (s1 , s2 , . . . , sm∗ ), ∀x ∈ X, i = 1, 2, . . . , m
and consider the following auxiliary game (X, A, {Xi }i=1,m , {ci }i=1,m , p, x),
where
cix,y = cix,y − ωxi , ∀x, y ∈ X, i = 1, 2, . . . , m.
Then according to Lemma 1 the auxiliary game has the same Nash equilibrium s∗ =
∗
∗
(s1 , s2 , . . . , sm ∗ ) as initial one. Moreover, this equilibrium is a Nash equilibrium
for an arbitrary starting position of the game and
∗
i
∗
F x (s1 , s2 , . . . , sm∗ ) = 0, ∀x ∈ X, i = 1, 2, . . . , m.
Therefore, according to Corollary 1, for the auxiliary game there exist the functions
εi : X → R,
i = 1, 2, . . . , m
and the values ω 1 , ω 2 , . . . , ωm (ω i = 0, i = 1, 2, . . . , m), that satisfy the conditions
of Theorem 4, i.e.
P a i
1) µix,a +
px,y εy − εix − ωix ≥ 0, ∀x ∈ Xi , ∀a ∈ A(x), i = 1, 2, . . . , m,
y∈X
P a i
where µix,a =
px,y cx,y ;
y∈X
2)
min {µix,a +
a∈A(x)
P
y∈X
pax,y εiy − εix − ω ix } = 0,
∀x ∈ Xi , i = 1, 2, . . . , m;
∗
3) on each position set Xi , i ∈ {1, 2, . . . , m} there exists a map si : Xi → A such
that
o
n
X
∗
si (x) = a∗ ∈ Arg min µix,a +
pax,y εiy − εix − ω i
a∈A(x)
and
µjx,a∗ +
X
y∈X
∗
pax,y εjy − εjx − ω j = 0,
y∈X
∀x ∈ Xi , j = 1, 2, . . . , m.
Taking into account that ω ix = 0, and µix,a = µix,a − ωxi (because cix,y = cx,y − ωxi )
we obtain conditions 1 − 3 of the theorem.
4.
Nash Equilibria Conditions for Stochastic Positional Games with
Discounted Payoffs
Now we formulate Nash equilibria conditions in the terms of bias equations for
stochastic positional games with discounted payoffs.
Theorem 6. Let a stochastic positional game (X, A, {Xi }i=1,m , {ci }i=1,m , p,
γ, x) with a discount factor 0 < γ < 1 be given. Then there exist the values
σxi , i = 1, 2, . . . , m, for x ∈ X that satisfy the following conditions:
210
Dmitrii Lozovanu, Stefan Pickl
1) µix,a + γ
P
pax,y σyi − σxi ≥ 0, ∀x ∈ Xi , ∀a ∈ A(x), i = 1, 2, . . . , m,
P a i
=
px,y cx,y .
y∈X
where µix,a
y∈X
P a i
2) min µix,a + γ
px,y σy − σxi = 0,
a∈A(x)
y∈X
∀x ∈ Xi , i = 1, 2, . . . , m;
∗
3) on each position set Xi , i ∈ {1, 2, . . . , m} there exists a map si : Xi → A such
that
X
a
i
i
i∗
∗
i
px,y σy − σx , ∀x ∈ Xi
s (x) = a ∈ Arg min µx,a + γ
a∈A(x)
and
µjx,a∗ + γ
X
y∈X
∗
y∈X
∗
pax,y σyj − σxj = 0, ∀x ∈ Xi ,
j = 1, 2, . . . , m.
∗
The set of maps s1 , s2 , . . . , sm ∗ determines a Nash equilibrium situation s∗ =
∗
∗
(s1 , s2 , . . . , sm∗ ) for the stochastic positional game with discounted payoffs, where
∗
∗
Fbxi (s1 , s2 , . . . , sm∗ ) = σxi , ∀x ∈ X,
1∗
2∗
Moreover, the situation s = (s , s , . . . , s
trary starting position x ∈ X.
∗
m∗
i = 1, 2, . . . , m.
) is a Nash equilibrium for an arbi-
Proof. According to Theorem 3 for the discounted stochastic positional game
(X, A, {Xi }i=1,m , {ci }i=1,m , p, γ, x) there exists a Nash equilibrium s∗ =
∗
∗
(s1 , s2 , . . . , sm∗ ) which is a Nash equilibrium for an arbitrary starting position
x ∈ X of the game. Denote
∗
∗
σxi = Fbxi (s1 , s2 , . . . , sm ∗ ), ∀x ∈ X, i = 1, 2, . . . , m.
∗
∗
∗
∗
Let us fix the strategies s1 , s2 , . . . , si−1 , si+1 , . . . , sm∗ of the players 1, 2, . . . , i−
1, i + 1, . . . , m and consider the problem of determining the expected total discounted cost with respect to player i. Obviously, the optimal stationary strat∗
egy for this problem is si . Then according to the properties of the bias equations for this Markov decision problem with discounted costs there exist the values
σxi , i = 1, 2, . . . , m, for x ∈ X that satisfy the conditions:
P a i
1) µix,a + γ
px,y σy − σxi ≥ 0, ∀x ∈ Xi , ∀a ∈ A(x), i = 1, 2, . . . , m;
y∈X
P a i
2) min µix,a + γ
px,y σy − σxi = 0,
a∈A(x)
y∈X
∗
∗
∀x ∈ Xi i = 1, 2, . . . , m.
∗
∗
∗
Moreover, for fixed strategies s1 , s2 , . . . , si−1 , si , si+1 , . . . , sm ∗ of the corre∗
sponding players 1, 2, . . . , i − 1, i + 1, . . . , m we can select the strategy si of the
player i where
X
∗
si (x) ∈ Arg min µix,a + γ
pax,y σyi − σxi
a∈A(x)
and
y∈X
∗
∗
Fbxi (s1 , s2 , . . . , sm ∗ ) = σxi , ∀x ∈ X, i = 1, 2, . . . , m.
This means that the conditions 1)–3) of the theorem hold.
Nash Equilibria Conditions for Stochastic Positional Games
5.
211
Saddle Point Conditions for Antagonistic Stochastic Positional
Games
The antagonistic stochastic positional game with the average payoff corresponds to
the case of the game from Section 2 in the case m = 2 when c = c1 = −c2 . So, we
have a game (X, A, X1 , X2 , c, p, x) where the stationary strategies s1 and s2 of the
players are defined as two maps
s1 : x → a ∈ A1 (x) for x ∈ X1 ; s2 : x → a ∈ A1 (x) for x ∈ X2 .
and the payoff function Fx (s1 , s2 ) of the players is determined by the values of average costs ωxs in the Markov processes with the corresponding probability matrices
∗
∗
P s induced by the situations s = (s1 , s2 ) ∈ S. For this game saddle points s1 , s2
always exists (Lozovanu and Pickl, 2014) , i.e. for a given starting position x ∈ X
holds
∗
∗
Fx (s1 , s2 ) = min
max2 Fx (s1 , s2 ) = max
min1 Fx (s1 , s2 ).
1
1 2
2
2 1
s ∈S s ∈S
s ∈S s ∈S
Theorem 7. Let (X, A, X1 , X2 , c, p, x) be an arbitrary antagonistic stochastic
positional game with an average payoff function Fx (s1 , s2 ). Then the system of
equations

P a


εx + ωx = max µx,a +
px,y εy , ∀x ∈ X1 ;



a∈A(x)
y∈X



P a



min µx,a +
px,y εy ,
 εx + ωx = a∈A(x)
y∈X
∀x ∈ X2 ;
has solution under the set of solutions of the system of equations

P a


ω
=
max
p
ω
∀x ∈ X1 ;

x
x,y x ,


a∈A(x)
y∈X



P a



px,y ωx ,
 ωx = min
a∈A(x)
y∈X
∀x ∈ X2 ,
i.e. the last system of equations has such a solution ωx∗ , x ∈ X for which there
exists a solution ε∗x , x ∈ X of the system of equations

P a

∗

ε
+
ω
=
max
µ
+
p
ε
∀x ∈ X1 ;
 x
x,a
x
x,y y ,


a∈A(x)
y∈X



P a


∗

px,y εy ,
 εx + ωx = min µx,a +
a∈A(x)
y∈X
∀x ∈ X2 .
The optimal stationary strategies of the players
s1 ∗ : x → a1 ∈ A(x) f or x ∈ X1 ;
s2 ∗ : x → a2 ∈ A(x) f or x ∈ X2
in the antagonistic stochastic positional game can be found by fixing arbitrary maps
s1 ∗ (x) ∈ A(x) for x ∈ X1 and s2 ∗ (x) ∈ A(x) for x ∈ X2 such that
212
Dmitrii Lozovanu, Stefan Pickl
s1 ∗ (x) ∈
P a ∗
T
P a ∗
Arg max
px,y ωx
Arg max µx,a +
px,y εy
,
a∈A(x)
y∈X
a∈A(x)
y∈X
∀x ∈ X1
and
P a ∗
T
P a ∗
px,y ωx
s2 (x) ∈ Arg min
Arg min µx,a +
px,y εy
.
∗
a∈A(x)
y∈X
a∈A(x)
y∈X
∀x ∈ X2
∗
∗
∗
∗
For the strategies s1 , s2 the corresponding values of the payoff function Fx (s1 , s2 )
coincides with the values ωx∗ for x ∈ X and
∗
∗
Fx (s1 , s2 ) = min
max2 Fx (s1 , s2 ) = max
min1 Fx (s1 , s2 ) ∀x ∈ X.
1
1 2
2
2 1
s ∈S s ∈S
.
s ∈S s ∈S
Based on the constructive proof of this theorem (see Lozovanu and Pickl, 2014)
an algorithm for determining the saddle points in antagonistic stochastic positional
games has been elaborated. The saddle point conditions for antagonistic stochastic
positional games with a discounted payoff can be derived from Theorem 6.
6.
Conclusion
Stochastic positional games with average and discounted payoffs represent a special class of Shapley stochastic games that extends deterministic positional games.
For the considered class of games Nash equilibria conditions have been formulated
and proven. Based on these results new algorithms for determining the optimal
stationary strategies of the players can be elaborated.
References
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96(2), 203–224.
Ehrenfeucht, A., Mycielski, J. (1979). Positional strategies for mean payoff games. International Journal of Game Theory, 8, 109–113. 8–113.
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Nash Equilibria Conditions for Stochastic Positional Games
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Pricing in Queueing Systems M/M/m with Delays⋆
Anna V. Melnik
St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
Universitetskii pr. 35, St.Petersburg, 198504, Russia
E-mail: [email protected]
Abstract A non-cooperative m-person game which is related to the queueing system M/M/m is considered. There are n competing transport companies which serve the stream of customers with exponential distribution
with parameters µi , i = 1, 2, ..., m respectively. The stream forms the Poisson process with intensity λ. The problem of pricing and determining the
optimal intensity for each player in the competition is solved.
Keywords: Duopoly, equilibrium prices, queueing system.
1.
Introduction
A non-cooperative n-person game which is related to the queueing system M/M/m
is considered. There are n competing transport companies, which serve the stream
of customers with exponential distribution with parameters µi , i = 1, 2, ..., m respectively. The stream forms the Poisson process with intensity λ. Suppose that
m
P
λ<
µi . Let companies declare the price for the service. Customers choose the
i=1
service with minimal costs. This approach was used in the Hotelling’s duopoly
(Hotelling, 1929; D’Aspremont, Gabszewicz, Thisse, 1979; Mazalova, 2012) to determine the equilibrium price in the market. But the costs of each customer are
calculated as the price for the service and expected time in queue. Thus, the incoming stream is divided into m Poisson flows with intensities λi , i = 1, 2, ..., m,
m
P
where
λi = λ. So the problem is following, what price for the service and
i=1
the intensity for the service is better to announce for the companies. Such articles as (Altman, Shimkin, 1998; Levhari, Luski, 1978; Hassin, Haviv, 2003), and
(Mazalova, 2013; Koryagin 2008; Luski, 1976) are devoted to the similar gametheoretic problems of queuing processes.
2.
The competition of two players
Consider the following game. There are two competitive transport companies which
serve the stream of customers with exponential distribution with parameters µ1
and µ2 respectively. The transport companies declare the price of the service c1 and
c2 respectively. So the customers choose the service with minimal costs, and the
incoming stream is divided into two Poisson flows with intensities λ1 and λ2 , where
λ1 + λ2 = λ. In this case the costs of each customer will be
ci +
⋆
λi
,
µi (µi − λi )
i = 1, 2,
This work was supported by the St. Petersburg State University under grants No.
9.38.245.2014
215
Pricing in Queueing Systems M/M/m with Delays
where λi /µi (µi − λi ) is the expected time of staying in a queue (Taha, 2011). So,
the balance equations for the customers for choosing the service are
c1 +
λ1
λ2
= c2 +
.
µ1 (µ1 − λ1 )
µ2 (µ2 − λ2 )
So, the payoff functions for each player are
H1 (c1 , c2 ) = λ1 c1 ,
H2 (c1 , c2 ) = λ2 c2 ,
We are interested in the equilibrium in this game.
Nash equilibrium. For the fixed c2 the Lagrange function for finding the best
reply of the first player is defined by
λ1
λ2
L1 = λ1 c1 + k c1 +
− c2 −
+ γ(λ1 + λ2 − λ). (1)
µ1 (µ1 − λ1 )
µ2 (µ2 − λ2 )
For finding the local maxima by differentiating (1) we get
∂L1
= λ1 + k = 0
∂c1
∂L1
k
kλ1
= c1 +
+
+γ =0
∂λ1
µ1 (µ1 − λ1 ) µ1 (µ1 − λ1 )2
∂L1
k
kλ2
=−
−
+γ =0
∂λ2
µ2 (µ2 − λ2 ) µ2 (µ2 − λ2 )2
from which
c1 = λ1
1
1
λ1
λ2
+
+
+
µ1 (µ1 − λ1 ) µ2 (µ2 − λ2 ) µ1 (µ1 − λ1 )2
µ2 (µ2 − λ2 )2
,
Symmetric model. Start from the symmetric case, when the services are the
same, i. e. µ1 = µ2 = µ. It is obvious from the symmetry of the problem, that in
equilibrium c∗1 = c∗2 = c∗ and λ1 = λ2 = λ2 . So
!
λ
2
λ
∗
c =
+
.
(2)
2 µ(µ − λ2 ) µ(µ − λ2 )2
So, if one of the players uses the strategy (2), the maximum of payoff of another
player is reached at the same strategy. That means that this set of strategies is
equilibrium.
Asymmetric model. Assume now, that transport services are not equal, i. e.
µ1 6= µ2 , suppose that µ1 > µ2 . Find the equilibrium in the pricing problem in this
case. The system of equations that determine the equilibrium prices of transport
companies is
λ1
λ2
c∗1 +
= c∗2 +
µ1 (µ1 − λ1 )
µ2 (µ2 − λ2 )
1
1
λ1
λ2
c∗1 = λ1
+
+
+
,
µ1 (µ1 − λ1 ) µ2 (µ2 − λ2 ) µ1 (µ1 − λ1 )2
µ2 (µ2 − λ2 )2
216
Anna V. Melnik
1
1
λ1
λ2
c∗2 = λ2
+
+
+
,
µ1 (µ1 − λ1 ) µ2 (µ2 − λ2 ) µ1 (µ1 − λ1 )2
µ2 (µ2 − λ2 )2
λ1 + λ2 = λ.
In Table 1 the values of the equilibrium prices with different µ1 , µ2 at λ = 10
and are given.
Table 1: The value of (c∗1 , c∗2 ), (p∗1 , p∗2 ) and (λ1 , λ2 ) at λ = 10
µ2
6
µ1
7 (c1 ;c2 )
(λ1 ;λ2 )
8 (c1 ;c2 )
(λ1 ;λ2 )
9 (c1 ;c2 )
(λ1 ;λ2 )
10 (c1 ;c2 )
(λ1 ;λ2 )
3.
7
(5,41;5,1)
(5,15;4,85)
(4,04;3,64)
(5,25;4,75)
(3,4;2,98)
(5,33;4,67)
(3,06;2,62)
(5,39;4,61)
(2,5;2,5)
(5;5)
(1,75;1,65)
(5,14;4,86)
(1,4;1,26)
(5,27;4,73)
(1,21;1,04)
(5,36;4,64)
8
9
10
(1,11;1,11)
(5;5)
(0,87;0,82) (0,625;0,625)
(5,14;4,86)
(5;5)
(0,73;0,66) (0,52;0,59) (0,4;0,4)
(5.26;4,74) (5,13;4,87)
(5;5)
The competition of m players.
Let us increase the number of players. There are m competitive transport companies
which serve the stream of customers with exponential distribution with parameters
µi , i = 1, 2, ..., m respectively. The transport companies declare the price of the service ci , i = 1, 2, ..., m and the customers choose the service with minimal costs. The
incoming stream is divided into n Poisson flows with intensities λi , i = 1, 2, ..., m,
m
P
where
λi = λ. Thus, the balance equations for the customers for choosing the
i=1
service are
c1 +
λ1
λi
= ci +
,
µ1 (µ1 − λ1 )
µi (µi − λi )
i = 1, ..., m.
The payoff functions for each player are
Hi (c1 , ..., ci ) = λi ci ,
i = 1, ..., m.
Find the equilibrium in this game.For the fixed ci , i = 2, ..., m the Lagrange function
for finding the best reply of the first player is defined by
L1 = c1 λ1 +
m
X
i=2
ki c1 +
λ1
λi
− ci −
µ1 (µ1 − λ1 )
µi (µi − λi )
Differentiating (3),we find
m
X
∂L1
= λ1 +
ki = 0,
∂c1
i=2
m
X
+ γ(
λi − λ).
i=1
(3)
217
Pricing in Queueing Systems M/M/m with Delays
m
P
m
P
ki λ1
∂L1
i=2
= c1 +
+ i=2
+ γ = 0,
∂λ1
µ1 (µ1 − λ1 ) µ1 (µ1 − λ1 )2
from which
ki
∂L1
ki
ki λi
=−
−
+ γ = 0,
∂λi
µi (µi − λi ) µi (µi − λi )2
c∗i
c∗i +
= λi
i = 2, ..., m.
1
1
Pm
+
2
(µi − λi )2
j=0,j6=i (µj − λj )
λi
λi+1
= c∗i+1 +
,
µi (µi − λi )
µi+1 (µi+1 − λi+1 )
m
X
!
,
i = 0, ..., m − 1
(4)
λi = λ.
i=1
4.
The competition of 2 players on graph.
Fig. 1: Competition of 2 players on graph G1
Consider competition on the graph G1 , which is equivalent to a part of the
Helsinki Metro. Let’s define the game as Γ =< I, II, G1 , Z1 , Z2 , H1 , H2 >, where I,
II are 2 competitive transport companies which serve the stream of customers with
exponential distribution with parameters µi , i = 1, 2 on graph G1 =< V, E >. V =
{1, 2, 3, 4} is the set of vertices, E = {e12 , e23 , e24 } - the set of edges. Zi = {R1i , R2i }
218
Anna V. Melnik
is the set of routes of player i. Each rout is a sequence of vertices. So there are two
routs R1i = {1, 2, 3} and R2i = {1, 2, 4}, i = 1, 2. The stream of passengers forms the
Poisson process with intensity Λ, where
0
Λ= 0
0
0
λ12
0
0
0
λ13
λ23
0
0
λ14
λ24
0
0
The transport companies declare the price of the service cikj , i = 1, 2, k = 1, 2,
j = 2, 3, 4, j 6= k and the customers choose the service with minimal costs. The
incoming stream Λ is divided into two Poisson flows with intensities λkj = λ1kj +λ2kj ,
k = 1, 2, j = 2, 3, 4, j 6= k. We are interested in equilibrium in this game.
The balance equations are
c112 + a11 = c212 + a21 ,
c123 + a12 = c223 + a22 ,
c124 + a13 = c224 + a23 ,
c113 + a11 + a12 = c213 + a21 + a22 ,
c114 + a11 + a13 = c214 + a21 + a23 ,
λkj = λ1kj + λ2kj ,
where
ai1 =
k = 1, 2,
j = 2, 3, 4,
j 6= k,
λi12 + λi13 + λi14
,
− λi12 − λi13 − λi14 )
µi (µi
ai2 =
ai3 =
λi13 + λi23
µi µi
2 ( 2
− λi13 − λi23 )
λi14 + λi24
µi µi
2 ( 2
The payoff functions for each player are
Hi ((c1R1 , c1R2 , c2R1 , c2R2 ) =
− λi14 − λi24 )
2
4
X
X
,
.
λikj cikj ,
i = 1, 2.
k=1 j=2,j6=k
The Lagrange function for finding the best reply of the first player is defined by
L1 =
2
4
X
X
k=1 j=2,j6=k
λikj cikj + k1 c112 + a11 − c212 − a21 + k2 c123 + a12 − c223 − a22 +
219
Pricing in Queueing Systems M/M/m with Delays
+k3 c124 + a13 − c224 − a23 + k4 c113 + a11 + a12 − c213 − a21 − a22 +
+k5 c114 + a11 + a13 − c214 − a21 − a23 .
Differentiating this equation we find
∂L1
= λ112 + k1 = 0,
∂c112
∂L1
= λ123 + k2 = 0,
∂c123
∂L1
= λ124 + k3 = 0,
∂c124
∂L1
= λ113 + k4 = 0,
∂c113
∂L1
= λ114 + k5 = 0.
∂c114
Since λ1kj = λkj − λ2kj , k = 1, 2, j = 2, 3, 4, j 6= k, we get
∂L1
= c112 + (k1 + k4 + k5 )
∂λ112
∂L1
= c123 + (k2 + k4 )
∂λ123
∂L1
= c124 + (k3 + k5 )
∂λ124
∂a11
∂a21
+
1
∂λ12
∂λ212
∂a12
∂a22
+
1
∂λ23
∂λ223
∂a13
∂a23
+
∂λ124
∂λ224
∂L1
= c113 + (k1 + k4 + k5 )
∂λ113
∂a11
∂a21
+
1
∂λ13
∂λ213
∂L1
= c114 + (k1 + k4 + k5 )
∂λ114
∂a11
∂a21
+
∂λ114
∂λ214
,
,
,
+ (k2 + k4 )
∂a12
∂a22
+
1
∂λ13
∂λ213
,
+ (k3 + k5 )
∂a12
∂a22
+
∂λ114
∂λ214
,
Symmetric model. Consider symmetric case, when the services are the same, i.
e. µ1 = µ2 = µ. It is obvious from the symmetry of the problem, that in equilibrium
λkj
2∗
∗
1
2
c1∗
kj = ckj = ckj and λkj = λkj = 2 , k = 1, 2, j = 2, 3, 4, j 6= k. So
c∗12 =
λ12 + λ13 + λ14
(λ12 + λ13 + λ14 )2
+
2
λ12 +λ13 +λ14
µ µ−
2µ µ − λ12 +λ213 +λ14
2
c∗23 =
c∗24 =
c∗13 =
µ
2
µ
2
λ23 + λ13
(λ23 + λ13 )2
+
2
µ
λ23 +λ13
13
µ µ2 − λ23 +λ
2 −
2
2
λ24 + λ14
(λ24 + λ14 )2
+
2
µ
λ24 +λ14
14
µ µ2 − λ24 +λ
2 −
2
2
(λ12 + λ13 + λ14 )2
λ12 + λ13 + λ14
+
2 +
λ12 +λ13 +λ14
µ µ−
2µ µ − λ12 +λ213 +λ14
2
220
Anna V. Melnik
+µ
2
c∗14 =
λ23 + λ13
(λ23 + λ13 )2
+
2
µ
λ23 +λ13
13
µ µ2 − λ23 +λ
2 −
2
2
λ12 + λ13 + λ14
(λ12 + λ13 + λ14 )2
+
2 +
λ12 +λ13 +λ14
µ µ−
2µ µ − λ12 +λ213 +λ14
2
λ24 + λ14
(λ24 + λ14 )2
+
2
µ
λ24 +λ14
14
µ µ2 − λ24 +λ
2 −
2
2
In Table 2 the values of the equilibrium prices with different µ, at λ12 = 1,
λ23 = 1, λ24 = 2, λ13 = 3, λ14 = 1 are given.
µ
2
Table 2: The value of equilibrium prices at λ12 = 1, λ23 = 1, λ24 = 2, λ13 = 3, λ14 = 1
µ
5.
prices
10
11
12
13
14
15
c∗12
c∗23
c∗24
c∗13
c∗14
0,089
0,44
0,24
0,53
0,33
0,069
0,327
0,188
0,396
0,258
0,055
0,25
0,15
0,305
0,204
0,045
0,198
0,12
0,243
0,165
0,038
0,16
0,089
0,199
0,137
0,032
0,13
0,083
0,16
0,115
Conclusion
It is seen from the table, that the higher the intensity of service is, the lower price
this transport company declare. But the prices c23 and c24 , that correspond to the
edges, where the pass is divided on two roads, are greater, that c12 , because after
this division the intensity of service is divided too.
References
Hotelling, H. (1929). Stability in Competition. Economic Journal, 39, 41–57.
D’Aspremont, C., Gabszewicz, J., Thisse, J.-F. (1979). On Hotelling’s “Stability in Competition”. Econometrica, 47, 1145–1150.
Mazalova, A. V. (2012). Hotelling’s duopoly on the plane with Manhattan distance. Vestnik
St. Petersburg University, 10(2), 33–43. (in Russian).
Altman, E., Shimkin, N. (1998). Individual equilibrium and learning in processor sharing
systems. Operations Research, 46, 776–784.
Levhari, D., Luski, I. (1978). Duopoly pricing and waiting lines. European Economic Review, 11, 17–35.
Hassin, R., Haviv, M. (2003). To Queue or Not to Queue / Equilibrium Behavior in Queueing Systems, Springer.
Luski, I. (1976). On partial equilibrium in a queueing system with two services. The Review
of Economic Studies, 43, 519–525.
Koryagin, M. E. (1986). Competition of public transport flows. Autom. Remote Control,
69:8, 1380–1389.
Taha, H. A. (2011). Operations Research: An Introduction, ; 9th. Edition, Prentice Hall.
Mazalova, A. V. (2013). Duopoly in queueing system. In: Vestnik St. Petersburg University,
10(4), 32–41. (in Russian).
How to arrange a Singles’ Party: Coalition Formation in
Matching Game⋆
Joseph E. Mullat
Tallinn Technical University,
Faculty of Economics, Estonia
E-mail: [email protected]
http://datalaundering.com/author.htm
Residence: Byvej 269, 2650 Hvidovre, Denmark
Abstract The study addresses important issues relating to computational
aspects of coalition formation. However, finding payoffs−imputations belonging to the core−is, while almost as well known, an overly complex, NPhard problem, even for modern supercomputers. The issue becomes uncertain because, among other issues, it is unknown whether the core is nonempty. In the proposed cooperative game, under the name of singles, the
presence of non-empty collections of outcomes (payoffs) similar to the core
(say quasi-core) is fully guaranteed. Quasi-core is defined as a collection
of coalitions minimal by inclusion among non-dominant coalitions induced
through payoffs similar to super-modular characteristic functions (Shapley,
1971). As claimed, the quasi-core is identified via a version of P-NP problem
that utilizes the branch and bound heuristic and the results are visualized
by Excel spreadsheet.
Keywords: stability; game theory; coalition formation.
1.
Introduction
It is almost a truism that many university and college students abandon schooling
soon after starting their studies. While some students opt for incompatible education programs, the composition of students following particular programs may not
be optimal; in other words, students and programs are mutually incompatible. Indeed, so-called mutual mismatches of priorities were among the reasons (Võhandu,
2010) behind the unacceptably high percentage of students in Estonian universities and colleges dropping out of schools, wasting their time and the entitlement
to government support. However, matching students and education programs more
optimally could mitigate this problem.
Similar problems have been thoroughly studied (Roth, 1990; Gale, 1962; Berge,
1958...) leading, perhaps, L. Võhandu to propose a way, in this wide area of research, to solve the problem of students and programs mutual incompatibility by
introducing “matching total” as the sum of duplets−priorities selected within two
directions—horizontal priorities of students towards programs, and vertical priorities of programs towards students. The best solution found among all possible horizontal and vertical duplet assignments, according to LV, is where the sum reaches
its minimum.
Finding the best solution, however, is a difficult task. Instead, LV proposed a
greedy type workaround. In the author’s words, the best solution to the problem
⋆
A thesis of this paper was presented at the Seventh International Conference on Game
Theory and Management (GTM2013), June 26-28, 2013, St. Petersburg, Russia.
222
Joseph E. Mullat
of matching students and programs will be close enough (consult with Carmen et
al., 2001) to a sum of duplets accumulated while moving along ordering of duplets
in non-decreasing direction. It seems that LV’s proposal to the solution is a typical
approach in the spirit of classical utilitarism, when the sum of utilities has to be
maximized or minimized (Bentham, The Principals of Morals and Legislation, 1789;
Sidgwick, The Methods of Ethics, London 1907).
As noted by Rawls in "Theory of Justice", the main weakness of utilitarian
approach is that, when the total max or min has been reached, those members of
society at the very low utility levels will still be receiving very low compensations
for incapacity, such as transfer payments to the poor. Arguing for the principal of
"maxima of the lowest", referred to as the "Second Principal of Justice", Rawls
suggested an alternative to the utilitarian approach. The motive driving this study
is similar. We address by example an alternative to conventional core solution in
cooperative games, along the lines of monotonic game (Mullat, 1979), whereby the
lowest incentive/compensation should be maximized. The reader studying matching
problems can also find useful information about these issues, where a number of ways
of constructing an optimal matching strategy have been discussed (Veskioja, 2005).
Learning by example is of high value because the conventional core solution in
cooperative games cannot be clearly explained unless the readers are sufficiently
familiar with utopian reality−a reality that sometimes does not exist. Thus, a rigorous set up of a simple game will be presented here, aiming to explain the otherwise
rather complicated intersection of interests. More specifically, we hope to shed light
on what we call a Singles-Game. It should be emphasized that, even though the game
primitives represent an independent mathematical object in a completely different
context, we have still “borrowed” the idea of LV duplets to estimate the benefits of
matching. For this reason, we changed the nomenclature of duplets to mutual risks
in order to justify the scale of payoffs−the incentives and compensations.
The rest of the paper is organized as follows. We start with the preliminaries,
where the game primitives are explained. In Section 3, we introduce the core concept of conventional stability in relation to the Singles-game. In Section 4, the reader
will come across an unconventional theory of kernel coalitions, and nuclei coalitions,
minimal by inclusion in accordance with the formal scheme. In Section 5, we continue explaining our techniques and procedures used to locate stable outcomes of
the game. The study ends with conclusions and suggestions for future work, which
are presented in Section 6. Appendix contains a visualization, which brings to the
surface the theoretical foundation of coalition formation. Finally, interested readers
would benefit from exploring the Excel spreadsheet, which helps visualize a "realistic" intersection of interests of 20 single women and 20 single men. The addendum
provides a sketched outline for the evidence of some propositions.
2.
Preliminaries
Five single women and five single men are ready to participate in the Singles Party.
It is assumed that all participants exhibit risk-averse behavior towards dating. To
cover dating bureau expenses, such as refreshments, rewards, etc., the entrance fee
is set at 50 e 1 . Thus, the cashier will be at the disposal of an amount of 500 e. All
the guests have been kindly asked to take part in a survey, helping determine the
attributes they look for in their prospective partner. Those who choose to provide
1
Note that red colour points at negative number.
How to arrange a Singles’ Party: Coalition Formation in Matching Game
223
this information have been promised to collect a Box of Delights 2 and are hereafter
referred to as participants, while others are labeled as dummies, by default, and
cannot participate in the game. In addition to the delights, promised to those willing
to reveal their priorities, we continue setting the rules of payoffs in the form of
incentives and mismatch compensations. However, if all participants decide to date,
as no reasonable justification exists for incentives and compensations, the game
terminates immediately.
We use index i for the women, and an index j for the men taking part in the dating party. Assuming that all the guests have agreed to participate in the game,
there are {1, ..., i, ...5} women and {1, ..., j, ...5} men, resulting in 2 × 5 × 5 combinations. Indeed, when priorities have been revealed, they can form two 5 × 5 tables,
W = kwi,j k, and M = kmi,j k, indicating that each woman i, i = 1, 5 revealed her
priorities positioned in the rows of table W towards men as horizontal permutations
wi of numbers h1, 2, 3, 4, 5i. Similarly, each man j, j = 1, 5, revealed his priorities
positioned in columns of the table M towards women
mj .
as vertical permutations
As can be seen in Table-1, priorities wi,j (numbers 1, 5 = 1, 2, 3, 4, 5 ) might repeat
in both the columns of the table W and in the rows of the table M . To be sure,
more than one man may prefer the same woman at priority level wi,j , and many
women, accordingly, may prefer the same man at the level mj,i . Thus, duplets or
mutual risks ri,j = wi,j + mi,j occupy the cells in table R = kri,j k.
Table 1.
Noting the assumption that all participants are risk-averse, some lucky couples
with lower level of mutual risks start dating. These lucky couples will receive an
incentive, such as a prepaid ticket to an event, free restaurant meal, etc. On the
other hand, unlucky participants—i.e., those that did not find a partner—may claim
a compensation, as only high-level mutual risk partners remained, given that the
eligible participants at the low level of mutual risk have been matched.
If no one has found a suitable partner, the question is—should the party continue?
Apparently, given that the original data that failed to produce matches might have
not been completely truthful, it would be unwise to offer compensation in proportion to mutual risks ri,j . Nonetheless, let us assume that the compensation equals
1/2r .10 e. In that case, couple’s [5, 5] profit may reach 50 e! Instead, the dating
i,j
bureau decides to organize the game, encouraging the players to follow Rawls second
principle of justice. In Table-1, the minimum−the lowest mutual risk among all
participants−is r1,4 = 3. Following the principle, the compensation to all unlucky
participants will be equal to 1/2r1,4 .10=15 e. This setting is also fiscally reasonable
from the cashier’s point of view. The balance of payoffs for all participants, will be
2
In case the Box is undesirable it will be possible to get 10 e in return.
224
Joseph E. Mullat
25 e, as 50 e paid as entrance fee will be reduced by 15 e compensation amount,
and additionally by 10 e, i.e., inclusive of the cost of collected delights. Further
on, we assume that each member of a dating couple will receive an incentive that
is offered to all dating couples and is equal to double the compensation amount.
What happens when the couple [1, 4] decides to date? The entire table R should be
dynamically transformed to reflect the fact that the participants [1, 4] are matched.
Indeed, as the women {2, 3, 4, 5} and men {1, 2, 3, 5} can no longer count on [1, 4]
as their potential partners, the priorities will decline, whereby the scale h1, 2, 3, 4, 5i
dynamically shrinks to h1, 2, 3, 4i3 . To reflect this, Table-1 transforms into Table-2:
Table 2.
The minimum mismatch compensation did not change and is still equal to 15
e. However, couple’s [1, 4] potential balance 50 e+10 e+2.15 e=10 e of payoffs
improves (W1 and M4 each receive 30 e as an incentive to date, based on the rule
that it is equal to twice the mismatch compensation). For those not yet matched,
the balance remains negative (in deficit) and equals 15 e. On the other hand, if, for
example, the couple [3, 5] decides to date, the balance of payoffs improves as well.
The party is over and the decisions have been made about who will date and who
will leave the party without a partner. The results are passed in writing to the dating
bureau. What would be the best collective decision of the participants based on the
principle of "maxima of the lowest" in accord with the rules of singles-game?
3.
Conventional stability4
In this section, the aim is to present the well-established solution to the singlesgame by utilizing the conventional concept, called the core. First, without any warranty, it is helpful to focus on the core stability.
In order to meet this aim, the original dating party arrangement is expanded to
a more general case. The game now has n × m participants, of whom n are single women h1, ..., i, ...ni and m are single men h1, ..., j, ...mi. Some of the guests
expressed their willingness to participate in the game and have revealed their priorities. Those who refused, in line with the above, are referred to as dummy players.
All those who agreed to play the game will be arranged by default into the grand
4
3
4
To highlight theoretical results of mutual risks, incitements or compensations, or whatever the scales we use, the dynamic quality of monotonic scales is the only feature
fostering the birth of MS − the "monotone system." Otherwise, the MS terminology, if
used in any type of serialization methods applied for data analysis, will remain sterile.
Terminology, which we shall use below, is somewhat conventional but mixed with our
own.
How to arrange a Singles’ Party: Coalition Formation in Matching Game
225
coalition P, |P| 6 n + m. Thus, indices i, j and labels α, ..., σ ∈ P are used to annotate the guests participating in the game. Only the guests in P are regarded as
participants, whereas couples [i, j] are referred to as α, ..., σ. This differentiation not
only helps make notations short, when needed, but can also be used in reference to
an eventual match or a couple without any emphasis on gender.
In the singles-game, we focus on the participants D ⊆ P who are matched. Having
formed a coalition, we suppose that coalition D has the power and is in a position
to enforce its priorities. It is assumed that participants in D can persuade all those
in X = P\D, i.e., participants that are not yet matched, to leave the party without
a partner and thus receive compensation. However, it is realistic to assume that the
suppression of interests of participants’ in X is not always possible. It is conceivable
that, those in the coalition D′ ⊆ X, whose interests would be affected (suppressed),
will still be capable to receive as much as the participants in D. However, we exclude
this opportunity, as it is better that no one expects that coalition D′ can be realized
concurrently with D and act as its direct competition.
Insisting on this restriction, however, we still assume that others−those participants
suppressed in X−have not yet found their suitable partners and have agreed to form
their own coalition, even though they could receive compensation equal to 50% of
the incentives in D. A realistic situation may occur when all participants in P are
matched, D = P, or, in contrast, no one decides to date, D = ∅. It is also reasonable
that, after revealing their priorities, some individuals might decide not to proceed
with the game and will, thus, be labeled as a dummy player δ ∈
/ P.
Among all coalitions D, we usually distinguish rational coalitions. Couple α, joining
the coalition D, extracts from the interaction in the coalition a benefit that satisfies
α ∈ D. In the singles-game, we anticipate that the extraction of benefits, i.e., the incentives and mismatch compensations, strictly depend on the membership−couples
in D or participants of coalition X. Using the coalition membership D ⊆ P, we can
always construct a payoff x to all participants P, i.e., we can quantify the positions
of all participants. The inverse is also true. Given a payoff x, it is easy to establish
which couple belongs to the coalition D and identify those belonging to the coalition
X = P\D. We label this fact as Dx . Recall that couples of the coalition Dx receive
an incentive to date, which is equal to the double amount of the mismatch compensation. Thus, the allocation Dx may provide an opportunity for some participants
σ ∈ P to start, or initiate, new matches, thus moving to better positions. We will
soon see that, while the best positions induced by special coalitions N, called the
nuclei, have been reached, this movement will be impossible to realize. 5
The inability of players to move to better positions by "pair comparisons" is an
example of stability. In the work "Cores of Convex games", convex games have
been studied (Shapley, 1971); these are so-called games with a non-empty core,
where similar type of stability exists. The core forms a convex set of end-points
(imputations) of a multidimensional octahedron, i.e., a collection of available payoffs
to all players. Below, despite the players’ asymmetry with respect to Dx = P\X,
we focus on their payoffs driving their collective behavior as participants P to form
a coalition Dx , Dx ⊆ P; here, X = Dx is called an anti-coalition to X.
In contrast to individual payoffs improving or worsening the positions of participants, when playing a coalition game, the total payment to a coalition X as
a whole is referred to the characteristic function v(X) > 0. In classical coop5
Our terminology is unconventional in this connection.
226
Joseph E. Mullat
erative game theory, the payment v(X) to coalition X is known with certainty,
whereby the variance v(X) − v(X\ {σ}) provides a marginal utility π(σ, X). Inequality π(α, X\ {σ}) 6 π(α, X) of the scale of risks expresses a monotonic decrease
(increase) in marginal utilities of the membership for α ∈ X.
S The monotonicity
T
is equivalent to the supermodularity v(X1 ) + v(X2 ) 6 v(X1 X2 ) + v(X1 X2 )
(Nemhauser et al., 1978). Thus, any characteristic function v(X), payment on which
is built according to the scale of risks, is supermodular. The inverse submodularity
was used to find solutions of many combinatorial problems (Edmonds, 1970; Petrov
and Cherenin, 1948). In general, such a warranty cannot be given.
Recall that we eliminated all rows and columns in tables W = kwi,j k, M = kmi,j k
in line with X = Dx . Table R = kπ(α, X) = wi,j (X) + mi,j (X)k, α = [i, j] ∈ X
reflects the outcome of shrinking priorities wi,j , mi,j when some σ ∈ X have found
a match and have formed a couple. Priorities wi,j , mi,j are consequently
P decreasing.
Given in the form of characteristic function, e.g., the value v(X) = α∈X π(α, X)
sets up a coalition game.6 An imputation
for the game ν(X) is defined by a |P|P
vector fulfilling two conditions: (i)
α∈P xα = v(P), (ii) xα > v({α}), for all
α ∈ P. Condition (ii) clearly stems from repetitive use of monotonic inequality
π(α, X\ {σ}) 6 π(α, X).
A significant shortcoming of the cooperative theory given in the form of the characteristic function stems from its inability to specify a particular imputation as a
solution. However, in our case, such imputation can be defined in an intuitive way.
In fact, the concept of risk scale determines a popularity index of players. More
specifically, the lower the risk of engagement π(α, X) of σ ∈ X, the more reliable
the couple’s α coexistence is. Therefore, we set up a popularity
index pi of a woman
P
i among men in the coalition X as number pi (X) = j∈X mi,j . P
The index number
pj of a man j among women, accordingly, is given by pj (X) = i∈X wi,j . We intend to redistribute the total payment v(X) in proportion to the components of the
vector p(X) = hpi (X), pj (X)i, or as the vector p(X). Hereby we can prove, owing to
monotonicity of the scale of priorities, that the payoffs in imputation p(P) cannot be
improved by any coalition X ⊂ P. Therefore, the game solution, among popularity
indices, will be the only imputation p(P). In other words, popularity indices core of
the cooperative game v(X) consists of only one point p(P).
In line with the terminology used above, we draw the readers’ attention to the
fact that the singles-game considered next is not a game given in the form of a
characteristic function. The above discussion was presented as the foundation for
the course of further investigation only.
4.
Concept of a kernel
In the view of "monotone system" (Mullat, 1971-1995) exactly as in Shapley’s convex games, the basic requirement of our model validity emerges from an inequality
of monotonicity π(α, X\ {σ}) 6 π(α, X). This means that, by eliminating an element σ from X, the utilities (weights) on the rest will decline or remain the same.
In particular, a class of monotone systems is called p-monotone (Kuznetsov et al.,
1982, 1985), where the ordering hπ(α, X)i on each subset X of utilities (weights)
follows the initial ordering hπ(α, W)i on the set W. The decline of the utilities
on p-monotone system does not change the ordering of utilities on any subset X.
6
ν(X) = |X|2 · (|X| + 1) . Check that v(P) = 150 for 5 × 5 -game, or use the Table-1.
How to arrange a Singles’ Party: Coalition Formation in Matching Game
227
Thus, serialization (greedy) methods on p-monotone system might be effective. Behind a p-monotone system is the fact that an application of Greedy framework
can simultaneously accommodate the structure of all subsets X ⊂ W. Perhaps, for
different reasons, many will argue that p-monotone systems are rather simplistic
and fail to compare to the serialization method. Nonetheless, many economists, including Narens and Luce (1983), almost certainly, will point out that subsets X of
p-monotone systemsSperform on interpersonally compatible scales.
An inequality F (X1 X2 ) > min hF (X1 ), F (X2 )i holds for real valued set function
F (X) = minα∈X π(α, X), referred to as quasi-convexity (Malishevski, 1998). We
observed monotone systems, which we think is important to distinguish. The system
is non quasi-convex when two coalitions contradict the last inequality. We consider
such systems as non-quasi-convex, which applies to the singles-game case.
The ordering of priorities in singles-games−i.e., what men look for in women, and
vice versa−remain intact within an arbitrary coalition X. However, in these systems,
the ordering of mutual risks kri,j k on grand coalition P does not necessarily hold
for some X ⊂ P. Contrary to initial ordering on R(P) = kπ(α, P) = ri,j k, the
ordering of mutual risks on R(X) = kπ(α, X)k may be inverse of the ordering on
R(P) for some couples. In that case, e.g., the ordering of two couples’ mutual risks
can turn "upside down" while the risk scale is shrinking compared to the original
ordering on the grand coalition P. Thus, in general, the mutual risks scale is not
necessarily interpersonally compatible. In other words, interpersonal incompatibility
of this risk scale radically differs from the p-monotone systems. This difference
became apparent when it was no longer possible to find a solution using Greedy
type framework of so-called defining chain algorithm−i.e., the monotone system was
non-quasi-convex. Before proceeding with the formal side of these processes, it is
informative to understand the nature of the problem.
Definition 1. By kernel coalition we call a coalition K ∈ arg maxX⊆P F (X); {K}
is the set of all kernels.
Recalling the main quality of defining a chain−a sequence of elements of a monotone
system—it is possible to arrange the elements α ∈ W, i.e., the couples α ∈ P of
players by a sequence hα1 , ..., αk i, k = 1, n. The sequence follows the lowest risk
ordering in each step k corresponding to sequence of coalitions hHk i, H1 = P,
Hk+1 ← Hk \ {αk }, αk = arg minα∈Hk π(α, Hk ). Given any arbitrarily coalition
X ⊆ P, we say that the defining sequence obeys the left concurrence quality if
there exists a superset Ht such that Ht ⊇ X, t = 1, k, where the first element
αt ∈ Ht to the left in the sequence hα1 , ..., αk i belongs to the set X, αt ∈ X
as well.
S On the condition that the element αt is not a member of the superset
H = {K ∈ arg maxX⊆P F (X)} including all kernels K, αk ∈
/ H, we observe that
π(αt , X) < π(αt , Ht ). Hereby, we can conclude that F (X) 6 π(αt , Ht ) is strictly
less than the global maximum of the set function F (X) = minα∈X π(α, X). The left
concurrence quality guarantees that the sequence can potentially be used for finding
the largest kernel H. Due to non-quasi-concavity, the left concurrence quality is no
longer valid. Eliminating a couple αk = [i, j], see above, we delete the row i and
the column j in the mutual risks table R. Thus, the operation Hk+1 ← Hk \ {αk } is
not an exclusion of a couple αk ∈ Hk , given that the couple αk = [i, j] is about to
start dating, but rather an exclusion of adjacent couples α in [i, ∗]-row and [∗, j]column. We annotate the engagement as Hk+1 ← Hk − αk or as an equal notation
Dk+1 ← Dk + αk .
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Joseph E. Mullat
In conclusion, note, once again, that, despite the properties of monotone system
remaining intact, the chain algorithm, assembling the defining sequence of elements
α ∈ P, cannot guarantee the extraction of the supposedly largest kernel H, particularly in the form given by Kempner et al. (2008). Thus, we need to employ special
tools for finding the solution. To move further in this direction, we are ready to
formulate some propositions for finding kernels K by branch and bound algorithm
types.
The next step will require a modified variant of imputation (Owen, 1982). We define
an imputation as the outcome connected to the singles-game in the form of a |P|vector of payoffs to all participants. More specifically, the outcome is a |P|-vector,
where each partner in a couple σ ∈ X receives the lowest mismatch compensation
F (X), whereas each partner in the couple σ ∈
/ X belonging to the anti-coalition
X = Dx receives the incentive to date, which is equal to twice that amount, i.e.,
2 · F (X), cf. Tables 3,4. The concept of outcome (imputation) in this form is not
common because the amount to be claimed by all participants is not fixed and
equals |P| + F (X) · |X| + 2 · X . Thus, it is likely that participants will fail to
reach an understanding, and will claim payoffs obtaining less than available total
amount (n + m)·50 e. The situation, in contrast, when participants will claim more
than total amount, is also conceivable.
Any coalition X induces a |P|-vector x = hxσ i as an outcome x:7
xσ =
X
2 + F (X) if σ ∈ X,
→
xσ = |P| + F (X) · |X| + 2 · X .
2 · (1 + F (X)) if σ ∈
/ X.
σ∈P
In this case, xσ is a quasi-imputation.
This definition of outcome is used later, adapting the concept of the quasi-imputation
for the purpose of the singles-game. We say that an arbitrary coalition X induces an
outcome x. Computed and prescribed by coalition X, the components of x consist
of two distinct values F (X) and 2 · F (X). Participants σ ∈ X could not form a
couple, while participants σ ∈ Dx were able to match. Recall that the notation for
X is also used as a mixed notation for dating couples Dx .
Before we move further, we will try to justify our mixed notation X. Although a
coalition X = Dx uniquely defines both those Dx among participants P who went
on dating, and those X = P\Dx who did not, the coalition X does not specifically
indicate matched couples. In contrast, using the notation Dx , we indicate that
all participants in Dx are matched, whereas a couple σ ∈ Dx also indicates an
individual decision how to match. More specifically, this annotation represents all
men and all women in Dx standing in line facing one member of the opposite
sex, with whom they are matched. However, any matching or engagement among
couples belonging to Dx , or whatever matches are formed in Dx , does not change
the payoffs xσ valid for the outcome x. In other words, each particular matching
Dx induces the same outcome x. Decisions in Dx with respect to how to match
provide an example of individual rationality, while the coalition Dx formation, as a
whole, is an example of collective rationality. Therefore, in accordance with payoffs
x, the notation Dx subsumes two different types of rationality−the individual and
the collective rationality. In that case, the outcome x accompanying Dx represents
7
Further, we follow the rule that capital letters represent coalitions X, Y, ..., K, H, ... while
lowercase letters x, y, ..., k, h, ... represent outcomes induced by these coalitions.
How to arrange a Singles’ Party: Coalition Formation in Matching Game
229
the result of a partial matching of participants P. Propositions below somehow bind
the individual rationality with the collective rationality.
One of the central issues in the coalition game theory is the question of the possible
formation of coalitions or their accessibility, i.e., the question of coalition feasibility.
While it is traditionally assumed that any coalition X ⊆ P is accessible or available
for formation, such an approach is generally unsatisfactory. We will try to associate
this issue with a similar concept in the theory of monotone systems. The issue of
accessibility of subsets X ⊂ W in the literature of monotone systems has been
considered not only in the context of the totality 2W of its subsets X ∈ 2W but also
with respect to special collections of subsets F ⊂ 2W . A singleton chain αt adding
elements step-by-step, starting with the empty set ∅, can, in principle, access any
set X ∈ F, or access the set X by removing the elements starting with the grand
set W−so called upwards or downwards accessibility.
Definition 2. Given coalition X ⊆ P, where P is the grand coalition, we call the
collection of pairs C(X) = {arg minα∈X π(α, X)} naming C(X) as best potential
couples, capable of matching with the lowest mutual risk, within the coalition X.
Consider a coalition Dx , generated by the formation by a chain of steps Dk+1 ←
Dk + hαk i. Let X1 = P, Xk = P\Dk , where Dk are participants trying to match
during the step k; C(Xk ) are couples in Xk with the lowest mutual risk among
couples not yet matched in steps k = 1, n, Xn+1 = ∅. Coalitions in the chain
Xk+1 = Xk − αk are arranged after the rows and columns, indicated by couple
αk , have been removed from W , M and R. Mutual risks R have been recalculated
accordingly.
Definition 3. Given the sequence hα1 , ..., αk i of matched couples, where X1 = P,
Xk+1 = Xk − αk , we say that coalition Dx = X = P\X of matched (as well as X of
not yet matched) participants is feasible, when the chain
T hX1 , ..., Xk+1 = Xi complies with the rational succession C(Xk+1 ) ⊇ C(Xk ) Xk+1 . We call the outcome
x, induced by sequence hα1 , ..., αk i, a feasible payoff, or a feasible outcome.
Proposition 1. The rational succession rationality necessarily emerges from the
condition that, under the coalition Dx formation a couple αk does not decrease the
payoffs of couples hα1 , ...αk−1 i formed in previous steps.
The accessibility or feasibility of coalition Dx formation offers convincing interpretation. In fact, the feasibility of coalition Dx means that the coalition can be
formed by bringing into it a positive increment of utilities to all participants P, or
by improving the position of existing participants having already formed a coalition
when new couples enter the coalition in subsequent steps. We claim that, in such
a situation, coalitions are formed by rational choice. The rational choice C(X) satisfies so-called heritage or succession rationality described by Chernoff (1954), Sen
(1970), and Arrow (1959). Below, we outline the heritage rationality in the form
suitable for visualization.
The proposition states that, in matches, the individual decisions are also rational
in a collective sense only when all participants in Dx individually find a suitable
partner. We can use different techniques to meet the individual and collective rationality by matching all participants only in Dx , which is akin to the stable marriage
procedure (Gale & Shapley, 1962). In contrast, the algorithm below provides an
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Joseph E. Mullat
optimal outcome/payoff accompanied by partial matching only−i.e., only matching
some of participants in P as participants of Dx ; once again, this is in line with the
Greedy type matching technique.
Proposition 2. The set {K} of kernels in the singles-game arranges feasible coalitions. Any outcome κ induced by a kernel K ∈ {K} is feasible.
At last, we are ready to focus on our main concept.
Definition 4. Given a pair of outcomes x and y, induced by coalitions X and Y ,
an outcome y dominates the outcome x, x ≺ y:
(i) ∃S ⊆ P| ∀σ ∈ S → xσ < yσ , (ii) the outcome y is feasible.
Condition (i) states that participants/couples σ ∈ S ⊂ P receiving payoffs xσ can
break the initial matching in Dx and establish new matches while uniting into Dy .
Alternatively, some members of X, i.e., not yet matched participants in S, can find
suitable partners amid participants in Dy , or, even their compensations in Y may be
higher than their incentives in x. Thus, by receiving yσ instead of xσ the participants
belonging to S are guaranteed to improve their positions. The interpretation of the
condition (ii) is obvious. Thus, the relation x ≺ y indicates that participants in S
can cause a split (bifurcation) of Dx , or are likely to undermine the outcome x.
Definition 5. A kernel N ∈ {K} minimal by inclusion is called a nucleus−it does
not include any other proper kernel K ⊂ N: K 6⊂ N is true for all K 6= N.
Proposition 3. The set {n} of outcomes, induced by nuclei {N}, arranges a quasicore of the singles-game. Outcomes in {n} are non-dominant upon each other, i.e.,
n ≺ n′ , or n ≻ n′ is false. Thus, the quasi-core is internally stable.
The proposition above clearly indicates that the concept of internal stability is
based on "pair comparisons" (binary relation) of outcomes. The traditional solution
of coalition games recognizes a more challenging stability, known as NM solution,
which, in addition to the internal stability, demands external stability. External
stability ensures that any outcome x of the game outside NM -solution cannot be
realized because there is an outcome n ∈ {n}, which is not worse for all, but it
is necessarily better for some participants in x. Therefore, most likely, only the
outcomes n that belong to NM -solution might be realized. The disadvantage of
this scenario stems from the inability to specify how it can occur. In contrast, in
the singles-game, we can define how the transformation of one coalition to another
takes place, namely, only along feasible sequence of couples. However, it may happen
that for some coalitions X outside the quasi-core {N}, feasible sequence may stall
unable to reach any nucleus N ∈ {N}, whereby starting at X the quasi-core is
feasibly unreachable. This is a significant difference with respect to the traditional
NM -solution.
5.
Finding the quasi-core
In general, when using Greedy type algorithms, we gradually improve the solution by
a local transformation. In our case, a contradiction exists because nowhere is stated
that local improvements can effectively detect the best solution−the best outcome
or payoffs to all players. The set of best payoffs, as we already established above,
How to arrange a Singles’ Party: Coalition Formation in Matching Game
231
arranges a quasi-core of the game. Usually, finding the core in the conventional sense
is a NP-hard task, as the number of "operations" increases exponentially, depending
on the number of participants. In the singles-game, or in almost all other types of
coalition games, we observe an extensive family of subsets constituting traditional
core imputations. Even if it is possible to find all the payoff vectors in the core,
it is impractical to do so. We thus posit that it is sufficient to find some feasible
coalitions belonging to the quasi-core and the payoffs induced by these coalitions.
This can be accomplished by applying a procedure of strong improvements of payoffs, and several gliding procedures, which do not worsen the players’ positions under
coalition formation. Indeed, based on rationality, known as the rational succession,
Definition 3, it is not rational in some situations to use the procedure of strong
improvements, as these do not exist. However, using gliding procedures, we can
move forward in one of the promising directions to find payoffs not worsening the
outcome. Experiments conducted using our polynomial algorithm show that, while
using a mixture of improvement procedure and gliding procedures, combined with
the succession condition, one can take the advantage of backtracking strategy, and
might find feasible payoffs of the singles-game belonging to the quasi-core.
We use five procedures in total—one improvement procedure and four variants of
gliding procedures. Combining these procedures, the algorithm below is given in
a more general form. While we do not aim to explain in detail how to implement
these five procedures, in relation to rational succession, it will be useful to explain
beforehand some specifics of the procedures because a visual interaction is best way
to implement the algorithm.
In the algorithm, we can distinguish two different situations that will determine in
which direction to proceed. The first direction promises an improvementTin case the
couple α ∈ X decides to match. We call the situation when C(X T
− α) C(X) = ∅
as a potential improvement situation. Otherwise, when C(X − α) C(X) 6= ∅, it is
a potential gliding direction. Let CH(X) be the set of rows C(X), the horizontal
routes in the table R, which contain the set C(X). By analogy CV (X) represents the
vertical routes, the set of columns, C(X) ⊆ CH(X)×CV (X). To apply our strategy
upon X, we distinguish four cases of four non-overlapping blocks in the mutual risk
table R:CH(X) × CV (X); CH(X) × CV (X); CH(X) × CV (X); CH(X) × CV (X).
Proposition 4. An improvement in payoffs for all participants in the singles-game
may occur only when a couple α ∈ X complies with the potential
improvement
T
situation in relation to the coalition X, the case of C(X − α) C(X) = ∅. The
couple α ∈ X is otherwise in a potential gliding situation.
The following algorithm represents a heuristic approach to finding a nucleus n among
nuclei {N} of the singles-game.
Input Build the mutual risks table, R = W + M −a simple operation in Excel
spreadsheet. Recall the notation P of players as the game participants. Set k ← 1,
X ← P in the role of not yet matched participants, i.e., as players available for
potential matching. In contrast to the set X, allocate indicating by Dx ← ∅ the
initial status of matched participants.
Do Step up: S Find a match αk ∈ CH(X) × CV (X), Dx ← Dx + αk , such that
F (X) < F (X − αk ), X ← X − αk , Xk = X, k = k + 1, otherwise Track Back.
Gliding: D Find a match αk ∈ CH(X) × CV (X), Dx ← Dx + αk , such that
F (X) = F (X − αk ), X ← X − αk , Xk = X, k = k + 1, otherwise Track Back.
232
Joseph E. Mullat
F Find a match αk ∈ CH(X) × CV (X), Dx ← Dx + αk , such that F (X) =
F (X − αk ), X ← X − αk , Xk = X, k = k + 1, otherwise Track Back.
G Find a match αk ∈ CH(X) × CV (X), Dx ← Dx + αk , such that F (X) =
F (X − αk ), X ← X − αk , Xk = X, k = k + 1, otherwise Track Back.
H Find a match αk ∈ CH(X) × CV (X), Dx ← Dx + αk , such that F (X) =
F (X − αk ), X ← X − αk , Xk = X, k = k + 1, otherwise Track Back.
Loop Until no couples to match can be found in accordance with cases S, D, F,
G and H.
Output The set Dx has the form Dx = hα1 , ..., αk i. The set N = P\Dx represents
a nucleus of the game while the payoff n induced by N belongs to the quasi-core.
In closing, it is worth noting that a technically minded reader would likely observe
that coalitions Xk are of two types. The first case is X ← X − αk operation when
the mismatch compensation increases, i.e., F (Xk ) < F (Xk − αk ). The second case
occurs when gliding along the compensation F (Xk ) = F (Xk − αk ). In general, independently of the first or the second type, there are five different directions in
which a move ahead can proceed. In fact, this poses a question—in which sequence
couples αt should be selected in order to facilitate the generation of the sequence
Dx = hα1 , ..., αk i? We solved the problem for singles-games underpinning our solution by backtracking. It is often clear in which direction to move ahead by selecting
improvements, i.e., either a strict improvement by s) or gliding procedures though
d), f ), g) or h). However, a full explanation of backtracking is out of the scope of our
current investigation. Thus, for more details, one may refer to similar techniques,
which effectively solve the problem (Dumbadze, 1989).
6.
Conclusions
The uniqueness of singles-game lies in the dynamic nature of priorities. As the
construction of the matching sequence proceeds, priorities dynamically shrink, and
finally converge at one point. Dynamic transformation, or the monotonic (dynamic)
nature of priorities, enabled constructing a game based on so-called monotone system, or MS. One disadvantage behind the use of the MS-system is its drawback
in the respective interpretation of the analysis results. More specifically, when the
process of extracting the core terminates, the interpretation requires further corrections. However, with regards to the choice of the best variants, i.e., the choice of
the best matches in the singles-game, the paper reports a scalar optimization in line
with "maxima of the lowest" principle, or rather an optimal choice of partial
matching. This view opens the way to consider the best partial matching as the
choice of the best variants−alternatives—and to explore the matching process from
the perspective of a choice problem.
Usually, when trying to analyze the results, a researcher must rely on the common
sense. Therefore, applying the well-known and well thought out concepts and categories that have been successfully applied in the past, we can move forward in
the right direction. Our advantage was that this relation was found, and was transformed into a shape similar to the core, which is known concept in the theory of
stability of collective behavior, e.g., in the theory of coalitional games.
Irrespective of the complexity of intersections in the interests of players, deftly
twisted rules for compensations in unfortunate circumstances, incitements, etc.,
singles-game, as it seems, makes a point. However, this is not enough in social
sciences, especially in economics, when a formal scheme rarely depicts the reality,
How to arrange a Singles’ Party: Coalition Formation in Matching Game
233
e.g., the difference in political views and positions of certain groups of interest, etc.
Perhaps, the individual components of the game will still be helpful in moving closer
to answering the question of what is right or wrong, or what is good and what is
bad, which would be a fruitful path to explore in future studies of this type.
Appendix
Visualization
Recall that, in the singles-game, the input to the algorithm presented in the main
paper contains two tables: W = wi,j −priorities wi the women specify with the
respect to the characteristics the men should possess, in the form of permutations of
numbers 1, n in rows, and the table M = mj,i −priorities mj the men specify with the
respect to the characteristics the women should possess, in the form of permutations
of numbers1, m in columns. These tables, and tabular information in general, are
well-suited for use in Excel spreadsheets that feature calculation, graphing tools,
pivot tables, and a macro programming language called VBA−Visual Basic for
Applications.
A spreadsheet was developed in order to present our idea visually, i.e., the search
for nuclei of the singles-game, and the stable coalitions with outcomes belonging to
the quasi-core induced by these coalitions. The spreadsheet takes for granted the
Excel functions and capabilities. Tables W , M and R of 20 × 20 dimensions can be
downloaded from http://www.datalaundering.com/download/singles-game.xls. We
first provide the user with the list of macros written in VBA. Then, we supply tables
W , M and R extracted from the spreadsheet by comments. We also hope that the
spreadsheet exercise will be useful in enhancing the understanding of our work. In
particular, we focus on the technology of backtracking, given by macros TrackR
and TrackB.
The list of macro-programming routines is in line with the steps of the algorithm
presented in Section 5.
• CaseS. Ctrl+s Trying to move by improvement along the block CH(X) ×
CV (X) of cells [i,j] by"<" operator in order to find a find a new match at the
strictly higher level. 8
• CaseD. Ctrl+d Trying to move while gliding along the block CH(X)×CV (X)
of cells [i,j] by "<=" operator in order to find a new match at the same or higher
level.
• CaseF. Ctrl+f Trying to move while gliding along the block CH(X)× CV (X)
of cells [i,j] by "<=" operator in order to find a new match at the same or higher
level.
• CaseG. Ctrl+g Trying to move while gliding along the block CH(X)×CV (X)
of cells [i,j] by "<=" operator in order to find a new match at the same or higher
level.
• CaseH. Ctrl+h Trying to move while gliding along the block CH(X)×CV (X)
of cells [i,j] by "<=" operator in order to find a new match at the same or higher
level.
V1. Spreadsheet layout
There are 20 single women and 20 single men attending the party, i.e., n, m = 20.
Three tables are thus available: The Pink table W −women’s priorities; The Blue
8
CH − cells in horizontal direction, CV − cells in vertical direction
234
Joseph E. Mullat
table M −men’s priorities, and the Yellow table R−the mutual risks table. The column to the right of the table R lists all women i = 1, 20 showing minj=1,20 ri,j level
of risk of couples [i, ∗]. The row down of the bottom of table R lists all men j = 1, 20
showing mini=1,20 ri,j level of risk of couples [∗, j]. In the right hand bottom corner
cell, the lowest mini=1,20,j=1,20 ri,j = F (X) level of risk over the whole table R
is given. Notice that the green cells in the table R visually represent the effect of
arg mini=1,20,j=1,20 ri,j operation. Actually, the green cells visualize the choice operator C(X). Arrays V24:AO25 and V26:AO26 will be implemented in the process
of generating the matching sequence together with the levels of risk associated
by this sequence. The players’ balance of payoffs occupies the array V31:AO32.
Some cells reflecting the state of finances of cashier are located below, in the array
AP34:AP44. Cells in row-1 and column-A contain the guests’ labels. We use these
labels in all macros.
V2. Functional test
The spreadsheet users are invited first to perform a functional test, in order to
become familiar with the effects of ctrl-keys attached to different macros. Calculations in Excel can be performed in two modes, automatic and manual . However,
it is advisable to choose properties and set the calculus in the manual mode, as this
significantly speeds up the performance of our macros.
The actions that can be taken if something goes wrong are listed below.
1. Originate. [Ctrl+o] Perform the macro by Ctrl+o, and then use Ctrl+b. This
macro restores the original status of the game saved by the BacKup, i.e., saved
by ctrl-k.
• RandM. [Ctrl+m] Perform the macro by Ctrl+m. It randomly rearranges
columns of Men’s priority table M by random (permutations). Notice the effect
upon men’s priority table M.
• RandW. [Ctrl+w] Perform the macro by Ctrl+w. It randomly rearranges rows
of Women’s priority table W by random (permutations). Notice the effect upon
women’s priority table W.
• Proceed. [Ctrl+e] While procEeding with macros RandM and RandW, the
macro is using random permutations for men and women until it generates the
priority tables M and W with minimum mutual risk equal to 4.
• Dummy. [Ctrl+u] This macro is removing from the list of participants those
guests that do not wish to play the game, or who decide not to pursue the dating.
We call them dUmmy players. Activate the row-1, or column-A by pointing at
man m##, or woman w## and then perform Ctrl+u excluding the chosen
guests from playing the game.
• MCouple. [Ctrl+a] Try to mAtch [ctrl+a] a couple by pointing at the cell
in the upper block: pink color to the left (or yellow to the right) in the row wi
(corres- ponding to a woman) and the column mj (corresponding to a man).
• TrackR. [Ctrl+r] Visualizes Tracking forwaRd. Memorizes the status of WomenW and Men-M priorities to be restored by TrackB macro. The effect of this
macro is invisible. It can be used whenever it is appropriate to save the active
status of all tables and all the arrays necessary to restore the status by TrackB
macro. Only when the search for quasi-core coalitions is performed manually,
the effect of macros is visible.
• TrackB. [Ctrl+b] Visualizes Tracking Back. Restores the status of Women-W
and Men-M priorities memorized by TrackR macro.
How to arrange a Singles’ Party: Coalition Formation in Matching Game
235
• Happiness [Ctrl+p] The macro calculates an index of haPpiness using the
initial risks table.
• Coalition [Ctrl+n] The macro rebuilds the matching coalitioN following the
coalition matching list previously transferred into area "AV24:AO25".
• Chernoff [Ctrl+q] Useful when indicating by red font the status of the Choice
Operator C(X) = {arg min}. Using this macro will help to confirm the validity
of the Succession Operator. To clear the status, use Ctrl+l.
V3. Extracting nuclei of the game
We came closer to the goal of our visualization, where we visually demonstrate the
main features of the theoretical model of the game by example. Generating the
matching sequence, which is performed in a stepwise fashion, constitutes the framework of the theory. At each step, to the right side of the sequence generated in the
preceding steps, we add a couple found by one of the macros CaseS, CaseD,. . . ,
CaseH, i.e., a couple that has decided to date. This process is repeated until all participants are matched, and the sequence is complete. One can easily verify that, the
levels of risk initially increase, and decline towards the end. This single ∩-peakedness
is a consequence of the levels of mutual risk monotonicity π(α, H\ {σ}) 6 π(α, H).
Indeed, recall that risk levels are recalculated after each match. With the proviso
of recommendations in our heuristic algorithm, see above, due to the recalculation,
the priority scales will "shrink" or "pack together", as only not yet matched participants remain. Let us try to generate a Matching Sequence using macros: CaseS,
CaseD, CaseF,. . . . The data, e.g., will occupy the array V24:O28.
Table 1:
Observe that, starting with the couple no. 14, we can no longer use macros of
our heuristic algorithm. Couples no. 1-13 represent a nucleus n of the game. Thus,
we can continue generating the sequence only by manual macro MCouple−Ctrl+a.
In Table-3, in the Matching Sequence of length 20, k = 1, 20, we labeled couple
[i, j] by α using notation αk . Together with levels of mutual risks in row 3, the rows
1,2 correspond to the sequence hαk i. Compensations and incentives for dating are
not payable at all, and only the costs of delights (each worth 10 e) occupy rows
4,5. Notice that, in accordance with single ∩-peakedness, the lowest levels of risk
first increase starting at 3, and after reaching 6, starting at couple no. 13, they
start declining down to 0. For couple no. 3, risks jump from 4 to 5, while, for couple
no. 4, they increase from 5 to 6.
Let us look at Table-4, where only 13 matches are accomplished, i.e., all columns
to right including the couple no. 14 are empty. Table-4 visualizes the nucleus below.
Pink and Blue colors mark those who decided to date, while Yellow marks those
who have not yet taken their decisions. Hereby, Yellow participants occupying rows
4-5 will mark the participants of a nucleus coalition−a coalition inducing payoffs
as incentives and mismatch compensations to all 40 participants—20 women and
236
Joseph E. Mullat
20 men. The payoffs 40 e and 70 e corresponding to the nucleus make up the
outcome. The balance of the outcome −the total amount of 2000 e as entrance fees
minus payoffs 2380 e −is not in cashier’s favor.
Table 2:
Addendum
We deem that it is necessary to provide a full proof of all propositions.
Proposition 1 Presented in terms of graph theory, the proposition would be obvious. Treating the formation of coalitions as a chain of sets Xk , 1, k, the proposition
may be explained in the form of a chain of graphs C(Xk ), whereby the lowest risk
F (Xk ) is assigned to couples α ready to match in the list hα = arg minσ∈Xk π(σ, Xk )i.
The list represents a graph C(Xk ) with edges h[i, j] = αi. Suppose that a couple
σ ∈ Xk , not necessarily listed in C(Xk ), decides to date. The couple σ leaves the
game. As a result, some less risky couples α ∈ C(Xk ) must reconsider whom they
prefer to date, as their preferred partners, while the chain Xk is under formation,
are no longer available. There are two possibilities. First, all partners, who are yet
unmatched and are present in couples α ∈ C(Xk ), preferred at least one of two
partners in σ, i.e., all these couples α are adjacent to σ in the graph C(Xk ). Second, because for some couples α′ ∈ C(Xk ) not adjacent to couple σ, the partners
of σ do not appear for α′ in the list C(Xk ). The proposition presupposed that, in
the process of coalitions’ Xk formation, the lowest risk function F (Xk ) does
T not
decrease. Therefore, the statement of the proposition C(Xk+1 ) ⊇ C(Xk ) Xk+1
holds in both situations.
Proposition 2 The proof is explained in the basic terms. The idea is to apply a
mathematical induction scheme. We claim that, starting from the initial state P of
the game, where nobody has been matched yet, it is possible to reach an arbitrary
coalition X by sequence hα1 , ..., αk i, X1 = P, Xk+1 = Xk − αk X = Xk , 1, k. The
sequence will improve the payoffs xk previous steps hα1 , ..., αk−1 i in accordance with
non-decreasing values F (Xk ). First, the statement of the proposition can be verified
by observation of all preference tables and all coalitions X that emerged from all
n × m tables, when both n and m are small integers. For higher n and m values, it
is NP-hard problem. Second, consider an arbitrary coalition X of the n × m-game.
While the coalition X = Dx includes all matched couples, in order to arrange a
new couple, all participants in X are still unmatched. We can thus always find a
couple α0 ∈ X such that F (P) 6 F (P − α0 ). Consider (n − 1) × (m − 1)-game,
which can be arranged from n × m-game by declaring the partners of the couple α0
as dummy players δ ∈
/ P. By the induction scheme, there exists a sequence of pairs
hα1 , ..., αk i with required quality of improving the payoffs xk starting from X1 =
P − α0 . Restoring the dummy couple α0 to the role of players in the n × m-game, we
can, in particular, ensure the required quality of the sequence hα0 , α1 , ..., αk i. The
statement of the proposition is obviously the corollary of the claim above. However,
How to arrange a Singles’ Party: Coalition Formation in Matching Game
237
it is clear that, ensured by its logic, the claim is a more general statement than the
statement of the proposition.
Proposition 3 The first part of the statement is self-explanatory. The coalition N
stops being a proper subset among kernels {K} as soon as the payoff function F (N)
allows improving the outcome n. The second part of the proposition is the same
statement, worded differently.
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Evolution of Agents Behavior in the Labor Market
Maria A. Nastych1 and Nikolai D. Balashov2
St.Petersburg State University,
Economic Faculty,
62 Chaikovskogo St., St.Petersburg, 190000, Russia
Faculty of Liberal Arts and Sciences,
58–60 Galernaya St., St.Petersburg, 190000, Russia
Center for Market Studies and Spatial Economics (CMSSE),
National research university "Higher school of economics",
B.47, pr.Rimskogo–Korsakova, Saint Petersburg, Russia, 190068
E-mail: [email protected]
2
St.Petersburg State University,
Economic Faculty,
62 Chaikovskogo St., St.Petersburg, 190000, Russia
E-mail: [email protected]
1
Abstract The paper studies imbalance of labor supply and labor demand
relative to qualifications. Every person faces a problem of choosing a right
path for his future career. On the other hand, employers have a dilemma
either to hire insufficient qualified personnel in a particular field and train
until he or she reaches required level of qualification, or seek an opportunity
to hire skilled personnel. We model these choices trough the evolutionary
game theory approach.
Keywords: labor market, qualification, evolution.
1.
Introduction
One of the leading resources of Russian economic internal growth in modern conditions is the labor force. Currently the focus is on the problem of the economic growth
in Russia associated with the energy dependency, whereas the problems connected
with the dependence of the Russian economic system on the cost and quality of the
labor force, unfairly in the shadows. The study of the reasons for the current state
of the labor market is definitely important applied problems as well. However, the
problem of explaining patterns of the labor market evolution is more significant on
the theoretical level in the long term.
Urgent problem of the Russian economy is the lack of qualified personnel and
imbalance of labor supply and labor demand relative to qualifications. Thus, there is
excess of labor demand over its supply for the market of rare and needed professions,
which causes the price of labor assignment by the candidates themselves. For the
low-skilled labor market it can be observed the reverse situation. An enormous
low-skilled labor supply leads to the situation where employers dictate most of the
conditions.
Every person faces a problem of choosing a right path for his future career. In
most of the cases decisions about profession and the quality of education are made
based on the imperfect information and incorrect definition of objectives. Employees
make decisions based not on the market indicators of demand and wages, but on
the basis of subjective factors. In other words, candidates take into account not
quantitative but qualitative indicators.
240
Maria A. Nastych, Nikolai D. Balashov
On the other hand, employers have a dilemma either to hire insufficient qualified
personnel in a particular field and train until he or she reaches required level of
qualification, or seek an opportunity to hire skilled personnel. In this case, firms
should be guided by the motives for the optimization of personnel costs. Therefore,
the objective of employers is the choice of their behavior, based on current market
conditions and adequate quantitative assessment of possible alternatives.
Described problems reveal the need to study the behavior of agents in the labor
market in order to find an equilibrium strategy in the long term. It is worth noting
that the labor market agents also play one of the major roles in this area of Russian
economy. Their recruiting strategy based on the appropriate quantitative indicators
through their access to information could be an integral part of the process of
equilibrium achieving.
2.
The model
The labor market can be divided into three sectors according to the level of personnel qualification: low-skilled, skilled and highly skilled. Each of these sectors has
different sets of rules and conditions. Selection of an optimal strategy for responding to the behavior of other similar participants can facilitate a potential conflict of
interests for both employers and candidates. Conflict character of participants’ behavior stipulates the use of mathematical tools of game theory to the labor market
study.
Each candidate has an alternative of either accept a job offer, which meets his
current qualifications, or look for a job which will offer him or her an opportunity
of getting trained by qualified personnel and in the long run give possibility of a
career growth. Therefore, employees have two appropriate strategies: to hire already
qualified personnel or to spend money on training non-qualified personnel.
We introduce further notations for the set Zw = {l, s, h} of possible workers’
pure strategies, and notations for the average wage according to three distinguished
sectors: wl , ws , wh for the low-skilled, skilled and highly skilled workforce, respectively, 0 < wl < ws < wh at that. In terms of labor demand the set Ze = {q, t}
of employers’ pure strategies includes two alternatives: to hire qualified or to train
unqualified personnel correspondingly. Assume the cost of meeting the skilled and
highly skilled level to be equal to cs and ch , respectively, for both employees and
employers, and 0 < cs < ch .
Let us consider firstly low-skilled labor market where is no need to train employees. Here and further we assume that in non-crisis economic environment, individuals will not compete for employment in the market with the need for minor
qualification then he or she has. Then in terms of employees the payoffs ulw (zw )
could be described as
(
wl , zw = l,
l
uw (zw ) =
0, zw ∈ {s, h}
Obviously, the solution of this game according to Nash equilibrium (NE) is the
strategy l. Otherwise, in crisis economic environment all of the payoffs ulw (zw ) in
this evolution game could be equal to wl regardless of the strategies. This case,
however, is of interest in neither the theoretical nor the practical point of view. A
similar situation is in the game in terms of employers, where there is no need for
any qualification level and, therefore, all of the payoffs ule (ze ) are designated as
Evolution of Agents Behavior in the Labor Market
241
ule (ze ) = f (zw ) − wl ,
where f (zw ) is the effect from the employee with type zw work, 0 < f (l) < f (s) <
f (h) at that. Here we assume zw = l in non-crisis economic environment. The
invariance of wages in low-skilled labor market is determined by the excess of labor
supply over labor demand, as mentioned above.
For modeling the games in the skilled and highly skilled labor markets it is necessary to introduce a parameter of propensity to education for workers 0 6 θw 6 1
and parameter of propensity to training for employers 0 6 θe 6 1. These parameters can be interpreted as the probability of interest coincidence for the players of
different types. Thus, in the skilled labor market the payoffs usw (zw ) in terms of
employees could be described as
 l e

 w θ , zw = l,
s
uw (zw ) = (ws − cs ) (1 − θe ) , zw = s,


0, zw = h.
It can be seen that there is exists dependence of the evolutionarily stable strategies on the training propensity parameter. Thus, strategy l is NE if
θe >
and strategy s is NE if
θe <
w s − cs
w l + w s − cs
wl
w s − cs
+ w s − cs
Otherwise, NE is determined by the mixed strategies l and s with training
propensity parameter
θe =
w s − cs
w l + w s − cs
In terms of employers skilled labor market could be defined by the payoffs use (ze )
as follow
(
w
(f (s) − ws )(1 − θ ) , ze = q,
s
ue (ze ) =
f (l) − wl − cs θw , ze = t.
NE here is defined by the strategy q if
θw >
and by the strategy t if
θw <
f (s) − ws
s
f (s) − w + f (l) − wl − cs
f (s) − ws
s
f (s) − w + f (l) − wl − cs
NE is determined by the mixed feasible strategies with education propensity
parameter
242
Maria A. Nastych, Nikolai D. Balashov
θw =
f (s) − ws
f (s) − w s + f (l) − wl − cs
Considering further the highly skilled market it is worth noting that the lowskilled employee is unlikely to be hired here. In this case the evolutionary game in
terms of employees is defined by the payoffs uhw (zw ) as


 0, zw = l,
s
s e
h
uw (zw ) = (w − c )θ , zw = s,

 h
(w − ch ) (1 − θe ) , zw = h.
Moreover, the average wage wh is dictated largely by the employees due to excess
of labor demand over labor supply, at least in accordance with the current state of
the labor market. NE for workers in the highly skilled labor market is the strategy
s if
θe >
and the strategy h if
θe <
w h − ch
w s − cs + w h − ch
ws
w h − ch
− cs + w h − ch
As before, NE is determined by the mixed strategies s and h with training
propensity parameter
θe =
ws
w h − ch
− cs + w h − ch
The opposite situation may occur in the entrepreneurial activity, for example,
and cause the existence of other stable strategies. But price formation in this kind of
employment market has another rules and conditions then we are aimed to consider
here.
Finally the highly skilled market in terms of employers is defined by the payoffs
use (ze ) as follow
(
(f (h) − wh )(1 − θw ), ze = q,
s
ue (ze ) =
s
(f (s) − ws − ch + c )θw , ze = t
NE here is specified by the strategy q if
θw <
and by the strategy t if
θw >
f (s) −
ws
f (h) − wh
+ cs + f (h) − wh − ch
f (h) − wh
f (s) − ws + cs + f (h) − wh − ch
NE is specified by the mixed feasible strategies with education propensity parameter
243
Evolution of Agents Behavior in the Labor Market
θw =
f (h) − wh
f (s) − ws + cs + f (h) − wh − ch
At the same time each of employees as well as each of employers takes a decision
in favor of only one employer and only one employee respectively. Here it is possible
to consider the situation where employers compete with each other for the same
employee, and employees compete for the same employer, in turn. The opportunity
to choose two players randomly from the population is available in the evolutionary
game theory.
Since availability of training candidates can improve their skills and employers can improve the skills of their workers. Employers also can evolve by gaining
experience of hiring new employees under conditions of turnover. Training and development are the crucial factors in the evolutionary game theory. Thus, propensity
for trainings can be considered as one of the main factors which determines an evolutionarily stable strategy (ESS, (Maynard Smith, 1982)) in the game theory model
for both employees and employers.
The main difference in the evolutionary models building is dependence of each
player’s gains upon a type of the opposing player. If competitors have the same
strategy, the gain goes to some one of them with equal probability. Here the above
assumption that individuals will not compete for employment in the market with the
need for minor qualification then he or she has remains. The zero gain of overqualified employee in this case can be interpreted as employers’ projections about stuff
turnover growth.
Let us return to the low-skilled labor market first. The payoff matrix in terms
of employees could be described as shown in the Table 1. ESS here coincides with
NE of this game and is obviously the same situation (l, l) . The employers do not
take any decision as well.
Table 1: Low-skilled labor market: the payoff matrix in terms of employees
l
l
wl wl
; 2
2
s
l
h
s
(w ; −c )
l
l
(w ; −ch )
l
s (−cs ; wl ) ( w2 − cs ; w2 − cs ) (wl − cs ; −ch )
l
l
h (−ch ; wl ) (−ch ; wl − cs ) ( w2 − ch ; w2 − ch )
In the skilled employment market the game is determined as shown in the Table
2. The amount of NE’s in pure strategies in this game depends on the propensity
parameter θe It can be shown that there are no other cases.:

w s − cs

e

(s,
s)
,
θ
6


2wl



2ws − wl + 2cs
w s − cs
e
NE :
(s, s) ; (l, l) ,
6
θ
6

2ws
2wl



s
l
s


 (l, l) , 2w − w + 2c < θe
2ws
NE in mixed strategies here is nothing more but the vector (θw ; 1 − θw ; 0):
244
Maria A. Nastych, Nikolai D. Balashov
w
w
(θ ; 1 − θ ; 0) =
ws − 2cs − 2wl θe wl − 2ws (1 − θe ) + 2cs
;
;0
(1 − 2θe ) (wl − ws )
(1 − 2θe ) (wl − ws )
Table 2: Skilled labor market: the payoff matrix in terms of employees
l
s
h
wl θe ; ws (1 − θe ) − cs
wl ; −ch
ws
s
s ws (1 − θe ) − cs ; wl θe
− cs ; w2 − cs
ws − cs ; −ch
2
s
ws
h
−ch ; wl
−ch ; ws − cs
− ch ; w2 − ch
2
l
wl
2
;
wl
2
In terms of ESS it means that in the skilled employment market the strategy s
is optimal if:
or

e
l
s
(1
−
2θ
)
w
−
w
<0



 s
w − 2cs − 2wl θe < 0


ws − 2cs − 2wl θe

 θw <
(1 − 2θe ) (wl − ws )

e
l
s
(1
−
2θ
)
w
−
w
>0






ws − 2cs − 2wl θe

 θw >
(1 − 2θe ) (wl − ws )
then θw → 0. Otherwise, if:

e
l
s

 (1 − 2θ ) w − w < 0


or


ws − 2cs − 2wl θe

 θw >
(1 − 2θe ) (wl − ws )

(1 − 2θe ) wl − ws > 0



 s
w − 2cs − 2wl θe > 0


ws − 2cs − 2wl θe

 θw <
(1 − 2θe ) (wl − ws )
then θw → 1 and strategy l is ESS.
ESS for the game in the terms of employers (Table 3) coincides with its game’
NE. Evolutionary games in the terms of employees and employers for all other types
of markets are presented.
245
Evolution of Agents Behavior in the Labor Market
Table 3: Skilled labor market: the payoff matrix in terms of employers
q
q
t
t
(f (s) − ws ) (1 − θw ) ; f (l) − wl − cs θw
f (l)−wl −cs f (l)−wl −cs
;
f (l) − wl − cs θw ; (f (s) − ws ) (1 − θw )
2
2
f (s)−ws f (s)−ws
;
2
2
Table 4: High-skilled labor market: the payoff matrix in terms of employees
l
l
wl
2
;
wl
2
s
h
(0; ws − cs )
0; wh − ch
s
s
w
s (ws − cs ; 0)
− cs ; w2 − cs
ws θe − cs ; wh (1 − θe ) − ch
2
h
h wh
h
w
h wh − ch ; 0 wh (1 − θe ) − ch ; ws θe − cs
−
c
;
−
c
2
2
Table 5: High-skilled labor market: the payoff matrix in terms of employers
q
q
t
f (h)−wh f (h)−wh
;
2
2
f (s) − ws − ch + cs θw ; f (h) − wh (1
−θw ))
t
f (h) − wh (1 − θw ) ; (f (s) − ws
−ch + cs θw
f (s)−ws −ch +cs f (s)−ws −ch +cs
;
2
2
Introduced propensity parameters in these employment market models play the
role of the population shares and do determine the process of equilibrium establishing in all types of labor market for both types of players. Therefore, their ratio to
the particular professions is of special interest.
3.
Conclusion
One of the main results of such evolutionary games is the necessity for player to
be guided by the unavailable information about the incentives of the opposite type
players. The solution to this lack of information is the involvement of recruitment
agencies that possess such data. Identified ESS’s dependence on the propensity
parameters, that is next conclusion, should be taken into account in this instance.
For practical application of this model, it is necessary to consider certain occupations for each of the sectors. Derived evolutionarily stable strategies determine
the equilibrium behavior of the labor market with certain allowable intervals for the
model parameters.
As a further development, we propose to introduce into the model additional
parameters that affect the behavior of economic agents in the labor market, such
as work experience. There are some occupations, in which more relevant is to get
practical experience rather than to receive theorize education, and vice versa.
References
Maynard Smith, J. (1982). Evolution and the Theory of Games. Cambridge University
Press
An Axiomatization of the Proportional Prenucleolus
Natalia Naumova
St.Petersburg State University, Faculty of Mathematics and Mechanics
Universitetsky pr. 28, Petrodvorets, St.Petersburg, 198504, Russia
E-mail: [email protected]
Abstract The proportional prenucleolus is defined on the class of all positive TU games with finite sets of players. The set of axioms used by Sobolev
(1975) for axiomatic justification of the prenucleolus is modified. It is proved
that the proportional prenucleolus is a unique value that satisfies 4 axioms:
efficiency, anonymity, proportionality, and proportional DM consistency. The
proof is a modification of the proof of Sobolev’s theorem.
For strictly increasing concave function U defined on (0, +∞) with range
equal to R1 , a generalization of the proportional prenucleolus is called U –
prenucleolus. The axioms proportionality and proportional DM consistency
are generalized for its justification.
Keywords: cooperative games; proportional nucleolus; prenucleolus; consistency.
1.
Introduction
For cooperative TU games the nucleolus was defined by Schmeidler, 1969. First it
was used for constructive proof of existence of the bargaining set M1i . The prenucleolus was defined in Sobolev, 1975 and later in Maschler, Peleg, Shapley, 1979. A
unique axiomatic justification of the prenucleolus was given by Sobolev, 1975. He
proved that the prenucleolus is a unique value that satisfies 4 axioms: efficiency,
anonymity, covariance, and Davis-Maschler consistency. Since that time there appeared only some weakening of his axioms (Orshan, 1993, Peleg and Sudholter,
2007).
For games with positive characteristic function, the proportional prenucleolus,
where excesses in the "classical" case are replaced by ratios of coalitional claims to
total shares of players in these coalitions, is natural. We propose a modiication of
Sobolev’s set of axioms, where covariance is replaced by proportionality and DavisMaschler consistency is replaced by proportional DM consistency. The proof is a
modification of the proof of Sobolev.
For strictly increasing concave function U , U –prenucleolus is a generalization
of the proportional prenucleolus, the generalization of the proportionality axiom is
called U –excess property, and the generalization of the proportional DM consistency
is called U –DM consistency. If U (t) = ln(t), then U –prenucleolus is the proportional
prenucleolus and the axioms for it justification coincide with axioms for justification
of the proportional prenucleolus.
The paper is organized as follows. Section 2 contains the main definitions and the
main statements of the paper. The results of Sobolev that will be used in the proof
are described in Section 3. The proof of axiomatic justification of U –prenucleolus is
given in Section 4.
247
An Axiomatization of the Proportional Prenucleolus
2.
Definitions and main theorems
Consider a class of positive TU games
G + = {(N, v) : |N | < ∞, v(S) > 0 for ∅ 6= S ⊂ N }.
|N |
Avalue on G + is a map that assigns to every (N, v) ∈ G + a vector x ∈ R++ .
P
|N |
A preimputation of (N, v) ∈P
G + is a vector x ∈ R++ such that i∈N xi = v(N ).
For S ⊂ N , denote x(S) = i∈S xi .
For preimputation z of (N, v) ∈ G + , let the collection of coalitions {S : S ⊂
N, S 6= ∅} be enumerated such that z(Si )/v(Si ) ≤ z(Si+1 )/v(Si+1 ). Denote
|N |
θ((N, v), z) = {z(Si )/v(Si )}2i=1 −1 .
The preimputation y of (N, v) belongs to the proportional prenucleolus of (N, v) iff
θ((N, v), y) ≥lex θ((N, v), z)
for all preimputations z
of (N, v).
For each (N, v) ∈ G + , the proportional prenucleolus of (N, v) is a singleton. It
follows from the results of Vilkov, 1974, Justman, 1977, Sobolev, 1975.
The following axiomatization of the proportional prenucleolus is a modification
of Sobolev’s axiomatization of the prenucleolus (Sobolev, 1975, see also Peleg and
Sudholter, 2007).
Let a value fPbe defined on G + . Consider the following properties of f .
Efficiency. i∈N fi (N, v) = v(N ).
Anonymity. Let for games (N, v) and (N ′ , w) there exists a bijection π : N →
N such that v(S) = w(πS) for all S ⊂ N . Then fi (N, v) = fπi (N ′ , w).
′
|N |
Proportionality. For any games (N, v), (N, w) ∈ G + , any x, y ∈ R++ ,
x(S)
y(S)
=
v(S)
w(S)
for all S ⊂ N
implies
x = f (N, v) iff
y = f (N, w).
The proportionality property means that the value f depends only on the values
of proportional excesses. It was used in Yanovskaya, 2002 for axiomatization of some
proportional solutions instead of covariance property.
Proportional DM consistency. Let x = f (N, v), then for each S ⊂ N , S 6= ∅,
xS = f (S, v x,S ), where
(
v(N ) − x(N \ S) for P = S,
x,S
v (P ) =
∪T )x(P )
for P ⊂ S, P 6= S.
maxT ⊂N \S v(Px(P
∪T )
The proportional DM consistency is a modification of Davis-Maschler consistency.
Theorem 1. The proportional prenucleolus is a unique value defined on G + that
satisfies efficiency, proportionality, anonymity, and proportional DM consistency
properties.
248
Natalia Naumova
The generalization of this theorem will be proved in Section 4.
Now consider a generalization of the proportional prenucleolus. Let U be a
strictly increasing concave function defined on (0, +∞) with U ((0, +∞)) = R1 .
For preimputation z of (N, v) ∈ G + , let the collection of coalitions {S : S ⊂
N, S 6= ∅} be enumerated such that
U (z(Si )) − U (v(Si )) ≤ U (z(Si+1 )) − U (v(Si+1 )).
Denote
|N |
θ((N, v), z) = {U (z(Si )) − U (v(Si ))}2i=1 −1 .
The preimputation y of (N, v) belongs to the U –prenucleolus of (N, v) iff
θ((N, v), y) ≥lex θ((N, v), z)
for all preimputations
z
of (N, v).
As U is a concave function on (0, +∞), the functions U (x(S)) are continuous
for all S, hence for each (N, v) ∈ G + , the U –prenucleolus of (N, v) is a nonempty
set. Moreover, it is a singleton. The proof is the same as in the case of "classical"
prenucleolus.
The proportionality and the proportional DM consistency properties are generalized as follows.
|N |
U –excess property. For any games (N, v), (N, w) ∈ G + , any x, y ∈ R++ ,
U (x(S)) − U (v(S)) = U (y(S)) − U (w(S))
for all S ⊂ N, S 6= ∅
implies
x = f (N, v) iff y = f (N, w).
The proportionality axiom is equvalent to ln–excess property.
U –DM consistency. Let x = f (N, v), then for each S ⊂ N , S 6= ∅, xS =
f (S, v x,S ), where

0 for P = ∅,



v(N ) − x(N \ S) for P = S,
v x,S (P ) =
U −1 U (x(P )) + maxT ⊂N \S [U (v(P ∪ T )) − U (x(P ∪ T ))]



for P ⊂ S, P 6∈ {S, ∅}.
It means that
U (v x,S (P )) − U (x(P )) = max [U (v(P ∪ T )) − U (x(P ∪ T ))].
T ⊂N \S
Note that v x,S is well defined. Indeed, since U is a strictly increasing and continuous
function with range equal to R1 , U −1 (t) is well defined for all t ∈ R1 .
U –DM consistency is a modification of Davis-Maschler consistency. The proportional DM consistency is equivalent to ln–DM consistency.
Theorem 2. Let U be a strictly increasing concave continuous function defined on
(0, +∞) with U ((0, +∞)) = R1 .
The U –prenucleolus is a unique value defined on G + that satisfies efficiency,
U –excess property, anonymity, and U –DM consistency conditions.
This theorem will be proved in Section 4.
An Axiomatization of the Proportional Prenucleolus
3.
249
Sobolev’s construction.
Let N be a finite set of players.
A collection D of coalitions is a balanced collection on N if there exist positive
numbers {δS }S∈D satisfying
X
δS = 1 for all i ∈ N.
S∈D:i∈S
The vector {δS }S∈D is called a vector of balancing weights of D.
A coalitional family is a pair (N, {Bl }l∈L ), where
1) N and L are finite nonempty sets,
2) Bl ⊂ 2N for all l ∈ L,
3) Bl ∩ Bt = ∅ for l, t ∈ L, l 6= t.
Let H = (N, {Bl }l∈L ) be a coalitional family. A permutation π of N is a symmetry of H if for every l ∈ L and every S ∈ Bl , π(S) ∈ Bl . H is transitive if for
every pair (i, j) ∈ N × N there exists a symmetry π of H such that π(i) = j.
If N is a finite set, i ∈ N , and B ⊂ 2N , then denote B i = {S ∈ B : i ∈ S}.
Let Hi = (Ni , {Bi,l }l∈Li ), i = 1, 2 be coalitional families. The product of H1 and
H2 is the coalitional family defined by
N ⋆ = N1 × N2 ,
L⋆ = {(1, l) : l ∈ L1 } ∪ {(2, l) : l ∈ L2 },
B(1,l) = {S ⊂ N ⋆ : S = T × N2 , T ∈ B(1,l) } for all l ∈ L1 ,
B(2,l) = {S ⊂ N ⋆ : S = N1 × T, T ∈ B(2,l) } for all l ∈ L2 .
Let N be a set of players. Let Bk , k = 1, . . . , p be balanced on N collections of
coalitions such that
Bi ⊂ Bi+1
for all i < p
and Bp = 2N \ {∅}.
Fix k ∈ {1, . . . , p}. The vector of balancing weights {δS }S∈Bk of Bk can be
chosen such that δS = mS /m for S ∈ Bk , where the numbers m and mS are natural
numbers. Hence
X
mS = m for all i ∈ N.
P
i
S∈Bk
Denote t = S∈Bk mS .
The players i, j ∈ N are equivalent with respect to Bk if Bki = Bkj .
Let Hi be the equivalence class of player i.
Denote r = max{|Hi | : i ∈ N }.
Sobolev, 1975 proved that there exists a coalitional family (Nk⋆ , Bk⋆ ) associated
with (N, Bk ) with the following properties.
N ⊂ Nk⋆ and |Nk⋆ | = rCtm .
⋆
Bk⋆ = {TS,q
: S ∈ Bk , 1 ≤ q ≤ mS }.
⋆
The sets TS,q
, S ∈ Bk , 1 ≤ q ≤ mS are distinct.
⋆
TS,q
∩N =S
for all S ∈ Bk , 1 ≤ q ≤ mS ;
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Natalia Naumova
|Bk⋆i | = m for each
|Hi⋆ | = r
where Hi⋆ = {j ∈ Nk⋆ : Bk⋆i = Bk⋆j }.
i ∈ Nk⋆ ;
for each i ∈ Nk⋆ ,
Denote the product of coalitional families (Nk⋆ , Bk⋆ ), k = 1, . . . , p by
b , {Bbl }l∈{1,...,p} .
H= N
b = Qp N ⋆ and for every k = 1, . . . , p,
Thus N
k=1 k
⋆
⋆
b : Sb = N1⋆ × . . . × Nk−1
Bbk = {Sb ⊂ N
× S × Nk+1
× . . . × Np⋆
for some
S ∈ Bk⋆ }.
Sp
b
Define Bbp+1 = 2N \ k=1 Bbk ,
b= N
b , {Bbl }l∈{1,...,p+1} .
H
b is a transitive coalitional family. (see Sobolev, 1975, pp.137–
Sobolev proved that H
145 or Peleg and Sudholter, 2007, pp.109-114.)
4.
Proof of the main theorem
4.1.
Auxiliary results
Let (N, v) be a cooperative TU game, x be a preimputation of (N, v), U be a
function defined on R1 . For every α ∈ R1 , denote
D(U, N, v, x, α) = {S ⊂ N : U (x(S)) − U (v(S)) ≤ α, S 6= ∅}.
We use the following modification of the theorem proved by Kohlberg, 1971.
Theorem 3. Let (N, v) be a cooperative TU game, U be a strictly increasing concave function defined on R1 or on (0, +∞). A preimputation x of (N, v) is the
U –prenucleolus of (N, v) if and only if each nonempty D(U, N, v, x, α) is a balanced
collection of coalitions on N .
Proof. Since U is a strictly increasing continuous concave function, the U –prenucleolus
exists and it is a singleton.
Formally, Kohlberg proved this theorem for the case U (t) = t, but the proof in
Maschler, Solan, Zamir, 2013 (pp.816-821) is valid for our case since U is a strictly
increasing function.
⊔
⊓
Lemma 1. Let U be a strictly increasing concave function defined on (0, +∞) with
U ((0, +∞)) = R1 .
Then the U –prenucleolus satisfies efficiency, U –excess property, anonymity, and
U –DM consistency conditions.
An Axiomatization of the Proportional Prenucleolus
251
Proof. Efficiency and anonymity properties of the U –prenucleolus are evident. U –
excess property follows from Kohlberg theorem. Let us check U –DM consistency.
Let x be U –prenucleolus of (N, v), S ⊂ N . For α ∈ R1 , denote
D(U, S, v x,S , xS , α) = {P ⊂ S : U (x(P )) − U (v x,S (P )) ≤ α, P 6= ∅}.
Let D(U, S, v x,S , xS , α) 6= ∅. By Kohlberg theorem, we need to prove that
D(U, S, v x,S , xS , α) is a balanced collection of coalitions on S. By the definition
of v x,S , D(α) = D(U, N, v, x, α) 6= ∅ and for each P ∈ D(U, S, v x,S , xS , α), there
exists Q ∈ D(α) such that Q ⊃ P . By Kohlberg theorem, D(α) is a balanced collection of coalitions on N . Let {δQ }Q∈D(α) be a vector of balancing weights of D(α)
on N . For P ∈ D(U, S, v x,S , xS , α), take
X
λP =
δQ ,
Q∈D(α): Q⊃P
then {λP }P ∈D(U,S,vx,S ,xS ,α) is a vector of balancing weights of D(U, S, v x,S , xS , α).
⊔
⊓
4.2.
Proof of Theorem 2.
Proof. By Lemma 1, U –prenucleolus satisfies 4 axioms.
Let f be a value defined on G + that satisfies efficiency, U –excess property,
anonymity, and U –DM consistency conditions. Let (N, v) ∈ G + and x be the U –
prenucleolus of (N, v). We have to prove that f (N, v) = x. Define (N, w) by
0 for S = ∅,
w(S) =
U −1 [U (|S|) + U (v(S)) − U (x(S))] for S 6= ∅.
Then
U (x(S)) − U (v(S)) = U (|S|) − U (w(S))
for all S ⊂ N, S 6= ∅.
In view of Kohlberg theorem, U –excess property of the U –prenucleolus implies that
the vector 1|N | = (1, 1, . . . , 1) ∈ R|N | is the U –prenucleolus of (N, w). By U –excess
property of f , it is sufficient to prove that f (N, w) = 1|N | .
Let {U (|S|) − U (w(S)) : S ⊂ N, S 6= ∅} = {µ1 , . . . , µp }. Denote
Bk = {S ⊂ N : U (|S|) − U (w(S)) ≤ µk , S 6= ∅} for all k = 1, . . . , p.
By Kohlberg’s theorem, Bk is a balanced collection on N . Take Sobolev’s construction and notations for Bk , k = 1, . . . , p.
b , w)
Define (N
b as follows.

0 if S = ∅,


 |N
b | if S = N
b,
w(S)
b
=
−1

U [U (|S|) − µk ]

 −1
U [U (|S|) − µp ]
b
if S ∈ Bbk \ {∅, N}
for k = 1, . . . , p,
b
if S ∈ Bbp+1 \ {∅, N}.
b is transitive, for each i, j ∈ N
b there exists a permutation π of N
b such that
As H
π(i) = j and S ∈ Bbk implies π(S) ∈ Bbk for all k = 1, . . . , p + 1. As |S| = |π(S)|,
252
Natalia Naumova
b , therefore by efficiency and anonymity properties of
w(S)
b
= w(π(S))
b
for all S ⊂ N
f , we get
b , w)
b.
f (N
b i = 1 for all i ∈ N
Let
b 0 = {bi = (i, . . . , i) ∈ N
b : i ∈ N }.
N
b 0, w
Consider (N
b0 ), where
w
b0 (P ) =

0 for P = ∅,



b 0 | for P = N
b 0,
 |N
U −1 U (|P |) + maxT ⊂Nb \Nb 0 [U (w(P
b ∪ T )) − U (|P ∪ T |)]




b 0 , P 6= N
b 0 , ∅.
for P ⊂ N
b 0, w
Then by U –DM consistency, f (N
b0 ) = (1, . . . , 1) ∈ R|N | .
b : i ∈ S}. We prove that
Let S ⊂ N , denote Sb = {bi = (i, . . . , i) ∈ N
b
w(S) = w
b0 (S).
(1)
If S ∈ {N, ∅} then (1) is valid. Let S 6= N, ∅.
b ≥ w(S). There exists k ≤ p such that µk =
Step 1. Let us prove that w
b0 (S)
U (|S|) − U (w(S)), i.e., S ∈ Bk . By the definition of Bk⋆ , there exists S ⋆ ∈ Bk⋆ such
that S ⋆ ∩ N = S. Take
⋆
⋆
b = N1⋆ × . . . × Nk−1
Q
× S ⋆ × Nk+1
× . . . × Np⋆ ,
h
i
b ∈ Bbk and w(
b = U −1 U (|Q|)
b − µk . As Sb ⊂ Q
b and U is a strictly increasthen Q
b Q)
ing function,
b ≥ U (|S|)
b + U (w(Q))
b = U (|S|) − µk = U (w(S)),
U (w
b0 (S))
b
− U (|Q|)
b ≥ w(S).
hence w
b0 (S)
b ≤ w(S). Let Tb ⊂ N
b \N
b 0 and
Step 2. We prove that w
b0 (S)
b = U (|S|)
b + U (w(
U (w
b0 (S))
b Sb ∪ Tb)) − U (|Sb ∪ Tb|).
(2)
Let k0 = min{k : S ∈ Bk }.
⋆
If Sb ∪ Tb ∈ Bp+1
then
U (|Sb ∪ Tb|) − U (w(
b Sb ∪ Tb)) = µp ≥ µk0 = U (|S|) − U (w(S)),
b ≤ w(S).
and, by (5), this implies w
b0 (S)
Now suppose that Sb ∪ Tb ∈ Bk⋆ , where k ≤ p. Then there exists S ⋆ ∈ Bk⋆ such
that
⋆
⋆
Sb ∪ Tb = N1⋆ × . . . × Nk−1
× S ⋆ × Nk+1
× . . . × Np⋆ .
Then S = S ⋆ ∩ N .
An Axiomatization of the Proportional Prenucleolus
253
Indeed, if i ∈ S then
bi = (i, . . . , i) ∈ Sb ⊂ Sb ∪ Tb,
hence i ∈ S ⋆ , so S ⊂ S ⋆ ∩ N .
b \N
b 0 implies b
If j ∈ S ⋆ ∩ N then b
j = (j, . . . , j) ∈ Sb ∪ Tb, but Tb ⊂ N
j 6∈ Tb, hence
⋆
b
j ∈ S and S ∩ N ⊂ S.
As S = S ⋆ ∩ N and S ⋆ ∈ Bk⋆ , we have S ∈ Bk , hence k0 ≤ k. Then
U (|Sb ∪ Tb|) − U (w(
b Sb ∪ Tb)) = µk ≥ µk0 = U (|S|) − U (w(S)),
b ≤ w(S).
and by (2) this implies w
b0 (S)
0 b
Thus, w(S) = w
b (S) for all S ⊂ N .
b 0 . π(i) = (i, . . . , i) ∈ R|N | . As was
Take the following bijection π : N → N
0
b 0, w
proved above, w(S) = w
b (πS) for all S ⊂ N Anonymity of f and f (N
b0 ) =
|N |
(1, . . . , 1) ∈ R
implies f (N, w)i = 1 for all i ∈ N . This completes the proof of
Theorem 2.
⊔
⊓
References
Justman, M. (1977). Iterative processes with nucleolar restrictions. International Journal
of Game Theory, 6, 189–212.
Kohlberg, E. (1971). On the nucleolus of a characteristic function game. SIAM Journal on
Applied Mathematics, 20(1), 62–66.
Maschler, M., Peleg, B., Shapley, L. S. (1979). Geometric properties of the kernel, the
nucleolus and related solution concepts. Math. Oper. Res. 4, 303–338.
Maschler, M., Solan, E., Zamir, S. (2013). Game theory. Cambridge University Press, 979
pp.
Orshan, G. (1993). The prenucleolus and the reduced game property: Equal treatment replaces anonymity. International Journal of Game Theory, 22, 241–248.
Peleg, B. and Sudholter, P. (2007). Introduction to the theory of cooperative games, Chap. 6.
Springer-Verlag, Berlin Heidelberg New York.
Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal on
Applied Mathematics, 17(6), 1163–1170.
Sobolev, A. I. (1975). The characterization of optimality principles in cooperative games
by functional equations. In: Mathematical Methods in the Social Sciences, (Vorobiev,
N. N., ed.), 6, 95–151, Vilnius,Academy of Sciences of the Lithuanian SSR, in Russian.
Vilkov, V. B. (1974). The nucleolus in cooperative games without side payments. Journal
of Computing Mathematics and Mathematical Physics, 14(5), 1327–1331.In Russian
Yanovskaya, E. (2002). Consistency for proportional solutions. International Game Theory
Review, 4(3), 343–356.
Competition Form of Bargaining
Tatyana E. Nosalskaya
Zabaikalsky Institute of Railway Transport,
Department of Higher Mathematics and Applied Informatics,
Magistralnaya str. 11, Chita, 672040, Russia
E-mail: [email protected]
Abstract We consider the noncooperative zero-sum game, related with the
competitions. Players submit the competition projects, that are characterized by a finite set of parameters. The arbitrator or arbitration committee
uses a stochastic procedure with the probability distribution to determine
the most preferred project. This distribution is known to all participants.
Payoff ot the winner depend on the parameters of his project. The threedimensional mathematical model of this problem is constructed, which is
then extended to the multi-dimensional case. The equilibria in the games
with four and n persons are found, as well as the corresponding payoffs are
computed.
Keywords: Model of competition, bargaining, stochastic procedure, n-person
game, Nash equilibrium.
1.
Introduction
Recently in the search for performer of any work or any service provider, tenders
are widely used. A lot of companies participate in it, both large and just developing. Tender means the competition form of bargaining. A competition presupposes
rivalry among participants for the right to carry out their projects. This means that
in the market there are several potential performers with similar capabilities, and
the initiator’s of tender offer is interesting for them. As a result, both sides stand to
benefit: the organizer receives the best performer, and performer gets a big contract
and good profit.
This paper presents a multi-dimensional game-theoretic model of the tender
as the competition of projects. The n persons participate in the competition. Their
proposals, or projects, are characterized by a finite set of parameters. As parameters
such project can include, for example, description of cost, time of implementation,
number of participants, etc.
2.
The model
2.1. Game with Four Players
We consider a noncooperative non-zero sum game, related with competition. Assume there are four persons, or players. They represent competition projects, characterized by a set of three parameters (x, y, z). Let the player I seeks for maximize
the amount of x + y + z, and players II, III and IV - for minimize the parameter x,
y or z, respectively.
An arbitrator or arbitration committee considers proposals received and selects
one of projects, using the stochastic procedure with the probability distribution
2
1
f (x, y, z) = g(x) · g(y) · g(z), where g(x) = √ · e−x /2 ,
2π
255
Competition Form of Bargaining
which is known to participants. In this case the winner receives payoff, which depends on the parameters of project. In the paper (Mazalov and Tokareva, 2010) an
equilibrium in competition of projects among three persons on the plane is obtained.
Because of the symmetry of the model, the optimal strategies of players will be
found in this form
player I: (c, c, c),
player II: (−a, 0, 0),
player III: (0, −a, 0),
player IV: (0, 0, −a).
Fix these proposals of players II, III and IV. Let player I submit the project
(x1 , y1 , z1 ), where x1 , y1 , z1 > 0. Then the space of projects is split into four subspaces, limited by the planes: α1 : y = x, α2 : z = x, α3 : z = y,
α4 : z = −
x1 + a
y1
x2 + y12 + z12 − a2
x− y+ 1
,
z1
z1
2z1
α5 : z = −
x1
y1 + a
x2 + y12 + z12 − a2
x−
y+ 1
,
z1
z1
2z1
α6 : z = −
x1
y1
x2 + y12 + z12 − a2
x−
y+ 1
.
z1 + a
z1 + a
2(z1 + a)
These planes intersect at the point with coordinates x = y = z = x0 , where
x0 =
x21 + y12 + z12 − a2
.
2(x1 + y1 + z1 + a)
Consider the subspace V1 , bounded by the planes α4 , α5 and α6 . Depict the
projection of lines, which considered planes intersect, on the plane XOY (Fig. 1) and
the mutual arrangement of all obtained regions in space (Fig. 2). If the arbitrator’s
decision is in the region V1 , player I wins, and his payoff is
H1 (x1 , y1 , z1 ) = (x1 + y1 + z1 ) · µ(V1 ),
(1)
where µ(V1 ) is measure of the set V1 , which is equal to
µ(V1 ) =
Zx0
g(x)dx
−∞
+
Zx0
−∞
Zx
g(y)dy
−∞
g(x)dx
Z∞
Z∞
g(z)dz +
g(y)dy
u(x)
g(z)dz +
+
g(x)dx
x0
There
u(x) = −
g(x)dx
Z∞
Z∞
v(x)
g(y)dy
g(y)dy
x
g(x)dx
v(x)
Z
Z∞
g(z)dz.
r(x,y)
x1 + z1 + a
x2 + y12 + z12 − a2
x+ 1
,
y1
2y1
Z∞
g(z)dz+
p(x,y)
g(y)dy
−∞
x0
r(x,y)
Z∞
u(x)
Z
−∞
q(x,y)
Z∞
Zx0
Z∞
g(z)dz+
q(x,y)
(2)
256
Tatyana E. Nosalskaya
✻y
y = u(x)
(x1 , y1 )
−a
0
x
✲
x0
y = v(x)
−a
y=x
Fig. 1: Projection on the plane XOY
v(x) = −
x2 + y12 + z12 − a2
x1
x+ 1
,
y1 + z1 + a
2(y1 + z1 + a)
p(x, y) = −
y1
x2 + y12 + z12 − a2
x1 + a
x− y+ 1
,
z1
z1
2z1
q(x, y) = −
x1
y1 + a
x2 + y12 + z12 − a2
x−
y+ 1
,
z1
z1
2z1
r(x, y) = −
x1
y1
x2 + y12 + z12 − a2
x−
y+ 1
.
z1 + a
z1 + a
2(z1 + a)
From equations (1) and (2) we obtain

H(x1 , y1 , z1 ) = (x1 + y1 + z1 ) · 1 −
+
u(x)
Z
x
−
Z∞
x0
Zx0
−∞
g(y) · G(p(x, y))dy +
g(x)dx 
Z∞
u(x)


g(x)dx 
v(x)
Z
−∞

Zx
−∞
g(y) · G(q(x, y))dy+


g(y) · G(r(x, y))dy  −
g(y) · G(q(x, y))dy +
Z∞
v(x)


g(y) · G(r(x, y))dy  ,
(3)
257
Competition Form of Bargaining
✻
z
V2
0
−a
V3
−a
x
y
−a
0
0
q
V1
✴
V4
Fig. 2: Schematic representation of the regions
where G(x) is the normal distribution function. The function (3) has a maximum,
depending on a, at the point x1 = y1 = z1 = c.
Suppose now that player I selects the strategy (c, c, c), while the offers of players
II and III remain the same: (−a, 0, 0) and (0, −a, 0), respectively, and player IV
offers (0, 0, −b). We find the best response of player II to the strategies of players
I, III and IV. The space of projects splits into subspaces. Consider the boundary
subspaces of V2 :
α1 : y = x,
a
b 2 − a2
x−
,
b
2b
c+a
3c2 − a2
α4 : z = −
x−y+
.
c
2c
The abscissa of the intersection point of these three planes is
2
3c − a2
b 2 − a2
1
m=
+
·
.
2c
2b
2 + a/b + a/c
α2 : z =
In the region under consideration player II wins, and his payoff is equal


u(x)
s(x,y)
Zm
Z
Z


H2 (a) = a · µ(V2 ) = a · 
g(x)dx
g(y)dy
g(z)dz  =
−∞
x
w(x)
258


=a·
Zm
−∞
where
g(x)dx
Tatyana E. Nosalskaya

u(x)
Z
x

(G(w(x)) − G(s(x, y))) · g(y)dy  ,
(4)
a
b 2 − a2
x−
,
b
2b
c+a
3c2 − a2
x−y+
.
s(x, y) = −
c
2c
Using symmetry, we can conclude that minimum of the function (4) is reached
when a = b. The optimal parameters a and c can be found approximately by the
methods of numerical simulation
w(x) =
a = b ≈ 1.5834,
c ≈ 1.3207.
In equilibrium the players get payoffs
H1 ≈ 0.9949,
H2 = H3 = H4 ≈ 0.3952
with probabilities, respectively
µ(V1 ) ≈ 0.2511,
µ(V2 ) = µ(V3 ) = µ(V4 ) ≈ 0.2496.
2.2. Game with n+1 Players
Suppose now that n + 1 players submit their projects for the competition, which are
characterized by a set of n parameters (u1 , u2 , ..., un ). Let player I still interested in
maximizing the amount of u1 + u2 + ... + un , and the other players, starting from
the second, seek for minimize the parameter ui−1 , where i = 1, n is the number of
player.
Present the stochastic procedure with the probability distribution for the arbitrator as follows
f (u1 , u2 , ..., un ) =
n
Y
2
1
g(ui ), where g(x) = √ · e−x /2 ,
2π
i=1
We assume that this distribution is known to all participants of the competition. According to the symmetry of the model, the optimal strategies of players
have the form of n-dimensional vector (u1 , u2 , ..., un ). Suppose that for player I the
components of this vector are the same u1 = u2 = ... = un = c, and for all other
players such vector has only one non-zero component ui−1 = −a, where i = 1, n is
the number of player.
Fix these proposals of players, from the second player. Let player I submit the
project (x1 , x2 , ..., xn ), where x1 , x2 , ..., xn > 0. Then the n-dimensional space of
projects is split into n + 1 subspaces, limited by hyperplanes:
!
n
1 X 2
2
π1 : (x1 + a)u1 + x2 u2 + ... + xn un =
x −a ,
2 i=1 i
!
n
1 X 2
π2 : x1 u1 + (x2 + a)u2 + ... + xn un =
x − a2 ,
2 i=1 i
259
Competition Form of Bargaining
...
1
πn : x1 u1 + x2 u2 + ... + (xn + a)un =
2
πn+1 : u1 = u2 ,
n
X
x2i
i=1
−a
2
!
.
πn+2 : u1 = u3 ,
...
n(n − 1)
.
2
All these hyperplanes intersect at the point with coordinates u1 = u2 = ... =
un = u0 , where
n
P
x2i − a2
i=1
.
u0 = n
P
2
xi + a
πn+k : un−1 = un , where k =
i=1
Consider the subspace V1 , bounded by the n-dimensional planes π1 , π2 , ..., πn .
If the arbitrator’s decision is in the region V1 , player I wins, and his payoff is
H1 (x1 , x2 , ..., xn ) = (x1 + x2 + ... + xn ) · µ(V1 ),
(5)
where mu(V1 ) is the measure of V1 , which is equal to
µ(V1 ) =
Zu0 Zu1
Z∞
Z∞
...
−∞ −∞ α3 (u1 ,u2 )
u1
Zu0 Zu1
+
−∞ −∞
+
u1
π2 (u1 ,...,un )
α3 (u
Z 1 ,u2 )
Z∞
...
u2
Zu0 Zu1 Zu2
−∞ −∞ −∞
+
Z∞ Z∞
Z∞
Z∞
u0 l2 (u1 ) α3 (u1 ,u2 )
...
...
π3 (u1 ,...,un )
Z∞
π4 (u1 ,...,un )
g(ui ) du1 ...dun + ...+
i=1
π1 (u1 ,...,un )
Zu0 l1Z(u1 ) α2 (u
Z 1 ,u2 )
+
...
−∞
n
Y
n
Y
g(ui ) du1 ...dun + ...+
i=1
n
Y
g(ui ) du1 ...dun + ...+
i=1
n
Y
g(ui ) du1 ...dun + ...+
i=1
ωn−1 (uZ
1 ,...,un−1 )
−∞
Z∞
πn (u1 ,...,un )
n
Y
g(ui ) du1 ...dun .
(6)
i=1
There
1
π1 ∩ π2 = ω1 : (x1 + x2 + a)u1 + x3 u3 + ... + xn un =
2
...
n
X
i=1
x2i
−a
2
!
,
260
Tatyana E. Nosalskaya
π1 ∩ πi+1 = ωi : x1 u1 + x2 u2 + ... + xi−1 ui−1 + (xi + xi+1 + a)ui +
!
n
1 X 2
2
+xi+2 ui+2 + ... + xn un =
x −a ,
2 i=1 i
...
π1 ∩ πn = ωn−1
1
: x1 u1 + x2 u2 + ... + (xn−1 + xn + a)un =
2
!
...
n
X
x2i
i=1
−a
2
!
,
!
n
1 X 2
2
α1 ∩ α2 = l1 :
xi + a u1 + xn un =
x −a ,
2 i=1 i
i=1
!
!
n
n
X
1 X 2
2
α1 ∩ α3 = l2 : x1 u1 +
xi + a un =
x −a
2 i=1 i
i=2
n−1
X
The function (6) has a maximum, depending on a, at the point x1 = x2 = ... =
xn = c. Suppose now that player I chooses a strategy (c, c, ..., c), while the offers of
the players with numbers i = 2, n remain the same: component ui of the vector for
player i is −a, the remaining components are equal to zero, and the player n + 1
offers (0, ..., 0, −b). We find the best response of player II to the strategies of players
with numbers i = 1 and i = 3, n + 1. The space of projects is splits into subspaces.
Consider the boundary subspaces of V2 :
π1 : (c + a)u1 +
n
X
cui =
i=2
nc2 − a2
,
2
πn+1 : u1 = u2 ,
...
π2n−2 : u1 = un−1 ,
a
b 2 − a2
u1 −
.
b
2b
The abscissa of the intersection point of these n planes is
2
nc − a2
b 2 − a2
1
m=
+
·
.
2c
2b
a/b + a/c + n − 1
π2n−1 : un =
In the region under consideration player II wins, and his payoff is equal
H2 (a) = a · µ(V2 ) =
Zm l1Z(u1 ) α1 (u
Z 1 ,u2 )
=a·
...
−∞
u1
u2
ω1 (u1Z
,...,un−2 ) ρ(u1Z,...,un )
u2
where
φ(u1 ) =
φ(u1 )
a
b 2 − a2
u1 −
,
b
2b
n
Y
i=1
g(ui ) du1 ...dun .
(7)
261
Competition Form of Bargaining
n−1
ρ(u1 , ..., un−1 ) = −
X
c+a
nc2 − a2
u1 −
ui +
.
c
2c
i=2
Using symmetry, we can conclude that the minimum of the function (7) is
reached when a = b. Thus, for large n the optimal parameters a and c can be
approximately estimated as follows:
a=b≈
n+1
+ ε, ãäå ε > 0,
n
c≈
n+1
.
n
In equilibrium the players get payoffs
H1 ≈ 1,
H 2 = H3 = H4 ≈
1+δ
,
n
δ>0
with probabilities, respectively
µ(V1 ) ≈ µ(V2 ) = µ(V3 ) = µ(V4 ) ≈
3.
1
.
n+1
Conclusion
We present an extension of the model, proposed in (Mazalov and Tokareva, 2010),
on the three-dimensional and multi-dimensional cases. The optimal solutions is
found by the methods of numerical modeling. A similar approach has been widely
used for solving zero-sum game problems on the line. In the papers (Mazalov et al.,
2012; Kilgour, 1994) equilibriums in games involving one arbitrator are obtained,
and in the paper (Mazalov, 2010) involving arbitration committee. Under real conditions of market the experts of competition committee act as the arbitrators. They
assess the expected project for each of the parameters, and on the basis of this
assessment the probability distribution, corresponding to the opinion of experts,
is formed. Then the players submit their proposals for the competition, and the
committee can immediately reject the projects, that are dominated by the other
projects.
References
Mazalov, V. V. (2010). Mathematical Came Theory and Applications. — St. Peterburg,
448 p.(in Russian).
Mazalov, V. V., Mentcher, A. E., Tokareva, J. S. (2012) Negotiations. Mathematical Theory
— St. Petersburg — Moscow — Krasnodar, 304 p. (in Russian).
Mazalov, V. V., Tokareva, J. S. (2010). Game-Theoretic Models of Tender’s Design. Mathematical Came Theory and its Applications, Vol. 2, 2, 66–78. (in Russian).
De Berg, M., Van Kreveld, M., Overmars, M., Schwarzkopf, O. (2000). Computational
Geometry. Springer.
Kilgour, M. (1994) Game-Theoretic Properties of Final-Offer Arbitration. Group Decision
and Negotiation, 3, 285–301.
Interval Obligation Rules and Related Results
Osman Palancı1 , Sırma Zeynep Alparslan Gök1 and Gerhald Wilhelm
Weber2
Suleyman Demirel University
Faculty of Arts and Sciences, Department of Mathematics,
32260 Isparta, Turkey. E-mail: [email protected]
E-mail: [email protected]
2
Middle East Technical University
Institute of Applied Mathematics,
06531 Ankara, Turkey.
E-mail: [email protected]
1
Abstract In this study, we extend the well-known obligation rules by using
interval calculus. We introduce interval obligation rules for minimum interval
cost spanning tree (micst) situations. It turns out that the interval obligation
rule and the interval Bird rule are equal under suitable conditions. Further,
we show that such rules are interval cost monotonic and induce population
monotonic interval allocation schemes (pmias). Some examples of pmias and
interval obligation rules for micst situations are also given.
Keywords: Graphs and networks, minimum cost spanning tree situations,
interval data, obligation rules, population monotonic allocation scheme.
1.
Introduction
A connection situation arises in the presence of a group of agents, each of which
needs to be connected directly or via other agents to a source. If connections among
agents are costly, then each agent will evaluate the opportunity of cooperating
with other agents in order to reduce costs. In fact, if a group of agents decides to
cooperate, a configuration of links which minimizes the total cost of connection is
provided by a minimum cost spanning tree (mcst).
The problem of finding an mcst can be easily solved by using different algorithms
proposed in literature (for example see Graham and Hell, 1985). However, finding an
mcst does not guarantee that it is going to be really implemented: agents must still
support the cost of the mcst and then a cost allocation problem must be addressed.
This cost allocation problem was introduced by Claus and Kleitman, 1973 and has
been studied with the aid of cooperative game theory since the basic paper of Bird,
1976.
The special case of a minimization problem where no network is initially presented is old problem for Operations Research (OR). In this context, algorithms to
construct a tree connecting every village to the source with minimal total cost is
provided in Borůvka, 1926. Later, (Dijksta, 1959; Kruskal, 1956; Prim, 1957)found
similar algorithms. A historic overview of this minimization problem can be found
in Graham and Hell, 1985. Further, (Claus and Kleitman, 1973) introduced the cost
allocation problem for the special case of minimum cost spanning tree problems, in
which no network is initially presented. In the sequel Bird, 1976 treated this problem with game-theoretic methods and proposed for each minimum cost spanning
Interval Obligation Rules and Related Results
263
tree a cost allocation associated with it. Furthermore, we note that discrete or combinatorial optimization embodies a vast and significant area of combinatorics that
interfaces many related subjects. Included among these are linear programming,
OR and game theory.
Since then, many authors have noted that this kind of cost allocation problems
may arise on many different physical networks such as telephone lines, highways,
electric power systems, computer chips, water delivery systems, rail lines etc. On
the other hand, numerous studies in the literature have shown that to retrieve the
information needed to assess the exact cost of all the links of a real network is a
very hard task (Janiak and Kasperski, 2008; Montemanni, 2006; Yaman et.al., 1999;
Yaman et.al., 2001). So, we argue that it is more realistic to imagine connection
situations where the costs of links are identifiable at a level of uncertainty, i.e., only
the range of the costs is known, and no probability information on the realization of
costs is given. Such connection situations with uncertain costs may be represented
using graphs where the costs associated to the edges are intervals of real numbers.
In this context a practical example of a cost allocation problem is studied in
Moretti et.al., 2011 which is inspired by the application suggested by Yaman et.al.,
2001. In this example, a design of a telecommunication network of users that want to
be connected with a service provider is considered. Here, the agents are the users, the
source is the service provider and the cost of a link is proportional to its traffic load.
Suppose that routing delays on links are not known with certainty. This uncertainty
is caused by the time varying nature of the traffic load of the network. It is then
desirable to develop a network that hedges against all possible configurations of the
costs, that we will call scenarios, which may occur. On the other hand, the cost
of the total traffic load must be shared among users and, consequently, incentives
to cooperation should be sustainable before and after the realization of an optimal
network.
As in the classical case, where edge costs are real numbers, also in the situation
where edge costs are intervals of real numbers, a cost allocation problem arises. With
the goal to study this kind of cost allocation problems, in this paper we extend
the notion of an obligation rule by using interval calculus, and we study some
cost monotonicity properties. It turns out that cost monotonicity, under interval
uncertainty, provides a population monotonic interval allocation scheme.
We note that Suijs, 2003 studied mcst problems in which the connection costs
are represented by random variables. In our paper, costs are not random variables,
but instead, they are closed and bounded intervals of real numbers.
We start with some preliminaries in the next section. In Section 3, interval
obligation rules are introduced; in the same section, the relation between the interval
obligation rules and the interval Bird rule are given. In Section 4, it is shown that
interval obligation rules are interval cost monotonic and induce pmias. A summary
on our work are given in Section 5.
2.
Preliminaries
In this section we give some terminology on graph theory, interval calculus and some
basic definitions and useful results from the theory of cooperative interval games
(Alparslan Gök, 2009; Alparslan Gök, 2010; Alparslan Gök et. al., 2009a; Alparslan
Gök et.al., 2011; Alparslan Gök et.al., 2009b; Diestel, 2000; Moretti et.al., 2011;
Tijs, 2003)
264
Osman Palancı, Sırma Zeynep Alparslan Gök, Gerhald Wilhelm Weber
An (undirected) graph is a pair < V, E >, where V is a set of vertices or nodes
and E is a set of edges e of the form {i, j} with i, j ∈ V , i 6= j. The complete graph on
a set V of vertices is the graph < V, EV >, where EV = {{i, j}|i, j ∈ V and i 6= j}. A
path between i and j in a graph < V, E > is a sequence of nodes i = i0 , i1 , . . . , ik = j,
k ≥ 1, such that all the edges {is , is+1 } ∈ E, for s ∈ {0, . . . , k − 1}, are distinct.
A cycle in < V, E > is a path from i to i for some i ∈ V . Two nodes i, j ∈ V are
connected in < V, E > if i = j or if there exists a path between i and j in < V, E >.
A connected component of V in a graph < V, E > is a maximal subset of V with
the property that any two nodes in this subset are connected in < V, E >.
A minimum interval cost spanning tree (micst ) situation is a situation where
N = {1, 2, . . . , n} is a set of agents who are willing to be connected as cheaply
as possible to a source (i.e., a supplier of a service) denoted by 0, based on an
interval-valued weight (or cost) function.
For each S ⊆ N , we also use the notation S0 = S ∪ {0}, and the notation W
for the interval weight function, i.e., a map which assigns to each edge e ∈ EN0
a closed interval W (e) ∈ I(R+ ). The interval cost W (e) of each edge e ∈ EN0
(N0 = N ∪ {0}) will be denoted by [W (e), W (e)]. No probability distribution is
assumed for edge costs. We denote an micst situation with set of users N , source 0,
and interval weight function W by < N0 , W > (or simply W ). Further, we denote
by IW N0 the set of all micst situations < N0 , W > (or W ) with node set N0 .
N0
is W (Γ ) =
P The cost of a network Γ ⊆ EN0 in an micst situation W ∈ IW
e∈Γ W (e). A network Γ is a spanning network on S0 = S ∪ {0}, with S ⊆ N , if
for every e ∈ Γ we have e ∈ ES0 and for every i ∈ S there is a path in < S0 , Γ >
from i to the source. For any micst situation W ∈ IW N0 it is possible to determine
at least one spanning tree on N0 , i.e., a spanning network without cycles on N0 ,
of minimum interval cost (such a network is also called an micst on N0 in W or,
shorter, an micst for W ). Note that the number of edges which form a spanning
tree on N0 is n. In the following, we will denote by TN0 the set of all spanning trees
N0
for N0 and by MW
.
N0 ⊆ TN0 the set of all micst for N0 in W , for each W ∈ IW
Let I (R) be the set of all closed intervals in R. A cooperative interval cost
game is an ordered pair < N, b
c >, where N = {1, 2, ..., n} is the set of players and
b
c : 2N → I (R) is the characteristic function with b
c (∅) = [0, 0] , which assigns to each
coalition S ∈ 2N a closed and bounded interval [b
c (S) , b
c (S)]. A classical cooperative
game < N, c > can be identified with < N, b
c >, where b
c (S) = [c (S) , c (S)] for each
S ∈ 2N . The family of all interval games with player set N is denote by IGN .
Instead of b
c ({i}) , b
c ({i, j}) , etc.,
c (i) , b
c (i, j), etc..
we often write b
Let I, J ∈ I (R) with I = I, I , J = [J , J], |I| = I − I and α ∈ R+ . Then,
I + J = [I + J, I + J] and αI = αI, αI .
In this paper we also need
a partial substraction operator. We define I − J, only
if |I| ≥ |J| , by I − J = I − J, I − J . We recall that I is weakly better than J,
which we denote by I < J, if and only if I ≥ J and I ≥ J. We also use the reverse
notation I 4 J, if and only if I ≤ J and I ≤ J. We say that I is better than J,
which we denote by I ≻ J, if and only if I < J and I 6= J.
Further, we use the notation I (R+ ) for the set of all closed nonnegative intervals
in R.
In this paper, n−tuples of intervals I = (I1 , ..., In ) where Ii ∈ I (R) for each
N
i ∈ N, will play a key role. For further use we denote by I (R) the set of all
n−dimensional vectors whose components are elements in I (R) . Let Ii = I i , I i
265
Interval Obligation Rules and Related Results
be the interval payoff of player i, and let I = (I1 , ..., In ) be an interval payoff vector.
Then, according to
1985, we have
PMoore,P
nolimitsi∈S Ii = [ i∈S I i , i∈S I i ] ∈ I (R) for each S ∈ 2N \ {∅} .
The interval core C (N, b
c) of the interval cost game b
c is defined by
(
)
X
X
N
C (N, b
c) := (I1 , ..., In ) ∈ I (R) |
Ii = b
c (N ) ,
Ii 4 b
c (S) , ∀S ∈ 2N \ {∅} .
i∈N
i∈S
The interval core consists of those interval payoff vectors which assure the distribution of the uncertain worth of the grand coalition such that each coalition of
players can expect a weakly better interval payoff than what that group can expect
any incentives to split off. Here,
P on its own, implying that no coalition hasP
I
=
b
c
(N
)
is
the
efficiency
condition
and
c (S) , S ∈ 2N \ {∅} , are
i
i∈N
i∈S Ii 4 b
the stability conditions of the interval payoff vectors.
Given an element a = (a1 , . . . , an ) ∈ (EN0 )n , we denote by a|j the restriction of
a to the first j components, that is a|j = (a1 , . . . , aj ) for each j ∈ N . Further, for
each j ∈ N , we denote by Π(a|j ) the partition of N0 defined as
Π(a|j ) = {T ⊆ N0 |T is a connected component in < N0 , {a1 , . . . , aj } >}.
In the following, we will use the notation Π(a|0 ) to denote the singleton partition
of N0 .
For each Γ ∈ TN0 and each W ∈ IW N0 , we denote by AΓ,W ⊆ (EN0 )n the set
of vectors a = (a1 , . . . , an ) of n distinct edges in Γ such that W (a1 ) 4 ... 4 W (an ),
Note that W (ai ) is monotonically increasing with respect to ” 4 ” :
AΓ,W = {a ∈ (Γ )n |W (a1 ) 4 . . . 4 W (an ), aj 6= ak for all j, k ∈ N }.
An micst game < N, b
cW > (or simply b
cW ) corresponding to an micst situation
W ∈ IW N0 is defined by
b
cW (T ) := min{W (Γ )|Γ is a spanning network on T0 }
for every T ∈ 2N \{∅}, with the convention that b
cW (∅) = [0, 0]. Also, an interval
solution is a map F : IW N0 → I(R)N assigning to every micst situation W ∈ IW N0
a unique allocation in I(R)N .
Finally, we give the notion of population monotonic interval allocation scheme
(pmias) for the game < N, b
c >. We say that for a cost game b
c, a scheme A =
N
(AiS )i∈S,S∈2N \{∅} with AiS ∈ I (R) is a pmias of b
c if
P
i) i∈S AiS = b
c (S) for all S ∈ 2N \ {∅} , and
ii) AiS < AiT for all S, T ∈ 2N and i ∈ N with i ∈ S ⊂ T.
3.
Interval obligation rules
P
N
Consider ∆(N ) to be an usual simplex on N , defined by ∆(N ) = {x ∈ R
P+ | i∈N xi =
1}. The sub-simplex ∆(S) of ∆(N ) given by ∆(S) = {x ∈ ∆(N )| i∈S xi = 1}
is called the set of obligation vectors of S. An obligation function is a map O :
2N \ {∅} → ∆(N ) assigning to each S ∈ 2N \ {∅} an obligation vector
O(S) ∈ ∆(S)
266
Osman Palancı, Sırma Zeynep Alparslan Gök, Gerhald Wilhelm Weber
in such a way that for each S, T ∈ 2N \ {∅} with S ⊂ T and for each i ∈ S it holds
Oi (S) ≥ Oi (T ).
Such an obligation function O on 2N \ {∅} induces an obligation map
Ô : Θ(N0 ) → RN such that
Ôi (θ) :=
X
Oi (S),
S∈θ,0∈S
/
for each i ∈ N and each θ ∈ Θ(N0 ); here, Θ(N0 ) is the family of partitions of N0 .
Note that if θ = {N0 }, then the resulting empty sum is assumed, by definition,
to be the n-vector of zeroes: Ô(θ) = 0 ∈ RN (for details see Tijs et.al., 2006). Obligation maps are basic ingredients for interval obligation rules. Now, we introduce
the notion of the interval obligation rule.
Definition 1. Let Ô be an obligation map on Θ(N0 ). The interval obligation rule
φÔ : IW N0 → I(R)N is defined by
φÔ (W ) :=
n
X
j=1
W (aj )(Ô(Π(a|j−1 )) − Ô(Π(a|j )))
Γ,W
, and where
for each micst situation W ∈ IW N0 , each Γ ∈ MW
N0 and a ∈ A
Π(a|j−1 )) and Π(a|j ), for each j = 1, . . . , n, are partitions of the set N0 .
Example 1. We consider an micst situation < N0 , W > with three agents denoted
by 1, 2, and 3 and the source 0. As depicted in Figure 1, to each edge e ∈ E{0,1,2,3}
is assigned a closed interval W (e) ∈ I(R+ ) representing the uncertain cost of edge
e. For instance, W (0, 1) = [20, 24], W (2, 3) = [10, 13] , etc.. Now we compute the
interval obligation rule In this micst situation, W, Γ = {(0, 1) , (1, 2) , (2, 3)} ∈ MW
N0
and
a = (a1 , a2 , a3 ) = ((2, 3) , (1, 2) , (0, 1)) ∈ AΓ,W .
Then,
2
[15, 20] +
3
1
φÔ
[10, 13] +
2 (W ) =
2
1
φÔ
[10, 13] +
3 (W ) =
2
φÔ
1 (W ) =
Hence,
φÔ (W ) =
1
50 64
[20, 24] =
,
,
3
3 3
1
1
85 107
[15, 20] + [20, 24] =
,
, and
6
3
6 6
1
1
85 107
[15, 20] + [20, 24] =
,
.
6
3
6 6
50 64
85 107
85 107
,
,
,
,
,
.
3 3
6 6
6 6
267
Interval Obligation Rules and Related Results
Fig. 1: An micst situation < N0 , W > .
Remark 1. It is obvious that if the cost of the edge connecting to source is the
cheapest cost, then the interval obligation rule equals the interval Bird rule which
is defined by Alparslan Gök et.al., 2014.
Example 2. Figure 2 corresponding to micst situation < N0 , W ′ >, the interval Bird
allocation is
IB(N, {0} , A, W ′ ) = ([10, 13] , [15, 20] , [20, 24]) (see Alparslan Gök et.al., 2014).
′
In this micst situation W ′ , Γ = {(0, 1) , (1, 2) , (2, 3)} ∈ MW
N0 and
′
a = (a1 , a2 , a3 ) = ((0, 1) , (1, 2) , (2, 3)) ∈ AΓ,W .
Then,
′
φÔ
1 (W ) = (1 − 0) [10, 13] + (0 − 0) [15, 20] + (0 − 0) [20, 24] = [10, 13] ,
′
φÔ
2 (W ) = (0 − 0) [10, 13] + (1 − 0) [15, 20] + (0 − 0) [20, 24] = [15, 20] ,
′
φÔ
3 (W ) = (0 − 0) [10, 13] + (0 − 0) [15, 20] + (1 − 0) [20, 24] = [20, 24].
It is clear that
φÔ (W ′ ) = IB(N, {0} , A, W ′ ) = ([10, 13] , [15, 20] , [20, 24]) .
Fig. 2: An micst situation < N0 , W ′ > .
268
4.
Osman Palancı, Sırma Zeynep Alparslan Gök, Gerhald Wilhelm Weber
Interval cost monotonicity and PMIAS
In this section, we discuss some interesting interval monotonicity properties for the
interval obligation rules. First, we provide the definition of interval cost monotonic
solutions for micst situations.
Definition 2. An interval solution F is an interval cost monotonic solution if for
all micst situations W, W ′ ∈ IW N0 such that W (e) 4 W ′ (e) for each e ∈ EN0 it
holds that Fi (W ) 4 Fi (W ′ ) for each i ∈ N .
We prove in Theorem 7 that interval obligation rules are interval cost monotonic;
the main step is the following lemma whose proof is straightforward.
Lemma 1. Let Ô be an obligation map on Θ(N0 ) and let W ∈ IW N0 . Let ē ∈ EN0
and let h ≻ W (ē) be such that there is no e ∈ EN0 with W (ē) ≺ W (e) ≺ h.
Define W̃ ∈ IW N0 by W̃ (e) := W̃ (e) if e ∈ EN0 \ {ē} , and W̃ (ē) = h. Then,
φÔ (W̃ ) < φÔ (W ) .
The proofs of the following theorems are straightforward (see Tijs et.al., 2006).
Theorem 1. Interval obligation rules are interval cost monotonic.
Theorem 2. Let Ô be an obligation map on Θ(N0 ) and let φÔ the interval obligation rule with respect to Ô, and W ∈ IW N0 . Then the table [φÔS (W|S0 )]S∈2N \{∅} is
a pmias for the micst game < N, b
cW > .
From Theorem 8 and the definition of a pmias, it follows that interval obligation
rules provide interval cost allocations which are interval core elements of the game
< N, b
cW > .
Now, we give an example of interval cost monotonicity and pmias.
Example 3. Consider again the micst situation < N0 , W > as depicted in Figure 1.
Then, as the interval obligation rule φÔ (W ) previously introduced, applied to each
micst situation < S0 , W|S0 >, provides the following population monotonic interval
allocation scheme:

1
2
3
S



4
2
1
5
1
5

(123)
[16
,
21
]
[14
,
17
]
[14


6
6
6
6
6 , 17 6 ]

1
1

(12) [17 2 , 22] [17 2 , 22]
∗



(13) [20, 24]
∗
[25, 28]
ÔS
.
[φ (W|S0 )]S∈2N \{∅} =
(23)
∗
[17, 21]
[17, 21]




(1)
[20, 24]
∗
∗




(2)
∗
[24,
29]
∗



(3)
∗
∗
[30, 33]
5.
Conclusions and outlook
This paper considers the class of interval obligation rules and studies their interval
cost monotonicity properties. The interval obligation rules are interval cost monotonic and induce a pmias. The most important result of this study is, as already
stated in Remark 3, if the cost of the edges connecting to source is the cheapest
Interval Obligation Rules and Related Results
269
cost, then the interval obligation rule equals the interval Bird rule which is defined
by Alparslan Gök et.al., 2014.
Before closing we note that the obtained results can be extended to Network
Steiner problem by using Kirzhner et.al., 2012. The Steiner tree problem is superficially similar to the minimum spanning tree problem. The difference between the
Steiner tree problem and the minimum spanning tree problem is that, in the Steiner
tree problem, extra intermediate vertices and edges may be added to the graph in
order to reduce the length of the spanning tree.
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Stable Cooperation in Graph-Restricted Games⋆
Elena Parilina and Artem Sedakov
Saint Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
Universitetsky prospekt 35, Saint Petersburg, 198504, Russia
E-mail: [email protected], [email protected]
Abstract In the paper we study stable coalition structures in the games
with restrictions on players’ cooperation and communication. Restriction
on cooperation among players is given by a coalition structure, whereas
restriction on their communication is described by a graph. Having both a
coalition structure and a graph fixed, a payoff distribution can be calculated
based on worth of each coalition of players. We use the concept of stability
for a coalition structure similar to Nash stability, assuming that the graph
structure is fixed. The results are illustrated with examples.
Keywords: cooperation, coalition structure, graph, characteristic function,
stability, Shapley value, Myerson value, ES-value.
1.
Introduction
In cooperative games with coalition structure it is supposed that players belonging to one coalition can communicate with each other. However, some interactions
among players in one coalition may be not possible due to several reasons. Such
restrictions lead us to consider games with coalition structure and constrained communication. In the paper, we associate constrained communication with an undirected graph showing the structure of players’ communication. In addition, not all
coalition structures are appropriate for all players. For this purpose we try to find
“proper” or “stable” in some sense coalition structures from the set of all coalition
structures. When we talk about stability of the coalition structure, we mean stability in the sense of players payoffs, prescribed by a single-valued solution—a rule
that assigns a single payoff distribution to each game. Specifically, we consider a rule
based on the Shapley value (Shapley, 1953) as a solution in such a game. Moreover,
the stable solution should satisfy the property of individual rationality.
In (Aumann and Drèze, 1974) games with a coalition structure were considered,
and the solution called the Aumann–Drèze value, based on the Shapley value, was
proposed. On the contrary, in graph-restricted games proposed in (Myerson, 1977),
the Myerson value was introduced as a solution, and it was also based on the
Shapley value. A solution, based on a combination of both types of restriction—
coalition structure and a graph, was introduced in (Vázquez-Brage et al., 1996).
Such a solution is a generalization of Owen value (Owen, 1977) and the Myerson
value. Other solutions, e.g. (Khmelnitskaya, 2010), were developed later.
In (Hart and Kurz, 1983) the concept of stability for a coalition structure in
a strategic setting was introduced. Other stability concepts were studied later in
(Haeringer G., 2001), (Tutic, 2010). In the present study we use a concept similar
⋆
The authors acknowledge Russian Foundation for Basic Research for grant 14-01-31141mol_a and Saint Petersburg State University for a research grant 9.38.245.2014.
272
Elena Parilina, Artem Sedakov
to Nash stability, proposed in (Bogomolnaia and Jackson, 2002), supposing that
players’ payoffs are defined based on some cooperative solution.
Characteristic function estimating the worth of any coalition, plays the key role
in the game. Note that in our discussion we do not claim the superadditivity of this
function. If characteristic function is superadditive in the game with restrictions
on players’ cooperation, it is obvious that the grand coalition is always stable, but
other coalition structures may be stable as well. However, in non-superadditive
case, we cannot a priori determine stable coalition structures without analyzing
properties of the characteristic function. It is worth mentioning that existence of
the stable coalition structure for an arbitrary characteristic function was proved in
(Sedakov et al., 2013) for at most three-person case, and it was shown that for more
than three players we cannot guarantee the existence of such structures. Stability
of the coalition structure was considered for both the Shapley value and the ESvalue1 (or the CIS-value2 ) (Driessen and Funaki, 1991). In the paper we find stable
coalition structures, provided that communication among players is restricted by
a fixed graph. Unlike the idea in (Caulier et al., 2013), where the stable network
is determined, provided that coalition structure is given, we use the opposite idea.
Specifically, we try to determine stable coalition stricture, assuming that players’
communication is restricted by an a priori given graph structure.
The paper is organized as follows. In Section 2 we introduce the game with coalition structure and restricted cooperation by a undirected graph. Then, in Section 3
we define the stable coalition structure, whereas in Section 4 we find stable coalition
structures in a game with the major player.
2.
Cooperative game with coalition structure restricted by graph
Consider the class of cooperative games with coalition structure determined as follows:
Definition 1. Cooperative game with coalition structure is a system Γ = (N, v, π),
where N = {1, . . . , n} is the set of players, v : 2N → R is a characteristic function
with v(∅) = 0 and π is a coalition structure π = {B1 , . . . , Bm }, i.e. B1 ∪ . . . ∪ Bm =
N , Bi ∩ Bj = ∅ for all i, j ∈ N, i 6= j.
Suppose that communication among players can be restricted by a graph g consisting of finite set of nodes which is the set of players N and the set of links. If players
i and j are linked, then {ij} ∈ g. Denote the complete graph by g N .
Given characteristic function v(S) and graph g which restricts communication
among players, determine “new” characteristic function using the following approach
(Myerson, 1977):
X
v(T ),
(1)
v g (S) =
T ∈S/g
where S/g is a unique partition of S s.t. S/g = {{i | i and j are connected in S by
g}|j ∈ S}. Construction of characteristic function v g using this approach has a useful
property: if characteristic function v is superadditive, then v g is also superadditive.
Denote a cooperative game with coalition structure π restricted by graph g as
Γ g = (N, v g , π).
1
2
The equal surplus division value.
Center of gravity of the imputation set.
273
Stable Cooperation in Graph-Restricted Games
Example 1. (Coauthor model) The model represents the work of researchers who
write papers. The link in the graph means players work on a joint paper. Coalition
structure can be interpreted as a partition of the players among research institutes.
In Fig. 1 we can see an example of coalition structure π = {{1, 2}, {3, 4, 5}} with a
given graph g = {12, 23, 34, 25} with the set of players N = {1, 2, 3, 4, 5}.
B1
B2
3
1
2
5
4
Fig. 1: Coalition structure with a given graph for Example 1.
Definition 2. An n-dimensional profile x = (x1 , . . . , xn ) ∈ Rn is a payoff distribution in game Γ g if it is efficient, i. e. for all Bj ∈ π:
X
xi = v g (Bj ).
i∈Bj
Definition 3. Payoff distribution x is an allocation in game Γ g with coalition
structure π if for all i ∈ N it is individually rational:
xi ≥ v g ({i}).
A solution (or a cooperative solution) is the rule which prescribes a subset of the
n-dimensional space for each game Γ g . If the prescribed subset consists of one point,
the solution is called the single-valued. We consider the Myerson value and the ESvalue as single-valued cooperative solutions. Let B(i) ∈ π be the coalition containing
player i ∈ N . The Myerson value is the payoff distribution µ = (µ1 , . . . , µn ) which
can be calculated by formula:
µi =
X
S⊆B(i),i∈S
(|B(i)| − |S|)!(|S| − 1)! g
[v (S) − v g (S \ {i})]
|B(i)|!
for all i ∈ N . The ES-value ψ = (ψ1 , . . . , ψn ) is defined as:
P g
v g (B(i)) −
v ({j})
ψi = v g ({i}) +
3.
j∈B(i)
|B(i)|
.
(2)
(3)
Stable coalition structure
Consider the problem of choosing stable coalition structure π, i.e. the structure
which is stable against the individual player’s deviations when other players do not
deviate from structure π.
274
Elena Parilina, Artem Sedakov
Definition of stable coalition structure with respect to a solution for cooperative
game Γ was proposed in (Parilina and Sedakov, 2012), and can be reformulated for
the game Γ g restricted by graph g in the following way:
Definition 4. Coalition structure π = {B1 , . . . , Bm } is stable with respect to a
solution if for each i ∈ N the following inequality holds:
xi ≥ x′i ,
where x, x′ are payoff distributions calculated according to the chosen solution for
games (N, v g , π), (N, v g , π ′ ) respectively, π ′ = {B(i) \ {i}, Bj ∪ {i}, π−B(i)∪Bj } for
any Bj ∈ π ∪ ∅, Bj 6= B(i), π−Bi = π \ Bi ⊂ π, i.e. B(i) ∈ π is the coalition which
contains i.
Contrary to the concept of network stability in the games with fixed coalition
structure (Caulier et al., 2013), we propose the model in which graph structure is
fixed and all possible coalition structures can be realized. The problem of stability
of a coalition structure arises. The main distinction in these two approaches is in
what should be chosen as an unchangeable property of a game, graph vs. coalition
structure.
If graph g is complete, i.e. g = g N , the following proposition directly follows
from (Sedakov et al., 2013).
Proposition 1. If graph g is complete and |N | 6 3, there always exists a stable
coalition structure with respect to the Myerson value and ES-value.
Unfortunately, stable coalition structure with respect to the ES-value may not exist
in a case of more than 3 players (Sedakov et al., 2013).
Now consider a general case of graph g, and superadditive characteristic function v.
Proposition 2. If characteristic function v g is superadditive, then coalition structure {N } is stable relative to the Myerson value and the ES-value.
Proof. Following Definition 4 coalition structure {N } turns into structure π ′ =
{{N \ i}, {i}} if player i ∈ N deviates. The ith component of the Myerson value
and the ES-value in coalition structure π ′ is equal to v g ({i}) which is not larger than
the ith component of the corresponding payoff distributions in coalition structure
π because of superadditivity of characteristic function v g .
⊔
⊓
Proposition 3. If characteristic function v g is superadditive and graph g is split
up into a finite number of connected components, i.e. g = g1 ∪ g2 ∪ . . . ∪ gk where
gi ∩ gj = ∅ for any i 6= j, then coalition structure {B1 , . . . , Bk }, where coalition
Bl contains all the players connected by component gl and no players else, is stable
with respect to the Myerson value and the ES-value.
Proof. Consider coalition structure π = {B1 , . . . , Bk } described in Proposition.
Prove that there are no players who can benefit deviating from structure π. If
player i ∈ B(i) ∈ π deviates, his deviation may lead to the coalition structure in
which he plays as an individual player, and his component of the Myerson value or
the ES-value will be equal to v g ({i}) which is not larger than his component calculated for coalition structure π because of superadditivity of characteristic function.
Stable Cooperation in Graph-Restricted Games
275
Player i may also deviate joining some coalition Bj ∈ π which is different from
coalition B(i). But his component of the Myerson value or the ES-value will be
v g ({i}) because he does not have any connections with players from Bj which follows from graph g definition. Therefore, coalition structure π is stable with respect
the Myerson or the ES-values.
⊔
⊓
4.
Game with the major player
Let the set of players be N = {1, . . . , n}, |N | = n > 2. Suppose there is the major
player referred as player 1, and all other players, i.e. players from set N \ {1}, are
supposed to be symmetric. It means that for any coalition S ⊆ N and any i, j ∈
N \{1}, i 6= j the following condition is satisfied: v(S)−v(S\{i}) = v(S)−v(S\{j}).
The worth of coalition S, v(S), is interpreted as its work efficiency and defined as


0,
if S = {i}, i ∈ N \ {1},


if s > 1, 1 ∈
/ S,
v(S) = γs,
(4)

β

α(s − 1) + , if s > 1, 1 ∈ S,
s
where s = |S| is a number of players in coalition S, and α, β, γ are positive parameters satisfying the conditions γ 6 α 6 β. The value of characteristic function
(4) for singleton S = {i}, i ∈ N \ {1} represents that this particular player gains
nothing working alone. Parameter β is the work efficiency of the major player. If s
players from the set N \ {1} cooperate, they could gain proportionally to the size of
coalition with coefficient γ, i.e. γs. Here parameter γ is an efficiency of one player if
he cooperates with players from N \ {1}. If the major player belongs to coalition S,
the efficiency of coalition S is directly proportional to the number of players from
N \ {1} in coalition S, and inversely proportional to the number of players in S
(since the major player works himself as well as controls the others).
Suppose that communication among players is restricted by graph g which is a
“star”: the major player can communicate with all other players, whereas any player
from N \ {1} communicates only with the major player (see Fig. 2).
Fig. 2: Graph g restricting communication in a game with the major player.
The value of characteristic function v g restricted by graph g becomes simpler
than v, and using (1) and (4) can be written as follows:

0,
if 1 ∈
/S
v g (S) =
(5)
β
α(s − 1) + , if 1 ∈ S.
s
276
Elena Parilina, Artem Sedakov
Proposition 4. If α ∈ [β/2, β], characteristic function (5) is superadditive.
Proof. Prove that for any disjoint coalitions S, T ⊂ N : v g (S ∪ T ) > v g (S) + v g (T ).
If both coalitions S and T do not contain the major player, this inequality holds.
Let coalition S contain the major player, i.e. 1 ∈ S and |S| = s, |T | = t. The
superadditivity condition can be rewritten into the system:

β
β

α(s − 1) + 6 α(s + t − 1) +
,
for any S, T : s > 2, t > 1, s + t 6 n,
s
s+t
β

β 6 αt +
,
for any T : 1 6 t 6 n − 1,
t+1
The system of inequalities above has a solution if
1
β
s
1
1
= .
α > β · max
,
= β · max
,
t
s>2,t>1, 1 + t s + t
s>2,t>1, t + 1 1 +
2
s
s+t6n
s+t6n
Taking into account restriction α 6 β, we obtain β/2 6 α 6 β.
⊔
⊓
Next proposition gives the explicit formulas for the Myerson value and the ESvalue.
Proposition 5. In the game with the major player determined by characteristic
function (5), the components of the Myerson value µ = (µ1 , . . . , µn ) have the form:

(n − 1)α Hn β


+
, if i = 1
n
(6)
µi = α 2(H − 1)β
n

 −
,
if i 6= 1,
2
n(n − 1)
Pn
where Hn = k=1 1/k.
The ES-value is

2

 (n − 1)α + (n − n + 1)β , if i = 1
n
n2
(7)
ψi =
(n
−
1)α
(n
−
1)β


−
,
if
i
=
6
1.
n
n2
Moreover, µi , i ∈ N \ {1} and ψi , i ∈ N are increasing functions of n.
Proof. First, calculate the Myerson value for the game with characteristic function
(5). The component µi of player i ∈ N \ {1} is:
n
X
(n − s)!(s − 1)! s−2
β
α (Hn − 1)β
µi =
Cn−1 α −
= −
.
n!
s(s − 1)
2
n(n − 1)
s=2
The component of the major player is
n
X
(n − s)!(s − 1)! s−1
β
(n − 1)α Hn β
µ1 =
Cn−1 α(s − 1) −
=
+
.
n!
s
2
n
s=2
Second, calculate the ES-value for this game. Following formula (3), we obtain:
ψi =
α(n − 1) + β/n − β
(n − 1)α (n − 1)β
=
−
,
n
n
n2
i ∈ N \ {1}
Stable Cooperation in Graph-Restricted Games
277
and for the major player:
ψ1 = β +
α(n − 1) + β/n − β
(n − 1)α (n2 − n + 1)β
=
+
.
n
n
n2
Now prove that µi , i ∈ N , i 6= 1 is an increasing function of n. For this purpose,
find the difference between components µi (n + 1) and µi (n) calculated for games
with n + 1 and n players respectively. We can easily show that for all n > 2 this
difference is positive:
µi (n + 1) − µi (n) = −
(2(Hn − 1) − n−1
(Hn+1 − 1)β
(Hn − 1)β
n+1 )β
+
=
>0
n(n + 1)
n(n − 1)
n(n + 1)(n − 1)
because Hn − 1 > 1/2 and (n − 1)/(n + 1) < 1. It proves that µi , i ∈ N \ {1} is an
increasing function on n.
Similarly, consider the difference between major player’s components of the ESvalue in games with n + 1 and n players respectively:
α
1
2n + 1
ψ1 (n + 1) − ψ1 (n) =
+β
+
> 0,
n(n + 1)
n(n + 1) n2 (n + 1)2
and the difference between components of the ES-value of any other player:
ψi (n + 1) − ψi (n) =
n2 − n − 1
α
+β 2
> 0.
n(n + 1)
n (n + 1)2
⊔
⊓
Examine the problem of stability of coalition structures with respect to the
Myerson value and ES-value. First, consider the case of the Myerson value, second,
the case of the ES-value.
2β
Proposition 6. If n−1
(1 − Hn /n) 6 α 6 β, coalition structure {N } is stable with
respect to the Myerson value. If α 6 β/2, coalition structure {{1}, B2, . . . , Bm } is
stable with respect to the Myerson value. No other stable coalition structures are
possible.
Proof. Coalition structure {N } is stable when the following system has a solution:

(n − 1)α Hn β


+
> β,
2
n
(8)
(H
−
1)β
α
n

 −
> 0.
2
n(n − 1)
The first (second) inequality corresponds to the condition that the major (any other)
player does not benefit deviating from coalition {N } and playing individually. The
system (8) is equivalent to the condition:
2β
Hn − 1
Hn
2β
Hn
α>
max
;1 −
=
1−
,
n−1
n
n
n−1
n
and given restriction α 6 β, we obtain the result of the proposition. It is easy to
2β
(1 − Hn /n) tends to zero as n tends to infinity.
show that the lower bound n−1
278
Elena Parilina, Artem Sedakov
Consider coalition structure {{1}, B2 , . . . , Bm }, where |B2 | 6 |B3 | 6 . . . 6 |Bm |.
This coalition structure is stable when the following system has a solution:

b α Hbj +1 β

β > j +
, j = 2, . . . , m
2
bj + 1
(9)

0 > α − (H2 − 1)β ,
2
2
where bj = |Bj |. The first inequality can be rewritten into the following:
Hbj +1
2β
α 6 min
1−
.
j=2,...,m bj
bj + 1
(10)
Since the right part of (10) is the increasing function of bj and it exceeds β/2, the
solution of (9) is
Hbj +1
β
2β
β 2β
Hb2 +1
β
, min
1−
= min
,
1−
= .
α 6 min
2 j=2,...,m bj
bj + 1
2 b2
b2 + 1
2
Consider coalition structure {B1 , B2 , . . . , Bm }, where |B1 | > 2 and |B2 | 6 |B3 | 6
. . . 6 |Bm |. Following Definition 4, write the system of constrains guaranteeing
stability of this coalition structure:

α(b1 − 1) βHb1


+
> β,


2
b1



βHbj +1
α(b1 − 1) βHb1
αbj


+
> max
+
,

j=2,...,m
2
b1
2
bj + 1
(11)
α β(Hb1 +1 − 1)


0
>
−
,


2
b1 (b1 + 1)




α β(Hb1 − 1)

 −
> 0.
2
b1 (b1 − 1)
From the last two inequalities we obtain condition:
06
α β(Hb1 − 1)
α β(Hb1 +1 − 1)
−
6 −
6 0,
2
b1 (b1 − 1)
2
b1 (b1 + 1)
which never holds. Therefore, the system (11) does not have solution. It proves that
2β
the only possible stable coalition structures are {N } when n−1
(1 − Hn /n) 6 α 6 β
and {{1}, B2, . . . , Bm }, where |B2 | 6 |B3 | 6 . . . 6 |Bm | if α 6 β/2.
⊔
⊓
Proposition 7. If β/n 6 α 6 β, coalition structure {N } is stable with respect
to the ES-value. If α 6 β/(bm + 1), coalition structure {{1}, B2 , . . . , Bm }, |B2 | 6
|B3 | 6 . . . 6 |Bm | = bm , is stable with respect to the ES-value. No other stable
coalition structures are possible.
Proof. Coalition structure {N } is stable when neither the major player nor any other
player benefits deviating and, therefore, becoming a singleton, i.e. the following
system has a solution

2

 α(n − 1) + β(n − n + 1) > β,
n
n2
(12)

 α(n − 1) − β(n − 1) > 0,
n
n2
279
Stable Cooperation in Graph-Restricted Games
if α > β/n. Taking into account the restriction α 6 β, we obtain β/n 6 α 6 β.
Consider coalition stricture {{1}, B2, . . . , Bm }, |B2 | 6 . . . 6 |Bm |, |Bj | = bj ,
j = 2, . . . , m. It is stable if neither the major player benefits joining any coalition
from the set {B2 , . . . , Bm }, nor any other player i, i 6= 1 benefits joining player 1:

αbj
β((bj + 1)2 − (bj + 1) + 1)

β > max
+
,
j=2,...,m bj + 1
(bj + 1)2

0 > α − β .
2
22
(13)
Using Proposition 5, i.e. that function ψ1 is an increasing function of the number
of coalition members, we rewrite system (13) as:

αbm
β((bm + 1)2 − (bm + 1) + 1)

β >
+
,
bm + 1
(bm + 1)2

α 6 β .
2
(14)
and it is equivalent to inequality
α 6 min
β
β
;
2 bm + 1
=
β
.
bm + 1
It proves the first part of the proposition.
Consider coalition structure {B1 , B2 , . . . , Bm }, where 1 ∈ B1 , |B1 | > 2 and
|B2 | 6 |B3 | 6 . . . 6 |Bm |. Following Definition 4, this structure is stable if

α(b1 − 1) β(b21 − b1 + 1)



+
> β,


b1
b21


2

αbj
β((bj + 1)2 − (bj + 1) + 1)
α(b1 − 1) β(b1 − b1 + 1)



+
> max
+
,
j=2,...,m bj + 1
b1
b21
(bj + 1)2
αb1
βb1


0>
−
,


2

b
+
1
(b
1
1 + 1)


 α(b1 − 1) β(b1 − 1)


−
> 0.

b1
b21
(15)
From the last two inequalities we obtain condition:
06
α(b1 − 1) β(b1 − 1)
αb1
βb1
−
6
−
6 0,
b1
b21
b1 + 1 (b1 + 1)2
which never holds. Therefore, the system (15) does not have solution, and any
coalition structure {B1 , B2 , . . . , Bm } s.t. 1 ∈ B1 , |B1 | > 2 and |B2 | 6 |B3 | 6 . . . 6
|Bm |, is always unstable.
⊔
⊓
In Figures 3 and 4 diagrams of all stable coalition structures with respect to both
the Myerson value and the ES-value in the game with the major are shown. Here
we see that in the game only two types of stable coalition structures are possible:
the grand coalition and any coalition structure in which the major player is the
singleton.
280
Elena Parilina, Artem Sedakov
Fig. 3: Diagram of stable coalition structures with respect to the Myerson value in the
game with the major player.
Fig. 4: Diagram of stable coalition structures with respect to the ES-value in the game
with the major player.
5.
Conclusion
We have considered cooperative games with coalition structures in which communication among players is restricted by graph. Assuming that any possible coalition
structure can be realised in the game we suggested a method of verification of a
coalition structure on stability. The idea of definition of a stable coalition structure
is close to Nash equilibrium. We have proposed a model of the game with the major
player in which communication among players is realised via a given “star” graph.
For the game we found all possible stable coalition structures. The type of stable
coalition structure depends on the parameters of the game.
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Power in Game Theory
Rodolfo Coelho Prates
UP - Universidade Positivo
E-mail: [email protected]
Abstract The main aim of this paper is to discuss power in game theory,
in order to model the asymmetries of forces among the players of the game.
The starting point is that games are strategic interactions between rational individuals in a social environment, and the players do not have equal
forces. Game theory has increased greatly in recent years, including several
new branches. However, the concept of power in game theory has not been
explored to any great extent. Indeed, power is a broad concept that has no
clear definition. This paper formalizes a model, taking account the asymmetric forces between players. Examples are given, using some well-known
games, to illustrate this relation involving power. As a result, this paper
presents an approach in two still distant scientific fields: sociology and game
theory.
1.
Introduction
Some words or concepts do not have a precise meaning. Power is one of these words.
It is said that the president of a nation has power, while civilians do not. On the
other hand, in a democratic environment, the coalition of people is a source of power
because it is precisely this coalition that will decide who will be the governor. The
director of a company has power, while the workers do not, but one of these workers
could be a union leader with as much power as the director. A teacher has power
over his students because he can assign them a grade.
Everyone knows that in recent times Europe and the United States have been
losing their economic power to China. In fact, like love, power is an abstract noun. It
is impossible to see, touch or measure it, but it is possible to feel it. Nye (2011) states
that it is very common to compare power with other things, such as energy (physics)
or money (economics). Both these comparisons are mistaken. It is impossible to
compare power with energy, as energy can be measured, as can money, which is
liquid and interchangeable.
To understand the "nature" of this word, different theorists and sciences, including sociology, psychology, politics and economics, have developed different or
complementary concepts of power and its relation to society. Power, in fact, is a
broad concept, involved in different sciences and with a long historical approach.
Hobbes, for instance, was one the first philosophers to analyze the meaning of power
in the modern ages as far back as the 17th Century. These ideas were published in the
Leviathan. To him, power is the ability to secure well-being or personal advantage
in order to achieve some apparent good in the future.
After Hobbes, many concepts were developed for understanding power, but in
all cases the meaning is the same: the asymmetry of forces. It is clear that power
has a different view and differs in accordance with time and space. Power, in fact,
is an important concept within society, and arises from the interaction between
Power in Game Theory
283
people. In this sense, power could be thought of in terms of different views, including
sociological, anthropological, internationally political and economic.
In this sense, the aim of this paper is to discuss power in game theory in order
to model the asymmetries of forces between the players of the game. The starting point is that games are strategic interactions between rational individuals in a
social environment and the players do not have equal forces. Game theory has increased greatly in recent years and several new branches have developed. However,
the concept of power in game theory has not been explored to any great extent.
Indeed, power is a broad concept with no clear definition. This paper formalizes a
model, taking into account the asymmetric forces between players. In addition to
this introduction, the paper discusses two specific topics: the concepts of power and
a game model that includes power. These are followed by the conclusions.
2.
The Concept of Power
Most people do not know the exact meaning of power, but they have an intuitive
notion regarding it. This difference arises because power is a universal phenomenon,
affecting people of different ages and in different places and times. Besides any kind
of definition, people feel it. A little child can identify the power of its father or
mother, an employee in an organization respects the power of his boss and, in
normal circumstances, all the people obey the president of the nation. On the other
hand, it is possible to identify its branches, such as political power, economic power
and international power.
It is very common to hear a conversation about power at different levels. Daily
newspapers publish news about power, who is involved and its consequences. But
what is power? The answer to this question is not simple and covers a long process
throughout the history of humanity. Some definitions of power are given below,
but this list is not exhaustive. Power is normally associated with glory, but this
association could distort the real meaning. Dahl (1957) p. 203 was led to think that
"A has power over B to the extent that he can get B to do something that B would
not otherwise do". This definition indicates that power exists over people. However,
it is also possible to have power over animals and objects.
Weber (1968), for instance, provided one of the most important definitions of
power: "the probability that one actor within a social relationship will be in a position to carry out his own will despite resistance". Foucalt (1978) p. 92 said that
"power must be understood in the first instance as the multiplicity of force relations immanent in the sphere in which they operate and which constitute their own
organization". In this sense, Foucault advocates the existence of numerous powers,
what he referred to as the micro-physics of power. Wartenberg (1990) distinguishes
the exercise and possession of power. The exercise of power has attracted the attention of theorists due empiricist assumptions, but it is fundamental to have a
strong concept of the possession of power. In this sense, the possession of power
can be defined as "a social agent A possesses power over another agent B if A controls B’s action-environment in a fundamental manner" Wartenberg (1990), p. 10.
Action-environment means the structure, in a whole sense, within which as agent
exists as a social actor. The action-environment supports, or provides conditions
for, action-alternatives. It can be understood as "a course of action that is available
to the agent in the situation" (p. 7). Power can be thought of as a capacity for
control that one agent has over the action-environment of another agent. The main
284
Rodolfo Coelho Prates
important notion is that one agent has power over another agent without having
exercised power in a practical sense. In this sense, if an agent has power, he could
exercise it if he chose to, even in situations where he chooses not to, but he possesses
that power nevertheless.
On the other hand, power has a practical meaning. Agents use power over other
agents to obtain material or immaterial elements or to gain any kind of advantage.
The use of power differs in intensity and type. The use is called the exercise of
power, which can be defined as an "agent A exercises power over an agent B if A
uses his control of B’s action-environment to change it in some fundamental manner"
Wartenberg (1990), p. 11. In this sense, power can be understood as the realization
of a capacity. At this juncture, it is important draw a distinction between possessing
and exercising power. As discussed above, possessing is capacity, and exercising is
the realization of that capacity. Wartenberg (1990) emphasizes that power is used
by restructuring the options of the other agent. This means that if agent A has
power over agent B, the options of B are changed by the power of agent A. "It
states that an agent who exercises power over another does so by changing the
circumstances within which the other agent acts and makes choices" (p. 12).
The exercise of power can take four forms: force, coercion, influence and manipulation. It is important to say that manipulation is a kind of influence. The relationship of force occurs when "A’s power over B is an instance of force if A physically
keeps B from pursuing an action-alternative that B has reason to pursue or makes
B’s body behave in a way that B would avoid if possible" (Wartenberg (1990), p.
13). The second form of power is coercion, which is related to threats, and these
threats must be recognized and accepted by the threatened agent. Coercion exists
when agent A exercises "power over social agent B if (1) A has the ability to affect
B in a significant way; (2) A threatens to do so unless B acts in a certain way; and
(3) B accedes to A’s threat and alters his course of action" (Wartenberg (1990),
p. 15). Influence, the third form, can be thought of as when agent A "influences
another agent B if A provides B with some putative information which results
in B altering his assessment of his action-environment in a fundamental manner"
(Wartenberg (1990), p. 21). The last form is manipulation, which can be understood
as "agent A manipulates another agent B if A influences B for purposes or ends
that he keeps concealed from B" " (Wartenberg (1990) p. 15). For the purpose of
this paper, the understanding of (Galbraith (1983), p.2) has been adopted. Indeed,
Galbraith uses the definition proposed by Weber: "power is the ability to impose
one’s will on the behavior of other persons". He goes on to divide it into three
categories: a) condign, b) compensatory, and c) conditioned power.
The first category is based on brute force, meaning that the person who does
not obey will be punished. In this situation, threat or intimidation is an essential
action. Condign is a kind of power which is based on "the ability to impose an
alternative to the preferences of the individual or group that is sufficiently unpleasant or painful so that these preferences are abandoned" (Galbraith (1983), p. 4).
Compensatory power means that a quantity of resources is used to exchange what
the owner wants. "Compensatory power, in contrast, wins submission by the offer of
affirmative reward by the giving of something of value to the person so submitting"
(p. 5.). Conditioned power refers to changing belief. Galbraith states that "persuasion, education, or the social commitment to what seems natural, proper, or right,
causes the individual to submit to the will of another or of others" (p.6).
285
Power in Game Theory
Galbraith also proposed three sources of power: personality, property or wealth
and organization. Personality is "the quality of physique, mind, speech, moral certainty or other personal trait" (Galbraith (1983), p. 23). It is the ability to create
or persuade. Belief, is the ability to change the mind or the way that people think.
Personality is associated with conditioned power.
Property is associated with compensatory power, and means that agent one
(the wealthy) can offer a quantity of money to another player, who accepts the
conditions proposed by the first agent. In this sense, property and income "provide the wherewithal to purchase submission" (Galbraith (1983), p. 23). The third
source of power is the organization, which is, in modern societies, the most important source of power. An organization is a number of persons or groups that
have united for some purpose. This means that an organization concentrates both
property and personality power. The State is a kind of organization that accesses
condign power (punishment). In some circumstances, it is possible to connect the
proscription of a company as a kind of punishment. (Nye (2006)) separated power
into two categories: soft power and hard power. To him, conditioned power is hard,
while compensatory power and condign power are soft. Figure 1, below, shows some
specifications concerning these two kinds of power.
Type of Power Behavior
Sources
Soft
Attract and co-opt Inherent qualities
Communications
Hard
Threaten and
Threats,
induce
intimidation. Payment
rewards
Examples
Charisma
Persuasion, example
Hire,fire,deote
Promotions,
compensation
Fig. 1: Soft and Hard Power.
Source: (Nye (2006))
(Balzer (1992)) has compared both game theory and power theory. Even power
theory has no status of generally acknowledged theory like game theory, and the
author has identified some particular converging aspects. First of all, the individuals
(players) in both models can be the same, as well the alternatives for the individuals
involved in the game. Nevertheless, the comparison described by Balzer requires two
distinct events, before and after, and this is a situation that is only present in power
theory. (Wiese (2009)) understands that compensatory power is a particular case
from co-operative games, because it means transferable utility. However, condign
power and conditioned power represent a non-transferable utility case, which plays
an important role in social relations.
Formalizing the Idea of Power-asymmetric Games
In this section the idea of power-asymmetric games is formalized in a broad sense.
It is important to emphasize that the power game does not invalidate the two main
postulates of game theory: developing criteria for rational behavior and the assumption that players maximize their own utility functions. It should be emphasized that
the only form of power is the exercise of power, not its mere possession. This means
that an agent with power will always exercise it, as generates better results for him.
It is also important to highlight that game theory usually encompasses different
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Rodolfo Coelho Prates
games when the action-environment changes. For instance, when power is incorporated into a specific game, the use of power by an agent generates a new game, which
transforms the options set of the player subjected by the power relation. However,
instead of different games, the model proposed in this paper understands it as a
single or unique game. This notion is important for capturing some elements of
changing, and allows for comparison ex ante and ex post action-environment when
the options are changed.
Definition 1. For an arbitrarily fixed n ∈ N, an n-player power-asymmetric game
ΓnP is given by the 5-uple:
Γnp = hIn , (Si )i∈In , (ui )i∈In , (pi )i∈In , Pi,
where In = 1, , n is the set of players, Si is the set of strategies of player i ∈ In , the
function
ui : ×(j∈In Sj → R
stands for the utility of player i, pi is the power of player i ∈ In and P is the power
map of the game.
The power map
P : Rn → 2×j∈In Sj
is a set-valued function. For each point (p1 , , pn ) it assigns a subset of ×j∈In Sj
so that the strategies in P(p1 , , pn ) are no longer available in the game. The power
function represents the power of player i over player j. If the power function is valid,
the strategic choices of player j are affected by the influence or force of player i, thus,
the power function restricts the strategic space of player j. Being a rational player
and having power over j, player i will influence the decisions of player j, rendering
invalid one or more strategies of j that result in a lower payoff to i. In other words,
player i constrains the space of strategy of player j to guarantee the maximization of
his utility. To explain the meaning of the power game, some examples are presented
below in which the main implications of the game when power relations are included
are shown.
Example 1: (generic game). Assuming a static game of complete and perfect
information, such as the normal-form representation below, where A and B are the
players, ai and bi are the strategy sets of A and B, and the number are the players’
utility.
A
a1
a2
a3
b1
2.4
2.3
1.5
B
b2
3.2
4.1
2.2
b3
2.1
3.0
1.5
Fig. 2: Generic Game
Source: the author.
First of all, it is necessary to define the existence of the power function and who
has the power, for instance by assuming that B has power over A. An inspection
reveals that the matrix can show that when player A chooses a2 , this results in a
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Power in Game Theory
lower payoff to player B. B, in turn, exerts his power and will preclude A choosing
a2 . By doing so, he avoids the possibility of having lower utilities. If player B did
not have power over A, A would be able to freely choose all available strategies, the
forces between them would be equal, and player B would have lower utilities.
Example 2 (Battle of the sexes - BoS). Ballet versus soccer. It is known that
there are two Nash equilibriums. Instead of the regular formulation that the woman
and man do not have power, it is supposed that the woman has a power pw and the
man has a power pm . Furthermore, it is assumed that the game is endowed with a
power function P that excludes soccer from the mans possible strategies whenever
pw > pm and excludes ballet from the woman‘s possible strategies if pm > pw . The
formalization of this game is:
P
ΓBoS
=
h{man = 1, woman = 2}, ({ballet, soccer}, {batter, soccer}), (u1, u2 ), (p1 , p2 ), Pi
Y
Si = {(B, B), (B, S), (S, B), (S, S)} = S
i∈In
The set of parts of S, 2S , is given by:


Φ, S, {(B, B)}, {(B, S)}, {(S, B)}, {(S, S)}, {(B, B), (B, S)},






{(B,
B),
(S,
B)},
{(B,
B),
(S,
S)},
{(B,
B),
(B,
S),
(S,
B)},
S
2 =
{(B, B), (B, S), (S, S)}, {(B, B), (S, B), (S, S)}, {(B, S), (S, B), (S, S)} 





{(B, S)}, {(S, B)}, {(B, S), (S, S)}, {(S, B)}, {(S, S)}
The power map P : R2 → 2S is:

 {(B, S), (S, S)} if x < y
P(x, y) = {(B, S), (B, B)} if x > y

φ if s = y
where x = p1 = man power, and y = p2 = woman power. Assuming that P
constrains the set of strategies, as shown below:
Y
Y
Y
Si |P(p1 ) =
Si |P(p2 ) =
Si \P(p)
i∈In
i∈In
i∈In
In the game of the BoS, the available strategies are:
Y
Si |P(x<y) = S\{(B, S), (S, S)} = {(S, B)}, {(B, B)}
i∈In
and
Y
i∈In
Si |P(x>y) = S\{(B, S), (B, B)} = {(S, B)}, {(S, S)}
Example 3 (Prisoners Dilemma). Supposing that criminal A is more powerful
than B, A will use his power to influence the decisions (set of strategies) of agent
B. For this particular situation, (Williamson (2010), p. 26) offered an explanation
regarding the relation between the prisoners that implicitly uses the concept of
power: "rather than assume that players are accepting of the coercive payoffs that
are associated with the prisoners’ dilemma - according to which each criminal is
288
Rodolfo Coelho Prates
induced to confess, whereas both would be better off if they could commit not to
confess - Transaction Costs Economy assumes that the criminals (or their handlers,
such as the mafia) can, upon looking ahead, take ex ante actions to alter the payoffs
by introducing private ordering penalties to deter defections. This latter is a governance move, variants of which can be introduced into many other bad games". The
ex ante actions are defined exactly by the power relations between the criminals,
which could make use of condign, compensatory or conditioned power.
Assuming asymmetric forces between both, the criminal with more power imposes the strategy not to confess on the man with less power, he chooses the strategy
not to confess, while the criminal with more power chooses the strategy to confess.
In this situation, the criminal with more power goes free while the man with less
power will be punished. Taking into account that power is present in the relationship, there is no more Nash equilibrium.
The examples above show that the powerful agent constrains the set of strategies
of the other agent. In this sense, the other agent does not have all the options
available to choose, only part of them. In this new condition, the equilibrium is not
the same that would be achieved if the options were unconstrained.
Due to this, it is possible to think of a measure or index of power which the
powerful player has over the other. In a two person game, and supposing that both
have mixed strategies, the measure of power of the powerful agent (player i) can be
described as:
ui |Pij
IPi =
ui
Where ui is the expected utility for the player i and uj |Pij is the expected utility
for the player i when he uses his power (Pij ) to constrain the strategy set of player
j. The power is the "force" that the powerful player uses to restrict the choices of
the other player. In fact, by the definition, IPi lies in the interval [1, ∞]. If IPi = 1,
it means that ui |Pij is equal to ui , and in this situation the function wij does not
restrict the strategy set of player j. Thus, player i does not have power over player
j. On the other hand, if ui |wij is greater than ui , IPi will be greater than one.
In this sense Pij acts and restricts the strategy set of player j. When the power is
acting, some of the options are not available in the strategy set, meaning that this
player does not choose the option that he would if all the options were available.
Example 4. The matrix below shows the same game as example one, but here the
strategies are mixed, with equal probabilities for both players and for all strategies.
Table 1: Example 4
A
a1
a2
a3
b1
2.4
2.3
1.5
B
b2
3.2
4.1
2.2
b3
2.1
3.0
1.5
The expected utility for player B is 2.55. Assuming again that player B has
power over player A, and that he uses this power, it is clear that the strategy a2
is restricted by the power. In this way, player A cannot choose this strategy (a2 )
because it is not available, and player B is free to choose any strategy in his set.
Power in Game Theory
289
Afterwards, the new expected utility for player B is 3.16, and the measure of power
is 1.23. This measure shows an increase in expected utility when the powerful player
exercises his power.
3.
Conclusions
Power is not a philosophical or remote subject, despite the fact that usually no one
can see or measure it. The understanding of this subject has been developed, over
many years, mainly by social scientists, including economists. Game theory is an applied branch of mathematics that analyses the rational interaction between agents,
but the models constructed have not embodied the asymmetric forces between them.
In this way, it is possible to understand in a formal model regarding game theory
how power can affect the relationship between players, assuming that one of the
agents has more forces than others, and uses his forces to guarantee that his will is
easily achieved. The model was constructed taking into account that power should
be analyzed in the strategic interaction between individuals.
In games where there are one or more Nash equilibria, the power relationship
between the players removes one or all of the equilibria due to the fact that one of
the players is not doing what is best, and does exactly is imposed on him. Finally,
this paper unites two different sciences that developed separately: sociology and
game theory.
References
Balzer, W. (1992). Game theory and power theory: a critical comparison. In: Wartenbert,
T. E. rethinking power. Albany, NY: Suny Press.
Dahl, R. A. (1957). The concept of power. Behavioral Science.
Foucault, M. (1978). The history of sexuality. New York: Pantheon Books.
Galbraith, J. K. (1983). The Anatomy of Power. John Kenneth Galbraith, Boston:
Houghton Mifflin.
Hobbes, T. (1968). (editor: MacPherson, Crawford Brow,) Leviathan, or The Matter,
Forme & Power of a Common-wealth Ecclesiastical and Civill, London, (original edition: London, 1651).
Nye, J. S. (2006). Soft Power, Hard Power and Leadership. Retrieved: March, 15,2013, from
http://www.hks.harvard.edu/netgov/files/talks/docs/11_06_06_seminar_Nye_HP
_SP_Leadership.pdf
Nye, J. S. (2011). The future of power. Public Affairs/Perseus Book Group.
Osborne, M., Rubinstein, A. (1994). A Course in Game Theory. Cambridge, MA: MIT
Press.
Wartenbaerg, T. E. (1990). The Forms of Power: From Domination to Transformation.
Philadelphia: Temple University Press.
Weber, M. (1968). Economy and Society. Bedminster Press, New York.
Wiese, H. (2009). Applying cooperative game theory to power relations. Quality and Quantity, 43(4), 519–533.
Williamson, O. E. (2010). Transaction cost economics: an overview. In: The Elgar Companion to Transaction Cost Economic (Peter G. Klein and Michael E. Sykuta, eds).
Completions for Space of Preferences
Victor V. Rozen
Saratov State University,
Astrakhanskaya St. 83, Saratov, 410012, Russia
E-mail: [email protected]
Abstract A preferences structure is called a complete one if it axiom linearity satisfies. We consider a problem of completion for ordering preferences
structures. In section 2 an algorithm for finding of all linear orderings of
finite ordered set is given. It is shown that the indicated algorithm leads to
construction of the lattice of ideals for ordered set. Further we find valuations for a number of linear orderings of ordered sets of special types. A
problem of contraction of the set of linear completions for ordering preferences structures which based on a certain additional information concerning
of preferences in section 4 is considered. In section 5, some examples for construction and evaluations of the number of all linear completions for ordering
preferences structures are given.
Keywords: preferences structure, ordering preferences structure, completion of preferences structure, a valuation for the number of linear completions.
1.
Introduction
A space of preferences (or preferences structure) can be defined as a triplet of the
form
hA, α, βi ,
(1)
where α and β are binary relations on a set A satisfying the following axioms:
1. α ∩ α−1 = ∅
−1
2. β = β
3. ∆A ⊆ β
4. α ∩ β = ∅
(asymmetry);
(symmetry);
(reflexivity);
(2)
(disjointness).
We mean
A as a set of alternatives;
α as a strict preference relation;
β as an indifference relation.
As usually we put ρ = α ∪ β and use the notation:
ρ
df
α
β
a . b ⇔ a < b or a ∼ b.
Then a space of preferences can be written as a pair hA, ρi, where the strict
preference relation and the indifference relation can be presented as
α = ρ\ρ−1 ,
β = ρ ∩ ρ−1 .
(3)
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Completions for Space of Preferences
The main special properties for preferences structures are the following ones:
Transitivity :
Antisymmetry :
Linearity :
ρ
ρ
ρ
a1 . a2 , a2 . a3 ⇒ a1 . a3 ;
(Tr)
a1 . a2 , a2 . a1 ⇒ a1 = a2 ;
(Antsym)
ρ
ρ
a1 . a2
ρ
ρ
or a2 . a1 .
(Lin)
Definition 1. Preferences structure satisfying the conditions (Tr) and (Antsym) is
called an ordering preferences structure and satisfying the conditions (Tr), (Antsym)
and (Lin) is called a linear (or complete) ordering preferences structure.
Definition 2. A preferences structure hA, α1 , β1 i is called a completion of a preferences structure hA, α, βi if inclusions
α ⊆ α1 , β ⊆ β1
(4)
hold and at least once of these inclusions is strict.
Remark 1. A preferences structure hA, α, βi has not completions if and only if it
is a linear one.
Thus the most interesting are completions of a preferences structure to a linear
preferences structure. In this paper, we study some questions concerning of completions for ordering preferences structure. The main problems of our investigation
are:
(PI) The problem of description of all completions for ordering preferences structure to a linear one and
(PII) The problem of contraction of the set of linear completions based on certain
additional information concerning of these completions.
2.
2.1.
Linear orderings of ordered sets
An algorithm for finding of all linear orderings
It is well known the following classical result (Birkhoff, 1967).
Szpilrajn Theorem. Any partial ordering can be enlarged to a linear ordering.
Thus in terms of our paper, any ordering preferences structure has a completion
to a linear one. However, Szpilrajn theorem is not a constructive propositional since
it does not indicate a method for construction of linear completions.
Consider an ordering preferences structure which on a set of alternatives A is
given. In algebra terminology, such a structure can be presented as an ordered set
(A, ≤) (i.e. ≤ is a binary relation on A satisfying conditions reflexivity, antisymmetry
and transitivity). In this notations, the strict preference relation α coincides with
strict order < and the indifference relation β is identity relation.
We now state an algorithm for finding of all linear orderings of a finite ordered set
that is an algorithm for finding of all linear completions of an ordering preferences
structures. Remark that formally a linear ordering of k-element subset B ⊆ A can
be represented as one-one isotonic function ϕ from B into {1, . . . , k}, where ϕ (a)
is a number of element a ∈ B under this linear ordering. The required algorithm is
based on the following lemma.
292
Victor V. Rozen
Lemma 1. Suppose an ordered set A contains n elements and a∗ is a maximal
element. Assume that we have a linear ordering ϕ of a subset A\a∗ (by numbers
1, 2, . . . , n − 1). Preserve the function ϕ for elements of A\a∗ and put ϕ (a∗ ) = n
then ϕ becomes a linear ordering of all set A.
Thus we can obtain all completions of the set A, having completions of subsets
which are a result of extraction of maximal elements. Further we use the same
method for these subsets until the empty set ∅ appears. To realize this algorithm
we need in the following steps.
Step 1. Define an auxiliary graph γ by the following rule. Vertexes of graph γ
γ
are some subsets of A and for two subsets A1 , A2 ⊆ A put A1 ≺ A2 if and only if A2
is a result of extraction of some maximal element belonging to A1 . Then starting of
the set A, we construct some sequence of conjugate subsets with respect to graph
γ. It is evident that in graph γ the length of any path is equal to n.
Step 2. For each one element subset which is a vertex of graph γ write its single
linear ordering.
Step 3. Let B be a k-element subset (k = 2, . . . , n) which is a vertex of graph
γ. Assume we have a linear ordering for each subset of the form B\a, where a
is a maximal element of B. Then we preserve these linear orderings for elements
belonging B\a and set ϕ (a) = k.
Step 4. As the final step of this algorithm we obtain all linear orderings for set
A which is a vertex of graph γ.
An example for finding of all completions of ordered set in section 5 is given.
2.2.
Ideals of ordered set
Definition 3. Let hA, ≤i be an arbitrary ordered set. A subset B ⊆ A is called an
ideal in hA, ≤i if the following condition
a ∈ B, a′ ≤ a ⇒ a′ ∈ B
holds. For any subset X ⊆ A we define a set of its minorants X ↓ by setting
X ↓ = {a ∈ A : (∃x ∈ X) a ≤ x} .
(5)
For any X ⊆ A, subset X ↓ is the smallest (under inclusion) ideal which contains
X; if X is an ideal then X ↓ = X. It is said that X ↓ is the ideal generated by
subset X. Particularly an ideal generated by one element subset {a} is called a
main ideal and denoted by a↓ . A mapping X → X ↓ which every subset X ⊆ A put
in correspondence the ideal generated by this subset is a closure operation, hence
the set Id (A) of all ideals of ordered set hA, ≤i forms (under inclusion) a complete
lattice in the sense (Birkhoff, 1967). Since the intersection and the union of any
family of ideals is an ideal also then the lattice of ideals Id (A) is distributive. We
now indicate some method for construction of the lattice Id (A).
Theorem 1. Let hA, ≤i be a finite ordered set. Then
1. A subset which is a result of extraction from ideal its maximal element is an
ideal also;
2. Any ideal can be realized from ideal A with help of procedure of extraction of
maximal elements by a finite number steps.
Completions for Space of Preferences
293
Proof (of theorem 1). 1. Let B ⊆ A be an ideal of the ordered set hA, ≤i and b∗ ∈ B
a maximal element of B. Show that B\b∗ also is an ideal. Indeed suppose a ∈ B\b∗
and a′ ≤ a; since subset B is an ideal and a ∈ B then a′ ∈ B. Assumption a′ = b∗
implies b∗ ≤ a. The equality b∗ = a is false since we obtain b∗ ∈ B\b∗ . Then b∗ < a
that is impossible for maximal element b∗ ∈ B. Thus a′ 6= b∗ hence a′ ∈ B\b∗ .
We now state the following lemma.
Lemma 2. Let B ⊆ A be an ideal in ordered set hA, ≤i and B 6= A. Then there
exists such element a1 ∈ A\B that
α) subset B ∪ {a1 } is an ideal and
β) the element a1 is a maximal one in B ∪ {a1 }.
Proof (of lemma 2). Consider any ideal B in ordered set hA, ≤i where B 6= A. Fix
some minimal element a1 of non-empty set A\B. Check that B ∪ {a1 } is an ideal.
Suppose a ∈ B ∪ {a1 } and a′ < a. If a ∈ B then a′ ∈ B by definition of ideal hence
a′ ∈ B∪{a1 }. In the case a = a1 assume a′ ∈
/ B. Then a′ ∈ A\B and we have a1 > a′
that is false since element a1 is minimal in A\B. Thus a′ ∈ B ⊆ B ∪ {a1 } and α)
is proved. Show β). The assumption b > a1 for some b ∈ B implies by definition of
ideal the inclusion a1 ∈ B in contradiction with a1 ∈ A\B and lemma 2 is proved.
⊔
⊓
We now prove the proposition 2 of theorem 1. Let B ⊆ A be an ideal in ordered
set hA, ≤i and B 6= A. By lemma 2 there exists such element a1 ∈ A\B that the
subset B ∪ {a1 } is an ideal and a1 is a maximal element in B ∪ {a1 }. If B ∪ {a1 } = A
then the ideal B is a result of extraction of maximal element a1 from A and our
proposition is proved. If B ∪ {a1 } 6= A then using lemma 2 once more we obtain
that there exists such a2 ∈ A\ (B ∪ {a1 }) that the subset B ∪ {a1 , a2 } is an ideal
and a2 is a maximal element in B ∪ {a1 , a2 }. Consider two cases: B ∪ {a1 , a2 } = A
and B ∪ {a1 , a2 } 6= A etc. Since the set A is finite, we have a sequence of the kind
{a1 , a2 , . . . , at } where as ∈ A\ (B ∪ {a1 , . . . , as−1 }) and the following conditions
hold (s = 1, . . . , t):
α∗ ) B ∪ {a1 , . . . , as } is an ideal;
β ∗ ) as is a maximal element in B ∪ {a1 , . . . , as };
γ ∗ ) B ∪ {a1 , . . . , at } = A.
Thus the ideal B is a result of extraction of maximal elements {at , at−1 , . . . , a1 }
from a chain of ideals starting of A which was to be proved. Finally for B = A
the proposition 2 of Theorem 1 is evident since in this case the required number of
extractions of maximal elements is equal to zero.
⊔
⊓
According to theorem 1, we remark that vertexes of an auxiliary graph γ are
precisely ideals of ordered set hA, ≤i. Hence we obtain
Corollary 1. We can identify auxiliary graph γ of ordered set hA, ≤i with lattice
Id (A) of its ideals Id (A). Namely, the set of vertexes of graph γ coincides with the
set of ideals and the canonical order relation of lattice Id (A) can be presented as
following: B1 ⊇ B2 if and only if there exists a path from B1 to B2 in graph γ.
We now remark that for finite ordered set hA, ≤i the procedure of finding its
linear orderings can be reduced to finding of maximal chains in the lattice of ideals
Id (A). Further using the indicated algorithm for construction of linear orderings,
we have
294
Victor V. Rozen
Corollary 2. For finite ordered set hA, ≤i, there exists one-one correspondence
between its linear orderings and maximal chains in the lattice Id (A) of its ideals.
Hence the number of linear completions of an ordering preferences structure hA, ≤i
coincides with the number of maximal chains in the lattice Id (A).
Remark 2. Indicated correspondence can be realized in the following manner. For
linear ordering {ai1 < ai2 < . . . < ain } of A, the corresponding maximal chain in
the lattice of ideals is
{∅ ⊂ {ai1 } ⊂ {ai1 , ai2 } ⊂ . . . ⊂ {ai1 , ai2 , . . . , ain }} .
3.
A valuation for a number of linear orderings
3.1.
A finding of the number of linear ordering with help of auxiliary
graph γ
Using inductive algorithm for construction of auxiliary graph γ (see 2.1.), we can
find the number N (A) of all linear orderings of a finite ordered set without of finding
of these orderings. As a first step, we need to construct the auxiliary graph γ. Denote
by N (B) the number of all linear orderings for arbitrary set B which is a vertex of
graph γ. Since linear orderings of B are extensions of linear orderings of conjugate
with B vertexes then we obtain the following recurrent formula:
X
N (B) =
N (B\a) ,
(6)
where a is an arbitrary maximal element of subset B. Since every subset which is
a final vertex in graph γ is one element hence it has a single linear ordering. Using
formula (6) we can find the number of all linear orderings for any vertex of graph γ.
In particular we can find the required number N (A). An example for count of the
number of all linear orderings of ordered set in section 5 will be given.
3.2.
A valuation of the number of linear orderings for some special
cases
Remark that formula (6) for finding of N (A) can be used only in the case the graph γ
(i.e. the lattice of ideals of ordered set) is given. However a practical construction
of the graph γ for ordered set which contains some tens of elements is very hard.
Further we consider certain methods for finding N (A) in some special cases. Let
hAk , ωk i (k = 1, . . . , r) be a family of ordered sets and hA, ωi is the discrete sum of
this family. Denote by Ndis the number of all linear orderings for hA, ωi. Then we
have the following formula (see Rozen, 2013):
Ndis =
n!
N1 · N2 · . . . · Nr ,
n1 !n2 ! . . . nr !
where nk = |Ak | (k = 1, . . . , r) , n =
r
P
k=1
(7)
nk .
This formula is proved by induction on r. For r = 1 the right part of (7) is
equal to N1 = Ndis . Let us show that (7) is truth for r = 2. Indeed, consider
two ordered sets A and B, where the first set contains n1 elements and the second
set n2 elements. Let (a1 , a2 , . . . , an1 ) and (b1 , b2 , . . . , bn2 ) be their linear orderings,
respectively. Then we can obtain a linear ordering for discrete sum A ∪ B in the
following manner. Fix a subset {i1 , i2 , . . . , in1 } in the set {1, 2, . . . , n1 + n2 } and
Completions for Space of Preferences
295
let {in1 +1 , . . . , in1 +n2 } be its complement (suppose these sequences are increasing).
Then by setting ϕ (as ) = is (s = 1, . . . , n1 ) and ϕ (bt ) = in1 +t (t = 1, . . . , n2 ) we
obtain a linear ordering of discrete sum A ∪ B. Hence every pair of linear orderings
2 )!
of A and B generates Cnn11+n2 = (nn11+n
!n2 ! of linear orderings for their discrete sum
A ∪ B. Denote by N1 the number of linear orderings of A and by N2 the number of
linear orderings of B, then the number of pairs of linear orderings is equal to N1 ·N2 ;
2 )!
thus we obtain (nn11+n
!n2 ! N1 · N2 of linear orderings for A ∪ B. For r = 2 formula (7)
is shown. Remark now that discrete sum of r ordered sets can be represented as a
discrete sum of two ordered sets: A1 ∪ . . . ∪ Ar−1 ∪ Ar = (A1 ∪ . . . ∪ Ar−1 ) ∪ Ar .
Using our assumption for r = 2, we obtain the required proposition in general case,
that is (7).
As a corollary of formula (7) we now obtain a valuation for N (A) in the case A
is a tree ordered set. Consider a tree T with a root a0 . Then we can define on the
set A of tree vertexes the tree order by the rule: a1 ≤ a2 if and only if there exists
a path from a1 to a2 . Remark that a0 is the greatest element under order ≤. For
each ak ∈ A the set Tak consisting of vertexes a ≤ ak forms a tree with root ak ; it
is called subtree with root ak . Particularly, T = Ta0 .
Corollary 3. Let Ta0 be a tree and {Ta0 , Ta1 , . . . , Tar } all its subtrees having not
less than two vertexes. Then a number N (Ta0 ) of all linear orderings of tree Ta0 is
defined by formula:
|Ta0 |!
NTa0 =
,
(8)
|Ta0 | · . . . · |Tar |
where |Tak | denotes a number of elements of subtree Tak .
Proof of corollary 3 is given by induction on numbers of levels of tree. To prove
induction step one can use that if to eliminate the greatest element of tree then we
obtain a discrete sum of tree orders for which formula (7) is true, and the number
of linear orderings for these tree orders can be founded by assumption (8).
4.
A contraction of the set of linear completions
We now consider a problem of contraction of the set of linear completions for ordering preferences structures which based on some additional information concerning
of preferences. Suppose an ordering preferences structure in the form hA, ωi is given
where ω is an order relation on the set of alternatives A.
We consider here additional information of the following types.
Type 1: Information under strict preferences
This information with binary relation δ ⊆ A2 can be given where the assertion
(a1 , a2 ) ∈ δ means that alternative a2 is strict better than alternative a1 . Such
information does not contradict with an ordering ω if and only if the relation ω ∪ δ
is acyclic; in this case ω1 = tr (ω ∪ δ) is an ordering of A which contains previous
ordering ω and the relation δ also. Further finding linear completions of ordering ω1
we obtain some part of all linear completions for ordering ω. Completions of ordering
ω1 are completions for ordering ω which conform with additional information in the
form of binary relation δ.
Type 2: Information under indifference relation
In this case, additional information in the form of an equivalence relation ε ⊆ A2
is given.
296
Victor V. Rozen
Definition 4. Let ϕ be an isotonic function from ordered set A in some chain C.
Put εϕ = {(a1 , a2 ) : ϕ (a1 ) = ϕ (a2 )}. An equivalence ε ⊆ A2 is said to be ranged
equivalence if ε = εϕ for some isotonic function ϕ.
We assume here that ε is a ranged equivalence. Then we factorize relation ω
under equivalence ε and obtain factor-ordering T r (ω/ε) on the factor-set A/ε. Let
C1 , . . . , Cr be all classes of equivalence ε and ωk is a restriction of ω on subset
Ck (k = 1, . . . , r); put Nk the number of linear completions for ωk and N (ω) the
number of linear completions for hA, ωi. Then we have the following evaluation for
the number N (ω, ε) of linear completions for the factor-structure:
N (ω, ε) ≤
N (ω)
.
N1 · N2 · . . . · Nr
(9)
The inequality (9) shows that additional information of type 2 implies a strong
contraction for the number of linear completions.
Remark 3. Conditions concerning of equivalence ε ⊆ A2 under which there exists
the unique linear completions for factor-structure A/ε (i.e. N (ω, ε) = 1) is given in
(Rozen, 2011).
5.
Examples
Example 1. Finding of all linear completions for ordering preferences structures
Consider an ordering preferences structure consisting of 6 alternatives A =
{a, b, c, d, e, f } presented by a diagram (fig. 1).
Fig. 1
To construct all its linear orderings, we need in the following steps.
Step 1. Using a procedure of extraction of maximal elements (see 2.1.), we
obtain an auxiliary graph γ (fig. 2).
297
Completions for Space of Preferences
Fig. 2
Step 2. We now construct Table 1 whose rows are vertexes of graph γ (i.e.
ideals) and for each ideal all its linear completions are given. Starting of one element
ideals, we receive at last linear orderings of ideal A = {a, b, c, d, e, f } in lower block
of Table 1.
298
Victor V. Rozen
Table 1.
a
{a}
{d}
{a, d}
{a, b}
{a, d, e}
{a, b, d}
{a, b, c}
{a, b, d, e}
{a, b, c, d}
b
c
d
e
1
1
2
1
1
2
1
2
1
1
1
2
1
2
1
1
2
1
1
f
a
b
c
d
e
{a, b, c, d, e}
1
2
1
2
1
1
2
1
1
4
4
3
3
2
3
3
2
2
5
5
5
5
5
4
4
4
3
2
1
2
1
3
2
1
3
4
3
3
4
4
4
5
5
5
5
A = {a, b, c, d, e, f }
1
2
1
2
1
1
2
1
1
4
4
3
3
2
3
3
2
2
5
5
5
5
5
4
4
4
3
2
1
2
1
3
2
1
3
4
3
3
4
4
4
5
5
5
5
1
2
1
2
3
3
2
2
4
4
3
3
2
3
3
2
2
2
1
2
1
3
3
3
2
1
2
1
3
2
1
3
4
3
3
4
4
4
3
4
4
4
3
f
6
6
6
6
6
6
6
6
6
Step 3. All linear orderings which are completions of the ordering preferences
structure in fig. 1 on the following diagram are given (fig. 3)
Fig. 3
Example 2. A count of the number of all linear completions for ordering
preferences structure
Consider an ordering preferences structure in the fig. 1. To count all its linear
completions we need in construction of auxiliary graph γ only. Since each subset
299
Completions for Space of Preferences
which is a final vertex of graph γ consists of one element, it has single linear completion, hence we write 1 near every final vertex of graph γ (fig. 4). Further we write
a number N (B) near others vertexes B of graph γ in accordance with formula
(6). The number N (B) indicates a number of all linear completions for subset B.
In particularly, N (A) is a number of all completions for the set of all alternatives
A = {a, b, c, d, e, f }.
Fig. 4
References
Birkhoff, G. (1967). Lattice theory. Amer. Math. Soc., Coll. Publ., Vol. 25.
Rozen, V. (2011). Decision making under many quality criteria.. Contribution to game
theory and management, vol. 5. /Collected papers presented on the Fifth International
Conference Game Theory and management-SPb.: Graduate School of Management,
SPbU, 2012, p.257-267.
Rozen, V. (2013). Decision making under quality criteria (in Russian). Palmarium Academic Publishing. Saarbrucken, Deutschland.
Bridging the Gap between the Nash and Kalai-Smorodinsky
Bargaining Solutions
Shiran Rachmilevitch
Department of Economics, University of Haifa,
Mount Carmel, Haifa, 31905, Israel
E-mail: [email protected]
Web: http://econ.haifa.ac.il/∼shiranrach/
Abstract Bargaining solutions that satisfy weak Pareto optimality, symmetry, and independence of equivalent utility representations are called standard. The Nash (1950) solution is the unique independent standard solution and the Kalai-Smorodinsky (1975) solution is the unique monotonic
standard solution. Every standard solution satisfies midpoint domination on
triangles, or MDT for short. I introduce a formal axiom that captures the
idea of a solution being “at least as independent as the Kalai-Smorodinsky
solution.” On the class of solutions that satisfy MDT and independence
of non-individually-rational alternatives, this requirement implies that each
player receives at least the minimum of the payoffs he would have received
under the Nash and Kalai-Smorodinsky solutions. I refer to the latter property as Kalai-Smorodinsky-Nash robustness. I derive new axiomatizations of
both solutions on its basis. Additional results concerning this robustness
property, as well as alternative definitions of “at least as independent as the
Kalai-Smorodinsky solution” are also studied.
Keywords: Bargaining; Kalai-Smorodinsky solution; Nash solution.
1.
Introduction
Nash’s (1950) bargaining problem is a fundamental problem in economics. Its formal description consists of two components: a feasible set of utility allocations, each
of which can be achieved via cooperation, and one special utility allocation—the
disagreement point—that prevails if the players do not cooperate. A solution is a
function that picks a feasible utility allocation for every problem. The axiomatic
approach to bargaining narrows down the set of “acceptable” solutions by imposing
meaningful and desirable restrictions (axioms), to which the solution is required to
adhere.
Weak and common restrictions are the following: weak Pareto optimality—the
selected agreement should not be strictly dominated by another feasible agreement;
symmetry—if the problem is symmetric with respect to the 45◦ -line then the players
should enjoy identical payoffs; independence of equivalent utility representations—
the selected agreement should be invariant under positive affine transformations of
the problem. I will call a solution that satisfies these three restrictions standard.
Two additional restrictions that will be considered in the sequel are the following.
Midpoint domination requires the solution to provide the players payoffs that are at
least as large as the average of their best and worst payoffs. Every standard solution
satisfies midpoint domination on triangular problems, which are the simplest kind
of bargaining problems: each such problem is a convex hull of the disagreement
point, the best point for player 1, and the best point for player 2. Independence of
Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions 301
non-individually-rational alternatives requires the solution to depend only on those
options that provide each player at least his disagreement utility; in other words,
outcomes that clearly cannot be reached by voluntary behavior should not matter
for bargaining.
Two other known principles are independence and monotonicity. Informally, the
former says that if some options are deleted from a given problem but the chosen
agreement of this problem remains feasible (it is not deleted) then this agreement
should also be chosen in the problem that corresponds to the post-deletion situation; the latter says that if a problem “expands” in such a way that the set of
feasible utilities for player i remains the same, but given every utility-payoff for i
the maximum that player j 6= i can now achieve is greater, then player j should
not get hurt from this expansion. Within the class of standard solutions, these principles are incompatible: Nash (1950) showed that there exists a unique standard
independent solution, while Kalai and Smorodinsky (1975) showed that a different
solution is the unique standard monotonic one. The reconciliation of independence
and monotonicity in bargaining, therefore, is a serious challenge.
One possible response to this challenge is to give up the restriction to standard
solutions.1 Here, however, I consider only standard solutions. When attention is
restricted to standard solutions, the aforementioned challenge can, informally, be
expressed in the form of the following question:
How can we “bridge the gap” between the Nash and Kalai-Smorodinsky solutions?
Motivated by this question, I introduce an axiom that formalizes the idea of a solution being “at least as independent as the Kalai-Smorodinsky solution.” 2 I denote
this axiom by IIA KS.3 I also consider the following requirement, which refers
directly to both solutions: in every problem each player should receive at least
the minimum of the payoffs he would have received under the Nash and KalaiSmorodinsky solutions. I call this property Kalai-Smorodinsky-Nash robustness, or
KSNR. On the class of solutions that satisfy midpoint domination on triangles and
independence of non-individually-rational alternatives, IIA KS implies KSNR.
KSNR, in turn, captures much of the essence of the Nash and Kalai-Smorodinsky
solutions: the Nash solution is characterized by KSNR and independence and the
Kalai-Smorodinsky solution—by KSNR and monotonicity. Both characterizations
hold on the class of all solutions, not only the standard ones, not only the ones that
satisfy midpoint domination on triangles.4
The rest of the paper is organized as follows. Section 2 describes the model. The
axiom IIA KS is defined in Section 3. KSNR is introduced and discussed in Section 4. Section 5 elaborates on midpoint domination and its non-trivial connections
to the Nash and Kalai-Smorodinsky solutions. Section 6 considers an alternative
definition of “at least as independent as the Kalai-Smorodinsky solution.” Section 7
concludes with a brief discussion.
1
2
3
4
For example, the egalitarian solution (due to Kalai (1977)) satisfies all the above mentioned requirements except independence of equivalent utility representations. A formal
definition of this solution will be given in Section 4 below.
The precise meaning of this term will be given in Section 3.
The rationale behind this notation will be clarified in Sections 2 and 3.
The class of solutions that satisfy midpoint domination on triangles contains the class
of standard solutions.
302
2.
Shiran Rachmilevitch
Preliminaries
A bargaining problem is defined as a pair (S, d), where S ⊂ R2 is the feasible set,
representing all possible (v-N.M) utility agreements between the two players, and
d ∈ S, the disagreement point, is a point that specifies their utilities in case they do
not reach a unanimous agreement on some point of S. The following assumptions
are made on (S, d):
– S is compact and convex;
– d < x for some x ∈ S.5
Denote by B the collection of all such pairs (S, d). A solution is any function
µ : B → R2 that satisfies µ(S, d) ∈ S for all (S, d) ∈ B. Given a feasible set S,
the weak Pareto frontier of S is W P (S) ≡ {x ∈ S : y > x ⇒ y ∈
/ S} and the
strict Pareto frontier of S is P (S) ≡ {x ∈ S : y x ⇒ y ∈
/ S}. The best that
player i can hope for in the problem (S, d), given that player j obtains at least
dj utility units, is ai (S, d) ≡ max{xi : x ∈ Sd }, where Sd ≡ {x ∈ S : x ≥ d}.
The point a(S, d) = (a1 (S, d), a2 (S, d)) is the ideal point of the problem (S, d). The
Kalai-Smorodinsky solution, KS, due to Kalai and Smorodinsky (1975), is defined
by KS(S, d) ≡ P (S) ∩ [d; a(S, d)].6 The Nash solution, N , due to Nash (1950), is
defined to be the unique maximizer of (x1 − d1 ) × (x2 − d2 ) over Sd .
Nash (1950) showed that N is the unique solution that satisfies the following
four axioms, in the statements of which (S, d) and (T, e) are arbitrary problems.
Weak Pareto Optimality (WPO): µ(S, d) ∈ W P (S).
Let FA denote the set of positive affine transformations from R to itself.7
Independence of Equivalent Utility Representations (IEUR): f = (f1 , f2 ) ∈
FA × FA ⇒ f ◦ µ(S, d) = µ(f ◦ S, f ◦ d).8
Let π(a, b) ≡ (b, a).
Symmetry (SY): [π ◦ S = S]&[π ◦ d = d] ⇒ µ1 (S, d) = µ2 (S, d).
Independence of Irrelevant Alternatives (IIA): [S ⊂ T ]&[d = e]&[µ(T, e) ∈
S] ⇒ µ(S, d) = µ(T, e).
Whereas the first three axioms are widely accepted, criticism has been raised regarding IIA. The idea behind a typical such criticism is that the bargaining solution
could, or even should, depend on the shape of the feasible set. In particular, Kalai
and Smorodinsky (1975) noted that when the feasible set expands in such a way
that for every feasible payoff for player 1 the maximal feasible payoff for player 2
increases, it may be the case that player 2 loses from this expansion under the Nash
5
6
7
8
Vector inequalities: xRy if and only if xi Ryi for both i ∈ {1, 2}, R ∈ {>, ≥}; x y if
and only if x ≥ y & x 6= y.
Given two vectors x and y, the segment connecting them is denoted [x; y].
i.e., the set of functions f of the form f (x) = αx + β, where α > 0.
If fi : R → R for each i = 1, 2, x ∈ R2 , and A ⊂ R2 , then: (f1 , f2 ) ◦ x ≡ (f1 (x1 ), f2 (x2 ))
and (f1 , f2 ) ◦ A ≡ {(f1 , f2 ) ◦ a : a ∈ A}.
Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions 303
solution. Given x ∈ Sd , let giS (xj ) be the maximal possible payoff for i in S given
that j’s payoff is xj , where {i, j} = {1, 2}. What Kalai and Smorodinsky noted, is
that N violates the following axiom, in the statement of which (S, d) and (T, d) are
arbitrary problems with a common disagreement point.
Individual Monotonicity (IM):
[aj (S, d) = aj (T, d)]&[giS (xj ) ≤ giT (xj ) ∀x ∈ Sd ∩ Td ] ⇒ µi (S, d) ≤ µi (T, d).
Furthermore, they showed that when IIA is deleted from the list of Nash’s axioms
and replaced by IM, a characterization of KS obtains.9 Following Trockel (2009), a
solution that satisfies the common axioms—namely WPO, SY, and IEUR—will be
referred to in the sequel as standard.
Most solutions from the bargaining literature (standard or not) also satisfy the
following axiom, in the statement of which (S, d) is an arbitrary problem.
Independence of Non-Individually-Rational Alternatives (INIR): µ(S, d) =
µ(Sd , d).10
3.
Relative independence
Consider the following partial order on the plane, . Given x, y ∈ R2 , write x y
if x1 ≤ y1 and x2 ≥ y2 . That is, x y means that x is (weakly) to the north-west
of y.
Let µ be a solution and let (S, d) ∈ B be a problem such that µ(S, d) = KS(S, d).
Say that µ is at least as independent as KS given (S, d) if the following is true for
every (Q, e) ∈ B such that e = d, Q ⊂ S, and KS(S, d) ∈ Q:
1. KS(Q, e) KS(S, d) ⇒ KS(Q, e) µ(Q, e) KS(S, d), and
2. KS(S, d) KS(Q, e) ⇒ KS(S, d) µ(Q, e) KS(Q, e).
That is, µ is at least as independent as KS given (S, d) if in every relevant “subproblem” the solution point according to µ is between the solution point of KS
and the solution point “of IIA.” A solution, µ, is at least as independent as KS
if it is at least as independent as KS given (S, d), for every (S, d) such that
µ(S, d) = KS(S, d). Denote this property (or axiom) by IIA KS.
There is no shortage of standard solutions satisfying this property. Obviously,
KS is such a solution and every standard IIA-satisfying solution is such a solution. However, there are many others. For describing such a solution, the following
notation will be useful (it will also turn out handy in the next Section). For each
(S, d) ∈ B and each i, let:
mi (S, d) ≡ min{Ni (S, d), KSi (S, d)}.
9
10
In many places throughout the paper I refer to “individual monotonicity” and “independence of irrelevant alternatives” simply as “monotonicity” and “independence.” (see, e.g.,
the Introduction). The longer names for these axioms, and their respective abbreviations
IM and IIA, are presented here in order to distinguish them from other monotonicity
and independence axioms from the literature.
To the best of my knowledge, the earliest paper that utilizes this axiom is Peters (1986).
304
Shiran Rachmilevitch
Now consider the following solution, µ∗ :
∗
µ (S, d) ≡
m(S, d)
if N (S, d) = KS(S, d)
KS({x ∈ S : x ≥ m(S, d)}, m(S, d)) otherwise
It is easy to see that µ∗ is a standard solution which is at least as independent as
KS, it is different from KS, and it violates IIA.
The property IIA KS is, of course, not the only possible formalization of the
idea “at least as independent as KS.” Moreover, it is not immune to criticism. I
will discuss one of its drawback and propose an alternative formal definition for “at
least as independent as KS” later in the paper, in Section 6.
Finally, it is worth noting that one may very well question the validity of taking the monotonic solution and employing it as the “measuring stick” for the extant to which independence can be violated. Why not prefer the analogous criterion, where N is taken to be the measuring stick for the degree to which monotonicity can be compromised? In principle, this alternative approach expresses
a sensible consideration, but in practice it is problematic. To see this, consider
S ≡ conv hull{0, (0, 1), (1, 1), (2, 0)} and T ≡ {x ∈ R2+ : x ≤ (2, 1)}.11 Note that
when we move from S to T the feasible set “stretches” in the direction of coordinate
1 and hence, by IM, player 1 should not get hurt from this expansion. Accordingly,
an “at least monotonic as N ” relation would naturally impose that player 1’s benefit
from the change (S, 0) 7→ (T, 0) would be at least as large as the one he would have
obtained under N . However, even KS fails this test.
4.
Kalai-Smorodinsky-Nash robustness
Consider the following axiom, in the statement of which (S, d) is an arbitrary problem.
Midpoint Domination (MD): µ(S, d) ≥ 12 d + 12 a(S, d).
This axiom is due to Sobel (1981), who also proved that N satisfies it. The idea
behind it is that a “good” solution should always assign payoff that Pareto-dominate
“randomized dictatorship” payoffs. Anbarci (1998) considered a weakening of this
axiom, where the the requirement µ(S, d) ≥ 21 d + 12 a(S, d) is applied only to problems (S, d) for which S = Sd and is a triangle. I will refer to this weaker axiom
as midpoint domination on triangles, or MDT.12 It is easy to see that every standard solution satisfies MDT. Next, consider the following axiom, in the statement
of which (S, d) is an arbitrary problem.
Kalai-Smorodinsky-Nash Robustness (KSNR): µ(S, d) ≥ m(S, d).
KSNR implies MD since both N and KS satisfy MD.13 Therefore, the following
implications hold:
11
12
13
0 ≡ (0, 0).
Anbarci calls it midpoint outcome on a linear frontier.
It is straightforward that KS satisfies MD; as mentioned above, the fact that N satisfies
it was proved by Sobel (1981).
Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions 305
KSNR⇒ MD⇒ MDT.
(1)
Now recall that the big-picture goal in this paper is to offer a reconciliation of
monotonicity and independence within the class of standard solutions; this class, in
turn, is a subclass of the MDT-satisfying solutions, so MDT essentially expresses no
loss of generality for our purpose. With the additional (weak) restriction of INIR,
the following theorem says that the aforementioned reconciliation, as expressed by
IIA KS, implies KSNR.
Theorem 1. Let µ be a solution that satisfies independence of non-individuallyrational alternatives and midpoint domination on triangles. Suppose further that
it is at least as independent as the Kalai-Smorodinsky solution. Then µ satisfies
Kalai-Smorodinsky-Nash robustness.
Proof. Let µ satisfy INIR, MDT and IIA KS. Let (S, d) be an arbitrary problem. By INIR we can assume S = Sd . Let x ≡ µ(S, d), k ≡ KS(S, d), and
n ≡ N (S, d). We need to prove that xi ≥ min{ni , ki }. Let T = conv hull{d, (2n1 −
d1 , d2 ), (d1 , 2n2 − d2 )}. By MDT, µ(T, d) = KS(T, d) = N (T, d) = n. If k = n, then
by IIA KS it follows that x = k = n. Now consider k 6= n; wlog, k1 < n1 . In this
case IIA KS dictates that k x n, which implies that xi ≥ min{ni , ki }.
Let A = {INIR, MDT, IIA KS, KSNR} and let A denote a generic axiom in A.
Theorem 1 says that A =KSNR is implied by A\{A}. From (1) we see that the same
is trivially true for A =MDT. On the other hand, for A =IIA KS or A =INIR,
an analogous conclusion cannot be drawn.
Consider first A =IIA KS. Let Q ≡ conv hull{0, (0, 1), (1, 1), (2, 0)}, Q′ ≡
{x ∈ Q : x1 ≤ KS1 (Q, 0)}, and consider the following solution, µ∗∗ . For (S, d)
such that S = Sd = S0 , µ∗∗ (S, d) = N (Q′ , 0)(= (1, 1)) if S = Q′ and µ∗∗ (S, d) =
KS(S, d) otherwise; for other problems the solution point is obtained by a translation of d to the origin and deletion of non-individually-rational alternatives. It
is obvious that µ∗∗ satisfies MDT, INIR, and KSNR. However, in the move from
(Q, 0) to (Q′ , 0), µ∗∗ violates IIA KS.
Regarding A =INIR, consider the following solution, µ∗∗∗ :
µ∗∗∗ (S, d) ≡
N (S, d) if {x ∈ S : x < d} 6= ∅
KS(S, d) otherwise
It is immediate to see that µ∗∗∗ satisfies KSNR and MDT, and it is also not hard
to check that it also satisfies IIA KS.
Finally, A =KSNR is not implied by any strict subset of A\{A}. The egalitarian
solution (due to Kalai (1977)), E, which is defined by E(S, d) ≡ W P (Sd ) ∩ {d +
(x, x) : x ≥ 0}, satisfies IIA (and thereforeIIA KS) as well as INIR, but does
not satisfy MDT or KSNR. The Perles-Maschler solution (Perles and Maschler
(1981)), P M , is an example of a solution that satisfies MDT and INIR, but not
IIA KS or KSNR. For (S, d) with d = 0 and P (S) = W P (S), it is defined to be
Ru
R (a ,0) √
√
the point u ∈ P (S) such that (0,a2 ) −dxdy = u 1
−dxdy, where a = a(S, d);
for other problems it is extended by IEUR and continuity is an obvious fashion.
This is a standard solution, and therefore it satisfies MDT; in fact, it actually
306
Shiran Rachmilevitch
satisfies MD.14 It is easy to see that it violates IIA KS; to see that it violates KSNR, look at S ∗ = conv{0, (0, 1), ( 34 , 0), ( 34 , 14 )}: it is easily verified that
P M1 (S ∗ , 0) = 38 < 37 = KS1 (S ∗ , 0) < 36 = N1 (S ∗ , 0). Finally, the following solution, µ∗∗∗∗ , satisfies IIA KS and MDT, but not INIR or KSNR.15
µ
∗∗∗∗
(S, d) ≡
KS(S, d) if d ∈
/ intS
1
K(S,
d)
otherwise
2
KSNR captures much of the essence of the Nash and Kalai-Smorodinsky solutions.
This is expressed in the following theorems.
Theorem 2. The Nash solution is the unique solution that satisfies Kalai-SmorodinskyNash robustness and independence of irrelevant alternatives.
Theorem 3. The Kalai-Smorodinsky solution is the unique solution that satisfies
Kalai-Smorodinsky-Nash robustness and individual monotonicity.
The proofs of Theorems 2 and 3 follow from the combination of two results from
the existing literature. Before we turn to these results, one more axiom needs to be
introduced. This axiom, which is due to Anbarci (1998), has a similar flavor to that
of MD, but the two are not logically comparable.
Balanced Focal Point (BFP): If S = d + conv hull{0, (a, b), (λa, 0), (0, λb)} for
some λ ∈ [1, 2], then µ(S, d) = d + (a, b).16
The justification for this axiom is that the equal areas to the north-west and southeast of the focal point d+(a, b) can be viewed as representing equivalent concessions.
Similarly to MD, BFP is implied by KSNR and implies MDT:
KSNR⇒ BFP⇒ MDT.
(2)
The first implication is due to the fact that every standard solution satisfies BFP,
and both N and KS are standard. The second implication follows from setting
λ = 2 in BFP’s definition.
Anbarci (1998) showed that KS is characterized by IM and BFP. His work was
inspired by that of Moulin (1983), who in what is probably the simplest and most
elegant axiomatization of N , proved that it is the unique solution that satisfies
IIA and MD. Combining the results of Moulin (1983), Anbarci (1998), and the
implication KSNR⇒ [MD, BFP], one obtains a proof for the theorems.
It is worth noting that whereas the implication KSNR⇒ [MD, BFP] is true,
not only the converse is not true, but, moreover, even the combination of MD and
“standardness”(and, therefore, the combination of MD and BFP) does not imply
KSNR. The Perles-Maschler solution, P M , is an example.
14
15
16
See, e.g., Salonen (1985).
In the definition of this solution, int stands for “interior.”
Anbarci assumes the normalization d = 0; the version above is the natural adaptation
of his axiom to a model with an arbitrary d.
Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions 307
5.
Midpoint domination
Both N and KS are related to MD. Regarding N , we already encountered the
results of Moulin (1983) and Sobel (1981). Additionally, a related result has been
obtained by de Clippel (2007), who characterized N by MD and one more axiom—
disagreement convexity.17 As for KS, Anbarci (1998) characterized it by BFP and
IM, and we already noted the relation between BFP and MD—each can be viewed
as an alternative strengthening of MDT which is also a weakening of KSNR. More
recently, I characterized KS by MD and three additional axioms (see Rachmilevitch
(2013)). Finally, Chun (1990) characterized the Kalai-Rosenthal (1978) solution—
which is closely related to KS—on the basis of several axioms, one of which is a
variant of MD.
Below is an example for another non-trivial link between MD and N/KS. In it,
KSNR takes center stage.
The following family of standard solutions is due to Sobel (2001). For each
number a ≤ 1 corresponds a solution, which is defined on normalized problems—
(S, d) for which d = 0 and a(S, d) = (1, 1). Given a ≤ 1, this solution is:
1
1
1
W (S, a) ≡ arg maxx∈S [ xa1 + xa2 ] a .18
2
2
On any other (not normalized) problem, the solution point is obtained by appropriate utility rescaling. In light of the resemblance to the well-known concept from
Consumer Theory, I will call these solutions normalized CES solutions. Both N and
KS are normalized CES solutions: N corresponds to lima→0 W (., a) and KS corresponds to lima→−∞ W (., a).19 Thus, Sobel’s family offers a smooth parametrization
of a class of solutions, of which N and KS are special members. It turns out that
on this class KSNR and MD are equivalent.
Theorem 4. A normalized CES solution satisfies Kalai-Smorodinsky-Nash robustness if and only if it satisfies midpoint domination.
6.
An alternative “at least as independent as KS” relation
Recall the basic definition from Section 3: µ is at least as independent as KS given
(S, d) if for every relevant “sub-problem” (Q, e) the solution point according to µ is
between the solution point of KS and the solution point “of IIA.” Underlying this
definition is a notion of “betweenness,” a notion which is not immune to criticism.
Specifically, it suffers the following drawback. Consider the case where KS(Q, e) is
to the left (and north) of KS(S, d), µ(Q, e) is to the right (and south) of KS(S, d),
but the distance between µ(Q, e) and KS(S, d) is only a tiny ǫ > 0; namely, the
solution µ hardly changed its recommendation in the move from (S, d) to (Q, e),
17
18
19
See his paper for the definition of the axiom. de Clippel’s result is related to an earlier
characterization which is due to Chun (1990).
The maximizer is unique for a < 1; a = 1 corresponds to the relative utilitarian solution,
which (in general) is multi-valued.
Maximizing W (S, a) describes a well-defined method for solving arbitrary bargaining
problems, not necessarily normalized. In this more general case, lima→−∞ W (., a) corresponds to the egalitarian solution (Kalai (1977)). See Bertsimas et al (2012) for a recent
detailed paper on the matter.
308
Shiran Rachmilevitch
but this change is opposite in its direction to that of KS. According to the aforementioned definition, such a solution µ is not at least independent as KS given
(S, d). Therefore, it is not at least as independent as KS even if it coincides with
KS (or with an IIA-satisfying solution) on all other problems.
Thus, one may argue that an appropriate definition of “more independent than”
should not rely on direction (and hence should not rely on betweenness) and should
take distance into consideration. This definition, therefore, should be based on an
appropriate metric of “IIA violations.” Here is one such definition.
Say that µ is metrically at least as independent as KS given (S, d), if µ(S, d) =
KS(S, d) ≡ x and for every relevant sub-problem (Q, e) it is true that:
maxi∈{1,2} |µi (Q, e) − xi | ≤ maxi∈{1,2} |KSi (Q, e) − xi |.
Metrically at least as independent as KS means that the corresponding property
holds for every (S, d) such that µ(S, d) = KS(S, d). In Theorem 1, “at least as independent as KS” can be replaced by “metrically at least as independent as KS”
provided that attention is restricted solutions that (i) are efficient, and (ii) satisfy
the following axiom, in the statement of which (S, d) is an arbitrary problem.
Weak Contraction Monotonicity (WCM): For every i and every number r,
µi (V, d) ≤ µi (S, d), where V ≡ {x ∈ S : xi ≤ r}.20
Note that WCM is implied (separately) by IIA and by IM.
Theorem 5. Let µ be solution that satisfies weak Pareto optimality, midpoint domination on triangles, independence of non-individually-rational alternatives, and
weak contraction monotonicity. Suppose that µ is metrically at least as independent as the Kalai-Smorodinsky solution. Then µ satisfies Kalai-Smorodinsky-Nash
robustness.
Finally, we also have the following result.
Theorem 6. A normalized CES solution that satisfies weak contraction monotonicity also satisfies Kalai-Smorodinsky-Nash robustness.21
7.
Discussion
Motivated by the goal to reconcile independence and monotonicity in bargaining
within the class of standard solutions, I have introduced the requirement that the
bargaining solution be “at least as independent as KS.” That is, I took the monotonic standard solution as the measuring stick for how far one can depart from
independence.
Weaker versions of IIA have previously been considered in the literature. The
best known axiom in this regard is Roth’s (1977) independence of irrelevant alternatives other than the disagreement point and the ideal point, which applies IIA only
to pairs of problems that, in addition to a common disagreement point, share the
20
21
Implicit here is the assumption that (V, d) ∈ B. Obviously, V = ∅ for a sufficiently small
r.
It is an open question whether the converse is true; namely, whether WCM and KSNR
are equivalent on the class of normalized CES solutions.
Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions 309
same ideal point. Thomson (1981) generalized Roth’s axiom to a family of axioms,
parametrized by a reference function. Such an axiom applies IIA only to pairs of
problems that share the same reference-function-value. Though these type of weakening of IIA are more common in the literature, they seems not be a good fit for the
task of reconciling independence and monotonicity: Roth’s axiom seems too weak
and Thomson’s seems too abstract. The property “at least as independent as KS,”
or IIA KS, though having a somewhat special structure, seems more well-suited
for the goal of the current paper.
This property, in turn, when considered in combination with MDT and INIR (a
combination which is almost without loss of generality within the class of standard
solutions), implies KSNR. KSNR, in turn, is of interest in its own right. On its basis
I have derived axiomatizations of the two solutions: N is characterized by KSNR
and IIA and KS is characterized by KSNR and IM. These new characterizations are
not a consequence of the known theorems of Nash (1950) and Kalai-Smorodinsky
(1975), since they hold on the entire class of bargaining solutions, not only on the
class of the standard ones. Note also that the fact that a solution satisfies KSNR
does not imply that it is standard. For example, the following is a non-standard solution that satisfies KSNR: E KSN R (S, d) ≡ m(S, d) + (e, e), where e is the maximal
number such that the aforementioned expression is in S.
Appendix
Proof of Theorem 4 : It is known that MD is equivalent to a ≤ 0 (see Sobel (1981)).
I will prove that KSNR is also equivalent to a ≤ 0. It is enough to prove this
equivalence for a < 0 (because a = 0 corresponds to the Nash solution). For simplicity (and without loss), I will consider only (normalized) strictly comprehensive
problems—those for which S = {(x, f (x)) : x ∈ [0, 1]}, where f is a differentiable
strictly concave decreasing function; the case of an arbitrary problem follows from
standard limit arguments.
The parameter a corresponds to the solution
1
1
1
W (S, a) ≡ arg maxx∈S0 [ xa1 + xa2 ] a .
2
2
In the case of a normalized strictly comprehensive problem, the object of interest is
1
1
1
arg max0≤x≤1 [ xa + f (x)a ] a ≡ W (x, a).
2
2
d
Therefore dx
W (x, a) = e[ 2a xa−1 + a2 f (x)a−1 f ′ (x)], where e is a shorthand for a
strictly positive expression.
At the optimum, the derivative of this expression is zero (i.e., an FOC holds).
[
f (x(a)) 1−a
]
= −f ′ (x(a)),
x(a)
where x(a) is player 1’s solution payoff given the parameter a. The derivative of the
RHS with respect to a is −f ′′ (x(a))x′ (a), hence the sign of x′ (a) is the same as the
sign of the derivative of the LHS. To compute the latter, recall the formula
Ψ ′ (a) = Ψ (a) × {h′ (a)log[g(a)] +
h(a)g ′ (a)
},
g(a)
310
Shiran Rachmilevitch
where Ψ (a) ≡ [g(a)]h(a) .
Taking g(a) ≡ f (x(a))
x(a) and h(a) ≡ 1 − a, we see that the signs of the derivative
of the LHS is the same as the sign of
−log[
(1 − a)
f (x(a))
f (x(a))
]+
x′ (a)(f ′ (x(a))x(a)−f (x(a))) ≡ −log[
]+Zx′ (a),
x(a)
f (x(a))x(a)
x(a)
where Z is a shorthand for a negative expression. Therefore, the sign of x′ (a) is the
′
same as that of −log[ f (x(a))
x(a) ] + Zx (a).
Let k ≡ KS1 (S, 0) and n ≡ N1 (S, 0).
f (x(a))
Case 1: x(a) < k. In this case, f (x(a))
x(a) > 1, hence −log[ x(a) ] < 0. This means
that x′ (a) < 0. To see this, assume by contradiction that x′ (a) ≥ 0. This means
′
′
that sign[−log[ f (x(a))
x(a) ] + Zx (a)] = −1 = sign[x (a)], a contradiction. Therefore
n < x(a) < k, so KSNR holds.
Case 2: x(a) > k. In this case, f (x(a))
< 1, hence −log[ f (x(a))
x(a)
x(a) ] > 0. This
′
means that x (a) > 0. To see this, assume by contradiction that x′ (a) ≤ 0. This
′
′
means that sign[−log[ f (x(a))
x(a) ]+ Zx (a)] = 1 = sign[x (a)], a contradiction. Therefore
k < x(a) < n, so KSNR holds. ⊓
⊔
Proof of Theorem 5 : Let µ be a solution that satisfies the axioms listed in the
theorem and let (S, d) ∈ B. Assume by contradiction that KSNR is violated for
(S, d). Wlog, we can assume that Sd is not a rectangle and that d = 0. Let T ≡
conv hull{(2n1 , 0)(0, 2n2 ), 0}, where n ≡ N (S, 0). By MDT, µ(T, 0) = N (T, 0) =
KS(T, 0) = n. Let k ≡ KS(S, 0). Wlog, suppose that x1 < min{n1 , k1 }. Let simply
denote the “metrically at least as independent as KS” relation by . Note that
because of , n1 = k1 is impossible. Therefore n1 6= k1 .
Case 1: k1 < n1 . Here we have |x1 − n1 | > |k1 − n1 | and hence, because of ,
|k2 − n2 | > |k1 − n1 |.
Case 1.1: |x1 − n1 | ≥ |x2 − n2 |. In this case |k2 − n2 | ≤ |x2 − n2 | ≤ |x1 − n1 | ≤
|k2 − n2 |. The first inequality follows from the fact that x (which, by WPO, is on
the boundary) is to the north-west of k, the second inequality is because we are in
Case 1.1, and the third is by .
Case 1.2: |x1 − n1 | < |x2 − n2 |. By , |x2 − n2 | ≤ |k2 − n2 |.
indent Whatever the case—1.1 or 1.2—it follows that |x2 − n2 | = |k2 − n2 |, which
is impossible, since k is in the relative interior of P (S).
Case 2: k1 > n1 . Let r ∈ (n1 , k1 ) be such that KS1 (V, 0) < n1 , where V ≡
{x ∈ S : x1 ≤ r}. Invoking WCM and applying the arguments from Case 1 to V
completes the proof. ⊓
⊔
Proof of Theorem 6 : I will prove that if a normalized CES solution violates KSNR—
that is, if its defining parameter is a > 0—then it also violates weak contraction
monotonicity. Fix then 0 < a < 122 and let µ denote the corresponding solution.
22
The case a = 1 can be easily treated separately (i.e., it is easy to show that the relative
utilitarian solution violates WCM; for brevity, I omit an example); focusing on a < 1
allows for “smooth MRS considerations” as will momentarily be clear.
Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions 311
Claim: There is a unique k = k(a) ∈ (0, 1) such that:
(3)
1 − k = k 1−a .
Proof of the Claim: The RHS is decreasing in k, the LHS is increasing, and opposite
strict inequalities obtain at k = 0 and k = 1. QED
From here on, fix k = k(a).
Consider S = conv hull{0, (1, 0), (0, 1), (1, k)}. Due to (3), a and k are such that
the slope of the line connecting (0, 1) and (1, k) (i.e., the slope of the strict Pareto
frontier of S) is |k − 1|, which equals the “MRS” of the objective that µ maximizes,
when this MRS is evaluated at the corner (1, k). Therefore, µ(S, 0) = (1, k).
Now let t ∈ (0, 1). Note that (t, (k−1)t+1) ∈ P (S). Let us “chop” S at the height
(k − 1)t + 1; namely, consider V = V (t) ≡ {(a, b) ∈ S : b ≤ (k − 1)t + 1}. To show
a violation of WCM, I will find a t ∈ (0, 1) for which µ2 (V (t), 0) > k = µ2 (S, 0).
To this end, let us consider the normalized feasible set derived from V = V (t),
b
) : (a, b) ∈ V (t)}. By
call it Q = Q(t). Given t ∈ (0, 1), Q = Q(t) = {(a, (k−1)t+1
SINV, µ2 (Q(t), 0) =
µ2 (V (t),0)
(k−1)t+1 ,
or µ2 (V (t), 0) = [(k − 1)t + 1]µ2 (Q(t), 0). Therefore,
k
I will show that [(k − 1)t + 1]µ2 (Q(t), 0) > k, or µ2 (Q(t), 0) > (k−1)t+1
. Note that
k
the south-east corner of P (Q) is (1, (k−1)t+1 ). Therefore, it is enough to show that
the MRS at this corner is smaller than the slope of P (Q). Namely, that
[
(k − 1)t + 1 a−1
1−k
]
<
,
k
(k − 1)t + 1
for a suitably chosen t. Note that for t ∼ 1 the LHS is approximately one, so we are
1
1
done if 1 < 1−k
k , or k < 2 . Hence, it is enough to prove that k(a) < 2 for a > 0. To
this end, let us re-write (3) as
G ≡ k 1−a + k − 1 = 0.
Since k(0) = 12 , it is enough to prove that k ′ < 0. By the Implicit Function Theorem,
∂G
∂G
−a
the sign of k ′ is opposite to the sign of [ ∂G
+1 > 0
∂k ]/[ ∂a ]. Note that ∂k = (1−a)k
and that the sign of
∂G
∂a
is the same is that of
∂logk1−a
∂a
= −logk > 0. ⊓
⊔
Acknowledgments: I am grateful to Nejat Anbarci, Youngsub Chun, Uzi Segal,
Joel Sobel, William Thomson, and an anonymous referee for helpful comments.
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Unravelling Conditions
for Successful Change Management
Through Evolutionary Games of Deterrence
Michel Rudnianski1 and Cerasela Tanasescu2
LIRSA, CNAM, Paris
E-mail: [email protected]
2
ESSEC Business School
Av. B. Hirsch, 95000 Cergy Pontoise, France
E-mail: [email protected]
1
Abstract The paper proposes analyze the conditions for successful change
management requiring information transmission and transformation of the
information received into change implementation. To that end, starting from
an elementary standard matrix game considering only information transmission, the paper will extend the case by considering that stakeholders have to
simultaneously take decisions concerning the two above dimensions. A dynamic approach supported by the Replicator Dynamics model will then be
proposed, aiming at analyzing asymptotic behaviors. The difficulties often
met when trying to solve differential systems will be pointed out. Therefore
a new method will be developed, leaning on a bridging in the evolutionary context between standard games and a particular type of qualitative
games, called Games of Deterrence, and which object is to analyze strategies playability. Through the equivalence between the two types of games,
the methodology will enable to remove some question marks in the analysis of asymptotic behaviors, thus contributing to a better understanding of
conditions fostering change pervasion, and in particular of the role played
by incentives.
Keywords: change, deterrence, evolution, incentives, playability, Replicator
Dynamics, stability.
1.
Introduction
The ever increasing pace of ICT development and globalization generates a dramatic
shortening of the products’ life cycle, which in turn decreases the possibility of sustainable competitive advantages for the firms. Richard D’Aveni (D’Aveni, 1994) has
analyzed this phenomenon distinguishing different arenas of hyper-competition. One
of the major elements highlighted is the core capacity of managing breakthroughs, in
particular through the mastering of timing and knowhow associated with products
and services. In this respect, there is no doubt that change management is a core
competency for a firm which aim is sustainable development. Now implementation of
change management may present a variety of difficulties, among which reluctance to
share information and to change (Fichman and Kemener, 1997; Rogers, 1983). The
works developed in Experimental Psychology, and especially Kahneman and Tversky’s Prospect Theory (Kahneman and Tversky, 1979), have highlighted decision
biases like anchoring, procrastination, sensitivity to loss, stubbornness, mirroring,
or status quo. All of them may lead the individual under consideration to take
inappropriate decisions, which most of the time have as hidden objective to comfort his/her position and hence not accept to change. Whence the necessity for the
314
Michel Rudnianski, Cerasela Tanasescu
firm’s management to develop an accurate cost-benefit analysis of change versus status quo for each decision maker. As a result of this analysis the firm’s management
may decide to allocate incentives to the personnel concerned.
A game theoretic model of the issues at stake has already been developed, considering a firm structured in departments, each one having a relative autonomy in terms
of information sharing and change adoption (Rudnianski and Tanasescu, 2012). This
model has considered several assumptions about the consequences for a department
of receiving information relevant from the company’s global perspective. The starting point was to consider that a department i can decide to send or not to send to a
neighboring department j an information pertaining to a possible change in the conduct of affairs. Similarly, department j, when receiving the information, may decide
or not to act accordingly and especially to implement change that might be triggered
by the information received. At the most elementary level, the problematic can be
analyzed through a series of standard 2x2 games in which the players’ strategic sets
could refer, either to information sending or to change adoption. Various cases have
been considered, depending on the respective values for each player of costs, incentives and benefits received from the company’s general management. At a second
level, the model used the Replicator Dynamics to define conditions under which
cooperation between connected departments can prevail. At a third level, the initial
model was extended to matrix games in which each party should simultaneously consider whether to send information or not, and whether to adopt changes stemming
from the information received or not. A general evolutionary analysis of these games
was not performed due to the difficulties to find an analytical solution of the dynamic
system. Now this obstacle can be removed, thanks to the results recently found by
Ellison and Rudnianski, about the existence of equivalence relations between standard quantitative games and a particular type of qualitative games called Games
of Deterrence (Ellison and Rudnianski, 2009; Ellison and Rudnianski, 2012). More
precisely these equivalence relations enable to translate standard evolutionary games
into evolutionary Games of Deterrence which display identical asymptotic properties. In turn, it has been shown (Ellison and Rudnianski, 2012) that the asymptotic
properties of these Evolutionary Games of Deterrence may be derived from the
playability properties of the players’ strategies in the corresponding matrix Games
of Deterrence. There is then no need to solve the original dynamic system.
On these bases, the present paper will in a first part recall the results available
in the analysis of conditions required for successful change management through the
standard game theoretic approach. In a second part, after having recalled the core
properties of matrix Games of Deterrence, the paper will develop the equivalences
between evolutionary standard games and evolutionary Games of Deterrence. A
third part will then use these equivalences to analyze the conditions of success in
non-elementary issues of change pervasion. In particular, success of change pervasion
will be associated with the playability properties of the Games of Deterrence under
consideration.
2.
Conditions for successful change management through the standard
game theoretic approach
The present global context can be characterized as highly dynamic with high failure
rates. The continuous increase in the rythm of technological innovations translates
into a dramatic shortening of products life cycles and a higher frequency of orga-
Unravelling Conditions for Successful Change Management
315
nizational change. One of the consequences being that the windows of opportunity
to make profit from innovation and change open more frequently, but for a shorter
time.
In order to overcome these difficulties one idea could be to develop a set of
incentives, such that the personnel of the organization accepts and contributes to
the implementation of the change.
In this section we start with an elementary model which will help to better
understand the context, and then we shall develop the general model of information
exchange between two departments.
2.1.
An introductive elementary example
Let us consider two departments i and j of the firm such that each one can decide
to send (S) or not to send (S) information to the other. Let us furthermore assume
that for each of the two departments:
– receiving information generates a profit of 3
– sending information generates a cost of 2.
The question is : should a department send information (strategy S) or not
(strategy S)? To find the answer, one may resort to a game theoretic approach
characterized by the matrix hereunder:
S - sending
S - not sending
S - sending S - not sending
(1,1)
(-2,3)
(3, -2)
(0,0)
Fig. 1
It can be easily seen that the game displays a unique Nash equilibrium (S, S).
In other words the example shows that despite the fact that the two departments
would benefit from information exchange, this exchange cannot occur. In fact, this
paradoxical conclusion just reflects the fact that the game is a Prisonner’s Dilemma.
Whence the perpetual question: what conditions could make cooperation prevail and
result into information exchange between the two parties?
A possible answer is to develop a set of incentives that will push each department
to send information to the other one. In our example, if the company rewards the
sending of information by an incentive of 3, the game is then represented by the
matrix hereunder:
S
S
S (4,4) (1,3)
S (3, 1) (0,0)
Fig. 2
The unique Nash Equilibrium here is (S, S). Thus by the use of incentives the
company is sure that information exchange will pervade.
2.2. General case: results from a standard approach
Let us generalize the case here above by considering a company’s department with
k employees, and which wants to implement a change possibly stemming from a
technological innovation. Let us furthermore assume that :
316
Michel Rudnianski, Cerasela Tanasescu
– the department’s incentives policy is decided by its manager
– the case is symmetric: benefits, incentives and costs resulting from information
sharing and change adoption are the same for all employees.
– success is an increasing function of information sharing and the resulting change
adoption
– the value for the department of change adoption by k employees is an increasing
function of k.
Each employee i generating change or possessing relevant information about
such change can decide to send or not to send this information to other employees.
Likewise each employee i receiving information from employee j can decide to adopt
or not to adopt the change made possible through reception of this information.
For each employee i there are four possible sets of actions, which are given by
the table hereunder:
S S
A SA SA
A SA S A
Fig. 3
For employee i, the results of the different possible interactions with employee
j are given by the following table:
Employee i
Benefit Cost Incentive
Sending information to j
bs
cs
βs
n
n
Not sending information to j
b
c
βn
a
a
Adopting change
b
c
βa
Not adopting change
0
0
0
Fig. 4
This means that for employee i, the payoff resulting from:
– sending an information to another employee, is: s = β s + bs − cs
– not sending information, is: n = bn − cn
– adopting change made possible by the information received from another employee, is:a = (β n + bn − cn ). In case employee j doesn’t send information, the
result for employee i of adopting is the same than the result of not adopting.
– not adopting change made possible by the information received from another
employee, is 0. Of course this is just an assumption. One could consider situations in which not adopting is associated with a non-zero payoff, for instance a
negative one, meaning that by adopting this attitude employee i is a loser.
We shall follow two approaches:
– the static one based on non-repeated standard matrix games
– the dynamic one based on the Replicator Dynamics.
2.3.
Static Approach
The starting point is the matrix of table 2.3.
Unravelling Conditions for Successful Change Management
317
SA
SA
SA
SA
SA (a+s,a+s) (a+s,s) (s, a+n) (s,n)
SA (s,s+a)
(s,s) (s, a+n) (s,n)
SA (a+n,s) (a+n,s) (n,n) (n,n)
SA
(n,s)
(n,s)
(n,n) (n,n)
Fig. 5
Conditions
n < s < n+a < s+a
A1
A2
A3
n < n+a < s < s+a
a>0
s < s+a < n < n+a
A4
B1
s <n<s+a<n+a
n+a < s+a < n < s
B2
n+a<n<s+a<s
B3
a<0
B4
s+a<s<n+a<n
s+a < n+a < s < n
Nash Equilibria
s>n
(SA,SA)
s < n (SA,SA), (SA,SA),(SA,SA),(SA,SA)
s>n
( SA, SA)
s < n (SA,SA), (SA,SA),(SA,SA),(SA,SA)
Fig. 6
To analyze the corresponding game, we need to compare and order the values of n,
s, n + a and s + a. It can be easily seen that 8 cases need to be distinguished, each
one associated with a specific set of Nash equilibria (see table 2.3. here below):
It follows from the above table that change will pervade if the following two quite
common sense conditions are satisfied :
– Adoption provides a benefit
– Sending information relative to change provides a payoff superior to the one
stemming from not sending that information.
Thus the value of adoption should be positive and independent from the decision
to send or not to send information.
2.4. Recalling the core properties of the Replicator Dynamics
The Replicator Dynamics is a classical dynamic system describing the evolution
of a population broken down into several species. The outcome of the interaction
between two individuals is given by a symmetric matrix game G.
Let us consider a population comprised of n species 1, 2, . . . , n, each one characterized by a particular behavior. Let θ = (θ1 , θ2 , . . . , θn ) define the population’s
profile, i.e. the proportion of each species in the population. Individuals may interact, whether they belong to the same species or not. The payoffs resulting from
these interactions are given by a matrix. Thus, with any pair (i, j) of interacting
individuals, one can associate a pair (uij , vij ) of payoffs. Let furthermore:
P
– ui = θk uik define the fitness of species i
k
P
– uT = θi ui define the fitness of the population.
i
The Replicator Dynamics is then defined by the following system of differential
equations:
∀i ∈ {1, 2, . . . , n}, θi′ = θi (ui − uT )
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Michel Rudnianski, Cerasela Tanasescu
According to this system, the evolution of the proportion θi of species i in the
entire population depends on its fitness with respect to the population’s fitness: if
the fitness of i is greater than the population’s fitness, then the proportion of species
i in the population will increase, while on the opposite if i’s fitness is smaller than
the average fitness of the population, then the proportion of species i will decrease.
It can be seen from the above set of equations that he Replicator Dynamics does
not take into account the possibility for new species to emerge during the evolution.
This means between other things that all species present at time t, whatever that
time is, were already present in the population at initial time.
Last, let us note that if θ represents the space of population’s profiles, and f is
a vector field on θ such that θ = f (θ) with fi (θ) = θi (ui − uT ), an equilibrium of
the Replicator Dynamics is then defined as a fixed point of f .
2.5. The dynamic approach of change
It then stems from table 2.3. that the average payoffs associated with the various
strategies
are given by the following set of equations :

u
= a(θSA + θSA ) + s

SA



 uSA = s
uSA = a(θSA + θSA ) + n


uSA = n



uT = a(θSA + θSA )(θSA + θSA ) + s(θSA + θSA ) + n(θSA + θSA )
Whence:

uSA − uT = a(θSA + θSA )(θSA + θSA ) + (s − n)(θSA + θSA )



uSA − uT = −a(θSA + θSA )(θSA + θSA ) + (s − n)(θSA + θSA )
u

SA − uT = a(θSA + θSA )(θSA + θSA ) − (s − n)(θSA + θSA )


uSA − uT = −a(θSA + θSA )(θSA + θSA ) − (s − n)(θSA + θSA )
The above system of equations enables to categorize the cases to be considered,
as indicated on table 2.5. hereunder.
A1,A2
s>n
A3,A4
B1,B2
a>0
B3,B4
a<0
uSA − uT uSA uSA − uT uSA − uT
+
?
?
−
s<n
s>n
?
?
−
+
+
−
?
?
s<n
−
?
?
+
Fig. 7
So we see that the relatively coarse granularity of the available information does
not enable to determine precisely the evolution of behaviors.
3.
The Games of Deterrence approach
Games of Deterrence consider only two possible states of the world:
– those which are acceptable for the player under consideration (noted 1)
– those which are unacceptable for that same player (noted 0)
Each player’s objective is to be in an acceptable state of the world. Therefore, Games
of Deterrence will not look for optimal strategies but for strategies that the player
under consideration can play, and which will therefore be called playable strategies.
Unravelling Conditions for Successful Change Management
319
3.1. Recalling core properties of matrix Games of Deterrence
Let E and R be two players with respective strategic sets SE (|SE | = n) and SR
(|SR | = p). Given any strategic pair (i, k) ∈ SE × SR , let:
– (uik , vik ) be the corresponding binary outcome pair
– U and V be the sets of binary outcome pairs of the two players associated with
the set of possible strategic pairs
A strategy i of E is said to be safe iff ∀k ∈ SR , uik = 1. A strategy which is not
safe will be termed dangerous.
Given i ∈ SE , let JE (ei ) be strategy i’s positive playability index defined as
follows:
– If i is safe, then JE (i) = 1
Q
Q
– If not JE (i) = (1−jE )(1−jR )
[1−JR (k)(1−uik )], with jE =
(1−JE (i))
i∈SE
k∈SR
Q
and jR =
(1 − JR (k))
k∈SR
If JE (i) = 1, strategy i ∈ SE is said to be positively playable. If there are no
positively playable strategies in SE , that is if jE = 1, all strategies i ∈ SE are said
to be playable by default. A strategy in SE ∪ SR is playable iff it is either positively
playable or playable by default.
The system P of all equations of JE (i), i ∈ SE , JR (k), k ∈ SR , jE and jR is called
the playability system of the game. The playability system P may be considered as
a dynamic system J = fˆ(J) on the playability set. A solution of the matrix Game
of Deterrence is a fixed point of fˆ. It has been shown (Fichman and Kemener, 1997)
that any matrix Game of Deterrence has at least one solution.
Given a strategic pair (i, k) ∈ SE × SR , i is said to be deterrent vis-a-vis k iff the
three following conditions apply:
– i is playable
– vik = 0
– ∃k ′ ∈ SR : JR (k ′ ) = 1
It has been shown (Fichman and Kemener, 1997) that a strategy k ∈ SR is
playable iff there is no strategy i ∈ SE deterrent vis-a-vis k. Thus, the study of
deterrence amounts to analyzing the strategies’ playability properties
A symmetric Game of Deterrence is a Game of Deterrence (SE , SR , U, V ) such that
2
SE = SR and U = V t (i.e. ∀(i, k) ∈ SE
, uik = vki ). In the case of symmetric games,
the strategic set will be noted S.
A symmetric solution is a solution in which ∀i ∈ S, JE (i) = JR (i). It has been
shown (D’Aveni, 1994) that in a symmetric Game of Deterrence, jE = jR
3.2. Evolutionary Games of Deterrence
It has been shown (Ellison and Rudnianski, 2009) that for a symmetric Game of
Deterrence G with playability system P and Replicator Dynamics D(G), if:
– P has a symmetric solution for which no strategy is playable by default
– at t = 0, the proportion of each positively playable strategy is greater than the
sum of the proportions of the non-playable strategies,
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Michel Rudnianski, Cerasela Tanasescu
then:
– the proportion of each non-playable strategy decreases exponentially towards
zero
– the proportion of each playable strategy has a non-zero limit
This result can be interpreted as follows: each symmetric solution of the playability
system is associated with an Evolutionarily Stable Equilibrium Set of the Replicator
Dynamics, i.e. a set of equilibria such that the union of the attraction basins of all
the equilibria is a neighbourhood of the set.
3.3.
Bridging Games of Deterrence with standard games
Two strategies i and j are equivalent if ∀k ∈ S, uik = ujk .
If i and j are equivalent, then:
1. θθji is constant in very solution of the Replicator Dynamics
2. i and j have the same playability in every solution of the playability system.
The case analyzed in section 2.2. has shown that the analysis of standard evolutionary games properties may not always be successful, while on the opposite
section 3.2. here above has displayed a significant set of results concerning evolutionary properties of Games of Deterrence. Moreover a correspondance has been
established between evolutionary standard games and evolutionary Games of Deterrence (Ellison and Rudnianski, 2012).
More precisely, let:
e be a symmetric matrix game
– Let G
e
– M and m be respectively the maximal and minimal payoffs in G
e by:
– G the game derived from G
• applying the affine transformation which replaces every payoff u by (u −
m)/(M m).
• for any strategic pair (p, q) such that vp q = z with 0 < z < 1, splitting
species p into two sub-species p1 and p2 differing only by the fact that
vp1 q = 1 whilevpq = 0, and with respective proportions z and 1 − z within
species p.
It has been proved that [ibid] that if the solution set of G has the properties described
in section 3,2 here above, then:
e corresponding to a non-playable strategy
– the proportion of each strategy of G
in G decreases exponentially towards 0
e corresponding to a playable strategy in G
– the proportion of each strategy of G
has a non-zero limit.
e can be deIn other words the asymptotic properties of the dynamic system D(G)
termined through analyzing the playability system associated with G.
Unravelling Conditions for Successful Change Management
4.
321
Application to change pervasion
Let us transform the standard game matrix of Fig. 5 into a binary matrix on the
basis of the affine transformation introduced in section 3.3. Between other things,
this transformation enables to gather the 8 cases of table 6 into 4 groups of two,
each one being characterized by a specific pair (minimum, maximum), as shown on
table 8.
Let us consider two cases belonging to the same group, for instance A1 and A2.
The minimum and the maximum being the same in the two cases, after application
of the affine transformation, the distribution of 0s and 1s will also be the same.
What may differ is the distribution of non-binary payoffs. But it stems from the
method defined in section 3.3, that the strategy of the other player associated with
each non binary number can be replaced by two strategies generating respectively
payoffs 1 and 0, and differing only by their proportion. As this proportion doesn’t
intervene in the resulting binary matrix supporting the game of deterrence, the matrix will be the same in the two cases, and so will be the conclusions concerning the
strategies’ playabilities. Now such invariance property doesn’t apply to two cases
such that the minimum in one case is the maximum in the other case and vice-versa.
Indeed, by definition, unlike for two cases belonging to the same group, the 0s and
1s in the matrices will not be the same. Thus if we consider for instance cases A1
and B3, we see that the payoff pair associated with strategic pair (SA, SA) is (1,
1) in case A1 and (0,0) in case A3. So, on the whole we have to analyze four cases
corresponding each one to a specific pair of extremal values.
CASE A1, A2: a > 0 and s > n
Let us for instance consider the case A1 as defined on table 2.3.. The maximum
payoff is s + a, and the minimum payoff is n. We decrease each member of the
matrix by n thus fixing the minmum to 0. If we apply the affine transformation and
let x = a/(a + s − n) and y = (s − n)/(a + s − n), we then get the matrix of figure
4. hereunder.
Fig. 8
Let us first consider the strategic pair (SA, A). Here z = y with 0 < y < 1.
Everything else being the same, we can then split species SA into two sub-species
SA1 and SA2 such that the respective proportions of sub-species SA1 and SA2 in
SA are y and 1y. Let us then consider the strategic pair (SA, SA). Here z = x with
0 < x < 1. We now can split SA1 into SA11 and SA12, and SA2 into SA21 and
SA22.
Likewise:
– SA can be replaced by SA11, SA12, SA21, SA22
– SA can be replaced by SA1, SA2
– SA can be replaced by SA1, SA2
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Michel Rudnianski, Cerasela Tanasescu
Fig. 9
Whence the new binary matrix:
It can be established that the game displays a solution in which strategies SA11 ,
SA12 , SA21 , SA22 are positively playable, and all other strategies are non playable.
Hence, if at t = 0, the proportion of each positively playable strategy is greater than
the proportions of the positively playable strategies tend toward a non âĂŞzero
limit, while the proportions of the non-playable strategies tend toward 0. In terms
of change analysis, this means that change will pervade through selection by more
and more employees of the decisions to send information and to adopt the changes
possibly stemming from information received. Thus the conclusion confirms the one
already reached with the standard approach, according to which the proportion of
employees who will send and adopt increases, while the proportion of those who
neither send nor adopt decreases. But the conclusion goes one step further. While
the standard approach led to a question mark with respect to the two intermediate
behaviors (send but not adopt and not send but adopt), we see here that, every
thing else being the same, those two behaviors will vanish with time.
CASE A3,A4: a > 0 and n > s
Let us for instance consider case A3.. By applying the same method than in case
A1, we get the matrix of figure 4.:
Fig. 10
In turn, by breaking down the species into sub-species when necessary, we get
the matrix of figure 4.:
Fig. 11
Unravelling Conditions for Successful Change Management
323
It can be seen that the game of deterrence associated with the above matrix
displays a solution in which SA1 and SA2 are positively playable and all other
strategies are not playable. If at initial the proportions of SA1 and SA2 are greater
than the sum of proportions of the non playable strategies, then it stems from the
properties recalled in section 3.2, that the dynamics tends toward a limit for which
all employees will decide to adopt but not to send. If we compare that conclusion,
with the one obtained through the standard approach, we see that the game of
deterrence approach has enabled to remove the question marks about the evolution.
The result seems to be paradoxical at first sight. Indeed how can people decided
not to send information, when they want in turn to adopt the changes possibly
stemming from the information they receive?
The answer simply stems from the characteristics of the case: while change adoption brings a benefit (a > 0), the payoff resulting from not sending information is
greater than the one resulting from sending. Somehow, the structure considered for
the model provides an arbitration between two possible results. Undoubtedly, to
foster change adoption, the management plays a crucial role. Indeed by allocating
appropriate incentives to the employees when they send information, the management can efficiently change the attitude of the personnel in that respect, with the
result that the situation switches from case A3 to case A1, for which change is
adopted.
CASE B1,B2: a < 0 and n > s
Adopting the previous notations and proceeding to the affine transformation
leads to the following matrix (Figure 4. ).
Fig. 12
In turn, by breaking down the species when necessary, we get the matrix of
figure 4. hereunder.
Fig. 13
In the associated Game of Deterrence, strategies SA11, SA12, SA21 and SA22
are safe, hence positively playable, while all other strategies are not playable. It follows that in this case, provided that the initial profile of the employees population
satisfies the condition stated in section 3.2, the employees will choose to send information but not adopt the change that might possibly stem from the information
324
Michel Rudnianski, Cerasela Tanasescu
they receive. This conclusion is consistent with the fact that sending information is
more rewarding than not sending, while on the opposite adoption generates a loss.
It is also consistent with the conclusion obtained in the standard approach, and
enable again to remove the question marks to which the standard analysis has led.
CASE B3,B4: a < 0 and n < s
Applying again the previous method leads first the matrix of figure 4. and then
to the matrix of figure 4..
Fig. 14
Fig. 15
In the associated Game of Deterrence, strategies SA1 and SA2 are safe, hence
positively playable, and all other strategies are non-playable. If the condition on
the initial population’s profile is satisfied, then with time the whole population of
employees will choose not to send information and not to adopt change possibly
stemming from the information they receive. Again this conclusion is consistent
with both the conclusion of the standard approach and the assumptions according
to which the payoff for adoption is negative and not sending information is more
rewarding than sending. Likewise we see that the conclusion obtained through the
evolutionary Games of Deterrence approach enables to remove question marks to
which the standard approach led.
5.
Conclusion
Starting from an elementary representation of change pervasion through information transmission, the paper has first shown that the issue could be seen as a
Prisoner’s Dilemma : while it would be in the common interest of all parties to
exchange information, this exchange can’t take place due to exchange information,
such exchange can’t take place dude to the cost-benefit structure. The paper has
then extended the issue to the case where information received might be used to
adopt change, situation in which employees have to take simultaneously two decisions: to send or not send information to other employees, and to adopt or not
to adopt change that could possibly stem from the information received. It has
been shown that incentives play a crucial role in change pervasion. From a more
Unravelling Conditions for Successful Change Management
325
technical point of view, the paper has extended the static approach to a dynamic
one, supported by the Replicator Dynamic. It has been shown that in some cases,
the difficulties of determining the asymptotic properties of the system within the
framework of standard evolutionary games, did not allow conclusions about change
pervasion. To solve that difficulty it has been proposed to use the results of recent
research work bridging standard evolutionary games with evolutionary Games of
Deterrence, for which the determination of asymptotic properties doesn’t require
the resolution of the dynamic system, but can be based only on the playability
properties of the players’ strategies. This bridging has enabled to draw conclusions
that could not be found through the standard approach. Nevertheless, there is still
place for improvement. On the one hand the condition about the initial population’s
profil is quite strong, and one may ask whether it would be possible to alleviate it
in order to extend the field of application. On the other hand, the condition about
existence of solutions with no playable by default strategies also limits the field of
application. But at the same time it paves the way for future work based on an
extension of the games of deterrence used here, i.e. fuzzy games of deterrence in
which the playability indices may take any value comprise between 0 and 1.
References
D’Aveni, R. (1994). Hypercompetition. Simon and Schuster, New York.
Ellison, D., Rudnianski, M. (2009). Is Deterrence Evolutionarily Stable. Annals of the
International Society of Dynamic games, vol 10, Bernhard P., Gaitsgory V., Pourtaillier
O. Eds, pp 357–375, Birkhauser. Berlin.
Ellison, D., Rudnianski, M. (2012). Playability Properties in Games of Deterrence and
Evolution in the Replicator Dynamics. In: Contribution to Game Theory and Management vol VI, collected papers of the sixth international conference Game Theory
and Management (Leon A. Petrosyan and Nikolay Zenkevich, eds), pp. 115–133, StPetersburg.
Fichman, R. G., Kemener, C. F. (1997). The Assimilation of Software Process Innovations:
An Organizational Learning Perspective. Management Science, 1345–1363.
Kahneman, D., Tversky, A. (1979). Prospect Theory: An Analysis of Decision under Risk.
Econometrica, 47(2), 263–291.
Rogers E. (1983). Diffusion of Innovations. Free Press, New York.
Rudnianski, M., Tanasescu, C. (2012). Knowledge Management and Change Pervasion:
a Game Theoretic Approach, paper presented at Cross-Border Innovation and Entrepreneurship Symposium, 3-5 December 2012 Singapore, under current publication.
Applying Game Theory in Procurement.
An Approach for Coping with Dynamic Conditions in
Supply Chains
Günther Schuh and Simone Runge
Institute for Industrial Management
at RWTH Aachen University
Aachen, Germany
Abstract Producing companies are facing continually changing conditions
accompanied by higher requirements with respect to the flexible configuration of their supply chain. The challenge resulting from this initial situation
is to develop systems that have the availability of adjusting their planning
procedures and aims depended on the situation and therefore accommodate
the increasing demand for flexibility. To address this challenge game theory
seems to be a new and promising approach.
The aim and added-value of the research work described here is to develop
a decision model for the area of procurement using solutions concepts of
game theory. Especially in times of high volatility such a decision model can
support material requirements planners better than today’s common selective planning logics. In this paper the model to be solved by game theoretic
solution concepts is presented. A research study has been conducted which
proved the need for combining existing methods of procurement quantity
calculation by means of game theoretic solution concepts. Some of the results of this study are presented in this paper. In the last part of the paper
a structure for classifying game theoretic models is presented. This structure should support in selecting the appropriate solution concept for real-life
decision-situations and is able to support in any practical application-field
finding out the most appropriate game theoretic solution concept.
Keywords: Supply Chain dynamics, procurement quantity calculation, flexibility, game theory.
1.
Introduction
Today, companies face increasing dynamic conditions and volatile markets. Varying
demands, shorter product lifecycle times and increasing diversity of products as well
as unsteady economic situations force enterprises to react more flexible compared
to some years ago (cf. Daxböck et al., 2011, p. 7; Schuh et al., 2012, p. 3; Stich et
al., 2012, p. 123).
Central planning approaches regarding production and logistics processes are
matched to a particular moment of the enterprises conditions. These planningoriented systems are not able to react spontaneously on changing conditions (cf.
Schmitt et al., 2011, p. 748). But exactly the ability to react flexibly becomes more
and more a key factor of success for enterprises. Other approaches reduce central
planning in favour of decentralised activities, which provide better possibilities to
flexibly react according to the current situation (cf. Schmitt et al., 2011, p. 749).
Thus integration into the value-added process is given in decentralised approaches.
Nevertheless decentralised approaches have a big disadvantage as well. Central planning approaches often result in best solutions for the considered scope, a so called
Applying Game Theory in Procurement
327
global optimum. Decentralised activities are performed with a smaller perspective
as the decision maker takes into consideration for example only the processes he is
responsible for. Hence in decentral approaches a global optimum will often not be
achieved.
Therefore it is necessary to find an optimal way between detailed central planning and spontaneous decentral reactions to occurred changes for enabling successful management of production and logistics processes. As a suitable approach for
solving this problem, self-optimising mechanisms could be integrated into the production planning and control as well as in supply chain processes (cf. Behnen et al.,
2011, p. 103). Self-optimising systems are “systems that are able to effect independent (“endogenous”) changes of their inner states or structure based on varying input
conditions or interferences” (Wagels and Schmitt, 2012, p. 162). Self-optimising systems have the capability to react autonomous and flexible to changing boundary
conditions. They continuously carry out three activities (cf. Brecher et al., 2011, p.
13):
1. analysis of the current situation,
2. determination of objectives and
3. adjustment of the behaviour of the system to reach the defined objectives.
The project “Cognition-enhanced, Self-Optimising Production Networks” which
is part of the Aachen Cluster of Excellence (CoE) founded by the German Research
Foundation (DFG, Deutsche Forschungsgemeinschaft) focusses on self-optimising
production planning and control from the level of machine control up to the level
of supply chain management. In this regard both, human decision making as well
as integrating the perspectives of production and quality management will be considered. Test-beds for experimental research in a real production environment to
validate and enhance the outcome will be build. The objective is to develop prototypes of cybernetic solution components based on self-optimising feedback loops.
2.
Problem statement
The described need for reacting more flexible is proven by a study which has been
published by Daxböck et al. 2011. The results of this study show that in 2011 costefficiency has been the most-important target for producing companies, but in future
flexibility will be the most important factor of success (cf. Daxböck et al., 2011, p.
7). Nevertheless, only 43% of the respondents suppose that their company is able to
react sufficiently flexible. Regarding the functional division – namely stock-keeping,
distribution, production and procurement – the division of procurement has been
identified to be the one with the highest potential for improvement (cf. Daxböck et
al., 2011, p. 9).
When performing central approaches in supply chains with different companies,
information barriers occur (cf. Qing-min and Lin, 2009, p. 1457). Thus decentral approaches seem to be more promising for achieving more flexibility in supply chains.
In the following an approach for achieving more flexibility with its focus on procurement, particularly procurement quantity calculation, is presented.
Today planning processes in the area of procurement quantity calculation are
generally matched to a particular moment of the enterprises conditions. Thus these
planning processes are applied continuously regardless of changes in the conditions.
If at all modifications are conducted, these are time-consuming and costly for the
328
Günther Schuh, Simone Runge
enterprises (cf. Schmitt et al., 2011, p. 748-749). Need for action exists not only
regarding the selection of appropriate planning processes, but also in terms of the
selection of appropriate parameter-settings for the planning processes. Procurement
managers generally rely on the parameters and methods that are set in their enterprise resource planning systems and they are overextended with the selection and
adaption of the parameter-settings (cf. Schmidt, 2012, p. 45).
Solution concepts of game theory can help in supporting procurement managers
in this challenge. Thus this constitutes a promising approach for obtaining more
flexibility in the area of procurement. By applying game theoretic solution concepts
the continuous verification of the planning procedures and parameter-settings, which
are set in the enterprise resource planning system of the company, could be supported and changes could be carried out. Thus by applying game theoretic solution
concepts it is possible to receive an improved decentral coordination of the entities
in a supply chain. This approach derives benefit both for supply chains with legally
independent companies and for deliveries between different locations of the same
company. The approach could be applied for any industrial sector.
3.
State of the art
Already in 1944 von Neumann and Morgenstern, two of the pioneers of game theory, realised the applicability of game theory for analysing economic issues (von
Neumann and Morgenstern, 1944). The development of game theory was inter alia
based on the work of other key players, such as John Nash (Nash, 1953), Reinhard
Selten (Selten, 1978) or John Harsanyi (Harsanyi, 1967).
Game theory has been widely studied in the application of supply chain management, but however, has been used in most cases only as an analytical tool (see for
example Hennet and Arda, 2008; Chen et al., 2006; Leng and Parlar, 2005; Abad,
1994; Kohli and Park, 1989; Jordan et al., 2007; Viswanathan and Wang, 2003; Li et
al., 1996). Thereby by means of game theoretic models it is shown, that cooperation
brings benefit for all participants in a supply chain. In other surveys the influence
of the transfer of information is analysed. Additionally rules for distributing savings
are developed and it is analysed which conditions have to be met for coalitions to
remain stable. Cooperative games, where negotiating processes are analysed, are
best developed (cf. Herbst, 2007, p. 85). Drozak describes that game theory is used
in purchasing nowadays, but he points out the need for a method that is tailored
to the qualifications of purchasers (cf. Drozak, 2013, p. 32). Drozak refers with this
requirement to netotiating processes as well.
In game theoretic literature a lack of application of game theoretic solution
concepts for improving decisions that are adopted over time could be recognised
(cf. Fischer, 1997, p. 10). This application area has been identified by Leng and
Parlar already in 2009 as an important field for future research works (cf. Leng
and Parlar, 2009, p. 212). Likewise the situation of imperfect information in game
theoretic applications is so far not sufficiently pervaded (cf. Herbst, 2007, p. 87).
Based on the current state of the art concerning the application of game theoretic
solution concepts a deficit could be seen especially with focus on purchasing and
inventory management. Indeed Wang and Parlar have already asked in 1989 for
more attention in game theoretic applications especially in this field (cf. Wang
and Parlar, 1989, p. 17), but up to now this claim has not been satisfied. Moreover
existing surveys using game theory in purchasing focus on deterministic demand (cf.
Applying Game Theory in Procurement
329
Sarmah et al., 2006, S. 13). In contrast to this in the approach, which is presented
in this paper, fluctuating demand will be considered.
In addition there is a need for dynamic methods for procurement quantity calculation and dynamic calculation of batch sizes. That is why an adaption of parametersettings and adaption of methods have to be done manually by procurement managers when boundary conditions such as sales fluctuations occur (cf. Stumvoll et
al., 2013, p. 570; Schmitt et al. 2011, p. 748 – 749).
In the context of the research activities presented in this paper, the focus lays
on the process-related aspects of game theory and players, who are in successive
value-added steps of a supply chain. Table 1 summarizes the results of the survey
of the state of the art in comparison to the research described in this paper. This
points out deficits in this area of the research activities.
Table 1: State of the art and the research activity presented in this paper
4.
Problem formulation
In the following the model which is solved by the methods of procurement quantity
calculation is presented. First some assumptions for the setting will be depicted:
Only one product in the supply chain will be considered and is procured independent of other products. Only one distributor delivers the product. The demand
of the product is known and is given as input of the model per single period t for the
whole period under review. This means that the result of the demand calculation is
not in focus in the model itself. Over all single periods for the whole period under
review the demand can be constant (stable, static demand) or fluctuating (dynamic
demand).
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Günther Schuh, Simone Runge
Further assumptions for the models are:
•
•
•
•
•
•
•
Stock-outs are not allowed
Fixed costs for procurement (Kf) are constant
Each order arrives directly after triggering and has direct effect to the stock
The procurement price is constant
Storage costs in percentage (l) and the interest rate (z) are constant
Inventory S0 at the beginning of the first period under review is 0
Inventory sn at the end of the whole period under review is 0
Let K ∗ denote the objective function and xt the quantity to be procured in
period t. The demand for period t is denoted by bt and let M be an arbitrary big
figure (‘big M ’). Furthermore let µt be a binary variable with µt = 1, if an order
is put in period t and 0 otherwise. Then the objective function for procurement
quantity calculation for a period under review from period i, . . . , n is:
Minimise
K∗ =
n
X
t=i
(Kf ∗ µt + (z + 1) ∗ st )
(1)
The side conditions for the above given objective function are:
st = st−1 + xt − bt
(2)
xt − M ∗ µt 6 0
(3)
xt > 0
(4)
st > 0
(5)
sn = 0
(6)
µt ∈ {0, 1}
(7)
t = i, i + 1, i + 2, ..., n
(8)
si ,
with:
For the problem definition see for example Tempelmeier (Tempelmeier, 2006, p.
138) or Neumann (Neumann, 2004, p. 594 – 595).
The objective function K ∗ is composed of the ordering costs and the storage
costs. The procurement costs as a product of the quantity to be procured and the
price do not have to be considered in more detail in the procurement quantity
calculation and do not need to be a component of the objective function K ∗ since
they are a constant over the whole period under review. Consequently they have
no influence on minimising the overall costs K (see for example Neumann, 2004, p.
594). In case of static demand, bt , is equal to the mean demand b.
The first side condition (equation 2) is also called storage-balance-equation. It
says that the inventory st at the end of period t is a result of the inventory st−1
Applying Game Theory in Procurement
331
at the end of period t − 1, the quantity to be procured in period t (xt ) and the
demand for period t (bt ) (see for example Neumann, 2004, p. 593–594). The binary
variable µt is 1, if an order is put in period t and 0 otherwise. This is achieved by
equation (3) in connection with the minimising provision of the objective function
(cf. Tempelmeier, 2006, p. 139). Thereby let M be an arbitrary big figure, which
has to be at least so big that the quantity to be procured in each period t (xt ) is
not restricted (cf. Tempelmeier, 2006, p. 139). The third side condition (equation
4) states that there are no negative quantities to be procured. By equation (5) is
stated that inventory cannot be negative.
The model presented in equation (1) to (8) describes the procurement quantity
calculation and is called single-level uncapacitated lot sizing problem (SLULSP) (cf.
Tempelmeier, 2006, p. 138). For the purpose of analysing the influence of different
logistics parameters and random incidents the above described assumptions have to
be extended as in reality for example stock-outs occure. These and other criterions
which have to be considered but are not incorporated in the SLULSP have been
taken into account by use of an additional model.
5.
Research study: Applicability of methods for procurement quantity
calculation under different conditions
In the literature exist a lot of different heuristic methods and one optimal method for
solving the presented SLULSP. As the SLULSP describes the problem of procurement quantity calculation, these heuristics are methods for procurement quantity
calculation. The most established methods are the method from Wagner and Within
(WW), the Least Unit Cost Method (LUC), the heuristic from Silver and Meal
(SM), the heuristic from Groff (Groff), the Least Total Cost Method (LTC), the
Incremental Order Quantity Method (IOQ), the method of Periodic Order Quantity
(POQ) and the McLaren Order Moment (MOM).
Existing surveys regarding the applicability of the heuristics do not analyse
all possible types of demand (static demand, seasonal fluctuating demand, sporadic
demand and trend in the demand) and especially uncertainty in logistics parameters
such as
• the demand calculation was not right
• the supplier delivered less items than ordered
• the delivery arrives delayed
have not been in focus in existing surveys, but exactly these uncertainties become
more relevant in practice nowadays. Game theoretic solution concepts can help
especially under dynamic conditions like these as it was pointed out at the beginning
of this paper. For setting up a decision model thus in a first step the influence of
these logistics parameters was investigated. As the assumptions of the SLULSP as
described in the previous section are not sufficient for these investigations, the model
had been expanded to analyse the influence of different logistics parameters and
random incidents. As mentioned above for example stock-outs need to be integrated
into the model by use of a further model which displays the inventory.
By use of these two models for the approach presented in this paper, the above
given methods of procurement quantity calculation have been implemented to investigate the influence of the logistics parameters for the different methods on the
332
Günther Schuh, Simone Runge
Fig. 1: Impact of the type of demand per method of procurement quantity calculation
result of the objective function of the SLULSP. First of all it could be seen, that
the types of demand have considerably impact on the results (see Figure 1).
The logistics parameters under consideration are listed in the following. Additionally
the range for the variation is given for each parameter (see table 2).
Table 2: Logistics parameters under review in the survey and their range
Logistics parameters
minimum Maximum
price (monetary units)
1
50
storage costs (percentage)
12
35
fixed costs for procurement (monetary units)
437,5
5292
stock-out costs (percentage)
50
500
replacement time (periods)
1
21
planning interval (periods)
25
90
Length of the planning horizon (periods)
140
365
delay in delivery (probability; periods)
0; 0
0.5; 10
shortshipment (probability; percentage)
0; 0
0,5; 0,5
deviation of demand (expected value; standard 0; 0
0; 10
deviation)
The results of the study confirm the relevance of combining these well-known
methods of procurement quantity calculation by use of game theoretic solution
concepts. The sequence of applicability of the methods could be seen with regard
to the result in the objective function expressed in monetary units. For example
considering different length of the planning horizon from 140 periods up to 365
periods when the demand is stationary, seasonal or has a trend the objective value
of the IOQ-Method becomes higher (see Figure 2). In contrast the objective value
becomes lower when the methods LUC, SM, LTC, Groff, POQ or MOM are applied
under a seasonal demand. Considering a trend in demand it could be seen, that
the method POQ causes the second highest costs when the length of the planning
horizon is 365 periods. With a shorter planning horizon of 140 periods the methods
WW, Groff and SM all cause higher costs than the method POQ. So in this case
Applying Game Theory in Procurement
333
when all the other parameters remain stable and only the length of the planning
horizon is varied the applicability of the methods POQ, Groff and SM is inverted.
Fig. 2: Influence of the length of the planning horizon on the objective value in different
demand situations
In real-life situations in most cases more than only one parameter changes. For this
purpose the correlations between different logistics parameters have been investigated as well. In Figure 3 for example the correlation of the planning interval and
the deviation of demand could be seen. A deviation of demand is given if the demand of the product was incorrect beforehand. When the method WW is applied
and there is a trend in demand the correlation of these two parameters is marginal
(see Figure 3, left side) but if the method IOQ is applied in the same situation
the correlation of these two parameters is much bigger (see Figure 3, right side). In
the second case when there is no deviation in demand (e.g. the demand was right
beforehand) the objective value is much higher with a bigger planning interval. In
contrast in case the demand was not calculated right beforehand the spread of the
objective value is way smaller.
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Günther Schuh, Simone Runge
Fig. 3: Correlation Diagrams: Influence of length of the planning interval combined with
devation of demand with a given trend in demand
As it could be seen by these two examples, the objective values generated by the
methods differ depending on the given environmental conditions. A sophisticated
analysis of the influence of the logistics parameters is thus an essential finding for
the decision model which will be developed and later on solved by game theoretic
solution concepts.
By means of the conducted investigations it became clear that none of the wellknown methods for procurement quantity calculation provides the best solution for
all combinations of the logistics parameters under review. Thus these parameters,
their combination as well as uncertainties have got an effect on the appropriate
method for procurement quantity calculation. Furthermore it could be seen that
the influence of the logistics parameters under consideration differs from method to
method. The influence on the objective value of the parameters differs from method
to method. This could be seen for example in Figure 2.
Thus through the investigations it was approved that game theoretic solution
concepts should support in choosing the appropriate method under given conditions,
as it will be done by the approach presented in this paper.
6.
A structure for classifying game theoretic models for choosing
appropriate solution concepts
Nowadays in game theory there exist a lot of different alternative models that are
suited for representing diverse real-life-situations. It is beyond dispute that the type
of game (i.e. the game theoretic model) has essential influence on the appropriate
game theoretic solution concept (see for example Schiml, 2008, p. 26). Thus when
applying game theory to real-life decision-problems the first very important step is
to decide which solution concept is the most suitable for the decision-problem at
hand.
To reach this goal, a scheme which can help classifying game theoretic solution
concepts is needed. This scheme has to be based on the different attributes of game
theoretic models as these models have essential influence on the solution concepts.
It has been found out, that no such scheme which is comprehensive exists in game
theoretic literature. A lot of authors list different attributes of game theoretic models
with some corresponding characteristics. In the majority of cases these explanations
take place only in a textual way (see for example Herbst, 2007, p. 84 – 86; Kaluza,
Applying Game Theory in Procurement
335
1972, p. 21 – 49). Other authors give some game theoretic models in form of a list
(see for example Lasaulce and Tembine, 2011, p. 15; Kuhn, 2007, p. 50). For each of
the considered types of games Vogt lists two different corresponding characteristics
(see Vogt, 2013, p. 300). Only two authors describe the different game theoretic
models and their corresponding schemes in form of a structured scheme: Marchand
gives such a scheme but for each type he depicts only two different corresponding
characteristics (see Marchand, 2012, p. 36). Pickel et al. sum up different game
theoretic models with their characteristics in a diagram, but this diagram is not
structured consistently (see Pickel et al., 2009, p. 67).
Independent from the form of representation none of the above mentioned compositions give an exhaustive description of all existing game theoretic models and
their corresponding characteristics. Thus to obtain a comprehensive and well-structured
description of the attributes and corresponding characteristics of game theoretic
models a morphology has been developed in the context of the research work presented in this paper. The morphology is depicted in Figure 4 and is described in
the following.
Fig. 4: Structure for classifying game theoretic models for choosing appropriate solution
concepts
Ten attributes have been identified which differentiate game theoretic models.
By the number of decision makers it is meant how many people participate in
the decision. If there is only one decision maker the model could rather be assigned
to decision theory than game theory. But as some authors assign these models to
game theory, this characteristic is listed in the morphology presented here. In game
336
Günther Schuh, Simone Runge
theoretic models not in all cases only two decision makers are incorporated in the
model. There are a lot of situations with more than two decision makers as well.
Another attribute when analysing decision situations is the distribution of
profits and losses, which could be expected as a result of the decision. The corresponding characteristics of this attribute were derived from the types of games
zero-sum-games, nonzero-sum-games, constant-sum-games and strictly competitive
games as all of these models could be differentiated by the attribute of how the
profits and losses of the decision makers are distributed.
The existence of agreements is the third attribute which has been identified
as an essential attribute when classifying game theoretic models and choosing the
most suitable game theoretic solution concept. Looking at cooperative game theoretic models, binding agreements between the decision makers are declared. On the
other hand in non-cooperative game theory these agreements could be non-binding
or the decision makers make no agreements at all.
The number of repetitions in a game corresponds to the attribute frequency
of decisions in the morphology given in Figure 4. This attribute, which refers to
repeated games, has considerable impact on the applicability of the solution concept
as well. If the game will not be repeated the decision maker will always chose the best
choice for himself. But if a game is repeated all decision makers will be more willing
to find best solutions for all of them as they are afraid of retaliation otherwise. In
game theory, games with singular decision, a known finite number > 1 of games to
be played, a unknown finite number of games to be played > 1 as well as games
which have a countable finite number of repetitions could be differentiated.
Another important attribute to distinguish game theoretic models is the sequence of decisions. Meant by this attribute is if the decisions of the players in a
game are performed simultaneously or successive. This refers to simultaneous and
sequential move games.
Moreover game theoretic models could be classified by the information available
to the players. In game theory there exist games with perfect or imperfect information. This is meant by the recognition of prior actions. Does every player
know exactly which situation is on hand (i.e. past and current decisions of the other
decision makers are completely known to all players) this is a game with perfect
information. If at least one decision maker does not exactly know past decisions of
the other players, this is called a game with imperfect information. Further characteristics of this attribute could be that past decisions of the other decision makers
are completely known to all players or that no information of past decisions are
know. Another deficit in information in game theoretic models could be caused by
the availability of the background of other decision makers. As a first corresponding characteristic it is possible that the payoff and possible strategies of all
decision makers are known to all players. These are games with complete information where every rational decision maker is able to calculate the best strategy for
himself exactly. In contrast to these there are games with incomplete information.
The corresponding graduation could be seen in Figure 4.
The strategic scope it the set of available strategies for a player. The size of
the strategic scope can be distinguished by the criterion if the possible strategies
could be depicted reasonable as a decision-matrix or game-tree. Theoretically every
game-tree could be depicted unless it is infinite. Surveys show that up to nine
alternatives could be processed by humans. Thus for a strategic scope of ten or
Applying Game Theory in Procurement
337
more alternatives it does not seem to be reasonable to depict them as a decisionmatrix or game-tree. For these cases as well as if the strategic scope is countably or
even uncountably infinite reaction-functions from game theory could help solving
such decision problems.
In evolutionary game theory no longer individual decision makers are in focus.
Instead these models focus on populations and the members of a population are
able to decide only in the way their genetic code allows. Thus regarding the attribute individuality of the decision makers it could be distinguished if the
individuality is existent or not.
The last attribute in the morphology for classifying game theoretic models is
the consideration of random incidents. In contrast to all other attributes this
attribute could not be derived from the types of games directly. This attribute could
be originary derived from decision-theory but is relevant in game theoretic models
as well. If random incidents have to be considered in game theory, a decision maker
has to determine the probability for the random incidents if possible. Therefore
the corresponding characteristics of this atribute in the morphology are: random
incidents are not relevant, random incidents are relevant and the probability for
these could be determined and random incidents are relevant but the probability
for random incidents could not be determined.
The developed morphology can support in carving out which game theoretic
solution concept will help to solve the described problem statement of combining
existing heuristics for the procurement quantity calculation for getting more flexibility in supply chains. Therefore the corresponding characteristic per attribute has
to be identified for each game theoretic solution concept which is generally suited
for solving the problem on hand. The same has to be done for the real-life-situation
under review. Then structural similarities could be seen. This gives a structured
basis for deciding which solution concept should be applied for solving the given
problem later on.
7.
Conclusion and further research-steps
In this paper at first a short introduction into the idea of applying game theoretic solution concepts for appropriate use of well-known methods of procurement quantity
calculation had been given. In the next part in the state of the art in could be seen
that there exists a lack in using game theory for improving and adapting decisions
over time. Additionally a lack in applying game theory especially in purchasing and
inventory had been identified. Solving the problem of procurement quantity calculation means to solve a SLULSP. Therefore the model was introduced in the next
part of this paper. Some results of the conducted study regarding the applicability
of the methods of procurement quantity calculation have as well been presented in
this paper. This study made clear that changing parameters impact the objective
value of the different methods for procurement quantity calculation significantly.
Thus the relevance of the work presented here is approved. To enable choosing the
most suitable method under given conditions the survey will be pursued in more
detail to deduce concrete recommendations for switching the methods from these
findings.
As a last step in this paper a structure for classifying game theoretic models to
enable choosing the appropriate solution concept was presented. This morphology
can help to answer the question of which game theoretic solution concepts can be
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Günther Schuh, Simone Runge
applied in an application area – for example in procurement quantity calculation as
regarded in this paper. For this purpose the solution concepts have to be opposed to
the application area by the use of this morphology. By this it is ensured that exactly
the solution concepts which are relevant in the application area of procurement are
considered in the following research steps.
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An Axiomatization of the Myerson Value
Özer Selçuk1 and Takamasa Suzuki2
1
2
CentER, Department of Econometrics & Operations Research, Tilburg University,
P.O. Box 90153, 5000 LE Tilburg, The Netherlands
E-mail: [email protected]
CentER, Department of Econometrics & Operations Research, Tilburg University,
P.O. Box 90153, 5000 LE Tilburg, The Netherlands
E-mail: [email protected]
Abstract TU-games with communication structure are cooperative games
with transferable utility where the cooperation between players is limited
by a communication structure represented by a graph on the set of players.
On this class of games, the Myerson value is one of the most well-known
solutions and it is the Shapley value of the so-called restricted game. In
this study we give another form of fairness axiom on the class of TU-games
with communication structure so that the Myerson value is uniquely characterized by this fainess axiom with (component) efficiency, a kind of null
player property and additivity. The combination is similar to the original
characterization of the Shapley value.
Keywords: Cooperative TU-games, communication structure, Myerson value,
Shapley value
Cooperative game theory describes situations of cooperation between players.
A cooperative game with transferable utility, TU-game for short, expresses such
situations by a finite set of players and a characteristic function that assigns a
worth to any subset of players, a coalition. Players within a coalition can freely
divide the worth of the cooperation among themselves. The main focuses of TUgames are investigating under which conditions the players cooperate to form the
grand coalition of all players and how to divide the worth of this grand coalition
into a payoff for each player.
A single-valued solution on a class of games assigns as an allocation a payoff
vector to each game which belongs to the class. Shapley (1953) introduces one of
the most well-known single-valued solution. The solution, the Shapley value, is the
average of all marginal vectors of a TU-game, where a marginal vector corresponds
to a payoff vector for a permutation on the player set. Each permutation can be seen
as an ordering of the players joining to from the grand coalition, and in the marginal
vector associated with a permutation each player gets as payoff the difference in
worth of the set of players preceding him in the permutation with and without him.
While being introduced, the Shapley value is characterized as the unique solution
on the class of TU-games that satisfies efficiency, additivity, null player property
and symmetry in Shapley (1953).
TU-games assume that any coalition can be formed to cooperate and gain its
worth of their cooperation, but in many economic situations there exist restrictions
which prevent some coalitions from cooperating. A TU-game with this kind of situation is firstly introduced by Myerson (1977) as a TU-game with communication
structure. It arises when the restriction is represented by an undirected graph in
342
Özer Selçuk, Takamasa Suzuki
which the vertices represent the players and a link between two players shows that
these players can communicate and are able to cooperate by themselves.
One of the most well-known single-valued solutions on the class of TU-games
with communication structure is the Myerson value (Myerson (1977)), defined as
the Shapley value of the so-called Myerson restricted game. By Myerson (1977),
the Myerson value is characterized by (component) efficiency and fairness, fair in
the sense that if a link is deleted between two players, the Myerson value imposes
the same loss on payoffs for each of these two players. Other characterizations of
the Myerson value are given in Borm et al. (1992), Brink (2009) for the class of
TU-games with cycle-free communication structure.
In this study we give an alternative axiomatization of the Myerson value for
TU-games with communication structure. Our approach is to give another form of
fainess axiom so that the Myerson value is characterized by (component) efficiency,
a kind of null player property, additivity and a kind of fairness. The combination is
similar to the original characterization of the Shapley value by Shapley (1953).
This paper is organized as follows. Section 2 introduces TU-games with communication structure and the Myerson value. In Section 3 an axiomatic characterization
for the solution is given.
1.
TU-games with communication structure and the Myerson value
A cooperative game with transferable utility, or a TU-game, is a pair (N, v) where
N = {1, . . . , n} is a finite set of n players and v : 2N → R is a characteristic function
with v(∅) = 0. For a subset S ∈ 2N , being the coalition consisting of all players in S,
the real number v(S) represents the worth of the coalition that can be maximially
achived, and can be freely distributed among the players in S. Let GN denote the
class of TU-games with fixed player set N . We often identify a TU-game (N, v) by
its characteristic function v.
A special class of TU-games is the class of unanimity games. For T ∈ 2N , the
unanimity game (N, uT ) ∈ GN has characteristic function uT : 2N → R defined as
uT (S) =
1 if T ⊆ S,
0 otherwise.
It is well-known that any TU-game can be uniquely expressed as a linear combination of unanimity games. Let (N, 0) ∈ GN denote the zero game, i.e., 0(S) = 0 for
all S ∈ 2N .
A payoff vector x = (x1 , ..., xn ) ∈ Rn is an n-dimentional vector and it assigns
payoff xi to player i ∈ N . A single-valued solution on GN is a mapping ξ : GN → Rn
which assigns to every TU-game (N, v) a payoff vector ξ(N, v) ∈ Rn .
The most well-known single-valued solution on the class of TU-games is the
Shapley value, see Shapley (1953). It is the average of the marginal vectors induced
from the collection of all permutations of players. Let Π(N ) be the collection of all
permutations on N . Given a permutation σ ∈ Π(N ), the set of predecessors of any
element i ∈ N in σ is defined as
Pσ (i) = {h ∈ N | σ −1 (h) < σ −1 (i)}.
343
An Axiomatization of the Myerson Value
Given a TU-game (N, v) ∈ GN , for a permutation σ in Π(N ) the marginal vector
mσ (N, v) assigns payoff
mσi (N, v) = v(Pσ (i) ∪ {i}) − v(Pσ (i))
to agent i = σ(k), k = 1, . . . , n. The Shapley value of (N, v), Sh(N, v), is the average
of all n! marginal vectors, i.e.,
1 X
Sh(N, v) =
mσ (N, v).
n!
σ∈Π(N )
A graph on N is a pair (N, L) where N = {1, . . . , n} is a set of vertices and
L ⊆ LcN , where LcN = {{i, j} | i, j ∈ N, i 6= j} is the complete set of undirected
links without loops on N and an unordered pair {i, j} ∈ L is called an edge in
(N, L). A subset S ∈ 2N is connected in (N, L) if for any i ∈ S and j ∈ S, j 6= i,
there is a sequence of vertices (i1 , i2 , . . . , ik ) in S such that i1 = i, ik = j and
{ih , ih+1 } ∈ L for h = 1, . . . , k − 1. The collection of all connected coalitions in
(N, L) is denoted C L (N ). By definition, the empty set ∅ and every singleton {i},
i ∈ N , are connected in (N, L). For S ∈ 2N , the subset of edges L(S) ⊆ L is
defined as L(S) = {{i, j} ∈ L| i, j ∈ S}, being the subset of L of edges that can be
established within S. The graph (S, L(S)) is a subgraph of (N, L). A component of
a subgraph (S, L(S)) of (N, L) is a maximally connected coalition in (S, L(S)) and
bL (S). For a graph (N, L),
the collection of components of (S, L(S)) is denoted C
if {i, j} ∈ L, then i is called a neighbor of j and vice versa. Given (N, L) and
i ∈ N , the collection of neighbors of i is denoted by DiL , that is, DiL = {j ∈
N \ {i} | {i, j} ∈ L}. The collection of neighbors of S ∈ 2N is defined similarly as
DSL = {j ∈ N \ S | ∃i ∈ S : {i, j} ∈ L}.
The combination of a TU-game and an (undirected) graph on the player set
is a TU-game with communication structure, introduced by Myerson (1977) and
denoted by a triple (N, v, L) where (N, v) is a TU-game and (N, L) is a graph on
N . A link between two players has as interpretation that the two players are able to
communicate and it is assumed that only a connected set of players in the graph is
able to cooperate to obtain its worth to freely transfer as payoff among the players in
cs
the coalition. Let GN
denote the class of TU-games with communication structure
cs
cs
and fixed player set N . A single-valued solution on GN
is a mapping ξ : GN
→ Rn
cs
which assigns to every TU-game with communication structure (N, v, L) ∈ GN
a
n
payoff vector ξ(N, v, L) ∈ R .
The most well-known single-valued solution on the class of TU-games with communication structure is the Myerson value, see Myerson (1977). It is the Shapley
value of the so-called Myerson restricted game. Following Myerson (1977), the restricted characteristic function v L : 2N → R of (N, v, L) is defined as
X
v(K), S ∈ 2N .
v L (S) =
b L (S)
K∈C
The pair (N, v L ) is a TU-game and is called the Myerson restricted game of (N, v, L),
cs
and the Myerson value of a game (N, v, L) ∈ GN
is defined as
µ(N, v, L) =
1
n!
X
σ∈Π(N )
mσ (N, v L ).
344
2.
Özer Selçuk, Takamasa Suzuki
An axiomatic characterization of the Myerson value
Most of the single-valued solutions proposed in the literature are characterized by
axioms which state desirable properties a solution possesses. The most well-known
characterization of the Shapley value for TU-games is given by Shapley (1953) as
the unique solution on the class of TU-games that satisfies efficiency, additivity, the
null player property and symmetry. Other characterizations of the Shapley value
are proposed in for example Young (1985) and Brink (2002). While introducing the
class of TU-games with communication structure, Myerson (1977) characterizes the
Myerson value by component efficiency and fairness axioms.
cs
Definition 1. A solution ξ : GN
→ Rn satisfies component efficiency if for any
P
cs
bL (N ).
(N, v, L) ∈ GN it holds that i∈Q ξi (N, v, L) = v(Q) for all Q ∈ C
A solution on the class of TU-games with communication structure satisfies component efficiency if the solution allocates to each component as the sum of payoff
among its members the worth of the component.
cs
cs
Definition 2. A solution ξ : GN
→ Rn satisfies fairness if for any (N, v, L) ∈ GN
and {i, j} ∈ L it holds that
ξi (N, v, L) − ξi (N, v, L \ {i, j}) = ξj (N, v, L) − ξj (N, v, L \ {i, j}).
A solution on the class of TU-games with communication structure satisfies fairness
if the deletion of an edge from the game results in the same payoff change for the
two players who own the edge.
cs
Theorem 1. (Myerson, 1977) The Myerson value is the unique solution on GN
that satisfies component efficiency and fairness.
For the class of TU-games with cycle-free communication structure, which is a
subclass of TU-games with communication structure, other characterizations of the
Myerson value are given by Borm et al. (1992) and Brink (2009). The axioms we
propose in this study are modified versions of the four axioms used in Shapley
(1953), i.e., an efficiency axiom (component efficiency), an additivity axiom, a null
player property and a fairness axiom.
For any two TU-games v and w in GN , the game v + w is well defined by
(v + w)(S) = v(S) + w(S) for all S ∈ 2N .
cs
Definition 3. A solution ξ : GN
→ Rn satisfies additivity if for any (N, v, L),
cs
(N, w, L) ∈ GN it holds that ξ(N, v + w, L) = ξ(N, v, L) + ξ(N, w, L).
Additivity of a solution means that if there are two TU-games with the same communication structure, the resulting payoff vectors coincide when applying the solution
to each of the two games and adding the two vectors and when applying the solution
to the game which is the sum of the two games.
A player i ∈ N is a restricted null player in a TU-game with communication
cs
structure (N, v, L) ∈ GN
if this player never
Pcontributes whenever he joins to form
a connected coalition, that is, v(S ∪ {i}) − K∈CbL (S) v(K) = 0 for all S ∈ 2N such
that i ∈
/ S and S ∪ {i} ∈ C L (N ). The restricted null player property says that this
player must get zero payoff.
345
An Axiomatization of the Myerson Value
cs
Definition 4. A solution ξ : GN
→ Rn satisfies the restricted null player property
cs
if for any (N, v, L) ∈ GN and restricted null player i ∈ N in (N, v, L) it holds that
ξi (N, v, L) = 0.
Note that a restricted null player of a TU-game with communication structure is a
null player of its Myerson restricted game. The last axiom replaces symmetry.
cs
Definition 5. A solution ξ : GN
→ Rn satisfies coalitional fairness if for any
′
cs
two TU-games (N, v, L), (N, v , L) ∈ GN
and Q ∈ 2N it holds that ξi (N, v, L) −
′
′
ξi (N, v , L) = ξj (N, v, L) − ξj (N, v , L) for all i, j ∈ Q whenever v(S) = v ′ (S) for all
S ∈ 2N , S 6= Q.
Coalitional fairness of a solution implies that given a TU-game with communication
structure, if the worth of a single coalition changes, then the payoff change should
be equal among all players in that coalition. From additivity and the restricted null
player property we have the following lemma.
cs
Lemma 1. Let a solution ξ : GN
→ Rn satisfy additivity and the restricted null
player property. Then for any two TU-games with the same communication structure
cs
(N, v, L), (N, v ′ , L) ∈ GN
it holds that ξ(N, v, L) = ξ(N, v ′ , L) whenever v(S) =
′
L
v (S) for all S ∈ C (N ).
Proof. Consider the game (N, w, L) where w = v − v ′ . Then every player is a
restricted null player in this game because w(S) = 0 for all S ∈ C L (N ). Therefore
every player must receive zero payoff, that is, ξ(N, w, L) = 0. From additivity and
v = w + v ′ it follows that ξ(N, v, L) = ξ(N, w, L) + ξ(N, v ′ , L) = 0 + ξ(N, v ′ , L) =
ξ(N, v ′ , L).
⊔
⊓
This lemma says that the worth of an unconnected coalition does not affect the
outcome of a solution that satisfies additivity and the restricted null player property,
which leads to the following corollary.
cs
Corollary 1. If a solution ξ : GN
→ Rn satisfies additivity and the restricted null
cs
player property, then ξ(N, v, L) = ξ(N, v L , L) for any (N, v, L) ∈ GN
.
To prove that on the class of TU-games with communication structure the axioms above uniquely define the Myerson value, we consider Myerson restricted unanimity games. Given a unanimity game with communication structure (N, uT , L) ∈
cs
GN
with T ∈ 2N , the Myerson restricted unanimity game (N, uL
T ) ∈ GN is given by
b L (S), T ⊆ K,
1 if ∃ K ∈ C
uL
T (S) =
0 otherwise.
L
Given a graph (N, L) and S ∈ 2N , let C (S) denote the collection of connected
coalitions which minimally contain S, that is,
L
C (S) = {K ∈ C L (N ) | S ⊆ K, K \ {i} ∈
/ C L (N ) ∀ i ∈ K \ S}.
Lemma 2. For a unanimity TU-game with communication structure (N, uT , L) ∈
cs
GN
with T ∈ 2N , it holds that
 X
L

(−1)|J|+1 u∪j∈J Qj if C (T ) = {Q1 , . . . , Qk },

uL
J⊆{1,...,k}
T =

L
0
if C (T ) = ∅.
346
Özer Selçuk, Takamasa Suzuki
L
Proof. First consider the case when C (T ) = ∅. This implies that there exists no
b L (N ) which contains T , and from the definition of uL it follows that uL (S) =
K∈C
T
T
P
L
0 for all S ∈ 2N . Next, let v = J⊆{1,...,k} (−1)|J|+1 u∪j∈J Qj when C (T ) 6= ∅. If
L
T ∈ C L (N ), then C (T ) = {T } and therefore it holds that v = uT = uL
T . Suppose
N
T ∈
/ C L (N ). It is to show that v(S) = uL
T (S) holds for every S ∈ 2 . First take
b L (S) satisfying T ⊆ K. This implies that
S ∈ 2N such that there is no K ∈ C
L
Q 6⊂ S for any Q ∈ C (T ), and thus we have u∪j∈J Qj (S) = 0 for all J ⊆ {1, . . . , k},
N
which results in v(S) = 0 = uL
such that there exists
T (S). Next, take any S ∈ 2
L
b
K ∈ C (S) satisfying T ⊆ K. This K is unique and let M ⊆ {1, . . . , k} be such
that Qj ⊆ K for all j ∈ M and Qj 6⊂ K for all j ∈
/ M . Among all J ⊆ {1, . . . , k}, it
holds that u∪j∈J Qj (S) = 1 only when J ⊆ M , and otherwise u∪j∈J Qj (S) = 0. Let
P
Pk=m
|M | = m. Then v(S) = J⊆M (−1)|J|+1 u∪j∈J Qj (S) = k=1 (−1)k+1 m
k = 1 =
Pk=m
uL
the binominal theorem that k=0 (−1)k m
T (S), since it is known from
k = 0 and
Pk=m
P
k=m
m
k+1 m
k m
therefore k=1 (−1)
⊔
⊓
k=1 (−1)
k = −
k = 0 = 1.
Note that for any J ⊆ {1, . . . , k}, it holds that ∪j∈J Qj is connected, since for
each j ∈ J, the set Qj itself is connected and it also contains T . This lemma
shows that any restricted unanimity TU-game with communication structure can
be uniquely expressed as a linear combination of unanimity TU-games with the
same communication structure for connected coalitions.
On the class of unanimity TU-games with communication structure, we have
the following expression, which is well known and we present without proof.
cs
Lemma 3. For any TU-game with communication structure (N, cuT , L) ∈ GN
with
L
T ∈ C (N ), T 6= ∅, and c ∈ R, it holds that
c/|T | if j ∈ T,
µj (N, cuT , L) =
0
if j 6∈ T.
This lemma says that the Myerson value of a unanimity TU-game with communication structure with a connected coalition assigns the allocation which gives zero
payoffs to the players who do not belong to the connected coalition and the worth
of the connected coalition is shared equally among those who belong to it. Next, we
give a characterization of the Myerson value in the following theorem.
cs
Theorem 2. The Myerson value is the unique solution on GN
that satisfies component efficiency, additivity, the restricted null player property, and coalitional fainess.
Proof. First, we show that the Myerson value satisfies all properties. Component
efficiency follows from the fact that all marginal vectors are component efficient by
construction. Since all marginal vectors of a TU-game with communication structure
are linear in the worths of the connected coalitions and the Myerson value is the
average of these vectors, the Myerson value satisfies additivity. If a player is a
restricted null player, this player has marginal contribution equal to zero at any
permutation and therefore the average is also zero. Finally, suppose there are two
cs
TU-games with the same communication structure (N, v, L), (N, v ′ , L) ∈ GN
and
Q ∈ C L (N ) such that v(S) = v ′ (S) for all S ∈ C L (N ), S 6= Q, and take any i ∈ Q.
It holds that mσi (N, v, L) = mσi (N, v ′ , L) for any σ ∈ Π(N ) unless Pσ (i) = Q \ {i}.
An Axiomatization of the Myerson Value
347
There are (|Q| − 1)!(n − |Q|)! permutations σ such that Pσ (i) = Q \ {i} and for
each such σ the marginal contribution of i changes by mσi (N, v, L) − mσi (N, v ′ , L) =
(v L (Q)−v L (Q\{i}))−(v ′L (Q)−v L (Q\{i})) = v L (Q)−v ′L (Q), which is independent
of i. Therefore every player in Q receives the same change the same number of times
and so the change in the Myerson value is the same among all players in Q.
cs
Second, let ξ : GN
→ Rn be a solution which satisfies all four axioms. Since
ξ satisfies additivity and the restricted null player property, with Corollary 1 and
Lemma 2, it suffices to show that for any graph (N, L) it holds that ξ(N, cuT , L) =
µ(N, cuT , L) for any T ∈ C L (N ) and c ∈ R. Let (N, L) be any graph on N .
cs
First consider the zero game (N, 0, L) ∈ GN
. In this game all players are restricted
null players and therefore it follows from the restricted null player property that
ξi (N, 0, L) = 0 = µi (N, 0, L) for all i ∈ N . Next consider the game (N, cuN , L) ∈
cs
GN
with N ∈ C L (N ). Between the games (N, cuN , L) and (N, 0, L) it holds that
cN (N ) = c and cuN (K) = 0(K) = 0 for all K ∈ 2N , K 6= N . From efficiency,
coalitional fairness, and Lemma 3, we have
ξi (N, cuN , L) =
c
= µi (N, cuN , L) ∀ i ∈ N.
n
cs
Now consider a game (N, cuT , L) ∈ GN
with T ∈ C L (N ), |T | = n − 1. It follows
from the restricted null player property that player i ∈
/ T receives zero payoff, since
this player yields zero marginal contribution when joining to any set of players to
form a connected coalition. For the games (N, cuT , L) and (N, cuN , L), it holds that
cuT (K) = cuN (K) for all K ∈ 2N , K 6= T . Coalitional fairness then implies that
ξi (N, cuT , L) − ξi (N, cuN , L) = ξj (N, cuT , L) − ξj (N, cuN , L) ∀ i, j ∈ T,
which, with efficiency and Lemma 3, results in
ξi (N, cuT , L) =
c
= µi (N, cuT , L) ∀ i ∈ T.
|T |
Next, suppose ξ(N, cuT , L) = µ(N, cuT , L) holds for all T ∈ C L (N ), |T | > m > 1.
cs
Consider (N, cuT , L) ∈ GN
with T ∈ C L (N ), |T | = m. For i ∈
/ T , it follows from
the restricted null player
property
that
ξ
(N,
cu
,
L)
=
0.
By
comparing
(N, cuT , L)
i
T
P
and (N, v, L) with v = ℓ∈DL cuT ∪{ℓ} − (k − 1)cuN where k = |DTL | is the number
T
of neighbors of T in (N, L), it holds that cuT (S) = v(S) for all S ∈ 2N , S 6= T , and
cuT (T ) = c while v(T ) = 0. Then coalitional fairness implies
ξi (N, cuT , L) − ξi (N, v, L) = ξj (N, cuT , L) − ξj (N, v, L) ∀ i, j ∈ T.
From additivity and the supposition that ξ(N, cuS , L) = µ(N, cuS , L) for all connected S with |S| > m, it follows that
X
ξi (N, v, L) =
ξi (N, cuT ∪{ℓ} , L) − (k − 1)ξi (N, cuN , L) =
X
L
ℓ∈DT
X
L
ℓ∈DT
L
ℓ∈DT
µi (N, cuT ∪{ℓ} , L) − (k − 1)µi (N, cuN , L) =
µj (N, cuT ∪{ℓ} , L) − (k − 1)µj (N, cuN , L) = ξj (N, v, L)
348
Özer Selçuk, Takamasa Suzuki
for all i, j ∈ T , and therefore
ξi (N, cuT , L) = ξj (N, cuT , L) ∀ i, j ∈ T.
By efficiency it holds that ξi (N, cuT , L) = c/|T | for all i ∈ T , which implies
ξ(N, cuT , L) = µ(N, cuT , L). When |T | = 1, efficiency and the restricted null player
cs
property imply that ξ allocates the Myerson value to (N, cuT , L) ∈ GN
. Therefore for a multiple of any unanimity TU-game with communication structure for
a connected coalition, the four axioms uniquely give the allocation of the Myerson
value. Since ξ satisfies additivity and the restricted null player property, it follows
cs
from Corollary 1 that ξ(N, v, L) = ξ(N, v L , L) for any (N, v, L) ∈ GN
. By Lemma
L
2 it holds that v can be expressed as a unique linear combination of unanimity
cs
games for connected coalitions. That is, given any (N, v, L) ∈
PGN there exist unique
L
L
numbers cT ∈ R for T ∈ C (N ), T 6= ∅, such that v = T cT uT . The proof is
cs
completed since for any (N, v, L) ∈ GN
it holds from additivity that
X
ξ(N, v, L) = ξ(N, v L , L) = ξ(N,
cT uT , L) =
X
T ∈C L (N ),T 6=∅
ξ(N, cT uT , L) =
X
T ∈C L (N ),T 6=∅
µ(N, cT uT , L) = µ(N, v, L).
T ∈C L (N ),T 6=∅
⊔
⊓
ToP
show the independence of the four axioms,
consider the linear solution ξ(N, v,
P
L) = T ∈C L (N ) f (N, cT uT , L) where v = T ∈C L (N ) cT uT and f (N, cT uT , L) allocates c to the player in T who has the smallest index and 0 to any other player. It
only fails coalitional fairness. Next, consider the solution ξ(N, v, L) that allocates
payoff vector ξ(N, v, L) as follows. When N = {1, 2}, L = {1, 2}, v L (S) 6= 0 for
all S ∈ C L (N ), and further v(S) 6= v(T ) for all distinct T, S ∈ 2N , then it gives
ξj (N, v, L) = v(N )/2, and in any other case it gives ξ(N, v, L) = µ(N, v, L). This
solution satisfies all axioms except additivity. The equal sharing solution, where
each agent receives v(N )/n, satisfies every axiom except the restricted null player
property. Finally, the solution where each agent receives zero payoff only fails efficiency.
References
Borm, P., G. Owen, and S. Tijs (1992). On the position value for communication situations.
SIAM Journal of Discrete Mathematics, 5, 305–320.
Brink, R. van den (2002). An axiomatization of the Shapley value using a fairness property.
International Journal of Game Theory, 30, 309–319.
Brink, R. van den (2009). Comparable axiomatizations of the Myerson value, the restricted
Banzhaf value, hierarchical outcomes and the average tree solution for cycle-free graph
restricted games. Tinbergen Institute Discussion Paper 2009–108/1, Tinbergen Institute, Amsterdam.
Myerson, R. B. (1977). Graphs and cooperation in games, Mathematics of Operations Research, 2, 225–229.
Shapley, L. (1953). A value for n-person games. In: Kuhn, H.W. and A.W. Tucker (eds.),
Contributions to the Theory of Games, Vol. II, Princeton University Press, Princeton,
pp. 307–317.
Young, H. P. (1985). Monotonic solutions of cooperative games. International Journal of
Game Theory, 14, 65–72.
Multi-period Cooperative Vehicle Routing Games
Alexander Shchegryaev and Victor V. Zakharov
St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
Universitetsky Prospect 35, St.Petersburg, Peterhof, 198504, Russia
E-mail: [email protected], [email protected]
Abstract In the paper we treat the problem of minimizing and sharing
joint transportation cost in multi-agent vehicle routing problem (VRP) on
large-scale networks. A new approach for calculation subadditive characteristic function in multi-period TU-cooperative vehicle routing game (CVRG)
has been developed. The main result of this paper is the method of constructing the characteristic function of cooperative routing game of freight
carriers, which guarantees its subadditive property. A new algorithm is proposed for solving this problem, which is called direct coalition induction
algorithm (DCIA). Cost sharing method proposed in the paper allows to
obtain sharing distribution procedure which provides strong dynamic stability of cooperative agreement based on the concept of Sub-Core and time
consistency of any cost allocation from Sub-Core in multi-period CVRG.
Keywords: VRP, vehicle routing problem, vehicle routing games, heuristics,
multi-period cooperative games, dynamic stability, time consistency.
1.
Introduction
When we study collaboration in cargo transportation and routing we have to address
the following questions partly discussed in (Agarwal et al., 2009):
– How does one evaluate the maximum potential benefit from collaboration of
carriers forming coalitions? However, to obtain such a benefit value is not easy
because the underlying computational problem is NP hard.
– How should a membership mechanism be formed to be stable during sufficiently
long period of time, and what are the desired properties that such a mechanism
should possess? For logistics applications, this involves issues related to the
design of the service network and utilization of assets, such as the allocation of
ship capacity among collaborating carriers, assignment and scheduling vehicles
on routes.
– How should the benefits achieved by collaborating be allocated among the members in a fair way? In the cargo transportation routing setting we investigate
what does a fair allocation mean and how such an allocation may be achieved in
the context of day-to-day operations to be time consistent during transportation
process?
– How to overcome these disadvantages?
Dynamic cooperative game theory can provide us with models of coordination
carrier’s actions in order to reduce transportation costs. Cooperation issues in vehicle routing models are still an insufficiently studied problem. Possible applications
of the cooperative game theory for such problems are demonstrated in the papers
350
Alexander Shchegryaev, Victor V. Zakharov
(Ergun et al., 2007; Krajewska et al., 2008). The most important object under investigation of the cooperative game theory is the characteristic function of the game
which reflects assessment of guaranteed values of total costs of participants united
in a coalition. If one constructs a mathematical model of cooperation in practical
tasks, it is important to select the method of such function building. Computational
difficulties of finding the values of the characteristic function in a cooperative vehicle routing game (CVRG) are caused by the large size of the problem, which makes
it unacceptable to use exact methods for solving wide class of routing problems
with a comparatively small number of customers to be served (Baldacci et al., 2012;
Kallehauge, 2008). At the same time, using of heuristic algorithms in the general
case does not allow to guarantee fulfillment of the subadditive property of the characteristic functions, which has crucial importance for achievement of cooperative
agreements and total cost reduction. Considering dynamic cooperation models, it
is expedient to use imputation distribution procedures (IDP) which were first proposed by L.A.Petrosyan, as well as cooperation stability principles formulated by
L.A.Petrosyan and N.A.Zenkevich (2009).
In our paper we propose mathematical setting of the freight carriers cooperation
problem, a new approach to building the characteristic function of the multi-period
CVRG and algorithm for constructing cost sharing scheme providing strong dynamic stability of the Sub-Core to meet condition of time consistency (dynamic
stability) of cooperative agreements.
2.
General Problem Statement
In this paper it is presumed that in the transportation service market there are several agents (companies) engaged in cargo transportation on a network. Each agent
has a great number of customers located in nodes of network and its own resources,
such as a depot and a non-empty fleet of vehicles. These companies consider various options of cooperation to reduce transportation costs. Each coalition meets the
demand of customers for transportation services of all companies involved in cooperation using consolidated resources. Thus, within cooperative service, customers
can be redistributed between participants in each coalition. In its turn, customers
exchange between agents within a coalition would extend the set of feasible routes of
consolidated fleet and provide additional possibility to improve current solution in
comparison to non-cooperative case. On the other hand, when agents cooperate, the
total number of customers that has to be dispatched at once to vehicles substantially
increases along with the computational complexity of finding routes minimizing the
total transportation costs of the coalition. Therefore, in operative decision-making
environment there is a lack of time for quick assignment of customers to optimal
routes, since this problem belongs to the class of NP-hard problems.
To find a good solution for vehicle routing problem with several depots the
adaptation of well-known metaheuristic algorithm proposed by Ropke and Pisinger
(2006) may be used for each coalition. Once the routes with minimum transportation costs for each possible coalition are found, the characteristic function value of
the cooperative routing game can be calculated. To ensure that the agents have the
motivation to form a coalition, the characteristic function has to satisfy subadditivity condition. In general, heuristic algorithms that find minimum of transportation
costs of a coalition do not guarantee this property. Therefore, a special metaheuris-
Multi-period Cooperative Vehicle Routing Games
351
tic algorithms providing subadditive property of the characteristic function has to
be proposed for VRP.
The solution of VRP is a set of vehicle routes, such that all customers are
visited exactly once, each route starts and ends in a depot, the length of each route
is limited to predetermined value. Additionally, in the vehicle routing problems with
time windows each customer has specified service time and must be visited within
the specified time interval.
Generally, the objective of such problems is to minimize the total length of
routes. In real-life cases the number of used vehicles has more significant impact on
the total transportation costs, because the cost of using additional vehicle appears
to be much higher than benefit from shorter routes.
3.
Mathematical Model of Static CVRG
Let N be a set of companies engaged in transportation service in the same transport
network. Each company i ∈ N provides transportation service to the given set
of customers Ai . Each customer is served by only one company. Companies are
considering possibilities of cooperation to reduce total transportation costs. Let
S ⊆ N be a proper coalition of companies (players or agents in the static CVRG
with transferable utilities) to be formed. The total cost of the coalition S consists
of two parts: costs of used vehicles and direct transportation costs. In this paper
two assumptions are made concerning costs:
– direct transportation cost is linear function of the total length of routes;
– fleet of vehicles of coalition S includes homogenous vehicles of all companies from
this coalition, and each vehicle has fixed utilization price. It is also assumed that
each coalition has unlimited number of identical vehicles and pays only for those
that are used in transportation service.
Thus, the cost function may be represented as follows:
cost(S, pS ) = aS · N T (S, pS ) + bS · T T C(S, pS ),
where
pS ∈ PS — feasible routing plan for the vehicles of the coalition S, PS — the
finite set of feasible routing plans of the collation S;
aS — the cost of one vehicle utilization for the coalition S;
N T (S, pS ) — number of vehicles used by the coalition S at the particular routing
plan pS ;
bS — cost of one unit of the length for the coalition S;
T T C(S, pS ) — the total length of routes of the coalition S at the particular
routing plan pS .
For the sake of simplicity, it is assumed that each company has only one depot.
It is also assumed that companies may redistribute transportation costs among the
collaborators using some cost sharing procedure.
In order to design subadditive characteristic function of CVRG consider for
coalition S ⊆ N the costs minimization problem over the set of feasible vehicle
routes
minpS ∈PS cost(S, pS )
(1)
352
Alexander Shchegryaev, Victor V. Zakharov
Suppose the exact minimum value of the problem (1) is equal to copt (S). In the
case of using heuristic algorithm for solving this problem the obtained value of the
minimum ch (S) will be not less than copt (S), that is
copt (S) ≤ ch (S)
(2)
For two disjoint coalitions S ⊆ N and T ⊆ N , for any pair of feasible routing plans
pS ∈ PS , pT ∈ PT , the routing plan (pS , pT ) consisting of the union of routing plans
pS and pT is feasible in the routing problem for the joint coalition of carriers S ∪ T ,
that is (pS , pT ) ∈ PS∪T , moreover PS ∪ PT ⊆ PS∪T and AS ∪ AT = AS∪T , then it
is clear that the following inequality holds
copt (S ∪ T ) ≤ copt (S) + copt (T )
Taking into account the inequality (2) we have
copt (S ∪ T ) ≤ ch (S) + ch (T )
(3)
Last inequality can be rewritten for the arbitrary coalition L ⊆ S and the corresponding values ch (L) and ch (S/L)
copt (S) ≤ ch (S/L) + ch (L)
(4)
We define the value of characteristic function c(S) in cooperative CVRG in the
following way
c(S) = min{minL⊂S {c(S/L) + c(L)}, ch (S)}
(5)
One can notice that if we start calculation of the characteristic function c(S) with
one-element coalitions and then gradually increase the size of coalitions by 1 until
we obtain the value for the grand coalition N , the characteristic function designed
by using (5) would fulfill the condition of the subadditive, i.e.
c(S ∪ T ) ≤ c(S) + c(T ), S ⊆ N, T ⊆ N, S ∩ T = ∅
(6)
We call this algorithm for constructing characteristic function of TU-cooperative
CVRG in the form (5) the direct coalition induction algorithm (DCIA). Thus, the
following theorem holds.
Theorem 1. The characteristic function c(S) defined by (5) of static TU-cooperative
VRG and calculated using direct coalition induction algorithm satisfies subadditivity
condition (6).
4.
Example of Cooperative Routing
To illustrate the algorithm implementation we consider one artificial problem of
cooperation with four transport companies D1, D2, D3, D4 having demand for
cargo transportation from 54, 49, 44, 53 customers. The example has been generated
using one benchmark (R2_2_1) proposed by Gehring and Homberger to compare
heuristic algorithms that solve vehicle routing problems with time windows.
Thus, in case of full cooperation, the transportation companies together have to
service 200 customers. Clients of each company are distributed evenly throughout
the nodes of network where the servicing is provided. And they have wide time
353
Multi-period Cooperative Vehicle Routing Games
service windows (the time interval during which servicing is possible) which allows
to use less vehicles, but at the same time increases the computational complexity
of the problem. Use of a little number of vehicles is also facilitated by big carrying
capacity thereof as compared to the customers’ demand. The total costs will be
calculated assuming that the value of use of one vehicle is 5000 condition monetary
units, and the value of one unit of the route length is 5 conditional monetary units.
The algorithm proposed by Ropke and Pisinger (2006) was used for solving corresponding routing problems. As a basic problem, this algorithm considers the more
general problem, the particular case of which is the problem in question. In order
to find efficient routes several basic heuristics were united in one algorithm with
the help of the simulated annealing. One part of these heuristics removes several
customers from the solution, and the other inserts them into the solution again. The
adaptive mechanism tracks the performance of basic heuristics and chooses at each
step of iterations two certain heuristics using obtained statistics of their previous
effectiveness. Such mechanism is based on the special genetic algorithm. To diversify search process and enhance algorithm robustness the noise value is added to
the value of objective function. Table 1 shows the solution of respective costs minimization problems for each coalition and values of the game characteristic function
calculated using the direct coalition induction algorithm.
Table 1: Solutions of costs minimization problems and values of characteristic function
Coalition
Vehilces Length Characteristi function value
(D1)
3
2043,4
25217,05
(D2)
3
2013,3
25066,46
(D3)
2
1852,8
19263,80
(D4)
2
2245,9
21229,39
(D1, D2)
3
3879,6
34398,07
(D1, D3)
4
2750,9
33754,39
(D1, D4)
3
3573,3
32866,71
(D2, D3)
3
2949,2
29745,90
(D2, D4)
3
3130,7
30653,36
(D3, D4)
4
2502,3
32511,41
(D1, D2, D3)
4
4127,5
40637,45
(D1, D2, D4)
4
4166,2
40830,99
(D1, D3, D4)
5
3569,8
42848,97
(D2, D3, D4)
4
3570,8
37853,79
(D1, D2, D3, D4)
5
4575,6
47878,11
As one can see in Table 1, the sum of minimum costs of companies, if there is
no cooperation, is equal to 90776.70. The minimum total costs in the case of cooperation are equal to 47878.11. Thus, the savings from cooperation in this example
are about 47 per cent.
354
Alexander Shchegryaev, Victor V. Zakharov
To share the total costs among players Shapley value is used. As Table 2 shows,
considerable reduction of costs of the companies can be achieved in comparison to
their minimum costs prior to cooperation.
It should be noted that reduction of costs of each of the companies under cooperation after redistribution of the total costs using the Shapley value was 43 to 54
percent.
Table 2: Solution for maximum coalition and cost sharing using Shapley value
Coalition Shapley Minimum costs Cost reduction
value without cooperation coefficient
(D1)
(D2)
(D3)
(D4)
5.
14382,5
11630,3
10571,1
11294,1
25217,0
25066,5
19263,8
21229,4
0,43
0,54
0,45
0,47
Dynamic Model of CVRG
Suppose CVRG has duration from 0 to T . Let the interval [0, T ] be divided by
periods t0 , t1 , . . . , tm . That is [0, T ] = (t0 , t1 , . . . , tm ). Cost functions of players for
the period [0, T ] and set of feasible routing plans (strategies) are determined in
the same way like in section 3. It is assumed that for CVRG starting from origin
t0 the characteristic function c(S, 0) is calculated by the direct coalition induction
algorithm.
For further calculations the following notation will be used:
phN (0) — is the optimal routing plan of grand coalition N in original game, which
calculated by direct coalition induction algorithm and minimizes the total costs of
the coalition for the periods t0 , t1 , . . . , tm ;
phS (0) — is the optimal routing plan of coalition S in origin game, which is
calculated by direct coalition induction algorithm and minimizes the total costs of
the coalition for the periods t0 , t1 , . . . , tm , S ⊂ N ;
phS (tk ) — profile of the optimal routing plan of the coalition S in period tk ,
S ⊆ N , k = 1, 2, . . . , m;
phN (0) = (phN (t0 ), . . . , phN (tm )) — vector of profiles of optimal routing plan;
phN,i (tk ) — optimal routing plan for vehicles of company i ∈ N in period tk as
part of optimal plan phN (tk );
c(S, k, phN (t0 ), . . . , phN (tk−1 )) — value of minimal total costs of coalition S ⊆
N after implementation the optimal routing plan calculated by direct coalition
induction algorithm.
One of the important issues of successful implementation of the routing plan
phN (0) = (phN (t0 ), . . . , phN (tm )) during all periods of the game is optimality of each
restriction of the original optimal plan phN (0) on the set of periods tk , tk+1 , . . . , tm
for k = 1, 2, . . . , m. We denote this restriction of the plan phN (0) by phN (k)) =
(phN (tk ), . . . , phN (tm )). When restriction of original optimal plan phN (0) appears to
be not optimal for at least one period tk , k = 1, 2, . . . , m, we call this plan time inconsistent. Notice that unlike Bellman optimality principle it might be happened in
routing optimization because of using heuristics instead of exact methods. To make
355
Multi-period Cooperative Vehicle Routing Games
an attempt to overcome time inconsistency of the plan formed by heuristic algorithm
we propose to realize along originally calculated plan phN (0) = (phN (t0 ), . . . , phN (tm ))
the following iterative coalition induction algorithm (ICIA).
Iterative coalition induction algorithm.
Step 1. Put k = 0.
Step 2. Assume that plan phN (k) has been implemented within the period tk .
We exclude nodes visited in the period tk from the set of customerâĂŹs nodes
and consider current CVRG(tk+1 ) under new conditions for the set of customers
to be served in periods tk+1 , . . . , t( m) and new depots location taking into account
current positions of vehicles at the end of routs executed in period tk . If heuristic
algorithm proposes new routing plan (pN (k + 1)) = (pN (tk+1 ), . . . , pN (tm )) in current CVRG(tk ) which gives less total costs for the grand coalition N for periods
tk+1 , . . . , tm than the plan phN (k + 1) = (phN (tk+1 ), . . . , phN (tm )) we make the following substitution to improve the plan considered for implementation before period
tk+1 :
phN (0)
=
(
(phN (t0 ), . . . , phN (tk )) – within the periods t0 , t1 , . . . , tk
(pN (tk+1 ), . . . , pN (tm ) – within the periods tk+1 , . . . , tm
(7)
And move to step 2 putting k = k + 1 and, if k < m. If plan ((pN (k + 1)))
in current CVRG(tk ) proposed by heuristic algorithm coincides with phN (k + 1) or
gives bigger value of total costs for grand coalition put k = k + 1, we do not make
substitution (7) and move to step 2, if k < m. In any case, if k = m go to step 3.
Step 3. Stop the procedure and use for implementation plan
h
pN (0) = (phN (t0 ), . . . , phN (tm )) which has been gotten on the last iteration.
Let phN (0) = (phN (t0 ), . . . , phN (tm )) be the optimal routing plan obtained by adjustment of the initial optimal plan with the help of the ICIA. For each period
t1 , . . . , tm along optimal routing plan phN (0) = (phN (t0 ), . . . , phN (tm )) we can calculate
values of characteristic function c(S, k, phN (t0 ), . . . , phN (tk−1 )) for current CVRG(tk )
using DCIA. Characteristic function for CVRG(t0 ) is c(S, 0). We can represent value
of the characteristic function for the grand coalition in CVRG(t0 )
c(N, 0) =
n X
m
X
cost(i, phN,i (tk )) = c(N, phN (0))
i=1 k=0
When all players form grand coalition N , for optimal routing plan phN (0) the set
of imputations in the cooperative game c(S, 0, phN (0)) = c(S, 0) will be determined
as follows
I(0, phN (0))
= {α = (α1 , α2 , . . . , αn ) : αi ≤ c({i}, 0), i = 1, . . . , n,
n
X
αi = c(N, 0)}
i=1
In this paper the Sub-Core was used as a solution of the cooperative game (Zakharov
and Kwon, 1999; Zakharov and Dementieva, 2004).
Definition 1. Sub-Core of the cooperative game c(S, 0, phN (0)) is called the set
[
SC(c(S, 0, phN (0))) =
SC(c(S, 0, phN (0)), c0 (0))
(8)
c0 (0)∈C0 (0)
356
Alexander Shchegryaev, Victor V. Zakharov
where
SC(c(S, 0, phN (0)), c0 (0)) =
!
n
n
X
0
0
h
= α=c −λ
ci (0) − C(N, 0, pN (0)) ,
i=1
λ = (λ1 , λ2 , . . . , λn ) :
n
X
i=1
λi = 1, λi ≥ 0, i = 1, 2, . . . , n
o
and C0 is the set of solutions of the following maximization problem
max
n
X
ci
i=1
provided that
X
i∈S
ci ≤ c(S, 0, phN (0)), S ⊂ N
We shall call the set C0 (0) as the basis of Sub-Core, any vector
c0 (0) = (c01 (0), c02 (0), . . . , c0n (0)) ∈ C0 (0) — as the basis imputation of the cooperative game c(S, 0, phN (0)).
By the structure the Sub-Core is not empty if and only if the Core of cooperative
game with the characteristic function c(S, 0, phN (0)) is not empty, and necessary
and sufficient condition for the Sub-Core (and hence the Core) to be not empty is
fulfillment the following inequality
X
c0i (0) ≥ c(N, 0, phN )
(9)
i∈N
Sub-Core in current CVRG(tk ) is determined by the same way. Presume that
the Sub-Core SC(c(S, k, phN (t0 ), . . . , phN (tk−1 )), c0 (k)) is not empty for k = 1, . . . , m.
Let αk = (αk1 , αk2 , . . . , αkn ) ∈ SC(c(S, k, phN (t0 ), . . . , phN (tk−1 ))), k = 0, 1, . . . , m, be
vectors of cost sharing in the current games c(S, k, phN (t0 ), . . . , phN (tk−1 )). In this
case costs of any coalition determined in the current game in accordance with the
vector αk will not exceed the values of the characteristic function for this coalition
for any value k = 0, 1, . . . , m. Thus, there is no coalition interested in leaving the
agreement at any stage of the game, which means strong dynamic stability of the
Sub-Core. By analogy with the imputation distribution procedures (IDP) discussed
e.g. in paper (Petrosyan and Zenkevich, 2009), the cost sharing procedure (CSP)
βk = (β1k , β2k , . . . , βnk ) can be considered in the multistage cooperative game, where
βik = αki − αk+1
, k = 0, 1, . . . , m − 1, i ∈ N
i
(10)
The crucial property of such procedure is fulfillment for any player i at any stage
of the game of the condition
m
X
βij = αki , k = 0, 1, . . . , m,
j=k
that we call condition of individual costs balance of the player i ∈ N .
357
Multi-period Cooperative Vehicle Routing Games
According to the definition of the Sub-Core, the following equation is valid for
the vectors αk = (αk1 , αk2 , . . . , αkn ) of cost sharing in the current games
n
X
αki = c(N, k, phN (0)), k = 0, 1, . . . , m
i=1
and taking into account (10), the following equation can be obtained
n
X
i=1
βik = c(N, k, phN (t0 ), . . . , phN (tk−1 )) − c(N, k + 1, phN (t0 ), . . ., phN (tk )),
k = 0, 1, . . . , m − 1
This condition will be called as condition of collective balance of coalition costs in
the multistage cooperative game.
Presume that the numerical value βik determines the size of payoff of the player
i within the period tk to a Costs Clearing Center (CCC), which accumulates funds
for covering the costs of all players in the process of implementation of the routing
plan phN (0) selected for implementation by coalition N . Then the economic meaning
of the condition of individual costs balance will be, that the sum of payoffs of any
player to Costs Clearing Center during the entire game will be equal to the size of
costs, which player have to pay in accordance with the selected optimal distribution
α0 = (α01 , α02 , . . . , α0n ). And collective balance of coalitional costs will provide the
possibility of covering the costs of participants of the coalition N within the same
period, when these costs are made.
6.
Example of Multi-Period CVRP
As illustration of the dynamic case, consider the static problem described earlier,
but assume now that the entire servicing time is divided into 3 equal periods.
All vehicles that are maintained in previous period by the grand coalition, begin
their movement in current period from the last serviced customer’s node. Each
company participating in one or another coalition may use additional vehicles which
begin their movement from the depot belonging to the company. The same heuristic
algorithm as in the static case (Ropke and Pisinger, 2006) is used for finding efficient
routes for CVRG in each period. To calculate characteristic function values given
in Table 3 we apply algorithms DCIA and ICIA.
Using the obtained values of the characteristic function find the basis of SubCore for each period. In this case all three maximization tasks have the unique
solution, and thus the set C0 for each period contains only of one element.
It should be noted that Sub-Core will not be empty within each period due to
fulfillment of the condition (9). In order to find certain imputation belonging to
Sub-Core within each period, the value of each component of the vector λ was set
to 0.25. After that, values of vectors βk using the obtained sharing vectors were
calculated. The calculation results are given in Table 5.
Negativity of payment values means that a company does not make payment to
CCC within the respective period, but receives in this period compensation from
CCC. Analyzing the data of Table 5, it is become clear that conditions of individual
costs balance and collective costs balance in the three-period CVRG have been
fulfilled.
358
Alexander Shchegryaev, Victor V. Zakharov
Table 3: Values of characteristic function for three periods
Coalition
The characteristic function
c(S, 0) c(S, 1) c(S, 2)
(D1)
(D2)
(D3)
(D4)
(D1, D2)
(D1, D3)
(D1, D4)
(D2, D3)
(D2, D4)
(D3, D4)
(D1, D2, D3)
(D1, D2, D4)
(D1, D3, D4)
(D2, D3, D4)
(D1, D2, D3, D4)
25217,05
25066,46
19263,80
21229,39
34398,07
33754,39
32866,71
29745,90
30653,36
32511,41
40637,45
40830,99
42848,97
37853,79
47878,11
22268,11
15418,88
20099,82
22199,77
29613,92
33180,69
34810,43
32592,22
33699,61
37683,95
40119,31
36572,44
37133,72
44562,17
38552,26
13367,79
12544,99
12327,52
12707,63
19046,92
24390,32
24110,87
19223,78
19324,51
23736,35
24834,71
20308,93
29854,66
24723,40
30364,98
Table 4: Basis of Sub-Core for three periods
Sub-Core basis
Period 1 Period 2 Period 3
Company 1
Company 2
Company 3
Company 4
All companies
16203
11208
13226
13420
54057
8720
15419
15980
12433
52552
9725
2782
12328
7802
32637
Table 5: Values of imputations and vectors βk
Period 1
α0
β0
Company
Company
Company
Company
1
2
3
4
14658
9663
11681
11875
9438
-2256
-799
2942
Period 2
α1
β1
Period 3
α2
β2
5220 -3937 9157 9157
11919 9705 2214 2214
12480 720 11760 11760
8933 1699 7234 7234
Multi-period Cooperative Vehicle Routing Games
7.
359
Conclusions
The main result of this paper is the method of constructing the characteristic function of cooperative routing game of freight carriers, which guarantees its subadditive
property. A new algorithm is proposed for solving this problem, which is called direct coalition induction algorithm (DCIA). To upgrade optimal routing plan and
values of characteristic function of grand coalition we develop iterative coalition
induction algorithm (ICIA) for dynamic CVRP. Both algorithms were built on the
basis of the combination of various heuristic algorithms which are appropriate for
solving large-scale VRP. For implementation of algorithms a special software has
been developed and used for solving sample examples.
Proposed cost sharing method allow to obtain sharing distribution procedure
which provide strong dynamic stability of cooperative agreement based on this
Sub-Core optimality principle and time consistency of the Sub-Core in multi-period
CVRG.
References
Agarwal, R., Ö. Ergun, L. Houghtalen and O. O. Ozener (2009). Collaboration in Cargo
Transportation. In Optimization and Logistics Challenges in the Enterprise. Springer
Optimization and Its Applications, 30, 373–409.
Baldacci, R., A. Mingozzi and R. Roberti (2012).Recent exact algorithms for solvingthe
vehicle routing problem under capacity and time window constraints. European Journal
of Operational Research, 218, 1–6.
Ergun, Ö., G. Kuyzu and M. W. P. Savelsbergh (2007). Shipper collaboration. Computers
& Operations Research, 34, 1551–1560.
Kallehauge B. (2008). Formulations and exact algorithms for the vehicle routing problem
with time windows. Computers & Operations Research, 35, 2307–2330.
Krajewska M. A., H. Kopfer, G. Laporte, S. Ropke and G. Zaccour (2009). Horizontal
cooperation among freight carriers: request allocation and profit sharing. Journal of the
Operational Research Society, 59, 1483–1491.
Petrosyan, L. A. and N. A. Zenkevich (2009). Principles of dynamic stability, Mat. Teor.
Igr Pril.„ 1:1, 106–123 (in Russian).
Ropke S. and D. Pisinger (2006). An adaptive large neighbourhood search heuristic for the
pickup and delivery problem with time windows. Transportation Science, 40, 455–472.
Zakharov, V., O-Hun Kwon (1999). Selectors of the core and consistency properties. Game
Theory and Applications, 4, 237–250.
Zakharov V., M. Dementieva (2004). Multistage cooperative games and problem of timeconsistency. International Game Theory Review 6, 1, 1–14.
Mechanisms of Endogenous Allocation of Firms and Workers
in Urban Area: from Monocentric to Polycentric City⋆
Alexandr P. Sidorov1
National Research University the Higher School of Economics,
Center for Market Studies and Spatial Economics,
Room K-203, b.20, Myasnitskaya st., Moscow, 101000 , Russia,
and
Sobolev Institute of Mathematics,
4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia,
E-mail: [email protected]
Abstract The purpose of paper is to investigate how the interplay of trade,
commuting and communication costs shapes economy at both inter-regional
and intra-urban level. Specifically, we study how trade affects the internal
structure of cities and how decentralizing the production and consumption
of goods in secondary employment centers allows firms located in a large
city to maintain their predominance. The feature of approach is using of
two-dimensional city pattern instead of the “long narrow city” model.
Keywords: city structure, secondary business center, commuting cost, trade
cost, communication cost.
1.
Introduction
Spatial economics has acquired new life since publication of Krugman’s (1991) pioneering paper. Combined increasing returns, imperfect competition, commodity
trade and the mobility of production factors Krugman has formed his now famous “core-periphery” model. Such a combination contradicts to the mainstream
paradigm of constant returns and perfect competition, which has dominated in
economic theory for a long time. Furthermore, to the trade-off between increasing
returns and transport costs Krugman (1980) has added a third factor: the size of
spatially separated markets. The main achievement of New Economic Geography
(NEG) was to show how market size interacts with scale economies internal to firms
and transport costs to shape the space-economy.
In NEG, the market outcome arises from the interplay between a dispersion
force and an agglomeration force operating within a general equilibrium model. In
Krugman (1991) and Fujita et al. (1999), the dispersion force ensures from the spatial immobility of farmers. As for the agglomeration force, Krugman (1991, p.486)
noticed that circular causation a la Myrdal (1957) takes place because the following
two effects reinforce each other: “manufactures production will tend to concentrate
where there is a large market, but the market will be large where manufactures
production is concentrated.”
In this framework, however, the internal structure of regions was not accounted
for. In the present paper we consider NEG models which allows for the internal
structure of urban agglomerations through the introduction of a land market. To be
⋆
This work was supported by the Russian Foundation for Fundamental Researches under
grants No.99-01-00146 and 96-15-96245.
Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area
361
precise, we start by focusing on the causes and consequences of the internal structure
of cities, because the way they are organized has a major impact of the well-being
of people. In particular, housing and commuting costs, which we call urban costs,
account for a large share of consumers’ expenditures. At this point we are agree
with Helpman (1998) for whom urban costs are the main dispersion force at work
in modern urbanized economies. In our setting, an agglomeration is structured as a
monocentric city in which firms gather in a central business district. Competition
for land among consumers gives rise to land rent and commuting costs that both
increase with population size. In other words, our approach endows regions with an
urban structure which is absent in standard NEG models.
As a result, the space-economy is the outcome of the interaction between two
types of mobility costs: the transport costs of commodities and the commuting costs
borne by workers. Evolution of commuting costs within cities, instead of transport
costs between cities, becomes the key-factor explaining how the space-economy is organized. Moreover, despite the many advantages provided by the inner city through
an easy access to highly specialized services, the significant fall in communication
costs has led firms or developers to form enterprise zones or edge cities (Henderson
and Mitra 1996). We then go one step further by allowing firms to form secondary
business centers. This analysis shows how polycentricity alleviates the urban of urban costs, which allows a big city to retain its dominant position by accommodating
a large share of activities.
Creation of subcenters within a city, i.e. the formation of a polycentric city,
appears to be a natural way to alleviate the burden of urban costs. It is, therefore,
no surprise that Anas et al. (1998) observe that “polycentricity is an increasingly
prominent feature of the landscape.” Thus, the escalation of urban costs in large
cities seems to prompt a redeployment of activities in a polycentric pattern, while
smaller cities retain their monocentric shape. However, for this to happen, firms set
up in the secondary centers must maintain a very good access to the main urban
center, which requires low communication costs.
Trying to explain the emergence of cities with various sizes, our framework, unlike Helpman (1998), Tabuchi (1998) and others, allows cities to be polycentric.
Moreover, in contrast to Sullivan (1986), Wieand (1987), and (Helsley and Sullivan, 1991), in our treatment, there are no pre-specified locations or numbers of
subcenters, and our model is a fully closed general equilibrium spatial economy. As
mentioned above, emergence of additional job centers is based on the urge towards
decreasing of urban costs, rather than consumer’s “propensity to big malls”, as suggested by Anas and Kim (1996). Our approach, that takes into account various
types of costs (trade, commuting, and communication) is similar to Cavailhès et al.
(2007) with one important exception. We drop very convenient (yet non-realistic)
assumption on “long narrow city.” Our analysis is extended to the two-dimension
because the geographical space in the real world is better approximated by a twodimensional space.
2.
2.1.
Model overview
Spatial structure
Consider an economy with G ≥ 1 regions, separated with physical distance, one
sector and two primary goods, labor and land. Each region can be urbanized by
accommodating firms and workers within a city, and is formally described by a two-
362
Alexandr P. Sidorov
dimensional space X = R2 . Whenever a city exists, it has a central business district
(in short CBD) located at the origin 0 ∈ X.
Firms are free to locate in the CBD or to set up in the suburbs of the metro where
they form secondary business districts, SBD in short. Both the CBD and SBDs are
assumed to be dimensionless. In what follows, the superscript C is used to describe
variables related to the CBD, whereas S describes the variables associated with a
SBDs. We consider the case where the CBD of urbanized region g is surrounded by
mg ≥ 0 SBDs; mg = 0 corresponds to the case of monocentric city. Without loss of
generality, we focus on the only one of SBDs, because all SBDs are assumed to be
identical.
Even though firms consume services supplied in each SBD, the higher-order
functions (specific local public goods and non-tradable business-to-business services such as marketing, banking, insurance) are still located in the CBDs. Hence,
for using such services, firms set up in a SBD must incur a communication cost,
K > 0. In paper of Cavailhès et al. (2007) more general communication cost function
K(xS ) = K + k · ||xS || was used, where k > 0, and ||xS || is a distance between CBD
and SBD. This generalization does not change the nature of our results, though
analytical calculation became more tedious. Both the CBD and the SBD are surrounded by residential areas occupied by workers. There is no overlapping between
residence zones. Furthermore, as the distance between the CBD and SBD is small
compared to the intercity distance, we disregard the intra-urban transport cost of
goods. Note that using the more general type of communication cost with k > 0
leads to consequence that in equilibrium Central and any Secondary residence zones
should be adjacent to each other. This condition is non-necessary for fixed communication cost, although the real SBD can not be placed too far from City Center.
Under those various assumptions, the location, size and number of the SBDs
as well as the size of the CBD will be endogenously determined. In other words,
apart from the assumed existence of the CBD, the internal structure of each city is
endogenous.
2.2.
Workers/Consumers
The economy is endowed with L workers, distributed across the regions, where
G
X
population of city g is lg , i.e.,
lg = L. In this paper our primary focus is on
g=1
the intra-city cost effects and on the trade, therefore the distribution of labor is
considered as exogenous. The welfare of a worker depends on her consumption of
the following three goods. The first good is unproduced and homogeneous. It is
assumed to be costlessly tradable and chosen as the numéraire. The second good is
produced as a continuum n of varieties of a horizontally differentiated good under
monopolistic competition and increasing returns, using labor as the only input. Any
variety of this good can be shipped from one city to the other at a unit cost of τ > 0
units of the numéraire. The third good is land; without loss of generality, we set the
opportunity cost of land to zero. Each worker living in city 1 ≤ g ≤ G consumes a
residential plot of fixed size chosen as the unit of area. The worker also chooses a
quantity q(i) of variety i ∈ [0, n], and a quantity q0 of the numéraire. She is endowed
with one unit of labor, which is supplied absolutely inelastically.
Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area
363
Preferences over the differentiated product and the numéraire are identical across
workers and cities and represented by Ottaviano’s quasi-linear utility function
U (q0 ; q(i), i ∈ [0, n]) = α
Zn
0
q(i)di −
β
2
Zn
0
 n
2
Z
γ
[q(i)]2 di −  q(i)di + q0
2
(1)
0
where α, β, γ > 0. Demand for these products (provided that job and location
are already chosen) is determined by maximizing of utility subject to the budget
constraint
Zn
ALRg
p(i)q(i)di + q0 + Rg (x) + Tg (x) = wg (x) +
,
(2)
lg
0
where Rg (x) is the land rent prevailing
at location x, Tg (x) is commuting cost,
Z
wg (x) is the wage, and ARLg =
Rg (x)dx is an aggregated land rent in the city
x∈X
g. This form of the budget constraint suggests that there are no landlords, who
appropriate the land rent, moving it out of city budget. In other words, land is in
a joint ownership of all citizen.
Each worker commutes to her employment center – without cross-commuting –
and bears a unit commuting cost given by t > 0, so that for the worker located at x
the commuting cost, Tg (x), is either t||x|| or t||x−xSg || according to the employment
center. Moreover, the wage wg (x) depends only on type of employment center and
takes one of two possible values: wage in CBD, wgC , or wage in SBD, wgS , which is
uniform across all SBDs. Thus, the budget constraint of an individual working in
the CBD is as follows
Zn
0
p(i)q(i)di + q0g + RgC (x) + t||x|| = wgC +
ALRg
,
lg
(3)
while for individuals working in the SBD, located at xSg , it takes the form
Zn
0
2.3.
p(i)q(i)di + q0g + RgS (x) + t||x-xSg || = wgS +
ALRg
.
lg
(4)
Firms
Our basic assumption on the manufacturing technology is that producing q(i) units
of variety i requires a given number ϕ of labor units. One may assume that producing
one unit of variety i requires additionally c ≥ 0 units of numéraire. This is not
significant generalization, however, because this model is technically equivalent to
one with c = 0 (see Ottaviano et al., 2002).
There is no scope economy so that, due to increasing returns to scale, there is
a one-to-one relationship between firms and varieties. Thus, the total number of
firms is given by n = L/ϕ. Labor market clearing implies that the number of firms
located (or varieties produced) in city g is such that ng = λg n, where λg = lg /L
stands for the share of workers residing in g. Denote by ΠgC (respectively ΠgS ) the
364
Alexandr P. Sidorov
profit of a firm set up in the CBD of city g (respectively the SBD). When the firm
producing variety i is located in the CBD, its profit function is given by:
ΠgC (i) = Ig (i) − ϕ · wgC ,
where
Ig (i) = pgg (i) · Qgg (i) +
X
f 6=g
(5)
(pgf (i) − τ ) · Qgf (i)
stands for the firm’s revenue earned from local sales Qgg (i) and from exports Qgf (i)
from city g to various cities f . When the firm sets up in the SBD of the same city,
its profit function becomes:
ΠgS (i) = Ig (i) − ϕ · wgS − K.
(6)
The firm’s revenue is the same as in the CBD because shipping varieties within the
city is costless, so that prices and outputs do not depend on firm’s location in the
city.
3.
Urban Costs and Decentralization within a City
A city equilibrium is such that each individual maximizes her utility subject to her
budget constraint, each firm maximizes its profits, and markets clear. Individuals
choose their workplace (CBD or SBD) and their residential location with respect to
given wages and land rents. In each workplace, the equilibrium wages are determined
by a bidding process in which firms compete for workers by offering them higher
wages until no firm can profitably enter the market. Given such equilibrium wages
and the location of workers, firms choose to locate either in the CBD or in the SBD.
At the city equilibrium, no firm has an incentive to change place within the city. To
ease the burden of notation, we drop the subscript g.
3.1. Land rents and Wage wedge
Let Ψ C (x) and Ψ S (x) be the bid rent at x ∈ X of an individual working, respectively, in the CBD and in the representative SBD. Land is allocated to the highest
bidder. An opportunity cost of land (e.g., for agricultural use) is assumed to be
zero. Urban costs (commuting and communication) increase with Euclidean distance, thus “efficient” shapes of both Central and Secondary residence zones are
circles. All locations with the same distance to the corresponding Business District
(Central or Secondary) are equivalent with respect to urban cots. Because there is
only one type
of labor, at the
city equilibrium it must be that the housing rent
R(x) = max Ψ C (x), Ψ S (x), 0 . Within each city, a worker chooses her location so
as to maximize her utility U (q0 , q(i); i ∈ [0, n]) under the corresponding budget
constraint, (3) or (4).
Because of the fixed lot size assumption, at the city equilibrium the value of the
equilibrium consumption of the nonspatial goods
Zn
p(i)q(i)di + q0 = E
(7)
0
is the same regardless of the worker’s location:
wC +
ALR
ALR
−RC (x′ )−t||x′ || = E C (x′ ) ≡ E S (x′′ ) = wS +
−RS (x′′ )−t||x′′ −xS ||
l
l
Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area
365
for all x′ , x′′ , belonging to CBD and SBD residence zones, respectively. To ensure
this, we assume for now, that the share of firms located in the CBD, θ, is given,
then (1 − θ)/m is the share of firms in each SBD.
Proposition 1. For any given city population l, SBD number m, and CBD share
of firms θ:
i) Central zone radius rC and SBD zone radius rS are as follows:
r
r
θl
(1 − θ)l
C
S
r =
, r =
.
(8)
π
mπ
ii) The following land rent function equalizes the disposable income E for all
central and suburb residence locations x:
)
( r
r
θl
(1 − θ)l
S
− ||x||,
− ||xk − x|| ,
(9)
R(x) = t · max 0,
1≤k≤m
π
mπ
m
where xSk k=1 is a set of all SBD locations.
iii) Redistributed aggregated land rent:
r Z
ALR
1
t
l 3/2 (1 − θ)3/2
√
=
R(x)dx = ·
θ
+
.
l
l
3
π
m
(10)
X
iv) In equilibrium there exists the positive wage wedge between CBD and SBD
!
r
r
θl
(1 − θ)l
C
S
w −w = t·
−
(11)
π
mπ
which is non-negative for all θ ∈
1
,1 .
1+m
For analytical proof see Appendix. Figure 1 presents the plot of function R(x)
for m = 4.
3.2. Urban Costs
Let’s define urban cost function as a sum of rent and commuting costs minus the
ALR 1
individual share of aggregated land rent
. Due to (9) and (10) these urban
l
costs are as follows
r
r ALR
t
θl
l 3/2 (1 − θ)3/2
C
C
√
Cu = Ψ (x) + t||x|| −
=t
− ·
θ
+
,
l
3
π r
m
rπ
ALR
(1 − θ)l
t
l 3/2 (1 − θ)3/2
S
S
S
√
Cu = Ψ (x) + t||x − x || −
=t
− ·
θ
+
.
l
π
3
π
m
(12)
The city equilibrium implies that the identity wC − CuC = wS − CuS holds. In these
terms, the wage wedge identity may be rewritten as a difference between urban costs
in CBD and SBD: wC − wS = CuC − CuS .
1
For technical reasons it is convenient to treat ALR
as some kind of rent compensation,
l
subtracting it from costs rather adding to wage.
366
Alexandr P. Sidorov
Fig. 1: Rent function R(x)
3.3.
Equilibrium city structure
Regarding the labor markets, the equilibrium wages of workers are determined by
the zero-profit condition. In other words, operating profits are completely absorbed
by the wage bill. Hence, the equilibrium wage rates in the CBD and in the SBDs
must satisfy the conditions Π C (wC∗ ) = 0 and Π S (wS∗ ) = 0, respectively. Thus,
setting (5) (respectively (6)) equal to zero, solving for wC∗ (respectively wS∗ ), we
get:
I −K
I
(13)
wC∗ = , wS∗ =
ϕ
ϕ
K
> 0, due to (8). Comparing the previous formula with (11)
ϕ
we obtain that CBD share of firms, θ satisfies the identity
p
√
√
ϕt mθl = K mπ + ϕt (1 − θ)l.
(14)
Hence wC∗ − wS∗ =
Admissible solution θ∗ of equation (14) will be referred as equilibrium CBD share.
πK 2
then the unique solution of equation (14) is θ∗ = 1
ϕ2 t2
with m = 0, i.e. city is monocentric;
πK 2
ii) Let l > 2 2 then for each m ≥ 1 equation (14) has unique solution θ∗ ∈
ϕ t
1
, 1 , i.e. there exists a unique equilibrium SBD share of firms.
1+m
iii) The CBD share of firms θ∗ decreases with respect to population l, number of
SBDs m and commuting costs t. Moreover, θ∗ increases with respect to communication cost K and
Proposition 2. i) Let l ≤
lim θ∗ = lim θ∗ = lim θ∗ =
l→∞
t→∞
For analytical proof see Appendix.
K→0
1
.
1+m
(15)
Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area
367
r
l
and note that it is in fact a radius of monocentric
π
πK 2
city with population l. Inequality l < 2 2 holds if and only if ϕt · rM (l) > K. The
ϕ t
left-hand side of this inequality is total commuting costs of firm’s workers, residing at
periphery of monocentric city in case of firm’s location at CBD. To hire ϕ workers
from periphery, firm should compensate their commuting costs in wage. On the
other hand, locating the firm at the periphery causes the lesser communication cost
K. Thus, producing on periphery (in SBD) is more efficient for new firm entering
the industry. For any given K we obtain minimum polycentric city population:
πK 2
lP = 2 2 . If city population l ≤ lP the corresponding central share θ∗ ≡ 1, i.e.
ϕ t
city pattern is monocentric. It is not surprising that increasing in commuting costs
t leads to lager dispersion of firms and workers. For very large magnitude of t,
communication costs K become negligible and the distribution of production across
all business centers is almost uniform.
Remark 1. Let r (l) =
M
Substituting equilibrium SBD share θ∗ (m, l, t) into the urban cost function
#
r
r "
r
θl
t
l
1−θ
C
3/2
Cu = t
− ·
θ
+ (1 − θ)
π
3
π
m
and taking into account that
r
√
√
1 − θ∗
K π
∗
√ ,
= θ −
m
ϕ·t l
which follows from equation (14), we obtain that the urban cost function
r
2t θ∗ (l, m, t) · l
K
C
Cu (l, m, t) =
+
· (1 − θ∗ (l, m, t)).
3
π
3ϕ
(16)
In particular,
r
2
l
= t
for m = 0 and l ≥ 0
π
r3
2
l
πK 2
CuC (l, m, t) = t
for all m > 0 and l ≤ 2 2 ,
3
π
ϕ t
because in these cases θ∗ ≡ 1.
CuC (l, 0, t)
Proposition 3. Function CuC (l, m, t) is continuous for all m ≥ 0, l ≥ 0, t ≥ 0 and
continuously differentiable function for m > 0, l > 0, t > 0. Moreover, CuC (l, m, t)
strictly increases with respect to l and t, strictly decreases with respect to m for all
l > lP .
For analytical proof see Appendix. Figure 2 represents results of Proposition 3 in
visual way as simulation in Wolfram’s Mathematica 8.0.
Remark 2. Note that urban cost function Cu is concave with respect to l. It may
reflect the fact that the housing price at periphery of residence zone increases with
l sufficiently slow. The newcomers reside at the periphery, where the housing rent is
very small. Moreover, unlike the linear model, in two-dimensional case this periphery
enlarges as the city population grows. Though immigration increases competition
for housing, an increment of the per capita urban costs Cu is less than before.
368
Alexandr P. Sidorov
Fig. 2: Comparative statics of urban costs
4.
Inter-City Equilibrium
Until now we studied equilibrium decentralization within the city, or Intra-City
equilibrium. Let’s turn to Inter-City equilibrium assuming that the city populations
lg and numbers of SBD mg are given for each city g. This paper focuses mainly on
trade aspects, putting aside labor migration, therefore this assumption is consistent.
Some considerations on endogenezation of SBD number are discussed at the end
of this Section. Equilibrium shares of firms, θg∗ , located at CBD, may be obtained
independently, as solutions of equation (14) for each city g. These shares, in turn,
allow to determine the urban costs Cug , which do not depend on inter-city trade
(and even on existence of other cities). On the contrary, wage


X
1
wgC = pgg (i) · Qgg (i) +
(pgf (i) − τ ) · Qgf (i) ,
ϕ
f 6=g
substantially depends on trade, as well as consumer’s utility U (q0 ; q(i), i ∈ [0, n]).
Moreover, if trade costs are too large, e.g., τ ≥ pgf (i), export is non-profitable and
firms choose the domestic sales only, which implies
wgC =
pgg (i) · Qgg (i)
.
ϕ
Now we split the study of equilibrium into two sub-cases: Equilibrium under
Autarchy and Equilibrium with Bilateral Trade.
4.1. Equilibrium under Autarchy
This case suggests that equilibrium is separately established for each city, hence we
may drop subscript g and consider the city with population l and SBD number m.
Moreover, assume that the number of firms n is given and condition wC − CuC > 0
holds. What determines n and how to provide this consumers’ “surviving condition”
will be discussed at the end of this subsection.
Representative consumer maximizes utility
U (q0 ; q(i), i ∈ [0, n]) = α
Zn
0
β
q(i)di −
2
Zn
0
 n
2
Z
γ
[q(i)]2 di −  q(i)di + q0
2
0
Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area
subject to
Zn
0
369
p(i)q(i)di + q0 = wC − CuC ,
First of all, recall some well-known results concerning consumer’s problem with this
form of utility.
Lemma 1. Consumer’s demand is linear function
q(i) =
where P =
Rn
α
1
γ
− p(i) +
· P,
β + γn β
(β + γn)β
p(i)di is price index. Equilibrium prices and demand of representative
0
are uniform by goods
p∗ (i) ≡ p∗ =
αβ
α
, q ∗ (i) ≡ q ∗ =
.
2β + γn
2β + γn
Consumer’s surplus at equilibrium is equal to
CS =
α2 n(β + γn)
.
2(2β + γn)2
For analytical proof see Ottaviano et al. (2002). Using this lemma and taking into
l
account that n =
we obtain the terms of equilibrium wage at CBD
ϕ
wC∗ =
l · p∗ · q ∗
α2 βϕl
=
2
ϕ
(2βϕ + γl)
and consumer’s surplus
α2 (βϕ + γl)l
,
2(2βϕ + γl)2
which does not depend on consumer residence. Moreover sum of wage and consumer
surplus (urban gains, for short) is
CS =
C∗
GC
=
u = CS + w
α2 (3βϕ + γl)l
.
2(2βϕ + γl)2
Finally, consumer’s welfare in CBD is a difference of urban gains and urban costs
V C = CS + wC∗ − CuC .
Similar to CBD we may calculate the corresponding SBD’s characteristics: wage
!
r
r
α2 βϕl
θ∗ l
(1 − θ∗ )l
S∗
w =
−
,
2 −t
π
mπ
(2βϕ + γl)
urban gains
GSu = CS + wS∗
α2 (3βϕ + γl)l
=
−t
2(2βϕ + γl)
r
θ∗ l
−
π
r
(1 − θ∗ )l
mπ
where θ∗ is solution of equation (14). Note that indirect utility
V S = CS + wS∗ − CuS ≡ CS + wC∗ − CuC = V C .
!
,
370
Alexandr P. Sidorov
Proposition 4. Wage function wC∗ (l) strictly increases for all 0 ≤ l <
strictly convex for l >
2βϕ
. Moreover,
γ
lim wC∗ (l) = 0, wC∗ (0) = 0,
l→+∞
2βϕ
and
γ
∂wC∗
α2
(0) =
< +∞.
∂l
2
Urban gains GC
u (l) strictly increase for all l ≥ 0,
lim GC
u (l) =
l→+∞
α2
, GC
u (0) = 0.
2γ
Proof of this proposition is straightforward from the formulas of wC∗ (l) and GC
u (l).
“Surviving” condition It is obvious that city equilibrium is consistent only if disposable income wC∗ (l) − CuC (l, m, t) ≥ 0, which is called Surviving condition. Feasibility of this condition depends on magnitude of commuting cost t: wage function
wC∗ is bounded and does not depend on t, while urban cost CuC (l, m, t) increases
unrestrictedly with t. As result, very large commuting cost makes the city formation
impossible.
K
3α2
Proposition 5. Let inequality
<
holds, then for any commuting cost t ∈
ϕ
16γ
r
K
πγ
0,
and any given SBD number m ≥ 0 there exist numbers 0 < lmin (m, t) <
ϕ 2βϕ
lmax (m, t) < ∞, such that inequality wC (l) − CuC (l, m, t) ≥ 0 holds if and only if
lmin (m, t) ≤ l ≤ lmax (m, t). Moreover, if m′ > m, then lmin (m′ , t) ≡ lmin (m, t) <
lmax (m, t) ≤ lmax (m′ , t) and lP < l∗ ⇒ lmax (m, t) ≤ lmax (m′ , t).
For analytical proof see Appendix.
K
3α2
<
is equivalent to
ϕ
16γ
r
α2
2t lP
2K
C∗
C P
= max w (l) > Cu (l , m, t) =
≡
,
l≥0
8γ
3
π
3ϕ
Remark 3. Note that inequality
which implies that the maximum possible wage exceeds the urban costs in the city
with minimum polycentric city population lP . The lack of this condition means that
the production transfer to SBD is ineffective, because per employee communication
K
cost
is too large.
ϕ
Increasing of m broadens interval [lmin (m, t), lmax (m, t)] (to be more precise, lmin
is not affected by changes in SBD number, while lmax increases with respect to
m). Moreover, disposable income wC (l) − CuC (l, m, t) and welfare V C = GC
u (l) −
CuC (l, m, t) both increases with respect to m for all l > lP . Figure 3 illustrates the
equilibrium existence under autarchy and comparative statics of lmax with respect
to m using simulation in Wolfram’s Mathematica 8.0.
Remark 4. Previous considerations show that autarchy may be very restrictive
to the city sizes: city survives only if its size exceeds the lower threshold lmin > 0
Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area
371
Fig. 3: Comparative statics of the population limits
and does not exceed the upper one lmax . It is not surprising, because self-sufficient
settlement of industrial type may exists only if its population is sufficiently large.
Moreover, unrestrictedly growing urban costs (in particular, commuting cost) eventually stop the city growth. Developing of the city infrastructure (i.e. increasing in
m) shifts up the upper bound lmax , but cannot affect the lower critical point lmin .
4.2. Endogenous SBD number
The concluding remark concerns the question: How to endogenize SBD number?
There is no simple and unambiguous answer, because in practice it depends on many
factors. One of the main questions is “Who can afford the building of additional
suburb?” If answer is “None”, we find ourself in setting with predefined number of
SBDs (like model of Cavailhès et al., 2007). Otherwise, we assume that decision is
up to ‘City Developer’, who takes into account the social welfare considerations.
For example, when city population reaches the upper bound lmax , an increasing the
number of subcenters is urgently needed. Let’s determine the following “compelled”
SBD number for given population l and commuting cost t:
m∗ (l, t) = min {m | l ≤ lmax (m, t)} .
Proposition 6. SBD number m∗ is non-decreasing function with respect to the
city population l and commuting costs t, i.e., for all l′ > l, t′ > t the following
inequalities hold:
m∗ (l′ , t) ≥ m∗ (l, t), m∗ (l, t′ ) ≥ m∗ (l, t).
Proof. The statement concerning city population l is obvious: city is monocentric (m∗ = 0) until population l exceeds lmax (0, t). By Proposition 5 upper bound
lmax (1, t) > lmax (0, t), thus while l ≤ lmax (1, m) the current SBD number m∗ = 1,
until l exceeds this upper bound,
e.t.c. Increasing in commuting cost leads to
decreasing of lmax (m, t) = sup l | wC∗ (l) ≥ CuC (l, m, t) , because CuC (l, m, t) increases with respect to t by Proposition 3. Therefore, if l > lmax (m, t′ ) for t′ > t then
to recover surviving condition we need to increase SBD number until lmax (m′ , t′ ) ≥ l.
372
Alexandr P. Sidorov
Remark 5. Although this mechanism of endogenezation is not perfect, that theoretical comparative statics is fully supported by empirical evidences (see MacMillen
and Smith, 2003). Anyway, it determines rather the endogenous minimum of SBD,
which may be increased by some another reason, for example, to increase social
welfare, i.e., total indirect utility of the city population.
Fig. 4: Disposable income and Welfare
Parametric Example Consider the numerical example of how may change the
inner structure of city under increasing of population size. Parameter values are
chosen as follows: ϕ = 5, K = 4, t = 1, α = 6, β = 4, γ = 1. Under these
assumptions the lower bound of the city population lmin ≈ 0.75 is very small and
once the city is grounded it starts to attract people, e.g., from rural neighborhood.
Moreover, disposable income w − Cu increases very quickly with respect to city size
Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area
373
l at early stage, then it reaches the maximum and go down to zero when population
size is close to upper population bound lmax (m). Its magnitude depends on city
structure, i.e., number of SBD. For example, monicentric city reaches its maximum
at lmax (0) ≈ 97.5, while for m = 3 the upper bound (or city capacity) is much larger,
lmax (3) ≈ 156.5. The plots of disposable income for m = 0, 1, 2, 3 are presented at
Figure 4a.
However, taking into account Consumer’s Surplus along with Disposable Income we obtain that the resulting Consumer’s Welfare (i.e., Indirect utility) V =
CS + w − Cu tends to grow further with respect to population size. It implies that
there is a strong incentive for City Developer to increase the SBD number m, which
in turn raises the city capacity. Of course, this expansion could be done “in advance”,
i.e., before the population size reaches the maximum. It is not so easy to predict,
however, when it happens, thus the “cautious strategy” of City Developer is presented at Figure 4b by bold line, i.e., an additional SBD appears only if capacity of
the city is exhausted. Moreover, we have assumed that the building of new SBD is
costless, but this is not the case in real world. Thus, the expansion m → m + 1 will
be well-grounded when per capita effect (welfare gap) reaches the maximum, i.e., at
current lmax (m). It can be easily observed that this welfare leap quickly decreases
with any next expansion of city structure, which eventually stops the increasing of
the city population.
4.3. Bilateral Trade Equilibrium
The current subsection tell us what changes if trade comes to the place. To simplify
description, assume that there are two cities, Home and Foreign. Let λ be the share
of workers residing in Home city, then populations of both cities are lH = λL
and lF = (1 − λ)L, respectively. Moreover, the equilibrium masses of firms are
nH = lH /ϕ = λ · n, nF = lF /ϕ = (1 − λ) · n, where n = L/ϕ is a total mass of firms
in the world. Demands of Home representative consumer for domestic and imported
differentiated goods, qHH (i) and qF H (i) respectively, are determined as solution of
consumer problem
max U (q0 ; q(i), i ∈ [0, nH + nF ])
subject to
ZnH
pHH (i)qHH (i)di +
nH
Z+nF
nH
0
C
C
pF H (i)qF H (i)di + q0 = EH = wH
− CH
.
(17)
Similarly demands of Foreign representative consumer, qF F (i) and qHF (i), are determined as solution of
subject to
ZnF
0
max U (q0 ; q(i), i ∈ [0, nH + nF ])
pF F (i)qF F (i)di +
nH
Z+nF
nF
pHF (i)qHF (i)di + q0 = EF = wFC − CFC .
Facing these demands, firms maximize profits
IH (i) = λL · pHH (i) · qHH (i) + (1 − λ)L · [pHF (i) − τ ] · qHF (i)
IF (i) = (1 − λ)L · pF F (i) · qF F (i) + λL · [pF H (i) − τ ] · qF H (i)
(18)
374
Alexandr P. Sidorov
and obtain optimal (equilibrium) prices and quantities. Zero-profit condition (13)
determines equilibrium wages. It should be mentioned that bilateral trade is profitable only if trade costs τ are sufficiently small: pHF (i) > τ and pF H (i) > τ . The
following results are well-known, see, for example, original papers of Ottaviano et
al. (2002) and Cavailhès et al. (2007).
Lemma 2. Trade equilibrium prices are uniform by goods
p∗HH (i) ≡ p∗HH =
2αβ + τ γnF
2αβ + τ γnH
, p∗F F (i) ≡ p∗F F =
,
2(2β + γn)
2(2β + γn)
p∗HF = p∗F F +
τ
τ
, p∗F H = p∗HH + ,
2
2
as well as equilibrium demands
∗
qHH
(i)
≡
∗
qHH
qF∗ F (i) ≡ qF∗ F
Consumer’s surplus
1
τγ
τ
∗
∗
α − pHH +
nF , qF∗ H = qHH
−
=
β + γn
2β
2β
1
τγ
τ
∗
=
α − p∗F F +
nH , qHF
= qF∗ F −
β + γn
2β
2β
α2 n
α
−
· [p∗HH · nH + p∗F H · nF ] +
2(β + γn) β + γn
i
1 h ∗ 2
γ
2
2
+
· (pHH ) · nH + (p∗F H ) · nF −
· [p∗HH · nH + p∗F H · nF ]
2β
2β · (β + γn)
CSH =
Bilateral trade is profitable if τ < τtrade =
2αβ
.
2β + γn
For analytical proof see Appendix.
λL
(1 − λ)L
L
Substituting
for nH ,
for nF and for n we obtain the equilibrium
ϕ
ϕ
ϕ
prices and quantities for the Bilateral Trade Equilibrium. We focus on the Home
city only, considerations for Foreign city are similar, mutatis mutandis. Without
loss of generality, we may assume that L ≤ lmax (mH ), which implies, in particular,
C∗
wH
(1) ≥ CuC (1). It allow us to consider the whole unit interval (0, 1) as a set of
admissible values for λ instead of truncation (0, lmax (mH )/L).
Bilateral trade changes magnitudes of wage, consumer’s surplus and indirect
utility in comparison to autarchy case. To discriminate these cases, we add τ to
notions of values, which are affected by trade. Recall that urban costs Cu (λ) does
not depend on τ . The following results are well-known (see, for example, Ottaviano
et al. (2002) and Cavailhès et al. (2007)).
Lemma 3. Home Equilibrium wage
C∗
wH
(λ, τ ) =
βϕL
(2βϕ + γL)2
"
#
2
2
τ γL
τ γL
α+
(1 − λ) · λ + (α − τ ) −
(1 − λ) · (1 − λ)
2βϕ
2βϕ
(19)
Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area
375
is strictly concave function, increasing at λ = 0.
Home Consumer’s Surplus
CSH (λ, τ ) =
α2 L
αL
−
· [p∗HH · λ + p∗F H · (1 − λ)] +
2(βϕ + γL) βϕ + γL
i
L h ∗ 2
γL2
2
2
· (pHH ) · λ + (p∗F H ) · (1 − λ) −
·[p∗ · λ + p∗F H · (1 − λ)]
2βϕ
2βϕ · (βϕ + γL) HH
(20)
is strictly increasing and concave function of λ.
+
Proof is straightforward (though tedious) from Lemma 2.
Proposition 7. i) There exists 0 < τ ∗ < τtrade such that for all τ ∈ (0, τ ∗ ) inequality wC∗ (λ) > Cu (λ) holds for all λ ∈ (0, 1).
ii) There exists 0 < τ ∗∗ < τtrade such that for all τ ∈ (0, τ ∗∗ ) indirect utility
with trade
C∗
C
VH (λ, τ ) = CSH (λ, τ ) + wH
(λ, τ ) − CuH
(λ)
C∗
exceeds the corresponding utility under autarchy VH (λ) = CSH (λ) + wH
(λ) −
C
CuH (λ) for all λ ∈ (0, 1).
For analytical proof see Appendix. Typical results of simulation are presented at
Figure 5.
Fig. 5: Autarchy and Trade
Remark 6. Proposition 7(i) implies that sufficiently free trade cancels the lower
bound of city size lmin , i.e. small cities could survive, trading with the larger ones.
It looks like small city became quasi-SBD for large one, replacing communication
cost with trade cost. On the other hand, trade cannot cancel the upper bound,
or maximum city capacity. Thus, all considerations endogenous SBD number from
subsection 3.2 are still valid. This proposition cannot be generalized for all τ ∈
(0, τtrade ). Computer simulations show that for τ sufficiently close to τtrade both
statements, (i) and (ii), are violated.
376
5.
Alexandr P. Sidorov
Conclusion
Paradigm of linear city is well suited for both actual “long narrow cities” and monocentric “two-dimensional”, because in this case location may be characterized by
scalar value – distance from Central Business District. In case of polycentricity –
especially, with multiple Secondary Business Districts – linear model can’t include
all range of possibilities, being limited at most by two SBDs. Two-dimensional
polycentric model, presented in this paper, lacks this disadvantage, while it is still
tractable and intuitive. The results obtained in presented paper are of two kinds:
some of them are common for both linear and two-dimensional models, while other
are specific for two-dimensional model with several Secondary Business Districts.
We discuss here these results, focusing on the specific ones.
Proposition 2 on Existence and Uniqueness of equilibrium CBD share implies
that polycentric structure may exists only if population of city exceeds the certain
threshold, i.e., too small city cannot bear the burden of polycentricity. This natural
result is not 2D specific, nevertheless, it contains the statement that city with population beyond this threshold, could have any number of SBDs. Moreover, increasing
in this number implies that per capita urban costs strictly decrease (see Proposition
3). It results in increasing (ceteris paribus) of disposable income and indirect utility
of the city residents, therefore, developing of the inner city structure may be an
important policy instrument.
It is obvious, that positiveness of disposable income is necessary condition for city
residents. One of results obtained in this paper is that disposable income is positive
if and only if city population is not less than strictly positive lower threshold a do
not exceeds the finite upper bound (see Proposition 5). It means that the effective
production (with increasing return to scale) cannot be developed on the base of too
small settlement, and, vice versa, very large city cannot survive because of too heavy
burden of urban costs. Increasing in SBD number shifts up the upper threshold (i.e.,
increases city capacity), therefore, extensive development of the city structure can
be an effective policy instrument for sufficiently large cities (see Proposition 5). It
cannot help, however, small cities to survive as industrial settlements.
Changes in city structure is mainly an instrument of inner policy, while change
in trade openness may results outwards. Moreover, sufficiently high level of trade
openness (i.e., sufficiently small trade costs) shifts down to zero the lower threshold of city population (see Proposition 6). It means that under condition of almost
free trade, small cities could survive as satellites of large ones. Another benefit
of sufficiently free trade is that real wage (indirect utility) increases for residents
in all cities, not depending on their sizes (see Proposition 6), although this effect
is more significant for small cities. It increases the relative attractiveness for the
labor inflow. This inflow may result in overpopulation of city with given number
of SBDs. To avoid this overpopulation, City Developer may increase the current
SBD number, which increases city capacity. Mechanism of determining of endogenous minimum SBD number was suggested in Section 3.3, which is consistent with
empirical evidences (see Proposition 7).
Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area
377
APPENDIX
Proof of Proposition 1
The land supply in equilibrium should equalize (inelastic) land demand
π · rC
2
+ m · π · rS
2
= l · 1,
where rC is radius of central zone, rS is radius of single suburb. On the other hand,
for given CBD’s share of firms, θ, the labor market clearing in CBD (without cross2
commuting) implies π · rC = θl. Therefore,
r
r
r
r
θl
(1 − θ)l
θl
(1 − θ)l
C
S
S
C
S
, r =
, ||x || = r + r =
+
.
y=r =
π
mπ
π
mπ
The budget constraint of an individual residing at point x and working in the CBD
implies that
ALR
E C (x) = wC +
− Ψ C (x) − t||x||,
l
whereas the budget constraint of an individual working in the SBD is
E S (x) = wS +
ALR
− Ψ S (x) − t||x − xS ||.
l
Note that equalizing condition E C (x) ≡ E S (x) ≡ const implies Ψ C (x) = A1 − t||x||,
Ψ S (x) = A2 − t||x − xS ||, where A1 , A2 do not depend on x. On the other hand,
worker living at the border of the CBD residential area (i.e., at the point y = rC
of the SBD residential area closest to CBD, see Figure 1b) is indifferent to the
decisions of working in the CBD or in the SBD. Moreover, for the border location
y an identities Ψ C (y) = Ψ S (y) = 0 hold, because there is no difference for landlord
where to rent out this plot of land: to Central city, to Suburb or for agricultural
use. Therefore,
r
r
θl
(1 − θ)l
S
A1 = ty = t
, A2 = t · (x − y) = t
.
π
mπ
As result, we obtain
ALR
1
=
l
l
Z
X
t
R(x)dx = ·
3
r
l 3/2 (1 − θ)3/2
√
θ
+
.
π
m
Note that no need to integrate actually this function. We may simply apply the
1
well-known formula of the cone volume V = πh · r2 , where h is a hight and r is a
3
radius of the base of cone.
C
S
C
S
Moreover,
! E (y) − E (y) = 0 implies w − w = A1 − A2 =
ran identity
r
θl
(1 − θ)l
t·
−
. It means that the difference in the wages paid in the
π
mπ
CBD and in the SBD compensates exactly the worker for the difference in the
corresponding commuting costs. The wage wedge wC − wS is positive as long as
1
θ>
, thus implying that the size of the CBD exceeds the size of each SBD.
1+m
378
Alexandr P. Sidorov
Proof of Proposition 2
There is one-to-one correspondence between θ ∈ [0, 1] and α ∈ [0,
θ = cos2 α. Substituting it into equation
r
r
θl
(1 − θ)l
= K + ϕt
ϕt
π
mπ
π
] given by
2
we obtain, after simple transformations, the following one
√
sin α K π
F (α, l, m, t) := cos α − √ − √ = 0.
m
ϕt l
Note that,
and
(21)
√
K π
πK 2
√ < 1 ⇐⇒ l > lP = 2 2
ϕ t
ϕt l
∂F
cos α
= − sin α − √ < 0.
∂α
m
Consider three possible cases:
√ α < 0 and equation (21) has no
i) l < lP then F (α, l, m, t) < (cos α − 1) − sin
m
roots.
√ α = 0 if and only if cos α = 1,
ii) l = lP then F (α, l, m, t) = (cos α − 1) − sin
m
∗
which implies θ = 1.
√
√
K π
1
K π
π
P
iii) l > l then F (0, l, m, t) = 1−0− √ > 0 and F ( , m, l) = 0 − √ − √ <
2
m
ϕt l
ϕt π l
∗
∗
0. Thus there exists unique root α ∈ 0, 2 of equation(21) and θ = cos2 α∗ .
Accordingly to Theorem on Implicit Function Derivative, we obtain
√
3
∂α∗
∂F/∂l
K π · l− 2
> 0.
=−
=
√ α)
∂l
∂F/∂α
2ϕt(sin α + cos
m
It implies that θ∗ (l) = cos2 (α∗ (l)) is decreasing function. Similarly,
3
∂F
∂α∗
sin α · m− 2
=
= − ∂m
> 0,
∂F
√ α)
∂m
2(sin α + cos
m
∂α
thus θ∗ (m) = cos2 (α∗ (m)) is also decreasing function. Furthermore,
√
1
∂α∗
∂F/∂t
K π · l− 2
> 0,
=−
=
∂t
∂F/∂α
√ α
ϕt2 sin α + cos
m
thus θ∗ (t) = cos2 (α∗ (t)) is also decreasing function with respect to t. Finally,
√
∂α∗
∂F/∂t
π
< 0,
=−
=− √ ∂K
∂F/∂α
√ α
ϕt l · sin α + cos
m
thus θ∗ (t) = cos2 (α∗ (t)) increases with respect to t.
Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area
379
To obtain the limit value of θ∗ is sufficient to note that equation (21) for l → ∞,
t → ∞, K → 0 transforms into
sin α
cos α − √ = 0
m
which is equivalent to
m · cos2 α = sin2 α = 1 − cos2 α,
1
. On the other hand, K → ∞ implies lP → ∞,
1+m
therefore m = 0 and θ∗ = 1 is a unique outcome.
implying θ∗ = cos2 α∗ =
Proof of Proposition 3
Let y(l) = θ∗ (l) · l, then y is an implicit function defined by equation
√
√
1 p
K π
=0
G(y, l) = y − √
l−y−
m
ϕt
which is equivalent to equation (15). Thus
√
y
∂(θ∗ (l) · l)
∂G ∂G
=−
=p
√ > 0,
∂l
∂l
∂y
m(l − y) + y
moreover
∂θ∗
< 0 by Proposition 2. It implies that function
∂l
r
2t θ∗ (l, m, t) · l
K
C
Cu (l, m, t) =
+
· (1 − θ∗ (l, m, t)).
3
π
3ϕ
increases with respect to l. Let’s prove that CuC (l) is continuously differentiable
at
r
2
2t
l
πK
l = lP = 2 2 . Indeed, for all l < lP the urban cost function CuC (l) =
, hence
ϕ t
3 π
∂CuC P
ϕt2
(l − 0) =
.
∂l
3πK
Note that θ∗ (lP ) = 1 and
√
∂(θ∗ (l) · l) P
lP
√ = 1,
p
(l + 0) =
∂l
m(lP − lP ) + lP
on the other hand,
∂(θ∗ (l) · l) P
∂θ∗ P
(l + 0) = lP
(l + 0) + θ∗ (lP ).
∂l
∂l
∂θ∗ P
(l + 0) = 0, therefore
∂l
r !
θl
∂
C
π
∂Cu P
2t
ϕt2
∂CuC P
(l + 0) =
·
(lP + 0) =
=
(l − 0).
∂l
3
∂l
3πK
∂l
It implies that
380
Alexandr P. Sidorov
Recall that for l ≤ l
P
the urban costs
CuC
Moreover, for l > lP
2t
=
3
r
θl
∂CuC
, therefore
≡ 0.
π
∂m
∂CuC
∂CuC ∂θ∗
=
,
∂m
∂θ ∂m
where
∂θ∗
< 0 by Proposition 2 and
∂m
r √
∂CuC
l
πK √
t
= √
1− √ θ >0
∂θ
3 θ π
ϕt l
because l > lP =
πK 2
and θ < 1. Therefore
ϕ2 t2
∂CuC
∂CuC ∂θ∗
=
·
< 0.
∂m
∂θ
∂m
Moreover, θ∗ (lP ) = 1, hence
∂CuC P
t
(l + 0) =
∂θ
3
r
lP
π
√
πK
= 0,
1− √
ϕt lP
which implies that
∂CuC P
∂θ∗ ∂CuC P
(l + 0) =
·
(l + 0) = 0,
∂m
∂m ∂θ
i.e. the urban cost function is continuously differentiable with respect to m.
Let y(t) = θ∗ (t) · t2 , then y is an implicit function defined by equation
H(y, t) =
√
p
K mπ
√
√ =0
my − t2 − y −
ϕ l
which is equivalent to equation (15). Moreover,
therefore
√
1
∂H
t
∂H
m
= √ + p
> 0,
=− p
< 0,
2
∂y
2 y 2 t −y
∂t
2 t2 − y
∂y
∂H ∂H
=−
> 0.
∂t
∂t
∂y
p
p
It implies that function t · θ∗ (t) = y(t) increases with respect to t, as well as
1 − θ∗ (l, m, t). Therefore, urban costs function
2t
CuC (l, m, t) =
3
r
θ∗ (l, m, t) · l
K
+
· (1 − θ∗ (l, m, t))
π
3ϕ
also increases with respect to t, increase with respect to t.
Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area
381
Proof of Proposition 5
Note that wage wC∗ (l) is bounded function, while urban costs increase unrestrictedly with respect to l, hence, wC∗ (l) − CuC (l, m) < 0 for all sufficiently large l.
∂wC∗
α2
∂CuC
(0) =
<
(0, m) = +∞, thus
Moreover, wC (0) = CuC (0, m) = 0, while
∂l
2
∂l
C∗
C
w (l) − Cu (l, m) < 0 for all sufficiently small l > 0. It implies that set of l guaranteeing “the surviving condition” wC∗ (l) − CuC (l,
m, t) ≥ 0 is subsetC of some interval
[lmin (m, t), lmax (m, t)], where lmin (m, t) = inf l > 0 | wC∗
(l) − Cu (l, m, t) ≥ 0 >
0 and lmax (m, t) = sup l > 0 | wC∗ (l) − CuC (l, m, t) ≥ 0 < ∞. It remains to prove
C∗
that this subset is nonempty and inequality
(l) − CuC (l, m, t) ≥ 0 holds for all
w r
K
πγ
l ∈ [lmin (m, t), lmax (m, t)], at least for t ∈ 0,
.
ϕ 2βϕ
Note that
r
r
K
πγ
πγ
3α2
t<
⇒t<
⇐⇒
ϕ 2βϕ
16γ 2βϕ
r
πγ
α2
2t
C∗ ∗
>
= CuC (l∗ , 0, t) ≥ CuC (l∗ , m, t)
w (l ) =
8γ
3 2βϕ
2βϕ
is the “maximum wage” population size. It implies
γ
that “surviving” set of city population is non-empty and lmax (m, t) > l∗ . Moreover,
K
3α2
inequality
<
ensures that equation
ϕ
16γ
for all m ≥ 0, where l∗ =
wC∗ (l) =
α2 βϕl
2
(2βϕ + γl)
=
2K
3ϕ
has two real positive roots
l1,2 =
3α2 ϕ2 β
2K
r 2 2
2
3α ϕ β
− 4βγϕ ∓
−
4βγϕ
− 16β 2 γ 2 ϕ2
2K
2γ 2
2K
2βϕ
if and only if l1 < l < l2 . In particular, l∗ =
∈ (l1 , l2 )
3ϕ
γ
because wC∗ (l∗ ) = max wC∗ .
Note that
r
r
r
πγ
K π
K π
πK 2
K
P
t<
=
⇒
t
<
⇐⇒
l
=
> l1 .
ϕ 2βϕ
ϕ l∗
ϕ l1
ϕ2 t2
and wC∗ (l) >
2K
= CuC (lP , m) holds, which implies
3ϕ
2K
=
that lmin (m, t) < lP . On the other hand, if lP ≥ l2 > l∗ then wC∗ (l∗ ) >
3ϕ
CuC (lP , m) > CuC (l∗ , m), which also implies that lmin (m, t) < lP .
Assume at first that m = 0 and consider set of positive roots of equation
r
α2 βϕ · l
2t l
C∗
C
w (l) =
= Cu (l, 0, t) =
.
3 π
(2βϕ + γl)2
Let l1 < lP < l2 then inequality wC∗ (lP ) >
382
Alexandr P. Sidorov
√
√
Dividing both sides by l and substituting x = l we obtain the equivalent equation
2
√
√
3α2 βϕ π · x = 2t 2βϕ + γx2
⇐⇒ 8tβ 2 ϕ2 − 3α2 βϕ π · x + 8tβγϕx2 + γ 2 x4 = 0.
Sign of coefficients changes twice, hence, this equation has either 2, or 0 positive
roots, due to Descartes’ rule of signs. On the other hand, wC∗ (l∗ ) > CuC (l∗ , 0, t) and
wC∗ (l) < CuC (l, 0, t) for sufficiently large t, i.e., there is at least one positive root.
It implies that these roots are lmin (0, t) and lmax (0, t), respectively, and wC∗ (l) −
CuC (l, 0, t) ≥ 0 if and only if l ∈ [lmin (0, t), lmax (0, t)]. Moreover, it was proved that
lmax (0, t) > l∗ and lmin (0, t) < lP .
Now let m > 0, then CuC (l, m) ≡ CuC (l, 0) for all l ∈ [0, lP ] and CuC (l, m) >
C
Cu (l, 0) for all l > lM by Proposition 2. Let’s show that equation wC∗ (l) =
CuC (l, m, t) also has two positive roots lmin (m, t) and lmax (m, t), such that wC∗ (l) ≥
CuC (l, m, t) if and only if l ∈ [lmin (m, t), lmax (m, t)]. Indeed, CuC (l, m, t) ≡ CuC (l, 0, t)
for all l ≤ lP , thus lmin (m, t) ≡ lmin (0, t) ∈ (0, lP ). There is no roots in interval
(lmin (0, t), lmax (0, t)), because CuC (l, m, t) ≤ CuC (l, 0, t) < wC∗ (l). Therefore, there
is a unique root of equation wC∗ (l) = CuC (l, m, t) on interval (lmax (0, t), +∞), because wC∗ (l) strictly decreases for all l > lmax (0, t) > l∗ , while CuC (l, m, t) strictly
increases on (0, +∞). This completes the proof of proposition.
Proof of Proposition 7
Note that
C∗
(0)
wH
α2 βϕL
=
(2βϕ + γL)2
for all τ < τtrade =
2
2αβϕ + γL
1−τ ·
> 0 = CuC (0)
2αβϕ
2αβϕ
. Moreover, substituting τ = 0 into (19) we obtain
2βϕ + γL
C∗
wH
(λ) ≡
α2 βϕL
C∗
C
C
= wH
(1) ≥ CuH
(1) > CuH
(λ)
(2βϕ + γL)2
for all λ ∈ (0, 1), because L < lmax (mH ). Thus, for all sufficiently small τ < τ ∗
C
C∗
inequality wH
(λ) > CuH
(λ) holds for all λ ∈ (0, 1).
Let
C∗
C∗
∆(λ, τ ) = VH (λ, τ ) − VH (λ) = wH
(λ, τ ) + CSH (λ, τ ) − wH
(λ) + CSH (λ) .
We are about to prove that ∆(λ, τ ) > 0 for all λ ∈ (0, 1) and sufficiently small
τ > 0. Note that ∆(1, τ ) = 0 and
L 2ϕLβγ(α − 3τ )(α − τ ) + 6ϕ2 β 2 (α − τ )2 + L2 γ 2 τ 2
∆(0, τ ) =
.
4ϕβ(2ϕβ + Lγ)2
Quadratic equation
2ϕLβγ(α − 3τ )(α − τ ) + 6ϕ2 β 2 (α − τ )2 + L2 γ 2 τ 2 = 0
has no real solutions with respect to τ , while ∆(0, 0) > 0. It implies that ∆(0, τ ) >
0 = ∆(1, τ ) for all τ . Now we are about to prove that ∆(λ, τ ) is decreasing function.
Note that
ϕLα2 β(6ϕβ + Lγλ)
∂∆
(λ, 0) = −
<0
∂λ
2(2ϕβ + Lγλ)3
∂∆
for all λ ∈ (0, 1). Thus, for sufficiently small τ < τ ∗∗ inequality
(λ, τ ) < 0 holds
∂λ
for all λ.
Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area
383
References
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259–271.
The Irrational Behavior Proof Condition for
Linear-Quadratic Discrete-time Dynamic Games with
Nontransferable Payoffs⋆
Anna V. Tur
St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
Universitetskii pr. 35, St.Petersburg, 198504, Russia
E-mail: [email protected]
Abstract The paper considers linear-quadratic discrete-time dynamic games
with nontransferable payoffs. Pareto-optimal solution is studied as optimality principle. The time consistency and irrational behavior proof condition
of this solution are investigated. As an example, the government debt stabilization game is considered.
Keywords: linear-quadratic games, discrete-time games, games with nontransferable payoffs, Pareto-optimal solution, time consistency, PDP, irrational behavior proof condition.
1.
Introduction
Consider N-person discrete-time dynamic game Γ (k0 , x0 ) which is described by the
state equation
n
X
x(k + 1) = A(k)x(k) +
Bi (k)ui (k),
(1)
i=1
k ≥ k0 ,
k0 ∈ K+ ,
x(k0 ) = x0 .
x is m-dimensional state of system, ui is a r-dimensional control variable of player i,
x(k0 ) = x0 is the arbitrarily chosen initial state of the system, A(k), Bi (k) ∈ Z(K+ )
are matrices of appropriate dimensions, K+ is the set of nonnegative integers, Z(K+ )
is the set of bounded real matrices. The payoff function of player i ∈ N is
Ji =
∞
X
wi (k, x(k), ui (k)),
k=k0
(2)
∀i = 1, . . . , n,
wi (k, x(k), ui (k)) = xT (k)Pi (k)x(k) + uTi (k)Ri (k)ui (k),
Pi (k), Ri (k) ∈ Z(K+ ),
Pi (k) = PiT (k),
Ri (k) = RiT (k)
∀i ∈ N.
Suppose that payoffs are nontransferable.
We will assume that the players use feedback strategies,
ui (k, x) = Mi (k)x(k),
to control the system.
⋆
This work was supported by the St. Petersburg State University under grants No.
9.38.245.2014
On The Irrational Behavior Proof Condition for Linear-Quadratic Games
385
Definition 1. A set of strategies
{ui (k, x) = Mi (k)x(k),
(3)
i = 1, . . . , n}
is called permissible if the following conditions are satisfied:
1. Mi (k) ∈ Z(K+ )
∀i = 1, . . . , n.
2. The resulting system described by
x(k + 1) = (A(k) +
n
X
(4)
Bi (k)Mi (k))x(k)
i=1
is uniformly asymptotically stable (when k → ∞).
Suppose that players agree to use a Pareto-optimal solution as optimality principle.
And suppose that players consent to use vector of weights
n
P
α = (α1 , . . . , αn ) :
αi = 1, 0 < αi < 1
i=1
on their payoffs to obtain a Pareto-optimal outcome.
Then the optimal cooperative strategies of players can be found by solving the
following control problem (Engwerda, 2005)
max
(u1 ,...,un )
n
X
(5)
αi Ji (k0 , x0 , u),
i=1
α
Let uα (k) = (uα
1 (k), . . . , un (k)) be the set of strategies solving this optimal
control problem:
α
(uα
1 , . . . , un ) = arg
Assume J α (k0 , x0 , u) =
n
P
max
(u1 ,...,un )
n
P
αi Pi (k),
i=1


α1 R1 (k)
O
...
O
 O

α2 R2 (k) . . .
O
,
Rα (k) = 
 ...
...
...
... 
O
O
. . . αn Rn (k)
J α (k0 , x0 , u) =
∞
X
(6)
αi Ji (k0 , x0 , u).
i=1
αi Ji (k0 , x0 , u), P α (k) =
i=1
Then
n
X
k ≥ k0 ,
k ≥ k0 .
(xT (k)P α (k)x(k) + u(k)Rα (k)u(k)).
(7)
k=k0
Finding of Pareto-optimal solution is reduced to linear-quadratic optimal control
problem (1)-(7) with one control variable u(k).
The unique control in class of admissible
α
{uα
i (k) = Mi (k)x,
i = 1, . . . , n},
maximizing J α (k0 , x0 , u) exists if and only if (Bertsekas, 2007) the following conditions are satisfied:
386
Anna V. Tur
1. The system of matrix equations

(A(k) + B(k)M α (k))T Θα (k + 1)(A(k) + B(k)M α (k)) − Θα (k)−




 − P α (k) − M α (k)T Rα (k)M α (k) = 0,

M α (k) = −(−Rα (k) + B T (k)Θα (k + 1)B(k))−1 B T (k)Θα k + 1)A(k),




k ≥ k0
(8)
has the solution {M α (k), Θα (k)} ∈ Z(K+ ), with dimensions rs × m and m × m
respectively, where Θα (k) – is symmetric for all k ≥ k0 .
2. The set of strategies
α
{uα
i (k) = Mi (k)x,
(9)
i = 1, . . . , n},
 α 
M1 (k)
α


M
2 (k)
where Miα (k) – i-th block of the matrix M α (k) = 
 . . .  , is admissible.
M α (k)
3. (−Rα (k) + B T (k)Θα (k + 1)B(k)) – positive definite matrices.
The cooperative state trajectory xα (k) one can find by substituting the cooperative strategies {uα
i (k)} in (1) and solving the system:
x(k + 1) = A(k)x(k) + B(k)uα (k).
(10)
And payoffs of players are:
Jiα (k0 , x0 , uα )
=
∞ X
α
T
α
(x (k)) Pi (k)x (k) +
T
α
(uα
i (k)) Ri (k)ui (k)
k=k0
. (11)
Here B(k) = B1 (k) B2 (k) . . . B( k) .
2.
Time-consistency
Suppose that there exists such α, that inequalities
Jiα (k0 , x0 , uα ) ≥ Vi (k0 , x0 ),
i = 1, . . . , n.
(12)
requiring for individual rationality in the cooperative game are satisfied at initial
time. Here Vi (k0 , x0 ) – is Nash outcome of player i in game Γ (k0 , x0 ).
But if there exists k > k0 such that for some i:
Jiα (k, xα (k), uα ) < Vi (k, xα (k)),
then time-inconsistency of the individual rationality condition is appear.
To overcome the time inconsistency problem in the game with nontransferable
payoffs the notion of Payoff Distribution Procedure (PDP) was introduced by L.A.
Petrosyan (1997). In this paper the PDP and time-consistency of Pareto-optimal
solution are detailed for linear-quadratic discrete-time dynamic games.
On The Irrational Behavior Proof Condition for Linear-Quadratic Games
Definition 2. Vector β(k) = (β1 (k), ..., βn (k)) is a PDP if
X
∞ ∞
X
T
α
(xα (k))T Pi (k)xα (k) + (uα
(k))
R
(k)u
(k)
=
βi (k),
i
i
i
k=k0
387
i = 1, . . . , n.
k=k0
Definition 3. Pareto-optimal solution is called time-consistent if there exists a
PDP such that the condition of individual rationality is satisfied
∞
X
k=l
βi (k) ≥ Vi (l, xα (l)),
∀l ≥ k0 ,
(13)
i = 1, . . . , n,
where Vi (l, xα (l)) – is Nash outcome of player i in subgame Γ (l, xα (l)).
Let for some Pareto-optimal solution the condition (12) is satisfied. Then there
exist such functions ηi (k) ≥ 0, that
Jiα (k0 , x0 , uα ) − Vi (k0 , x0 ) =
∞
X
(14)
ηi (k).
k=k0
In (Petrosyan, 1997) the formula for PDP, which guarantees a time-consistency
in cooperative differential game with nontransferable payoffs, is considered. The
following theorem gives an analog of this formula.
Theorem 1. Let inequalities
Jiα (k0 , x0 , uα ) ≥ Vi (k0 , x0 ),
i = 1, . . . , n,
are satisfied for some Pareto-optimal solution. Then PDP β(k) computed by formula
βi (k) = ηi (k) − Vi (k + 1, xα (k + 1)) + Vi (k, xα (k))
i = 1, . . . , n,
k > k0
(15)
guarantees time-consistency of this Pareto-optimal solution along the cooperative
trajectory xα (k) for k > k0 . Here ηi (k) ≥ 0 – are functions satisfying (14).
Proof. Show that β(k) is a PDP:
∞
X
βi (k) =
k=k0
∞
X
k=k0
ηi (k) − Vi (∞, xα (∞)) + Vi (k0 , x0 ) =
= Jiα (k0 , x0 , uα ) − Vi (k0 , x0 ) + Vi (k0 , x0 ) = Jiα (k0 , x0 , uα ). (16)
Here Vi (∞, xα (∞)) = lim Vi (k, xα (k)) = 0. So β(k) satisfies definition 2.
k→∞
Now show that the condition of individual rationality is satisfied. Using (15) we
obtain
∞
X
k=l
βi (k) =
∞
X
k=l
ηi (k) − Vi (∞, xα (∞)) + Vi (l, xα (l)) =
=
∞
X
k=l
ηi (k) + Vi (l, xα (l)) ≥ Vi (l, xα (l)). (17)
⊔
⊓
388
Anna V. Tur
2.1. Irrational Behavior Proof Condition
The condition under which even if irrational behaviors appear later in the game the
concerned player would still be performing better under the cooperative scheme was
considered in (Yeung, 2006). The irrational behavior proof condition for differential
games with nontransferable payoffs is proposed in (Belitskaia, 2012). In this paper
the irrational behavior proof condition is concretized for linear-quadratic discretetime dynamic games with nontransferable payoffs.
Definition 4. Pareto-optimal solution (J1α (k0 , x0 , uα ), . . . , Jnα (k0 , x0 , uα )) satisfies
the irrational behavior proof condition (Yeung, 2006) in the game Γ (k0 , x0 ), if the
following inequalities hold
l
X
k=k0
βi (k) + Vi (l + 1, xα (l + 1)) ≥ Vi (k0 , x0 ),
i = 1, . . . , n
(18)
for all l ≥ k0 , where β(k) = (β1 (k), . . . , βn (k)) is time-consistent PDP of (J1α (k0 , x0 , uα ),
. . . , Jnα (k0 , x0 , uα )).
So if for all i = 1, . . . , n the following inequalities holds
βi (k) + Vi (k + 1, xα (k + 1)) − Vi (k, xα (k)) ≥ 0,
k ≥ k0 ,
then the Pareto-optimal solution satisfies the irrational behavior proof condition.
Rewrite these inequalities using (8)
βi (k) + (xα (k))T (A(k) + B(k)M α (k))T Θi (k + 1)(A(k) + B(k)M α (k))−
!
Θi (k) xα (k) ≥ 0,
k ≥ k0
(19)
If we use formala (15), then
βi (k) + Vi (k + 1, xα (k + 1)) − Vi (k, xα (k)) = ηi (k),
k ≥ k0 ,
where ηi (k) ≥ 0 for all k ≥ k0 . It means that conditions (19) are always satisfied
in this case.
Let’s formulate these results.
Theorem 2. If in linear-quadratic discrete-time dynamic games with nontransferable payoffs for some Pareto-optimal solutions and its PDP the following inequalities
hold
βi (k) + Vi (k + 1, xα (k + 1)) − Vi (k, xα (k)) ≥ 0,
k ≥ k0
i = 1, . . . , n.
where Vi (l, xα (l)) – is Nash outcome of player i in subgame Γ (l, xα (l)), then the
irrational behavior proof condition for this Pareto-optimal solutions is satisfied.
Proposition 1. If the PDP β(k) of Pareto-optimal solution in linear-quadratic
discrete-time dynamic games with nontransferable payoffs is calculated using formula (15), then the irrational behavior proof condition for this Pareto-optimal solutions is satisfied.
On The Irrational Behavior Proof Condition for Linear-Quadratic Games
3.
389
Example
As an example consider the government debt stabilization game(van Aarle, Bovenberg and Raith, 1995). Pareto solution of this game is considered in (Engwerda,
2005). This paper shows the discrete-time case of this problem and time-consistency
of cooperative solution.
Assume that government debt accumulation, d(k), is the sum of interest payments on government debt, rd(k), and primary fiscal deficits, f (k), minus the
seignorage (i.e. the issue of base money) m(k). So,
d(k + 1) = rd(k) + f (t) − m(t),
d(0) = d0 ,
The objective of the fiscal authority is to minimize a sum of time profiles of the
primary fiscal deficit, base-money growth and government debt
J1 =
k
∞ X
1
((f (k) − f )2 + η(m(k) − m)2 + λ(d(k) − d)2 ).
1+ρ
k=0
The monetary authorities are assumed to choose the growth of base money such
that a sum of time profiles of base-money growth and government debt is minimized.
That is
k
∞ X
1
((m(k) − m)2 + γ(d(k) − d)2 ).
J2 =
1+ρ
k=0
Let
x1 (k) =
1
1+ρ
k2
(d(k) − d),
x2 (k) = (f − m + (r − 1)d)
1
1+ρ
k+1
2
,
k2
1
(f (k) − f ),
u1 (k) =
1+ρ
k2
1
u2 (k) =
(m(k) − m)
1+ρ
Then our system can be rewritten as
x(k + 1) = A(k)x(k) +
2
X
Bi (k)ui (k)
i=1
 
12
1
r
1
 1+ρ

A=
12  ,
1
0
1+ρ
The payoff function of player i
Ji =
∞
X
k=k0

B1 = 
(xT (k)Pi (k)x(k) +
2
X
j=1

12 
12 
1
 , B2 =  − 1+ρ  ,
0
0
1
1+ρ
uTj (k)Rij (k)ui (k)),
∀i = 1, 2
390
Anna V. Tur
P1 =
λ0
γ0
, P2 =
,
00
00
R11 = 1,
R12 = η,
R21 = 0,
R22 = 1.
Following (Basar and Olsder, 1999) to find the Nash equilibrium we solve the system

2
2
X
X


NE
T

(A(k)
+
B
(k)M
(k))
Θ
(k
+
1)(A(k)
+
Bi (k)MiN E (k))−

i
i
i



i=1
i=1

− Θi (k) + Pi (k) + MjN E (k)T Rij (k)MjN E (k) + MiN E (k)T Rii (k)MiN E (k) = 0,




MiN E (k) = −(Rii (k) + BiT (k)Θi (k + 1)Bi (k))−1 BiT (k)Θi (k + 1)×




× (A(k) + Bj (k)MjN E (k)), i = 1, 2, j 6= i.
Let λ = 12 , η = 1,
1
1+ρ
12
= 14 , s = 2, γ = 1. Then
E
uN
1 (k, x) = −0.073193 −0.166311 x(k),
E
uN
2 (k, x) = 0.142083 0.318188 x(k),
0.656174 0.354202
J1 =
x ,
0.354202 0.844156 0
1.273766 0.613087
T
J2 = x0
x .
0.613087 1.444844 0
0.656174 0.354202
T
V (1, x(k)) = x (k)
x(k),
0.354202 0.844156
1.273766 0.613087
T
V (2, x(k)) = x (k)
x(k),
0.613087 1.444843
xT0
According to (8) to find the Pareto Solution we solve the system

(A(k) + B1 M1α + B2 M2α )T Θα (k + 1)(A(k) + B1 M1α + B2 M2α )−




 − Θα (k) + P α (k) + M α (k)T Rα (k)M α (k) = 0,
 M α (k) = −(Rα (k) + B T (k)Θα (k + 1)B(k))−1 ×




× B T (k)Θα (k + 1)A(k).
αR11
O
Where P α (k) = αP1 (k) + (1 − α)P2 (k), Rα (k) =
,
O αR21 + (1 − α)R22
B(k) = B1 (k) B2 (k) .
For α = 0, 45
M1α = (−0.2272618408
M2α
− 0.5075099515)
= (0.1022678284 0.2283794781)
0.6808499028 0.4139353163
J1 (uα ) = xT0
x
0.4139353163 0.9409769084 0
On The Irrational Behavior Proof Condition for Linear-Quadratic Games
1.223914910 0.4917964794
J2 (uα ) = xT0
x0
0.4917964794 1.139011465
If, for example, x0 = −3 2 , then
391
J1α (k0 , x0 , uα ) − V1 (k0 , x0 ) = −0.107435164999999722
J2α (k0 , x0 , uα ) − V2 (k0 , x0 ) = −0.216497664600000528
So, conditions (12) are satisfied (we consider the minimization problem, that is why
we have an opposite sign in (12)).
But on the next step we have
J1α (k1 , x1 , uα ) − V1 (k1 , x1 ) = 0.0504046297943969643
It means, that time-inconsistency of the individual rationality condition is appear.
To avoid this problem, use PDP, calculated by formula (15)
−0.107435164999999722
0.537430998 0.0789600736 α
+ xαT (k)
x (k),
0.0789600736 0.16307449
k(k + 1)
−0.216497664600000528
1.060309389 0.144646529 α
β2 (k) =
+ xαT (k)
x (k).
0.144646529 0.35798954
k(k + 1)
(20)
β1 (k) =
Note, that ηi (k) < 0, because we consider the minimization problem now.
Sufficient condition for realization of irrational behavior proof condition has
form:
0.537430998 0.0789600736 α
x (k) ≤ 0
β1 (k) − xαT (k)
0.0789600736 0.16307449
1.060309389 0.144646529 α
αT
β2 (k) − x (k)
x (k) ≤ 0.
0.144646529 0.35798954
And they are satisfied for β(k), computed by formula (20).
References
Aarle, B. van, Bovenberg, L. and Raith, M. (1995). Monetary and fiscal policy interaction
and government debt stabilization. Journal of Economics, 62(2), 111–140.
Basar T. and Olsder G. J. (1999). Dynamic Noncooperative Game Theory, 2nd edition.
Classics in Applied Mathematics, SIAM, Philadelphia.
Belitskaia, A. V. The D.W.K. Yeung Condition for Cooperative Differential
Games with Nontransferable Payoffs. Graduate School of Management, Contributions
to game theory and management, 5, 45–50.
Bertsekas D. P. (2007). Dynamic Programming and Optimal Control, Vol I and II, 3rd
edition. Athena Scientific,
Engwerda, J. C. (2005). LQ Dynamic Optimization and Differential Games. Chichester:
John Wiley Sons, 497 p.
Markovkin, M. V. (2006). D. W. K. Yeung’s Condition for Linear Quadratic Differential Games. In: Dynamic Games and Their Applications (L. A. Petrosyan and A. Y.
Garnaev , eds.), St Petersburg State University, St Petersburg, 207–216.
Markovkina, A. V. (2008). Dynamic game-theoretic model of production planning under
competition. Graduate School of Management, Contributions to game theory and management, 2, 474–482.
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Petrosjan, L. A. (1997). The Time-Consistency Problem in Nonlinear Dynamics. RBCM J. of Brazilian Soc. of Mechanical Sciences, Vol. XIX, No 2. pp. 291–303.
Petrosyan, L. A. and N. N. Danilov (1982). Cooperative differential games and their applications. (Izd. Tomskogo University, Tomsk).
Yeung, D. W. K. (2006). An irrational-behavior-proofness condition in cooperative differential games. Intern. J. of Game Theory Rew., 8, 739–744.
Yeung, D. W. K. and L. A. Petrosyan (2004). Subgame consistent cooperative solutions
in stochastic differential games. Journal of Optimization Theory and Applications,
120(3), 651-666.
Von Neumann-Morgernstern Modified Generalized Raiffa
Solution and its Application
Radim Valenčík1 and Ondřej Černík2
The University of Finance and Administration Prague,
Faculty of Economic Studies,
Estonská 500, 101 00 Prague 10, Czech Republic E-mail: [email protected]
WWW home page: http://www.vsfs.cz/en
2
The University of Finance and Administration Prague,
Faculty of Economic Studies,
Estonská 500, 101 00 Prague 10, Czech Republic
E-mail: [email protected]
WWW home page: http://www.vsfs.cz/en
1
Abstract In this paper we would like to discuss one of the possible modifications of Raiffa’s unique point solution which has applications in the
analysis of social networks associated with investing in social position and
creating the structures based on mutual covering of violations of the generally accepted principles. These structures are formed on the base of games
of Tragedy of commons type when one player detects breaking the rules by
another player. Hence the first player begins bribing the other player and
simultaneously covering his back, one player is rejudiced in favour of another player. This gives a rise to social networks that significantly affect the
formation of coalitions in various areas of the social system, including institutions whose mission is to protect society against violations of the generally
accepted principles. We also consider an original theoretical concept. We
show that this concept can be used to implement the NM-modified Raiffa’s
solution for n = 3.
Keywords: three-person game, bribing, Nash bargaining problem; NMmodified Raiffa sequential solution; redistribution system; social networks
based on mutual covering violate the generally accepted principles.
1.
Introduction
Our approach comes from formal definition Nash bargaining problem for n players
as a set B settled pairs (S, d), where S is compact convex subset Rn and point
d belongs to S. The elements B of are called instance (examples) of the problem
B, elements S are called variants or vector of utility, point d is called the point of
disagreement, or status quo. Every example is called d-comprehensive. The theory
suggests for the one-point solution several concepts. The term “solution” is understood as function f from B to Rn that each example (S, d) from B assigns value f(S,
d) belonging to S. The most known concept of solution is Nash’s one (Nash, 1950),
the other is Kalai-Smorodinsky’s one. The egalitarian approach suggested by Kalai
(Kalai, 1977) can be also understood as the solution. All mentioned solutions can be
expressed by axioms. Kalai-Smorodinsky’s solution (Kalai and Smorodinsky, 1975)
is maximum point on the segment S connecting point and so called utopian point ,
whose coordinates are defined as Ui (S, d) = max{xi : x ∈ S a x ≥ d}
From the point of view that we develop it is interesting Raiffa’s solution that was
proposed in the early 1950’s. Raiffa (Raiffa, 1953) suggested dynamic procedures
394
Radim Valenčík, Ondřej Černík
for the cooperative bargaining in which the set S of possible alternatives is kept
unchanged while the disagreement point d gradually changes. He considers two
variants of such process – a discrete one and the continuous one. Discrete Raiffa’s
solution is the limit of so called dictated revenues. Diskin, A., Koppel, M., Samet
D. (Diskin et al., 2011) have provided an axiomatization of a family of generalized
Raiffa’s discrete solutions.
2.
Experimental Section
Let S is a nonempty, closed, convex, comprehensive, and positively bounded subset
of Rn whose boundary points are Pareto optimal. They propose a solution concept
which is composed of two solution functions. One solution function specifies an
interim agreement and the other specifies the terminal agreement. Such a step-bystep solution concept can formally be defined as follows. The pair (f, g) functions
is called step-by-step solution, if as f (S, d) as g(S, d) belongs to for each example
(S, d) from B. The set of generalized Raiffa’s solution is certain kind of step-bystep negotiation solution {(fp , gp )0<p<1 } where are fp a gp defined as: fp (S, d) =
d + p/n(U (S, d) − d), gp (S, d) = d∞ (S, d),
where d∞ (S, d) is the limit of progression {dk (S, d)} of points constructed by
induction follows: d0 (S, d) = d, dk+1 (S, d) = fp (S, dk ).
2.1.
NM-modified Raiffa’s solution
The solution that we suggest and that we called NM-minified discrete Raiffa’s solution for n = 3 can be obtained by stipulating: d0 (S, d), dk+1 (S, d) = fnm (S, dk ),
where fnm (S, d) = d + 2/3(N M (S, d) − −d) where N M (S, d) is point derived from
utilities (in our interpretation we will use more suitable term pay-offs) of players in the points of Neumann-Morgenstern discrete internally and externally stable
set on S. These points have coordinates: dxy = (d1 , d2 , 0), dxz = (d1 , 0, d3 ), dyz =
(0, d2 , d3 ).
Note: from discrete solution which expect full symmetry of possibilities of players
in the creation of two-person coalition exists also another NM sets having infinitive
many elements. They play also important role, but we do not concerned with them.
If we define S = S(x1 , x2 , x3 ), it means as the function of payoffs of players, then d =
(d1 , d2 , d3 ) is given as the solution of following systems of equations: S(x1 , x2 , 0) =
0; S(x1 , 0, x3 ) = 0; S(0, x2 , x3 ) = 0.
Here it is valid that pay-off of every player in coalition with each other player
(e.g. pay-off of first player with second player or with third player) is same. This
fact causes the condition that points (d1 , d2 , 0), (d1 , 0, d3 ), (0, d2 , d3 ) create discrete
three-points NM set. The generalized Raiffa’s solution and by us established NMmodified Raiffa’s solution are very similar by their logic of construction.
Figure 1 depicts NM-modified Raiffa’s solution dRmN M for n = 3 graphically.
However, they have some important differences, especially from the point of view
of interpretation. NM-modified Raiffa’s solution dRmN M in a certain way connects
two situations: In the first case the players (each of them) decide to create only a
two-person fully discriminated coalition, i.e. two players who form a coalition, can
give to the third player the smallest possible pay-off. In our case this pay-off equals
0.
Von Neumann-Morgernstern Modified Generalized Raiffa Solution
395
But the smallest possible pay-off can have also different value (including negative) what is important for some interpretations and with them connected application.
The simplest example is simple majority game described in Neumann and Morgenstern (Neumann and Morgenstern, 1953) and following game with coalition of
different power (Neumann and Morgenstern, 1953), § 22. The same is valid in our
case of the game with non-zero sum. In the second case the players form a great
coalition, i.e. three-player coalition. The connection between both the cases can be
interpreted as follows: Pay-offs of each player in the formation of fully discriminated
coalitions can be seen from his perspective as an opportunity cost to the possibility
of creating a great coalition. If players create a great coalition, for obvious reasons
they will require pay-off higher or at least equal to the one they would have required
in a two-person coalition. The problem is how to evaluate player’s pay-offs for the
creation of fully discriminated two-person coalitions.
2.2.
Characterization of NM-modified Raiffa’s solution
Here we use (introduced by us) the term average expected pay-off, which is a multiple of its pay-off in a situation where the player is the member of the winning
coalition, and the probability of this coalition i.e. 2/3di , where i = 1, 2 or 3. We
simplify as we do not distinguish between pay-off and utility from pay-off. If the
utility function of a player has degressive character, the risk aversion would play
its role. The players would in such situation prefers two-person coalition even if
the value of pay-offs is lower than 2/3di . The value depends on the degressivity of
utility function. But the example is not important for our future ideas. "Bridge" by
which we’re connecting both the cases (formation of two-player coalitions and the
great coalitions above), i.e. application of the principle of opportunity cost and the
introduction of the concept of expected average pay-off, implicitly contains input
"step-by-step" process, which results in a single point solution in the case of great
coalition.
396
Radim Valenčík, Ondřej Černík
The key importance of the presented concept consist in the fact that it enables us to
express external factors that affect various real system, referred to as redistribution
systems, in which the following applies:
– We have a group of people that operate within a certain system. They perform
some role and, based on the performance of such role, they are attributed specific
funds that are subsequently redistributed among them in a certain manner.
– Coalitions may be formed within the aforementioned group of people, with a
view to provide privileges to those who take part in the coalition at the costs
of those who do not.
– Such privileges are in the form of the funds the players may divide among them.
Two questions arise in this connection: 1. What defines (how to describe) the
amount of funds that the players would be able to divide among them; 2. How
(based on what rules or regularities) will they divide such funds.
– In case social networks operate within the given system, we will understand
them as one-sided or mutual affinity of certain players within the given system,
whereas one and the same network may operate within a number of systems of
this type. Generally speaking, a system will be referred to as a redistribution
system if funds are divided and redistributed within the system as a result of
certain external factors: Formation of coalitions within the given system; formation of social networks within the system; reflection of roles of such networks
between different redistribution systems into individual redistribution systems.
It is necessary to emphasize the fact that the aforementioned factors characterize, and not define, a redistribution system. The characteristics are used to give
us an idea about the types of objects, to which it is possible to apply the tools
developed by us.
A game, in which we do not consider any impact of external factors, shall be referred
to as the original game for the sake of explicitness. External factors shall refer to
anything that may be expressed by a change in the parameters of the original game,
that affects the conduct of players, and that concurrently exists as
an independent parameter, the creation/development of which is not directly
controlled by any of the players. The expression of the external factors through the
change of the original game parameters shall be referred to as the original game
extension.
3.
Results and Discussion
3.1. Results and effects
Affinity of one player to another shall refer to the benefits (utility) the player gets
just by forming a coalition with another player, whereas such benefits (utility) may
be expressed in denominations that are used for payoffs within the original game.
In case both players generate utility, it is referred to as mutual affinity; however,
the extent may vary for each of the players. Positive affinity may also be referred
to as sympathies of one player to another, with negative affinity being antipathy of
one player to another.
Affinity may be expressed as follows: a player, who forms coalition with another
player, generates specific benefits (utility) just by forming the coalition, whereas
such utility are expressed in the same denominations as their payoffs. The total
payoff of a player (referred to as xij ∗ ) within a coalition with another player, under
Von Neumann-Morgernstern Modified Generalized Raiffa Solution
397
a relationship of certain affinity, shall then equal to the player’s payoff in the original
game plus the player’s payoff corresponding to the benefits (utility) arising from
the formation of the coalition (the additional payoff shall be referred to as sij ):
xij ∗ = xi + sij .
The value of si may be both positive (positive affinity – i.e. sympathies) or
negative (negative affinity – i.e. antipathy). It would seem that if a one-sided or
mutual affinity exists between two players, with no affinity existing between either
of the players and a third player, the formation of coalition between the two players
is predetermined. However, this may not be the case and if the third player is
informed about the affinity of the other two, he may offset such positive affinity
through a lower payoff. Let us assume that all players are fully informed about all
affinities of the players. The original set of equations shall be modified as follows:
S(x12 ∗ , x21 ∗ , 0) = s12 + s21
S(x13 ∗ , 0, x31 ∗ ) = s13 + s31
S(0, x23 ∗ , x32 ∗ ) = s23 + s32
Right-hand side of equations shall be interpreted by saying that additional payoffs arise within the game on the basis of the relevant affinities. The following shall
then apply to the payoffs within the original game: x1 = 1/2(x12 ∗ +s12 +x13 ∗ +s13 ),
etc.
The original generalized Raiffa sequential solution does not allow the assessment
of the role of affinities, because it does not contain an alternative to the formation
of two-member coalitions with regard to the alternative of a three-member coalition
formation, measured by opportunity costs. For the same reason, other point solutions of the Nash bargaining problem do not make it possible to assess the role of
affinities. Therefore, we have created an original theoretical concept, on the basis
of which we are able to identify (ascertain and assess) how external factors – in the
form of affinities – affect any community or partnership of the redistribution system
type.
In line with the specified objectives, the project solution is aimed at expressing
(modeling, evaluating, and assessing) the impact of the following affinities and social
networks interconnected by such affinities:
– Those that arise by investments in social status and are associated with a creation of social networks (affinities between players) derived from investments in
social status.
– Those that arise by violations of principles generally accepted within the given
system as well as its social environment and that lead to the creation of social
networks relying on mutual covering, blackmail, and favoring of those entities
that violate the generally accepted principles. We will mainly strive to describe
the method of formation, development, operation, and anatomy of the structures
based on mutual covering of violations of the generally accepted principles in
terms of the potential elimination of their impact.
We will distinguish the following:
– Effects arising through investments in social status.
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Radim Valenčík, Ondřej Černík
– Effects arising as a result of activities of social networks derived from investments in social status.
– Effects arising as a result of violations of the generally accepted principles.
– Effects arising as a result of activities of social networks based on mutual covering, blackmail, and favoring of players that violate the generally accepted principles, i.e. as a result of what we call the structures based on mutual covering
of players that violate the generally accepted principles.
3.2.
Effects arising through investments in social status
We all have a supply of investment opportunities. If we apply a rational decision
making model, it is safe to assume that people will use investments funds available
to them (both their own funds as well as funds raised on the capital market) to
carry out such investment opportunities based on their respective rates of return.
They will thus carry out those investment opportunities that offer higher rate of
return compared to interest rate.
Now, let us assume that by investing in social status, it is possible to prevent
the utilization of an investment opportunity by those, who cannot afford such investment. The effect of such situation will be an increase of the return from the
investments in social status. Based on this, it is possible to draw three conclusions:
– An investment in social status has non-Pareto consequences for the bearers of
investment opportunities (i.e. those, who cannot utilize funds into investments
in social status, will be worse off).
– An investment in social status reduces social effectiveness (significant reduction
of effects generated within the economic environment in case of the utilization
of investment opportunities).
– An investment in social status will pay off to those, who make such investment,
provided the effect of such investment as a result of higher share in the return
of the given investment opportunity – as opposed to a situation, where the
investment does not limit the utilized investment opportunities – exceeds the
costs of the relevant investment in social status.
3.3.
Effects arising as a result of activities of social networks derived
from investments in social status
The scope of the primary effects is given by the difference between the expected average payoff, or between the payoff in the together acceptable equilibrium point (as
appropriate), and the payoff the relevant player gets within the winning coalition. In
order for a player to become (remain) a member of a winning coalition, he/she must
make certain effort or act in a certain manner, as appropriate. It is then necessary
to analyze, how the expected average payoff (payoff in the jointly acceptable equilibrium point) differs from the net payoff a player gets within the winning coalition
(i.e. payoffs within the winning coalition minus all costs of a player associated with
his/her participation in the winning coalition). In case it is possible to identify the
costs of a player associated with his/her participation in the winning coalition, it
is also possible to identify the ways of increasing such costs, through regulation or
organization, thus at least partly eliminating investments in social status and their
non-Pareto consequences.
Von Neumann-Morgernstern Modified Generalized Raiffa Solution
3.4.
399
Effects arising as a result of violations of the generally accepted
principles
In this case, it is possible to derive from the game Tragedy of the Commons. In
case there is a risk that a player might be detected and punished with a certain
probability, it is possible to analyze under what conditions players opt to violate the
principles. It is possible to use the existing literature - e.g. work of (Ostrom, 2008).
However, existing analyses usually do not consider the role of the structures based
on mutual covering of violations of the generally accepted principles. These analyses
rely on the premise that a player, who decides to violate (or already directly violates) the generally acceptable principles, compares the benefits (utility) and costs
associated with such violations, whereas the analyses that rely on the examination
of such benefits and costs sometimes include proposals for reducing such benefits
and increasing the costs. However, the analyses usually do not include, as one of the
potential benefits, the fact that – by violating the generally accepted principles - a
player wishes to take part in a structure based on such violations, because he/she
derives benefits from the participation in such structure.
Furthermore, the analyses do not really consider the possibility that other players
might actively seek out players, who wish to violate (or are already violating) the
generally accepted principles in order to create a social network with them or to
involve them in an existing network (also see the following section).
3.5.
Effects arising as a result of activities of social networks based on
mutual covering, blackmail, and favoring of players that violate
the generally accepted principles, i.e. as a result of what we call
the structures based on mutual covering
Let us first recall and further clarify the mechanism, on the basis of which the
structures based on mutual covering of violations of the generally accepted principles come into existence. A player, who finds out that another player violates the
generally accepted principles, has the following options:
– To spread the information about the violation of the generally accepted principles – i.e. to help to punish violating player.
– To overlook the conduct of the relevant player – i.e. no response.
– To start violating the generally accepted principles as well.
– To exploit the information – i.e. blackmail the relevant player. The higher sanctions are imposed for the violation of the generally accepted principles, the
higher effect might result from the blackmailing of the player, who violated the
generally accepted principles.
In case of an attempt to blackmail the player, who violated the generally accepted principles, the relevant player has several options:
– Refuse the blackmailing, even at the cost of being punished by the community.
– Notify the community of an attempted blackmail, which itself represents a certain form of violation of the generally accepted principles and, as such, may be
sanctioned by the community
– Submit to the player, who is blackmailing him/her, and allow to be blackmailed.
In this case, the blackmailed player compares the sanction to be imposed in
case he/she does not accept the proposal of the blackmailer and the benefits
generated if he/she accepts the blackmailer’s proposal.
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Radim Valenčík, Ondřej Černík
The higher sanctions are imposed for the violation of the generally accepted principles (in the form of uncooperative conduct in our case), the higher the protection
of the community against such behavior, whereas the blackmailing of one player by
another may prove to be more effective. As soon as the structures based on mutual
covering of violations of the generally accepted principles start to form on the basis
of this within the system, it may result in a significant reduction in the cooperative
conduct and subsequent losses in effectiveness. In case this concerns a larger system,
which creates its own institutional structure, it may result in serious dysfunctions
of the entire institutional structure. More detailed analysis of these issues assumes
the application and interpretation of other models, and this is the objective of the
project solution. However, just the description specified herein shows that a community, which wishes to prevent the violation of the generally accepted principles
that allow the development of such community (in general, principles of justice,
fairness, and equality), must be able to detect the structures that wish to violate
such principles. The given community’s system of regulations must then be set up
in a way that individual members of the community do not find it beneficial to
form (become engaged in) the structures based on mutual covering of violations of
the generally accepted principles. One of the project outputs will be the proposal
of a structure (design) of such system of regulations, which would lead to the fact
that the membership in the structures based on mutual covering of violations of
the generally accepted principles would not be beneficial for players (community
members).
It is already possible to describe the effects that arise on the basis of a player’s
involvement in a structure based on mutual covering of violations of the generally
accepted principles. It is possible to distinguish several effects of this type. They are
as follows: Effect of impunity, effect of predetermining coalitions, effect of favoring.
3.6.
Effect of impunity
Let us first recall and further clarify the mechanism, on the basis of which the
structures based on mutual covering of violations of the generally accepted principles
come into Similarly as various communities create mechanisms and institutions
that make it possible to detect and punish those players, who violate agreements
or generally accepted principles, the structures based on mutual covering create
various mechanisms and even institutions that allow effective covering of violations
of the generally accepted principles. In case a player allows to be blackmailed, he/she
gets under the protection of the relevant structure, which considerably or – to be
precise – fundamentally reduces the risk that his/her conduct would be detected
and punished by the community.
3.7.
Effect of predetermining coalitions
Affinity between players given by an inclusion in the same structure based on mutual
covering of violations of the generally accepted principles significantly predetermines
the formation of coalitions. Players, who are not involved in such structures, virtually do not have a chance to compensate, through their concessions (reduction of
the required payoff), the equalizing of chances for the participation in the winning
coalition. Since the relevant affinities are covert by nature, they are not informed
about them. Furthermore, this concerns very strong affinities, also associated with
investments in social status.
Von Neumann-Morgernstern Modified Generalized Raiffa Solution
3.8.
401
Effect of favoring
We have not yet prepared a suitable model here. These are effects associated with
principal-agent problems, when the structures based on mutual covering have the
ability to appoint those, who are involved in such structures, to important positions
in a organizations (for instance in a police structure to hinder or to stop police
investigation of members such structure or in parliament in order to accept such
bills that are convenient to the structures). There are two types of effects that arise
in this manner: In the form of financial and nonfinancial returns from the prominent position within the given organization. In the form of significant expansion
of the possibilities to violate the generally accepted principles with minimum risk
of punishment. The above mentioned verbal description of effects gives the basic
conceptual process for solution. From the methodological perspective, the solution
will take place in the form of the application of the NM-modified Raiffa solution
for the drafting models that allow the assessment of effects arising as a result of the
activities of social networks based on mutual covering, blackmail, and favoring of
players, who violate the generally accepted principles, i.e. of what we call the structures based on mutual covering. On the basis of such models, we will also look for
answers to the following questions: What is the role of the players’ knowledge of the
existence of affinities? What possibilities has a player to compensate the impact of
affinities if he/she is informed about such affinities? The elaboration of the relevant
models within the project is associated with an analysis of real social situations –
for example, events of indiscretion and absorption of indiscretion, as disclosed by
public sources. From this perspective, the creation of theoretical models may play
an important role in understanding the real social events.
4.
Conclusions
1. A question arises, whether a three-player model is sufficient to describe events
in the area of social reality. It is possible to consider two alternative extensions
of the model. The first one consists in the creation of models for more players,
which is associated with certain fundamental theoretical problems. The other
one (which we believe to be more in line with the real life) is the possibility to
examine, under what conditions a player line-up might change within a specific
environment or, alternatively, what environmental change may result in the
change of the player line-up and particularly the players’ objectives. This is a
problem area, which is taken into account by the project team; however, it is
currently not the focal point of the project – so as not to make the project too
ambitious, among other things. Therefore, when describing the effects arising
as a result of the activities of the structures based on mutual covering, we will
only confine to some of them.
2. From the perspective of the description of the proposed conceptual and methodological procedures, it is also necessary to mention another important aspect.
The project solution assumes a wide range of theoretical outputs, from the application of the axiomatic approach, design of mathematical models on the basis
of the game theory, to the creation of suitable concepts associated with the conceptual description of the social reality and analysis of real situations. On the
one hand, this makes the project solution extremely challenging (and it may be
rightfully pointed out whether the proposed team has sufficient qualifications);
on the other hand, it offers the opportunity to demonstrate the possible appli-
402
Radim Valenčík, Ondřej Černík
cation of an exact theory in solving pressing social issues. The team expects
to speak to certain leading experts and perform certain required theoretical
outputs in cooperation with such experts in the course of the project.
This concerns, for example, the axiomatization of the NM-modified Raiffa solution. Using examples, it is possible to demonstrate that it is different from
the generalized Raiffa solution and, therefore, should have a different system of
axioms corresponding thereto. In case it is possible to explicitly express such differences, the results could be published in one of the leading international journals. However, it would particularly be of a considerable practical importance
and demonstrate the possible interconnection of the theoretical (mathematical)
bases of the game theory with the solution of pressing social issues, which are
currently being discussed in the Czech Republic(corruption, ineffective public
administration, operation of the system of political parties, etc.).
3. If it is possible to develop, without any serious problems, a suitable model for
the effects of impunity and effect of predetermining coalitions, that would allow
their assessment, it is more complicated for the effect of favoring.
Acknowlegments We thank the audience at the Seventh International Conference on Game Theory and Management (GTM2013) which is held in St. Petersburg
University and organized by the Graduate School of Management (GSOM) in collaboration with the Faculty of Applied Mathematics & Control Processes and the
International Society of Dynamic Games (Russian Chapter). for their helpful comments and suggestions. This article represents one of the outputs under the research
project supported by the funds from Specific University Research, allocated to the
University of Finance and Administration in 2013.
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Institute of Mathematical Economics (IMW), Bielefeld University.
Subgame Consistent Cooperative Solution of Stochastic
Dynamic Game of Public Goods Provision ⋆
David W.K. Yeung1,2 and Leon A. Petrosyan3
Department of Business Administration, Hong Kong Shue Yan
University, Hong Kong
Center of Game Theory, St Petersburg State University, St Petersburg, Russia
E-mail: [email protected]
3
Faculty of Applied Mathematics-Control Processes, Saint Petersburg
State University, St Petersburg, Russia
E-mail: [email protected]
1
2
Abstract The provision of public goods constitutes a classic case of market
failure which calls for cooperative optimization. However, cooperation cannot be sustainable unless there is guarantee that the agreed-upon optimality
principle can be maintained throughout the planning duration. This paper
derives subgame consistent cooperative solutions for public goods provision
by asymmetric agents with transferable payoffs in a stochastic discrete-time
dynamic game framework. This is the first time that dynamic cooperative
game in public goods provision is analysed.
Keywords: Public goods, stochastic dynamic games, dynamic cooperation,
subgame consistency.
1.
Introduction
Public goods, which are non-rival and non-excludable in consumption, are not uncommon in today’s economy. Examples of public goods include clean environment,
national security, scientific knowledge, accessible public capital, technical know-how
and public information. The non-exclusiveness and positive externalities of public
goods constitutes major factors for market failure in their provision. In many contexts, the provision and use of public goods are carried out in an intertemporal
discrete time-period framework under uncertainty. Cooperation suggests the possibility of socially optimal solutions in public goods provision problem. A discrete-time
game framework is developed for theoretical analysis and practical applications.
Problems concerning private provision of public goods are studied in Bergstrom
(1986). Static analysis on provision of public goods are found in Chamberlin (1974),
McGuire (1974) and Gradstein and Nitzan (1989). In many contexts, the provision
and use of public goods are carried out in an intertemporal framework. Fershtman
and Nitzan (1991) and Wirl (1996) considered differential games of public goods
provision with symmetric agents. Wang and Ewald (2010) introduced stochastic elements into these games. Dockner et al. (2000) presented a game model with two
asymmetric agents in which knowledge is a public good. These studies on dynamic
game analysis focus on the noncooperative equilibria and the collusive solution that
maximizes the joint payoffs of all agents.
In dynamic cooperation, the solution scheme would offer a long-term solution only if there is guarantee that participants will always be better off throughout the entire cooperation duration and the agreed-upon optimality principle be
⋆
This research was supported by the HKSYU Research Grant.
405
Subgame Consistent Cooperative Solution of Stochastic Dynamic Game
maintained from the beginning to the end. To enable a cooperation scheme to be
sustainable throughout the agreement period, a stringent condition is needed – that
of subgame consistency. This condition requires that the optimality principle agreed
upon at the outset must remain effective in any subgame starting at a later starting
time with a realizable state brought about by prior optimal behaviour. Hence the
players do not possess incentives to deviate from the cooperative scheme throughout
the cooperative duration. The notion of subgame consistency in stochastic cooperative differential games was originated in Yeung and Petrosyan (2004) in which a
generalized theorem for the derivation of an analytically tractable "payoff distribution procedure" (PDP) leading to subgame-consistent solutions has been developed. A discrete time version of the analysis is provided in Yeung and Petrosyan
(2010). Yeung and Petrosyan (2013) presented subgame consistent cooperative solutions for public goods provision by asymmetric agents with transferable payoffs
in a continuous-time stochastic differential game framework.
In this paper, an analytical framework entailing the essential features of
public goods provision in a discrete-time stochastic dynamic paradigm is set up.
The noncooperative game outcome is characterized and dynamic cooperation is
considered. Group optimal strategies are derived and subgame consistent solutions
are characterized. A ?payoff distribution procedure? leading to subgame-consistent
solutions is derived. Illustrative examples are presented to demonstrate the derivation of subgame consistent solution for public goods provision game.
The paper is organized as follows. Section 2 provides the analytical framework and the non-cooperative outcome of public goods provision in a discrete-time
stochastic dynamic framework. Details of a subgame consistent cooperative scheme
are presented in Section 3. Illustrative examples are given in Section 4. Section 6
concludes the paper.
2.
Analytical Framework and Non-cooperative Outcome
Consider the case of the provision of a public good in which a group of n agents carry
out a project by making continuous contributions of some inputs or investments to
build up a productive stock of a public good. The game horizon consists of T stages.
We use Kt denote the level of the productive stock and Iti denote the contribution
to the public capital or investment by agent i at stage t ∈ {1, 2, · · · , T }. The stock
accumulation dynamics is then
Kt+1 =
n
X
j=1
Itj − δKt + ϑt , K 1 = K 0 , for t ∈ {1, 2, · · · , T },
(2.1)
where ϑt is a sequence of statistically independent random variables and δ is the
depreciation rate.
The payoff of agent i at stage t is
Ri (Kt ) − C i (Iti ), i ∈ {1, 2, · · · , n} = N,
(2.2)
where Ri (Kt ) is the revenue/payoff to agent i, C i (Iti ) is the cost of investing Iti ∈ X i .
The objective of agent i ∈ N is to maximize its expected net revenue over the
planning horizon, that is
Eϑ1 ,ϑ2 ,··· ,ϑT
T
X
s=1
[Ri (Ks ) − C i (Isi )](1 + r)−(s−1) + q i (KT +1 )(1 + r)−T
(2.3)
406
David W.K. Yeung, Leon A. Petrosyan
subject to the stock accumulation dynamics (2.1),
where r is the discount rate, and q i (KT +1 ) > 0 is an amount conditional on the
productive stock that agent i would received at stage T .
Acting for individual interests, the agents are involved in a stochastic dynamic
game. In such a framework, a feedback Nash equilibrium has to be sought. Let
{φis (K)∈ Isi , for i ∈ N and s ∈ {1, 2, · · · , T }} denote a set of feedback strategies
that brings about a feedback Nash equilibrium of the game (2.1) and (2.3). Invoking
the standard techniques for solving stochastic dynamic games, a feedback solution
to the problem (2.1) and (2.3) can characterized by the following set of discretetime Hamilton-Jacobi-Bellman equations (see Basar and Olsder 1995; Yeung and
Petrosyan 2012):
V i (t, K) = max Eϑt
Iti
+V i
t + 1,
n
X
j=1
j 6= i
[Ri (K) − C i (Iti )](1 + r)−(t−1)
φjt (K) + Iti − δK + ϑt
, for t ∈ {1, 2, · · · , T },
V i (T + 1, K) = q i (KT +1 )(1 + r)−T , for i ∈ N.
(2.4)
(2.5)
A Nash equilibrium non-cooperative outcome of public goods provision by
the n agents is characterized by the solution of the system of equations (2.4)− (2.5).
3.
Subgame Consistent Cooperative Scheme
It is well-known problem that noncooperative provision of goods with externalities, in general, would lead to dynamic inefficiency. Cooperative games suggest the
possibility of socially optimal and group efficient solutions to decision problems involving strategic action. Now consider the case when the agents agree to cooperate
and extract gains from cooperation. In particular, they act cooperatively and agree
to distribute the joint payoff among themselves according to an optimality principle. If any agent disagrees and deviates from the cooperation scheme, all agents
will revert to the noncooperative framework to counteract the free-rider problem in
public goods provision. In particular, free-riding would lead to a lower future payoff
due to the loss of cooperative gains. Thus a credible threat is in place. In particular,
group optimality, individual rationality and subgame consistency are three crucial
properties that sustainable cooperative scheme has to satisfy.
3.1.
Pareto Optimal Provision and Individual Rationality
To fulfill group optimality the agents would seek to maximize their expected joint
payoff. To maximize their expected joint payoff the agents have to solve the stochastic dynamic programming problem
max
{Isj f or j∈N }
Eϑ1 ,ϑ2 ,··· ,ϑT
n X
T
X
j=1 s=1
+
n
X
j=1
[Rj (Ks ) − C i (Isj )](1 + r)−(s−1)
q j (KT +1 )(1 + r)−T
(3.1)
Subgame Consistent Cooperative Solution of Stochastic Dynamic Game
407
subject to the stock dynamics (2.1).
Invoking the standard stochastic dynamic programming technique an optimal
solution to the stochastic control problem (2.1) and (3.1) can characterized by the
following set of equations (see Basar and Olsder (1995) and Yeung and petrosyan
(2012)):
W (t, K) =
+W
{Itj
max
f or j∈N,}
t + 1,
n
X
j=1
Eϑt
n
X
j=1
Itj − δK + ϑt
W (T + 1, K) =
n
X
[Rj (K) − C i (Itj )](1 + r)−(t−1)
, for t ∈ {1, 2, · · · , T },
q j (KT +1 )(1 + r)−T .
(3.2)
(3.3)
j=1
Let ψs∗ (K) ={ψs1∗ (K),ψs2∗ (K), · · · , ψsn∗ (K)}, for s ∈ {1, 2, · · · , T } denote a set
of strategies that brings about an optimal cooperative solution. A group optimal
solution of public goods provision by the n agents is characterized by the solution
of the equation (3.2)-(3.3).
The optimal cooperative path can be derived as:
Kt+1 =
n
X
j=1
ψtj∗ (Kt ) − δKt + ϑt , K 1 = K 0 , for t ∈ {1, 2, · · · , T },
(3.4)
We use Xs∗ to denote the set of realizable values of Ks generated by (3.4) at
stage s and use Ks∗ ∈ Xs∗ to denote an element in the optimal set.
Let ξ(·, ·) denote the agreed-upon imputation vector guiding the distribution of
the total cooperative payoff under the agreed-upon optimality principle along the
T
cooperative trajectory { Ks∗ }s=1 . At stage s and if the productive stock is Ks∗ , the
imputation vector according to ξ(·, ·) is
ξ(s, Ks∗ ) = [ξ 1 (s, Ks∗ ), ξ 2 (s, Ks∗ ), · · · , ξ n (s, Ks∗ )], for s ∈ {1, 2, · · · , T }.
(3.5)
A variety of examples of imputations ξ(s, Ks∗ ) can be found in Yeung and Petrosyan (2006 and 2012). For individual rationality to be maintained throughout all
stages, it is required that:
ξ i (s, Ks∗ ) ≥ V i (s, Ks∗ ), for i ∈ N and s ∈ {1, 2, · · · , T }.
To satisfy group optimality, the imputation vector has to satisfy
W (s, Ks∗ ) =
n
X
j=1
ξ i (s, Ks∗ ), for s ∈ {1, 2, · · · , T }.
408
David W.K. Yeung, Leon A. Petrosyan
3.2.
Subgame Consistent Solutions and Payoff Distribution Procedure
Under a subgame consistent situation, an extension of the solution policy to a subgame starting at a later stage with a state brought about by previous optimal
behaviour would remain optimal. For subgame consistency to be satisfied, the imputation ξ(·, ·) according to the original agreed-upon optimality principle in (3.5)
T
has to be maintained along the cooperative trajectory { Ks∗ }s=1 .
Following the analysis of Yeung and Petrosyan (2010 and 2012), we formulate
a Payoff Distribution Procedure so that the agreed-upon imputations (3.5) can be
realized.
Let Bki (Kk∗ ) denote the payment that agent i will received at stage k under the
cooperative agreement if Kk∗ is realized at stage k ∈ {1, 2, · · · , T }.
The payment scheme involving Bki (Kk∗ ) constitutes a PDP in the sense that if
∗
Kk is realized at stage kthe imputation to agent i over the stages from k to T can
be expressed as:
k−1
1
∗
i
∗
i
ξ (k, Kk ) = Bk (Kk )
1+r
+Eθk+1 ,θk+2 ,··· ,θζ
T
X
Bζi (Kζ∗ )
ζ=k+1
1
1+r
ζ−1
=
i
∗
(Kk+1
)
Bk+1
1
1+r
+ q i (KT +1 )(1 + r)−T
for i ∈ N and k ∈ κ.
Using (3.6) one can obtain
i
ξ (k +
+Eθk+2 ,θk+3 ,··· ,θζ
T
X
∗
1, Kk+1
)
Bζi (Kζ∗ )
ζ=k+2
ζ−1
1
1+r
+Eθk
ξ i [k + 1,
n
X
j=1
+ q i (KT +1 )(1 + r)−T
1
1+r
,
(3.6)
.
(3.7)
k
Upon substituting (3.7) into (3.6) yields
ξ i (k, Kk∗ ) = Bki (Kk∗ )
k−1
ψtj∗ (Kk∗ ) − δKk∗ + ϑk ]
(3.8)
,
for i ∈ N and k ∈ κ.
Theorem 3.1. Given that the public capital stock is Kk∗ in stage k a payment
equalling
Bki (Kk∗ ) = (1 + r)k−1 ξ i (K, x∗k )
−Eθk
ξ i [k + 1,
n
X
j=1
ψtj∗ (Kk∗ ) − δKk∗ + ϑk ]
,
(3.9)
for i ∈ N , be paid to agent i at stage k ∈ {1, 2, · · · , T } would lead to the realization
of the imputation {ξ(k, Kk∗ ), for k ∈ {1, 2, · · · , T }}.
409
Subgame Consistent Cooperative Solution of Stochastic Dynamic Game
Proof. From (3.8), one can readily obtain (3.9). Theorem 4.1 can also be verified
alternatively by showing that from (3.6)
ξ
+Eθk+1 ,θk+2 ,··· ,θζ
=
+
T
X
i
(k, Kk∗ )
T
X
ζ=k+1
ξ i (k, Kk∗ ) − Eθk
Eθk+1 ,θk+2 ,··· ,θζ
ζ=k+1
=
1
1+r
k−1
1
1+r
ζ−1
+ q i (KT +1 )(1 + r)−T
Bki (Kk∗ )
Bζi (Kζ∗ )
ξ i [k + 1,
n
X
j=1
ξ i (ζ, Kζ∗ ) − Eθζ
ψtj∗ (Kk∗ ) − δKk∗ + ϑk ]
ξ i [ζ + 1,
n
X
j=1
ψtj∗ (Kk∗ ) − δKk∗ + ϑk ]
= ξ i (k, Kk∗ );
given that ξ i (T + 1, KT∗ +1 ) =q i (KT +1 )(1 + r)−T .
⊔
⊓
Note that the payoff distribution procedure in Theorem 3.1 would give rise to
the agreed-upon imputation in (3.5) and therefore subgame consistency is satisfied.
When all agents are using the cooperative strategies, the payoff that agent i will
directly receive at stage s is
Ri (Ks∗ ) − C i [ψsi∗ (Ks∗ )].
However, according to the agreed upon imputation, agent i is supposed to receive
Bsi (Ks∗ ). Therefore a transfer payment (which could be positive or negative)
̟i (s, Ks∗ ) = Bsi (Ks∗ ) − {Ri (Ks∗ ) − C i [ψsi∗ (Ks∗ )]}
(3.10)
will be allotted to agent i ∈ N at stage s to yield the cooperative imputation
ξ i (k, Kk∗ ).
4.
An Illustration
In this section, we provide an illustration with an application in the build-up of
public capital by multiple asymmetric agents which is a discrete time counter-part
of example in Yeung and Petrosyan (2013). Consider an economic region with n
asymmetric agents. These agents receive benefits from an existing public capital
stock K(s). The accumulation dynamics of the public capital stock is governed by
Kt+1 =
n
X
j=1
Itj − δKt + ϑt , K 1 = K 0 , for t ∈ {1, 2, · · · , T },
(4.1)
where δ is the depreciation rate of the public capital, Iti is the investment made
by the ith agent in the public capital in stage t, and ϑt is an independent random
t
variable with non-negative range
{ϑ1t , ϑ2t , · · · , ϑω
t } and corresponding probabilities
P
ω
ω
t
{λ1t , λ2t , · · · , λt t }. Moreover h=1 λht ϑht = ϑ̄t > 0.
410
David W.K. Yeung, Leon A. Petrosyan
Each agent gains from the existing level of public capital and the ith agent seeks
to maximize its expected stream of monetary gains:
Eϑ1 ,ϑ2 ,··· ,ϑT
T
X
[αi Ks − ci (Isi )2 ](1 + r)−(s−1) + (q1i KT +1 + q2i )(1 + r)−T , (4.2)
s=1
subject to (4.1);
where αi , ci , q1i and q2i are positive constants.
In particular, αi gives the gain that agent i derives from the public capital,
i i
c (Is (s))2 is the cost of investing Isi in the public capital, and (q1i KT +1 + q2i ) is the
terminal valuation of the public capital at stage T + 1. The noncooperative market
outcome of the industry will be explored in the next subsection.
4.1.
Noncooperative Market Outcome
Invoking the analysis in (2.1)-(2.5) in section 2 we obtain the corresponding HamiltonJacobi-Bellman equations
V i (t, K) = max Eϑt
Iti
+V i
t + 1,
n
X
j=1
j 6= i
[αi K − ci (Iti )2 ](1 + r)−(t−1)
φjt (K) + Iti − δK + ϑt
, for t ∈ {1, 2, · · · , T },
V i (T + 1, K) = (q1i KT +1 + q2i )(1 + r)−T , for i ∈ N.
(4.3)
(4.4)
Performing the maximization operator in (4.3) yields:
φit (K) =
ωt
X
h=1
λht
n
X
1 i
V
[
t
+
1,
φjt (K) − δK + ϑht ] (1 + r)(t−1) , for i ∈ N. (4.5)
K
2ci t+1
j=1
To solve the game (4.1)-(4.2) we first obtain the value functions as follows.
Proposition 4.1. The value function of agent i can be obtained as:
V i (t, K) = (Ait K + Cti )(1 + r)−(t−1) ,
for t ∈ {1, 2, · · · , T + 1} and i ∈ N ;
where AiT +1 = q1i and CTi +1 = q2i ,
Ait = (αi − Ait+1 δ) and Cti = −
for t ∈ {1, 2, · · · , T }.
Proof. See Appendix A.
(Ait+1 )2
i
4ci +At+1
Pn
Ajt+1
h
j=1 2cj +ϑ̄t
(4.6)
i
+Ct+1
,
⊔
⊓
Using Proposition 4.1 and (4.5) the game equilibrium strategies can be obtained
to characterize the market equilibrium. The asymmetry of agents brings about different payoffs and investment levels in public capital investments.
Subgame Consistent Cooperative Solution of Stochastic Dynamic Game
411
4.2. Cooperative Provision of Public Capital
Now we consider the case when the agents agree to act cooperatively and seek
higher gains. They agree to maximize their expected joint gain and distribute the
cooperative gain proportional to their non-cooperative expected gains. To maximize
their expected joint gains the agents maximize
Eϑ1 ,ϑ2 ,··· ,ϑT
n X
T
X
j=1 s=1
+
n
X
[αj Ks − cj (Isj )2 ](1 + r)−(s−1)
(q1j KT +1 + q2j )(1 + r)−T
j=1
,
(4.7)
subject to dynamics (4.1).
Following the analysis in (3.2)-(3.3) in Section 3, the corresponding stochastic dynamic programming equation can be obtained as:
W (t, K) =
+W
max
{Itj f or j∈N }
t + 1,
n
X
ℓ=1
Eϑt
n
X
j=1
Itℓ − δK + ϑt
W (T + 1, K) =
n
X
[αj K − cj (Itj )2 ](1 + r)−(t−1)
, for t ∈ {1, 2, · · · , T },
(q1j KT +1 + q2j )(1 + r)−T .
(4.8)
(4.9)
j=1
Performing the maximization operator in (4.8) yields:
ψti (K) =
ωt
X
h=1
n
λht
X j
1
WKt+1 [ t+1,
ψt (K)−δK +ϑht ] (1+r)(t−1) , for i ∈ N. (4.10)
i
2c
j=1
Proposition 4.2. The value function W (t, K) can be obtained as
W (t, K) = (At K + Ct )(1 + r)−(t−1) ,
(4.11)
for t ∈ {1, 2, · · · , TP+ 1};
P
where AT +1 = nj=1 q1j and CT +1 = nj=1 q2j ,
Pn
Pn (At+1 )2
+At+1 ϑ̄ht +Ct+1 ,
At = j=1 αj − At+1 δand Ct = j=1 4c
j
for t ∈ {1, 2, · · · , T }.
Proof. Follow the proof of Proposition 4.1.
⊔
⊓
Using (4.10) and Proposition 4.2 the optimal investment strategy of public capital stock can be obtained as:
ψti (K) =
At+1
, for i ∈ N and t ∈ {1, 2, · · · , T }.
2ci
(4.12)
Using (4.1) and (4.12) the optimal trajectory of public capital stock can be
expressed as:
412
David W.K. Yeung, Leon A. Petrosyan
Kt+1 =
n
X
At+1
− δKt + ϑt , K 1 = K 0 , for t ∈ {1, 2, · · · , T },
j
2c
j=1
(4.13)
We use Xs∗ to denote the set of realizable values of Ks generated by (4.13) at
stage s. The term Ks∗ ∈ Xs∗ is used to denote and element in Xs∗ .
4.3.
Subgame Consistent Payoff Distribution
Next, we will derive the payoff distribution procedure that leads to a subgame
consistent solution. With the agents agreeing to distribute their gains proportional
to their non-cooperative gains, the imputation vector becomes
V i (s, Ks∗ )
ξ i (s, Ks∗ ) = Pn
W (s, Ks∗ )
j (s, K ∗ )
V
s
j=1
= Pn
Ais Ks∗ + Csi
j ∗
j
j=1 (As Ks + Cs )
(As Ks∗ + Cs )(1 + r)−(s−1) ,
(4.14)
for i ∈ N and s ∈ {1, 2, · · · , T } if the public capital stock is Ks∗ ∈ Xs∗ .
To guarantee dynamical stability in a dynamic cooperation scheme, the solution
has to satisfy the property of subgame consistency which requires the satisfaction
of (4.14) at all stages s ∈ {1, 2, · · · , T }. Invoking Theorem 3.1 we can obtain:
Proposition 4.3. A PDP which would lead to the realization of the imputation
ξ(s, Ks∗ ) in (4.14) includes a terminal payment (q1i KT∗ +1 + q2i ) to agent i ∈ N at
stage T + 1 and an payment at stage s ∈ {1, 2, · · · , T }:
Bsi (Ks∗ ) = Pn
Ais Ks∗ + Csi
j ∗
j
j=1 (As Ks + Cs )
(As Ks∗ + Cs )
ωs
X
i
Ai K ∗ (ϑh ) + Cs+1
∗
λhs Pn s+1j s+1∗ s
[As+1 Ks+1
(ϑhs ) + Cs+1 ](1 + r)−1 , for i ∈ N,
h) + C j ]
[A
K
(ϑ
s+1 s+1 s
s+1
j=1
h=1
(4.15)
Pn As+1
∗
∗
h
(ϑhs ) =
−
δK
+ϑ
.
⊔
⊓
where Ks+1
s
s
j=1 2cj
Finally, when all agents are using the cooperative strategies, the payoff that
agent i will directly receive at stage s is
−
αj Ks∗ −
(AS+1 )2
.
4cj
However, according to the agreed upon imputation, agent i is to receive Bsi (Ks∗ ) in
Proposition 4.3. Therefore a transfer payment (which can be positive or negative)
equalling
(A
)2
̟ii (s, Ks∗ ) = Bsi (Ks∗ ) − αj Ks∗ − S+1
(4.16)
4cj
will be imputed to agent i ∈ N at stage s ∈ {1, 2, · · · , T }.
413
Subgame Consistent Cooperative Solution of Stochastic Dynamic Game
5.
Concluding Remarks
This paper presented subgame consistent cooperative solutions for stochastic discretetime dynamic games in public goods provision. The solution scheme guarantees
that the agreed-upon optimality principle can be maintained in any subgame and
provides the basis for sustainable cooperation. A "payoff distribution procedure"
(PDP) leading to subgame-consistent solutions is developed. Illustrative examples
are presented to demonstrate the derivation of subgame consistent solution for public goods provision game. This is the first time that subgame consistent cooperative
provision of public goods is analysed in discrete time. Various further research and
applications, especially in the field of operations research, are expected.
Appendix A. Proof of Proposition 4.1.
Using the value functions in Proposition 4.1 the optimal strategies in (4.5) becomes:
φit (K) =
Ait+1
, for i ∈ N and t ∈ {1, 2, · · · , T }.
2ci
(A.1)
Using (A.1) the Hamilton-Jacobi-Bellman equations (4.4)-(4.5) reduces to:
ω
Ait K + Cti = αi K −
t
(Ait+1 )2 X
+
λht Ait+1
i
4c
h=1
n
X
Ajt+1
j=1
2cj
− δK + ϑht
i
+ Ct+1
,
(A.2)
for i ∈ N and t ∈ {1, 2, · · · , T },
AiT +1 K + CTi +1 = q1i K + q2i , for i ∈ N.
(A.3)
For (A.3) to hold it requires
AiT +1 = q1i and CTi +1 = q2i .
(A.4)
Re-arranging terms in (A.2) yields:
Ait K + Cti = (αi − Ait+1 δ)K −
(Ait+1 )2
+ Ait+1
4ci
n
X
Ajt+1
j=1
2cj
+ ϑ̄ht
i
+ Ct+1
, (A.5)
for i ∈ N and t ∈ {1, 2, · · · , T }.
For (A.5) to hold it requires
Ait = (αi − Ait+1 δ) and Cti = −
(Ait+1 )2
+ Ait+1
4ci
n
X
Ajt+1
j=1
2cj
+ ϑ̄ht
i
+ Ct+1
. (A.6)
Note that Ait and Cti depend on the model parameters and the succeeding values
i
of Ait+1 andCt+1
. Using (A.4) all Ait and Cti , for i ∈ N and t ∈ {1, 2, · · · , T }, are
explicitly obtained.
Hence Proposition 4.1 follows. Q.E.D.
414
David W.K. Yeung, Leon A. Petrosyan
References
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Dockner, E., Jorgensen, S., Long, N. V., Sorger, G. (2000). Differential games in economics
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Wirl, F. (1996). Dynamic voluntary provision of public goods: extension to nonlinear strategies. Eur J Polit Econ, 12, 555–560.
Yeung, D. W. K., Petrosyan, L. A. (2004). Subgame consistent cooperative solution in
stochastic differential games. J Optim Theory Appl, 120(2.3), 651-666.
Yeung, D. W. K., Petrosyan, L. A. (2006). Cooperative stochastic differential games.
Springer-Verlag, New York
Yeung, D. W. K., Petrosyan, L. A. (2010) Subgame Consistent Solutions for cooperative
Stochastic Dynamic Games. Journal of Optimization Theory and Applications 145(3):
579-596.
Yeung, D. W. K., Petrosyan, L. A. (2012). Subgame consistent economic optimization: an
advanced cooperative dynamic game analysis. Boston, Birkhauser
Yeung, D. W. K., Petrosyan, L. A. (2013). Subgame Consistent Cooperative Provision
of Public Goods. Dynamic Games and Applications, forthcoming in 2013. DOI:
10.1007/s13235-012-0062-7.
Joint Venture’s Dynamic Stability with Application to the
Renault-Nissan Alliance
Nikolay A. Zenkevich and Anastasia F. Koroleva
St.Petersburg State University,
Graduate School of Management,
Volkhovskiy per. 3, St.Petersburg, 199004, Russia
E-mail: [email protected]
E-mail: [email protected]
Abstract The cooperative dynamic stochastic multistage game of joint venture is considered. We suggest a payoff distribution procedure (PDP), which
defines a time consistent imputation. Based on the results obtained, we conduct a retrospective analysis of dynamic stability of the Renault-Nissan
alliance. It is shown that partners within the alliance have divided their
cooperative payoffs according to the suggested PDP.
Keywords:strategic alliance, joint venture, dynamic stochastic cooperative
games, dynamic stability, normalized share.
1.
Introduction
In the recent decades economic globalization continuously increases at a rapid pace.
There are constant and strong changes in competitive environment and markets
structure. Moreover, customers become more and more informed and seek better
quality of products and services. Under such conditions companies are confronted
with the increasing challenges of providing themselves with the resources, technologies, competences, skills and information, necessary for achieving competitive
advantage. Thus, strategic alliances, and, in particular, joint ventures (JV), are
considered to become a necessary condition for company to survive in a violent
competitive world. For this reason during the recent decades a number of strategic alliances and JVs shows steadily growth (Meschi and Wassmer, 2013). Indeed,
strategic alliances allow companies expanding their geography, entering new markets, getting access to new knowledge, information, technologies, skills and competencies rather quickly (Barringer and Harrison, 2000; Bucklin and Segupta, 1993;
Inkpen and Beamish, 1997). Hence, it is not surprising that numerous companies
across the world view strategic alliances and JVs as a source of competitive advantage that allows them managing challenges that arise under conditions of markets
globalization (Kumar, 2011; Smith et al., 1995).
Strategic alliances have attracted much academic attention in the recent decades.
In particular, due to strategic alliances high failure rates (according statistics, more
than 50% of strategic alliance and JV agreements dissolve (Kale and Singh, 2009),
researchers are especially interested in the issues of strategic alliances and JVs stability (Das and Teng, 2000; Inkpen and Beamish, 1997; De Rond and Buchikhi, 2004).
However, despite considerable interest in the academic community to the issue
of strategic alliance and JV stability, common view on this topic has not yet been
reached. Moreover, there are several major unresolved issues that require solutions,
among which are:
416
Nikolay A. Zenkevich, Anastasia F. Koroleva
1. Problem of measuring stability of strategic alliances. The question of how to
measure the degree of stability of alliance remains unanswered. In general, studies
are focusing on identifying factors that may affect stability or instability of strategic
alliance ( Deitz et al., 2010; Gill and Butler, 2003; Jiang et al., 2008).
2. Strategic alliance stability evaluation. Currently, existing research examining
the stability of the strategic alliance has not offered a method of assessing the
strategic alliance stability. This can be explained by the existence of various factors
of different nature that can influence overall alliance stability in numerous ways.
Hence, it becomes a challenge to assess all the components of alliance stability. For
instance, there are external factors that affect stability, such as institutional and
competitive environment, but there are internal factors as well - trust, opportunistic
behaviour, distribution of cooperative benefits, etc. It is clear, that methods of
stability evaluation of stability components (e.g. external and internal), probably,
should differ due to the different nature of factors, that determine stability.
The important problems associated with the concept of a strategic alliance and
its’ stability warrant further theoretical and methodological research in this area. In
this paper, we attempt to address this gap by implementing and testing game theory
methodology for evaluating alliance and JV stability component, that is determined
by cooperative benefits allocation factor.
Despite the existence of different factors that affect alliance and JV stability,
the factor of allocation of cooperative benefits between the partners during the
whole period of alliance realization can be considered as one of the most important
(Dyer et al., 2008). It is obvious that when one or several alliance participants do
not agree on the distribution of cooperative benefits their motivation for participation decreases which affects stability. Hence, it would be highly useful for alliance
partners to know in what way they should design the part of their cooperative
agreement concerning benefits allocation for alliance to be stable and to have some
instrument that will allow them assessing alliance stability during its realization
phase.
In this paper we make an attempt develop an approach for solving these tasks.
The approach is based on the concept of dynamic stability in dynamic cooperative
games (Petrosjan, 2006).
The paper organized as follows: the first section presents the model of joint
venture and suggests a way for cooperative benefits allocation among partners; in
the second section the model is applied to a case of Renault-Nissan JV to analyze
it’s stability; in the conclusions we summarize the main results of the article.
2.
Model of Joint Venture
In order to model a JV, a cooperative stochastic multistage dynamic game is considered (Petrosjan, 2006). In particular, a multistage game with the infinite duration
and random time closure is used due to the fact that most of the agreements on
strategic cooperation between the companies do not have a predetermined end date
of the alliance. Dependent on the circumstances in which alliance partners are, they
can only make assumptions on when strategic partnership will come to an end. Multistage principle of the game means that players make decisions at certain discrete
points of time which correspond to the steps of the game. The stochastic multistage
game with random duration G(xt0 ) = (N ; V (S, xt0 )) is considered:
417
Joint Venture’s Dynamic Stability
1. players together take a decision on their cooperative strategy in order to
obtain the highest overall benefits;
2. players agree on the allocation mechanism of jointly received benefits between
partners.
The game G(xt0 ) that we are considering is described as follows:
N = 1, ..., n – is a number of players (members of JV).
Z = 0, ..., ∞ – is a set of steps in the game G(xt0 ).
tm , m = 0, z − 1 – is time, during which the game evolves.
X – is a set of all possible states in the game, such that:
∞
[
Xtk ∩Xtl = ∅ ,
Xtm = X ,
m=0
k 6= l ,
t0 < t1 < ... < tl <, · · · , < t∞ .
In other words, for any point in time tm in the game G(xt0 ) corresponds definite
step m in the game, on which a set of possible states Xtm of a strategic alliance is
given.
F – is a multivalued mapping, that:
Ftm : Xtm → Xtm+1 ,
tm
m = 0, z − 1
forx ∈ Xtz .
Mapping F defines a set of possible states of the game at each step.
Utim = {uitm } – is a set of possible controls for the player i at the game moment
(step m).
Utm =
Y
Utim ,
U=
i∈N
Y
Utm ,
m=0,z−1
where i = 1, n, m = 0, z − 1, uitm – control
Vector ui = (uit0 , ..., uitm , ..., uitz−1 ) is called
of player i in the game moment tm .
a strategy of the player i in the game
G(tt0 ):
u i ∈ Ui ,
i∈N .
Vector u = (u , · · · , u ) – is a situation in the game.
Vector utm = u1tm , ..., uitm , ..., untm , i = 1, n, m = 0, z − 1 – is a control vector at
the time tm . utm is such that:
utm : ∀xtm ∈ Xtm → xtm+1 ∈ Xtm+1 .
It is assumed that being in the state xtm ∈ Xtm players do not know for sure
what state in xtm+1 ∈ Ftm (x) ⊂ Xtm+1 they will reach using control vector utm .
But in each state xtm ∈ Xtm , m = 0, z − 1 probabilities of reaching states at the
next step of the game, that are dependent on the control vector utm are given:
xtm+1 ∈ Ftm (x) ⊂ Xtm+1 :
1
n
p(xtm , xtm+1 ; u1tm , · · · , untm ) = p(xtm , xtm+1 ; utm ) > 0 ,
X
p(xtm , xtm+1 ; utm ) = 1 ,
xtm+1 ∈Ftm (x)
where p(xtm , xtm+1 ; utm ) is the probability that at the step m + 1 the xtm+1 state is
realized , provided that at the step m was implemented control utm . In each possible
418
Nikolay A. Zenkevich, Anastasia F. Koroleva
state x ∈ X is given a probability qm , 0 < qm 6 1, m = 0, z − 1, that the game
will end at step m.
Now, let us consider only those states in the game, that have positive probability
of being reached by implementation of control vectors utm , m = 0, z − 1:
CX = {xtm : p(xtm , xtm+1 ; utm > 0, ∀xtm ∈ X, m = 0, ∞}. CX ⊂ X.
Value function of the game G(N ; V (S, xt0 )) is defined as a lower value of a zerosum game between two players – coalition S and coalition N \ S, assuming that
the players use only pure strategies. Details on the value function of a cooperative
game can be found in (Zenkevich et al. 2009). Let us define it.
Coalition N acts as one decision making center and will try to maximize their total benefits in the game. Suppose, that a sequence of control vectors ut0 , ut1 , · · · , utm ,
· · · , utz−1 was implemented.
Then the payoff of player i will be determined by the formula:
Ki (xt0 ; ut0 , ut1 , · · · , utm , · · · , utz−1 ) = Ki (xt0 ) =


!
j
∞
X
Y
X
tm


=
qm
(1 − qm )
Ki (utm .
j=0
m<j,j>0
k=0
Due to the fact that the game has a random nature, it is reasonable to consider
the expected payoff of the alliance, that players try to maximize in the game G(xt0 )):
V (N, xt0 ) = maxutm
"
X
i∈N
#
Ei (xt0 ; utm , · · · , utz−1 )
(1)
.
Vector ū = (ū1 , · · · , ūn ) is called a cooperative solution.
Maximum of (1) is found by solving the corresponding Bellman equation
V (N, xt0 ) = maxui (xt0 )∈Ui (xt0 ), i∈N
"
X
i∈N
(1 − q0 )
=
X
i∈N
Kit0 (ūt0 ) + (1 − q0 )
X
xt1 ∈F (xt0 )
X
Kit0 (ut0 )+

p(xt0 , xt1 ; ut0 V (N, xt1 ) =
p(xt0 , xt1 ; ūt0 )V (N, xt1 )
(2)
xt1 ∈F (xt0 )
with the boundary condition
V (N, xtm ) = maxui (xtm )∈Ui (xtm ),i∈N
X
i∈N
Kitm (utm ) ,
x ∈ {x : F (x) = ∅} .
(3)
419
Joint Venture’s Dynamic Stability
In the case when coalition S 6= N and S 6= ∅, value function is described by the
following equation
maxuS (xtm )∈US (xtm ) minuN \S (xtm )∈UN \S (xtm )
"
X
V (S, xtm ) =
Kitm (uS (xtm ), uN \S (xtm ))+
i∈S
(1 − qm )
X
xtm+1 ∈F (xtm )

p xtm , xtm+1 ; uS (xtm ), uN \S (xtm ) V (S, xtm+1 )
(4)
with the boundary condition
maxuS (xtm )∈US (xtm ) minuN \S (xtm )∈UN \S (xtm )
X
V (S, xtm ) =
Kitm (uS (xtm ), uN \S (xtm )),
i∈S
x ∈ {x : F (x) = ∅} ,
(5)
where i1 , · · · , ik ∈ S, ik+1 , · · · , in ∈ N \S and
uS (xtm ) = (uit1m , · · · , uitkm );
i
uN \S (xtm ) = (utk+1
, · · · , uitnm ) .
m
For the case when S = ∅ it is assumed that its’ payoff is 0 :
V (∅, xtm ) = 0 .
(6)
Thus, the game G(xt0 ) is defined by the pair (N ; V (S, x0 )), where
1. Value function V (S, xt0 ) is determined by the formula (2) with the boundary
condition (3) for S = N ;
2. Value function V (S, xt0 ) is determined by the formula (4 ) with the boundary
condition (5) with S 6= ∅ ;
3. Value function V (S, xt0 ) is determined by the formula (6) with S = ∅.
The main objective of alliance members is a division of the benefits derived by
joint efforts. In the game theory terminology, payoffs of players at the end of the
game are called imputation.
Definition 1 (Petrosjan et al., 2004). Vector ξ(xt0 ) = (ξ1 (xt0 ), · · · , ξn (xt0 ))
is called imputation in a cooperative stochastic game with the random duration
G(xt0 ), if :
P
1.
i∈N ξi (xt0 ) = V (N, xt0 ) ;
2. ξi (xt0 ) > V ({i}, xt0 ), for all i ∈ N ,
where V ({i}, xt0 ) is a winning coalition S in a zero-sum game against the coalition
V ({i}, xt0 ) when coalition S consists of only one player i.
The set of all possible imputations in the cooperative stochastic game G(xt0 ) is
denoted as I(xt0 ).
Definition 2 (Petrosjan et al., 2004). Solution of a cooperative stochastic game
is any fixed subset of C(xt0 ) ⊂ I(xt0 ).
420
Nikolay A. Zenkevich, Anastasia F. Koroleva
Value function definition and definitions 1-2 are also valid for any subgame
G(xtm ) of the original game G(xt0 ), that starts at time tm from the state xtm .
Thus, having introduced the cooperative stochastic game G(xt0 ) and having
defined the concept of sharing the benefits of cooperation, we defined the stochastic
model of strategic alliance.
The main issue of cooperative game theory is the study of the dynamic stability
of the division of benefits from cooperation. So let us move to the results obtained
in the game theory in the area of the stability of cooperative behaviour.
Definition 3 (Petrosjan et al., 2004). Vector function β(xtm ) = (β1 (xtm ), · · · ,
βn (xtm )), where xtm ∈ CX, is called payoff distribution procedure (PDP) at a vertex
xtm , if
X
X
X
βi (xtm ) =
Kitm (ū1tm , · · · , ūntm ) =
Kitm (ūtm ) ,
i∈N
i∈N
(ūt1m , · · ·
i∈N
where ūtm =
is the situation at the time tm in the game element
G(xtm ) that was realized under cooperative solution ū = (ū1 , · · · , ūn ) in the game
G(xt0 ).
, ūtnm )
Definition 4 (Zenkevich et al. 2009). Imputation ξ(xt0 ) ∈ C(xt0 ) is called time
consistent in a cooperative stochastic game G(xt0 ), if for each vertex xtm ∈ CX ∩
(F (xt0 ))k there exists a nonnegative PDP β(xtm ) = (β1 (xtm ), · · · , βn (xtm )) such
that
X
p(xtm , xtm+1 , ūtm ξi (xtm+1 )
(7)
ξi (xtm ) = βi (xtm ) + (1 − qm )
xtm+1 ∈F (xtm )
and
ξi (xtm ) = βi (xtm ), xtm ∈ {xtm : F (xtm ) = ∅} ,
where xtm ∈ (F (xtm ))k , ξ(xtm+1 ) = (ξ1 (xtm+1 ), · · · , ξn (xtm+1 )) is some imputation,
that belongs to a solution C(xtm+1 ) of cooperative subgame G(xtm+1 ).
Definition 5 (Zenkevich et al. 2009). Cooperative stochastic game with random duration G(xt0 ) is a time consistent solution C(xt0 ), if all imputations ξ(xt0 ) ∈
C(xt0 ) are time consistent.
Now, based on definitions 1-5, we introduce a normalized share.
Consider normalized shares for imputation ξ(xtm ) in the subgame G(xtm ), where
θi (xtm ) =
ξi (xtm )
, i∈N .
V (N, xtm )
(8)
According to equation (7)
θi (xtm ) = ai (xtm )+(1−qm )
X
p(xtm , xtm+1 , ūtm )
xtm+1 ∈F (xtm )
where
ai (xtm ) ≡
θi (xtm+1 )V (N, xtm+1 )
,
V (N, xtm )
(9)
βi (xtm )
,
V (N, xtm )
(10)
421
Joint Venture’s Dynamic Stability
X
ai (xtm ) =
i∈N
P
βi (xtm )
=
V (N, xtm )
i∈N
P
Kitm (ūtm )
<1.
V (N, xtm )
i∈N
Let us verify the normalization condition:
P
X
X
βi (xtm )
+(1−qm )
θi (xtm ) = i∈N
V (N, xtm )
i∈N
p(xtm , xtm+1 , ūtm )
xtm+1 ∈F (xtm )
V (N, xtm+1 )
.
V (N, xtm )
that is
1=
P
Kitm (ūtm )
+ (1 − qm )
V (N, xtm )
X
i∈N
p(xtm , xtm+1 , ūtm )
xtm+1 ∈F (xtm )
V (N, xtm+1 )
V (N, xtm )
or
V (N, xtm ) =
X
X
Kitm (ūtm )+(1−qm )
i∈N
p(xtm , xtm+1 , ūtm )V (N, xtm ) ,
xtm+1 ∈F (xtm )
Thus we came up to the equation (2).
Let us consider constant normalized share:
θi (xtm ) = θi (xtm+1 ) = θi = const .
(11)
Proposition. If normalized shares θi , i ∈ N (8) are constant (11) for any subgame
G(xtm ), m = 0, z − 1, then the imputation ξ(xt0 ) is time consistent in the game
G(xt0 ).
Proof. From (9) it follows that


X
V (N, xtm+1 )
 = ai (xtm ) .
θi 1 − (1 − qm )
p(xtm , xtm+1 , ūtm )
V (N, xtm )
xtm+1 ∈F (xtm )
Taking into account equation (2), it is easy to obtain
!
P
V (N, xtm ) − i∈N Kitm (ūtm )
θi 1 −
= ai (xtm ) .
V (N, xtm )
Simplifying equation (12) we derive
ai (xtm ) = θi
P
Kitm (ūtm )
V (N, xtm )
i∈N
or, what is the same,
θi = ai (xtm ) P
V (N, xtm )
.
tm
i∈N Ki (ūtm )
Finally, with the use of (10)
θi =
βi (xtm )
V (N, xtm )
βi (xtm )
·P
=P
,
tm
tm
V (N, xtm )
i∈N Ki (ūtm )
i∈N Ki (ūtm )
(12)
422
Nikolay A. Zenkevich, Anastasia F. Koroleva
that is
βi (xtm ) = θi
X
Kitm (ūtm ) .
i∈N
Thus, it is shown that when normalized share is constant in the sense of (11), it
means that ξ(xt0 ) is time consistent in a game G(xt0 ).
3.
Analysis of Renault-Nissan JV
In this section, the Renault-Nissan JV is analysed. This alliance is considered to
be one of the most successful and stable alliances in the world. Renault and Nissan
companies started their informal collaboration in 1999. At that time Nissan had
strong engineering experience, car design that did not attract much customers and
serious financial problems. Renault had good design and administration practices
but was not strong in engineering. It was seen that companies had complementary
resources and could improve each others positions. Hence, companies decided to
start to cooperate. Renault bought 36.8% shares of Nissan company and increased
the amount up to 44.4% in 2001, and Nissan bought 15% of Renault company shares
the same year. At that time there was no formal agreement on cooperative activity
of the Renault and Nissan companies, however both of them were interested in
developing of the partners’ company due to the ownership of partners equity shares.
Finally, companies formed a strategic alliance in a form of JV in 2003. This step
was the initiation of formal cooperative activity.
Fig. 1 shows the strategic alliance’s structure (Renault official website).
Fig. 1: Renault-Nissan strategic alliance structure
Within JV companies cooperate in a broad range of areas. First of all, Renault
and Nissan use the same distribution channels for both companies, which allowed
Nissan gaining positions on European market and made possible for Renault enter
Japan and South American markets. Secondly, most of the Renault and Nissan cars
have the same production platforms. This means that Renault can produce its cars
at Nissan plants and vice versa. That leads to significant cost reductions. Thirdly,
companies cooperate in innovations and technology areas. They jointly provide research and development activities and jointly produce engines, accumulators and
other car components. For instance, partners concentrate on development of engines
423
Joint Venture’s Dynamic Stability
with zero gas emission rate to the atmosphere. Finally, companies have unified supply chain of components. In 2010 companies announced that collaborative initiatives
led to 1.5 billion Euro of cost reduction that year. In 2012 the sales of Renault-Nissan
alliance reached the level of 8.1 million units across the world. That showed a 1%
increase in sales comparing to the previous period and continuing growth. In order
to analyse whether the one of the long lasting alliance is dynamically stable, we
calculated their payoffs for the realization period of the alliance.
The analysis starts from 2004, because 2003 is considered as a period of alliance formation phase in accordance with strategic management theory. At this
phase alliance coordinates it’s operations and companies adapt to new conditions
(De Rond and Buchikhi, 2004; Styles and Hersch, 2005).
The goal of the analysis is to check whether Renault and Nissan companies
use such PDP, that their imputations are time consistent. Therefore, we are going
to check whether normalized shares during the alliance realization phase remain
constant or not.
First, it is necessary to calculate companies’ payoffs, taking into account the
complicated structure of their relationship. Hence we consider companies’ financial data. Because Nissan is a Japanese company, Nissan’s reports provide the
numbers in Japanese Yens. Hence, the convertion from Japanese Yens to Euros
was necessary. Table 1 shows financial data of Nissan company in Japanese Yens
(Nissan official website) and exchange rates, that were used to make the conversion
(Renault official website).
Table 1: Financial data of Nissan company, U million
Year
Nissan
shareholders’
equity
Nissan
dividends
Exchange
rate
e/ U
2004
2465.75
94.24
134.00
2005
3087.99
105.66
136.80
2006
3586.62
131.06
146.10
2007
3868.14
151.73
161.20
2008
3556.48
126.30
152.30
2009
3598.97
0.00
129.40
2010
3981.51
20.92
116.50
2011
4269.83
62.75
111.00
The financial data necessary to make the calculations is presented in Table 2 and
is obtained from the official sources (Renault official website; Nissan official website).
Columns with the Renault and Nissan net incomes report only the income received
by the companies from the Renault-Nissan JV.
Table 2 represents the data in unified form in Euro currency.
424
Nikolay A. Zenkevich, Anastasia F. Koroleva
Table 2: Financial data of Renault and Nissan companies, e million
Year
Renault
net
income
Nissan
net
income
Renault
shareholders’
equity
Nissan
shareholders’
equity
Renault
dividends
Nissan
dividends
2004
1.35
3.90
15.86
18.40
1.80
0.70
2005
1.18
5.19
19.49
22.57
2.40
0.77
2006
1.07
4.26
21.07
24.55
3.10
0.90
2007
1.45
2.95
22.07
24.00
3.80
0.94
2008
0.25
1.00
19.42
23.35
0.00
0.83
2009
-2.17
-1.91
16.47
27.81
0.00
0.00
2010
2.41
2.61
22.76
34.18
0.30
0.18
2011
0.88
3.29
24.57
38.47
1.16
0.57
In order to explain how we calculated companies payoffs, let us introduce the following notation: P ayof fR – Renault’s payoff; P ayof fN – Nissan’s payoff; IncomeR
– Renault net income from participating in JV; IncomeN – Nissan net income from
participating in JV; ShEqR – Renault shareholders’ equity; ShEqN – Nissan shareholders’ equity; DivR – dividends paid by Renault company; DivN – dividends paid
by Nissan company.
To get payoffs it is not enough to consider only the net income the companies
earned from JV. As was mentioned earlier, companies exchanged shares with each
other. During the whole period of alliance realization Nissan owned 15% shares of
Renault and this percentage remained constant. However, the percentage of Nissan’s shares owned by Renault differed during the alliance period and amounted to
43.4%, 44.3% and 44.4%. Possibly, the differences were caused by different methods
used by company to evaluate its’ share. For this reason we decided to consider the
average percent of three numbers listed above, which is equal to 44.03%.
The logic for companies’ payoff calculation is the following: if Renault company
owns 44.03% shares of Nissan company, than we should add the value of 44.03%
shares in Nissan company to Renault payoff. Also, we should not forget to incorporate the value of Renault company, which has the value of 85% shares. Moreover,
the payoff should include 44.03% of all the dividends that were distributed by Nissan company, as well as dividends that were distributed in Renault company. The
same logic applies to Nissan company payoff calculation.
Thus, the formula for Renault company payoff calculation is:
P ayof fR = IncomeR + 0, 85ShEqR + 0.4403ShEqN + 0.85DivR + 0.4403DivN .
Let us show how it works by analysing Renault payoff for 2004. In this case
IncomeR = 1.35, ShEqR = 15.86, ShEqN = 18.40, DivR = 1.80, DivN = 0.70.
Renault company owns only 85% of its’ shares. Hence, it has 0, 85ShEqR and
0.85DivR , but also it owns 44.4% of Nissan shares, that yields 0.444ShEqN and
0.444DivN . Finally, we should sum up all the components with IncomeR .
We got that P ayof fR = 24, 77.
425
Joint Venture’s Dynamic Stability
Following the same methodology, equation for Nissan company is
P ayof fN = IncomeN + 0.55977ShEqN + 0.15ShEqR + 0.5597DivN + 0.15DivR.
To calculate Nissan payoff in 2004 we should take the following numbers: IncomeN =
3.90, ShEqR = 15.86, ShEqN = 18.40, DivR = 1.80, DivN = 0.70.
Hence, P ayof fN = 17.24.
Payoffs for years 2005-2011 are calculated using the same formulas.
Total alliance benefits are the sum of the partners payoffs:
Benef itsJV = P ayof fR + P ayof fN .
Renault’s share and Nissan’s share are calculated as follows:
ShareR =
P ayof fR
,
Benef itsJV
ShareN =
P ayof fN
.
Benef itsJV
The results of the calculations of companies’ parameters are represented at Table
3.
Table 3: Renault and Nissan shares of alliance cooperative benefits, e millions
Year
Renault
payoff
Nissan
payoff
Total
alliance
benefits
Renault
share
Nissan
share
2004
24.77
17.24
42.01
0.59
0.41
2005
30.06
21.53
51.59
0.58
0.42
2006
32.82
22.12
54.94
0.60
0.40
2007
34.41
20.78
55.20
0.62
0.38
2008
27.40
17.44
44.84
0.61
0.39
2009
24.08
16.12
40.21
0.60
0.40
2010
37.13
25.29
62.42
0.59
0.41
2011
39.86
28.98
68.84
0.58
0.42
Here columns with Renault and Nissan benefits represent players’ payoffs at
game stages. Columns with Renault and Nissan shares correspond to considered in
the paper normalized shares.
Figure 2 represents the dynamics of Renault and Nissan shares during
426
Nikolay A. Zenkevich, Anastasia F. Koroleva
Fig. 2: Shares of Renault and Nissan companies of cooperative benefit
It is seen from the graph that payoff shares of Renault and Nissan companies
are approximately stable across the realization stage of the strategic alliance in the
form of JV. There are multiple reasons for slight variations of values. The major
one refers to the evaluation methodology used for annual report preparation.
To assess these variations, standard deviation σ = 0.01 was calculated. Thus,
companies’ shares can be considered to be equal during the whole period of alliance
realization.
We showed that one of the most successful and stable alliances in the world has a
time consistent imputation. This fact can be considered as a first argument towards
adequacy of implementation of dynamic stability concept to the investigation of JV
and alliance stability in terms of cooperative benefits distribution among alliance
partners.
It is worth mentioning that the reverse problem to those, that usually handle
game theory, was solved. The task was not to evaluate future payoffs of the players using the model of dynamically stable behaviour principle, but rather to check
whether the dynamic stability took place in terms of implementation of time consistent imputation principle. Thus, it is possible to provide retrospective analysis of
JV and alliance stability based on historical data of alliance/JV performance.
4.
Conclusion
In this paper we attempted to apply the cooperative game theory methodology in order to evaluate JV stability. We showed, that constant normalized share guarantees
imputation to be consistent. This fact allows companies developing a cooperative
agreement in a way, when all partners receive constant normalized share during the
whole period of alliance realization. Also, it enables retrospective checking of JV
stability as it was shown on the case of Renault-Nissan alliance. It appeared that
the alliance most known in the world for it’s success, durability and stability uses
imputation principle with constant normalized shares.Of course, the instrument developed and presented in the paper is applicable to a restricted range of problems
in the sense that it allows considering only one type of imputation. However, the
paper can be considered as a first step towards developing instrumental apparatus
Joint Venture’s Dynamic Stability
427
for JV stability evaluation. We believe that game theory methodology has a great
potential for solving the problem of strategic alliance and JV stability evaluation.
It can serve as a basis for developing instruments of practical assessment of different stability components, which would provide companies across the world with
a valuable strategic tool in designing and managing their alliance agreements in a
"stable" manner.
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Symmetric Core of Cooperative Side Payments Game
Alexandra B. Zinchenko
Southern Federal University,
Faculty of Mathematics, Mechanics and Computer Science,
Milchakova, 8 a, Rostov-on Don, 344090, Russia
E-mail: [email protected]
Abstract This paper concerns cooperative side payments games (with transferable utility and discrete) where at least two players are symmetric. The
core and symmetric core properties are compared. The problem of symmetric
core existence is considered.
Keywords: cooperative TU game, discrete game, core, symmetric core, balancedness.
1.
Introduction
In many practical situations some of participants have the identical power (prestige,
influence, resources, capitals). They are substitutes in associated cooperative game.
Moreover, non-symmetric in the underlying problem agents may become substitutes
in corresponding game. Player’s status can also changes in the zero-normalization
of a game. It seems reasonable to require that symmetric players should receive the
same payoff. However, almost no set-valued solution concepts (including the core,
core-based solutions, von Neumann-Morgenstern stable sets, the bargaining set)
that satisfy the equal treatment property. It is not difficult to provide the examples
of cooperative games, where the core allocations assign to symmetric players vastly
different payoffs. Even multi-solutions based on a concept of egalitarianism cannot
satisfy the equal treatment property (see for instance (Dutta and Ray, 1989)).
The symmetric core is a subset of core satisfying the equal treatment property.
This notion has been introduced in (Norde et al., 2002) for TU games with special
structure: the airport game, generalized airport game, maintenance cost game, infrastructure cost game. The symmetric core was used to get a minimal collection of
conditions that are equivalent to balancedness. In (Hougaard et al., 2001) the symmetric core was used for calculation of Lorenz-solution of a production economy
with a landowner and peasants. To the best of our knowledge, the symmetric core
was not yet discussed for general TU game.
Next section recalls some definitions and notations. The core and the symmetric
core properties are compared in section 3. It will be shown that symmetric core satisfies the most core axioms. The last section is devoted to the problem of symmetric
core existence.
2.
Preliminaries
A cooperative game with transferable utility (TU game) is given as GT = (N, ν),
where N = {1, ..., n}, n > 2, ν : 2N → R, ν(∅) = 0. So-called discrete game GD
differs from GT that ν is integer-valued function and players payoffs must be integers
(Azamkhuzhaev, 1991). In economic settings, the integer requirement reflects some
forms of indivisibility. Both games summarizes the possible outcomes to a coalition
Symmetric Core of Cooperative Side Payments Game
429
by one number, i.e. side payments are allowed. GT and GD can be also described
N
as a games with nontransferable utility (NTU games). Let GTN and GD
be the
N
N
N
sets of n-person TU and discrete games respectively, G = GT ∪ GD . Denote by
Ω = 2N P
\ {N, ⊘} the family of proper coalitions. Given x ∈ RN and ∅ 6= K ⊆ N :
x(K) = i∈K xi , x(∅) = 0. The cardinality of coalition ∅ 6= K ⊆ N is denoted by
|K|. When there is no ambiguity, we write ν(i), K \ i instead of ν({i}), K \ {i} and
so on.
Two players i, j ∈ N are called symmetric (substitutes, interchangeable) in a
game G ∈ G N if
ν(K ∪ i) = ν(K ∪ j) f or every K ∈ N \ {i, j}.
(1)
Player i ∈ N is veto player in a game G ∈ G N if ν(K) = 0 for all K 6∋ i. Denote
by veto(G) the set of veto players of G ∈ G N . A game GT is called convex if
ν(K) + ν(H) 6 ν(K ∪ H) + ν(K ∩ H) for K, H ⊆ N . A game GT is integer if
N
ν : 2N → Z, where Z denotes the set of integer numbers. The operator Ψ : GD
→ GTN
will be used to compare TU and discrete game solutions, i.e. Ψ (GD ) is an integer
TU game corresponding to GD .
The set of feasible payoff vectors X ∗ (GT ) and pre-imputation set X(GT ) of TU
game GT are defined by
X ∗ (GT ) = {x ∈ RN |x(N ) 6 ν(N )}, X(GT ) = {x ∈ RN |x(N ) = ν(N )}.
The related sets of discrete game GD are
X ∗ (GD ) = X ∗ (Ψ (GD )) ∩ ZN , X(GD ) = X(Ψ (GD )) ∩ ZN .
For any set G˜N ⊆ G N a set-valued solution (or multisolution) on G˜N is a mapping
ϕ : G˜N →→ RN which assigns to every G ∈ G˜N a set of payoff vectors ϕ(G) ⊆
X ∗ (G). Notice that the solution set ϕ(G) is allowed to be empty. A value of game
G is a function f : G˜N → X(G). The core of TU game and core of discrete game
are the sets
C(GT ) = {x ∈ X(GT )|x(K) > ν(K), K ∈ Ω}, C(GD ) = C(Ψ (GD )) ∩ ZN .
The formulas to obtain the CIS-value, ENSC-value, Shapey value and equal division
solution of a game GT are
P
ν(N ) − j∈N ν(j)
CISi (GT ) = ν(i) +
,
n
P
ν(N ) − j∈N ν ∗ (j)
EN SCi (GT ) = ν ∗ (i) +
,
n
X |K|!(n − |K| − 1)!
ν(N )
Shi (GT ) =
(ν(K ∪ i) − ν(K)), EDi (GT ) =
,
n!
n
K6∋i
∗
where i ∈ N , ν (K) = ν(N ) − ν(N \ K), K ⊆ N . The CIS-value is also called
the equal surplus division solution. Notice that CIS-value, ENSC-value and equal
division solution assign to every player some initial payoff and distribute the remainder of ν(N ) equally among all players. For CIS-value (the center of gravity
of imputation set I(GT ) = {x ∈ X(GT )|xi > ν(i), i ∈ N }) initial payoff to player
i ∈ N is equal to its individual worth ν(i). For ED-value and ENSC-value the initial
payoffs are equal to zero and player’s marginal contribution ν ∗ (i) to grand coalition
N , respectively. Thus, the ENSC-value assigns to any game GT the CIS-value of
dual game (N, ν ∗ ).
430
3.
Alexandra B. Zinchenko
Symmetric core properties
For a game G ∈ G N denote by ℑ(G) the family of coalitions each of which contains
only symmetric players
ℑ(G) = {K ∈ 2N ||K| > 2, every i, j ∈ K, i 6= j, are symmetric in G}.
Definition 1. A game G ∈ G N is called semi-symmetric if at least two players
are symmetric in G, i.e. ℑ(G) 6= ∅. A game G ∈ G N is (totally) symmetric if
ℑ(G) = {{N }}. A game G ∈ G N is non-symmetric if ℑ(G) = ∅.
N
N
Let SG N = SG N
T ∪ SG D be the set of semi-symmetric games G ∈ G .
Definition 2. The symmetric core SC(G) of a game G ∈ G N is the set of core
allocations for which the payoffs of symmetric players are equal
SC(G) = {x ∈ C(G)|xi = xj f or all i, j ∈ K, i 6= j, K ∈ ℑ(G)}.
Example 1. Let U H = (N, uH ) be n-person (n > 3) unanimity game for a coalition
H ∈ Ω: uH (K) = 1 for K ⊇ H, uH (K) = 0 otherwise. Since

if |H| = n − 1,
 {H}
ℑ(U H ) = {N \ H}
if |H| = 1,

{H, N \ H} else,
then the game U H is semi-symmetric. Well known that any unanimity game is
convex and C(U H ) = {x ∈ RN |xi = 0, i ∈ N \ H, x(H) = 1}. Therefore, the
symmetric core SC(U H ) consists of one point which is the Shapey value: SC(U H ) =
1
for i ∈ H, Shi (U H ) = 0 otherwise.
{Sh(U H )}, where Shi (U H ) = |H|
Example 2. Consider situation with four investors having the endowments 80, 60,
50, 50 units of money (m.u. for short). Assume the following investment projects
are available: a bank deposit that yields 10 interest rate whatever the outlay, two
production processes that require an initial investment of 100 ore 200 m.u. and
yields 15 ore 20 rate of return, respectively. The related four-person investment
game (de Waegenaere et al., 2005) GT ∈ GTN is given by

N = {1, 2, 3, 4}, ν(N ) = 284,




ν(1) = 88, ν(2) = 66, ν(3) = ν(4) = 55,

ν(1, 2) = 159, ν(1, 3) = ν(1, 4) = 148,


ν(2, 3) = ν(2, 4) = 126, ν(3, 4) = 115,



ν(1, 2, 3) = ν(1, 2, 4) = 214, ν(1, 3, 4) = 203, ν(2, 3, 4) = 181.
We obtain non-convex (ν(2, 4)+ν(3, 4) > ν(4)+ν(2, 3, 4)) balanced semi-symmetric
game with symmetric players 3 and 4, ℑ(GT ) = {{3, 4}}. The core of game GT has
16 extreme points whereas symmetric core is the convex hull of 4 points
SC(GT ) = co{x1 , x2 , x3 , x4 },
1
1
1
1
x1 = (100 , 68 , 57 , 57 ),
2
2
2
2
x3 = (98, 66, 60, 60),
1
1
1
1
x2 = (90 , 78 , 57 , 57 ),
2
2
2
2
x4 = (88, 76, 60, 60).
Symmetric Core of Cooperative Side Payments Game
431
Denote by G0T = (N, ν 0 ), where

 5 if |K| ∈ {2, 3},
ν 0 (K) = 20 if K = N,

0 else,
the zero-normalization of game GT . All players are substitutes in G0T , ℑ(G0T ) =
{{1, 2, 3, 4}}. The symmetric core of game G0T consists of one point
SC(G0T ) = {x0 }, x0 = (5, 5, 5, 5) = Sh(G0T ) = CIS(G0T ) = EN SC(G0T ) = ED(G0T ).
The payoff vector x0 corresponds to symmetric core allocation x6 = (93, 71, 60, 60)
3
4
, it is equal the Shapey value Sh(GT )
of original game GT . Notice, that x6 = x +x
2
of original game, but does not coincide with the barycenter (94 41 , 72 41 , 58 43 , 58 43 ) of
the symmetric core of game GT .
In game theory literature there exist two (equivalent) versions of TU game balancedness: a game GT ∈ GTN is called balanced if it has a nonempty core ore if it
satisfies the Bondareva-Shapley condition
X
X
λK ν(K) 6 ν(N ), λ : Ω → R+ ,
λK = 1, i ∈ N,
(2)
K∈Ω
K∈Ω, K∋i
see (Bondareva, 1963) and (Shapley, 1967). Since (2) is necessary but not sufficient
condition for the nonemptiness of core of discrete game, the unified definition is
required.
Definition 3. A game G ∈ G N with nonempty core is called balanced.
We need the following axiom to be satisfied by solution ϕ.
Axiom 3.1 (equal treatment). For all G ∈ G˜N , all x ∈ ϕ(G) and every symmetric
players i, j in G: xi = xj .
Known that Sh(GT ), CIS(GT ), EN SC(GT ) and ED(GT ) satisfy equal treatment.
From above definitions it straightforwardly follows that:
• the symmetric core of a game G ∈ G N may be empty;
• the symmetric core of TU game GT is a convex subset of its core;
• the symmetric core of non-symmetric game G ∈ G N coincides with its core,
therefore, apart from their different definitions the real difference is exposed for
semi-symmetric balanced games;
• the symmetric core of balanced symmetric TU game consists of one point
which is the equal division solution SC(GT ) = {ED(GT )};
• the symmetric core of balanced semi-symmetric TU game contains all core selectors satisfying equal treatment, in particular, the nucleolus that realizes a fairness
principle based on lexicographic minimization of maximum excess for all coalitions;
• if the Shapley value of semi-symmetric TU game satisfies the core inequalities
then it belongs to symmetric core, the Shapley value is always symmetric core
allocation on the domain of convex TU games;
• the CIS-value, the ENSC-value, the equal division solution which "have some
egalitarian flavour" (Brink and Funaki, 2009) and any convex combination of these
432
Alexandra B. Zinchenko
solutions cannot belong to symmetric core of balanced semi-symmetric TU game.
A nonempty core of NTU game (even 3-person) may contains no equal treatment
outcomes (Aumann, 1987). The following two propositions show that balancedness
of TU game is the necessary and sufficient condition for nonemptiness of its symmetric core, but the same is not true for balanced discrete game.
Proposition 1. Let GT ∈ SG N
T . Then SC(GT ) 6= ∅ iff C(GT ) 6= ∅.
Proof. If SC(GT ) 6= ∅ then C(GT ) 6= ∅ by inclusion SC(GT ) ⊆ C(GT ). Assume
now that C(GT ) 6= ∅ and take x1 ∈ C(GT ). If x1 ∈ SC(GT ) then SC(GT ) 6= ∅.
Otherwise, there exist a coalition K ∈ ℑ(GT ) and players i, j ∈ K such that
x1i < x1j . Construct x2 ∈ RN as follows: x2i = x1j , x2j = x1i , x2l = x1l for l ∈ N \ {i, j}.
1
2
∈ C(GT ). So,
Using (1) we see that x2 ∈ C(GT ). By core convexity, x3 = x +x
2
we get the core allocation x3 satisfying x3i = x3j , x3l = x1l for l ∈ N \ {i, j}. If
x3 ∈
/ SC(GT ) then by repeated application of above procedure one obtains the
payoff vector belonging to SC(GT ).
⊔
⊓
Proposition 2. There exist discrete games GD ∈ SG N
D such that C(GD ) 6= ∅ but
SC(GD ) = ∅.
Proof. Consider discrete games GsD , defined by set function ν s on N : ν s (K) ∈ {0, 1}
for K ⊂ N and ν s (N ) = 1. The associated TU game Ψ (GsD ) = (N, ν s ) is simple.
Assume |veto(Ψ (GsD ))| > 2. Then C(Ψ (GsD )) = co{ei ∈ ZN |i ∈ veto(Ψ (GsD ))} and
C(GsD ) = {ei ∈ ZN |i ∈ veto(Ψ (GsD )), where eij = 0 for i 6= j, eii = 1. Obviously, veto
players are substitutes in games Ψ (GsD ) and GsD . However xi 6= xj for all x ∈ C(GsD )
and every (i, j) ∈ veto(GsD ). Thus SC(GsD ) = ∅.
⊔
⊓
The core of TU game has been intensely studied and axiomatized. We shall formulate some convenient properties of a solution concept ϕ on G˜N ⊆ G N which has
been employed in the well-known core axiomatizations. The axiomatic characterizations of discrete game solutions are not yet provided.
Axiom 3.2 (efficiency). x(N ) = ν(N ) for all x ∈ ϕ(G) and all G ∈ G˜N .
Axiom 3.3 (symmetry). For all G ∈ G˜N and every symmetric players i, j in G: if
x ∈ ϕ(G) then there exists y ∈ ϕ(G) such that xi = yj , xj = yi and xp = yp for
p ∈ N \ {i, j}.
Axiom 3.4 (modularity). For any modular game G ∈ G˜N generated by the vector
x ∈ RN : ϕ(G) = {x} .
Axiom 3.5 (antimonotonicity). For any pair of games G1 , G2 ∈ G˜N defined by set
functions ν 1 , ν 2 on N such that ν 1 (N ) = ν 2 (N ) and ν 1 (K) 6 ν 2 (K) for all K ⊂ N ,
it holds that ϕ(G2 ) ⊆ ϕ(G1 ).
Axiom 3.6 (reasonableness (from above)). For all G ∈ G˜N , all x ∈ ϕ(G) and every
i ∈ N : xi 6 max {ν(K ∪ i) − ν(K)}.
K⊆N \i
Axiom 3.7 (covariance). For any pair of games G1 , G2 ∈ G˜N defined by set functions ν 1 , ν 2 such that ν 2 = αν 1 + β for some α >0 and some β ∈ RN it holds that
ϕ(G2 ) = αϕ(G1 ) + β.
Symmetric Core of Cooperative Side Payments Game
433
Axiom 3.8 (projection consistency (or reduced game property)). Let G ∈ G˜N , ∅ 6=
H ⊂ N and x ∈ ϕ(G), then RxH = (H, rxH ) ∈ G˜H and xH ∈ ϕ(RxH ), where xH =
(xi )i∈H ∈ RH and

if K = ∅,
0
rxH (K) = ν(K)
if ∅ 6= K ⊂ H,

ν(N ) − x(N \ H) if K = H,
is the projected reduced game with respect to H and x.
Known (Llerena and Carles, 2005) that the core is the only solution on GTN satisfying projection consistency, reasonableness (from above), antimonotonicity and
modularity. Notice that projection consistency is one of the fundamental principle
used in this field. By summarizing the statements formulated above we can say
that the symmetric core of balanced semi-symmetric TU and discrete games satisfies equal treatment, efficiency, symmetry, modularity, reasonableness (from above)
and many other core axioms based on only the original game. Theorem 1 (below)
shows that for the class of balanced semi-symmetric games the symmetric core is
in conflict with antimonotonicity, covariance and projection consistency. All these
properties involve the pairs of games.
Lemma 1. Let G ∈ SG N is a balanced game and G0 is its zero-normalization.
Then G0 ∈ SG N , SC(G0 ) ⊆ SC(G) and there exist games G ∈ SG N such that
SC(G0 ) 6= SC(G).
Proof. The zero-normalization G0 of any game G ∈ G N is uniquely determined by
set function ν 0 on N , where
X
ν 0 (K) = ν(K) −
ν(l), ∅ 6= K ⊆ N.
(3)
l∈K
Obviously, G0 ∈ SG N . Let i, j ∈ N , i 6= j, are symmetric players in G. The formulas
(1) and (3) imply that ν 0 (K ∪i) = ν 0 (K ∪j) for all K ⊆ N \ {i, j}. Thus, symmetric
players in G remain symmetric in G0 . Example 2 shows that non-symmetric in G
players can become symmetric in G0 . If G = GT then a linear system defining
SC(G0T ) contains the one for SC(GT ) and, perhaps, additional equality constraints.
So SC(G0T ) ⊆ SC(GT ). In view of Example 2 this inclusion can be strict. For
discrete game G = GD the final part of lemma is proved analogously.
⊔
⊓
Theorem 1. Let G ∈ SG N is a balanced game. Then SC(G) does not satisfy
(i) Axiom 3.5;
(ii) Axiom 3.7 even for α = 1 and β = (ν(1), ..., ν(n));
(iii) Axiom 3.8.
Proof. (i) Consider two balanced four-person TU games G1T , G2T defined by set
functions ν 1 , ν 2 such that

2 |K| = 2,


1

4 |K| = 3,
ν (K) + 1 = 5, K = {1, 3, 4},
1
ν (K) =
ν 2 (K) =
6
K
=
N,
ν 1 (K),
else.



0 else,
434
Alexandra B. Zinchenko
The games G1T and G2T are symmetric and semi-symmetric, respectively. ℑ(G1T ) =
{{1, 2, 3, 4}}, ℑ(G2T ) = {{3, 4}}, ν 1 (N ) = ν 2 (N ) and ν 1 (K) 6 ν 2 (K) for all K ⊂ N .
It holds that
1 1
1 1 1 1
SC(G2T ) = co{(2, 1, 1 , 1 ), (2, 0, 2, 2), (1, 1, 2, 2)} 6⊂ SC(G1T ) = {(1 , 1 , 1 , 1 )}.
2 2
2 2 2 2
Consider now discrete games G1D , G2D corresponding to given TU games. We have
SC(G2D ) = {(2, 0, 2, 2), (1, 1, 2, 2)} 6⊂ SC(G1T ) = ∅.
Thus, antimonotonicity is violated by SC(G).
(ii) This statement follows from lemma 1.
(iii) In four-person TU game GT defined by

N = {1, 2, 3, 4}, ν(N ) = 8, ν(i) = 0, i ∈ N,

ν(1, 2) = ν(1, 3) = ν(1, 4) = ν(2, 3) = ν(2, 4) = 2, ν(3, 4) = 3,

ν(1, 2, 3) = ν(1, 2, 4) = 6, ν(1, 3, 4) = 5, ν(2, 3, 4) = 4
players 3 and 4 are symmetric, ℑ(GT ) = {{3, 4}}. The symmetric core is the
convex hull of four points SC(GT ) = co{x1 , x2 , x3 , x4 }, where x1 = (4, 0, 2, 2),
x2 = (4, 1, 1 21 , 1 21 ), x3 = (1, 3, 2, 2) and x4 = (2, 3, 1 21 , 1 12 ). The projected reduced
game RxH2 = (H, rxH2 ) relative to H = {1, 2, 3} at x2 is defined by:
rxH2 (1, 2, 3) = 6 12 , rxH2 (i) = 0, i ∈ H, rxH2 (1, 2) = rxH2 (1, 3) = rxH2 (2, 3) = 2.
The reduced game is symmetric. Its symmetric core consists of one point (2 61 , 2 16 , 2 61 ).
The restriction of x2 to H, x2H = (4, 1, 1 21 ), does not belong to the symmetric core
of reduced game. For discrete game GD corresponding to last TU game GT we have
SC(GD ) = {x1 , x3 , x5 , x6 }, where x5 = (3, 1, 2, 2), x6 = (2, 2, 2, 2). The projected
reduced game RxH1 relative to H = {1, 2, 3} at x1 is defined by:
rxH1 (1, 2, 3) = 6, rxH1 (i) = 0, i ∈ H, rxH1 (1, 2) = rxH1 (1, 3) = rxH1 (2, 3) = 2.
Since the reduced game is symmetric SC(RxH1 ) = {(2, 2, 2)}. The restriction of x1
to H, x1H = (4, 0, 2), does not belong to SC(RxH1 ). So SC(G) does not provide
projection consistency.
⊔
⊓
It has been interesting to study the interrelation between the symmetric core of
a game G ∈ SG N and strongly egalitarian core allocations.
Definition 4. Let G ∈ G N , x ∈ C(G) and x ∈ RN is obtained from x by permuting
its coordinates in a non-decreasing order: x1 6 x2 6 ... 6 xn . A core allocation x
is Lorenz allocation (Lorenz maximal, strongly egalitarian ) iff itP
is undominated
in
P
the sense of Lorenz, i.e. there does not exist y ∈ C(G) such that pi=1 y i > pi=1 xi
for all p ∈ {1, ..., n − 1} with at least one strict inequality.
For a game G ∈ G N denote by LA(G) the set of its Lorenz allocations.
Example 3. Consider balanced

 7
ν(K) = 12

0
four-player TU game GT defined by
if (K = {1, 2}) ∨ (K = {1, 3}),
if K = N,
else.
In was proved (Arin et al., 2008) that the set of Lorenz allocations is of the form
LA(GT ) = {x ∈ C(GT )| x = (7 − µ, µ, µ, 5 − µ), 2
1
1)
6 µ 6 3 }.
2
2
435
Symmetric Core of Cooperative Side Payments Game
Taking µ = 3 12 , µ = 2 12 and µ = 3 yield the lexmax solution Lmax(GT ) =
(3 12 , 3 21 , 3 21 , 1 21 ), the lexmin solution Lmin(GT ) = (4 21 , 2 21 , 2 21 , 2 21 ) and least squares
solution LS(GT ) = (4, 3, 3, 2), respectively ((Arin et al., 2008, p.571)). By the formulas in section 2 one obtains Sh(GT ) = (4 16 , 3, 3, 1 65 ) 6∈ LA(GT ), CIS(GT ) =
EN SC(GT ) = ED(GT ) = (3, 3, 3, 3) 6∈ LA(GT ).
The next theorem states that the symmetric core of balanced semi-symmetric TU
game contains all Lorenz allocations. Besides, SC(GT ) is externally stabile with
respect to Lorenz domination, but internal stability does not hold.
Theorem 2. Let GT ∈ SG N
T is a balanced game. Then
(i) LA(GT ) ⊆ SC(GT ) and the inclusion can be strict;
(ii) SC(GT ) Lorenz dominates every other core allocation.
Proof. (i) LA(GT ) satisfies equal treatment and LA(GT ) ⊆ C(GT ). Therefore,
LA(GT ) ⊆ SC(GT ). The four-person TU game in Example 3 is semi-symmetric
ℑ(GT ) = {{2, 3}},
1 1 1 1
1 1 1 1
LA(GT ) = co{(3 , 3 , 3 , 1 ), (4 , 2 , 2 , 2 )}
2 2 2 2
2 2 2 2
⊂ SC(GT ) = co{(2, 5, 5, 0), (7, 0, 0, 5), (12, 0, 0, 0)}.
(ii) If C(GT ) = SC(GT ) then the statement is straightforward. Let C(GT ) 6=
SC(GT ) and take x0 ∈ C(GT )\SC(GT ). Then there exists K ∈ ℑ(GT ) and i, j ∈ K
such that x0j > x0i . By symmetry there is y ∈ C(GT ) with yi = x0j , yj = x0i , yl = x0l
0
for l ∈ N \ {i, j}. Consider x1 = x 2+y . By core convexity x1 ∈ C(GT ). Vector x1
Lorenz dominates x0 (x1 ≻L x0 ) since x1j = x1i = x0j − δ = x0i + δ, x1l = x0l for
l ∈ N \{i, j}, δ > 0. Repetition of this procedure gets the sequence x0 , x1 , ..., xp core
allocations, where xk ≻L xk−1 for all k ∈ {1, ..., p}, x0 6∈ SC(GT ), xp ∈ SC(GT ).
The transitive property of Lorenz domination completes the proof.
⊔
⊓
4.
Existence conditions
The balancedness condition (2) is derived by means of dual linear programming
problems associated with a game GT ∈ GTN
X
X
f (x) =
xi → min,
xi > ν(K), K ∈ Ω,
(4)
i∈N
g(λ) =
X
K∈Ω
i∈K
ν(K)λK → max,
X
K∈Ω,i∈K
n
λK = 1, i ∈ N, λ ∈ R2+ −2 .
(5)
The condition (2) can be as well written as
X
λK ν(K) 6 ν(N ), λ ∈ ext(M n ),
K∈Ω
where ext(M ) is the set of extreme points of problem (5) constraint set M n . The
number of extreme points and their explicit representation known only for small n
n
|ext(M 3 )| = 5, |ext(M 4 )| = 41, |ext(M 5 )| = 1291, |ext(M 6 )| = 200213.
We concentrate now on n-person non-negative semi-symmetric TU games in zero0
normal form (SG N
T )+ . The following example illustrates how the problem (4) is
modified by replacing the core by symmetric core.
436
Alexandra B. Zinchenko
0
Example 4. Consider two four-person games (G0T )1 , (G0T )2 ∈ (SG N
T )+ with two and
0 1
0 2
three symmetric players, ℑ((GT ) ) = {{3, 4}}, ℑ((GT ) ) = {{2, 3, 4}}. The explicit
representations of (4) and modified problems given in table 1. It is remarkable that
the number of extreme points of modified dual problems constraint sets Ms4 , where
s is the number of symmetric players, decreases as s increases: |ext(M24 )| = 21,
|ext(M34 )| = 6.
Table 1.
Original problem
Modified problem 1,
Modified problem 2,
ℑ((G0T )1 ) = {{3, 4}}
ℑ((G0T )2 ) = {{2, 3, 4}}
f (x) = x1 + x2 + x3 + x4 → min f (x) = x1 + x2 + 2x3 → min f (x) = x1 + 3x2 → min
xi > 0, i ∈ {1, 2, 3, 4}
xi > 0, i ∈ {1, 2, 3}
xi > 0, i ∈ {1, 2}
x1 + x2
> ν(1, 2)
x1 + x2
> ν(1, 2)
x1 + x2 > ν(1, 2)
x1
+ x3
> ν(1, 3)
x1
+ x3 > ν(1, 3)
x1
+ x4 > ν(1, 4)
x2 + x3
> ν(2, 3)
x2 + x3 > ν(2, 3)
2x2 > ν(2, 3)
x2
+ x4 > ν(2, 4)
x3 + x4 > ν(3, 4)
2x3 > ν(3, 4)
x1 + x2 + x3
> ν(1, 2, 3)
x1 + x2 + x3 > ν(1, 2, 3)
x1 + 2x2 > ν(1, 2, 3)
x1 + x2
+ x4 > ν(1, 2, 4)
x1
+ x3 + x4 > ν(1, 3, 4)
x1
+ 2x3 > ν(1, 3, 4)
x2 + x3 + x4 > ν(2, 3, 4)
x2 + 2x3 > ν(2, 3, 4)
3x2 > ν(2, 3, 4)
The symmetry of all players makes a game especially easy to handle. The criterion for existence of its core (and, by Proposition 1, for symmetric core too) contains
(n − 1) inequalities only
ν(K)
ν(N )
6
f or all K ∈ Ω.
|K|
n
It is then natural to focus the attention on games with (n − 1) symmetric players.
Notice that any such game is determined by 2(n − 2) numbers ν(K), K ∈ Ω1 ∪ Ω2 ,
where
Ω1 = {{2, 3}, {2, 3, 4}, ..., {2, ..., n}}, Ω2 = {{1, 2}, {1, 2, 3}, ..., {1, ..., n − 1}}.
A few of their applications:
• market with one seller and symmetric buyers;
• games with a landlord and landless workers;
• weighted majority game with one large party and (n − 1) equal sized smaller
parties;
• patent licensing game with the firms each producing an identical commodity
and a licensor of a patented technology (Watanabe and Muto, 2008);
• subclass of games related information collecting situations under uncertainty
(Branzei et al., 2000) where an action taker can obtain more information from other
agents;
• big boss games (Muto et al.,1988) with symmetric powerless players.
The characterization of such games and the sufficient conditions under which
the symmetric core is a singleton have been provided in (Zinchenko, 2012). Let
Symmetric Core of Cooperative Side Payments Game
437
0
0
0
G0T ∈ (SG N
T )+ , ℑ(GT ) = {{2, ..., n}} and n > 3. The symmetric core of game GT is
nonempty iff the system
ν 0 (T ) +
n − |T | 0
n−1 0
ν (H) 6 ν 0 (N ),
ν (H) 6 ν 0 (N ), H ∈ Ω1 , T ∈ Ω2
|H|
|H|
is consistent. Notice that system consists of (n − 1)(n − 2) inequalities. If G0T ∈
0
0
0
(SG N
T )+ is a balanced game, ℑ(GT ) = {{2, ..., n}}, n > 4 and ν satisfies at least
one of three equalities
n−1 0
n−2 0
ν (N \ {1, n}) = ν 0 (N ),
ν (N \ 1) + ν 0 (1, 2) = ν 0 (N ),
n−2
n−1
ν 0 (N \ 1)
+ ν 0 (N \ n) = ν 0 (N )
n−1
then SC(G0T ) consists of a unique allocation.
References
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Azamkhuzhaev, M. Kh. (1991). Nonemptiness conditions for cores of discrete cooperative
game. Computational Mathematics and Modeling, 2(4), 406–411.
Bondareva, O. N. (1963). Certain applications of the methods of linear programing to the
theory of cooperative games. Problemy Kibernetiki, 10, 119–139 (in Russian).
Branzei, R., S. Tijs and J. Timmer (2000). Collecting information to improve decision
making. International Game Theory Review, 3, 1–12.
van den Brink, R. and Y. Funaki (2009). Axiomatizations of a class of equal surplus sharing
solutions for cooperative games with transferable utility. Theory and Decision, 67, 303–
340.
Dutta, B. and D. Ray (1989). A concept of egalitarianism under participation constraints.
Econometrica, 57, 403–422.
Hougaard, J. L., B. Peleg and L. Thorlund-Petersen (2001). On the set of Lorenz-maximal
imputations in the core of a balanced game. International Journal of Game Theory, 30,
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Muto, S., M. Nakayama, J. Potters and S. Tijs (1988). On big boss games. The Economic
Studies Quarterly, 39, 303–321.
Norde, H., V. Fragnelli, I. Garcia-Jurado, F. Patrone and S. Tijs (2002). Balancedness of
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453–460.
de Waegenaere, A., J. Suijs and S. Tijs (2005). Stable profit sharing in cooperative investment. OR Spectrum, 27(1), 85–93.
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bargaining outcomes. International Journal of Game Theory, 37(4), 505–523.
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region. Natural science, 5, 10–14 (in Russian).
CONTRIBUTIONS TO GAME THEORY AND MANAGEMENT
Collected papers
Volume VII
presented on the Seventh International Conference Game Theory and
Management
Editors Leon A. Petrosyan, Nikolay A. Zenkevich.
УСПЕХИ ТЕОРИИ ИГР И МЕНЕДЖМЕНТА
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