Applied Mathematical Sciences, Vol. 9, 2015, no. 27, 1331 - 1340
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2015.519
The SM -Nucleolus in Glove Market Games
Svetlana Tarashnina
Saint-Petersburg State University
Universitetsky pr. 35, St.Petersburg
198504, Russia
Tatiana Sharlai
Saint-Petersburg State University
Universitetsky pr. 35, St.Petersburg
198504, Russia
c 2015 Svetlana Tarashnina and Tatiana Sharlai. This is an open access
Copyright article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Abstract
In the present paper we consider a special class of TU-games – glove
market games. Using both analytical and algorithmic approaches we
apply the SM -nucleolus – a comparably new solution concept that is
based on the idea of the blocking power of a coalition – to this kind
of games. We provide analytical formulae for calculation of the SM nucleolus. On the top of this, Potters’ algorithm for the prenucleolus
has been adapted for finding the SM -nucleolus as well. Some examples
are given and explained in the paper. Finally, we discuss similarity
and differences that the SM -nucleolus and different solution concepts
of TU-games demonstrate for a class of glove market games.
Mathematics Subject Classification: 91A12
Keywords: glove market games, solution concept, SM -nucleolus, prenucleolus
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1
Svetlana Tarashnina and Tatiana Sharlai
Introduction
In this paper we investigate a special class of cooperative TU-games — glove
market games. It is well known that there is a number of solution concepts
for these games and all of them have some pros and cons. We consider a
comparably new solution concept — the SM -nucleolus that was introduced in
[15]. The SM -nucleolus is based on the idea of using the blocking power of a
coalition as well as the constructive one and has pretty convenient properties.
Another solution concept possessing similar properties is the [0, 1]-nucleolus
introduced in [14] as a generalization of the considered solution.
Using both analytical and algorithmic approaches we apply the SM -nucleolus to glove market games and analyze its properties in comparison to the
well known and widely used solution concepts such as the prenucleolus [11]
and the Shapley value [12]. The analytical formulae that give a simple way for
finding the SM -nucleolus is justified in the paper. Moreover, we provide a way
for calculation of the pointed solution by using a programming realization of
Potters’ algorithm. Some significant examples are given and discussed in the
paper. The examples illustrate how the total payoff can be divided between
the players according to the SM -nucleolus as well as the core, the prenucleolus
and the Shapley value. Finally, we show that for such kind of games the SM nucleolus is a more reasonable solution than the core or the prenucleolus. All
the same, some similarities of the SM -nucleolus and the Shapley value are
noticed. In fact, they have cognate structures, and fully coincide for all three
person TU-games.
2
Preliminary Notes and Definitions
In this paper we deal with special glove market games which are a subclass
of cooperative games with transferable utility (TU-games). First of all, let us
give some basic definitions and notations from the cooperative game theory.
A cooperative TU-game is a pair (N, v), where N = {1, 2, . . . , n} is the set
of players and v : 2N 7→ R is a characteristic function with v(∅) = 0. Here
2N = {S ⊆ N } is the set of coalitions of (N, v).
Due to the classical cooperative approach we look for ways to distribute
the amount v(N ) over the members of the grand coalition.
Denote by X ∗ (N, v) the set of feasible payoff vectors of a game (N, v), i.e.
X ∗ (N, v) = {x ∈ RN : x(N ) ≤ v(N )}.
Here by x(S) we mean
P
xi (S ⊆ N ). Let GN be the set of TU-games.
i∈S
Definition 2.1 A solution on GN is a mapping f : GN 7→ RN obeying
f (N, v) ⊆ X ∗ (N, v).
The SM -nucleolus in glove market games
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Definition 2.2 An imputation in a game (N, v) is a payoff vector x =
(x1 , x2 , . . . , xn ) satisfying
x(N ) = v(N ) (group rationality),
(1)
xi ≥ v({i}) for all i ∈ N (individual rationality).
(2)
Definition 2.3 A preimputation in (N, v) is a group rational payoff vector,
i.e. condition (2) is not required.
We denote by X(v) and X 0 (v) the imputation and preimputation sets of
the game (N, v), respectively. In this paper the preimputation set X 0 (v) is
under consideration.
The SM -nucleolus is based on the prenucleolus approach, therefore, some
precise definitions are needed.
Let x be a preimputation in a game (N, v). The excess e(x, v, S) of a
coalition S at x is
e(x, v, S) = v(S) − x(S).
Let X be an arbitrary nonempty closed set in RN . For each x ∈ RN define
a mapping θ such that θ(x) = y ∈ RN , where y is a vector that arises from x
by arranging its components in a non-increasing order.
Definition 2.4 1 The nucleolus of X denoted by N (X), or N (N, v, X), is
the set of vectors in X whose θ(e(x, v, S)S⊆N )’s are lexicographically least, i.e.
N (X) = {x ∈ X : θ(e(x, v, S)S⊆N ) lex θ(e(y, v, S)S⊆N ) for all y ∈ X}.
If X = X(v), it is called the nucleolus of the game (N ).
If X = X 0 (v), it is called the prenucleolus of the game (PN ).
It follows from [11] that the nucleolus is never empty and consists of a
unique point as well as the prenucleolus of the game.
In continuation of the subject we refer to the definition of the SM -nucleolus.
Let (N, v) be a TU-game, then the dual game (N, v ∗ ) of (N, v) is defined
by
v ∗ (S) = v(N ) − v(N \S)
(3)
for all coalitions S.
The sum-excess of a coalition S ⊆ N at each x ∈ RN is defined as follows
e(x, v, S) = e(x, v, S) + e(x, v ∗ , S).
(4)
Formula (4) contains two summands — the excess e(x, v, S) w.r.t. characteristic function v and the excess e(x, v ∗ , S) w.r.t. characteristic function v ∗ .
1
The definition has been taken from [6].
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Svetlana Tarashnina and Tatiana Sharlai
The constructive power of S is the worth of the coalition, or exactly what
S can reach by cooperation. Under the blocking power of coalition S we
understand the amount v ∗ (S) which this coalition brings to N if the last is
formed. In words, the blocking power of a coalition can be judged as a measure
of necessity of this coalition for N — how much S contributes to N .
Definition 2.5 The simplified modified nucleolus, or shortly — the SM nucleolus, of a game (N, v) is the set
µ(v) = {x ∈ X 0 (v) : θ(e(x, v, S)S⊆N ) lex θ(e(y, v, S)S⊆N ) for all y ∈ X 0 (v)},
where θ(e(x, v, S)S⊆N ) is a vector whose components are the sum-excesses arranged in a non-increasing order.
It follows from Definition 2.5 and formula (4) that both the constructive and
the blocking powers are taken into account equally.
The following theorem is useful [15].
Theorem 2.6 Let (N, v) be a TU-game. The SM -nucleolus of (N, v) coincides with the prenucleolus of the constant-sum game (N, w), where
1
1
1
w(S) = v(S) + v ∗ (S) =
v(S) + v(N ) − v(N \S) ,
2
2
2
S ⊂ N.
Theorem 2.6 states a direct relationship between the SM -nucleolus of
a game (N, v) and the prenucleolus of a corresponding constant-sum game
(N, w). This tie confirms that the SM -nucleolus is a singleton, and gives a
number of ways for its finding.
3
3.1
Results and Discussion
Glove Market Games
L. Shapley and M. Shubik in [13] proposed to consider a glove market game as
a simple market model. This game describes the following situation: there are
two types of complementary products on the market and a finite set of firms
each of which can produce a product of one type. A customer needs both types
of products.
Let N consists of two types of players: N = P ∪ Q where P and Q are
disjunct sets of players. Each player of P owns a right-hand glove and each
player of Q owns a left-hand glove. If j members of P and k members of
Q form a coalition, they have min{j, k} complete pairs of gloves, each being
worth 1. Unmatched gloves are worth nothing.
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The SM -nucleolus in glove market games
The characteristic function for the game is defined by
v(S) = min{|P ∪ S|, |Q ∪ S|},
S⊂N.
(5)
Thus, the worth of coalition S is equal to the number of pairs of gloves the
coalition can assemble. Without loss of generality, assume |P | ≥ |Q|.
This model is quite popular in the cooperative game theory and its core as
well as other solution concepts have been already studied (see [1], [2], [3], [8]).
The core represents a payoff vector where the holders of the scarce commodity (the left-glove owners in our case) obtain a payoff of 1 and the other
players obtain nothing. This result holds for |P | = 100 and |Q| = 99 as well
as for |P | = 100 and |Q| = 1. Shapley and Shubik in [13] noticed violent
insensitivity of the core to the ratio of the dimensions of sets P and Q. In
contrast, the Shapley value is sensitive to the relative scarcity of the gloves
what is an attractive property of the Shapley value. The SM -nucleolus also
possesses this desirable sensitivity.
In this paper, we apply the SM -nucleolus to the glove market games. As a
main result, we present analytical formulae for finding the solution for arbitrary
finite sets P and Q.
3.2
Analytical Formulae for Finding the SM -Nucleolus
Let |P | = p, |Q| = q, P = {i1 , . . . , ip }, Q = {j1 , . . . , jq }. Taking into account
anonymity of the SM -nucleolus (see [15]) denote a player from P by ik (ik ∈ P )
and a player from Q by jl (jl ∈ Q). The SM -nucleolus is a payoff vector
µ(v) ∈ Rp+q with components µik for players from P and µjl for players from
Q. We justify analytical formulae for calculation of the SM -nucleolus for
arbitrary dimensions of sets P and Q.
Theorem 3.1 Let (N, v) be a glove market game with characteristic function (5), p ≥ q. Then the SM -nucleolus µ(v) is defined by the formulae
µi k =
q
,
4p − 2q
µjl =
3p − 2q
.
4p − 2q
(6)
where ik ∈ P and jl ∈ Q.
We avoid here the proof of the theorem since it is quite ponderous. The
proof is based on Theorem 2.6 and Kohlberg theorem (see [4], [7]). The presented formulae have been obtained constructively. We select possible sets
of coalitions with the minimal sum-excesses and check their for balance by
Kohlberg theorem. Going step by step (in fact, we consider consequently
cases |P | = p, |Q| = 1; |P | = p, |Q| = 2 and |P | = p, |Q| = q) we have found
out that coalitions T ∪ Q and P where T ⊂ P , |T | = |Q|, have the minimal
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Svetlana Tarashnina and Tatiana Sharlai
sum-excesses and form a balanced collection, so, the following equalities give
the necessary formulae for the SM -nucleolus.
e(x, v, T ∪ Q) = e(x, v, P ),
T ⊂ P,
|T | = |Q|.
We present here some significant examples to give the intuition of the considered solution concept. The examples demonstrate some similarities and
differences of the SM -nucleolus comparing with the well-known solution concepts of TU-games such as the core, the Shapley value, the prenucleolus.
Example 3.2 Suppose that P = {1, 2}, Q = {3}. The resulting payoff
vectors for this game are presented in the following table:
The core
C(v) = (0, 0, 1)
The prenucleolus
ν(v) = (0, 0, 1)
The Shapley value
ϕ(v) = ( 61 , 61 , 23 )
The SM -nucleolus
µ(v) = ( 61 , 16 , 32 )
The nucleolus as well as the core assigns one to player 3 and zero to the other
players. On the other hand, players 1 and 2 together can prevent player 3
from getting 1 by forming a coalition against him. Therefore, together they
have the same blocking power as player 3 does. The SM -nucleolus takes into
1
2
account the blocking power of coalition {1, 2}. It assigns to player 3 and
3
6
to players 1 and 2 each. Note that for this game the SM -nucleolus coincides
with the Shapley value.
At the same time, we have come across a rather interesting result which
indicates that the SM -nucleolus coincides with the Shapley value for an arbitrary three-person TU-game. That was proved in [15]. This gives some
insight that we have a solution concept with similar to the Shapley value
properties.
Example 3.3 Suppose that P = {1, 2, 3}, Q = {4}. The resulting payoff
vectors for this game are presented in the following table:
The core
C(v) = (0, 0, 0, 1)
The prenucleolus
ν(v) = (0, 0, 0, 1)
The Shapley value
1 1 1 3
, 12 , 12 , 4 )
ϕ(v) = ( 12
The SM-nucleolus
1 1 1 7
µ(v) = ( 10
, 10 , 10 , 10 )
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The SM -nucleolus in glove market games
The nucleolus does not assigns any positive payoff to the owners of a righthand glove, the whole amount of the total payoff goes to the holder of the
unique left-hand glove. The Shapley value assigns 25 percents of the total
payoff to players 1, 2, and 3 altogether, whereas the SM -nucleolus distributes
30 percents of the total payoff between the holders of a right-hand glove.
Example 3.4 Suppose that P = {1, 2, 3}, Q = {4, 5}. The resulting payoff
vectors for this game are presented in the table below.
The core
C(v) = (0, 0, 0, 1, 1)
The prenucleolus
ν(v) = (0, 0, 0, 1, 1)
The Shapley value
14 14 14 39 39
ϕ(v) = ( 60
, 60 , 60 , 60 , 60 )
The SM-nucleolus
µ(v) = ( 41 , 14 , 41 , 58 , 58 )
In this example the Shapley value assigns 35 percents of the total payoff
to players 1, 2, and 3 altogether, whereas the SM -nucleolus distributes 37,5
percents of the total amount between the holders of a right-hand glove.
As a result, we can notice that the SM -nucleolus distributes a bigger part
of v(N ) between the players with a non-scarce glove than the Shapley value.
Finally, let us give an example when p = q.
Example 3.5 Suppose that P = {1, 2, 3}, Q = {4, 5, 6}. The resulting
payoff vectors for this game are presented below.
The core
C(v) = ( 12 , 21 , 12 , 12 , 12 , 12 )
The prenucleolus
ν(v) = ( 21 , 12 , 21 , 12 , 12 , 12 )
The Shapley value
ϕ(v) = ( 12 , 12 , 21 , 12 , 12 , 12 )
The SM-nucleolus
µ(v) = ( 12 , 21 , 12 , 12 , 12 , 21 )
As a matter of fact, all players in this game are treated equally and all
considered solution concepts (the core, the prenucleolus, the Shapley value
and the SM -nucleolus) propose the same payoff vector as a solution of the
game.
These examples illustrate also what happens with the right- and left-hand
glove owners’ payoffs according to the SM -nucleolus when changing the dimensions of sets P and Q.
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Svetlana Tarashnina and Tatiana Sharlai
Conclusion
Following the theoretical results we can conclude that dividing the total payoff
according to the core (the nucleolus) is not acceptable for the most of players in
these games. It is worth to notice that both groups of players need each other.
Obviously, left-hand glove holders cannot obtain any positive payoff without
right-hand glove holders. In this sense, the SM -nucleolus and the Shapley
value propose to divide the total payoff between the players more ”fairly”.
At the same time, it is difficult to make a certain conclusion about differences between the SM -nucleolus and the Shapley value. We can only note
that the analytical formulae for finding the SM -nucleolus is considerably easier
than the ones for the Shapley value.
The obtained formulae for calculation of the SM -nucleolus are additionally
verified by means of algorithmic approach.
Computational complexity of the SM -nucleolus is comparable with that
of the prenucleolus due to Theorem 2.6. Therefore, the SM -nucleolus can
be computed by each of the presently known algorithms for the prenucleolus
(for example, [5], [9], [10]). In fact, any prenucleolus algorithm for TU-games
can be adapted to find the SM -nucleolus. We apply Potters’ algorithm for
this purpose. The computer program that realizes the algorithm has been
developed by Olga Dmitrenko2 . The SM -nucleolus has been calculated for
different glove market games with a help of the computer program. The results
are identical to the ones obtained by formulae (6) in the examples above.
In continuation of this work we shall consider a case when one of the players
(a right-hand or a left-hand glove owner) is additionally given one more glove.
The solution of the changed game might be a topic for our next paper.
Acknowledgements. I would like to thank Pr. Leon Petrosyan, Dr. Elena
Parilina and Dr. Nadezhda Smirnova for their helpful advises and support in
my scientific work.
References
[1] R. J. Aumann, L. S. Shapley, Values of non-atomic games, Princeton
University Press, 1974.
[2] L. J. Billera, J. Raanan, Cores of non-atomic linear production
games, Mathematics of Operations Research, 6 (1981), 420 - 423.
http://dx.doi.org/10.1287/moor.6.3.420
2
In a frame of her diploma thesis in Saint-Petersburg State University at faculty of
Applied Mathematics and Control Processes
The SM -nucleolus in glove market games
1339
[3] E. Einy, R. Holzman, D. Monderer, B. Shitovitz, Core and Stable Sets of
Large Games Arising in Economics, J. Economic Theory, 68 (1996), 200
- 211. http://dx.doi.org/10.1006/jeth.1996.0010
[4] E. Kohlberg, The nucleolus as solution of a minimization problem, SIAM
J. Appl. Math, 23(1) (1972), 34 - 39. http://dx.doi.org/10.1137/0123004
[5] A. Kopelowitz, Computation of the kernel of simple games and the nucleolus of n-person games, Program in Game Th and Math Economics, Dept
of Math, The Hebrew University of Jerusalem RM, 31 (1967).
[6] M. Maschler, The bargaining set, kernel, and nucleolus: a survey, In:
Aumann, R.J., Hart, S. (eds.) Handbook of Game Theory 1, Elsevier
Science Publishers BV, (1992), 591 - 665.
[7] M. Maschler, B. Peleg, and L. S. Shapley, Geometric properties of the kernel, nucleolus and related solution concepts, Mathematics of Operations
Research, 4 (1979), 303 - 338. http://dx.doi.org/10.1287/moor.4.4.303
[8] G. Owen, On the core of linear production games, Mathematical Programming, 9 (1975), 358 - 370. http://dx.doi.org/10.1007/bf01681356
[9] J. Potters, J. Reijnierse, and M. Ansing, Computing the nucleolus by solving a prolonged simplex algorithm, Mathematics of Operations Research,
21(3) (1996), 757 - 768. http://dx.doi.org/10.1287/moor.21.3.757
[10] J. K. Sankaran, On Finding the Nucleolus of an N-Person Cooperative Game, Int. J. Game Theory, 19 (1991), 329 - 338.
http://dx.doi.org/10.1007/bf01766424
[11] D. Schmeidler, The nucleolus of a characteristic function game, SIAM J.
Appl. Math, 17 (1969), 1163 - 1170. http://dx.doi.org/10.1137/0117107
[12] L. S. Shapley, A value for n-person games, In: Kuhn and Tucker (eds.)
Contributions to the Theory of Games II, Princeton University Press,
(1953), 307 - 317.
[13] L. Shapley, M. Shubik, Pure competition, coalitional power, and fair
division, International Economic Review, 10(3) (1969), 337 - 362.
http://dx.doi.org/10.2307/2525647
[14] N. Smirnova, S. Tarashnina, On a generalization of the prenucleolus in cooperative games, Discrete Analisys and Operation Research, 18(4) (2011),
77 - 93 (In Russian).
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[15] S. Tarashnina, The simplified modified nucleolus of a cooperative TUgame, TOP, 19(1) (2011), 150 - 166. http://dx.doi.org/10.1007/s11750009-0118-z
Received: January 16, 2015; Published: February 23, 2015