Nash Equilibrium Computation in Various
Games
Submitted in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
by
Ruta Mehta
(Roll No. 07405005)
Under the guidance of
Prof. Milind Sohoni
and
Prof. Bharat Adsul
DEPARTMENT OF COMPUTER SCIENCE & ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY BOMBAY
2012
In fond memory of my uncle
Mr. Bipin K. Joshi.
Declaration
I declare that this written submission represents my ideas in my own words and where
others’ ideas or words have been included, I have adequately cited and referenced the original sources. I also declare that I have adhered to all principles of academic honesty and
integrity and have not misrepresented or fabricated or falsified any idea/data/fact/source
in submission. I understand that any violation of the above will be cause for disciplinary
action by the Institute and can also evoke penal action from the sources which have thus
not been properly cited or from whom proper permission has not been taken when needed.
(Signature)
(Name of the student)
(Roll No.)
Date:
Abstract
Non-cooperative Game Theory provides a framework to analyze the strategic interactions
of selfish agents in any given organization. In a pioneering work, John Nash (1951)
[49] proved that in any such (finite) organization, there exists a steady state where no
agent benefits by unilateral deviation, henceforth called Nash equilibrium (NE). Nash
equilibrium is perhaps the most well known and well studied solution concept in game
theory. In this thesis we study structural and computational aspects of Nash equilibria
in various games. The thesis may be divided into three parts.
The first part deals with analysis of NE structure for a special class of two player finite
(bimatrix) games. Given a rank-1 bimatrix game (A, B), i.e., where rank(A + B) = 1, we
construct a suitable linear subspace of the rank-1 game space and show that this subspace
is homeomorphic to its Nash equilibrium correspondence. Using this structure, we give
the first polynomial time algorithm for computing an exact NE of a rank-1 bimatrix game.
This settles an open question posed by Kannan and Theobald [38, 62]. In addition, we
give a novel algorithm to enumerate all the Nash equilibria of a rank-1 game and show
that a similar technique may also be applied to find a NE of any bimatrix game. Our
approach also provides new proofs of important classical results such as the existence and
oddness of Nash equilibria, and the index theorem for bimatrix games.
Further, we give two more homeomorphism results. The first result extends the
rank-1 homeomorphism result to a fixed rank game space, and provides a fixed point
formulation on [0, 1]k for solving a rank-k game. The homeomorphism maps and the fixed
point formulation are piece-wise linear but considerably simpler than classical constructions. The second result is a sum preserving homeomorphism, where we consider the
subspace where the sum A + B remains the same, and show that its NE correspondence
is homeomorphic to the subspace.
In the second part we define bilinear games, an extension of bimatrix games, reprei
sented by two payoff matrices and polytopal strategy sets. We show that bilinear games
are very general and capture many important classes of games like bimatrix games, two
player Bayesian games, polymatrix games, two-player extensive form games with perfect
recall etc. as special cases, and hence are hard to solve in general. For a bilinear game, we
define its best response polyhedra (BRPs) and characterize its NE as the fully-labeled pairs
of the BRPs. Rank of a game (A, B) is defined as rank(A + B). We give polynomialtime algorithms for computing: (i) exact NE of rank-1 games, (ii) approximate NE for
fixed-rank games, and (iii) exact NE for games having A or B of a constant rank; this
improves a result of [46] on bimatrix games.
The third part deals with games defined on a market mechanism. The concept of
market equilibrium is based on optimality, and assumes that there are infinitely many
agents (players) in the market, each too small to influence the equilibrium. However, in
the current day oligopolistic markets, players may be able to change the equilibrium and
achieve better payoffs by strategization. Motivated by this observation, we define Fisher
Market Game where buyers strategize by posing different utility functions. We show
that existence of a conflict-free allocation (acceptable to all the players), is a necessary
condition for a NE and also sufficient for the symmetric NE in this game. There are many
Nash equilibria with very different payoffs, and the Fisher equilibrium payoff is captured
at a symmetric NE. We show that there are Nash equilibria with sub-optimal payoffs as
well, and discuss the difficulties in the complete characterization of Nash equilibria; even
for markets with three buyers.
For market games with two buyers, we provide a complete polyhedral characterization of all the Nash equilibria. Surprisingly, all the NE of such a game turn out to be
symmetric (herd behavior) and the corresponding payoffs constitute a piecewise linear
concave curve. For every point on this curve, there is a convex set of NE, leading to a
different class of non-market behavior such as incentives. We also study the correlated
equilibria of general Fisher market game and show that third-party mediation does not
help to achieve a better payoff than NE payoffs.
ii
Contents
Abstract
i
List of Tables
v
List of Figures
vi
1 Introduction
1
1.1 Bimatrix Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2 Bilinear Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3 Fisher Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Organization
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Rank-1 Homeomorphism and a Polynomial Time Algorithm
18
2.1 Facts About Polyhedra and Notations . . . . . . . . . . . . . . . . . . . . . 19
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Game Space and its NE Correspondence . . . . . . . . . . . . . . . . . . . 21
2.3.1
Orientation of N and the Index Theorem . . . . . . . . . . . . . . . 26
2.4 Rank-1 Space: A Homeomorphism
. . . . . . . . . . . . . . . . . . . . . . 31
2.5 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.1
Rank-1 NE: A Polynomial Time Algorithm . . . . . . . . . . . . . . 35
2.5.2
Enumeration Algorithm for Rank-1 Games . . . . . . . . . . . . . . 38
2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Other Homeomorphism Results
41
3.1 Rank-k Homeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Sum Preserving Homeomorphism . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.1
Extract à from A′ . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
iii
3.2.2
Extract ᾱ, β̄ from A′ , Ã . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.3
Construction of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Efficient Algorithms for Rank Based Subclasses of Bilinear Games
52
4.1 Bilinear Games and Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . 53
4.1.1
Examples of Bilinear Games . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Rank-1 Games: A Polynomial Time Algorithm . . . . . . . . . . . . . . . . 61
4.2.1
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 FPTAS for Rank-k Games . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.1
FPTAS for Approximate NE . . . . . . . . . . . . . . . . . . . . . . 65
4.3.2
FPTAS for Relative Approximate NE . . . . . . . . . . . . . . . . . 66
4.4 Games With a Low Rank Matrix . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Fisher Market Game
70
5.1 The Fisher Market Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Nash Equilibria: A Characterization
. . . . . . . . . . . . . . . . . . . . . 73
5.2.1
Conflict Removal Procedure . . . . . . . . . . . . . . . . . . . . . . 74
5.2.2
Symmetric and Asymmetric NESPs . . . . . . . . . . . . . . . . . . 77
5.3 The Two-Buyer Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.1
Polyhedral Characterization of NESPs . . . . . . . . . . . . . . . . 80
5.3.2
The Payoff Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3.3
Incentives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4 Correlated Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
A Homeomorphism Maps for Two Player Game Space
89
B Regions in the Game Space
90
Bibliography
94
Publications
100
iv
List of Tables
1.1 Prisoner’s Dilemma
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2 Summary of Results for Bilinear Games . . . . . . . . . . . . . . . . . . . . 12
2.1 BinSearch Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 IsNE Procedure for the Degenerate Case . . . . . . . . . . . . . . . . . . . 38
2.3 Enumeration Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.1 Conflict Removal Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Bk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Bk′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 P (G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
v
List of Figures
5.1 The payoff curve of Example 5.19 market . . . . . . . . . . . . . . . . . . . 84
vi
Chapter 1
Introduction
Non-cooperative game theory [48] is one of the most important and vibrant mathematical fields. It was established to understand strategic interaction of selfish agents in a
given organization. The concepts from game theory are applicable in every aspect of decision/policy making, and analysis, at all levels, for example the decision to jump a signal
or not on individual level, and designing policies so that all the countries are compelled
to control pollution on international level.
In the language of game theory, a game refers to any social situation involving two
or more individuals. The individuals involved in a game are called players. Each player’s
objective is to maximizer his/her own happiness, which may be measured in some utility
(payoff) scale1 . A game is said to be common knowledge if everything is known to all the
players. The simplest form of game is a single shot finite game with common knowledge,
also called a normal form game. An equilibrium of a game is a steady state where no
player gains by unilateral deviation. Let us take a small (but famous) example of a two
player normal form game.
Example 1.1 : Prisoner’s Dilemma - Two men are accused of conspiring in a major
crime, for which they can be convicted only if at least one confesses. The prosecutor
promises that, if exactly one confesses, the confessor will be used as a witness and released,
but the other will go to jail for 6 years. If both confess, then they both go to jail for 5
years. If neither confesses then they will both go to jail for only 1 year, due to lack of
witness. So each player i has two possible pure-strategies: to confess (ci ) or to not confess
1
Players are allowed to play a probability density over their action set, and then the objective is to
maximize the expected payoff.
1
(ni ). The payoffs, measured in the number of years of freedom that the player will enjoy
over the next 6 years, are as shown in Table 1.1. The entries in the table gives utility
n2
c2
n1
5,5
0,6
c1
6,0
1,1
Table 1.1: Prisoner’s Dilemma
of both the players when the corresponding strategy profile is played, i.e. (0,6) in second
row, third column indicates that when first player plays n1 and second plays c2 then first
player’s payoff is 0 and second player’s payoff is 6. Observe that the only equilibrium state
of this game is where both the players confess.
The game in Example 1.1 has an equilibrium in pure strategies. This may not be
the case in general; an equilibrium state is achieved only when the players play some
probability distribution over their action set (mixed-strategy), like in case of matching
penny game (see [61]).
A game is said to be zero-sum if sum of the payoffs of all the players is zero in every
play. In 1928 John von Neumann [71] showed that any two-person zero-sum (normal
form) game has an equilibrium which is a min-max strategy. Later it was established that
computation of such a min-max strategy is equivalent to solving a Linear Programming
problem (LP) [15], and thus computationally tractable [39]. Further, the book by von
Neumann and Morgenstern (1944) [72] noted that the zero-sum games is a very special
case of general games with more than two players, and for that no existence theorem is
known. In 1951, John Nash [49] proved that there exists an equilibrium in any normal
form game. It has since been named Nash equilibrium (NE) and is perhaps the most
well known and well studied solution concept in game theory. Nash’s proof is notoriously
non-constructive and uses heavy machinery of the Brower’s fixed point theorem. In this
thesis we study structural and computational aspects of Nash equilibria in various games.
The question of finding a mechanism or an algorithm for arriving at a Nash equilibrium has been the subject of intense study within mathematical economics and theoretical
computer science alike, with major successes and catastrophic negative results. The algorithms by Scarf [57] and Smale [60] to approximate fixed points are known to perform
2
well in practice, however they are exponential time algorithms [33] in the worst case. The
classical Lemke-Howson (LH) algorithm [45] finds a Nash equilibrium of a game with two
players by solving a related linear complementarity problem, hence rationality of NE is
ensured in two player games. However, recently Savani and and von Stengel [56] showed
that it is not a polynomial time algorithm by constructing an example, for which the LH
algorithm takes an exponential number of steps in number of strategies available to the
players. An alternative to the LH algorithm was provided by the algorithms of Lemke
[44] and van den Elzen and Talman [64]. Generalization of these algorithms to n-person
games are given in [27, 32, 54, 75]. All these are finite time algorithms, see [31] for a
detailed survey. A difficulty to compute Nash equilibria in n-player games is posed by the
irrationality problem [49]. This makes it impossible to solve the problem exactly.
In 1994 Papadimitriou [51] introduced complexity class PPAD, Polynomial Parity
Arguments on Directed graphs, for problems where existence of a solution is guaranteed
but computing one seems difficult. It contains computation of approximate fixed point
problem (in turn approximate NE computation), an exact Nash equilibrium for games
with two players, Sperner’s lemma and many more. He proved that the approximate
fixed point problem is also complete for this class. After a decade, Daskalakis, Goldberg
and Papadimitriou (2006) [16] proved that computing an approximate Nash equilibrium,
in games with more than two players, is PPAD-complete. Within a few months of this
result, Chen et al. [13] showed even for two player games NE computation is PPADcomplete. They together with Teng also showed that the computation of even a
1
nΘ(1)
approximate Nash equilibrium remains PPAD-complete. These results suggest that a
polynomial time algorithm for Nash equilibrium computation in normal form games is
unlikely unless PPAD is in P, a complexity class for polynomial time algorithms. As
efficient computation of NE for general games seems less likely from these complexity
results, many efficient algorithms are designed to compute an exact or an approximate
NE for special classes of two player games [38, 43, 46].
There has been much work to understand structural aspects of Nash equilibria as
well. In 1974 Shapley [59] proved that every Nash equilibrium of a bimatrix game can be
assigned a sign (also called index) in a canonical manner. This sign was used crucially
to put Nash equilibrium computation problem for bimatrix games in PPAD [51, 56].
Shapley also proved that the number of Nash equilibria in any bimatrix game is odd.
3
In yet another classical work, Kohlberg and Mertens (1986) [40] showed that the Nash
equilibrium correspondence of normal form games is homeomorphic to the game space
itself. In their words - “The graph of NE correspondence is like a deformation of a
rubber sphere around the sphere of the normal form games”. This result paved way for
many interesting results for general games; index and oddness results were extended to
general games, and very importantly the existence and characterization of stable Nash
equilibria [40, 25, 28] were achieved. The homeomorphism result also provides a way to
validate homotopy based Nash equilibrium computation methods [27, 31]. This indicates
the tremendous importance of establishing such structural results.
The scope of this thesis is two fold, the structural analysis and the efficient computation of Nash equilibria in various games. The thesis may be divided into three parts.
A two player normal form game can be represented by two payoff matrices (A, B),
and hence is called a bimatrix game. The first part of the thesis is based on [2], and deals
with analysis of Nash equilibrium structure for a special class of bimatrix games. Using
this structure it derives an efficient algorithm (in P) for this class thereby settling an open
question of Kannan and Theobald [38]. The second part is based on [23], and extends
bimatrix games to bilinear games, where strategy sets of the players are polytopes. The
bilinear games turns out to be quite important since these include many important games
like polymatrix games, 2-player Bayesian games and 2-player extensive form games with
perfect recall as special cases. Further, we extend three efficient algorithms for bimatrix
games, including the one given in the first part, to bilinear games.
The third part is based on [1], and deals with games defined on a market mechanism. The concept of market equilibrium is optimality based [4, 6, 68], and assumes
that there are infinitely many agents (players) in the market, each too small to influence
the equilibrium. However, this is not the case in the current day oligopolistic markets,
where strategization may help to change the equilibrium and achieve better payoffs to the
players. Motivated by this observation, we define the Fisher Market Game and characterize its Nash equilibria, which in turn illustrates herd mentality, incentive offerings and
ineffectiveness of mediation in such markets.
Next three sections briefly describe the work covered in each of the three parts
while keeping the related literature in perspective. In what follows vectors are in column
representation by default, and transpose is used to denote a row vector.
4
1.1
Bimatrix Games
For a finite two-player game, let the strategy sets of the first and the second player be
S1 = {1, . . . , m} and S2 = {1, . . . , n} respectively. The payoff function of such a game
may be represented by the two payoff matrices (A, B), each of dimension m×n. The game
of Example 1.1 is a bimatrix game, where matrix A is formed by the first coordinates of
the cells of Table 1.1, and the second coordinates form matrix B.
The strategies in S1 and S2 are called pure strategies. A mixed strategy is a probability
distribution over the available set of strategies. The set of mixed strategies for the first
P
player is ∆1 = {(x1 , . . . , xm ) | xi ≥ 0, ∀i ∈ S1 ,
i∈S1 xi = 1} and for the second
P
player, it is ∆2 = {(y1 , . . . , yn ) | yj ≥ 0, ∀j ∈ S2 ,
j∈S2 yj = 1}. If the strategy profile
(x, y) ∈ ∆1 × ∆2 is played, then the payoffs of the first and second players are xT Ay and
xT By respectively. Recall that a strategy profile is said to be a Nash equilibrium strategy
profile (NESP) if no player achieves a better payoff by a unilateral deviation [49].
As we mentioned, computing a Nash equilibrium in zero-sum games (when B = −A)2
and solving an LP are equivalent [15]. More recently, Lipton et al. (2003) [46] gave a
polynomial time algorithms to solve games where both payoff matrices are of fixed rank
k. However, the expressive power of this restricted class of games is limited in the sense
that most zero-sum games are not contained in this class. Kannan and Theobald [38]
defined a rank-based hierarchy of bimatrix games, where rank of a game (A, B) is defined
as rank(A + B), and gave a fully polynomial time approximate scheme (FPTAS) for Nash
equilibrium computation in fixed rank games. The set of rank-k games consists of games
with rank at most k. Clearly, rank-0 games are the zero-sum games whose NE sets are
known to be convex and polyhedral.
The set of rank-1 games is the smallest extension of zero-sum games in the hierarchy, which strictly generalizes zero-sum games. For any given constant c, Kannan and
Theobald [38] construct a rank-1 game, for which the number of connected components of
Nash equilibria is larger than c. This shows that the expressive power of rank-1 games is
larger than the zero-sum games. Rank-1 games may also arise in practical situations, in
particular the multiplicative games between firms and workers in [8] are rank-1 games. A
polynomial time algorithm to compute an exact Nash equilibrium for rank-1 games was an
2
Note that, the total payoff in case of zero-sum games is always zero.
5
important open problem [38, 62]. Kontogiannis and Spirakis [43] defined the notion of mutual (quasi-) concavity of a bimatrix game and provided a polynomial time computation
of a Nash equilibrium for mutually concave games (FPTAS for mutually quasi-concave
games). However their classification and the games of fixed rank are incomparable.
On the structural side, the notion of best response polyhedra P and Q, and fullylabeled points3 of P × Q may be defined for a game (A, B), where P depends on A and
Q depends on B [73]. It is known that there is a one-to-one correspondence between
the fully-labeled points and the Nash equilibria of the game. The famous Lemke-Howson
algorithm [45] works in the modified best response polyhedra, containing exactly one fullylabeled point not related to any Nash equilibrium of the game, namely origin. Starting
from this dummy fully-labeled point, the algorithm traces a path that is guaranteed to
end at another fully-labeled point which has to be a Nash equilibrium of the game. The
LH scheme can be started from any fully-labeled point and the path traversed is called
an LH path (whose other endpoint is also fully-labeled). Shapley’s index theory (1974)
[59] assigns opposite signs to the two endpoints of a LH path. The signs of the endpoints
of LH paths provide a direction and in turn a “parity argument” that puts the Nash
equilibrium problem of a bimatrix game in PPAD (see [51, 56]).
The set of bimatrix games (Ω) forms a Euclidean space, i.e., Ω = {(A, B) ∈
Rmn × Rmn } = R2mn . Kohlberg and Mertens [40] showed that Ω is homeomorphic to
its Nash equilibrium correspondence EΩ = {(A, B, x, y) ∈ R2mn+m+n | (x, y) is a Nash
equilibrium of (A, B)} (See appendix A).
As discussed above, this structural result has been used extensively to understand
the index, degree and the stability of Nash equilibria [26], and provides validation for
homotopy based methods [27, 31]. Such a structural result has been established for more
general game spaces [53], however, to the best of our knowledge, no such result is known
for special subspaces of the bimatrix game space. The insights from such a result may be
useful to better understand the stability of Nash equilibria and the complexity of Nash
equilibrium computation problem.
3
Points satisfying a set of complementarity conditions.
6
Our Contribution
In Chapter 2, we establish a homeomorphism result for special subspaces, and design a
polynomial time algorithm for rank-1 games. The following theorem summaries the main
results of this chapter informally, see Theorems 2.18 and 2.19 for precise statements.
Theorem 1.2 There exists an m-dimensional affine subspace of the bimatrix game space,
such that the subspace and its Nash equilibrium correspondence are homeomorphic, where
the subspace consists of only rank-1 games. Further, there exists a polynomial time algorithm to compute an exact Nash equilibrium of a rank-1 game.
Theorem 1.2 establishes the first structural result for subspaces of the bimatrix game
space. The homeomorphism maps that we derive are very different from the ones given
by Kohlberg and Mertens for the bimatrix game space [40] and build on the structure of
the NE correspondence. We use this structure crucially to design the polynomial time
algorithm for rank-1 games, and settle an open question posed by Kannan and Theobald
[38], and Theobald [62]. Apart from these we also derive alternate proofs for a few classical
results pertaining to NE and an algorithm to enumerate all the NE of rank-1 games.
Next we briefly describe important technical ideas we develop to get these results.
For a given rank-1 game (A, B) ∈ Rmn × Rmn , the matrix (A + B) may be written as
α · β T , where α ∈ Rm and β ∈ Rn . Motivated by this fact, we define an m-dimensional
subspace Γ = {(A, C + α.β T ) | α ∈ Rm } of Ω, where A ∈ Rmn , C ∈ Rmn and β ∈
Rn are fixed and analyze the structure of its Nash equilibrium correspondence EΓ =
{(A′ , B ′ , x, y) | (x, y) is a Nash equilibrium of (A′ , B ′ ) ∈ Γ}. Note that the best response
polyhedron P is same for all the games in Γ since the payoff matrix of the first player
is fixed to A. However the payoff matrix of the second player varies with α, hence Q is
different for every game. We define a new polyhedron Q′ , which encompasses Q for all
the games in Γ. We show that the set of fully-labeled points of P × Q′ , say N , captures
all the Nash equilibria of all the games in Γ and in turn captures EΓ .
Surprisingly, N turns out to be a set of cycles and a single path on the 1-skeleton
of P × Q′ under the non-degeneracy assumption. We refer to the path in N as the fullylabeled path and show that it captures at least one Nash equilibrium of every game in Γ,
thereby proving existence of a NE in bimatrix games. The structure of N also proves that
the number of Nash equilibria of a game is odd. Moreover, an edge of N may be efficiently
7
oriented, and using this orientation, we determine the index of every Nash equilibrium
and prove the index theorem.
Solving a bimatrix game can be formulated as a quadratic program (QP), and for a
rank-1 game the program turns out to be rank-1 QP, which is N P-hard in general [29]. We
convert this QP into a parametrized LP by replacing a linear expression of the quadratic
cost term with a parameter. We show that if Γ contains only rank-1 games (i.e., C = −A)
then N can be captured exactly by the solutions of corresponding parametrized LP with
the parameter taking every value form R. This in turn proves that on any component of
N , there is a parameter which is monotone, and hence N can not contain cycles. Using
the monotonic nature of the path of N , we establish a homeomorphism map between Γ
and EΓ . This is the first structural result for a subspace of the bimatrix game space.
Using the above facts on the structure of N , we design two algorithms for rank1 games. For a given non-degenerate rank-1 game (A, −A + γ.β T ), we consider the
subspace Γ = {(A, −A + α.β T ) | α ∈ Rm }. Note that Γ contains the given game and
the corresponding set N is a path which captures all the Nash equilibria of the game.
The first algorithm (BinSearch) finds a Nash equilibrium of the given game in polynomial
time by applying binary search on the fully-labeled path using its monotonic nature,
where in each step it solves the parametrized LP for a specific value of the parameter.
The algorithm works for degenerate games as well with a minor modification. This is the
first polynomial time algorithm to find an exact Nash equilibrium of a rank-1 game.
The second (Enumeration) algorithm traces the fully-labeled path of N . Since the
set N consists of only the path in case of rank-1 space, the algorithm locates all the Nash
equilibria of any rank-1 game. For an arbitrary bimatrix game, we may define a suitable
Γ containing the game. Since the fully-labeled path of the corresponding N covers at
least one Nash equilibrium of all the games in Γ, the Enumeration algorithm locates at
least one Nash equilibrium of the given game. Theobald [62] also gave an algorithm to
enumerate all the Nash equilibria of a rank-1 game, however it may not be generalized to
find a Nash equilibrium of any bimatrix game.
In Chapter 3 we give two more homeomorphism results, informally stated as follows,
see Theorems 3.3 and 3.8 for formal statements.
Theorem 1.3
1. There exists km-dimensional affine subspace containing only rank-k
games, such that the subspace and its NE correspondence are homeomorphic.
8
2. For a fixed matrix C ∈ Rm×n , the set of games (A, B) with A + B = C and its NE
correspondence are homeomorphic.
The first result of Theorem 1.3 extends the rank-1 homeomorphism result to a fixed
rank game space, and leads to a possible approach to solve rank-k games in polynomial
time with k. The approach is as follows: For a given rank-k game (A, B), the matrix
P
T
(A + B) may be written as kl=1 γ l .β l , where ∀l, γ l ∈ Rm and β l ∈ Rn . We define
P
T
a km-dimensional affine subspace Γk = {(A, −A + kl=1 αl .β l ) | αl ∈ Rm , ∀l} of Ω,
and establish a homeomorphism between Γk and its Nash equilibrium correspondence
EΓk using techniques similar to the rank-1 homeomorphism. Further, to find a Nash
equilibrium of a rank-k game we give a piece-wise linear polynomial-time computable fixed
point formulation on [0, 1]k using the homeomorphism result and discuss the possibility
of a polynomial time algorithm. The fixed point formulation is considerably simpler than
classical constructions, and has some monotonicity properties.
The second result of Theorem 1.3 derives homeomorphism maps such that the summation matrix A + B is preserved. For a fixed matrix C ∈ Rmn , the game space in to
consideration is ΓC = {(A, −A + C) | A ∈ Rmn }. The homeomorphism maps between the
game space ΓC and its Nash equilibrium correspondence EΓC are different from the previous maps, and uses a technique from the result of Kohlberg and Mertens [40]. We observe
that similar approach works to get homeomorphism between {(C, −C + A) | A ∈ Rmn }
and its Nash equilibrium correspondence as well. The maps we derive are more intricate
and challenging than that of [40], because we need to encode relatively more values in
less space. We hope that these new homeomorphism result will help in designing efficient
algorithms for Nash equilibrium computation in fixed rank games.
1.2
Bilinear Games
Specifying the two payoff matrices of a bimatrix game requires a polynomial number of
entries in the numbers of pure strategies available to the players. This is adequate when
the set of pure strategies are explicitly given. However, there are situations where the
natural description gives the set of pure strategies implicitly, and as a result they may
be exponential in the description of the game. For example, normal form (bimatrix)
representation of two player extensive-form game may have exponentially many strategies
9
in the size of the extensive-form description [21]. In such a case, even if the resulting
bimatrix game has a fixed rank, the known efficient algorithms for bimatrix games are no
more efficient with respect to the original game.
Nevertheless, certain types of extensive-form games have some combinatorial structure which may be exploited. Koller, Megiddo and von Stengel [42] converted an arbitrary
two-player, perfect-recall, extensive form game into a payoff-equivalent two-player game
with continuous strategy sets. In this derived formulation, which they call the sequence
form, there is a pair of payoff matrices A and B, one for each player. Further, their strategy sets turn out to be compact polytopes in Euclidean space of polynomial dimension.
Given a pair of strategies (x, y), utilities of the players are xT Ay and xT By respectively.
Interestingly, the sequence form requires only a polynomial number of bits to specify.
Our Contribution
Motivated by the sequence form of Koller et al., we define and study bilinear games in
Chapter 4. Bilinear games are two-player, non-cooperative, single shot games represented
by two payoff matrices, say A and B, of dimension m × n and two polytopal compact
strategy sets X and Y in dimensions m and n respectively. If (x, y) ∈ X × Y is the
played strategy profile, then xT Ay and xT By are the utilities derived by the first and the
second player respectively. In other words, the payoffs are bilinear functions of strategies,
hence the name bilinear games. The concept of Nash equilibrium in bilinear games is
defined in similar way as normal form games; a strategy profile, from which no player
gains by unilateral deviation. The scope of bilinear games is large enough to capture
many important classes of games besides bimatrix games and two-player extensive form
games with perfect recall. For example, Bayesian games [35, 52], polymatrix games [34],
and various classes of optimization duals [36], have polynomial-sized payoff-equivalent
formulations which (either explicitly or implicitly) turn out to be bilinear games (see
Chapter 4 for details).
As many different types of games may be concisely described as bilinear games,
designing efficient algorithms for these games is an important problem with wide applicability. Since bimatrix games is a subclass of bilinear games, all the hardness results
of bimatrix games automatically apply to bilinear games as well. Therefore, the only
hope is to design efficient algorithms or FPTAS for special subclasses. There are many
10
similarities between bilinear and bimatrix games, for example, payoffs are represented by
two matrices, and utilities are bilinear functions of strategy vectors, hence it is natural
to try to adapt algorithms for bimatrix games to bilinear games. However, a technical
challenge is that the polytopal strategy sets of bilinear games are generally much more
complex than the bimatrix case; in particular the number of vertices may be exponential,
while the set of mixed strategies is just a simplex.
We first study Nash equilibria of bilinear games. Existence of a (symmetric) Nash
equilibrium in bilinear games follow directly from the known results. For any player, the
problem of computing utility maximizing strategy, given a strategy of the other player,
can be formulated as a primal-dual LP. Using the complementarity conditions of this
formulation we define Best Response polyhedra (BRP), and characterize Nash equilibria as
fully-labeled pairs in BRPs. This in turn gives a quadratic programming (QP) formulation
for the NE computation problem.
Next, we extend Kannan and Theobald’s [38] rank-based hierarchy for bimatrix
games to bilinear games, by defining the rank of a bilinear game (A, B) as the rank
of matrix (A + B). Zero-sum games are rank-0 games, and the linear programming
approach proposed by Koller et al.[41] to solve zero-sum sequence form, works for zerosum bilinear games as well (see also [10, 52]). The following theorem informally summaries
our algorithmic results, see Theorem 4.17, 4.21, 4.22 and 4.25 for formal statements.
Theorem 1.4 There exist polynomial time algorithms to compute, (1) an exact NE of
rank-1 bilinear games, (2) an approximate NE of rank-k bilinear games (FPTAS where k
is a constant), and (3) all the exact NE when rank of A or B is a constant.
Like bimatrix games, computing a NE in a rank-1 bilinear game can be formulated
as a rank-1 QP (recall it is N P-hard in general). We show that the QPs arising from
rank-1 bilinear games can be solved in polynomial time. We extend the approach designed
in the first part (for rank-1 bimatrix games) to solve rank-1 bilinear games efficiently, in
spite of the very general structure of the strategy sets in these games. Further, we design
two FPTAS algorithms for the fixed rank bilinear games, which are generalizations of the
algorithms by Kannan and Theobald [38] for the bimatrix games.
Finally, we obtain a polynomial time algorithm for the case when the rank of either
A or B is a constant. Given a vertex of a BRP one can check if it forms a fully-labeled
11
pair (or gives an NE) by solving an LP. Further, we show a strongly polynomial upper
bound on the number of vertices of the BRP corresponding to constant rank matrix.
Putting these two together gives a polynomial time algorithm. It improves upon a result
by Lipton et al. [46] for bimatrix games, where they require both A and B of constant
rank. This algorithm can also enumerate extreme equilibria of a bilinear game, in time
polynomial under the above assumptions and exponential in general.
Table 1.2 summarizes all the results while keeping the bimatrix games in perspective.
Results
Bimatrix games
Bilinear games
Existence of (symmetric) NE
Nash [49]
Follows from [49, 24]
LCP formulation for NE
Known [73]
Koller et al. [42]
NE as fully-labeled pairs of BRPs
Known [73]
This thesis
Zero-sum games
LP [15]
LP [41]
Rank-1 games
This thesis
This thesis
FPTAS for fixed rank games
Kannan and Theobald [38]
This thesis
Games with low rank matrices
Lipton et al. [46]
This thesis
Table 1.2: Summary of Results for Bilinear Games
1.3
Fisher Market
In the third part we study a market mechanism under strategic agents. The market
equilibrium theory started with a pioneering work by Leon Walras (1987) [74], where he
proposed the famous tatonnement process terminating only at a state where supply equals
demand, an equilibrium. However, the existence of an equilibrium and termination guarantee were not established. In 1951, Arrow and Debreu (AD) [4] gave a non-constructive
proof of existence under some mild conditions. Since then there has been numerous studies aimed at understanding the computability of market equilibrium [19, 37, 70]. A market
equilibrium yields fair and optimal resource allocation, under the assumption that there
are infinite number of agents in the market, each too small to influence the equilibrium
prices through strategization. However, many big markets are oligopolistic, like car, aircraft and oil markets, and do not satisfy this assumption. Therefore, it is essential to
12
study the effect of strategization, since markets being an integral part of our social system any change may lead to serious implications. We study strategic aspects of Fisher
market model, a special case of AD model [4].
Fisher market model, formulated by Fisher in 1891 [6], consists of a set B of buyers
(agents) and a set G of divisible goods, where |B| = m and |G| = n. Buyer i comes with a
positive endowment of money; mi is the money possessed by agent i, and a utility function
over a bundle of goods, ui : Rn → R. We consider the model where utility functions are
linear. In that case for each buyer i and good j we are specified a non-negative number
uij , which represents the utility agent i derives per unit of good j. Overall utility of
P
buyer i from a bundle x = (x1 , . . . , xn ) of goods is ui (x) = j∈G uij xj . Let µj denote the
quantity of good j available in the market.
Given prices p = (p1 , . . . , pn ) of goods, each buyer buys a bundle of goods that
maximizes her utility (optimal bundle), subject to her budget constraints. We say that p
is a market equilibrium price vector if there are choices of optimal bundles for the agents
such that after each agent is given such a bundle, there is no deficiency or surplus of any
good, and money of all the buyers is exhausted, i.e., the market clears. The utility buyer
i derives per unit money from good j at prices p is uij /pj ; call it bang-per-buck of agent
i for good j. Clearly, a buyer would like to spend money on the goods fetching highest
bang-per-buck. Using this observation the market equilibrium condition can be formally
stated as follows: Market equilibrium prices p and allocation [xij ]i∈G,j∈G satisfies,
• Optimal Bundle: xij > 0 ⇒ uij /pj = maxk∈G uik /pk .
• Market Clearing:
P
i∈B
xij = µj , ∀j ∈ G and
P
j∈G
xij pj = mi , ∀i ∈ B.
Without loss of generality (w.l.o.g.) we assume that for every good j, there is a buyer
i such that uij > 0, or else we can freely distribute the good. By scaling uij ’s appropriately,
we can assume that the quantity of every good is one unit, i.e., µj = 1, ∀j ∈ G. Moreover,
equilibrium prices are unique and the set of equilibrium allocations is a convex set [68];
each allocation fetching the same utility to every buyer. The following example illustrates
a small market.
Example 1.5 Consider a 2 buyers, 2 goods market with m1 = m2 = 10, q1 = q2 = 1,
hu11 , u12 i = h10, 3i and hu21 , u22 i = h3, 10i. The equilibrium prices of this market are
13
hp1 , p2 i = h10, 10i and the unique equilibrium allocation is hx11 , x12 , x21 , x22 i = h1, 0, 0, 1i.
The payoff of both the buyers is 10.
Recently, much work has been done on the computation of market equilibrium for
various utility functions [50, 14, 17, 22]. Note that, the payoff (i.e., utility) of a buyer
depends on the equilibrium allocation and in turn on the utility functions and initial
endowments of the buyers. A natural question to ask is: Can a buyer achieve a better
payoff by feigning a different utility function? It turns out that a buyer may indeed gain
by feigning! For example in the above market buyer 1 can force price change by posing a
different utility tuple, illustrated next.
Example 1.5 (Contd.). Suppose buyer 1 posts her utility tuple as h5, 15i instead of h10, 3i,
then coincidentally, the equilibrium prices hp1 , p2 i are also h5, 15i. The unique equilibrium
allocation hx11 , x12 , x21 , x22 i is h1, 31 , 0, 32 i. The payoff of buyer 1 is u11 ∗ 1 + u12 ∗
and that of buyer 2 is u22 ∗
2
3
=
20
.
3
1
3
= 11,
Note that the payoffs are still calculated w.r.t. the
true utility tuples.
The above example clearly shows that a buyer could gain by posting a different utility
tuple, hence the Fisher market is susceptible to gaming by strategic buyers. Therefore,
the equilibrium prices w.r.t. the true utility tuples may not be the actual operating point
of the market. The natural questions to investigate are: What are the possible operating
points of this market model under strategic behavior? Can they be computed? Is there a
preferred one? This motivates us to study the Nash equilibria of the Fisher market game,
where buyers strategize by posing different utility functions.
The strategic behavior of agents in the market has been studied earlier too, however
in different settings. Shapley and Shubik [58] consider a market game for the exchange
economy, where every good has a trading post, and the strategy of a buyer is to bid
(money) at each trading post. For each strategy profile, the prices are determined naturally so that market clears and goods are allocated accordingly, however agents may
not get the optimal bundles. Many variants [3, 18] of this game have been extensively
studied. Essentially, the goal is to design a mechanism to implement market equilibrium
(ME), i.e., to capture ME at a NE of the game. The strategy space of this game is tied to
the implementation of the market (in this case, trading posts). Our strategy space is the
utility tuple itself, and is independent of the market implementation. It is not clear that
14
bids of a buyer in the Shapley-Shubik game correspond to the feigned utility tuples. In
word auction markets as well, a similar study on strategic behavior of buyers (advertisers)
has been done [7, 20, 66].
Our Contribution
In Chapter 5, we formulate Fisher market game where buyers are the players and the
strategy set of a player is the set of all possible utility tuples (functions), and study its
Nash equilibria. In all we show the results stated informally in the following theorem (see
Theorem 5.5 and 5.26, Proposition 5.7 and Lemma 5.11), apart from many other results.
A strategy profile is said to be symmetric if all the players play the same strategy.
Theorem 1.6 If a strategy profile is NE then the corresponding allocation is conflictfree. A symmetric strategy profile is a NE iff it is conflict-free, and all the NE of a two
player market are symmetric. No third party mediation helps to achieve better payoffs
than symmetric NESPs.
Every (pure) strategy profile of the players defines a Fisher market, and therefore
market equilibrium prices and a set of equilibrium allocations, say X . Since the payoff
derived by a buyer is calculated with respect to her true utility tuple, it may not be the
same across all the allocations of X (see Example 5.1). Therefore, it is not clear which
allocation should be used to decide the payoffs in the game. Furthermore, there may not
exist an allocation in X preferred (fetching maximum possible payoff) by all the buyers.
This behavior causes a conflict of interest among buyers. A strategy profile is said to
be conflict-free, if there is an equilibrium allocation which gives the maximum possible
payoffs to all the buyers.
Suppose a strategy profile S is not conflict-free and let wi be the maximum possible
payoff to buyer i w.r.t. S. We show that buyer i can unilaterally deviate, say to S ′ with
allocation set X ′ , so that she gets almost wi payoff from every allocation of X ′ . This in
turn implies that whatever allocation we choose to calculate the payoffs in the game, if it
is not preferred by all, then someone would deviate and gain. Therefore, the social welfare
maximizing allocation of X is the right choice to calculate the payoffs in the game. It
also implies that Nash equilibrium strategy profiles (NESPs) are always conflict-free.
15
Using the equilibrium prices, we associate a bipartite graph to a strategy profile
and show that this graph must satisfy certain conditions when the corresponding strategy
profile is a NE. Next, we define symmetric strategy profiles, where all buyers play the same
strategy. We show that a symmetric strategy profile is a NESP iff it is conflict-free. It is
interesting to note that a symmetric NESP can be constructed for a given market game,
whose payoff is same as the Fisher payoff, i.e., payoff when all buyers play truthfully.
We show that all NESPs need not be symmetric and the payoff w.r.t. a NESP need not
be Pareto optimal (Example 5.10). However, the Fisher payoff is always Pareto optimal
(see First Theorem of Welfare Economics [65]). Characterization of all the NESPs seems
difficult; even for markets with only three buyers.
Two player games as well as markets have been given special attention [69, 45]
because they exhibit remarkable properties which are missing in their counterpart with
more players, in addition to having many applications. Same is the case with Fisher
market game. Some nice properties of two-buyer markets are:
• All NESPs are symmetric; a complete characterization.
• There is a nice ordering on the goods, given by precisely the ratio of the true utility
tuples, i.e.,
u1j
, ∀j
u2j
∈ G. This ordering helps us to compute all the NESPs and show
that it is the union of at most 2n convex sets.
• The set of NESP payoffs constitute a piecewise linear concave curve and all these
payoffs are Pareto optimal. In other words, the strategizing on utilities has the
same effect as increased initial money endowment for a buyer (see Second Theorem
of Welfare Economics [65]).
Some interesting observations about two-buyer markets are:
• A curious observation is that buyer i gets the maximum payoff among all Nash
equilibrium payoffs when they play (u−i, u−i), where u−i is the true utility tuple
of the other buyer. For example, when there are two types of buyers, say rich and
poor, both having different utility tuples for the goods in the market, then it is best
for the rich to imitate the poor and vice-versa.
• There may exist NESPs, whose welfare (i.e., sum of the payoffs of both the buyers)
is larger than the welfare from the Fisher payoff (Example 5.19).
16
• The behavior of the prices is even more curious. For a particular payoff tuple, we
show that there is a convex set of NESPs and hence convex set of equilibrium prices.
This motivates a seller to offer incentives to the buyers to choose a particular NESP
from this convex set, which fetches the maximum price for her good (Example 5.20
in Section 5.3.3 illustrates this behavior).
Most qualitative features of these markets may carry over to oligopolies, which arise
in numerous scenarios. For example, relationship between a few manufacturers of aircrafts
or automobiles and many suppliers. Finally, we show that the third-party mediation does
not help in the general Fisher market games, and conclude that it is highly unlikely that
buyers will act according to their true utility tuples in Fisher markets.
1.4
Organization
The first part constitute of Chapters 2 and 3. In Chapter 2 we discuss a homeomorphism
result for subspaces of bimatrix game space, and using this structural result derive an
efficient algorithm for Nash equilibrium computation in rank-1 bimatrix games. Further,
we provide new proofs of important classical results such as the existence and oddness
of Nash equilibria, and the index theorem for bimatrix games. In Chapter 3 we discuss
two more homeomorphism results, and formulate Nash equilibrium computation in rank-k
game as a fixed point finding problem in [0 1]k space with some monotonicity properties.
The second part constitutes Chapter 4, where we define bilinear games similar to
bimatrix game except that the strategy sets are now bounded polytopes. We show that
these games includes many important games like bimatrix games, polymatrix games, and
2-player Bayesian games. We characterize the Nash equilibria of these games and extend
three efficient algorithms of special classes of bimatrix games, including the one given in
Chapter 2.
Chapter 5 covers the third part, where we define Fisher market games motivated
by the observation that buyers may indeed gain by feigning their utility functions. We
characterize its Nash equilibria through the notion of conflict-free allocation, and derive
interesting insight in the markets like herd behavior, incentive offerings and Pareto optimality. Further, we show that the mediation does not help to achieve better outcome in
such games.
17
Chapter 2
Rank-1 Homeomorphism and a
Polynomial Time Algorithm
In Chapter 1 we briefly discussed the bimatrix games, its rank based hierarchy as defined
in [38], and its known structural and computational results. In this chapter we derive
a homeomorphism result for special subspaces containing only rank-1 games, and using
some structural insights from this result we design the first polynomial time algorithm for
rank-1 bimatrix games.
First we discuss the relation between Nash equilibria of a bimatrix game and its best
response polyhedron in Section 2.2. In Section 2.3 we define an m-dimensional subspace
and study its NE correspondence. This gives us alternate proofs of known classical results on existence [49], oddness and index theorem [59] for bimatrix games. As solving
a bimatrix game reduces to solving a QP, and specifically for rank-1 games it is rank-1
QP, we discuss a technique of considering it as a parametrized LP in Section 2.4. We
show that the solution set of such a parametrized LP has a close connection with the NE
correspondence of a subspace containing only rank-1 games. We use this insight crucially
to establish homeomorphism between such a subspace and its NE correspondence. Further, using this parametrized LP we design the first efficient algorithm for rank-1 games
in Section 2.5 thereby settling an open problem of [38]. We also design a path following
enumeration algorithm for rank-1 which finds one NE in case of a general game.
We start by describing some facts about polyhedra and a few notations to be used
in the rest of the thesis.
18
2.1
Facts About Polyhedra and Notations
An inequality is said to be tight at some point if it holds with equality at that point.
A polyhedron is non-degenerate if there is no linear dependency among the set of tight
inequalities at any given point of the polyhedron. Next we state some facts about a
k-dimensional non-degenerate polyhedron, say K, represented by a set of inequalities.
These are used extensively in our analysis. There are exactly k inequalities tight at a
vertex of K, and exactly k − 1 are tight on an edge. In general exactly k − i are tight
on an i-dimensional facet of K. The set of facets of dimension i or less of K is called its
i-skeleton. By relaxing a tight inequality at a vertex, we mean moving on an adjacent
edge by making that inequality untight.
Now some notations. For a matrix A = [aij ] ∈ Rmn of dimension m × n, Ai denotes
the ith row and Aj denotes the j th column of the matrix. For a vector α ∈ Rm , let αi be its
ith coordinate. Vectors are considered as column vectors by default, and transpose (i.e.,
xT ) is used to specify a row vector. For a vector x ∈ Rn and a scalar c ∈ R, by x ≤ c we
mean, ∀i ≤ n, xi ≤ c. A “0” in the block representation of a matrix, is the matrix with
all zero entries of appropriate dimension, and “1k ” is a vector of all 1s of length k. For a
given matrix X, |X| denotes the maximum absolute entry in X, i.e., |X| = maxij |Xij |.
For a set S, ∆(S) denotes the set of probability distribution vectors over the elements of
P
S, i.e., ∆(S) = {x ∈ R|S| | x ≥ 0,
i∈S xi = 1}.
2.2
Preliminaries
Recall the bimatrix game defined in Section 1.1 of Chapter 1. A bimatrix game is a two
player game, each player having finitely many strategies to play. The strategy set of the
first and the second players are denoted by S1 and S2 respectively, where m = |S1 | and
n = |S2 |. Their payoff matrices are A and B respectively each of m × n dimension. If the
played strategy profile is (i, j) ∈ S1 × S2 , then the payoffs of the first is aij and that of
second players is bij . Note that the rows of these matrices correspond to the strategies of
the first player and the columns to that of the second player, hence the first player is also
referred to as the row-player and second player as the column-player. The mixed strategy
sets of the row and the column players are ∆1 = ∆(S1 ) and ∆2 = ∆(S2 ) respectively. If
the strategy profile (x, y) ∈ ∆1 × ∆2 is played, then the payoffs of the row-player and the
19
column-player are xT Ay and xT By respectively.
A strategy profile is said to be a Nash equilibrium strategy profile (NESP) if no
player achieves a better payoff by a unilateral deviation [49]. Formally, (x, y) ∈ ∆1 × ∆2
is a NESP iff ∀x′ ∈ ∆1 , xT Ay ≥ x′T Ay and ∀y ′ ∈ ∆2 , xT By ≥ xT By ′ . These conditions
may also be equivalently stated as,
∀i ∈ S1 , xi > 0 ⇒
Ai y = maxk∈S1 Ak y
∀j ∈ S2 , yj > 0 ⇒
xT B j = maxk∈S2 xT B k
(2.1)
From (2.1), it is clear that at a Nash equilibrium, a player plays a pure strategy
with non-zero probability only if it gives the maximum payoff with respect to (w.r.t.) the
opponent’s strategy. Such strategies are called the best response strategies (w.r.t. the
opponent’s strategy). The polyhedron P in (2.2) is closely related to the best response
strategies of the row-player for a given strategy (y) of the column-player [73] and it is
called the best response polyhedron of the row-player. Similarly, the polyhedron Q is
called the best response polyhedron of the column-player. In the following expression, x
and y are vector variables, and π1 and π2 are scalar variables.
P = {(y, π1) ∈ Rn+1 | Ai y − π1 ≤ 0, ∀i ∈ S1 ;
yj ≥ 0,
∀j ∈ S2 ;
Pn
j=1 yj
Q = {(x, π2 ) ∈ Rm+1 |
xi ≥ 0,
∀i ∈ S1 ; xT B j − π2 ≤ 0, ∀j ∈ S2 ;
Pm
i=1
= 1}
(2.2)
xi = 1}
Note that for any y ′ ∈ ∆2 , a unique (y ′, π1′ ) may be obtained on the boundary of
P , where π1′ = maxi∈S1 Ai y ′ . Clearly, the pure strategy i ∈ S1 is in the best response
against y ′ only if Ai y ′ − π1′ = 0, hence indices in S1 corresponding to the tight inequalities
at (y ′ , π1′ ) are in the best response. Note that, in both the polyhedron the first set of
inequalities correspond to the row-player, and the second set correspond to the column
player. Since |S1 | = m and |S2 | = n, let the inequalities be numbered from 1 to m, and
m+1 to m+n in both the polyhedron. Let the label L(p) of a point p in the polyhedron be
the set of indices of the tight inequalities at p. A pair (p, q) ∈ P × Q is called fully-labeled
pair if L(p) ∪ L(q) = {1, . . . , m + n}. The next lemma follows easily using the fact that
every NESP satisfies (2.1).
Lemma 2.1 [73] A strategy profile (x, y) is a NESP of the game (A, B) iff ((y, π1 ), (x, π2 )) ∈
P × Q is a fully-labeled pair, for some π1 and π2 .
20
A game is called non-degenerate if at any NESP no best response strategy is played
with zero probability. It is easy to check that both the best response polyhedra of a
non-degenerate game are also non-degenerate. For a non-degenerate game, |L(p)| ≤ n
and |L(q)| ≤ m, ∀(p, q) ∈ P × Q, and the equality holds iff p and q are the vertices of P
and Q respectively. Therefore, a fully-labeled pair of a non-degenerate game has to be a
vertex-pair.
Let Ω = {(A, B) ∈ Rmn × Rmn } = R2mn be the bimatrix game space and EΩ =
{(A, B, x, y) ∈ Rmn × Rmn × ∆1 × ∆2 | (x, y) is a NESP of the game (A, B)} be its Nash
equilibrium correspondence. Kohlberg and Mertens [40] proved that EΩ is homeomorphic
to the bimatrix game space R2mn (Ω). No such structural result is known for a subspace of
the bimatrix game space R2mn . With the hope of establishing such a result for a subspace,
we define an m-dimensional affine subspace of R2mn and analyze the structure of its Nash
equilibrium correspondence in the next section.
2.3
Game Space and its NE Correspondence
Let Γ = {(A, C + α · β T ) | α ∈ Rm } be a game space, where A ∈ Rmn and C ∈ Rmn are
m × n dimensional non-zero matrices, and β ∈ Rn is an n-dimensional non-zero vector.
Note that for a game (A, B) ∈ Γ, there exists a unique α ∈ Rm , such that B = C + α · β T .
Therefore, Γ may be parametrized by α, and let G(α) be a game (A, C + α · β T ) ∈ Γ.
Clearly, Γ forms an m-dimensional affine subspace of the bimatrix game space R2mn . Let
EΓ = {(α, x, y) ∈ Rm × ∆1 × ∆2 | (x, y) is a NESP of the game G(α) ∈ Γ} be the Nash
equilibrium correspondence of Γ. We wish to investigate: Is EΓ homeomorphic to the
game space Γ (≡ Rm )?
For a game G(α) ∈ Γ, let the best response polyhedron of row-player and columnplayer be denoted by P (α) and Q(α) respectively. Since the row-player’s matrix is fixed
to A, P (α) is the same for all α and we denote it by P . However, Q(α) varies with α.
We define a new polyhedron Q′ with an extra variable λ, which encompasses Q(α) for all
G(α) ∈ Γ.
Q′ = {(x, λ, π2 ) ∈ Rm+2 | xi ≥ 0, ∀i ∈ S1 ;
T
j
x C + βj λ − π2 ≤ 0, ∀j ∈ S2 ;
m
X
i=1
21
xi = 1}
(2.3)
Note that the inequalities of Q′ may also be numbered from 1 to m + n in a similar
manner as in Q. For a game G(α), the polyhedron Q(α) may be obtained by replacing λ
Pm
P
′
′
with m
i=1 αi xi −
i=1 αi xi in Q . In other words, Q(α) is the projection of Q ∩{(x, λ, π2 ) |
λ = 0} on the (x, π2 )-space. Let N = {(p, q) ∈ P × Q′ | L(p) ∪ L(q) = {1, . . . , m + n}}
be the set of fully-labeled pairs in P × Q′ . Let Ψ : EΓ → N be such that,
Ψ(α, x, y) = ((y, π1), (x, λ, π1 )) where λ =
m
X
αi xi , π1 = xT Ay, π2 = xT (C + α · β T )y
i=1
Lemma 2.2 The map Ψ is well defined and surjective.
Proof : For a point (α, x, y) ∈ EΓ , the corresponding Ψ(α, x, y) is fully-labeled (clear
from Lemma 2.1) and hence lies in N .
Let (p, q) ∈ N be a fully-labeled pair with p = (y, π1) and q = (x, λ, π2 ). Let α ∈ Rm
P
be such that m
i=1 αi xi − λ = 0, then clearly (p, (x, π2 )) ∈ P (α) × Q(α) is a fully-labeled
pair. Therefore, (α, x, y) ∈ EΓ and (p, q) = Ψ(α, x, y).
⊓
⊔
We further strengthen the connection between EΓ and N with the following lemma.
Lemma 2.3 EΓ is connected iff N is a single connected component.
Proof : (⇒) Since Ψ is a continuous surjective function from EΓ to N (Lemma 2.2), if
EΓ is connected then N is connected as well.
(⇐) For a (p, q) ∈ N , where q = (x, λ, π2 ), all the points in Ψ−1 (p, q) satisfy
Pm
i=1
xi αi = λ, hence Ψ−1 (p, q) is homeomorphic to Rm−1 . Since N is connected, Ψ
is continuous and the fact that the fibers Ψ−1 (p, q), ∀(p, q) ∈ N are connected imply that
⊓
⊔
EΓ is connected.
Lemmas 2.2 and 2.3 imply that EΓ and N are closely related. Henceforth, we assume
that the polyhedra P and Q′ are non-degenerate. Recall that when the best response
polyhedra (P and Q) of a game are non-degenerate, all the fully-labeled pairs are vertex
pairs. However Q′ has one more variable λ than Q, which gives one extra degree of
freedom to form the fully-labeled pairs. We show that the structure of N is very simple
by proving the following proposition.
Proposition 2.4 The set of fully-labeled points N admits the following decomposition
into mutually disjoint connected components: N = P ∪ C1 ∪ · · · ∪ Ck , k ≥ 0, where P and
Ci s respectively form a path and cycles on 1-skeleton of P × Q′ .
22
The proof of Proposition 2.4 is based on simple facts about non-degenerate polyhedra
(stated in Section 2.2) and games with a dominant pure-strategy.
Since P and Q′ are non-degenerate and have no common variable, P × Q′ is also
non-degenerate. Polyhedron P ×Q′ is defined in m+n+3 variables (dimensions), however
there are two linearly independent equalities. Therefore, essentially P × Q′ is m + n + 1dimensional polyhedron. From the discussion in Section 2.1 it is clear that every vertex
of P × Q′ is formed by a pair of vertices of P and Q′ , i.e. v = (p, q) ∈ P × Q′ , where p
and q are vertices in P and Q′ respectively. Every edge is formed by a (vertex,edge) or
an (edge,vertex) pair, i.e. v, v ′ = (p, q, q ′) ∈ P × Q′ , where p is a vertex of P and q, q ′ is
an edge in Q′ . We represent an edge corresponding to (edge,vertex) pair by (p, p′ , q) in
general, where p, p′ is an edge of P and q is a vertex of Q′ . Edges of type (vertex, edge)
and (edge,vertex) are essentially edges of Q′ and P respectively.
Since N is a set of fully-labeled point, there are at least m + n inequalities tight
at every point of N . Hence, N is a subset of 1-skeleton of K. Further, at a vertex v of
N , there exists a label for which both the inequalities are tight (by pigeonhole principle).
Such a label is called the duplicate-label of v. Suppose i ≤ m is a duplicate-label of
v = (p, q), then xi = 0 and Ai y = π1 at v. The edge we get by relaxing xi = 0 at v is
of type (p, q, q ′), which is also in N since on this edge Ai y = π1 still holds. Similarly, by
relaxing Ai y = π1 at v we get an edge of type (p, p′ , q) in N .
From the above discussion we conclude that N is on 1-skeleton of P × Q′ . A vertex
of N has a duplicate-label, and by relaxing the corresponding inequalities (one at a time),
we get its two adjacent edges in N . In other words degree of every vertex is two in N .
This implies that, N is a set of paths, each with unbounded edges on both the ends and a
set of cycles. An edge is said to be unbounded if it has only one bounding vertex. Further,
edges alternate between type (vertex,edge) and (edge,vertex) in a component of N .
Next we show that there is exactly one path in N . To do this, we prove that there
are exactly two unbounded edges in N . First we prove this for edges of type (vertex,edge)
(of Q′ ), and later show that there are no unbounded edges of type (edge,vertex) (of P ).
Let js = arg minj βj , je = arg maxj βj , is = arg maxi Aji s and ie = arg max Aji e . Here, is is
the best response to js and ie is the best response to je . Let ps = (y s , π1s ) ∈ P be a such
that π1s = Ajiss , yjss = 1 and rest all are zero, and pe = (y e , π1e ) ∈ P be such that π1e = Ajiee ,
yjee = 1. It is easy to show that ps and pe are vertices.
23
For a player, a strategy is said to be dominant if she strictly prefers it over any other
strategy regardless of what the other player plays (i.e. ci for player i in Example 1.1). In
a (non-degenerate) bimatrix game if a player has a dominant pure-strategy then the game
has exactly one Nash equilibrium. We show the next lemma using the fact that there are
games in Γ with js or je being dominant strategy of the column player.
Lemma 2.5 There are exactly two unbounded edges of type (vertex,edge) in N , where
the vertex is either ps or pe .
Proof : Let qs = (xs , λs , π2s ) ∈ Q′ be such that xsis = 1. Now for qs to make fully-labeled
pair with ps we need Cijss + λs βjs = π2s and Cijs + λs βj ≤ π2s , ∀j 6= js . This implies that
λs = minj6=js (Cijss − Cijs )/(βj − βjs ) and π2s = Cijss + λs βjs . It is easy to check that qs is
a vertex of Q′ , and there is an unbounded edge adjacent to it with direction vector d,
having all zeros for x coordinates, −1 for λ and −βjs for π2 . Let this edge be denoted by
qs , . Clearly, (ps , qs , ) is in N , where λ varies from λs to −∞.
Similarly, let qe = (xe , λe , π2e ) ∈ Q′ be such that xeie = 1, λe = maxj6=je (Cije −
Cijee )/(βje − βj ) and π2e = Cijee + λe βje . The direction vector with 1 for λ, βje for π2 and
zero for the rest gives an unbounded edge qe , of Q′ . Further, (pe , qe , ) is in N , where λ
varies from λe to ∞.
Next, we construct a game with js being the dominant strategy of column player. For
this the game has to satisfy Cjis + αi βjs > Cji + αi βj , ∀i, ∀j 6= js . Let λ− = mini,j (Cijs −
Cij )/(βj − βjs ) and consider any G(γ s ) be s.t. γis < λ− , ∀i. Clearly, G(γ s ) satisfies
these conditions and has only one NE, where the row-player plays is and the column
P s
player plays js . Since,
γi xi < λ− ≤ λs , the NE of G(γ s ) is covered by edge (ps , qs , ).
Similarly, a game can be constructed with je being the dominant strategy. Let λ+ =
maxi,j (Cij − Cije )/(βje − βj ). A game G(γ e ) with γie > λ+ , ∀i has only one NE where the
row-player plays ie and the column player plays je , and it is covered by edge (pe , qe , ).
Now, if there is another unbounded edge (p, q, ) ∈ N , then λ has to be constant on it,
P
or else it contradicts the above games having only one NE. Further, x ≥ 0 and i xi = 1
P
forces x to be constant on it. Since some yj > 0 (as j yj = 1) at p, xT C j + λβj = π2
should hold. This forces π2 to be constant. A contradiction.
Now we are in a position to prove Proposition 2.4.
24
⊓
⊔
Proof of Proposition 2.4:
P
For i xi = 1 to hold at any (p, q) ∈ N , xi > 0 for some i and in turn Ai y = π1 . Recall
P
that, j yj = 1 and y ≥ 0 in P . Therefore, if (p, , q) is an unbounded edge of N , then
y is constant on it, and in turn π1 is also constant. A contradiction. This together with
Lemma 2.5 and the above discussion proves the proposition.
From Proposition 2.4, it is clear that N contains at least the path P. We show the
importance of P in the next two lemmas.
Lemma 2.6 For every a ∈ R, there exists a point ((y, π1), (x, λ, π2 )) ∈ P such that
λ = a.
Proof : Since P is a continuous path in P ×Q′ (Proposition 2.4), λ changes continuously
on P. Moreover, in the proof of Lemma 2.5, we saw that on the edge (ps , qs , ) ∈ P, λ
varies from −∞ to λs and on the edge (pe , qe , ) ∈ P it varies from λe to ∞. Therefore for
any a ∈ R, there is a point ((y, π1), (x, λ, π2 )) in P such that λ = a.
Consider a game G(α) ∈ Γ, and the corresponding hyper-plane H : λ−
⊓
⊔
Pm
i=1
αi xi = 0.
Note that, every point in N ∩ H corresponds to a NESP of the game G(α) and vice-versa.
Lemma 2.7 The path P of N covers at least one NESP of the game G(α).
Proof : If there are points in P on opposite sides of H, then the set P ∩H has to be nonempty. Let q1 = (x1 , λ1 , π21 ) and q2 = (x2 , λ2 , π22 ) in P be s.t. λ1 = mini∈S1 αi and λ2 =
P
1
maxi∈S1 αi . Note that w1 and w2 exist (Lemma 2.6) and they satisfy λ1 − m
i=1 αi xi ≤ 0
P
2
⊓
⊔
and λ2 − m
i=1 αi xi ≥ 0.
Remark 2.8 The proof of Lemma 2.7 in fact shows the existence of a Nash equilibrium
for a bimatrix game. It is also easy to deduce that the number of Nash equilibria of a
non-degenerate bimatrix game is odd from the fact that N contains a set of cycles and a
path (Proposition 2.4), simply because a cycle must intersect the hyper-plane H an even
number of times, and the path must intersect H an odd number of times.
Every vertex of N has a duplicate label and the two edges incident on a vertex may
be easily obtained by relaxing the inequality corresponding to the duplicate label in P
and in Q′ respectively. Therefore, given a point of some component of N , it is easy to
25
trace the full component by leaving the duplicate label in P and Q′ alternately at every
′
vertex. Let N P and N Q be projections of N on P and Q′ respectively. Clearly, such a
projection of the path P gives rise to a path in Q′ with unbounded edges qs , and qe , on
each end and a bounded path in P , and a cycle gives rise to a cycle in both.
For a p ∈ P , let N (p, ∗) be the set of points of Q′ making fully labeled pairs with p,
and similarly for a q ∈ Q′ define N (∗, q) as a set of points of P making fully labeled pairs
with q. Clearly, N (p, ∗), if non-empty, forms an edge of Q′ if p is a vertex, and forms
a vertex of Q′ if p is on an edge. Having defined these notations we refer the reader to
Appendix B For a more detailed structural description of EΓ . From Lemma 2.3, it is clear
that if N is disconnected, then EΓ is also disconnected. Example 2.9 shows that EΓ may
be disconnected in general by illustrating a disconnected N (i.e., N with a cycle).
Example 2.9 Consider the

0


A= 6

9
following A, C and


6
9 9




6 5 , C =  5


4
7 2
β.
8 6


9




,
β
=

 7
8 8


8
3 0





The set N of the corresponding game space Γ contains a path P and a cycle C1 . Here
we demonstrate the path P P and the cycle C1P of N P , and using them P and C1 of N
may be easily obtained. The path P P is ps , p1 , p1 , pe , where ps = ((0, 1, 0), 9), p1 =
((0.18, 0.82, 0), 7.36) and pe = ((1, 0, 0), 9). The cycle C1P is p2 , p3 , p3 , p4 , p4 , p2 , where
p2 = ((0.5, 0, 0.5), 5.5), p3 = ((0.38, 0.18, 0.44), 5.56) and p4 = ((0.4, 0, 0.6), 5.4).
Note that ps and pe correspond to the minimum and maximum coordinates of β respectively
(Lemma 2.5).
Since Γ (Rm ) is connected, EΓ can not be homeomorphic to Γ if it is disconnected.
Therefore the answer to our question, Is EΓ homeomorphic to Γ?, is “no” in general. In
Section 2.4 we show that it is “yes” if C = −A (i.e., rank-1 space). The results of next
section are not necessary to derive the final homeomorphism or algorithmic results, and
hence may be skipped if so wish. However, they are interesting from classical view point.
2.3.1
Orientation of N and the Index Theorem
In this section we show that the edges of N may be given canonical orientation, such that
every component gets an orientation. Consider a vertex u = (p, q) ∈ N , where p = (y, π1 ),
26
and q = (x, λ, π2 ). Let X = {i ∈ S1 | Ai y = π1 }, and Y = {j ∈ S2 | xT C j + βj λ = π2 }
be ordered sets. Note that X = L(p) ∩ S1 and Y = L(q) ∩ S2 . Let X̄ = S1 \ X, and
Ȳ = S2 \ Y be the complements. The duplicate label, say l, of vertex u is either in X
or in Y . Let l ∈ X, i.e., Al y = π1 and xl = 0 hold at u. Let −Ik be the negative of the
Y
k × k identity matrix, AX = [aij ]i∈X,j∈Y be the submatrix of A and similarly βY be the
subvector. The set of tight inequalities at p and q may be written as follows:

 


11×n
0
1
y

 Y  


 


Y
Ȳ
 AX
AX −1|X|×1   yȲ  =  0|X|×1 

 


0|Ȳ |×|Y | −I|Ȳ | 0|Ȳ |×1
0|Ȳ |×1
π1


0
βY
0
01×|X̄|




Y
i
CX
0|X|×|X̄|  h



λ xX xX̄ π2  1m×1
−el
 = 1 01×|Y | 0 01×|X̄|
Y

CX̄
−I|X̄| 


0
−11×|Y | 0
01×|X̄|
(2.4)
(2.5)
In the above expression −el is a negative unit vector of size m, with −1 in the
position corresponding to xl . Let the matrices of (2.4) and (2.5) be denoted by E(p)
and E(q) respectively. For the case l ∈ Y , E(p) and E(q) may be analogously defined.
It is easy
 to see that the
 coefficient matrix of tight equations at u may be written as
E(p)
0
. Using det(E(u)) = det(E(p)) ∗ det(E(q)), we define the sign
E(u) = 
T
0
E(q)
of vertex u as follows:
s(u) = sign(det(E(u)))
Since E(p) and E(q) are well-defined, s is a well-defined function. Using the function
s on the vertices of N , next we give direction to the edges of N .
−−→ ←−−
Lemma 2.10 Let E be the set of edges of N , and E ′ = {u, u′, u, u′ | u, u′ ∈ N } be the
set of directed edges. There exists a (efficiently computable) function →: E → E ′ such
that it maps a cycle of N to a directed cycle and the path P gets oriented from (ps , qs ) to
(pe , qe ).
Proof : Define the function → as follows: Let u = (p, q) be a vertex of N and let
up = (p′ , q) and uq = (p, q ′ ) be its adjacent vertices obtained by relaxing the inequalities
corresponding to its duplicate label in P and in Q′ respectively. If s(u) = +1 then →
27
(u, up ) = −
u,−→
up and → (u, uq ) = ←
u,−u−q , otherwise → (u, up) = ←
u,−u−p and → (u, uq ) = −
u,−→
uq .
In other words, if s(u) = +1 then direct the edges (p′ , p, q) and (p, q, q ′) away from u and
towards u respectively, otherwise give opposite directions.
Note that, → (u, u′) is polynomially computable using any of the s(u) and s(u′ ).
Further, in order to prove the consistency of function →, we need to show that u and
u′ have opposite signs. Next we show that, if u and u′ are adjacent vertices of N , then
s(u) ∗ s(u′ ) = −1.
Let u = (p, q). The proof may be easily deduced from the following facts:
• In a polyhedron, the coefficient matrix of tight inequalities for the adjacent vertices
have determinants of opposite signs if they are the same except for the row which
has been exchanged [59].
• Vertices u and u′ are fully-labeled and both have a duplicate label. Further, to obtain
u′ from u, the inequality corresponding to its duplicate label should be relaxed in
P or Q′ .
• Reordering of the elements in set X (Y ) does not change det(E(u)), since it enforces
a reordering of the corresponding rows (columns) in both E(p) and E(q).
• Reordering of the elements in set X̄ does not change det(E(u)), since in E(w), the
columns corresponding to the equations of type xi = 0 (except the one with the
duplicate label) should be written such that they form −I|X̄| . Similarly, reordering
of the elements in set Ȳ does not change det(E(u)).
Clearly, the function → maps a cycle of N to a directed cycle. Therefore, we get the
directed traversal of a component of N by leaving the duplicate label in P if the current
vertex has positive sign otherwise leaving the duplicate label in Q′ . Further, it is easy to
check that the sign associated with the vertex of the extreme edge (ps , qs , ) is positive.
Therefore, the path P gets oriented from (ps , qs , ) to (pe , qe , ).
⊓
⊔
The above lemma may be seen as an implication of [59] or [63], however that may
require involved mapping. This is because, the polyhedra (P and Q′ ) we consider are
different from that of [59] and the role of λ as an extra variable does not directly match
the extra variable in Lemke’s algorithm for the linear complementarity problem [63].
28
The direction of the edges of N , defined by the function →, may be used to determine
the index of every Nash equilibrium for a game in Γ. The definition of index requires the
game to be non-negative, i.e., A > 0, B > 0 [59]. Note that if a game is not non-negative,
then it may be modified to an equivalent non-negative game by adding a positive constant
to its payoff matrices. Let (x, y) be a NESP of a non-degenerate non-negative bimatrix
game (A, B). Let I = {i ∈ S2 | xi > 0} and J = {j ∈ S1 | yj > 0}. Then the index of
(x, y) is defined as [59]1
J
J
(−1)|I| sign(det(AI ) ∗ det(BI ))
Let α ∈ Γ be a non-degenerate non-negative game and let H : λ −
and H + be the corresponding hyper-plane and half-spaces.
Pm
i=1
αi xi , H −
Proposition 2.11 Let u, u′ ∈ N intersect H at a NESP (x, y) of G(α), and let →
−−→
(u, u′) = u, u′. If u ∈ H − and u′ ∈ H + then the index of (x, y) is −1, otherwise it is +1.
Proof : Suppose, edge u, u′ intersects H. Since, the coordinates of y and π1 are zero in
H, u, u′ has to be of type (vertex,edge). Let u, u′ = (p, q, q ′ ). Clearly, s(u) = −1 since
−−→
→ (u, u′) = u, u′. Let the ordered sets X, Y and their complements be as defined above.
Let l be the duplicate label of u. Clearly, either l ∈ Y or l ∈ X.
Suppose l ∈ Y . Let d be the direction obtained by relaxing the inequality l at u in
Q′ , which leads to the vertex u′ . The dot product of d with the normal vector of H may be
obtained by replacing the column corresponding to xT C l + βl λ − π2 = 0 with the normal
vector in E(q). If u ∈ H − and u′ ∈ H + then this dot product is positive, otherwise it is
negative. Next we show that the expression of the dot product and the expression of the
index of (x, y) are the same except that they have opposite signs.
Let the payoff matrix of the column player in G(α) be denoted by B, i.e., B =
C + α · β T and let a = xT Ay and b = xT By. Since A > 0 and B > 0, a and b are
positive. Clearly, the sets I and J associated with the NESP (x, y) are such that I = X
and J = Y \ l. Let k = |I| = |J|. We reorder the elements in set Y such that Y = [J l].
Note that this does not change the sign of det(E(u)), since it forces the similar reordering
1
In [59], the index is defined as determinant of a matrix (on pg 10), which simplifies to the given
expression.
29
of the columns of both E(p) and E(q). The expression for the dot product is:

11×n
0

l
Ȳ
 J
 AX AX AX −1|X|×1
−1

∗ det 
det(E(u))
 01×k −1 01×|Ȳ |
0

0|Ȳ |×(k+1) −I|Ȳ | 0|Ȳ |×1


0
βJ
1
01×|X̄|




J
CX
−αX 0|X|×|X̄|


 ∗ det  1m×1


J


CX̄
−αX̄ −I|X̄|


0
−11×|J |
0
01×|X̄|








Since I = X and |I| = |J| = k, we get,


0
βJ
1


11×k
0
(−1)1+(n−k)(2k+4)


J
 ∗ (−1)(m−k)(2k+6) ∗ det  1
∗ det  J

C
−α
k×1
I
I
det(E(u))


AI −1k×1
0 −11×k
0





0
01×k
1



11×k
0
−1


J
T


=
∗ det  1k×1 (C + α · β )I −αI 
∗ det
J


det(E(u))
AI −1k×1
0
−11×k
0
J
J
Since B = C + α · β T , AI yJ = a ∗ 1|I|×1 and xTI BI = b ∗ 11×|J | , we get,

1
a


J

11×k
1
BI
−1
 ∗ (−1)k+3  k×1

∗ det  J
det(E(u))
1
AI 0k×1
01×k
b
k+2(k+2)
(−1)k
(−1)
J
J
J
J
∗ det(AI ) ∗ det(BI ) =
∗ det(AI ) ∗ det(BI )
=
det(E(u)) ∗ a ∗ b
det(E(u)) ∗ a ∗ b
When the duplicate label l is in X, we may derive the same expression for the dot
product by similar reductions. Since s(u) = sign(det(E(u))) = −1, a > 0 and b > 0, the
sign of the above expression is same as (−1)∗ index of (x, y).
⊓
⊔
From Proposition 2.11, it is easy to see that in a component, the index of the Nash
equilibria alternates2 . Further, both the first and the last Nash equilibria, on the path
P, have index −1. This proves that the number of Nash equilibria with index −1 is one
more than the number of Nash equilibria with index +1, which is an important known
result [59, 56].
2
The two endpoints of a LH path also have opposite index [59].
30
2.4
Rank-1 Space: A Homeomorphism
From the discussion of the last section, we know that Γ and EΓ are not homeomorphic in
general (illustrated by Example 2.9). Surprisingly, they turn out to be homeomorphic if
Γ consists of only rank-1 games, i.e., C = −A. Recall that EΓ forms a single connected
component iff N has only one component (Lemma 2.3). First we show that when C = −A,
the set N consists of only a path.
For a given matrix A ∈ Rmn and a vector β ∈ Rn , we fix the game space to Γ =
{(A, −A+α·β T ) | α ∈ Rm }. We assume that A and β are non-zero and the corresponding
polyhedra P and Q′ are non-degenerate. Lemma 2.12 shows that the set N may be easily
identified on the polyhedron P × Q′ .
Lemma 2.12 For all (p, q) = ((y, π1 ), (x, λ, π2 )) in P ×Q′ , we have λ(β T ·y)−π1 −π2 ≤ 0,
and the equality holds iff (p, q) ∈ N .
Proof : Recall that C = −A, hence from (2.2) and (2.3), we get xT · (A · y − π1 ) ≤ 0
and (xT · (−A) + β T λ − π2 ) · y ≤ 0. By summing up these two inequalities, we get
λ(β T · y) − π1 − π2 ≤ 0. If (p, q) ∈ N , then ∀i ≤ m, xi > 0 ⇒ Ai · y − π1 = 0 and
∀j ≤ n, yj > 0 ⇒ xT (−Aj ) + βj λ − π2 = 0, hence λ(β T · y) − π1 − π2 = 0.
If (p, q) ∈
/ N , then at least one label 1 ≤ r ≤ m + n is missing from L(p) ∪ L(q). Let
r ≤ m (wlog), then xr > 0 and Ar · y − π1 < 0, which imply that xT · (A · y − π1 ) < 0.
Therefore, λ(β T · y) − π1 − π2 < 0.
⊓
⊔
Motivated by the above lemma, we define the following parametrized linear program.
LP (δ) : max δ(β T · y) − π1 − π2
s.t. (y, π1 ) ∈ P ;
′
(x, λ, π2 ) ∈ Q ;
(2.6)
λ=δ
Note that the above linear program may be broken into a parametrized primal linear
program and its dual, with δ being the parameter. The primal may be defined on polyhedron P with the cost function maximize: δ(β T · y) − π1 and its dual is on polyhedron
Q′ with additional constraint λ = δ and the cost function minimize: π2 .
Remark 2.13 LP (δ) may look similar to the parametrized linear program, say T LP (ξ),
by Theobald [62]. However the key difference is that T LP (ξ) is defined on the best response
polyhedra of a given game (i.e., P (α) × Q(α) for game G(α)), while LP (δ) is defined on
a bigger polyhedron (P × Q′ ) encompassing best response polyhedra of all the games in Γ.
31
Let OP T (δ) be the set of optimal points of LP (δ). In the next lemma, we show that
∀δ ∈ R, OP T (δ) is exactly the set of points in N , where λ = δ.
Lemma 2.14 ∀a ∈ R, OP T (a) = {((y, π1 ), (x, λ, π2 )) ∈ N | λ = a} and OP T (a) 6= ∅.
Proof : Clearly the feasible set of LP (a) consists of all the points of P ×Q′ , where λ = a.
Therefore the set {((y, π1), (x, λ, π2 )) ∈ N | λ = a} is a subset of the feasible set of LP (a).
The set {((y, π1 ), (x, λ, π2 )) ∈ N | λ = a} is non-empty (Lemma 2.6). From Lemma 2.12,
it is clear that the maximum possible value, the cost function of LP (a) may achieve is 0,
and it is achieved only at the points of N . Therefore, OP T (a) = {((y, π1 ), (x, λ, π2 )) ∈
N | λ = a} and OP T (a) 6= ∅.
⊓
⊔
Next we show that N in fact consists of only one component.
Proposition 2.15 N does not contain cycles.
Proof : From Proposition 2.4, it is clear that there is always a path P in the set N .
Moreover, for every a ∈ R, there exists a point ((y, π1), (x, λ, π2 )) ∈ P with λ = a (Lemma
2.6). Further, OP T (a) is connected, since it is the solution set of LP (a). Therefore,
∀a ∈ R, OP T (a) is contained in the path P. Therefore N consists of only the path.
⊓
⊔
From Proposition 2.15, it is clear that N consists of only the path P, henceforth we
refer to N as a path. To construct homeomorphism maps between EΓ and Γ, we need
to encode a point (α, x, y) ∈ EΓ (of size 2m + n) into a vector α′ ∈ Γ (of size m), such
that α′ uniquely identifies the point (α, x, y) (i.e., a bijection). First we show that there
is a bijection between N and R and using this, we derive a bijection between Γ and EΓ .
Consider the function g : N → R such that
g((y, π1), (x, λ, π2 )) = β T · y + λ
(2.7)
Lemma 2.16 Each term of g, namely β T · y and λ, monotonically increases on the directed path N , and the function g strictly increases on it.
Proof : From the proof of Proposition 2.4, we know that the edges of type (vertex,edge)
and of type (edge,vertex) alternate in N . Clearly β T · y is a constant on an edge of type
(vertex,edge) and λ is a constant on an edge of type (edge,vertex). Now, consider the
32
two consecutive edges (p′ , p, q) and (p, q, q ′) with (p, q) being their common vertex. It is
enough to show that λ and β T · y are not constants on (p, q, q ′ and (p′ , p, q) respectively,
and β T · y increases from (p′ , q) to (p, q) (i.e., on (p′ , p, q)) iff λ also increases from (p, q)
to (p, q ′ ) (i.e., on (p, q, q ′ )).
Let q = (x, γ, π2 ), q ′ = (x′ , γ ′ , π2′ ), p = (y, π1) and p′ = (y ′ , π1′ ). Clearly, OP T (γ) =
(p′ , p, q) and (p, p′ ) ∈ OP T (γ ′) (Lemma 2.14). Further γ 6= γ ′ , since OP T (γ) contains
only one edge.
Claim 2.17 β T · y ′ 6= β T · y, and β T · y ′ < β T · y ⇔ γ < γ ′ .
Proof : Since the feasible set of LP (γ ′ ) contains all the points of P × Q′ with λ = γ ′ ,
the point (p′ , q ′ ) is a feasible point of LP (γ ′ ). Note that (p′ , q ′ ) is a suboptimal point of
LP (γ ′ ) otherwise N (∗, q ′) = p, p′ and N (p′ , ∗) = q ′ , q, which creates a cycle in N . Further,
(p, q ′) ∈ OP T (γ ′), hence γ ′ (β T · y) − π1 − π2′ > γ ′ (β T · y ′ ) − π1′ − π2′ . Since both (p′ , q) and
(p, q) are in OP T (γ), we get γ(β T · y ′) − π1′ − π2 = γ(β T · y) − π1 − π2 . Summing up these
two, we get γ(β T · y ′) + γ ′ (β T · y) > γ(β T · y) + γ ′(β T · y ′) ⇒ (β T · y − β T · y ′)(γ ′ − γ) > 0. ⊓
⊔
The above claim shows that β T · y is strictly monotonic on (p′ , p, q) and λ is strictly
monotonic on (p, q, q ′). Further, if β T · y increases on (p′ , p, q) from (p′ , q) to (p, q) then λ
increases on (p, q, q ′) from (p, q) to (p, q ′ ) and vice-versa.
Recall that on the directed path N , (ps , qs , ) is the first edge and (pe , qe , ) is the last
edge (Lemma 2.10). Further, λ varies from −∞ to λs on the first edge (ps , qs , ), and it
varies from λe to ∞ on the last edge (pe , qe , ) (proof of Lemma 2.5). Therefore, λ and
β T · y increase monotonically on the directed path N , and in turn g strictly increases from
−∞ to ∞ on the path.
⊓
⊔
Lemma 2.16 implies that g is a continuous, bijective function with a continuous
inverse g −1 : R → N . Now consider the following candidate function f : EΓ → Γ for the
homeomorphism map.
f (α, x, y) = (β T · y + αT · x, α2 − α1 , . . . , αm − α1 )T
(2.8)
Using the properties of g, next we show that f indeed establishes a homeomorphism
between Γ and EΓ . This settles the first part of Theorem 1.2 mentioned in Chapter 1.
Theorem 2.18 EΓ is homeomorphic to Γ.
33
Proof : The function f of (2.8) is continuous because it is a quadratic function. To show
the homeomorphism we need to show that it has a continuous inverse. Define function
f −1 : Γ → EΓ using g of (2.7) as follows. Given α′ ∈ Γ, let (p, q) = ((y, π1 ), (x, λ, π2 )) =
g −1 (α1′ ) be the corresponding point in N . This gives the values of x, y and λ. Using these
values, we solve the following equalities with the variable vector a = (a1 , . . . , am ).
m
X
xi ai = α1′ − β T · y;
∀i > 1, ai − a1 = αi′ ;
i=1
It is easy to see that the above equations have a unique solution, which gives a unique
value for the vector a and a unique point (a, x, y) ∈ EΓ . Let, f −1 (α′ ) = (a, x, y), then
clearly f −1 ◦ f and f ◦ f −1 are identity maps.
Function f −1 is continuous as well. Hence, f and f −1 establishes homeomorphism
⊓
⊔
between Γ and EΓ .
2.5
Algorithms
In this section, we present two algorithms to find Nash equilibria of a rank-1 game using
the structure and monotonicity of N . First we discuss a polynomial time algorithm to find
a Nash equilibrium of a non-degenerate rank-1 game and later extend it for degenerate
games. It does a binary search on N using the monotonicity of λ. Later we give a pathfollowing algorithm which enumerates all Nash equilibria of a rank-1 game, and finds at
least one for any bimatrix game (Lemma 2.7).
Recall that only vertex pairs of the best response polyhedra P and Q (of (2.2)) form
Nash equilibria in non-degenerate game and hence the Nash equilibrium set is finite in
such games. Consider a non-degenerate rank-1 bimatrix game (A, B) ∈ R2mn such that
A + B = γ · β T , where γ ∈ Rm and β ∈ Rn . We assume that β is a non-zero and
non-constant3 vector, and both A and B are rational matrices. Let c be the LCM of the
denominators of the aij s, βi s and γi s. Note that multiplying both A and B by c2 makes
A, γ and β integers, and the total bit length of the input gets multiplied by at most
O(m2 n2 ), which is a polynomial increase. Since scaling both the matrices of a bimatrix
game by a positive integer does not change the set of Nash equilibria, we assume that
entries of A, γ and β are integers.
3
If β is a constant vector, then the game (A, B) may be converted into a zero-sum game without
changing its Nash equilibrium set, by adding constants in the columns and rows of A and B respectively.
34
Now consider the game space Γ = {(A, −A + α · β T ) | α ∈ Rm }. Clearly, G(γ) =
(A, B) ∈ Γ and the corresponding polyhedra P and Q′ of (2.3) are non-degenerate. Let
N be the set of fully-labeled points of P × Q′ as defined in Section 2.3. By Lemma 2.2,
we know that for every Nash equilibrium of the game G(γ), there is a unique point in N .
P
Consider the hyper-plane H : λ − m
i=1 γi xi = 0 in (y, π1 , x, λ, π2 )-space and the
P
P
m
corresponding half spaces H + : λ − i=1 γi xi ≥ 0 and H − : λ − m
i=1 γi xi ≤ 0. It is easy
to see that the intersection of N with the hyper-plane H gives all the Nash equilibria of
G(γ). If the hyper-plane H intersects an edge of N , then it intersects the edge exactly at
one point, because G(γ) is a non-degenerate game.
Let γmin = mini∈S1 γi and γmax = maxi∈S1 γi . Since ∀x ∈ ∆1 and in turn γmin ≤
Pm
i=1
γi xi ≤ γmax , for the points of N corresponding to Nash equilibria of G(γ), the value
of λ is between γmin and γmax . From Proposition 2.15, we know that N contains only a
path. If we consider this path from the first edge (ps , qs , ) to the last edge (pe , qe , ), then
λ monotonically increases from −∞ to ∞ on it (Lemmas 2.16). Therefore all the points,
corresponding to the Nash equilibrium of G(γ) on the path N , lie between OP T (γmin)
and OP T (γmax ) (Lemma 2.14).
2.5.1
Rank-1 NE: A Polynomial Time Algorithm
Recall that finding a Nash equilibrium of the game G(γ) is equivalent to finding a point
in the intersection of N and the hyper-plane H. Since λ increases monotonically on N
and all the points in the intersection are between λ = γmin and λ = γmax , the BinSearch
algorithm of Table 2.1 applies binary search on N between λ = γmin and λ = γmax to
locate a point in the intersection.
The IsNE procedure of Table 2.1 takes a δ ∈ R as the input, and outputs a NESP if
possible, otherwise it indicates the position of OP T (δ) with respect to the hyper-plane H.
First it finds the optimal set OP T (δ) of LP (δ) and the corresponding edge u, v containing
OP T (δ). Next, it finds a set H, which consists of all the points in the intersection of u, v
and the hyper-plane H if any, i.e., Nash equilibria of G(γ). Since the game G(γ) is nondegenerate, H is either a singleton or empty. In the former case, the procedure outputs
H and returns 0 indicating that a Nash equilibrium has been found. However in the
latter case, it returns 1 if u, v ∈ H + and returns −1 otherwise, indicating the position of
OP T (δ) w.r.t. the hyper-plane H.
35
BinSearch(γmin , γmax )
a1 ← γmin ; a2 ← γmax ;
if IsNE(a1 ) = 0 or IsNE(a2 ) = 0
IsNE(δ)
then return;
Find OP T (δ) by solving LP (δ);
u, v ← The edge containing OP T (δ);
while true
a←
a1 +a2
;
2
flag ← IsNE(a);
H ← {w ∈ u, v | w ∈ H};
if flag = 0 then break;
if H =
6 ∅ then Output H; return 0;
else if flag < 0 then a1 ← a;
else if u, v ∈ H + then return 1;
else a2 ← a;
else return −1;
endwhile
return;
Table 2.1: BinSearch Algorithm
The BinSearch algorithm maintains two pivot values a1 and a2 of λ such that the
corresponding OP T (a1) ∈ H − and OP T (a2) ∈ H + , i.e., always on the opposite sides of
the hyper-plane H. Clearly N crosses H at least once between OP T (a1) and OP T (a2).
Since OP T (γmin ) ∈ H − and OP T (γmax ) ∈ H + , the pivots a1 and a2 are initialized to γmin
and γmax respectively. Initially it calls IsNE for both a1 and a2 separately and terminates
if either returns zero indicating that a NESP has been found. Otherwise the algorithm
repeats the following steps until IsNE returns zero: It calls IsNE for the mid point a
of a1 and a2 and terminates if it returns zero. If IsNE returns a negative value, then
OP T (a) ∈ H − implying that OP T (a) and OP T (a2) are on the opposite sides of H, and
hence the lower pivot a1 is reset to a. In the other case OP T (a) ∈ H + , the upper pivot
a2 is set to a, as OP T (a1) and OP T (a) are on the opposite sides of H.
Note that, the index of the Nash equilibrium obtained by BinSearch algorithm is
always −1, since a1 < a2 is an invariant (Proposition 2.11). For X ∈ Rkl , let X̃ =
max |xij |. Since the column-player’s payoff matrix is represented by −A + γ · β T of the
i,j
game G(γ), let |B| = max{Ã, β̃, γ̃}. Let ∆ = (m + 2)! (|B|)(m+2) .
Theorem 2.19 Let L be the bit length of the input. The BinSearch algorithm takes
poly(L, m, n) time.
Proof : Clearly, the algorithm terminates when the call IsNE(a) outputs a NESP of
36
G(γ). Let the range of λ for an edge (p, q1 , q2 ) ∈ N be [λ1 λ2 ].
Claim 2.20 λ2 − λ1 ≥
1
.
∆2
Proof : The λ1 and λ2 correspond to the vertices q1 and q2 of Q′ . Since Q′ is in a
(m + 2)-dimensional space, there are m + 2 equations tight at every vertex. Hence both
λ1 and λ2 are rational numbers with denominator at most ∆ and λ2 −λ1 is at least
When a2 − a1 ≤
1
,
∆2
1
.
∆2
⊓
⊔
OP T (a1) and OP T (a2) are either part of the same edge or
adjacent edges. In either case, the algorithm terminates after one more call to IsNE.
γmax − γmin
Clearly a2 − a1 =
after l iterations of the while loop. Let k be such that
2l
1
γmax − γmin
= 2 ⇒ k = 2 log ∆ + log(γmax − γmin )
k
2
∆
≤ O(m log m + m log |B| + log(γmax − γmin ))
BinSearch makes at most ⌈k⌉ calls to the procedure IsNE, which is poly(L, m, n).
The procedure IsNE solves a linear program and computes a set H, both may be done in
poly(L, m, n) time. Therefore the total time taken by BinSearch is poly(L, m, n).
⊓
⊔
The above theorem settles the second part of Theorem 1.2.
Degeneracy. For a degenerate rank-1 game (A, B) the corresponding polyhedron P × Q′
may be degenerate as well. However, the BinSearch algorithm, with a small modification,
works with the same polynomial time bound. First we make a few observations and then
state a minor modification to the algorithm.
Note that Lemma 2.14 (i.e. ∀a ∈ R, OP T (a) = N (a)) holds in case of degeneracy
as well. Therefore, the set of fully-labeled points N ∈ P × Q′ is a connected set, and λ
′
varies continuously on this set. Recall that N Q = {q ∈ Q′ | (p, q) ∈ N for some p ∈ P }.
′
Lemma 2.21 Let f ∈ N Q be a maximal face, then λ does not take a unique value on f .
Proof : Suppose λ takes a unique value a on f . Let S = ∩q∈f N (∗, q) (note that S is
a vertex of P ). Clearly, there exists a vertex p ∈ f such that S ⊂ N (p, ∗). In that case,
{p×N (p, ∗)}∪{f ×S} ∈ OP T (a), which makes OP T (a) non-convex, a contradiction. ⊓
⊔
It is clear from the above lemma that on every alternate maximal face of N (i.e.,
similar to (p, q, q ′) edges in non-degenerate case), increase in λ is lower bounded by
1
.
∆2
It is easy to check that BinSearch algorithm with the IsNE procedure given in Table 2.2
works for degenerate cases as well.
37
IsNE(δ)
Find OP T (δ) by solving LP (δ);
f ← The minimal face of P × Q′ containing OP T (δ);
(Note that f is a maximal face of N )
H ← {w ∈ f | w ∈ H};
if H =
6 ∅ then Output H; return 0;
else if f ∈ H + then return 1;
else return −1;
Table 2.2: IsNE Procedure for the Degenerate Case
2.5.2
Enumeration Algorithm for Rank-1 Games
As all the Nash equilibria of the game G(γ) lies between OP T (γmin) and OP T (γmax ) on
N , the Enumeration algorithm of Table 2.3 simply follows the path N between these two
P
points and outputs the NESPs whenever it hits the hyper-plane H : λ − m
i=1 γi xi = 0.
Enumeration(u1 , v1 , u2 , v2 )
u, u′ ← u1 , v1 ;
if u, u′ of type (vertex,edge) then flag ← 1;
else flag ← 0;
while true
H = {w ∈ u, u′ | w ∈ H}; Output H;
if u, u′ = u2 , v2 then break;
u ← u′; i ← duplicate-label of u′ ;
if flag = 1 then u′ ← relax inequality i in P ; flag ← 0;
else u′ ← relax inequality i in Q′ ; flag ← 1;
endwhile
return;
Table 2.3: Enumeration Algorithm
Obtain OP T (γmin) and OP T (γmax ) on the path N by solving LP (γmin ) and LP (γmax )
respectively. Let the edges u1 , v1 and u2 , v2 contain OP T (γmin ) and OP T (γmax ) respectively. The call Enumeration(u1, v1 , u2 , v2 ) enumerates all the Nash equilibria of the G(γ).
38
The Enumeration algorithm initializes u, u′ to the edge u1 , v1 . Recall that a vertex
u′ = (p, q) of N has a duplicate label, say i, and we can obtain its two adjacent edges
(p, N (p, ∗)) and (N (∗, q), q) by relaxing inequality i in Q′ and P respectively. The value
of flag indicates the type of edge to be considered next, or in other words the polyhedron
in which the inequality to be relaxed to get the next vertex. It is set to one if we need to
relax inequality i in polyhedron P otherwise it is set to zero. In the while loop, it first
outputs the intersection of the edge u, u′ with the hyper-plane H, if any. After that u is
set to u′ and u′ is set to the next vertex and we get the next edge u, u′ of N . Further, since
the edges alternate between the type (vertex,edge) and (edge,vertex) on N , we need to
relax the inequality corresponding to the duplicate label in P and Q′ alternately. Hence
the flag is toggled in every iteration. The algorithm terminates when u, u′ = u2 , v2 .
Every iteration of the loop takes time polynomial in L, m and n. Therefore, the
time taken by the algorithm depends on the number of edges between u1 , v1 and u2 , v2 .
For a general (non-degenerate) bimatrix game (A, B), we may obtain C, γ and β
such that B = C + γ · β T , and define the corresponding game space Γ and the polyhedra P
and Q′ accordingly (Section 2.3). There is a one-to-one correspondence between the Nash
equilibria of the game (A, B) and the points in the intersection of the fully-labeled set N
P
and the hyper-plane λ − m
i=1 γi xi = 0. Recall that the set N contains one path (P) and
a set of cycles (Proposition 2.4). The extreme edges (ps , qs , ) and (pe , qe , ) of P may be
easily obtained as described in the proof of Lemma 2.5. Since P contains at least one Nash
equilibrium of every game in Γ (Lemma 2.7), hence the call Enumerate((ps , qs , ), (pe , qe , ))
outputs at least one Nash equilibrium of the game (A, B). Note that the time taken by
the algorithm again depends on the number of edges on the path P.
Comparison with Earlier Approaches
The Enumeration algorithm may be compared to two previous algorithms. One is the
Theobald algorithm [62], which enumerates all Nash equilibria of a rank-1 game, and the
other is the Lemke-Howson algorithm [45], which finds a Nash equilibrium of any bimatrix game. The Enumeration algorithm enumerates all the Nash equilibria of a rank-1
game and for any general bimatrix game it is guaranteed to find one Nash equilibrium.
All three algorithms are path following algorithms. However, the main difference is that
both the previous algorithms always trace a path on the best response polyhedra of a
39
given game (i.e., P (γ) × Q(γ)), while the Enumeration algorithm follows a path on a
bigger polyhedron P × Q′ which encompasses best response polyhedra of all the games
of an m-dimensional game space. Therefore, for every game in this m-dimensional game
space, the Enumeration follows the same path. Further, all the points on the path followed by Enumeration algorithm are fully-labeled, and it always hits the best response
polyhedron of the given game at one of its NESP. However the path followed by previous
two algorithms is not fully-labeled and whenever they hit a fully-labeled point, it is a NE.
In every intermediate step, the Theobald algorithm calculates the range of a variable
(ξ) based on the feasibility of primal and dual, and accordingly decides which inequality
to relax (in P or Q) to locate the next edge. While Enumeration algorithm simply leaves
the duplicate label in P or Q′ (alternately) at the current vertex to locate the next edge.
Further, for a general bimatrix game, the Enumeration algorithm locates at least one
Nash equilibrium, while Theobald algorithm works only for rank-1 games.
2.6
Discussion
Given A, C and β such that rank(A + C) = k, it is clear that the corresponding subspace
contains only rank-(k + 1) games. Does this additional information help in understanding
the structure of N ? Another interesting structural problem would be to characterize A, C
and β for which N is a single path. Note that, even when N is a single path, λ need not
be monotonic on it, and in that case the BinSearch algorithm does not guarantee a NE. It
would be interesting to know if there is an efficient way to find a NE whenever N consists
of only a path. Recall that N of a rank-1 game space (C = −A) is simply a path with λ
being monotonic on it. It would be nice to have an effective bound on the number of edges
of this path. Note that, a strongly polynomial bound will imply a strongly polynomial
algorithm for linear programs (LPs). Further, answers to any of these questions may help
devise efficient algorithm for special classes of games, such as rank-k games.
40
Chapter 3
Other Homeomorphism Results
As discussed in Chapter 1, the homeomorphism result by Kohlberg and Mertens [40] has
been utilized extensively to understand the index, degree and the stability of a Nash
equilibrium of a bimatrix game [26]. Moreover, the homeomorphism result also validates
the homotopy methods devised to compute a Nash equilibrium [27, 31]. In Chapter 2 we
derived a homeomorphism for rank-1 game space, which in turn gave us the first polynomial time algorithm to find a Nash equilibrium in rank-1 games. Here we present two
more homeomorphism results, which may lead to deeper insights and better algorithms.
The first result is an extension of rank-1 homeomorphism presented in Chapter 2, to
a fixed rank game space (Section 3.1). Using this structure, for Nash equilibrium computation in rank-k games we derive an alternative fixed point formulation over [0, 1]k space
with some monotonicity properties. The second result is a sum preserving homeomorphism (Section 3.2) and uses some ideas from the construction by Kohlberg and Mertens
[40]. However, the maps we derive are more intricate and challenging than that of [40],
because we need to encode more values in relatively less space.
In what follows we use same notations as in Chapter 2.
3.1
Rank-k Homeomorphism
It turns out that the approach used to show the homeomorphism between the subspace of rank-1 games and it’s Nash equilibrium correspondence may be extended to
the subspace with rank-k games. Given a bimatrix game (A, B) ∈ R2mn of rank-k, the
P
T
matrix A + B may be written as kl=1 γ l · β l , using the linearly independent vectors
41
γ l ∈ Rm , β l ∈ Rn , 1 ≤ l ≤ k. Therefore, the column-player’s payoff matrix B may
P
T
be written as B = −A + kl=1 γ l · β l , where {β l }kl=1 are linearly independent. Let
P
T
α = (α1 , . . . , αk ), and G(α) denote the game (A, −A + kl=1 αl · β l ). Consider the
corresponding game space Γk = {G(α) | αl ∈ Rm , ∀l ≤ k}. This space is an affine
km-dimensional subspace of the bimatrix game space R2mn , and it contains only rankk games. The Nash equilibrium correspondence of the space Γk is EΓk = {(α, x, y) ∈
Rkm × ∆1 × ∆2 | (x, y) is a NESP of G(α)}.
For all the games in Γk , again the row-player’s payoff matrix remains constant, hence
for all G(α) the best response polyhedron of the row-player P (α) is P of (2.2). However,
the best response polyhedron of the column player Q(α) varies, as the payoff matrix of
the column-player varies with α. Consider the following polyhedron (similar to (2.3)).
Q′k = {(x, λ, π2 ) ∈ Rm+k+1 | xi ≥ 0, ∀i ∈ S1 ;
Pm
P
xT (−Aj ) + kl=1 βjl λl − π2 ≤ 0, ∀j ∈ S2 ;
i=1 xi = 1}
Note that λ = (λ1 , . . . , λk ) is a variable vector.
(3.1)
The column-player’s best re-
sponse polyhedron Q(α), for the game G(α), is the projection of the set {(x, λ, π2 ) ∈
Pm l
We assume that the polyheQ′k |
i=1 αi xi −λl = 0, ∀l ≤ k} on (x, π2 )-space.
dra P and Q′k are non-degenerate. Let the set of fully-labeled pairs of P × Q′k be
N k = {(p, q) ∈ P × Q′k | L(p) ∪ L(q) = {1, . . . , m + n}}. For a p ∈ P let N k (p, ∗) denote
the set of points of Qk making fully-labeled pair with p, and similarly define N k (∗, q) for
a q ∈ Qk . The following facts regarding the set N k may be easily derived.
• For every point in EΓk there is a unique point in N k , and for every point in N k
there is a point in EΓk (Lemma 2.2). Further the set of points of EΓk mapping to a
point (p, q) ∈ N k , is equivalent to k(m − 1)-dimensional space.
• Since there are k more variables in Q′k , namely λ1 , . . . , λk compared to Q of (2.2),
N k is a subset of the k-skeleton of P × Q′k . If a point p ∈ P is on a d-dimensional
face (d ≤ k), then the set N k (p, ∗) is either empty or it is a (k − d)-dimensional
face, where N k (p, ∗) = {w ∈ Q′k | (p, q) ∈ N k } (Observations of Section 2.3).
• ∀(p, q) = ((y, π1 ), (x, λ, π2 )) in P × Q′k ,
holds iff (p, q) ∈ N k .
Pk
42
l=1 λl (β
lT
· y) − π1 − π2 ≤ 0, and equality
For a vector δ ∈ Rk , consider the following parametrized linear program LP k (δ).
LP k (δ) : max
Pk
l=1 δl (β
lT
· y) − π1 − π2
(x, λ, π2 ) ∈ Q′k ;
s.t. (y, π1 ) ∈ P ;
(3.2)
λl = δl , ∀l ≤ k
Let OP T k (δ) be the set of optimal points of LP k (δ). Note that for any a ∈ Rk ,
all the points on N k with λ = a may be obtained by solving LP k (a). In other words,
{((y, π1), (x, λ, π2 )) ∈ N k | λ = a} = OP T k (a) (Lemma 2.14). Using this fact we show
T
T
that the tuple (λ1 + β 1 · y, . . . , λk + β k · y) uniquely identifies a point of N k . For a vector
T
a ∈ Rk , let S(a) = {((y, π1), (x, λ, π2 )) ∈ N k | λl + β l · y = al , ∀l ≤ k}.
Lemma 3.1 For any a ∈ Rk , the set S(a) contains exactly one element, i.e., |S(a)| = 1.
T
Proof : First we show that S(a) 6= ∅. Let S1 (a) = {((y, π1), (x, λ, π2 )) ∈ N k | λl + β l ·
y = al , ∀l > 1}. Using the similar analysis as in Lemmas 2.5 and 2.6 of Chapter 2, it may
T
be easily shown that for every b ∈ R there is a point in S1 (a) such that λ1 + β 1 · y = b.
Therefore S(a) 6= ∅.
Now, suppose |S(a)| > 1 implying that there are at least two points (p1 , q1 ) and
(p2 , q2 ) in S(a). Let pi = (y i, π1i ), q1 = (x1 , c, π21 ) and q2 = (x2 , d, π22). Clearly, (p1 , q1 )
and (p2 , q1 ) are feasible points of LP k (c) and (p1 , q1 ) ∈ OP T k (c). Similarly, (p2 , q2 ) and
(p1 , q2 ) are feasible points of LP k (d) and (p2 , q2 ) ∈ OP T k (d). Therefore the following
holds.
Pk
l=1 cl (β
Pk
lT
· y 1) − π11 − π21 ≥
lT
· y 2 ) − π12 − π22 ≥
l=1 dl (β
Pk
l=1 cl (β
Pk
l=1
lT
· y 2 ) − π12 − π21
T
dl (β l · y 1) − π11 − π22
T
Using the fact that β l · y = al − λl , ∀l ≤ k and the above equations, we get
Pk
l=1 cl (al
P
− cl ) + dl (al − dl ) ≥ kl=1 cl (al − dl ) + dl (al − cl )
P
⇒ − kl=1 (cl − dl )2 ≥ 0 ⇒ ∀l ≤ k, cl = dl
T
T
⇒ c = d ⇒ ∀l ≤ k, β l · y 1 = β l · y 2
The above expressions and the fact that
Pk
l=1 λl (β
lT
· y) − π1 − π2 evaluates to zero at
both (p1 , q1 ) and (p2 , q2 ) imply that π11 = π12 and π21 = π22 . Note that, S(a) ⊂ OP T k (c).
Claim 3.2 The set {q ∈ Q′k | (p, q) ∈ OP T k (c), p ∈ P } is a singleton.
Proof : Suppose the set {q ∈ Q′k | (p, q) ∈ OP T k (c), p ∈ P } contains two distinct
points q and q ′ . In that case, λ takes value c on the 1-dimensional line L ⊂ Q′k containing
43
both q and q ′ . Note that the points corresponding to the end-points of L are on the lower
dimensional face (< k) of Q′k and both these points make separate convex sets of fully
labeled pairs with the points of P . Further the convex hull of these two convex sets is not
contained by N k , however both these sets are contained in OP T k (c) and OP T k (c) ⊂ N k .
It implies that OP T k (c) is not convex, which is a contradiction.
⊓
⊔
The above claim implies that q = q1 = q2 . Now it is enough to show that p1 = p2 to
prove the lemma. In the extreme case, q is a vertex of Q′k and makes a fully-labeled pair
with a k-dimensional face of P . Let M(q) = {1, . . . , m+n}\L(q). Clearly, |M(q)| ≥ n−k,
M(q) ⊆ L(p1 ) and M(q) ⊆ L(p2 ). Suppose p1 6= p2 , then on the line joining p1 and p2 , the
P
following equations are tight: β l · y = al − cl , ∀l ≤ k; nj=1 yj = 1 and all the equations
corresponding to M(q). Clearly, there are at least n + 1 equations tight on this line,
hence they are not linearly independent. This contradicts the fact that A and β l s are
⊓
⊔
generic.
Motivated by Lemma 3.1, we consider the function g k : N k → Rk such that,
T
T
g k ((y, π1), (x, λ, π2 )) = (λ1 + (β 1 · y), . . . , λk + (β k · y))
The function g k is continuous and bijective (Lemma 3.1), and the inverse g k
−1
: Rk →
N k is also continuous, since N k is a closed and connected set. Now consider a function
f k : EΓk → Γk similar to (2.8).
T
T
f k (α, x, y) = (α′1 , . . . , α′k ), where α′l = ((αl · x) + (β l · y),
l
α2l − α1l , . . . , αm
− α1l )T , ∀l ≤ k
The following theorem establishes homeomorphism between Γk and EΓk by constructing
−1
a continuous inverse of f k using g k . and settles the first part of Theorem 1.3.
Theorem 3.3 Function f k establishes a homeomorphism between NE correspondence EΓk
and the game space Γk .
Proof : First we show that function f k is bijective. Consider an α′ = (α′1 , . . . , α′k ) ∈
Rmk . Construct a point (α, x, y) ∈ EΓk such that f k (α, x, y) = α′ . Let ((y, π1), (x, λ, π2 )) =
−1
g k (α1′1 , . . . , α1′k ). Now solve the following system of equations to get α.
∀l ≤ k,
m
X
T
xi αil = λl − β l · y;
i=1
44
αil − α1l = αi′l , ∀i > 1
It is easy to see that we get a unique α by solving the above equations, and f k (α, x, y) = α′
holds. From the above analysis it is clear that f k is a continuous bijective function. The
inverse function f k
−1
: Γk → EΓk is also continuous, since g k
−1
is continuous and the set
⊓
⊔
EΓk is closed and connected.
Given a game G(γ) ∈ Γk , we know that all its Nash equilibria are the points in the
P
intersection of N k and hyper-planes i γil xi − λl = 0, ∀l ≤ k. Using this fact, next we
1
k
give a fixed point formulation for Nash equilibria of G(γ). Let γmin = (γmin
, . . . , γmin
)
1
k
l
l
and γmax = (γmax
, . . . , γmax
), where ∀l ≤ k, γmin
= mini∈S1 γil and γmax
= maxi∈S1 γil .
Consider the box B ⊂ Rk s.t. B = {a ∈ Rk | γmin ≤ a ≤ γmax }1 . For the rank-1 case, B
is an interval.
Lemma 3.4 Finding a Nash equilibrium of G(γ) reduces to finding a fixed point of a
polynomially computable piece-wise linear function on B.
Proof : Clearly, ∀x ∈ ∆1 , (
Pm
1
i=1 γi xi ,
...,
Pm
i=1
γik xi ) ∈ B. Now, consider the function
f : B → B such that,
Pm k
P
1
f (a) = ( m
i=1 γi xi ),
i=1 γi xi , . . . ,
where (x, λ, π2 ) = {q ∈ Q′k | (p, q) ∈ OP T k (a), p ∈ P }
The function f is a piece-wise linear function. For every a ∈ B, the corresponding x is well
defined in the above expression (Lemma 3.1), and may be obtained in polynomial time by
solving LP k (a). Since, Nash equilibria of G(γ) are exactly the points in the intersection
P
of N k and hyper-planes i γil xi − λl = 0, ∀l ≤ k, fixed points of f correspond to the Nash
⊓
⊔
equilibria of game G(γ) and vice-versa.
Analysis similar to Lemma 2.16 shows that, for a given a ∈ Rk , there is a way to trace the
points in the intersection of N k and λl = al , l 6= i, such that λi increases monotonically.
A question: Is there a way to locate a fixed point of f in polynomial time using this
observation and the simple structure of N k , even though finding a fixed point in general
is PPAD-complete [51].
1
For any two vectors a, b ∈ Rn , by a ≤ b we mean ai ≤ bi , ∀i ≤ n.
45
3.2
Sum Preserving Homeomorphism
In this section we derive a homeomorphism map for a subspace, where the summation
matrix A + B is preserved. Our approach is more on the lines of Kohlberg and Mertens
construction, which is given in Appendix A for the reference.
Fix a matrix C ∈ Rmn . Let ΓC = {(A, −A+C) ∈ R2mn | A ∈ Rmn } be the game space
into consideration. Note that, the set ΓC forms an mn-dimensional affine subspace in the
2mn-dimensional bimatrix game space. Let the Nash equilibrium correspondence of the
game space ΓC be EΓC = {(A, x, y) ∈ Rmn+m+n |(x, y) is a NE of the game (A, −A + C)}.
Next we construct the maps to establish homeomorphism between ΓC and EΓC . We
note that similar maps work to establish homeomorphism between {(C, −C + A) | A ∈
Rmn } and its Nash equilibrium correspondence. The maps we derive are more intricate
and challenging than that of [40], because we need to encode mn + m + n values in mn
space, while they have 2mn space to encode 2mn + m + n values. First we define the
forward map f : EΓC → ΓC . Let (A, x, y) ∈ EΓC , and
li =
kj =
Pm
Pn
j=1 aij ,
Pm
i=1
1≤i≤m
(3.3)
aij , 1 ≤ j ≤ n.
Pn
Construct a matrix à ∈ Rmn , such that
Pm
li
li kj
ãij = aij − −
+ i=1 , 1 ≤ i ≤ m, 1 ≤ j ≤ n
(3.4)
n m
mn
P
P
From (3.3) and (3.4) it is clear that nj=1 ãij = 0, ∀i and ni=1 ãij = 0, ∀j. Now,
Note that,
i=1 li
=
j=1 kj .
create vectors ᾱ and β̄, one for each player, using (A, x, y) as follows.
1≤i≤m
αi = Ai y + xi
(3.5)
βj = xT (−A + C)j + yj 1 ≤ j ≤ n
Next, construct a matrix A′ = [a′ij ] using Ã, ᾱ and β̄: a′ij = ãij +
αi
n
+
βj
.
m
The function f maps the tuple (A, x, y) to the game (A′ , −A′ + C). Since f is a
polynomial map, it is continuous. To prove that f establishes homeomorphism we need
to show that it has a continuous inverse. For that, first we show that f is a bijective
map by constructing an inverse map f −1 : ΓC → EΓC such that f ◦ f −1 : ΓC → ΓC , and
f −1 ◦ f : EΓC → EΓC are identity maps. Later we show that f −1 is also continuous. To
define f −1 we need to construct (A, x, y) given an A′ so that f (A, x, y) = A′ . There are
three main steps involved in this process.
46
1. Extract à from A′
2. Construct ᾱ, β̄ using à and A′ . In this process, we also construct the x and y.
3. Finally, construct A using x, y, ᾱ, β̄ and Ã.
In general, given an ᾱ calculated using Equation (3.5), we use the following trick by
Kohlberg and Mertens to extract x from it.
Let vx = min{v |
X
(αi − v)+ ≤ 1, v ∈ R}, where (αi − v)+ = max{0, αi − v}
i
xi = max{0, αi − vx }
(3.6)
By construction of ᾱ and β̄ (Eqn (3.5)) following equality should be satisfied.
xT Ay
= xT (ᾱ − x)
T
T
T
⇒ xT (ᾱ − x) = −(β̄ T − y T − xT C)y
T
−x Ay = (β̄ − y − x C)y
(3.7)
Next we describe each of the three steps of f −1 .
3.2.1
Extract à from A′
Let H = [hij ] ∈ Rm+n be the intermediate matrix obtained by subtracting the row-average
from the elements of A′ , i.e. H = A′ − (A′ ∗ n1 [1]n×n ).
hij = a′ij −
= ãij +
= ãij +
Pn
k=1
a′ik
n
αi
n
βj
m
+
−
βk
αi
k=1 (ãik + n + m
Pn
βj
−
m
Pn
)
n
k=1 βk
mn
It is easy to check that by subtracting column-average of H from its elements we get
Ã. Formally, Ã = H − ( m1 [1]m×m ∗ H).
3.2.2
Extract ᾱ, β̄ from A′ , Ã
Substituting values of a′ij and ãij in the equation a′ij = ãij +
equations
because if
αi
n
αi
n
+
βj
,
m
we get mn linear
β
+ mj = a′ij − ãij , ∀(i, j). Out of these only m + n − 1 are linearly independent,
α1
n
+
βj
m
= a′1j − ã1j , ∀j and
αi
n
+
β1
m
= a′i1 − ãi1 , ∀i are given, then all other
are linear combination of these m + n − 1 equations. We construct the following system
47
of m + n linear equations in m + n variables from the mn equation by taking row-sums
and column-sums.
  P



n
1
′
Im×m
[1]
[
(a
−
ã
)]
[α
]
m×n
ik i≤m
m
 ≡ Z ∗ [ᾱT β̄ T ] = d¯

  i i≤m  =  Pk=1 ik
m
′
1
[βj ]j≤n
In×n
[1]n×m
l=1 (alj − ãlj ) j≤n
n
(3.8)
Claim 3.5 rank(Z) = m + n − 1
Proof : Clearly, vector nZ = [[1]1×m
m
[−1]1×n ]T
n
is in null space of Z. Hence rank(Z) <=
m + n − 1. Let ei be the vector of size m + n having 1 at ith position and 0 else where,
then ei − ei+1 , ∀i ≤ m + n − 1 are eigenvectors of Z with eigenvalue 1, and they are all
⊓
⊔
linearly independent.
Claim 3.5 implies that solution of equations in (3.8) form a one dimensional subspace.
From this solution space only one point correspond to actual values of ᾱ and β̄. Observe
that if any p = [ᾱ′T β̄ ′T ] is a solution of (3.8), then p+(λ∗nZ) is also a solution, where nZ
is in the null space of Z. Hence, for a given solution p, ∃λ ∈ R, s.t.[ᾱT β̄ T ] = p + (λ ∗ nZ).
Lemma 3.6 Let ᾱ′ = ᾱ − (λ ∗ [1]m×1 ), λ ∈ R. If x and x′ are constructed from ᾱ and ᾱ′
respectively using (3.2), then x = x′ .
Proof : The proof follows from the construction of x and x′ as given in (3.2) from α
and α′ , and the fact that αi − αi′ = λ, ∀i.
⊓
⊔
For the system (3.8) find a solution [ᾱ′T β̄ ′T ]. Now we need to calculate the λ corresponding to [ᾱT β̄ T ]. For that, first construct x′ and y ′ from ᾱ′ and β̄ ′ respectively using
(3.2). Because of Claim 3.6 we know that they should be same as x (of ᾱ) and y (of β̄)
respectively. Using the equality in (3.7), λ can be calculated as follows.
xT (ᾱ − x)
= −(β̄ T − y T − xT C)y
xT (ᾱ′ + λ[1]m×1 − x) = −((β̄ ′ −
λ(1 −
λ
m
)
n
mλ
[1]n×1 )T
n
− y T − xT C)y
= −β̄ ′T y − xT ᾱ′ + y T y + xT x + xT Cy
=
n
(−β̄ ′T y
n−m
⇒
⇒
⇒
− xT ᾱ′ + y T y + xT x + xT Cy)
For the correctness of the above procedure, we need to argue that the value of λ
is consistent with the choice of [ᾱ′T β̄ ′T ]. Suppose, p′ = [ᾱ′T β̄ ′T ], and p′′ = [ᾱ′′T β̄ ′′T ] =
48
p′ + γ ∗ nZ, γ ∈ R. If value of λ corresponding to p′ and p′′ are δ ′ and δ ′′ , then [ᾱT β̄ T ] =
p′′ + δ ′′ ∗ nZ = p′ + γ ∗ nZ + δ ′′ ∗ nZ = p′ + (γ + δ ′′ ) ∗ nZ. Hence, the above calculation
should give δ ′ = δ ′′ + γ.
δ ′′ =
n
(−β̄ ′′T y
n−m
=
n
(−(β̄ ′
n−m
=
n
(−β̄ ′T y
n−m
− xT ᾱ′′ + y T y + xT x + xT Cy)
−
mγ
[1]n×1 )T y
n
− xT (ᾱ′ + γ[1]m×1 ) + y T y + xT x + xT Cy)
− xT ᾱ′ + y T y + xT x + xT Cy +
m−n
γ)
n
= δ′ − γ
We have x, y, ᾱ, β̄ and Ã. The only remaining part is to construct A from the
available data, to complete the f −1 .
3.2.3
Construction of A
To construct A, first li s and kj s of Equation (3.3) should be calculated. Note that,
Ay = ᾱ − x and −AT x = β̄ − y − C T x (∵ Eqn (3.5)). Putting all the known values
in equations of Ay and −AT x after expanding A using (3.3), we get following system of
linear equations in li s and kj s.














..
.
...
..
.
yn
m
m−1
mn
y1
m
...
yn
m
..
.
...
..
.
xm
n
1−n
mn
..
.
...
..
.
1
mn
x1
n
...
xm
n
1
mn
...
1−n
mn
..
.
...
..
.
−1
mn
−1
mn
...
x1
n
m−1
mn
..
.
..
.
y1
m
..
.
..
.
















α1 − x1 − Ã1 y

l1
 
..  
.  
..
 
.
 
lm  

=
αm − xm − Ãm y

k1  
  β − y − xT (−Ã + C)1
1
1
..  
.  

..
 
.

kn
βn − yn − xT (−Ã + C)n

















(3.9)
Equivalently say W ∗ [¯lT k̄ T ]T = h̄
Matrix W is of (m + n) × (m + n)-dimension. If W is invertible then li s and kj s can be
obtained by doing [¯lT k̄ T ]T = W −1 ∗ h̄, and in-turn A can be constructed back using Ã, ¯l
and k̄ in Equation 3.3.
Pn
P
Lemma 3.7 det(W ) 6= 0. Let [¯lT k̄ T ]T = W −1 ∗ h̄, then m
j=1 kj .
i=1 li =
49
Proof : After a few row operations and column

0 0 ... 0
0


−1
 0 m−1
. . . mn
0
mn

..
..
..
 ..
..
.
 .
.
.
.


−1
 0 mn
. . . m−1
0
mn


1−n
 0 0 ... 0
mn

 ..
..
..
..
 .
.
...
.
.


1
 0 0 ... 0
mn

1 0 ... 0
0
operations W can be converted to:

... 0 1


... 0 0 

.. 
..
. 
... .


... 0 0 


1
. . . mn 0 

..
.. 
..
.
.
. 



. . . 1−n
0
mn

... 0 0
Clearly, determinant of this matrix is non-zero. Next, to show that
Pm
i=1 li =
Pn
j=1
kj ,
we construct the vector p = [[1]1×m [−1]1×n ] by taking linear combination of rows of W .
Let v = −mn ∗ [xT y T ]. It is easy to see that p = v T ∗ W and v T ∗ h̄ = 0.
⊓
⊔
From Lemma 3.7 it is clear that W −1 ∗ h̄ gives the values of li s and kj s and finally
we can construct A from these and Ã. This completes the description of the function f −1
such that f (A, x, y) = A′ if and only if f −1 (A′ ) = (A, x, y). The inverse function is also
continuous because the whole construction involves operations in polynomials. Hence,
maps f and f −1 gives the following theorem, and settles the second part of Theorem 1.3.
Theorem 3.8 The game space ΓC is homeomorphic to its NE correspondence EΓC .
3.3
Discussion
Using the structure developed in Section 3.1 one may hope to extend the BinSearch
algorithm of Chapter 2 to fixed rank games. However, we observe that even to solve a
rank-2 game, the obvious possible extensions of BinSearch algorithm, like nested binary
search and 2-dimensional binary search, fail. Given a rank-2 game (A, B) such that
T
T
A + B = γ 1 β 1 + γ 2 β 2 , consider the rank-2 game space formed by A, β 1 and β 2 , the set
of fully-labeled points N 2 , and hyper-planes H1 : xT γ 1 − λ1 = 0, H2 : xT γ 2 − λ2 = 0.
Recall that, the points in N 2 ∩ H1 ∩ H2 are NE of (A, B), and their values of (λ1 , λ2 ) ∈
1
1
2
2
B = [γmin
, γmax
] × [γmin
, γmax
].
2
2
The nested binary search algorithm does search on λ2 , in the range of [γmin
, γmax
],
to get to H2 , while probing points on N 2 ∩ H1 which is equivalent to solving a rank-1
50
game. The reason for failure is that λ2 is not monotonic on N 2 ∩ H1 , since N 2 ∩ H1 is
equivalent to N of A, C, β case with rank(A + C) = 1. The 2-dimensional binary search
tries to reduce the box B by a fraction such that it contains at least one NE. All standard
ways of reducing it to a quarter, a half or a three-quarter fail. The increase in complexity
even for rank-2 games forces us to wonder if these games are computationally as hard as
general games.
51
Chapter 4
Efficient Algorithms for Rank Based
Subclasses of Bilinear Games
As we discussed in Chapter 1, polynomial time NE computation algorithms are known
for many subclasses of bimatrix games, including zero-sum games [15], (quasi-) concave
games [43], and games with low rank payoff matrices [46]. A line of work focuses on games
of small rank ; defined as rank(A+B) [38]. A fully polynomial time approximation scheme
(FPTAS) was derived for fixed rank games [38]. Further, in Chapter 2 [2] we presented a
polynomial time algorithm for computing an exact Nash equilibrium for rank-1 games.
Specifying the two payoff matrices of a bimatrix game requires a polynomial number
of entries in the numbers of pure strategies available to the players. This is adequate
when the set of pure strategies are explicitly given. However, there are situations where
the natural description gives the set of pure strategies implicitly, and as a result they may
be exponential in the description of the game. For example, normal form (bimatrix) representation of two player extensive-form game may have exponentially many strategies in
the size of the extensive-form description [21]. In such a case, even if the resulting bimatrix
game has a fixed rank, the above results may not be applied for efficient computation.
Nevertheless, certain types of extensive-form games have some combinatorial structure which may be exploited. Koller, Megiddo and von Stengel [42] converted an arbitrary
two-player, perfect-recall, extensive form game into a payoff-equivalent two-player game
with continuous strategy sets. In this derived formulation, which they call the sequence
form, there is a pair of payoff matrices A and B, one for each player. Further, their strategy sets turn out to be compact polytopes in Euclidean space of polynomial dimension.
52
Given a pair of strategies (x, y), utilities of the players are xT Ay and xT By respectively.
Interestingly, the sequence form requires only a polynomial number of bits to specify.
Motivated by the sequence form of Koller et al., we define bilinear games, which are
two-player, non-cooperative, single shot games represented by two payoff matrices, say A
and B, of dimension m × n and two polytopal compact strategy sets X and Y of dimensions m and n respectively. If (x, y) ∈ X × Y is the played strategy profile, then xT Ay
and xT By are the utilities derived by player one and player two respectively. In other
words, the payoffs are bilinear functions of strategies, hence the name bilinear games.
The scope of bilinear games is large enough to capture many important classes of games
besides two-player extensive form games with perfect recall. For example, for two-player
Bayesian games [35, 52], polymatrix games [34], and various classes of optimization duels
[36], researchers have proposed polynomial-sized payoff-equivalent formulations which (either explicitly or implicitly) turn out to be bilinear games (see Section 4.1.1 for details).
Intuitively the polytopal strategy sets are concise representations of the original sets of
mixed strategies as marginal probabilities, and xT Ay and xT By express the expected
payoffs of the original game in terms of these marginal probabilities.
In this chapter we first study Nash equilibrium and its characterization for bilinear
games, and later give three efficient algorithms for their rank based subclasses. In Section
4.1, we formally define a bilinear game, characterize its Nash equilibria as fully-labeled
pairs in the best response polyhedra and demonstrates many different games as special
cases of bilinear games. Similar to bimatrix games, rank of a bilinear game (A, B) is
defined as the rank(A + B). Section 4.2 extends the polynomial time algorithm presented
in Chapter 2 to rank-1 bilinear games, in spite of a very general structure of the strategy
sets in bilinear games. In Section 4.3, we discuss two FPTAS algorithms for the fixed rank
bilinear games, which are generalization of the algorithms by Kannan and Theobald [38]
for the bimatrix games. Finally, in Section 4.4, using the structure of BRPs, we obtain a
polynomial time algorithm for the case when the rank of either A or B is a constant.
4.1
Bilinear Games and Nash Equilibria
As discussed above, bilinear games are two-player non-cooperative, single shot games. A
bilinear game is represented by two m × n dimensional payoff matrices A and B, one
53
for each player, and two compact polytopal strategy sets. Let S1 = {1, . . . , m} be the
set of rows and S2 = {1, . . . , n} be the set of columns of the matrices. Let E ∈ Rk1 ×m
and F ∈ Rk2 ×n be the matrices, and e ∈ Rk1 and f ∈ Rk2 be the vectors. The strategy
set of the first-player is X = {x ∈ Rm | Ex = e, x ≥ 0} and the second-player is
Y = {y ∈ Rn | F y = f, y ≥ 0}. Sets X and Y are assumed to be compact. From a
strategy profile (x, y) ∈ X × Y , the payoffs obtained by the first and the second player
are xT Ay and xT By respectively.
Remark 4.1 Note that our formulation may express arbitrary polytopes as strategy space,
for example if the strategy set of a player is expressed as {x : Gx ≤ g}, i.e., the intersection
of a set of half-spaces, it may be transformed to an equivalent game with strategy set {x′ :
Ex′ = e, x′ ≥ 0} using standard techniques (i.e., by adding slack variables, substituting
−
+
−
unbounded xi with x+
i − xi , xi , xi ≥ 0, and modifying the payoff matrices accordingly).
From a Nash equilibrium (NE) strategy profile, no player gains by unilateral deviation. Formally,
Definition 4.2 A strategy profile (x, y) ∈ X × Y is a NE of the game (A, B) iff xT Ay ≥
x′T Ay, ∀x′ ∈ X and xT By ≥ xT By ′, ∀y ′ ∈ Y .
As a direct corollary of Glicksberg’s [24] result that there always exists a Nash
equilibrium in a game whose players’ strategy spaces are convex and compact, and whose
utility function for each player i is continuous in all players’ strategies and quasi-concave
in i’s strategy, we have
Proposition 4.3 Every bilinear game has at least one Nash equilibrium.
A bilinear game is completely represented by a six-tuple (A, B, E, F, e, f ) in general.
However, for ease of notation we represent it by (A, B) fixing (E, F, e, f ). Given a strategy
y ∈ Y of the second-player, the objective of the first player is to play x ∈ X such that
xT (Ay) is maximized, i.e., solve the following linear program [42].
max : xT (Ay)
s.t.
Ex = e
min : eT p
Dual
−−→
s.t.
x≥0
54
E T p ≥ Ay
(4.1)
Note that pi is the dual variable of the equation Ei x = ei in the above program.
At the optimal point (x, p) of (4.1), we get xi > 0 ⇒ Ai y = pT E i , ∀i ∈ S1 from the
complementarity. A similar condition may be obtained for the second-player, given an
x ∈ X. At a Nash equilibrium both the conditions should be satisfied, which characterizes
the NE strategies as follows: A strategy pair (x, y) ∈ X × Y is a Nash equilibrium of the
game (A, B) iff it satisfies the following conditions.
∃p ∈ Rk1
∃q ∈ R
k2
s.t.
s.t.
Ay ≤ E T p
T
T
x B≤q F
and
and
∀i ∈ S1 , xi > 0 ⇒
∀j ∈ S2 , yj > 0 ⇒
Ai y = pT E i
T
j
T
x B =q F
j
(4.2)
The above characterization implies that, a player plays a strategy with non-zero
probability only if it gives the maximum payoff with respect to (w.r.t.) the opponent’s
strategy in some sense. Such strategies are called the best response strategies (w.r.t. the
opponent’s strategy). Using this fact, we define best response polyhedra (BRP), similar to
the best response polyhedra of a bimatrix game [73].
In the following expression, x, y, p and q are vector variables.
P = {(y, p) ∈ RN +k1 |Ai y − pT E i ≤ 0,∀i ∈ S1 ;
Q = {(x, q) ∈ RM +k2 |
xi ≥ 0,
yj ≥ 0,
∀j ∈ S2 ; F y = f }
∀i ∈ S1 ; xT B j − q T F j ≤ 0,∀j ∈ S2 ; Ex = e}
(4.3)
The polyhedron P in (4.3) is closely related to the best response strategies of the
first-player for any given strategy of the second-player and it is called the best response
polyhedron of the first-player. Similarly Q is called the best response polyhedron of the
second-player. Note that, in both the polyhedra the first set of inequalities corresponds to
the first-player, and the second set corresponds to the second-player. Since |S1 | = m and
|S2 | = n, let the inequalities be numbered from 1 to m, and m + 1 to m + n in both the
polyhedra. Let the label L(v) of a point v in the polyhedron be the set of indices of the
tight inequalities at v. If a pair (v, w) ∈ P × Q is such that L(v) ∪ L(w) = {1, . . . , m + n},
then it is called a fully-labeled pair. The proof of the next lemma follows using (4.2).
Lemma 4.4 A strategy profile (x, y) is a NE of the game (A, B) iff ((y, p), (x, q)) ∈ P ×Q
is a fully-labeled pair, for some p and q.
A game is called non-degenerate if both the polyhedra are non-degenerate. Note that
a fully-labeled pair of a non-degenerate game has to be a vertex-pair. Lemma 4.4 implies
55
that a NE strategy profile has to satisfy the following linear complementarity conditions
(LCP) over P × Q [42].
((y, p), (x, q)) ∈ P × Q corresponds to a NE
⇔
xT (Ay − E T p) = 0 and (xT B − q T F )y = 0
(4.4)
Clearly, xT (Ay − E T p) ≤ 0 and (xT B − q T F )y ≤ 0 over P × Q and hence xT (Ay −
E T p) + (xT B − q T F )y ≤ 0. Simplifying the expression using Ex = e and F y = f we get
xT (A + B)y − eT p − f T q ≤ 0 over P × Q and equality holds iff (x, y) is a NE (using (4.4)).
This gives the following QP formulation which captures all the NE of game (A, B) at its
optimal points.
max: xT (A + B)y − eT p − f T q
s.t.
(4.5)
(y, p) ∈ P ; (x, q) ∈ Q;
Symmetric Bilinear Games. Nash [49] proved that any symmetric finite game has a
symmetric Nash equilibrium. The concept of symmetry may be straightforwardly adapted
to the bilinear games: We say a bilinear game is symmetric if B = AT , E = F , and e = f .
A strategy profile (x, y) is symmetric if x = y. A straightforward adaptation of Nash’s
[49] proof yields the following proposition.
Proposition 4.5 Any symmetric bilinear game has a symmetric NE.
Note that for a symmetric game (A, AT ), strategy sets X and Y are the same. From
(4.2) and (4.4), it is clear that a symmetric NE x ∈ X must satisfy q = p, Ax ≤
E T p, xT (Ax − E T p) = 0. This gives the following QP formulation to capture all the
symmetric NE of a symmetric game (A, E, e).
max: xT Ax − eT p
s.t.
(4.6)
Ax ≤ E T p; Ex = e; x ≥ 0
A bilinear game (A, B, E, F, e, f ) may be converted to an equivalent symmetric game
(A′ , E ′ , e′ ), where


0 A
,
A′ = 
BT 0

E′ = 
E
0
0 F

,

e′ = 
56
e
f


 , with strategy vector z = 
x
y


It is easy to check that any symmetric NE of the derived game corresponds to a
NE of the original game and vice-versa. In the next section, we discuss reductions of
different games to polynomial size bilinear games, which do not seem to be possible with
the bimatrix games.
4.1.1
Examples of Bilinear Games
The simplest subclass of bilinear games is the set of two-player normal-form games (bimatrix games).
Example 4.6 Bimatrix Games
A bimatrix game (A, B) with A, B ∈ Rm×n may be straightforwardly transformed to the
bilinear game (A, B, E, F, e, f ) where E T = 1m , e = 1 and similarly F T = 1n , f = 1.
Many other interesting classes of finite games may be formulated as bilinear games.
In this section, we provide a few examples where the bilinear formulation are exponentially
smaller than a direct bimatrix formulation.
Example 4.7 Two-player Bayesian Games
In a Bayesian game [30], there is a type set associated with each player, which is her
private information. The nature draws the type for each player from a joint distribution,
which is a common knowledge, and each player gets to know only her own type before
choosing an action. The final payoffs of the game is determined by types of all the players,
and hence are uncertain.
Here we consider the two-player case, where Ti s are the type sets and Si s are the
strategy sets. The joint probability distribution is denoted by pts for the type profile (t, s) ∈
T1 × T2 . Let S = S1 × S2 , T = T1 × T2 , |Ti | = ti and |Si | = mi . The utilities are the
functions of actions and types, i.e., ui : S × T → R, hence for every type profile they may
be represented by the two matrices. For a type profile (t, s) let Ats and B ts denote the
respective m1 × m2 dimensional payoff matrices. The strategy of a player is to decide her
play for each of her type so that her expected payoff is maximized, i.e., x : T1 → ∆(S1 )
for player one and y : T2 → ∆(S2 ) for player two. For a t ∈ T1 , let xt denote the mixed
strategy given type t.
The induced normal form of this game is a bimatrix game in which each pure strategy
of a player prescribes an action for each of her types. Thus the size of the induced normal
57
form is exponential in the number of types. However, it may be formulated as a polynomial
sized bilinear game as follows

p A11 · · ·
 11
..

..
A=
.
.

pt1 1 At1 1 · · ·

1T 0
 m

E =  0 1Tm

..
.
1t2
p1t2 A
..
.
pt1 t2 At1 t2

···


,

..
.






, B = 


p11 B
..
.
11
···
..
.
p1t2 B
..
.
1t2
pt1 1 B t1 1 · · · pt1 t2 B t1 t2


T
1
0 ···

 n


F =  0 1Tn



..
..
.
.



,

and e = 1t1 , f = 1t2 . Given mixed strategies x1 , . . . , xt1 , y 1, . . . , y t2 of the Bayesian game,
T
T
T
T
define x = [x1 , · · · , xt1 ]T and y = [y 1 , · · · , y t2 ]T . Then xT Ay and xT By are exactly the
expected utilities of the Bayesian game. This transformation is implicit in Howson and
Rosenthal’s [35] adaptation of Lemke-Howson algorithm to two-player Bayesian games.
Example 4.8 Polymatrix Games [34]
A polymatrix game is an n-player game in which each player’s utility is the sum of the
utilities resulting from her bilateral interactions with each of the n − 1 other players. Let
Si be player i’s set of pure strategies. The game is represented by the payoff matrices
Aij ∈ R|Si|×|Sj | for all pairs of players (i, j). Let xi ∈ ∆(Si ) denote a mixed strategy of
player i. Given a strategy profile (x1 , . . . , xn ), the expected utility of player i is,
ui (x1 , . . . , xn ) =
X
(xi )T Aij xj
j6=i
We show that any polymatrix game may be transformed to a symmetric bilinear game
such that any symmetric NE of the bilinear game corresponds to a NE of the polymatrix
game. Our derivation is adapted from Howson’s [34] formulation of NE of polymatrix
games as solutions of an LCP. Formally, given a polymatrix game, we define the induced
symmetric bilinear game as (A, AT , E, E, e, e), where



1T
0
0 A12 · · · A1n
 |S1 |



 21

 0 1T|S2 |
 A
0
A2n 
, E = 
A=
 ..
 ..

..
.
 .
 .




n1
n2
0
0
A
A
0
T
T
and e = 1n . The space of strategy vectors is x = [x1 · · · xn ]T .
58
···
···
..
.
0
0
· · · 1T|Sn |




,



Proposition 4.9 Consider a polymatrix game of n players. The strategy (x1 , x2 , . . . , xn )
is an NE of the game if and only if (x, x) is a symmetric NE of its induced bilinear game.
Proof :
The proof is relatively straightforward, by observing that the respective incentive
constraints are equivalent. Thus the problem of finding a NE of a polymatrix game reduces
to the problem of finding a symmetric NE of a symmetric bilinear game. Note that an
asymmetric NE doesn’t correspond to a NE of the polymatrix game.
⊓
⊔
Immorlica et al. [36] analyzed several classes of games between two optimization
algorithms whose objectives are to outperform each other. The space of pure strategies
are the possible outputs of the algorithm which are exponential, however the authors were
able to formulate some of these games as zero-sum bilinear games (which they call bilinear
duels). We describe one example from [36].
Example 4.10 Ranking Duels [36]
Each player chooses a ranking over m elements. Thus the number of pure strategies is
exponential in m. Such a ranking may be represented as a m × m permutation matrix. By
the Birkhoff-von Neumann theorem, the space of mixed strategies corresponds to the space
of m × m doubly-stochastic matrices, whose each row and each column sum to 1. This
P
P
2
space may be described by the polytope {x ∈ Rm | x ≥ 0; j xij = 1, ∀i;
i xij = 1, ∀j}.
Viewing x and y as column vectors, the sizes of the corresponding E, e are polynomial in
2 ×m2
m. Immorlica et al. [36] constructed matrices A ∈ Rm
such that the players’ expected
utilities are equal to xT Ay and −xT Ay respectively.
Example 4.11 Two-player Perfect-recall Extensive-form Games
Extensive-form represents a dynamic game as a tree [47], where every pure strategy of
a player prescribes a move at each of the player’s information sets. As a result the
number of pure strategies may be exponential in the size of the extensive-form description. Fortunately, if we assume perfect recall—roughly, that each player remembers all
her past decisions and observations—then there always exists a Nash equilibrium in behavior strategies, where each player independently chooses a distribution over actions at
each of her information sets. Representation of a behavior strategy requires space linear in the extensive form. However, the expected utilities of the two-player perfect-recall
extensive-form games are not bilinear functions of the behavior strategies. Koller et al.
59
[42] proposed the sequence form, which is a bilinear game formulation for these games.
The number of rows and columns of the matrices A and B, in the bilinear form, are the
number of feasible sequences of plays of the first and second player respectively. If a play
sequence pair (i, j) leads to a leaf node then the ij th entry of A and B are the payoffs of
the first and second player at that leaf node, otherwise it is zero ∗ . A strategy x of the
first player is such that, x(root) = 1, and if a sequence σ ends at an information node
P
C, then a∈Actions(C) xσa − xσ = 0. Similar conditions hold for a strategy y of the second
player. Such a strategy can be transformed to a behavior strategy and vice versa. This
gives E, F, e and f . Note that the reduction is polynomial sized and xT Ay and xT By are
exactly the expected payoffs under the corresponding behavior strategies. We refer readers
to [42] for more details.
Remark 4.12 Lemke’s algorithm on the LCP formulation of bilinear games terminates
with a solution (i.e., not at a ray) if the only non-negative solutions x and y to Ex = 0
and F y = 0 are x = 0 and y = 0, and the payoff matrices are non-positive, i.e., A ≤ 0 and
B ≤ 0. This result directly follows from [42]. Note that games of all the above examples
satisfy these requirements, without loss of generality.
As the richness of bilinear games is apparent from the above examples, efficient
algorithms for these games are of great importance. Next we present efficient algorithms
for rank-based subclasses of bilinear games. The rank of a game (A, B) is defined as
rank(A + B), and we consider the rank based hierarchy of the bilinear games. The set
of rank-k games consists of all (A, B) such that rank(A + B) ≤ k. Zero-sum games are
rank-0 games, the smallest set in the hierarchy. Koller et al. [41] gave an LP formulation
for zero-sum bilinear games, derived from two-player extensive form games with perfect
recall. However, their formulation works for general bilinear games as well. Beyond rank-0
games, no polynomial time algorithm is known for NE computation (even for the reduction
specific formulations). In the next section, we extend the polynomial time solvability of
Nash equilibrium to the rank-1 bilinear games.
For all the algorithms that follow, we make the following assumptions (without loss
of generality): (1) The entries of A, B, E, F, e and f are integers, since scaling them by a
∗
Due to chance moves, the entry may correspond to multiple leaf nodes. In that case the entry stores
the expected payoff.
60
positive number does not change the set of NE. (2) The equalities Ex = e and F y = f
are all linearly independent because even if we discard the dependent equalities, X and
Y do not change. (3) The letter L denotes the bit length of the input game.
4.2
Rank-1 Games: A Polynomial Time Algorithm
The approach used in this section is motivated by Chapter 2 (also see [2]). Given a rank1 game (A, B), it is easy to find α ∈ Rm and β ∈ Rn such that A + B = α · β T , since
any two rows of A + B are multiple of each other. In that case B = −A + α · β T . Let
G(α) = (A, −A + α · β T ) be a parametrized game for a fixed A ∈ Rm×n and β ∈ Rn . For
any game G(α) the BRP of first-player is fixed to P (α) = P (of (4.3)) since A is fixed.
However, the BRP of second-player Q(α) changes with the parameter. Now, consider the
following polyhedron with x, q as vector variables and λ as a scalar variable:
Q′ = {(x, λ, q) ∈ Rm+1+k2 | xi ≥ 0, ∀i ∈ S1 ; xT (−Aj ) + λβj − q T F j ≤ 0,
∀j ∈ S2 ; Ex = e}
(4.7)
It is easy to see that Q(α) is the projection of {(x, λ, q) ∈ Q′ | λ = xT α} on
(x, q)-space. In other words, Q(α) is a section of Q′ obtained by intersecting it with
hyper-plane λ = xT α. Clearly, Q′ covers Q(α), ∀α ∈ Rm . Number the equations of Q′
in similar way as the equations of Q. Let N be the set of fully-labeled pairs of P × Q′ ,
i.e., N = {(v, w) ∈ P × Q′ | L(v) ∪ L(w) = {1, . . . , m + n}}. Using the definition of
fully-labeled pairs, it is easy to check that for a given ((y, p), (x, λ, q)) ∈ P × Q′ ,
((y, p), (x, λ, q)) ∈ N
⇔ xT (Ay − E T p) = 0 and (xT (−A) + λβ T − q T F )y = 0 (4.8)
Lemma 4.13 For every NESP (x, y) of a game G(α) there is a point ((y, p), (x, λ, q)) ∈
N satisfying λ = xT α. Further, for every point ((y, p), (x, λ, q)) of N , (x, y) is a NE of
G(α) for all α satisfying λ = xT α.
Proof : Given a (x, y) of G(α), from (4.2) and (4.4) it is clear that ∃p, q such that
xT (Ay − E T p) = 0 and (xT (−A + αβ T ) − q T F )y = 0. Let λ = xT α, then we get
(xT (−A) + λβ T − q T F )y = 0. Therefore, ((y, p), (x, λ, q)) ∈ N .
Since a point of N is fully-labeled it satisfies, xT (Ay−E T p) = 0 and (xT (−A)+λβ T −
q T F )y = 0. Let α be such that λ = xT α then we get (xT (−A) + (xT α)β T − q T F )y = 0 ⇒
61
(xT (−A + αβ T ) − q T F )y = 0. This implies that (x, y) is a NE of the game (A, −A + αβ T )
(i.e., G(α)), since it satisfies the complementarity condition of (4.4).
⊓
⊔
The above lemma establishes strong relation between the set of NE of all the G(α)s
and the set N . Next we discuss the structure of N , and later use it to design a polynomial
time algorithm to find a NE of a given game G(α).
The polyhedra P and Q′ are assumed to be non-degenerate, and let k1 = k2 = k
for simplicity. As there are k linearly independent equalities in P and Q′ , they are of
dimension n and m + 1 respectively. Therefore, ∀(v, w)Q′ , |L(v)| ≤ n and |L(w)| ≤ m + 1.
Since, m + n labels are required for a pair (v, w) to be part of N , N ⊂ 1-skeleton of
P × Q′ . Further, if (v, w) ∈ N is a vertex pair then |L(v) ∩ L(w)| = 1. Let the label in the
intersection be called the duplicate label of (v, w). Relaxing the inequality corresponding
to the duplicate label at (v, w) in P and Q′ respectively gives its two adjacent edges in
N . Therefore, every vertex of N has degree two. This implies that N is a set of cycles
and infinite paths. Here by, an infinite path, we mean a path with unbounded edges at
both the ends. We will show that N forms a single infinite path. The next lemma follows
directly from the definition of P (4.3) and Q′ (4.7), and expression (4.8).
Lemma 4.14 For all (v, w) = ((y, p), (x, λ, q)) ∈ P ×Q′ , we have λ(β T y)−eT p−f T q ≤ 0,
and the equality holds iff (v, w) ∈ N .
Lemma 4.14 implies that N is captured by λ(β T y) − eT p − f T q = 0 over P × Q′ .
Using this fact and Lemma 4.14, we define the following parametrized LP.
LP (δ) : max δ(β T y) − eT p − f T q
s.t.
(y, p) ∈ P ; (x, λ, q) ∈ Q′ ; λ = δ
For an a ∈ R, let OP T (a) be the set of optimal solutions of LP (a) and N (a) be the
set of points of N with λ = a, i.e., N = {(v, w) ∈ N | w = (x, λ, q) and λ = a}.
Lemma 4.15 For an a ∈ R, N (a) 6= ∅ and OP T (a) = N (a)
Proof : Consider a game G(α) where α = [a, . . . , a]. Clearly, for any Nash equilibrium
(x, y) of G(α) the corresponding point in N has λ = xT α = a (Lemma 4.13). Therefore,
N (a) 6= ∅. The feasible set of LP (a) is all points of P × Q′ with λ = a. Further, function
λ(β T y) − eT p − f T q achieves maximum only at points on N (Lemma 4.14). Therefore,
OP T (a) = N (a) follows.
⊓
⊔
62
Lemma 4.16 The set N forms an infinite path, with λ being monotonic on it.
Proof : To the contrary suppose there are cycles and multiple paths in N . Let C be
a cycle in N . The set N (a) is exactly the intersection of N and hyper-plane λ = a.
Therefore, ∃a ∈ R, such that either C is contained in λ = a or it cuts the cycle at exactly
two points. Both the cases contradicts convexity of the set N (a) (Lemma 4.15).
Now let P1 and P2 be two paths in N . Since, N (a) is a convex set ∀a ∈ R, λ is
monotonic on both the paths. Suppose, the range of λ covered by P1 and P2 be (−∞, a]
and (a, inf). However, this contradicts the fact that P2 is a closed set. Monotonicity of λ
follows from the convexity of N (a).
4.2.1
⊓
⊔
Algorithm
P
Let (A, B) be a given rank-1 game and A + B = γ · β T . Let γmin = minx∈X i∈S1 γi xi
P
and γmax = maxx∈X i∈S1 γixi . The γmin and γmax exists since X is a bounded polytope.
From Lemma 4.13 it is clear that every point in the intersection of the set N and hyperP
plane Hγ : λ− i∈S1 γi xi = 0 corresponds to a NE of the given game (A, B). Note that for
any point in the intersection, corresponding λ is between γmin and γmax . Let Hγ− and Hγ+
be the negative and positive half spaces of the hyper-plane Hγ , then clearly N (γmin ) ∈ Hγ−
and N (γmax ) ∈ Hγ+ . All the points in the intersection of N and Hγ are between N (γmin )
and N (γmax ). The following algorithm does binary search on N between N (γmin ) and
N (γmax ) to find a point in the intersection using the fact that λ monotonically increases
(similar to the algorithm in [2]).
S1 Initialize a1 = γmin and a2 = γmax .
S2 If the edge containing N (a1 ) or N (a2 ) intersects Hγ , then output the intersection
and exit.
S3 Let a =
a1 +a2
.
2
Let u, v be the edge containing N (a).
S4 If u, v intersects Hγ , then output the intersection and exit.
S5 Else if u, v ∈ Hγ− , then set a1 = a else set a2 = a and continue from step S3 .
63
Correctness. Since the feasible set of LP (a) is a section of P × Q′ , where λ = a, the
OP T (a) is on an edge of P × Q′ (assuming non-degeneracy of LP (a)). It is easy to construct this edge from the tight equations of P × Q′ at OP T (a). Clearly, this entire edge
should be part of the set N , hence if this edge intersects the hyper-plane Hγ then we get
a Nash equilibrium of the given game. Since, all the points in the intersection of N and
Hγ are between N (γmin ) and N (γmax ), and λ is monotonically increasing between these
two (Lemma 4.16), the algorithm does a simple binary search between γmin and γmax to
find an a, such that the edge containing N (a) = OP T (a) intersects N (Lemma 4.15).
Time Complexity. Recall that L is the bit length of the input game. Since, γmin and
γmax are optimal points of two LPs on set X = {Ex = e, x ≥ 0}, they may be represented
in poly(L, m, n) bits. Let Z = max{|A|, |E|, |F |, |e|, |f |, |γ|, |β|}, l = m + n + k1 + k2 + 1,
and ∆ = l!Z l . The following theorem settles the first part of Theorem 1.4.
Theorem 4.17 The above algorithm finds a NE of game (A, B) in time poly(L, m, n).
Proof : One round of steps S3 to S5 may be done in polynomial time since computation
of N (a) requires solving LP (a) (Lemma 4.15), and computation of u, v ∩ Hγ requires
checking the feasibility of a polyhedron. Now, to show polynomial time complexity, we
need to bound the number of rounds of steps S3 to S5 .
Note that the denominator of any co-ordinate of a vertex of P × Q′ is at most ∆,
and if λ is not constant on an edge of N , then the difference in its value between the
two end points of the edge is at least
terminates. After k rounds a2 − a1 =
1
.
∆2
Therefore, if a2 − a1 <
γmax −γmin
.
2k
1
∆2
In round k if a2 − a1 =
the algorithm
γmax −γmin
2k
>
1
,
∆2
then k < log(γmax − γmin ) + 2 log ∆. Therefore, the algorithm is guaranteed to terminate
after log(γmax − γmin ) + 2 log ∆ + 1 = poly(L, m, n) many rounds.
4.3
⊓
⊔
FPTAS for Rank-k Games
In this section, we discuss fully polynomial time approximation schemes for fixed rank
games (i.e., rank(A + B) is constant). The approximation notion in bilinear games may
be defined in a similar way to that of bimatrix games given by Kannan et al. [38]. Let
P
P
xmax = maxx∈X i xi , ymax = maxy∈Y j yj and D = |A + B|. Clearly the total payoff
64
derived from a strategy profile (x, y) ∈ X × Y is at most xmax Dymax . Using this we define
an ǫ-approximate NE for a bilinear game (A, B) as follows.
Definition 4.18 For a strategy profile (x, y) ∈ X × Y , let u = maxx′ ∈X x′T Ay and v =
maxy′ ∈Y xT By ′. Then (x, y) is an ǫ-approximate NE of the game (A, B) if u + v − xT (A +
B)y ≤ ǫ(xmax Dymax ).
For a bimatrix game xmax Dymax = D, since x and y are probability distributions,
which is compatible with the definition of [38]. Next we define a stronger notion of
ǫ-approximate NE called relative ǫ-approximate NE, where the error is relative to the
maximum achievable payoff from the given strategy.
Definition 4.19 For a strategy profile (x, y) ∈ X × Y , let u = maxx′ ∈X x′T Ay and v =
maxy′ ∈Y xT By ′. Then (x, y) is a relative ǫ-approximate NE of the game (A, B) if u + v −
xT (A + B)y ≤ ǫ(u + v), i.e., the total error is relatively small.
Since the value of u + v is at most xmax Dymax , if (x, y) is relative ǫ-approximate
NE, then it is also ǫ-approximate NE. For all the examples mentioned in Section 4.1.1,
a (relative) approximate NE of the bilinear game formulation may be straightforwardly
turned into a (relative) approximate NE of the corresponding finite game under standard
definitions. Without loss of generality we assume that A, B, E, F, e and f are integer
matrices, since scaling them by a positive value does not change the set of (relative) ǫapproximate NE. Next we discuss two FPTAS to solve QP of (4.5), one for each definition
of approximation (generalization of the approaches of [38]).
4.3.1
FPTAS for Approximate NE
We show that the result by Vavasis [67] may be applied to get an ǫ-approximate Nash
equilibrium (Definition 4.18). The following proposition states the result by Vavasis.
Proposition 4.20 Let min{ 21 xT Qx + q T x : Ax ≤ b} be a quadratic optimization problem
with compact polytope {x ∈ Rn : Ax ≤ b}, and let the rank of Q be a fixed constant.
If x∗ and x# denote points minimizing and maximizing the objective function f (x) =
1 T
x Qx
2
+ q T x in the feasible region, respectively, then one can find in time poly(L, 1ǫ ) a
point x⋄ satisfying
f (x⋄ ) − f (x∗ ) ≤ ǫ(f (x# ) − f (x∗ )).
65
Next we restate QP formulation of (4.5), capturing all the NE of (A, B) at its optimal.
min: eT p + f T q − xT (A + B)y
s.t.
Ay − E T p ≤ 0; F y = f ; y ≥ 0
xT B − q T F ≤ 0; Ex = e; x ≥ 0
Theorem 4.21 Let (A, B) be a rank-k game, then for every ǫ > 0, an ǫ-approximate
Nash equilibrium may be computed in time poly(L, 1ǫ ), where L is the bit length of the
game and k is a constant.
Proof : The objective function of the above QP may be easily transformed to the standard QP form 12 xT Qx + q T x, where rank(Q) = 2k. To apply Proposition 4.20 on this QP,
we need to bound its feasible set. Since, {x : Ex = e, x ≥ 0} and {y : F y = f, y ≥ 0} are
compact, the only variables to bound are ps and qs. Since, the maximum possible value
of xT (A + B)y for any (x, y) ∈ X × Y is xmax Dymax , the value of eT p + f T q is at most
xmax Dymax at any point of the polyhedron corresponding to NE (by (4.4)). Therefore,
we impose eT p + f T q ≤ xmax Dymax . However, this may not bound the ps and qs.
Let Z = max{|A|, |B|, |E|, |F |, |e|, |f |} and l = m + n + k1 + k2 . Recall that NE
of a non-degenerate game correspond to vertices of the polyhedron. It is easy to see
that maximum absolute value of a co-ordinate of any vertex in the polyhedron is at most
l!Z l . Further, the quantity l!Z l may be represented in poly(L) bits. Therefore, imposing
−l!Z l ≤ p ≤ l!Z l and −l!Z l ≤ q ≤ l!Z l in the above QP incur only a polynomial
increase in its representation and does not change its optimal set. The minimum and the
maximum objective values of this QP are zero (Lemma 4.14) and at most 2xmax Dymax
respectively. Let ((y ⋄, p⋄ ), (x⋄ , q ⋄ )) be the solution given by Vavasis algorithm for 2ǫ , then
from Proposition 4.20 we get,
eT p⋄ + f T q ⋄ − x⋄T (A + B)y ⋄ ≤ ǫ(xmax Dymax )
From the primal-dual formulation of (4.1) it is clear that maxx′ ∈X x′T Ay ⋄ ≤ eT p⋄ and
maxy′ ∈Y x⋄T By ′ ≤ f T q ⋄ . Therefore, we get maxx′ ∈X x′T Ay ⋄ + maxy′ ∈Y x⋄T By ′ − x⋄T (A +
B)y ⋄ ≤ ǫ(xmax Dymax ).
4.3.2
⊓
⊔
FPTAS for Relative Approximate NE
Let the rank of a game (A, B) be k, then A + B =
Pk
l=1 α
l lT
β , where αl ∈ Rm and
β l ∈ Rn , ∀l ≤ k. We assume that the game is such that αl s and β l s are positive vectors.
66
For all l ≤ k, let wl = minx∈X xT αl and wl′ = maxx∈X xT αl , similarly let zl = miny∈Y y T β l
and zl′ = maxy∈Y y T β l . Note that wl , wl′, zl and zl′ may be represented by poly(L, m, n)
bits, since X and Y are compact. Given an ǫ > 0, consider the sub-intervals [wl , (1+ǫ)wl ],
[(1+ǫ)wl , (1+ǫ)2 wl ] of [wl , wl′ ] and similarly of [zl , zl′ ]. All combinations of these intervals
form a grid in 2k-dimensional box B = ×l [wl , wl′ ] ×l [zl , zl′ ]. Let (x, y) ∈ X × Y be such
that xT αl ∈ [ul , (1 + ǫ)ul ] and y T β l ∈ [vl , (1 + ǫ)vl ] ∀l, then clearly,
k
X
ul vl ≤ xT (A + B)y ≤ (1 + ǫ)2
k
X
ul vl
(4.9)
l=1
l=1
For a fixed hyper-cube of the grid, consider the following LP based on (4.5).
min: eT p + f T q
s.t.
Ay ≤ E T p, F y = f, y ≥ 0
xT B ≤ q T F ; Ex = e; x ≥ 0
ul ≤ xT αl ≤ (1 + ǫ)ul , vl ≤ y T β l ≤ (1 + ǫ)vl , ∀l ≤ k
Algorithm. Run the above LP for each hyper-cube of the grid, and output an optimal
point of the one giving the best approximation. As the number of hyper-cubes in the grid
is poly(L, 1/ log(1 + ǫ)), the running time of the algorithm is poly(m, n, L, 1/ log(1 + ǫ)).
1
Correctness. Next we show that the above algorithm gives 1 − (1+ǫ)
2 -approximate
NE of the game (A, B). Let (x′ , y ′) be a NE of the given game, and (p′ , q ′ ) be such
that x′T Ay ′ = eT p′ and x′T By ′ = f T q ′ . Consider the hyper-cube containing (x′T α1 , . . . ,
x′T αk , y ′T β 1 , . . . , y ′T β k ) of the grid and corresponding LP. Clearly, (x′ , y ′, p′ , q ′ ) is a feasible
P
P
point of this LP and kl=1 ul vl ≤ eT p′ + f T q ′ ≤ (1 + ǫ)2 kl=1 ul vl , since eT p′ + f T q ′ =
x′T (A + B)y ′ . Therefore, at the optimal point (x̃, ỹ, p̃, q̃) of the LP we get eT p̃ + f T q̃ ≤
P
(1 + ǫ)2 kl=1 ul vl , and this gives,
x̃T (A + B)ỹ ≥
Pk
i=1
ul vl ≥
eT p̃+f T q̃
(1+ǫ)2
⇒ eT p̃ + f T q̃ − x̃T (A + B)ỹ ≤ 1 −
Let µ = 1 −
1
(1+ǫ)2
(using (4.9))
1
(eT p̃ + f T q̃)
2
(1+ǫ)
, ũ = maxγ∈X γ T Aỹ, and ṽ = maxγ∈Y x̃T Bγ. Clearly, eT p̃ ≥ ũ
and f T q̃ ≥ ṽ (using (4.1)). Let D = eT p̃ + f T q̃ − ũ − ṽ, then ũ + ṽ − x̃T (A + B)ỹ =
eT p̃ + f T q̃ − D − x̃T (A + B)ỹ ≤ µ(eT p̃ + f T q̃) − D ≤ µ(eT p̃ + f T q̃ − D) = µ(ũ + ṽ), since
µ ∈ (0, 1]. Therefore, (x̃, ỹ) is a relative µ-approximate NE of the given game (A, B)
(Definition 4.19).
67
Pk
l lT
l
l=1 α β , such that α s
1
and β l s are positive vectors. Then given an ǫ > 0, a relative 1 − (1+ǫ)
2 -approximate
Theorem 4.22 Let (A, B) be a rank-k game, and A + B =
NE may be computed in time poly(L, 1/ log(1 + ǫ)), where L is the input bit length.
For a symmetric game (B = AT , E = F, e = f ), an (relative) ǫ-approximate symmetric NE may be defined as an (relative) ǫ-approximate NE with the same strategies,
i.e., x = y. Note that, if we use the QP formulation of (4.6) instead of (4.5) in any of
these algorithms, then the output strategy is an (relative) ǫ-approximate symmetric NE
strategy. Theorems 4.21 and 4.22 settles the second part of Theorem 1.4.
4.4
Games With a Low Rank Matrix
In this section we show that if rank of even one payoff matrix (A or B) is constant, then
Nash equilibrium computation may be done in polynomial time. Recall the best response
polyhedra P and Q (4.3) for the bilinear game (A, B).
Lemma 4.23 Given a game (A, B), there exists a vertex pair ((y, p), (x, q)) ∈ P ×Q such
that (x, y) is a NE of (A, B).
Proof : All the solutions of (4.4) over P × Q are the NE of the game, and existence of
a solution is guaranteed (Proposition 4.3). Further, it is easy to check that if a solution
lies on a face of P × Q then the entire face is solution, which contains at least one vertex,
as P × Q is bounded from one side.
⊓
⊔
Lemma 4.24 If k1 = k2 = k and rank(A) = l, then P has at most O(nl+k ) vertices.
Proof : From (4.3), it is clear that P is in (n+ k)-dimensional Euclidean space, however
F y = f gives k linearly independent equalities. Therefore, P is of dimension n, and at a
vertex of P , n linearly independent inequalities must be tight. Since A is of rank l, rank([A
-E]) ≤ l + k. Therefore, ∃!S ⊂ S1 , |S| = l + k such that ∀i ∈ S1 \ S, Ai y − pT E i ≤ 0 are
not needed in defining the polyhedron P . At a vertex, if d inequalities are tight from S
then rest n − d must be of type yj = 0, hence for a fixed D ⊂ S, |D| = d, there are at
n
most n−d
= nd choices to form a vertex. Therefore, the total number of vertices are at
P
l+k n
≤ 2l+k nl+k .
⊓
⊔
most l+k
i=0
i
i
Note that if we remove the assumption k1 = k2 = k then the exponent of n turns
out to be a linear function of l, k1 and k2 . A similar proof may be worked out for Q.
68
Theorem 4.25 If rank of either A or B is constant then a Nash equilibrium of a bilinear
game (A, B) may be computed in polynomial time, assuming k to be a constant.
Proof : Suppose rank(A) = l (a constant) and v = (y, p) ∈ P be a vertex. Whether v
corresponds to a NE or not can be checked as follows: Let I = {i ∈ S1 | Ai y − pT E i = 0}
and J = {j ∈ S2 | yj > 0}. Consider all (x, q) ∈ Rm+k such that
Ex = e;
∀j ∈ J, xT B j − q T F j = 0; ∀i ∈ I, xi ≥ 0
∀j ∈
/ J, xT B j − q T F j ≤ 0; ∀i ∈
/ I, xi = 0.
Every such (x, q) lies in Q and makes a fully-labeled pair with v, and hence forms a NE
(Lemma 4.4). Note that such an (x, q) may be obtained in polynomial time by solving an
LP. The proof follows from Lemmas 4.23 and 4.24. A similar argument proves the other
⊓
⊔
case when rank(B) is a constant.
Theorem 4.25 settles the third part of Theorem 1.4. As the set of bimatrix games
is a subclass of the bilinear games (Example 4.6), where k1 = k2 = 1, Theorem 4.25
strengthens the results by Lipton, Markakis and Mehta [46] (Corollary 4), and Kannan
and Theobald [38] (Theorem 3.2), where they require that the rank of both A and B to be
constants. Note that k in Bayesian games depends on the number of types of players and
in the sequence form, it depends on the number of information sets of players. Therefore,
this result may be applied to these games if in their bilinear representation, a payoff
matrix has low rank and k is constant.
In fact Theorem 4.25 gives a polynomial time algorithm to enumerate all the extreme equilibria of a bilinear game with a constant rank matrix, and an exponential time
enumeration algorithm for any bilinear game. A similar (exponential time) algorithm was
given by Avis et al. [5] to enumerate all the Nash equilibria of a bimatrix game.
4.5
Discussion
In this chapter we extended various combinatorial and algorithmic results, pertaining to
the efficient computation of Nash equilibria, from bimatrix games to bilinear games. It
will be interesting to explore more subclasses of bilinear games and to extend other results
of bimatrix games like Lemke-Howson type algorithm for NE computation, and extending
other algorithms for computation of approximate equilibria.
69
Chapter 5
Fisher Market Game
Recall the Fisher market model [6] with linear utilities described in Section 1.3 of Chapter
1. The market consists of a set B of buyers and set G of divisible goods; let m = |B| and
n = |G|. Buyer i comes with a positive endowment of money and a utility function; mi
is the money possessed by agent i, and uij represents the utility she derives from a unit
amount of good j. Over all utility of buyer i from a bundle x = (x1 , . . . , xn ) of goods is
P
ui (x) = j∈G uij xj . W.l.o.g we assume that quantity of each good in the market is one,
and for every good j there is a buyer i with uij > 0.
At prices p = (p1 , . . . , pn ) the utility maximizing bundle x (optimal bundle) should
satisfy the following:
• Optimal Bundle: xij > 0 ⇒ uij /pj = maxk∈G uik /pk .
At p, after every buyer is assigned an optimal bundle, if the market clears then p is
called a market equilibrium price vector. For prices p and corresponding optimal bundles
[xij ]i∈B,j∈G , the market clearing condition may be formally stated as,
• Market Clearing:
P
i∈B
xij = 1, ∀j ∈ G and
P
j∈G
xij pj = mi , ∀i ∈ B.
It is known that in linear Fisher market equilibrium prices are unique and the set
of equilibrium allocations is a convex set [68]. Note that, the utility functions influence
the equilibrium prices, which in turn influences the utility derived by every buyer. This
makes it possible for a buyer to gain by feigning a different utility tuple as discussed in
Example 1.5 in Chapter 1, implying that the Fisher market is susceptible to gaming by
strategic buyers. Therefore, the equilibrium prices w.r.t. the true utility tuples may not
70
be the actual operating point of the market. The natural questions to investigate are:
What are the possible operating points of this market model under strategic behavior?
Can they be computed? Is there a preferred one?
Motivated by these questions we define Fisher market game (Section 5.1), where
buyers are the players and strategies are the utility tuples that they may pose, and study
its Nash equilibria. We show that existence of a conflict-free allocation (acceptable to all
the players), is a necessary condition for the Nash equilibria (NE) and also sufficient for
the symmetric NE in this game (Section 5.2). There are many NE with very different
payoffs, and the Fisher equilibrium payoff is captured at a symmetric NE. Further, we
show that there are NE with sub-optimal payoffs as well, and discuss the difficulties in
the complete characterization of Nash equilibria; even for markets with only three buyers.
For markets game with two buyers, we provide a complete polyhedral characterization of all the NE (Section 5.3). Surprisingly, all the NE of such a game turn out to
be symmetric and the corresponding payoffs constitute a piecewise linear concave curve.
For every point on this curve, there is a convex set of NE, leading to a different class
of non-market behavior such as incentives (Section 5.3.3). We also study the correlated
equilibria of this game and show that third-party mediation does not help to achieve a
better payoff than NE payoffs (Section 5.4).
5.1
The Fisher Market Game
As defined above, a linear Fisher market is defined by the tuple (B, G, (ui)i∈B , m), where
B is a set of buyers, G is a set of goods, ui = (uij )j∈G is the true utility tuple of buyer
i, and m = (mi )i∈B is the endowment vector. We assume that |B| = m, |G| = n and the
quantity of every good is one unit.
The Fisher market game is a one shot, non-cooperative game with a set of players
(B) who are the buyers in the market1 , and a set of goods (G) to be sold. For each player
i ∈ B, the money mi possessed by her is assumed to be known, while true utility tuple ui
is her private information.
The strategy set of player i comprise of all possible utility tuples that she may pose,
P
i.e., Si = {hsi1 , si2 , . . . , sin i | sij ≥ 0,
j∈G sij 6= 0}, ∀i ∈ B. Clearly, the set of all
1
Henceforth we use buyers and players interchangeably
71
strategy profiles is S = S1 × · · · × Sm . When a strategy profile S = (s1 , . . . , sm) is played,
where si ∈ Si , we treat s1 , . . . , sm as utility tuples of buyers 1, . . . , m respectively, and
compute the equilibrium prices and a set of equilibrium allocations w.r.t. S and m.
Using the equilibrium prices (p1 , . . . , pn ), we generate the corresponding solution
graph G as follows: Let V (G) = B ∪ G. Let bi be the node corresponding to the buyer
i, ∀i ∈ B and gj be the node corresponding to the good j, ∀j ∈ G in G. We place an
edge between bi and gj iff
sij
pj
= maxk∈G
sik
,
pk
and call the edges of the solution graph as
tight edges. Note that when the solution graph is a forest, there is exactly one equilibrium
allocation, however this is not so, when it contains cycles. In the standard Fisher market
(i.e., strategy of every buyer is her true utility tuple), all equilibrium allocations give the
same payoff to a buyer. However, this is not so when buyers strategize on their utility
tuples: Different equilibrium allocations may not give the same payoff to a buyer. The
following example illustrates this scenario.
Example 5.1 Consider the Fisher market of Example 1.5. Consider the strategy profile S = (h1, 19i, h1, 19i). Then, the equilibrium prices hp1 , p2 i are h1, 19i and the solution
graph is a cycle. There are many equilibrium allocations and the allocations [x11 , x12 , x13 , x14 ]
10
10
9
9
, 0, 19
] and [0, 19
, 1, 19
] respecachieving the highest payoff for buyers 1 and 2 are [1, 19
tively. The payoffs corresponding to these allocations are (11.42, 5.26) and (1.58, 7.74)
respectively. Note that there is no allocation, which gives the maximum possible payoff to
both the buyers.
Let p(S) = (p1 , . . . , pn ) be the equilibrium prices, G(S) be the solution graph, and
X(S) be the set of equilibrium allocations w.r.t. a strategy profile S. The payoff w.r.t.
P
X ∈ X(S) is defined as (u1(X), . . . , um (X)), where ui (X) = j∈G uij xij . Let wi(S) =
maxX∈X(S) ui(X), ∀i ∈ B.
Definition 5.2 A strategy profile S is said to be conflict-free if ∃X ∈ X(S), s.t.
ui (X) = wi (S), ∀i ∈ B. Such an X is called a conflict-free allocation.
When a strategy profile S = (s1 , . . . , sm) is not conflict-free, there is a conflict of
interest in selecting a particular allocation for the play. If a buyer, say k, does not get
the same payoff from all the equilibrium allocations, i.e., ∃X ∈ X(S), uk (X) < wk (S),
then we show that for every δ > 0, there exists a strategy profile S ′ = (s′1 , . . . , s′m), where
72
s′i = si, ∀i 6= k, such that uk (X ′ ) > wk (S) −δ, ∀X ′ ∈ X(S ′ ) (Section 5.2.1). The following
example illustrates the same.
Example 5.3 In Example 5.1, for δ = 0.1, consider S ′ = (h1.1, 18.9i, h1, 19i), i.e., buyer
1 deviates slightly from S. Then, p(S ′ ) = h1.1, 18.9i, and G(S ′ ) is a tree; the cycle of
Example 5.1 is broken. Hence there is a unique equilibrium allocation, and w1 (S ′ ) =
11.41, w2 (S ′ ) = 5.29.
Therefore, if a strategy profile S is not conflict-free, then for every choice of allocation
X ∈ X(S) to decide the payoff, there is a buyer who may deviate and assure herself a
better payoff. In other words, when S is not conflict-free, there is no way to choose an
allocation X from X(S) acceptable to all the buyers. This suggests that only conflict-free
strategies are interesting. Therefore, we may define the payoff function Pi : S → R for
each player i ∈ B as follows:
∀S ∈ S, Pi (S) = ui(X), where X = arg max
X ′ ∈X(S)
Y
ui (X ′ ).
(5.1)
i∈B
The payoff functions are well-defined, and when S is conflict-free, Pi (S) = wi (S), ∀i ∈ B.
5.2
Nash Equilibria: A Characterization
In this section, we prove some necessary conditions for a strategy profile to be a NESP
of the Fisher market game defined in the previous section. Nash equilibrium [49] is a
solution concept for games with two or more rational players. When a strategy profile is
a NESP, no player benefits by changing her strategy unilaterally.
For technical convenience, we assume that uij > 0 and sij > 0, ∀i ∈ B, ∀j ∈ G. The
boundary cases may be easily handled separately. Note that if S = (s1 , . . . , sm) is a NESP
then S ′ = (α1 s1 , . . . , αm sm), where α1 , . . . , αm > 0, is also a NESP. Therefore, w.l.o.g. we
P
consider only the normalized strategies si = hsi1 , . . . , sin i, where j∈G sij = 1∗ , ∀i ∈ B.
As mentioned in the previous section, the true utility tuple of buyer i is hui1, . . . , uin i.
P
P
For convenience, we may assume that j∈G uij = 1 and i∈B mi = 1 (w.l.o.g.).
We show that all NESPs are conflict-free, but not all conflict-free strategies are
NESPs. A symmetric strategy profile, where all players play the same strategy (∀i, j ∈
∗
For simplicity, we do use non-normalized strategy profiles in the examples.
73
B, si = sj ), is a NESP iff it is conflict-free. If a strategy profile S is not conflict-free, then
there is a buyer a such that Pa (S) < wa (S). The ConflictRemoval procedure in the next
section describes how she may deviate and assure herself payoff almost equal to wa (S).
5.2.1
Conflict Removal Procedure
Definition 5.4 Let S be a strategy profile, X ∈ X(S) be an allocation, and P = v1 , v2 , v3 , . . .
be a path in G(S). P is called an alternating path w.r.t. X, if the allocation on the
edges at odd positions is non-zero, i.e., xv2i−1 v2i > 0, ∀i ≥ 1. The edges with non-zero
allocation are called non-zero edges.
The ConflictRemoval procedure in Table 5.1 takes a strategy profile S, a buyer a and
a positive number δ, and outputs another strategy profile S ′ , where s′i = si, ∀i 6= a such
that ∀X ′ ∈ X(S ′ ), ua (X ′ ) > wa (S) − δ. Observe that if a buyer, say a, does not belong
to any cycle in the solution graph of a strategy profile S, then ua (X) = wa (S), ∀X ∈
X(S). The procedure essentially breaks all the cycles containing ba in G(S) using the
Perturbation procedure iteratively such that the payoff of buyer a does not decrease by
more than δ.
This is done by first picking a good b from J, which gives the maximum payoff per
unit of money to buyer a among all the goods in J, where J is the set of goods j, such
that the edge (ba , gj ) belongs to a cycle in G(S). Then, it picks an allocation X = [xij ]
such that ua (X) = wa (S) and xab is maximum among all allocations in X(S). It is easy
to check that such an X exists. Finally, using Perturbation procedure, it obtains another
strategy profile S ′ , where s′i = si, ∀i 6= a and wa (S ′ ) > wa (S) − nδ . The edge (ba , gb )
does not belong to any cycle in G(S ′ ) and E(G(S ′ )) ⊂ E(G(S)). Then, it repeats the
above steps for the strategy profile S ′ until ba belongs to a cycle in the solution graph.
Clearly, there may be at most n repetition steps and the final strategy profile S ′ is such
that ua (X ′ ) > wa (S) − δ, ∀X ′ ∈ X(S ′ ).
The Perturbation procedure in Table 5.1 takes a strategy profile S, a buyer a, a good
b, an allocation X ∈ X(S), where xab is maximum among all allocations in X(S) and a
positive number γ, and outputs another strategy profile S ′ such that s′i = si, ∀i 6= a and
wa (S ′ ) > ua (X) − γ. It essentially breaks the cycles containing the edge (ba , gb ) in G(S).
If (ba , gb ) does not belong to a cycle in G(S), then it outputs S ′ = S. Otherwise, let
74
ConflictRemoval(S, ba , δ)
while ba belongs to a cycle in G(S) do
(p1 , . . . , pn ) ← p(S);
J ← {j ∈ G | the edge (ba , gj ) belongs to a cycle in G(S)};
gb ← arg max
j∈J
uaj
;
pj
X ← an allocation in X(S) such that ua (X) = wa (S) and xab is maximum;
S ← Perturbation(S, X, ba , gb , nδ );
endwhile
return S;
Perturbation(S, X, ba , gb , γ)
S ′ ← S;
if (ba , gb ) does not belong to a cycle in G(S) then
return S ′ ;
endif
J1 ← {v | there is an alternating path from ba to v in G(S) \ (ba , gb ) w.r.t. X};
J2 ← {v | there is an alternating path from gb to v in G(S) \ (ba , gb ) w.r.t. X};
P
P
(p1 , . . . , pn ) ← p(S); l ← gj ∈J1 pj ; r ← gj ∈J2 pj ;
W.r.t. α, define prices of goods to be
∀gj ∈ J1 : (1 − α)pj ; ∀gj ∈ J2 : (1 +
lα
)pj ;
r
∀gj ∈ G \ (J1 ∪ J2 ) : pj ;
Raise α infinitesimally starting from 0 such that none of the three events occur:
Event 1: a new edge becomes tight;
Event 2: a non-zero edge becomes zero;
Event 3: payoff of buyer a becomes ua (X) − γ;
s′ab ← sab
)
(1+ lα
r
;
(1−α)
s′a ←
′
P sa
j∈G
s′aj
;
return S ′ ;
Table 5.1: Conflict Removal Procedure
J1 and J2 be the sets of buyers and goods to which there is an alternating path w.r.t. X
starting from ba and gb in G(S) \ (ba , gb ) respectively. Note that J1 ∩ J2 = φ, otherwise
there is an alternating path P from ba to gb in G(S)\(ba , gb ), and using P with (ba , gb ), xab
75
may be increased and another allocation X ′ ∈ X(S) may be obtained, where x′ab > xab ,
which contradicts the maximality of xab .
If the prices of goods in J1 are decreased and the prices of goods in J2 are increased,
then clearly all the cycles in G(S) containing the edge (ba , gb) break. Such a price change
may be forced by increasing sab infinitesimally. The procedure first finds such a price
change and then constructs an appropriate strategy profile. Let (p1 , . . . , pn ) = p(S), l =
P
P
′
gj ∈J1 pj , r =
gj ∈J2 pj and α be a variable. The prices are changed as pj = (1 − α)pj
for gj ∈ J1 , p′j = (1 +
lα
)pj
r
for gj ∈ J2 and p′j = pj for the remaining goods. When α is
increased continuously starting from 0, the corresponding changes in prices may trigger
any of the following three events:
Event 1: A new edge may become tight from a buyer outside J1 to a good in J1 or from
a buyer in J2 to a good outside J2 .
Event 2: To reflect the price change, the money has to be pulled out from the goods in
J1 and transferred to the goods in J2 through the edge (ba , gb). This may cause a non-zero
edge in J1 or J2 to become zero.
Event 3: Since the price of good b as well as the allocation on the edge (ba , gb) is
increasing, the payoff of buyer a may decrease and become equal to ua (X) − γ.
The procedure finds α > 0 such that none of the three events occur, and constructs
a new strategy profile S ′ , where s′ij = sij , ∀(i, j) 6= (a, b), and s′ab = sab
(1+ lα
)
r
.
(1−α)
Clearly,
p(S ′ ) = (p′1 , . . . , p′n ), and G(S ′ ) = (G[J1 ] ∪ G[J2 ] ∪ G[V (G) \ (J1 ∪ J2 )]) + (ba , gb ), where
G[U] denotes the induced graph on U ⊆ B ∪ G in G(S).
In the next theorem, we use the ConflictRemoval procedure to show that all the
NESPs in the Fisher market game are conflict-free. It settles a part of the informal
Theorem 1.6.
Theorem 5.5 If S is a NESP, then
(i) ∃X ∈ X(S) such that ui(X) = wi (S), ∀i ∈ B, i.e., S is conflict-free.
(ii) the degree of every good in G(S) is at least 2.
(iii) for every buyer i ∈ B, ∃ki ∈ Ki s.t. xiki > 0, where Ki = {j ∈ G |
maxk∈G
uik
},
pk
(p1 , . . . , pn ) = p(S) and [xij ] is a conflict-free allocation.
76
uij
pj
=
Proof : Suppose there does not exist an allocation X ∈ X(S) such that ui(X) =
wi (S), ∀i ∈ B, then there is a buyer k ∈ B, such that Pk (S) < wk (S). Clearly, buyer
k has a deviating strategy (apply ConflictRemoval on the input tuple (S, k, δ), where
0 < δ < (wk (S) − Pk (S))), which is a contradiction.
For part (ii), if a good b is connected to exactly one buyer, say a, in G(S), then
buyer a may gain by reducing sab , so that price of good b decreases and prices of all other
goods increase by the same factor.
For part (iii), if there exists a buyer i such that xiki = 0, ∀ki ∈ Ki , then she may
gain by increasing the utility for a good in Ki .
⊓
⊔
The following example shows that the above conditions are not sufficient.
Example 5.6 Consider a market with 3 buyers and 2 goods, where m = h50, 100, 50i,
u1 = h2, 0.1i, u2 = h4, 9i, and u3 = h0.1, 2i. Consider the strategy profile S = (u1 , u2 , u3 )
given by the true utility tuples. The payoff tuple w.r.t. S is (1.63, 6.5, 0.72). It satisfies
all the necessary conditions in the above theorem, however S is not a NESP because buyer
2 has a deviating strategy s′2 = h2, 3i and the payoff w.r.t. strategy profile (s1 , s′2 , s3 ) is
(1.25, 6.75, 0.83).
5.2.2
Symmetric and Asymmetric NESPs
Recall that a strategy profile S = (s1 , . . . , sm) is said to be a symmetric strategy profile
if s1 = · · · = sm, i.e., all buyers play the same strategy. The following theorem settles a
part of Theorem 1.6.
Proposition 5.7 A symmetric strategy profile S is a NESP iff it is conflict-free.
Proof : (⇒) is straightforward (Theorem 5.5). For the other direction, let S = (s, . . . , s)
be a conflict-free symmetric strategy profile. Clearly, G(S) is a complete bipartite graph
and Pi (S) = wi (S), ∀i ∈ B. If S is not a NESP, then there is a buyer, say k, who
may deviate and get a better payoff. Let S ′ = (s′1 , . . . , s′m) be a strategy profile, where
s′i = s, ∀i 6= k, such that Pk (S ′ ) > Pk (S). Let X ′ ∈ X(S ′) be such that uk (X ′ ) = Pk (S ′ ).
Let (p1 , . . . , pn ) = p(S) and (p′1 , . . . , p′n ) = p(S ′ ). If p(S ′ ) = p(S), i.e., pj = p′j , ∀j ∈
G, then uk (X ′ ) ≤ wk (S). For the other case p(S ′ ) 6= p(S), let J1 = {j ∈ G | p′j < pj },
J2 = {j ∈ G | p′j = pj }, and J3 = {j ∈ G | p′j > pj }. Note that all buyers except k will
77
have edges only to the goods in J1 in G(S ′ ), i.e., goods whose prices have been decreased.
From X ′ , we can construct an allocation X ∈ X(S), such that ∀j ∈ J2 ∪ J3 , xkj = x′kj
and ∀j ∈ J1 , xkj > x′kj . Hence uk (X ′ ) < uk (X) ≤ wk (S), which is a contradiction.
⊓
⊔
Let S f = [sij ] be a strategy profile, where sij = uij , ∀i ∈ B, ∀j ∈ G, i.e., true
utility functions. All allocations in X(S f ) give the same payoff to the buyers (i.e., ∀i ∈
B, ui (X) = wi (S f ), ∀X ∈ X(S f )), and we define Fisher payoff (uf1 , . . . , ufm ) to be the
payoff derived when all buyers play truthfully.
Corollary 5.8 A symmetric NESP can be constructed, whose payoff is the same as the
Fisher payoff.
Proof : Let S = (s, . . . , s) be a strategy profile, where s = p(S f ). Clearly S is a
⊓
⊔
symmetric NESP, whose payoff is the same as the Fisher payoff.
Proposition 5.9 There is exactly one symmetric NESP in a Fisher market game iff the
degree of every good in G(S f ) is at least two.
Proof : (⇒) If there is exactly one symmetric NESP in a Fisher market game, then it
has to be S = (s, . . . , s), where s = p(S f ). Let (p1 , . . . , pn ) = s. If the degree of a good,
say gl , in G(S f ) is one, then we may construct more symmetric NESPs. Let bk be the
buyer connected to gl in G(S f ). It is easy to check that there exist α < 1 and β > 1, such
P
that S ′ = (s′, . . . , s′), where s′ = (βp1, . . . , βpl−1 , αpl , βpl+1, . . . , βpn ) and j∈G s′j = 1, is
a symmetric NESP.
(⇐) If the degree of every good in G(S f ) is at least two, then we show that there is
exactly one symmetric NESP given by S = (s, . . . , s), where s = p(S f ). Let (p1 , . . . , pn ) =
p(S f ). Suppose there exists another symmetric NESP S ′ = (s′, . . . , s′), where s′ =
(p′1 , . . . , p′n ), then consider the set J = {gj ∈ G |
mink∈G
p′k
pk
p′j
pj
= mink∈G
p′k
}.
pk
Since s′ 6= s, hence
< 1. We show that the degree of every good in J is exactly one in G(S f ) to
get the contradiction. Let Γ(J) be the set of buyers connected to J in G(S f ). W.r.t.
S ′ , every buyer in Γ(J) gets the highest bang-per-buck from a good in J and the sum of
money of buyers in Γ(J) is more than the sum of prices of goods in J. Therefore, it is easy
to check that if the degree of a good in J is more than one, then it will cause a conflict
among the buyers connected to it in G(S f ). Hence, S ′ is not a symmetric NESP.
78
⊓
⊔
Later (Lemma 5.25) we show that the payoff w.r.t. a symmetric NESP is always
Pareto optimal. The characterization of all the NESPs for the general market game
seems hard; even for markets with only three buyers. The following example illustrates
an asymmetric NESP, whose payoff is not Pareto optimal.
Example 5.10 Consider a market with 3 buyers and 2 goods, where m = h50, 100, 50i,
u1 = h2, 3i, u2 = h4, 9i, and u3 = h2, 3i. Consider the two strategy profiles given by
S1 = (s1 , s2 , s3 ) and S2 = (s, s, s), where s1 = h2, 0.1i, s2 = h2, 3i, s3 = h0.1, 3i, and
s = h2, 3i. The payoff tuples w.r.t. S1 and S2 are (1.25, 6.75, 1.25) and (1.25, 7.5, 1.25)
respectively. Next we show that S1 and S2 are NESPs for the above market.
Clearly, S2 is a NESP because it is a symmetric strategy profile and there is a conflictfree allocation (Lemma 5.7). For S1 , it can be easily checked that buyers 1 and 3 have no
deviating strategy. Buyer 2 is essentially the price setter. However, no matter whatever
the strategy, buyer 2 plays, buyer 1 will buy only good 1 and buyer 3 will buy only good
3. Hence, let h2 =
4∗x
50+x
+
9∗(100−x)
150−x
be the payoff of buyer 2, when she gives x amount of
money to good 1. Note that h2 is a concave curve. We compute the maximum value of h2
w.r.t. x ∈ [0, 100], which turns out to be for x = 30. Hence buyer 2 also has no deviating
strategy at S1 .
5.3
The Two-Buyer Markets
A two-buyer market consists of two buyers and a number of goods. These markets arise in
numerous scenarios. The two firms in a duopoly may be considered as the two buyers with
a similar requirements to fulfill from a large number of suppliers, for example, relationship
between two big automotive companies with their suppliers.
In this section, we study two-buyer market game and provide a complete polyhedral
characterization of NESPs, all of which turn out to be symmetric. Next, we study how
the payoffs of the two buyers change with varying NESPs and show that these payoffs
constitute a piecewise linear concave curve. For a particular payoff tuple on this curve,
there is a convex set of NESPs, hence a convex set of equilibrium prices, which leads to a
different class of non-market behavior such as incentives. The next lemma settles a part
of Theorem 1.6.
Lemma 5.11 All NESPs for a two-buyer market game are symmetric.
79
Proof : If a NESP S = (s1 , s2 ) is not symmetric, then G(S) is not a complete bipartite
graph. Therefore there is a good, which is exclusively bought by a buyer, which is a
⊓
⊔
contradiction (Theorem 5.5, part (ii)).
5.3.1
Polyhedral Characterization of NESPs
In this section, we compute all the NESPs of a Fisher market game with two buyers.
Henceforth we assume that the goods are so ordered that
u1j
u2j
≥
u1(j+1)
,
u2(j+1)
for j = 1, . . . , n−1.
Chakraborty et al. [9] also use such an ordering to design an algorithm for the linear
Fisher market with two agents. Let S = (s, s) be a NESP, where s = (s1 , . . . , sn ) and
(p1 , . . . , pn ) = p(S). The graph G(S) is a complete bipartite graph. Since m1 + m2 = 1
P
and nj=1 sj = 1, we have pj = sj , ∀j ∈ G. In a conflict-free allocation X ∈ X(S), if
x1i > 0 and x2j > 0, then clearly
u1i
pi
≥
u1j
pj
and
u2i
pi
≤
u2j
.
pj
Definition 5.12 An allocation X = [xij ] is said to be a nice allocation, if it satisfies
the property: x1i > 0 and x2j > 0 ⇒ i ≤ j.
The main property of a nice allocation is that if we consider the goods in order, then
from left to right, goods get allocated first to buyer 1 and then to buyer 2 exclusively,
however they may share at most one good in between. Note that a symmetric strategy
profile has a unique nice allocation.
Lemma 5.13 Every NESP has a unique conflict-free nice allocation.
Proof : Let S be a NESP, which does not have a conflict-free nice allocation. Consider
a conflict-free allocation X ∈ X(S). Then w.r.t. X, there are goods i and j such that
x1i > 0, x2j > 0 and i > j.
Since we started with a conflict-free allocation,
Moreover, we assumed that
u1j
u2j
≥
u1i
,
u2i
hence
u1i
pi
=
u1j
pj
u1i
pi
and
≥
u1j u2i
, pi
pj
u2j
pj
=
u2i
.
pi
≤
u2j
pj
⇒
u1j
u2j
≤
u1i
.
u2i
Hence, buyer 1 may
take away some money from good i and spend it on good j and buyer 2 may take away
some money from good j and spend it on good i without affecting the payoffs. This gives
another conflict-free allocation. We may repeat this operation till we get a nice allocation.
There is exactly one nice allocation in X(S), hence the uniqueness follows.
⊓
⊔
The non-zero edges in a nice allocation either form a tree or a forest containing
two trees. We use the properties of nice allocations and NESPs to give the polyhedral
80
characterization of all the NESPs. The convex sets Bk for all 1 ≤ k ≤ n, as given in Table
5.2, correspond to all possible conflict-free nice allocations, where non-zero edges form a
tree, and the convex sets Bk′ for all 1 ≤ k ≤ n − 1, as given in Table 5.3, correspond
to all possible conflict-free nice allocations, where non-zero edges form a forest∗ . Let
n−1
B = ∪nk=1 Bk ∪k=1
Bk′ and S N E = {(α, α) | α = (α1 , . . . , αn ) ∈ B}. Note that S N E is a
connected set.
Pk−1
αi
<
m1
i=k+1 αi
Pn
i=1 αi
<
m2
=
m1 + m2
i=1
Pn
u1j αi − u1i αj
≤ 0 ∀i ≤ k, ∀j ≥ k
u2i αj − u2j αi
≤ 0 ∀i ≤ k, ∀j ≥ k
αi
Pk
i=1 αi
=
m1
i=k+1 αi
=
m2
Pn
u1j αi − u1i αj
≤ 0 ∀i ≤ k, ∀j ≥ k + 1
u2i αj − u2j αi
≤ 0 ∀i ≤ k, ∀j ≥ k + 1
αi
≥ 0 ∀i ∈ G
≥ 0 ∀i ∈ G
Table 5.3: Bk′
Table 5.2: Bk
Lemma 5.14 A strategy profile S is a NESP iff S ∈ S N E .
Proof : (⇐) is easy by the construction and Proposition 5.7. For the other direction, we
know that every NESP has a conflict-free nice allocation (Lemma 5.13), and B corresponds
⊓
⊔
to all possible conflict-free nice allocations.
5.3.2
The Payoff Curve
In this section, we consider the payoffs obtained by both the players at various NESPs.
Recall that whenever a strategy profile S is a NESP, Pi (S) = wi (S), ∀i ∈ B. Henceforth,
we use wi (S) as the payoff of buyer i for the NESP S. Let F = {(w1(S), w2 (S)) | S ∈ S N E }
be the set of all possible NESP payoff tuples.
Let X be the set of all nice allocations, and H = {(u1 (X), u2(X)) | X ∈ X }. For
α ∈ [0, 1], let t(α) = (hs1 , . . . , sn i, hs1 , . . . , sn i), where si = u1i + α(u2i − u1i ), and
G = {(w1 (S), w2(S)) | S = t(α), α ∈ [0, 1]}. Let Tk be the tree, where buyer 1 and
2 are adjacent to goods 1, . . . , k and k, . . . , n respectively, and Fk be the forest, where
buyer 1 and 2 are adjacent to goods 1, . . . , k and k + 1, . . . , n respectively. Let F0 be
∗
In both the tables αi ’s may be treated as price variables.
81
the forest where buyer 1 is not adjacent to any good and buyer 2 is adjacent to all the
goods, and Fn is defined similarly. Let G(X) = (B, G, E) be the bipartite graph, where
the edge (i, j) ∈ E iff xij > 0. Let ti = {(u1(X), u2 (X)) | G(X) = Ti , X ∈ X }, and
fi = {(u1 (X), u2(X)) | G(X) = Fi , X ∈ X }.
Next, we show that F is a piecewise linear concave (PLC) curve. For this, first we
show the same for H and then relate it to F.
Lemma 5.15 The set H is a PLC curve, whose end points are (0, 1) and (1, 0).
Proof : The proof is based on the following observations:
• H=
n
[
ti ∪
n
[
fi .
k=0
k=1
• fi is a point, for all 0 ≤ i ≤ n, and f0 = (0, 1), fn = (1, 0).
2i
• ti is a straight line with slope − uu1i
, and the limit of the end points of ti are fi−1
and fi for all 1 ≤ i ≤ n.
• Since
u1i
u2i
≥
u1(i+1)
,
u2(i+1)
for all i < n, hence slope of ti decreases as i goes from 1 to n.
⊓
⊔
Lemma 5.16 t(α) ∈ S N E , ∀α ∈ [0, 1].
Proof : The equilibrium prices w.r.t. S = t(α) are s1 , . . . , sn . Since
u1i
si
≥
u2i
si
≤
u1(i+1)
, ∀i
si+1
u2(i+1)
, ∀i
si+1
u1i
u2i
≥
u1(i+1)
u2(i+1)
⇒
< n. We can also view si = u2i + (1 − α)(u1i − u2i ), ∀i ≤ n, and hence
< n. It implies that there exists a conflict-free nice allocation w.r.t. S,
hence S ∈ S N E (Lemma 5.14).
⊓
⊔
Clearly, G ⊂ H and is a PLC curve with the end points (w1 (S 1 ), w2 (S 1 )) and (w1 (S 2 ),
w2 (S 2 )), where S i = (ui , ui ), i = 1, 2. Let X be an allocation, and X1 = [x1j ], X2 = [x2j ]
be the restrictions of X to buyers 1 and 2 respectively. For a NESP S, let X(S) be
the (conflict-free) nice allocation. Let X1 (S) and X2 (S) be the restrictions of X(S) to
buyers 1 and 2 respectively. The following lemma proves that F equals G as sets. As
a preparation towards this lemma, we introduce a notion of buyer i getting more goods
according to the allocation X than the allocation X ′ , by which we mean Xi′ ≥ Xi , i.e.,
xij ≥ x′ij , ∀j ∈ G, and Xi′ 6= Xi .
82
Lemma 5.17 As sets, F = G, i.e., if S ∈ S N E then (w1 (S), w2(S)) ∈ G.
Proof : Buyer 1 gets more goods according to the nice allocation w.r.t. S ∈ S N E than
the nice allocation w.r.t. S ′ ∈ S N E , i.e., X1 (S) ≥ X1 (S ′ ) and X2 (S) ≤ X2 (S ′ ) ⇔ w1 (S) >
w1 (S ′ ) and w2 (S) < w2 (S ′ ). Hence, to show that w.r.t. S ∈ S N E , w1 (S 1 ) ≤ w1 (S) ≤
w1 (S 2 ) and w2 (S 2 ) ≤ w2 (S) ≤ w2 (S 1 ), it is enough to show that X1 (S 1 ) ≤ X1 (S) ≤
X1 (S 2 ) and X2 (S 2 ) ≤ X2 (S) ≤ X2 (S 1 ).
Suppose there exists a NESP S such that w1 (S) < w1 (S 1 ). Clearly, p(S 1 ) = (u11 , . . . ,
u1n ). Let (p1 , . . . , pn ) = p(S). Consider the nice allocation w.r.t. S, i.e., X(S). Since
w1 (S) < w1 (S 1 ) ⇒ X1 (S) < X1 (S 1 ). This implies price of at least one good, say a,
allocated to buyer 1 w.r.t. X1 (S) is more than u1a , i.e., pa > u1a . Similarly, price of at
least one good, say b, allocated to buyer 2 w.r.t. X2 (S) is less than u1b , i.e., pb < u1b .
Since
u1a
pa
<
u1b
,
pb
buyer 1 prefers good b, allocated to buyer 2, over good a. A
contradiction, since it violates conflict-freeness. Further, X1 (S) ≤ X1 (S 2 ) may follow
from the similar argument. This implies that X(S) = X(t(α)) for some α ∈ [0, 1].
⊓
⊔
The next theorem follows from Lemmas 5.15, 5.16 and 5.17
Theorem 5.18 F is a piecewise linear concave (PLC) curve.
The next example demonstrates the payoff curve for a small market game.
Example 5.19 Consider a market with 3 goods and 2 buyers, where m = h7, 3i, u1 =
h6, 2, 2i, and u2 = h0.5, 2.5, 7i. The payoff curve for this game is given in Figure 5.1.
The first and the second line segment of the curve correspond to the sharing of good
2 and 3 respectively. The payoffs corresponding to the boundary NESPs S 1 = t(0) and
S 2 = t(1) are (7, 8.25) and (9.14, 3) respectively. Furthermore, the Fisher payoff (8, 7)
may be achieved by a NESP t(0.2). Note that in this example the social welfare (i.e., sum
of the payoffs of both the buyers) from the Fisher payoff is 15 which is lower than 15.25,
the welfare from the NESP S 1 .
5.3.3
Incentives
For a fixed payoff tuple on the curve F, there is a convex set of NESPs and hence a convex
set of prices, giving the same payoffs to the buyers, and these can be computed using the
83
Payoff of buyer 2
(7, 8.25)
(8, 7) Fisher Payoff
L1
L2
(9.14, 3)
Payoff of buyer 1
Figure 5.1: The payoff curve of Example 5.19 market
convex sets defined in Table 5.2 and 5.3. This leads to a different class of behavior, i.e.,
motivation for a seller to offer incentives to the buyers to choose a particular NESP from
this convex set, which fetches the maximum price for her good. The following example
illustrates this possibility.
Example 5.20 Consider a market with 2 buyers and 4 goods, where m = h10, 10i, u1 =
h4, 3, 2, 1i, and u2 = h1, 2, 3, 4i. Consider the two NESPs given by S1 = (s1 , s1 ) and
, 20 , 10
, 10 i and s2 = h 20
, 20 , 9 , 11 i. Both S1 and S2 gives
S2 = (s2 , s2 ), where s1 = h 20
3 3
3 3
3 3 3 3
the payoff (5.5, 8), however the prices are different, i.e., p(S1 ) = h 20
, 20
, 10 ,
3
3 3
10
i
3
and
p(S2 ) = h 20
, 20 , 9 , 11 i. In S2 , good 3 is penalized and good 4 is rewarded (compared to S1 ).
3 3 3 3
5.4
Correlated Equilibria
In general, the Fisher market game has many Nash Equilibria, with very different and
conflicting payoffs. This makes it difficult to predict how a particular Fisher market game
will actually be played out in practice, and if there is a different solution concept which
will yield an outcome liked by all the players.
We examine the correlated equilibria framework as a possibility. Recall that according to the correlated equilibria, the mediator decides and declares a probability distribution π on all possible pure strategy profiles (S1 , . . . , Sm ) ∈ S1 × · · · × Sm beforehand.
84
During the play, she suggests what strategy to play to each player privately, and no player
benefits by deviating from the advised strategy.
Let X be the set of all possible allocations, i.e., X = {[xij ]i∈B,j∈G |
P
i∈B
xij =
1, ∀j ∈ G, xij ≥ 0}. Let H(X) be the payoff tuple w.r.t. an allocation X ∈ X, i.e.,
P
P
( j∈G u1j x1j , . . . , j∈G umj xmj ), and let H be the set of all possible payoff tuples, i.e.,
H = {H(X) | X ∈ X}. Let G(X) = (B, G, E) be the bipartite graph, where E is the set
of all non-zero allocation edges in X.
Lemma 5.21 H is a convex set.
⊓
⊔
Proof : The proof is simple and is omitted.
Let Hpo ⊆ H be the Pareto optimal set of payoff tuples in H. We show that every
payoff tuple on Hpo is the Fisher payoff w.r.t. some money vector2 . Let H f (U, m) be the
Fisher payoff w.r.t. utility function matrix U and money vector m. Let Hf (U ) be the
set of all Fisher payoffs w.r.t. utility function matrix U and all possible money vectors,
P
whose sum is normalized to one, i.e., Hf (U ) = {H f (U, m) | m ∈ Rm
+,
i∈B mi = 1}.
Let G = (B, G, E) be a bipartite graph, where E is the set of edges in G. Let P (G)
be the polytope w.r.t. G as defined in Table 5.4.
Table 5.4: P (G)
∀i ∈ B, ∀j ∈ G, if E(i, j) = 1
:
uij yi = pj
(1)
∀i ∈ B, ∀j ∈ G, if E(i, j) = 0
:
uij yi < pj
X
pj = 1
(2)
yi ≥ 0, pj ≥ 0
(4)
:
(3)
j∈G
∀i ∈ B, ∀j ∈ G
:
Definition 5.22 We say G to be a feasible graph, if P (G) is non-empty.
Next lemma proves that every feasible graph corresponds to the Fisher market equilibrium for some money vector.
2
It is also evident from the Second Theorem of Welfare Economics
85
f
Lemma 5.23 . G(X) is feasible iff ∃m ∈ Rm
+ s.t. H (U, m) = H(X).
Proof : If G(X) is a feasible graph, then consider prices p, which is a feasible point in
P (G(X)) and compute the money vector m from X and p. Clearly, the payoff derived by
X is same as the Fisher payoff w.r.t. m, i.e., H f (U, m). The other direction is easy.
⊓
⊔
Now, we prove an important result that allocations with feasible graphs fetch Pareto
optimal payoffs, which in turn proves that payoffs from the Fisher market equilibrium are
Pareto optimal using Lemma 5.23.
Proposition 5.24 Hpo = Hf (U ) .
Proof : To show Hpo ⊂ Hf (U ) let H(X) ∈ Hpo . If H(X) ∈
/ Hf (U ) then by Lemma
5.23 G(X) is not feasible. In that case, there exists a set of buyers {bi1 , . . . , bik } and
a set of goods {gj1 , . . . , gjk } such that (bil , gjl ) ∈ E, 1 ≤ l ≤ k and ui1 j1 ui2 j2 . . . uik jk <
ui1 j2 ui2 j3 . . . uik−1 jk uik j1 . It is enough to show that there is an exchange possible among
these buyers such that no one loses. Let buyer bil exchanges xl amount of good gjl with
xl+1 amount of good gjl+1 , for 1 ≤ l ≤ k − 1, and buyer bik exchanges xk amount of good
gjk with x1 amount of gj1 . We want these xl ’s to satisfy the following equations:
−xl uiljl + xl+1 uil jl+1
≥ 0,
−xk uik jk + x1 uik j1
≥ 0
∀l ∈ {1, . . . , k − 1}
It is easy to check that there are many solutions to the above set of equations. Let X ′
be the allocation after such an exchange. Clearly, H(X) ≤ H(X ′ ) and H(X) 6= H(X ′ ).
To show Hf (U ) ⊂ Hpo , let X be a market equilibrium allocation of the Fisher market
(U, m), and G(X) be its graph which is feasible by Lemma 5.23. Suppose H(X) is not
a Pareto optimal point in H, and let X ′ be an allocation such that H(X ′ ) is a Pareto
optimal point in H and H(X ′ ) ≥ H(X). Let m and m′ be the money vectors w.r.t.
X and X ′ respectively such that H(X) = H f (U, m) and H(X ′ ) = H f (U, m′). Let
p = (p1 , . . . , pn ) and p′ = (p′1 , . . . , p′n ) be the Fisher market equilibrium prices w.r.t. m
and m′ respectively. Let αi =
m′i
, ∀i
mi
∈ B and βj =
p′j
, ∀j
pj
∈ G. Let βmin = minj∈G βj and
J = {gj ∈ G | βj = βmin }. We may assume that βmin < 1.
Let K and K ′ be the set of buyers, who have edges to the goods in J w.r.t. G(X)
and G(X ′ ) respectively. Let α0 = mini∈K αi . Since H(X ′) ≥ H(X), hence K ⊆ K ′ and
86
βmin ≤ α0 . This implies that the only solution is K ′ = K, βmin = α0 = αi , ∀i ∈ K, and
buyers in K are connected to goods in J only. Therefore, we can remove J and K from
G(X ′ ) and by induction on the remaining graph, we get the result.
⊓
⊔
Lemma 5.25 The payoff of a symmetric NESP is Pareto optimal.
Proof : The solution graph w.r.t. a symmetric NESP S is the complete bipartite graph.
Let X be a conflict-free allocation w.r.t. S. Clearly, G(X) has to be a feasible graph and
therefore using Lemma 5.23 and Proposition 5.24, H(X) is a Pareto optimal point in
⊓
⊔
H.
Theorem 5.26 The correlated equilibrium does not give better payoff than any symmetric
NESP payoff to all the buyers.
Proof : The payoff tuple w.r.t. any strategy profile lies in H. Since, the payoff tuple
derived by any probability distribution π on S1 ×· · ·×Sm is actually a convex combination
of these payoff tuples, it also lies within H, because H is a convex set. Further, by Lemma
5.25 the payoffs w.r.t. all symmetric NESPs are points on Hpo . Therefore, a correlated
equilibrium cannot give better payoff than a symmetric NESP to all the buyers.
⊓
⊔
In summary, a trusted third party does not help to derive better individual payoffs
for Fisher market game! This settles the remaining part of Theorem 1.6.
5.5
Discussion
The main conclusion is that Fisher markets in practice will rarely be played with true
utility functions. In fact, the utilities employed will usually be a mixture of a player’s
own utilities and her conjecture on the other player’s true utilities. Moreover, there seems
to be no third-party mediation which will induce players to play according to their true
utilities so that the true Fisher market equilibrium may be observed. Further, any notion
of market equilibrium should examine this aspect of players strategizing on their utilities.
This poses two questions: (i) is there a mechanism which will induce players into revealing
their true utilities? and (ii) how does this mechanism reconcile with the ”invisible hand”
of the market? The strategic behavior of agents and the question whether true preferences
may ever be revealed, has been of intense study in economics [55, 47, 65]. The main point
87
of departure for our work is that buyers strategize directly on utilities rather than market
implementation specifics, like trading posts and bundles. Hopefully, some of these analysis
will lead us to a more effective computational model for markets.
There is of course, another solution concept; that of repeated games. Even here,
unless players know each others true utilities, equilibrium strategies will deviate substantially from the preferred Fisher market equilibrium strategy. However, if the utilities are
public, then there are schemes such as the grim trigger for two buyers market which will
force both the buyers to play truthfully, by posing threats to each other.
On the technical side, the obvious next question is to completely characterize the
NESPs for the general Fisher market game. We assumed the utility functions of the buyers
to be linear, however Fisher market is gameable for the other class of utility functions
as well. It will be interesting to do a similar analysis for more general utility functions.
Building on our work, recently Chen et al. analyzed how profitable strategization is to a
buyer in markets with Leontief [12] and Cobb-Douglas [11] utilities.
88
Appendix A
Homeomorphism Maps for Two
Player Game Space
Kohlberg and Mertens [40] established a homeomorphism between a finite game space
and its NE correspondence. Here we discuss their homeomorphism maps for the two
player game space, where the number of strategies with the first and the second players
are m and n respectively. Let the game space be Ω = {(A, B) ∈ Rmn × Rmn } and its NE
correspondence be EΩ = {(A, B, x, y) ∈ R2mn+m+n | (x, y) is a NE of (A, B)}.
For the forward map f : EΩ → Ω consider a tuple (A, B, x, y) ∈ EΩ . Let αi =
Pm
α
a
,
1
≤
i
≤
m
and
β
=
ij
j
i=1 bij , 1 ≤ j ≤ n. Define à = A − ( n · 11×n ), and
j=1
Pn
B̃ = B − (1m×1 ·
βT
m
). Observe that a row of à and a column of B̃ sum to zero. Let
σ = (A · y) + x and µ = (xT · B) + y. The map f (NE to game) is as follows:
(A, B, x, y) → (Ã + (
σ
µT
· 11×n ), B̃ + (1m×1 ·
)) = (A′ , B ′ )
n
m
To get the inverse map f −1 : Ω → EΩ , we need to construct (A, B, x, y) back from
P
P
(A′ , B ′ ). Given (A′ , B ′ ), we know that σ = j A′j and µ = i Bi′ .
Let vσ = min{γ|
X
xi = σi − vσ
(σi − γ)+ ≤ 1},
=
i
0
if σi − vσ > 0
otherwise
Here by (σi − γ)+ we mean max{0, σi − γ}. Similarly y can be obtained from µ.
T
Further, Ã = A′ − ( nσ · 11×n ) and B̃ = B ′ − (1m×1 · µm ). From all the available information
α and β are constructed as follows to get back A and B.
α
= σ − x − (Ã · y)
n
and
β
= µ − y − (xT · B̃)
m
Since both f and f −1 are continuous maps, EΩ is homeomorphic to Ω.
89
Appendix B
Regions in the Game Space
We analyze the structure of EΓ in detail. For every vertex p ∈ N P , first we identify a
region in the game space and the points in EΓ corresponding to the region. Later we
combine them to get the complete structure of EΓ .
For a vertex p = (y, π1 ) of P , let R(p) = {α | (α, x, y) ∈ EΓ , for some (x, λ, π2 ) ∈
N (p, ∗)} be it’s region in the game space, i.e., the set of games with at least one NE
corresponding to p. Clearly, R(p) is non-empty only when p ∈ N P . For a q = (x, λ, π2 ) ∈
P
′
N (p, ∗), let Hq be the hyper-plane m
i=1 xi αi − λ = 0 in the game space. By α ∈ Hq
we mean Hq (α′ ) = 0. Recall that for a game G(α) ∈ Γ the row-player’s best response
polyhedron is P (α) = P , and column-player’s best response polyhedron is Q(α) which
P
′
may be obtained by replacing λ with m
i=1 αi xi in Q of (2.3).
Lemma B.1 Let p = (y, π1) ∈ N P be a vertex. α′ ∈ R(p) iff ∃q ∈ N (p, ∗) s.t. Hq (α′ ) =
0.
Proof : (⇒) Suppose α′ ∈ R(p) ⇒ (α′ , x′ , y) ∈ EΓ for some u′ = (x′ , π2 ) ∈ Q(α′ ).
(α′ , x′ , y) ∈ EΓ ⇒ (p, u′) makes a fully-labeled pair of P (α′) × Q(α′ ). Let q ′ = (x′ , λ′, π2 ),
P
′
′
′
′
′
′
′ ′
where λ′ = m
i=1 αi xi . Clearly, q ∈ Q and L(u ) = L(q ) ⇒ q ∈ N (p, ∗), and Hq ′ (α ) = 0.
(⇐) For a q = (x, λ, π2 ) ∈ N (p, ∗) consider an α′ ∈ Hq . Clearly, the Q(α′ ) of game
G(α′ ) has a vertex (x, π2 ) with the same set of tight equations as q, and it makes fullylabeled vertex pair with p. This makes (x, y) a NE of G(α′ ) ⇒ (α′ , x, y) ∈ EΓ ⇒ α′ ∈ R(p).
⊓
⊔
Lemma B.1 implies R(v) =
[
Hq . Next lemmas identify the boundary of R(p).
∀q∈N (p,∗)
A support pair of a point (y, π2) ∈ P is the tuple ({i | Ai y = π1 }, {j | yj > 0}).
90
Lemma B.2 Let (I, J) be the support-pair of p ∈ P with N(p, ∗) 6= ∅. (1.) If |I| = |J| ≥
2, then R(p) is a union of two convex sets, and it is defined by only two hyper-planes.
(2.) If |I| = |J| = 1, then R(p) is a convex-set. It has one defining hyper-plane if p is
either ps or pe , otherwise it has two parallel defining hyper-planes.
Proof : For the first part, let the bounding vertices of edge N (p, ∗) be q1 and q2 (Lemma
2.5). Every point q ∈ N (p, ∗) may be written as a convex combination of q1 and q2 .
Therefore, the corresponding hyper-plane Hq may be written as a convex combination of
the hyper-planes Hq1 and Hq2 . This implies that ∀q ∈ N (p, ∗), Hq1 ∩ Hq2 ⊂ Hq . Further,
it is easy to see that the union of convex sets {Hq1 ≥ 0, Hq2 ≤ 0} and {Hq1 ≤ 0, Hq2 ≥ 0}
forms the region R(p) (Lemma B.1), and hence the hyper-plane Hq1 and Hq2 defines the
boundary of R(p).
For the second part if p = ps or p = pe then the corresponding edge N (p, ∗) ∈ N Q
′
has exactly one vertex q1 (Lemma 2.5), and hence there is exactly one defining hyperplane of R(p), namely Hq1 . Moreover, for p = ps and p = ps the region R(p) is defined by
Hq1 ≤ 0 and Hq1 ≥ 0 respectively.
If p 6= ps and p 6= pe , then the edge N (p, ∗) has two bounding vertices q1 and q2 ,
and x remains constant on N (p, ∗) (Lemma 2.5). Therefore, the hyper-planes Hq1 and
Hq2 are parallel to each other. Further, since any point q ∈ N (p, ∗) may be written as a
convex combination of q1 and q2 , the hyper-plane Hq lies between Hq1 and Hq2 . Hence,
the hyper-planes Hq1 and Hq2 define the boundary of the region R(p) (Lemma B.1).
⊓
⊔
Lemma B.2 shows that the regions are very simple and they are defined by at most
two hyper-planes. Moreover, if Hq1 and Hq2 are the two defining hyper-planes of R(p)
then ∀q ∈ N (p, ∗), Hq1 ∩Hq2 ⊂ Hq . Next, we discuss how the adjacency of p in N P carries
forward to the adjacency of the regions through the corresponding defining hyper-planes.
′
Lemma B.3 If the edges N (p, ∗) and N (p′, ∗) share a common vertex q ∈ N Q , then the
hyper-plane Hq = 0 forms a boundary of both R(p) and R(p′ ).
Proof : Lemma B.2 establishes a one-to-one correspondence between the bounding vertices of N (p, ∗) and the defining hyper-planes of the region R(p). For every bounding vertex q of N (p, ∗), there is a defining hyper-plane Hq of R(p) and vice-versa, and Hq ⊂ R(p).
⊓
⊔
91
Clearly, R(p) and R(p′ ) are adjacent through a common defining hyper-plane Hq ,
where q = N (p, ∗) ∩ N (p′, ∗) is a vertex. Moreover, for every defining hyper-plane of R(p)
there is a unique adjacent region. Hence, every region has at most two adjacent regions
and there are exactly two regions with only one adjacent region (Lemmas B.2 and B.3).
In short adjacency of vertices of N P carries forward to the regions.
Let region graph be the graph, where for every non-empty region R(p) there is a
node in the graph, and two nodes are connected iff the corresponding regions are adjacent.
Clearly, the degree of every node in this graph is at most two and there are exactly two
nodes with degree one. The region graph consists of a path and a set of cycles, and it is
isomorphic to N P where a vertex v ∈ N P is mapped to the vertex R(p). Therefore, for
every component of N , we get a component of the region graph.
To identify a component of the region graph with a component of EΓ , first we distinguish the part of EΓ related to R(p). For an α ∈ R(p), let S(α) = {(α, x, y) ∈ EΓ | y ∈
∆2 and (x, λ, π2 ) ∈ N (p, ∗)}. Let p = (y, π1) ∈ N P be a vertex and N (p, ∗) = q1 , q2 . Let
qi = (xi , λi , π2i ), i = 1, 2, N (∗, q1 ) = p1 , p and N (∗, q2) = p, p2 . We easily deduce the
following.
1. Let q ′ = (x′ , λ′ , π2′ ) ∈ N (p, ∗) be a non-vertex point and α′ ∈ Hq′ \ (Hq1 ∩ Hq2 ), then
S(α′ ) = {(α′ , x′ , y)}.
2. For α′ ∈ Hq1 \Hq2 , S(α′ ) = {(α′ , x1 , y ′) | (y ′, π1′ ) ∈ p1 , p}. Similarly, for α′ ∈ Hq2 \Hq1 ,
S(α′ ) = {(α′ , x2 , y ′ ) | (y ′ , π1′ ) ∈ p, p2 }.
3. For α′ ∈ Hq1 ∩Hq2 , S(α′ ) = {(α′, x′ , y ′) | ((y ′, π1′ ), (x′ , λ′ , π2′ )) ∈ (p1 , p, q1 )∪(p, q1 , q2 )∪
(p, p2 , q2 )}. From Lemma B.2, ∀q ∈ Ep , Hq1 ∩ Hq2 ⊂ Hq . Therefore the projection of
edges (p1 , p, q1 ), (p, q1 , q2 ), (p, p2 , q2 ) on (y, π1 , x, π2 )-space is contained in P (α′ ) ×
Q(α′ ) and all the points on them are fully-labeled.
The above facts imply that (x, y) changes continuously inside the region (R(p)) as
well as on the boundary (Hq1 , Hq2 ), and their values come from the corresponding adjacent
edges of N ((p1 , p, q1 ), (p, q1 , q2 ), (p, p2 , q2 )). Moreover, the consistency is maintained
across the regions through the NESPs on the common defining hyper-plane.
Now it is clear that there is a path between two points of EΓ iff the corresponding
points in N lie on the same component of N . This reestablishes the fact that EΓ does
92
not form a single connected component if N has more than one component. From the
discussion in Section 2.3, we know that N contains at least a path and may contain cycles
(Proposition 2.4). Hence EΓ forms a single connected component iff N contains only the
path. Using this fact, Example 2.9 illustrates that EΓ is not connected in general.
93
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Publications
1. J. Garg, R. Mehta, and V. V. Vazirani. On computability of equilibria in markets
with production. Manuscript, 2012.
2. J. Garg, R. Mehta, M. Sohoni, and N. Vishnoi. Towards polynomial simplex-like
algorithms for market equilibria. To appear in SODA, 2013.
3. M. F. Balcan, F. Constantin, and R. Mehta. The weighted majority algorithm
does not converge in nearly zero-sum games. In ICML 2012 workshop on Markets
Mechanisms and Multi-Agent Models, 2012.
4. J. Garg, R. Mehta, M. Sohoni, and V. V. Vazirani. A complementary pivot algorithm for market equilibrium under separable piecewise-linear concave utilities. In
Proceedings of the 44th ACM Symposium on Theory of Computing (STOC), pages
1003-1016, 2012.
5. J. Garg, R. Mehta, and A. X. Jiang. Bilinear games: Polynomial time algorithms
for rank based subclasses. In Proceedings of the 7th international conference on
Internet and Network Economics (WINE), pages 399-407, 2011.
6. B. Adsul, J. Garg, R. Mehta, and M. Sohoni. Rank-1 bi-matrix games: A homeomorphism and a polynomial time algorithm. In Proceedings of the 43rd ACM
Symposium on Theory of Computing (STOC), pages 195-204, 2011.
7. B. Adsul, Ch. Sobhan Babu, J. Garg, R. Mehta, and M. Sohoni. Nash equilibria
in fisher market. In Proceedings of the 3rd international conference on Algorithmic
Game Theory (SAGT), pages 30-41, 2010.
8. B. Adsul, Ch. Sobhan Babu, J. Garg, R. Mehta, and M. Sohoni. A simplex-like
algorithm for fisher markets. In Proceedings of the 3rd international conference on
Algorithmic Game Theory (SAGT), pages 18-29, 2010.
(Invited to the special issue of Current Science on Game Theory).
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