Approximate Well Supported Nash
Equilibria in Win-Lose Bimatrix Games
Yogesh Anbalagan
School of Computer Science
McGill University
Montreal, Canada
July 2014
A thesis submitted to McGill University in partial fulfillment of the requirements for the
degree of Master of Science.
c Yogesh Anbalagan 2014
i
Abstract
In this thesis, we study -well supported Nash equilibria (-WSNE) in two player normalform games. In an -WSNE, every pure strategy that is played with positive probability
must have an expected payoff within of the best response payoff. These can be viewed as
combinatorial relaxations of exact Nash equilibria, and this thesis examines the structure of
-WSNE via the use of a graphical representation of win-lose games. Our main result states
that to obtain -WSNE with an approximation guarantee better than 23 the players must
use mixed strategies whose supports have polylogarithmic cardinality. In addition, we show
that, to obtain any non-trivial approximation guarantee requires supports of cardinality at
least three. Finally, we use the graphical representation to provide a characterization of
when a win-lose game has an -WSNE with both good approximation guarantee and small
cardinality supports.
ii
Abrégé
Dans cette thèse, nous étudions les -Equilibres de Nash bien supportés (E-ENBS) dans des
jeux de forme-normale à deux joueurs. Dans un -ENBS, chaque stratégie pure jouée avec
probabilité positive doit avoir une récompense attendue à l’intérieur d’un intervalle de de
la meilleure récompense de réponse. Ceux-ci peuvent être considérés comme des relaxations
combinatoires d’équilibres de Nash exacts; et cette thèse examine la structure d’-ENBS
à travers l’utilisation d’une représentation graphique des jeux gagnant-perdant. Notre
résultat principal stipule que, pour obtenir des -ENBS avec une garantie d’approximation
meilleure que 2/3, les joueurs doivent utiliser des stratégies mixtes dont les supports ont une
cardinalité polylogarithmique. En outre, nous montrons que, pour obtenir toute garantie
d’approximation non-triviale, nous avons besoin d’au moins trois supports de cardinalité.
Enfin, nous utilisons la représentation graphique pour fournir une caractérisation du cas
ou un jeu gagnant-perdant a un -ENBS avec à la fois une bonne garantie d’approximation
et des petits supports de cardinalité.
iii
Acknowledgements
First, my deepest and wholehearted gratitude goes to my supervisor, Adrian Vetta, who has
helped me persistently with his thoughtful insights and immense patience. I have enjoyed
his supervision completely. Our recurrent meetings were the greatest inspiration for my
research work. Without his diligent guidance this thesis would never be possible. Thanks
for everything Adrian.
It would not have been possible for me to study at McGill University without the generous financial support from MITACS and the Government of Canada. My sincere thanks
to them.
I would like to thank Luc Devroye, Hamed Hatami and Sergey Norin for the awesome
courses and interesting problems. My heartfelt gratitude to Bruce Shepherd for his fascinating course and words of inspiration.
Thanks to my undergrad friends Baggy, Dhinakaran, Jai, Sid and Subbu. Without their
love and encouragement, it would have been difficult for me to do research.
Thanks to my friends here at McGill, Balaji, Bundit, Gavin, Huining, James, Jeremy,
Kailash, Liana, Remi, Saravan and Swathi for the super happy funtimes.
I would also like to thank my parents and brother for their care and support. Finally,
Sadhu, for whom I have no words to express my gratitude.
iv
Contribution of Authors
The content of Chapter 4 was based on the joint work with Sergey Norin, Rahul Savani and
Adrian Vetta [3] and it appeared in Proceedings of 9th Workshop on Internet and Network
Economics (WINE), 2013.
v
Contents
1 Introduction
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
3
4
2 Preliminaries
2.1 A Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Win-Lose Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
6
8
3 -WSNE in Bimatrix Games
3.1 A Mapping between Bimatrix Games and Win-Lose Games
3.2 The Existence of -WSNE with Logarithmic Supports . . .
3.3 Efficient Algorithms for Finding -WSNE . . . . . . . . . .
3.3.1 The Linear Programming Method . . . . . . . . . .
3.3.2 The Best Response Mapping Algorithm . . . . . . .
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5 A Characterization of Win-lose Games with Small Support -WSNE
5.1 -WSNE with Constant Supports. . . . . . . . . . . . . . . . . . . . . . . .
5.2 A Conjecture on Non-Bipartite Graphs. . . . . . . . . . . . . . . . . . . . .
33
33
35
6 Conclusion
39
References
40
Contents
vi
A The Approximation Lemma
42
vii
List of Acronyms
-NE
-WSNE
PPAD
- Approximate Nash Equilibrium
- Approximate Well Supported Nash Equilibrium
Polynomial Parity Argument for Directed Graphs
1
Chapter 1
Introduction
A Nash equilibrium in a game is a collection of strategies for the players where no player has
a unilateral incentive to change strategy. This is one of the most important and engaging
concepts in game theory. Nash equilibria are of great interest to economists, biologists and
computer scientists because of their applicability in real-life phenomena such as networks,
auctions, markets and evolution. Moreover, in his seminal paper [15], Nash proved that
an equilibrium always exists in a finite k-person game. Recently, though, it was proven [7]
that there is no polynomial time algorithm to find a Nash equilibrium in finite k-players
game unless PPAD is in P.1
Due to the perceived hardness of finding exact Nash equilibria (even for bimatrix games
[6]), there has been a plethora of research concerning approximate notions of Nash Equilibria. Two fundamental notions of approximate Nash equilibria are of interest in this thesis.
The first and most well known are -approximate Nash equilibria (-NE). In an -NE, every
player must get an expected payoff within of their best response payoff, where is some
positive constant less than 1. Observe that -NE are a numeric relaxation of Nash equilibria. In particular, no restriction is placed upon the collection of strategies (support) that
a player may use. For example, in an -NE, a player may place a positive probability on
a pure strategy with very low payoff value against the opponents’ strategies provided the
overall expected payoff satisfies the constraint. Such an occurrence may be undesirable
in certain games and this, in part, motivates the study of -well supported Nash equilibria
(-WSNE). In an -WSNE, every pure strategy played with positive probability must have
1
PPAD is a subset of the complexity class TFNP. A parity argument on a directed graph guarantees
the existence of a solution in any instance of a problem in PPAD. See [7] for more details.
1 Introduction
2
payoff within of the best response payoff. Consequently, -WSNE can be viewed as a
combinatorial relaxation of Nash equilibria. Structurally they are more closely related to
Nash equilibria and, in practice, are induced by the player’s natural inclination to play
strategies with better payoff values. The focus of this thesis is to examine the structural
properties of -WSNE using techniques that exploit a correspondence between games and
directed bipartite graphs.
1.1 Background
Daskalakis, Goldberg and Papadimitriou [7] introduced -WSNE while proving the PPADcompleteness of finding a Nash Equilibrium. Their usefulness in PPAD-reductions arose
because of the close structural relationship with exact Nash Equilibria. This is best illustrated by the seminal work of Chen, Deng and Teng [6] who used -WSNE in proving
PPAD-hardness for computing an exact Nash Equilibrium in a bimatrix game.
In another paper [5], Chen et al. presented an important relationship between approximate Nash equilibria and approximate well-supported Nash equilibria. For any bimatrix
2
-NE, we can construct an -WSNE in polynomial time.
game, given an 8n
Much recent work has considered efficiently finding -WSNE with as small an approximation factor as possible. One approach is to focus upon the special class of win-lose
games – games where all payoffs are in {0, 1}. Daskalakis, Mehta and Papadimitriou [8]
showed that an algorithm for -WSNE in win-lose games induces an algorithm for 1+
2
WSNE in general bimatrix games. This can be shown by examination of the win-lose
game produced by rounding the payoffs of the general game – we detail this in Section 3.1.
Daskalakis et al. incorporated this transformation into a best response mapping procedure
1
and showed that a (1− 2k
)-WSNE with support sizes at most k exists in any bimatrix game
provided a graph theoretic conjecture is true. Furthermore, this existence is unconditional
for the case k = log n.
Two player win-lose games can be represented by directed bipartite graphs. This observation was used by Addario-Berry et al. [1] to obtain a polynomial time algorithm for
finding an exact Nash equilibria in planar win-lose games. This graphical representation
will lie at the heart of this thesis.
As stated, for constant performance guarantees, the results of Daskalakis et al. [8]
are dependent upon the validity of a graph theoretic conjecture. The first unconditional
1 Introduction
3
constant -WSNE for win-lose games was given by Kontogiannis and Spirakis [11]. They
achieved this via a reduction to finding exact equilibria in zero-sum games. Such games can
be solved efficiently via linear programming and the method produced 12 -WSNE for win-lose
games. The method produces 23 -WSNE for general bimatrix games. A Nash equilibrium
in a zero-sum game may, however, require the use of a large number of strategies (large
support). Kontogiannis and Spirakis were able to prove the existence of -WSNE with
supports of moderate cardinality. Specifically, for any > 0, there exist -WSNE with
supports of cardinality O( log2 n ).
Currently, the best approximation guarantee for -WSNE is due to Fearnley et al. in [9].
They refine the linear programming approach of Kontogiannis and Spirakis [11] to produce
( 23 − δ)-WSNE for general bimatrix games, where δ = 0.00473. However, their refinements
do not improve the = 21 bound of [11] for the special case of win-lose games.
As alluded to, the numerical relaxation of -NE has been even more widely studied in
the past decade. Here we will discuss only the results that directly motivate our research.
Althöfer [2] and Lipton and Young [12] independently proved the existence of -WSNE with
supports of cardinality O( log2 n ), for any > 0. Indeed, Althöfer’s Approximation Lemma
[2] provides the basis for the analogous -WSNE result of Kontogiannis and Spirakis. On
the other hand, Althöfer showed the existence of zero-sum games in which every -NE, with
< 41 , require supports of cardinality at least log n. In contrast, there is a simple method
to find 12 -NE with supports of cardinality at most two, due to Daskalakis et al. [8].
1.2 Our Results
The most intriguing open problem in this area is the following. Is there a constant (strictly less than one), such that every bimatrix game has an -WSNE with supports of
constant cardinality? Recall that this is true in the case of -NE for supports of cardinality just two. For -WSNE, though, this question is completely open. Mathematically,
small support -WSNE are interesting objects in their own right. Computationally it is
also easy to search for equilibria with small supports. Moreover, in practice, the decision
making procedures used by players often focus upon strategies with small supports. These
observations motivate the work in this thesis.
Our main result, Theorem 4.1, is that there is limit to the quality of -WSNE achievable
when the players use strategies with constant size supports. Specifically, we prove that there
1 Introduction
4
√
exist win-lose games that require supports of cardinality Ω( 3 log n) in every -WSNE with
< 32 .
Whether small supports case be used to provide a weaker guarantee of ≥ 32 remains
open. We prove, however, that supports of cardinality two cannot provide any positive
result. In particular, we show in Theorem 4.6 that for any δ > 0, there exist win-lose
games for which every (1 − δ)-WSNE has supports of cardinality at least three. Observe
that this result shows that -WSNE differ structurally from -NE in at least one notable
way, as good guarantees are achievable for -NE with supports of cardinality two.
The main theorem is based upon a bipartite variant of the renowned Caccetta-Häggkvist
conjecture [4]. We obtain, in Theorem 4.4, a bound on the minimum in-degree required in
a directed bipartite graph to ensure the occurrence of a 4-cycle.
Finally, we prove a necessary and sufficient condition regarding when a win-lose game
has -WSNE with constant supports in Theorem 5.5. Specifically, we prove that a winlose game has an -WSNE with small supports if and only if the corresponding directed
bipartite graph has either a small cycle or a small set of uncovered vertices. We believe
this characterization may be useful in addressing the fundamental open problem above.
1.3 Thesis Overview
Chapter 2 contains the required game theoretic definitions and mathematical notations.
In Chapter 3, we present relevant and important prior work. Specifically, we give an
existence result on -WSNE with logarithmic supports, due to Kontogiannis and Spirakis
[11], using Althöfer’s Approximation Lemma [2]. We also present two methods to find
-WSNE in bimatrix games in polynomial time. The first algorithm, based on a linear
programming approach, is due to Kontogiannis and Spirakis [11] and the second algorithm
is due to Daskalakis, Mehta and Papadimitriou [8]. With the background work complete, we
examine, in Chapter 4, the combinatorial structure of -WSNE. We exploit this structure
to prove that high quality -WSNE with constant supports need not exist in all games.
Then, in Chapter 5, we study the structural properties of -WSNE via a directed bipartite
graph representation of win-lose games. This leads to the characterization of -WSNE with
respect to directed cycles and covered sets. We conclude in Chapter 6 with open problems.
5
Chapter 2
Preliminaries
In this chapter, we present the mathematical notations that will be used in this thesis. We
also explain various game theoretic terms.
2.1 A Game
An m × n matrix A with rational entries is denoted by A ∈ Rm×n . The notation (A, B) ∈
Rm×n denotes a bimatrix, a matrix whose entries are ordered pairs of rational numbers. A
2-player game in normal form can be represented by a bimatrix (R, C) ∈ Rm×n . The row
player has m pure strategies – one for each row. Similarly, the column player has n pure
strategies – one for each column. If the row player plays row i and the column player plays
column j then the row player receives a payoff Rij and the column player receives a payoff
Cij .
The row player may also play a mixed strategy, that is, a probability distribution over
the pure strategies (rows). Formally, let ∆m be the set of all probability distributions over
m rows. Then x is a probability distribution over the rows (that is, x ∈ ∆m ) if
∀i ∈ [m] , xi ≥ 0
and
m
X
xi = 1
i=1
Observe that the pure strategy row i corresponds to the mixed strategy x = ei where the
vector ei has a 1 in the ith coordinate and a 0 elsewhere.
Analogously, the column player may play a distribution over the columns. When the
2 Preliminaries
6
players use mixed strategies, the payoffs received are extended linearly. Thus, if the row
and column player use the mixed strategies x and y, respectively, then the row player
receives payoff xT Ry and the column player receives payoff xT Cy. The pair (x, y) is called
a (mixed) strategy profile.
The support, denoted sup(x), of a probability distribution x ∈ ∆m is the set of coordinates with positive values. Thus, sup(x) := {i ∈ [m] : xi > 0}. For example, the support
of the pure strategy ei is simply {i}.
A bimatrix game is called a win-lose game if every payoff entry is either 0 or 1. A
bimatrix game (R, C) is zero-sum if the payoff matrices satisfy R = −C.
Throughout the thesis, we consider games where every payoff is in [0, 1]. Furthermore,
all payoff matrices will be m × n. As a result, we will omit m, n subscripts/superscripts
whenever the context is clear.
2.2 Nash Equilibria
The standard solution concept in a game is a a Nash equilibrium - a collection of strategies
under which no player has a unilateral incentive to deviate. Specifically, consider a bimatrix
game (R, C). Then a strategy pair (x0 , y 0 ) form a Nash equilibrium if and only if,
T
x0 Ry 0 = max xT Ry 0
x
T
T
x0 Cy 0 = max x0 Cy
y
∀x ∈ ∆m
∀y ∈ ∆n
(2.1)
Therefore, at a Nash equilibrium, the row and column players are playing mutual best
responses. The strategy x0 played by the row player gives the maximum achievable expected
payoff against the strategy y 0 used by the column player and, similarly, y 0 provides the
column player with the maximum possible payoff against the strategy x0 .
Unfortunately, computing an exact Nash equilibrium appears hard. As we have seen
earlier, even for bimatrix games there is no polynomial time algorithm to find exact Nash
Equilibrium unless PPAD is in P [6]. Consequently, researchers have also studied relaxations of equilibria, the predominent ones being -approximate Nash equilibria and -well
supported Nash equilibria.
2 Preliminaries
7
-Approximate Nash Equilibria
Consider Definition (2.1) of a Nash equilibrium. Rather than insist upon exact best response, a natural numeric way to relax this definition is to assume that each player plays
an approximate best response. To wit, we say that a strategy pair (x0 , y 0 ) form an approximate Nash equilibrium (-NE) if each player receives an expected payoff within of
the best response payoff. Formally,
T
x0 Ry 0 ≥ max xT Ry 0 − x
0T
T
0
x Cy ≥ max x0 Cy − (2.2)
y
-Well Supported Nash Equilibria
Note that -NE can be viewed as a numeric relaxation of Nash equilibria. To provide an
alternate combinatorial relaxation consider, again, Definition (2.1). Observe that, because
x0 maximizes expected payoff against y 0 , every pure strategy in the support of x0 must also
be a best response against y 0 . Similarly, every pure strategy in the support of y 0 is a best
response against y 0 . Hence, we can restate the Nash equilibrium conditions by:
∀i : x0i > 0 ⇒ ei T Ry 0 = max xT Ry 0
x∈∆m
∀i :
yi0
0T
T
> 0 ⇒ x Cei = max x0 Cy
y∈∆n
(2.3)
Viewed in this manner, given y 0 , we see that the row player’s support is combinatorially
constrained to include only rows that are pure best responses to y 0 . A similar observation
applies to the column player. Ergo, a combinatorial relaxation to Nash equilibria is to
allow the players’ supports to extend slightly to allow the use row/columns that are are
approximate best responses. Specifically, a strategy-pair (x0 , y 0 ) form an -well supported
Nash equilibrium (-WSNE) if
∀i : x0i > 0 ⇒ ei T Ry 0 ≥ max xT Ry 0 − x∈∆m
∀i :
yi0
0T
T
> 0 ⇒ x Cei ≥ max x0 Cy − y∈∆n
(2.4)
Clearly, these are both relaxations since 0-NE and 0-WSNE correspond to exact Nash
2 Preliminaries
8
equilibria. However, the combinatorial restriction is stronger. To illustrate this, consider
the following simple example. Take a bimatrix game (R, C) where
R=
0 1
1 0
!
C=
1 0
0 0
!
The strategy profile (x0 , y 0 ) with x0 = 12 (e1 + e2 ) and y 0 = e1 forms a 21 -NE for the game
(R, C). To see this, note that both players obtain an expected payoff of 12 ; since the
maximum possible payoff is 1, we have a 12 -NE.
This strategy pair does not, though, give a good well-supportedness guarantee. By
playing his first row, the row player gets 0 payoff against column one, the column player’s
strategy. But the best response (namely, row two) against column one produces a payoff of
1. Thus, the positive probability on row one ensures that (x0 , y 0 ) forms only a 1-WSNE –
this is the worst possible guarantee as all payoffs are in [0, 1]. This is his pure best response
payoff too. So he should not place any positive probability on his first row against column
player’s first column. But, he places a positive probability on his first row in the strategy
profile x0 against column player’s y 0 . Thus, the strategy profile (x0 , y 0 ) form a 1-WSNE, the
worst you can get.
This example shows that every -NE need not be a -WSNE. Clearly, though, any WSNE is also an -NE. Thus the combinatorial constraint can be stronger than the numeric
constraint.
Observe that, the approximation defined in (2.2) and (2.4) are additive. All the
results in this thesis are based on this additive bound. However, it can also be viewed
as a multiplicative approximation, provided all the payoffs were positive and scaled to be
between 0 and 1.
2.3 Win-Lose Games
Recall that, in a win-lose game (R, C) every payoff is in {0, 1}. A win-lose game has
an elementary graphical representation. For (R, C) we build a directed bipartite graph
G = (R ∪ C, E) as follows. There is a vertex ri ∈ R for each row and a vertex cj ∈ C for
each column. There is an arc (cj , ri ) ∈ E if and only if Aij = 1. So, ri is a best response
for the row player against the strategy cj of the column player. Similarly, there is an arc
2 Preliminaries
9
(ri , cj ) ∈ E if and only if Bij = 1.
For example, consider the win-lose bimatrix game (R, C) where,


0 1 0


R = 0 0 1
1 0 0


1 0 0


C = 0 0 0
0 1 0
This game can be represented using the directed bipartite graph G shown in Figure 2.1.
r1
c1
r2
c2
r3
c3
Fig. 2.1: A Graphical Representation of Win-Lose Game.
The pure best response moves for the row player are indicated in red, and the pure best
response moves for the column player are indicated in blue. Note that, if the row player
plays pure strategy r2 then the best response for the column player has payoff 0. This is
can be inferred from the graph as the out-degree of r2 is zero.
Given a graphical representation, certain combinatorial structures in the graph will have
important ramifications for the corresponding game. Two structures that are of interest in
this thesis are directed cycles and covered sets. The latter object may be unfamiliar so we
define it here.
Covered Sets
In a directed graph G = (V, A), a set S = {v1 , v2 , . . . , vk } ⊆ V is covered by a vertex u ∈ V
if there is an arc (vi , u), for all 1 ≤ i ≤ k. A set is uncovered if it is not covered by any
vertex.
2 Preliminaries
10
For example, consider the directed graph shown in Figure 2.2. Here the set of vertices
S = {b, c} is covered by a. The sets {a, b}, {a, c} and {a, b, c} are uncovered.
a
c
b
Fig. 2.2
In this thesis, we will be predominantly work with directed bipartite graphs. We specialize the definition of covered sets to the bipartite setting as follows. In a directed bipartite
graph, G = (R ∪ C, A), a set of vertices S on one side of the bipartition (that is, S ⊆ R or
S ⊆ C) is covered if there exists a vertex u on the other side of the partition that covers it.
11
Chapter 3
-WSNE in Bimatrix Games
In this chapter, we present results on -WSNE that are related to our work. All the results
stated in this chapter are from [8] and [11]. We present some of their proofs.
3.1 A Mapping between Bimatrix Games and Win-Lose Games
In [8], Daskalakis, Mehta and Papadimitriou presented a mapping that transforms a general
bimatrix game into a win-lose game. This mapping has the important property that WSNE in win-lose game corresponds to -WSNE in the original game, albeit with a loss
in the approximation guarantee. For our purposes, it then suffices to study -WSNE in
the special case of win-lose games. We will be able to analyse these games using the
graphical representation described in Chapter 2; specifically, -WSNE will correspond to
combinatorial structures in these graphs.
The mapping is simple. Given a bimatrix game (R, C) with payoffs in [0, 1], we create
a win-lose game (RW , C W ) by rounding the payoff entries. Thus, the new payoffs for the
row player are, ∀i ∈ [m], j ∈ [n],
(
W
Rij
=
1
0
if Rij ≥ 1/2
if Rij < 1/2
(3.1)
The new payoffs for the the column player are defined analogously.
Now the following lemma provides a simple relationship between -WSNE of the winlose game (RW , C W ) and the 2-player game (R, C) from which it was derived.
3 -WSNE in Bimatrix Games
12
Lemma 3.1. (Daskalakis et al. [8]) Let (x, y) be an -WSNE of the win-lose game
(RW , C W ). Then (x, y) is a 1+
-WSNE of the bimatrix game (R, C).
2
Proof. Let (x, y) form an -WSNE in the win-lose game (RW , C W ). We will show the wellsupportedness of the row player in the original game (R, C); a similar argument applies for
the column player.
From the rounding method (3.1), it can be seen that, ∀i ∈ [m], j ∈ [n],
W
W
Rij
1 Rij
≤ Rij ≤ +
2
2
2
It immediately follows from the above inequality that, ∀i ∈ [m],
1 1
1 T W
ei R y ≤ ei T Ry ≤ + ei T RW y
2
2 2
(3.2)
Furthermore, by Definition (2.4) of -WSNE, we have that, ∀k ∈ sup(x) and ∀i ∈ [m],
ei T R W y − ek T R W y ≤ (3.3)
We may now derive the well-supportedness of the row player in the game (R, C). Take any
k ∈ sup(x) and any i ∈ [m]. Then
T
1 1 T W
+ ei R y − ek T Ry
2 2
1 1 T W
1
+ ei R y − ek T RW y
2 2
2
1 1
+ · (ei T RW y − ek T RW y)
2 2
1+
2
T
ei Ry − ek Ry ≤
≤
=
≤
(3.4)
Here the first two inequalities follow directly from (3.2). The final inequality applies Inequality (3.3), that is, it uses the fact (x, y) be an -WSNE in (RW , C W ). But, Inequality
(3.4) implies that row player is -well-supported in the game (RW , C W ), as desired.
3 -WSNE in Bimatrix Games
13
3.2 The Existence of -WSNE with Logarithmic Supports
In 1994, Althöfer [2] and Lipton and Young [12] independently proved the existence of
-NE with logarithmic support sizes in zero-sum bimatrix games. Later, Lipton et al.
[13] generalized this result to non zero-sum bimatrix games and multiplayer games. As
a corollary, one can find an -NE in sub-exponential time by examining all supports of
cardinality at most log2 n , for every > 0.
Kontogiannis and Spirakis [11] proved a similar result for -WSNE: there exists a kuniform -WSNE with logarithmic support sizes in any bimatrix game. We will now present
a proof of this result which is based upon the following Approximation Lemma of Althöfer.
Lemma 3.2. (Approximation Lemma [2])
Let A ∈ Rm×n with entries in [0, 1]. For any m-probability vector, p ∈ ∆m and any
> 0,
log(2n)
there exists another k-uniform probability vector,1 p̂ ∈ ∆m (k) with |sup(p̂)| ≤ k ≡
22
and sup(p̂) ⊆ sup(p), such that, |pT Aej − p̂T Aej | ≤ , ∀j ∈ [n].
Proof. See Appendix A.
Theorem 3.3. (Kontogiannis and Spirakis [11])
Take a bimatrix game (R, C) with payoffs in [0, 1]. For any > 0, there is a 2-WSNE
(x∗ , y ∗ ), where x∗ ∈ ∆m (k), y ∗ ∈ ∆n (l) with k ≤ d log(2n)
e and l ≤ d log(2m)
e.
22
22
Proof. Let (x, y) be a Nash Equilibrium in the bimatrix game (R, C). By Definition (2.3),
we have that
∀i : xi > 0 ⇒ ei T Ry = max
xT Ry
0
x ∈∆m
∀i : yj > 0 ⇒ x Rej = max
xT Cy 0
0
T
y ∈∆n
(3.5)
Now, by the Approximation Lemma, there is a strategy-pair (x∗ , y ∗ ) with x∗ ∈ ∆m (k) and
y ∗ ∈ ∆n (l) with the properties that
|xT Cej − x∗T Cej | ≤ ∀j ∈ [n]
|eTi Ry − eTi Ry ∗ | ≤ ∀i ∈ [m]
1
(3.6)
A probability distribution x ∈ ∆m is called a k-uniform strategy if and only if each individual component has a value that is some multiple of k1 . It is denoted as ∆m (k).
3 -WSNE in Bimatrix Games
14
and that sup(x∗ ) ⊆ sup(x), sup(y ∗ ) ⊆ sup(y) where
log(2n)
|sup(x )| ≤ k ≡
22
log(2m)
∗
|sup(y )| ≤ l ≡
22
∗
We can now show the well-supportedness of the row player. Because (x, y) forms a Nash
Equilibrium, it follows from Inequality (3.5) that
ei T Ry ≥ ej T Ry,
∀i : xi > 0, ∀j ∈ [m]
ei T Ry ≥ ej T Ry,
∀i : x∗i > 0, ∀j ∈ [m]
(3.7)
Here the second inequality arises as sup(x∗ ) ⊆ sup(x). Now take any i with x∗i > 0. Then
for any other j ∈ [m] we obtain
ei T Ry ∗
≥ ei T Ry − ≥ ej T Ry − ≥ ej T Ry ∗ − 2
Here the first and third inequality follow from (3.6). The second inequality follows from
(3.7). Thus row player is 2-well supported in the game (R, C) with supports of logarithmic
cardinality. A similar argument applies for the column player using the inequalities. Hence
(x∗ , y ∗ ) is 2-WSNE for the game (R, C) with logarithmic supports.
3.3 Efficient Algorithms for Finding -WSNE
Theorem 3.3 implies that we can find an -WSNE in sub-exponential time by exhaustively
examining all strategy pairs with polylogarithmic supports. Moreover, this approach succeeds for any constant approximation guarantee .
This prompts the question: what can be achieved with polynomial time algorithms? It
turns out that -WSNE can be obtained in polynomial time only for weaker approximation
guarantees. In this section, we present two such algorithms. The first is a linear program
based method due to Kontogiannis and Spirakis [11]; the second method proposed by
3 -WSNE in Bimatrix Games
15
Daskalakis, Mehta and Papadimitriou [8] uses best response mapping to construct -WSNE.
3.3.1 The Linear Programming Method
The Minimax Theorem for zero-sum games was proven in 1928 by von Neumann [17].
Consequently, a Nash equilibrium in a zero-sum game can be computed efficiently using a
linear program. Kontogiannis and Spirakis [11] exploited this fact to obtain 12 -WSNE in
win-lose games in polynomial time. The idea is to construct a zero-sum game from a winlose game such that exact Nash equilibria in the zero-sum game correspond to 21 -WSNE in
the win-lose game. We now present a simplified version of their proof.
Theorem 3.4. (Kontogiannis and Spirakis [11])
In a win-lose bimatrix game a 12 -WSNE can be found in polynomial time.
Proof. Take a win-lose bimatrix game (R, C). If the game contains a (1, 1) entry, then we
are immediately done as this corresponds to a pure-strategy Nash equilibrium. Thus, we
may assume there are no such entries. Now define a matrix Z = 12 (R + C), and consider
the bimatrix game (R − Z, C − Z). This game is zero-sum game because
1
(R − Z) + (C − Z) = (R + C) − 2Z = (R + C) − 2 · ( (R + C)) = 0
2
Hence we can find a Nash equilibrium (x, y) in the game (R − Z, C − Z) via a linear
programming. By Definition (2.3) of a Nash equilibrium, we have
ei T (R − Z)y ≥ ej T (R − Z)y
∀i : xi > 0, ∀j ∈ [m]
(3.8)
Rearranging the Inequality (3.8), we obtain
ei T Ry ≥ ej T Ry − [ej T Z − ei T Z]y
∀i : x0i > 0, ∀j ∈ [m]
(3.9)
Recall that every entry in (R, C) is in {(0, 0), (0, 1), (1, 0)}. Ergo, every entry of Z =
1
(R + C) is in {0, 21 }. Then eTi Z, the ith row of Z, is a vector in {0, 12 }n . It follows, for
2
any rows i and j, that
1
1
(eTj Z − eTi Z)T y ≤ ( )T y =
(3.10)
2
2
Here the inequality arises as the maximum difference between any two entries of any two
3 -WSNE in Bimatrix Games
16
rows in Z is at most 12 . The equality follows from the fact that y is probability vector over
the columns. Plugging (3.10) into (3.9) produces
ei T Ry 0 ≥ ej T Ry 0 −
1
2
∀i : x0i > 0, ∀j ∈ [m]
(3.11)
Consequently, the pair (x, y) is 12 -well-supported with respect to the row player, in the winlose game (R, C). A similar argument applies to the column player. Thus, the strategy-pair
(x, y) is a 12 -WSNE for the original win-lose game (R, C).
An obvious question is whether or not we can improve the approximation guarantee
of Theorem 3.4 by a more judicious choice of the zero-sum game Z. Here we show that
the most natural extension of the method will not improve the guarantee. Specifically, let
α + β = 2 and consider the bimatrix game (R − α · Z, C − β · Z) where Z = 21 (R + C).
Again, this game is zero-sum because
1
(R − α · Z) + (C − β · Z) = (R + C) − (α + β) · Z = (R + C) − 2 · ( (R + C)) = 0
2
Now let (x, y) be a Nash equilibrium of this game. Analogous to Inequality (3.11), we can
show that
1
ei T Ry 0 ≥ ej T Ry 0 − ( · α)
2
1
0T
0T
x Cei ≥ x Cej − ( · β)
2
∀i : x0i > 0, ∀j ∈ [m]
(3.12)
∀i : yi0 > 0, ∀j ∈ [n]
(3.13)
Now, by Inequality (3.12), we can improve the guarantee for the row player by decreasing
the α. Similarly, by Inequality (3.13), we can improve the well-supportedness guarantee
for of the column player by decreasing the β value. Of course, we cannot achieve both of
these simultaneously as α and β must sum to two. Thus the choice of α = β = 1 in (3.12)
and (3.13) is optimal.
We remark that a similar linear programming approach produces a 23 -WSNE for general
bimatrix games [11]. But, simply combining Thoerem 3.4 with the bimatrix rounding
method from lemma 3.1 gives a weaker bound of 3/4 for general bimatrix games. Recently,
Fearnley et al. [9] refined this linear programming approach to give an improved bound of
( 32 − δ), where δ = 0.00473. Their method does not lead to an improvement in the factor
1
bound for win-lose games.
2
3 -WSNE in Bimatrix Games
17
3.3.2 The Best Response Mapping Algorithm
A major drawback of the linear programming approach is that it may produce strategy
pairs with very large supports, for example support sizes that are linear in m and n. On
the other hand, Theorem 3.3 shows that -WSNE exist with logarithmic supports, for any
constant > 0. So can we find -WSNE with small supports in polynomial time?
Daskalakis, Mehta and Papadimitriou [8] proposed a combinatorial technique to attempt
this task. First, they apply the rounding method of Lemma 3.1 to convert a bimatrix game
into a win-lose game. Second, they further transform this win-lose game to a new win-lose,
which we call the shuffled win-lose game. This latter transformation consists of moving
and duplicating the columns of payoff matrices based upon a best response mapping of the
column player. This transformation is applied such that, -WSNE in the shuffled win-lose
game can be mapped back to -WSNE in the original win-lose game.
The Shuffled Win-Lose Game
Given an m × n win-lose game (R, C), we create the shuffled win-lose game (Rf , C f ) as
follows. Let f be a best response function for the column player. That is, if the row player
plays row i, a best response for the column player is column f (i). Thus Cif (i) = maxj Cij .
The shuffled win-lose game (Rf , C f ) is then simply obtained by replacing the jth column
of R (respectively, C) by the f (j)th column. Formally, for all i, j ∈ [n] we set
f
Rij
= Rif (j)
Cijf = Cif (j)
(3.14)
To illustrate this consider the win-lose game


0 1 0 0


R = 1 0 0 1
1 0 0 0


0 0 1 1


C = 0 1 0 0
0 1 0 0
A best response function f for this game is:
C1f (1) = 1
f (1) = 4
3 -WSNE in Bimatrix Games
18
C2f (2) = 1
f (2) = 2
C3f (3) = 1
f (3) = 2
We now create the shuffled win-lose game (Rf , C f ) according to the transformation
(3.14). Since f (1) = 4, we replace the first columns in R and C by their fourth columns.
Similarly, because f (2) = f (3) = 2, we replace the second and third columns in both R
and C by their respective second columns. The fourth column remains unchanged as f (4)
is undefined. Therefore, the shuffled game is,


0 1 1 0


Rf = 1 0 0 1
0 0 0 0


1 0 0 1


C f = 0 1 1 0
0 1 1 0
Observe that, by construction, the diagonal entries in C f are the maximum entries in each
row. Consequently, a best response for the column player in the shuffled game is to play
column i if the row player plays row i.
Furthermore, we remark that the best response function f need not be unique as there
may be more than one column that is a best responses to row i. As an illustration, in the
example above we could have chosen f (1) = 3. Consequently the shuffled game (Rf , C f )
obtained from (R, C) also need not be unique.
Now Daskalakis et al. gave the following claim concerning the shuffled game. For a
proof, we refer the interested reader to [8].
Claim 3.5. A -WSNE of the shuffled game (Rf , C f ) corresponds to an -WSNE of the
game (R, C) with the same cardinality supports.
In the light of Claim 3.5, it suffices for us to search for -WSNE in the shuffled game
(R , C f ). The key to doing this is the following simple claim.
f
Lemma 3.6. Take the shuffled game (Rf , C f ) and suppose the row player plays the uniform
distribution x̄ on S ⊆ [m]. Then any strategy y for the column player, whose support is the
1
columns of S, is 1 − |S|
-well-supported against x̄.
1
for each row i in S. By the transformation (3.14), column i is the best
Proof. So x̄i = |S|
response for the column player, if the row player plays row i. This implies that Cilf ≤ Ciif
3 -WSNE in Bimatrix Games
19
for any other column l. Thus, for any l ∈ [n] we have
x̄T C f el =
1 X f
1 X f
Cil ≤
C
|S| i∈S
|S| i∈S ii
(3.15)
Trivially, for any k ∈ S, we also have
f
x̄T C f ek ≥ x̄k · Ckk
=
1 f
C
|S| kk
(3.16)
Combining Inequalities (3.16) and (3.15) gives, for any k ∈ S and any l ∈ [n], that
x̄T C f el − x̄T C f ek ≤
1 X f
1 f
Cii −
C
|S| i∈S
|S| kk
(3.17)
We now break the analysis up into two cases. First suppose that there exists a j ∈ S such
f
that Cjj
= 0. Therefore, Inequality (3.17) becomes
x̄T C f el − x̄T C f ek ≤
1
1 X f
Cii = 1 −
|S| i∈S
|S|
(3.18)
f
= 1 for all j ∈ S. Inequality (3.17) is then
On the other hand, suppose Cjj
x̄T C f el − x̄T C f ek ≤ 1 −
1
|S|
(3.19)
Inequalities (3.18) and (3.19)
prove
that any strategy y for the column player, whose support
1
is the columns of S, is 1 − |S|
-well-supported against x̄.
Note that Lemma 3.6 only provides a guarantee for the column player. Thus, to obtain
an -WSNE, we need to find a suitable choice y for the column player such that x̄ is wellsupported for the row player against y. Furthermore, our aim is to find a subset S of small
cardinality.
Our focus now is only upon the row player, so it suffices to consider just the payoff
matrix Rf . If Rf is an m × n matrix with m ≤ n, then we convert it into an m × m matrix
by retaining only the first m columns of Rf . Daskalakis et al. used this as the adjacency
matrix of a directed graph H on m vertices. Thus, we have an arc from vertex j to vertex i
f
in H, if Rij
= 1. Directed cycles and uncovered sets in H then correspond to good choices
3 -WSNE in Bimatrix Games
20
for S. Let’s see why.
Lemma 3.7. If there is a directed cycle of length k in H, then there is a (1 − k1 )-WSNE
with supports of size at most k.
Proof. Let S be (the vertices of) a directed cycle of length k in H. If the column player
plays a uniform strategy ȳ over S then each pure strategy in S is a k−1
-best response for
k
the row player. This is because each vertex in S has in-degree one from the cycle. Thus,
each such pure strategy guarantees the column player an expected payoff of at least k1
against ȳ. As the maximum payoff is one, each row is then a (1 − k1 )-best response against
ȳ. Therefore by Lemma 3.6 the strategy-pair (x̄, ȳ) form a (1 − k1 )-WSNE in (Rf , C f ).
As an example, consider the 3 × 3 matrix A and the corresponding directed graph H
shown in Figure 3.1. Then S = {r2 , r3 } is 2-cycle in H. Each strategy in S has an expected
payoff of 21 for the row player against the uniform strategy of the column player over the
same set S.
dummy
r1

dummy1
A

1 0 0

= 
 0 0 1 
1 1 0
r2
Fig. 3.1
r3
dummy1
3 -WSNE in Bimatrix Games
21
Lemma 3.8. If there is a set of l vertices that are uncovered in H, then there is a (1 − 1l )WSNE with supports of size at most l.
Proof. Take a set U of l uncovered vertices in H. Suppose the column player plays a
uniform strategy ȳ over the set U . Now a pure strategy in U for the row player may only
generate a payoff 0 against ȳ. This is sufficient for our purposes! To see this, observe that
because U is uncovered there is no row in Rf that has a payoff of one against every column
in U . Thus a best response against y ∗ has expected payoff at most 1 − 1l . So the uniform
distributions x̄ and ȳ on U form a (1 − 1l )-WSNE in (Rf , C f ).
For example, consider the 3×3 matrix B and the corresponding directed graph H shown
in Figure 3.2. Note that the set U = {r1 , r3 } is uncovered. So every strategy with support
U is a 21 -best response for the row player against the uniform strategy of the column player
over U .
dummy2
dummy
r1

dummy1
B

0 1 0

= 
 0 0 1 
1 1 0
r2
r3
dummy1
Fig. 3.2
Therefore, if the graph H contains a small cycle or a small set of uncovered vertices then
this method can be applied to give an -WSNE with small supports. But does a directed
graph always contain a small cycle or uncovered set? Daskalakis, Mehta and Papadimitriou
conjectured that it does. Specifically,
3 -WSNE in Bimatrix Games
22
Conjecture 3.9. (Daskalakis et al [8]):
There exist integers k and l such that every digraph either has a cycle of length at most k
or an uncovered set of l vertices.
If the conjecture holds then the previous discussions shows that we obtain the following
guarantees.
Theorem 3.10. (Daskalakis et al [8])
Suppose Conjecture 3.9 holds for a specific k and l. Then for any win-lose game we can find
in polynomial time a max{1 − k1 , 1 − 1l }-WSNE with supports size of cardinality max{k, l}.
Proof. Take any win-lose game (R, C) and transform it to the shuffled game (Rf , C f ) using
the transformation (3.14). This can be done in polynomial time. We have seen that the
payoff matrix Rf can be considered as the adjacency matrix of a directed graph H on m
vertices. Since Conjecture 3.9 holds for k and l, H contains either a cycle of length at most
k or an uncovered set of l vertices. In the former case, we can easily find such a cycle using
depth first search. In the latter case, we can exhaustively search for the uncovered set in
polynomial time because there are only ml subsets of cardinality l. The theorem then
follows via Lemmas 3.6, 3.7 and 3.8.
Corollary 3.11. Suppose Conjecture 3.9 holds for a specific k and l. Then in any bimatrix
1
game we can find in in polynomial time a max{1 − 2k
, 1 − 2l1 }-WSNE with supports of
cardinality size of max{k, l}.
Proof. To apply this method to a general bimatrix games we must first round it to a winlose game using the rounding method (3.1). By Lemma 3.1, an bound in the win-lose
game induces a 21 (1 + ) bound in the bimatrix game. Thus, a bound of 1 − 1t converts to
a bound of
1
1
1
1 + (1 − ) = 1 −
2
t
2t
The result then follows by Theorem 3.10.
So, what is the state of Conjecture 3.9? This seems to be a very hard graph theoretic
problem. It is interesting even for very small values of k and l. For example, the case
k = 3, l = 2 was stated as a separate conjecture by Myers in his doctoral thesis [14]. Graph
theoretically, this can be interpreted as follows. In any directed graph, in which every
3 -WSNE in Bimatrix Games
23
two vertices have a common in-neighbour, there is a directed triangle. Myers stated this
conjecture while solving a special class of the Caccetta-Häggkvist Conjecture. This case
{k = 3, l = 2} remains open. The conjecture is known to be false, however, if both k and l
are at most two.2 Daskalakis, Mehta and Papadimitriou [8] opined that the case k = l = 3
is true. For win-lose games this corresponds to the following conjecture.
Conjecture 3.12. (Daskalakis et al [8]):
Every win-lose game has a 23 -WSNE with supports of cardinality at most three.
It is known that every digraph has a set of log n uncovered vertices. Thus, Conjecture
3.9 is true for the case l = log n. We conclude this chapter with a proof of this fact. As a
consequence, we have the following theorem.
Theorem 3.13. Every win-lose game has a (1 − log1 n )-WSNE with supports of logarithmic
cardinality.
Before proving Theorem 3.13, we present two lemmas that we will need.
Lemma 3.14. Every tournament has a set of vertices of cardinality at most dlog ne that
covers every other vertex in the tournament.
Proof. Consider a tournament T = (V, A) with n vertices. Let deg − (v) denote the in-degree
of a vertex v in T . Let S be an empty set. As a tournament is an oriented complete graph,
there exists an arc between every pair of vertices in the tournament. So, the total number
of arcs in T is exactly n2 . Thus
X
v∈V
deg − (v) =
n · (n − 1)
2
It follows that the vertex of maximum in-degree, vmax , has deg − (vmax ) ≥ d (n−1)
e. Hence,
2
(n−1)
vmax covers at least d 2 e vertices in T . Add vmax to the set S and remove vmax and
the vertices it covers from T . Now recurse on the remaining tournament which contains at
most b (n−1)
c vertices. Upon termination, the set S contains at most dlog ne vertices that
2
covers every other vertex in the tournament.
2
There is a simple counterexample on 7 vertices. Namely, for each vertex i ∈ [7], let i have three
outgoing arcs to vertices (i + 1) mod 7, (i + 2) mod 7 and (i + 4) mod 7.
3 -WSNE in Bimatrix Games
24
Lemma 3.15. Every oriented digraph has an uncovered set of vertices of cardinality at
most dlog ne.
Proof. Take any oriented digraph G. Since it is oriented, it contains at most one arc
between any pair of vertices. Transform G into a tournament T by adding an arc (oriented
at random) between every pair of vertices that currently share no arc. From Lemma 3.14,
we know that T contains a set S of at most dlog ne vertices that covers every other vertex
in T . But this implies that no vertex can cover S. Thus S is uncovered in T . Since T is a
supergraph of G, it follows that S is also uncovered in G. Thus we have an uncovered set
of cardinality at most dlog ne.
Proof of Theorem 3.13.
Take a win-lose game (R, C). Recall, the payoff matrix R can be considered as the adjacency matrix of a directed graph H on n vertices. If there is a self loop in H then, by
Theorem 3.10, we have a 0-WSNE with supports of cardinality at most one. Thus, we
have a pure Nash Equilibrium. If H contains a directed cycle of length 2 then we have a
1
-WSNE by Theorem 3.10. Thus, we may assume H is an oriented directed graph. The
2
result then follows directly from Theorem 3.10 and Lemmas 3.14 and 3.15.
We will revisit Conjecture 3.9 in Chapter 5.
25
Chapter 4
Polylogarithmic Supports are
Required for -WSNE for any < 32
In this chapter, we prove our main result on the structure of -WSNE. We construct win√
lose games for which every -WSNE with < 23 has supports of cardinality at least 3 log n.
Recall, from Chapter 3, that Daskalakis, Mehta and Papadimitriou [8] conjectured that
every win-lose bimatrix game has a 23 -WSNE with supports of cardinality at most three.
Our result does not solve this conjecture, but it does imply that if the conjecture is true
then the 32 guarantee is tight and cannot be improved. We also prove a significant structural
difference between -NE and -WSNE. Namely, in contrast to -NE, supports of cardinality
two are insufficient to guarantee -WSNE with non-trivial approximation bounds.
First, we state our main theorem.
Theorem 4.1. For any < 23 , there exist win-lose bimatrix games for which every -WSNE
√
has supports of cardinality Ω( 3 log n).
To prove this theorem, first we represent win-lose games graphically as in Chapter 2.
Now, we need to construct a directed bipartite graph for which the corresponding game
must have no high quality -WSNE with small supports. We will prove the existence of
such a graph using probabilistic arguments. We construct our graph from a random tournament as follows.
The Construction.
Let T = (V, E) be a random tournament on N nodes. We now build from T an auxiliary
4 Polylogarithmic Supports are Required for -WSNE for any <
2
3
26
bipartite graph G(T ) = (R ∪ C, A) corresponding to a 2-player win-lose game:
Vertices of G(T ). The auxiliary graph has a vertex-bipartition R ∪ C where there is a
vertex of R for each node of T and there is a vertex of C for each set of k distinct nodes of
T . (Observe that, for clarity, we will refer to nodes in the tournament T and to vertices in
the bipartite graph G(T ).)
Arcs of G(T ): There are two types of arc in G(T ),
• Arcs oriented from R to C. For arcs of this type, each vertex X ∈ C will have
in-degree exactly k. Specifically, let X correspond to the k-tuple {v1 , . . . , vk } where
vi ∈ V (T ), for all 1 ≤ i ≤ k. Then there are arcs (vi , X) in G for all 1 ≤ i ≤ k.
Observe that, these arcs are not influenced by the orientation of the arcs in the
tournament T .
• Arcs oriented from C to R. For each node u ∈ R there is an arc (X, u) in G if and
only if u dominates X = {v1 , . . . , vk } in the tournament T , that is if (u, vi ) are arcs
in T for all 1 ≤ i ≤ k.
This completes the construction of our auxiliary bipartite graph G.
Coverage property of a bipartite graph:
We saw in Chapter 2 that, a set of vertices W = {w1 , . . . , wt } is said to be covered if there
exists a vertex y such that (wj , y) ∈ A, for all 1 ≤ j ≤ t. Moreover, a bipartite graph is said
to be k-covered if every collection of k vertices that lie on the same side of the bipartition
is covered. Now the following lemma shows that, with positive probability the auxiliary
graph G(T ) is k-covered.
Lemma 4.2. For all sufficiently large n and k ≤
whose auxiliary bipartite graph G(T ) is k-covered.
√
3
log n, there exists a tournament T
Proof. Consider the the auxiliary bipartite graph G(T ). We see that, there are N vertices
on one side of the bipartition (the R side of the vertex bipartition) and Nk vertices on the
other side of the bipartition (the C side). So the payoff matrices that correspond to G(T )
must have m = N rows and n = Nk columns. Moreover, from the construction we know
4 Polylogarithmic Supports are Required for -WSNE for any <
2
3
27
that, any set of k vertices in R is covered. If we can ensure that any set of k vertices in C
is also covered, then the graph G(T ) will be k-covered.
So consider a collection X = {X1 , . . . , Xk } of k vertices in C. Since each Xi ∈ C
corresponds to a k-tuple of nodes of T , we see that X corresponds to a collection of at
most k 2 nodes in T . Thus, for any node u ∈
/ ∪i Xi , we have that u has an arc in T to every
2
2
node in ∪i Xi with probability at least 2−k . Thus with probability at most (1 − 2k12 )N −k
the subset X of C is not covered in G(T ). Applying the union bound we have that there
exists the desired tournament if
N −k2
n
1
· 1 − k2
<1
k
2
1
1
(4.1)
2
Now set k = log 3 n. Therefore log n k = log 3 n = k 2 .
1
1
In addition, because n = Nk , we have that N ≥ ke · n k . Hence, N − k 2 > n k . (Note
that, since N ≥ k this implies that G(T ) is defined.) Consequently,
N −k2
n k1
n
1
1
· 1 − k2
≤ nk · 1 − k2
k
2
2
k
≤ n ·e
≤ nk · e
1
−
1
2
2k
−
1
1
·n k
2
ek ·log 2
·n k
Thus, taking logarithms, we see that Inequality (4.1) holds if
ek
1
2 ·log 2
1
· k · log n < n k
(4.2)
2
But n k = ek , so Inequality (4.2) clearly holds for large n. The result follows.
The following lemma shows that the auxiliary graph G(T ) contains no cycles with less
than six vertices.
Lemma 4.3. The auxiliary graph G(T ) contains no digons and no 4-cycles.
Proof. Suppose G(T ) contains a digon {w, X}. So, it has both the arcs (w, X) and (X, w).
The arc (w, X) implies that X = {x1 , . . . , xk−1 , w}. On the other-hand, the arc (X, w)
implies that w dominates X in T and, thus, w ∈
/ X, which is a contradiction. Hence G(T )
contains no digons.
4 Polylogarithmic Supports are Required for -WSNE for any <
2
3
28
Suppose G(T ) contains a 4-cycle {w, X, z, Y } where w and z are in R and where X =
{x1 , . . . , xk−1 , w} and Y = {y1 , . . . , yk−1 , z} are in C. Then z must dominate X in T and
w must dominate Y in T . But then we have a digon in T as (w, z) and (z, w) must be arcs
in T . This contradicts the fact that T is a tournament.
In light of Lemma 4.3, we will be interested in the minimum in-degree required to
ensure that a bipartite graph contains a 4-cycle. The following theorem plays a vital role
in proving our main theorem and it also resolves a variant of the well-known CaccettaHäggkvist conjecture [4] for bipartite digraphs. For Eulerian graphs, a related but different
result is due to Shen and Yuster [16].
Theorem 4.4. Let H = (L ∪ R, A) be a directed k × k bipartite graph. If H has minimum
in-degree λ · k then it contains a 4-cycle, whenever λ > 31 .
Proof. To begin, by removing arcs we may assume that every vertex has in-degree exactly
λ · k. No new cycles can be formed by removing arcs. Now take a vertex v with the
maximum out-degree in H, where without loss of generality, v ∈ L. Let A1 be the set of
out-neighbours of v, and set α1 · k = |A1 |. Similarly, let Bt be the set of vertices with paths
to v that contain exactly t arcs, for t ∈ {1, 2}, and set βt · k = |Bt |. Finally, let C1 be the
vertices in R that are not adjacent to v, namely C1 = L \ (A1 ∪ B1 ). Set γ1 · k = |C1 |.
These definitions are illustrated in Figure 4.1.
B2
v
B1
A1
C1
Fig. 4.1
Observe that we have the following constraints on α1 , β1 and γ1 . There are k vertices
4 Polylogarithmic Supports are Required for -WSNE for any <
2
3
29
on either side of the partition. Thus, we have,
(γ1 · k) + (α1 · k) + (β1 · k) = k
γ1 + α1 + β1 = 1
γ1 = 1 − α1 − β1
(4.3)
By assumption, we can see that β1 = λ. So, γ1 = 1 − α1 − λ. Moreover, by the choice of
v, we have α1 ≥ λ, since the maximum out-degree must be at least the average in-degree.
Note that if there is an arc from A1 to B2 then H contains a 4-cycle. So, let’s examine
the in-neighbours of B2 . We know B2 has exactly λ · k · |B2 | incoming arcs. We may assume
all these arcs emanate from B1 ∪ C1 . On the other-hand, there are exactly λ · k · |B1 | arcs
from B2 to B1 . Thus, there are at most |B1 | · (|B2 | − λ · k) arcs from B1 to B2 . So the
number of arcs from C1 to B2 is at least,
λ · k · |B2 | − |B1 | · (|B2 | − λ · k) = λ · k · β2 · k − β1 · k · (β2 · k + λ · k)
= λ · k · β2 · k − λ · k · (β2 · k + λ · k)
= λ2 · k 2
Since the maximum out-degree is α1 · k, the number of arcs emanating from C1 is at most
γ1 · α1 · k 2 . Thus we have,
λ2 · k 2 ≤ γ1 · α1 · k 2
Using inequality (4.3) we get,
λ2 ≤ (1 − α1 − λ) · α1
Rearranging, we obtain the quadratic inequality,
α12 − α1 (1 − λ) + λ2 ≤ 0
The discriminant of this quadratic is 1 − 2λ − 3λ2 . But we have,
1 − 2λ − 3λ2 = (1 − 3λ)(1 + λ)
(4.4)
4 Polylogarithmic Supports are Required for -WSNE for any <
2
3
30
The right hand side of Equality (4.4) is non-negative if and only if λ ≤ 13 . This completes
the proof.
We can also observe that the minimum-in-degree requirement of 13 · k in the above
theorem is tight. To see this, take a directed 6-cycle C and replace each vertex in C by
1
· k copies. Thus each arc in C now corresponds to a complete k3 × k3 bipartite graph with
3
all arc orientations in the same direction. The graph H created in this fashion is bipartite
with all in-degrees (and all out-degrees) equal to 13 · k. Clearly the minimum length of a
directed cycle in H is six.
We now prove our main result, that, no approximation guarantee better than 32 can be
√
achieved unless the well-supported equilibria has supports with cardinality Ω( 3 log n).
Proof of Theorem 4.1. Take a tournament T whose auxiliary bipartite graph is kcovered. By Lemma 4.2, such a tournament exists. Consider the win-lose game corresponding to the auxiliary graph G(T ), and take strategy vectors x and y with supports of
cardinality k or less. Without loss of generality, we may assume that x and y are rational.
Denote these supports as S1 ⊆ R and S2 ⊆ C, respectively. As G(T ) is k-covered, there
is a pure strategy c∗ ∈ C that covers S1 and a pure strategy r∗ ∈ R that covers S2 . Thus,
in the win-lose game, c∗ ∈ C has an expected payoff of 1 against x and r∗ ∈ R has an
expected payoff of 1 against y.
Suppose x and y form an -WSNE for some < 32 . Then it must be the case that
each ri ∈ S1 has expected payoff at least 1 − > 13 against y. Similarly, each cj ∈ S2
has expected payoff at least 1 − > 31 against x. But this cannot happen. Consider the
subgraph of G(T ) induced by S1 ∪ S2 where each ri ∈ S1 has weight wi = x(ri ) and each
cj ∈ S2 has weight wj = y(cj ). We convert this into an unweighted graph H by making
L · wv copies of each vertex v, for some large integer L. Now H is an L × L bipartite graph
with minimum in-degree (1 − ) · L > 31 · L. Thus, by Theorem 4.4, H contains a 4-cycle.
This is a contradiction, by Lemma 4.3.
It is not hard to observe that the 23 in Theorem 4.1 cannot be improved using this proof
technique, as we showed that the minimum in-degree requirement of 13 · k in Theorem 4.4 is
tight. Also, we saw in Theorem 3.3 that, there exist -WSNE with supports of cardinality
O( 12 · log n). So, the polylogarithmic cardinality bound, as proved in this chapter is the
best we can expect.
4 Polylogarithmic Supports are Required for -WSNE for any <
2
3
31
A Structural difference between -NE and -WSNE
Here we use Lemmas 4.2 and 4.3 to show that -WSNE differ structurally from -NE in one
crucial manner. Daskalakis et al. [8] proved that, in any bimatrix game, there is a 12 -NE
with supports of cardinality at most two. In sharp contrast we prove that, for supports of
cardinality at most two, no constant approximation guarantee can be obtained for -WSNE.
We now present both these results.
Theorem 4.5. (Daskalakis et al. [8]) Every bimatrix game has a 12 -NE with supports
of cardinality at most two.
Proof. Take a game (A, B) with payoffs in [0, 1]. (The game need not be a win-lose game.)
Select an arbitrary row i. Let column j be a best response against row i. Thus, Bij =
maxj 0 Bij 0 . Now, let row k be a best response against column j. That is, Akj = maxk0 Ak0 j .
We claim that the strategy profile (x̂, ŷ) is a 12 -WSNE, where, x̂ = 12 ei + 12 ek and ŷ = ej .
Consider first the row player. The expected payoff of the row player, under (x̂, ŷ), is
T
x̂ Aŷ = 12 Aij + 12 Akj . On the other hand, the response against ŷ (column j) is row k. The
would give a payoff of Akj . Thus, the incentive for the row player to deviate is
1
1
1
1
Akj − ( Aij + Akj ) =
Akj − Aij
2
2
2
2
1
Akj
≤
2
1
≤
2
Here the final inequality follows from the fact that all payoffs are at most one.
Now consider the column player. The expected payoff of the row player, under (x̂, ŷ),
is x̂T B ŷ = 12 Bij + 21 Bkj . Assume that, a (pure) best response for the column player against
x̂ is column j 0 . This would produce a payoff of 12 Bij 0 + 12 Bkj 0 . Thus, the incentive for the
column player to deviate is
1
1
1
1
1
1
( Bij 0 + Bkj 0 ) − ( Bij + Bkj ) =
(Bij 0 − Bij ) + (Bkj 0 − Bkj )
2
2
2
2
2
2
1
≤ 0 + (Bkj 0 − Bkj )
2
1
≤
2
4 Polylogarithmic Supports are Required for -WSNE for any <
2
3
32
Here the first inequality follows as column j is a best response to row i. The second
inequality follows as every payoff is at most one.
Consequently, (x̂, ŷ) is a 12 -NE for the game (A, B).
Theorem 4.6. For any δ > 0, there exist win-lose games for which no pair of strategy
vectors with support sizes at most two is a (1 − δ)-WSNE.
Proof. Take the auxiliary win-lose game G(T ) from Lemma 4.2 for the case k = 2. Now
consider any pair of strategy vectors x and y with supports of cardinality 2 or less. Let
these supports be S1 ⊂ R and S2 ⊂ C. Since G(T ) is 2-covered, the best responses to
x and y both generate payoffs of exactly 1. Suppose x and y form an (1 − δ)-WSNE.
Then it must be the case that each ri ∈ S1 has expected payoff at least δ > 0 against y.
Similarly, each cj ∈ S2 has expected payoff at least δ > 0 against x. Thus (x, y) can be a
(1 − δ)-WSNE only if each strategy in the support of x is a best response to at least one
of the pure strategies in the support of y, and vice versa. Therefore, in the subgraph H of
G(T ) induced by the supports of x and y, each vertex has in-degree at least one. Thus, H
contains a directed cycle. But G(T ) has no digons or 4-cycles, by Lemma 4.3. This is a
contradiction as H contains at most four vertices.
Theorem 4.6 shows that, for any δ > 0, there exist win-lose games that require supports
of cardinality at least three in any (1 − δ)-WSNE. It remains open whether supports of
cardinality three are sufficient to produce a (1 − )-WSNE in any win-lose game, for some
constant < 1.
33
Chapter 5
A Characterization of Win-lose
Games with Small Support -WSNE
In Chapter 3 we studied the best response mapping algorithm of Daskalakis, Mehta and
Papadimitriou [8]. Recall that they constructed a directed non-bipartite graph H from
the shuffled payoff matrix Rf of the row player. The original win-lose game then had a
(1 − k1 )-WSNE with supports of cardinality at most k if H contained a uncovered set of
vertices of cardinality k (Lemma 3.8) or contained a cycle of length k (Lemma 3.7).
In this chapter, we will prove that a similar result holds if, rather than H, we use
the bipartite graphical representation of win-lose games described in Chapter 2. More
importantly, however, we show that the converse also holds. If a win-lose game has a
(1 − k1 )-WSNE with supports of cardinality at most k then the graph contains a small
uncovered set or small cycle. Thus we obtain a characterization of win-lose games that
have -WSNE with small cardinality supports.
5.1 -WSNE with Constant Supports.
Let (A, B) ∈ Rm×n (assume, m ≤ n) be a win-lose bimatrix game, and let G = (R ∪ C, E)
be its corresponding directed bipartite graph representation. We will assume that every
vertex in G has non-zero in-degree. The following lemmas show the combinatorial structure
of -WSNE in win-lose games.
5 A Characterization of Win-lose Games with Small Support -WSNE
34
Lemma 5.1. If G contains an uncovered set of cardinality k then there is a (1 − k1 )-WSNE
with supports of cardinality at most k.
Proof. Without loss of generality, let U = {r1 , ..., rk } be the uncovered set. Let the row
player play a uniform strategy x̄ on these k rows. Since U is uncovered, any column has
expected payoff at most 1− k1 against x̄. Therefore every column cj is a (1− k1 )-approximate
best response against x̄.
By assumption, each row vertex ri has in-degree at least one. Let cf (i) be an in-neighbour
of ri (possibly f (i) = f (j) for j 6= i). Now let the column player play a uniform strategy
ȳ on the cf (i) . Because ȳ has supports of cardinality at most k, each pure strategy ri ∈ U
has an expected payoff at least k1 against ȳ.
Thus, these ri ’s are all (1 − k1 )-approximate best responses for the row player against ȳ.
So {x̄, ȳ} is a (1 − k1 )-WSNE with supports of cardinality at most k.
Lemma 5.2. If G contains a cycle of length 2k then there is a (1− k1 )-WSNE with supports
of cardinality k.
Proof. Let W be a cycle of length 2k in G. Since G is bipartite, k of the vertices in the
cycle are row vertices and k are column vertices. Let x̄ be the uniform strategy on the
rows in W and let ȳ be the uniform strategy on the columns in W . We claim that x̄ and ȳ
form a (1 − k1 )-WSNE. To prove this, consider the subgraph F induced by the vertices of
W . Every vertex in H has in-degree (and out-degree) at least one since W ⊆ F . So, every
pure strategy in x̄, gives the row player an expected payoff of at least k1 against ȳ. Thus,
every pure strategy in x̄ is a (1 − k1 )-best response for the row player against ȳ. Similarly,
every pure strategy in ȳ is a (1 − k1 )-best response for the column player against x̄.
Lemma 5.1 and Lemma 5.2 immediately give the following corollary.
Corollary 5.3. Let G be a win-lose game with minimum in-degree at least one. If G
contains a cycle of length 2k or an uncovered set of cardinality k then the win-lose game
has (1 − k1 )-WSNE with supports of cardinality at most k.
Importantly, the converse also holds for games with minimum in-degree at least one.
Lemma 5.4. If there is an -WSNE (for any < 1) with supports of cardinality at most k
then the corresponding graph G either contains an uncovered set of cardinality k or contains
a cycle of length at most 2k.
5 A Characterization of Win-lose Games with Small Support -WSNE
35
Proof. Take a win-lose game G = (R ∪ C, E) and let x and y be an -WSNE. Suppose the
supports of x and y, namely X ⊆ R and Y ⊆ C, have cardinality at most k.
We may assume that every set of cardinality every set of k (on the same side of the
bipartition) is covered; otherwise we are already done. In particular, both X and Y are
covered. Consequently, the row player has a best response with expected payoff 1 against
y. Similarly, the column player has a best response with expected payoff 1 against x. Thus,
for the -WSNE {x, y}, we have:
∀i : xi > 0 ⇒
ei T Ry ≥ 1 − > 0
∀j : yj > 0 ⇒
xT Cej ≥ 1 − > 0
Here the strict inequalities follow because < 1. Therefore, in the subgraph H induced by
X ∪ Y , every vertex has an in-degree at least one. But then H contains a cycle W . Since
H contains at most 2k vertices, the cycle W has length at most 2k.
Corollary 5.3 and Lemma 5.4 then give the following characterization for win-lose games
with -WSNE with small cardinality supports
Theorem 5.5. Let G be a win-lose game with minimum in-degree at least one. Take any
constant k and any such that 1− k1 ≤ < 1. The game contains an -WSNE with supports
of cardinality at most k if and only if G contains an uncovered set of cardinality k or a
cycle of length at most 2k.
5.2 A Conjecture on Non-Bipartite Graphs.
Recall the conjecture of Daskalakis et al. [8] discussed in Chapter 3.
Conjecture 5.6. (Daskalakis et al. [8]): There is an integer k such that every digraph
either has a cycle of length at most k or an uncovered set of k vertices.
We have seen that if the conjecture holds then there is an (1 − k1 )-WSNE with supports
of cardinality at most k. But what if the conjecture is false? We wish then to exploit the
characterization of Theorem 5.5 and conclude that the game has no -WSNE with small
supports. The conjecture refers to non-bipartite graphs so we first need to understand what
a counter-example would imply about win-lose games, that is, non-bipartite graphs.
5 A Characterization of Win-lose Games with Small Support -WSNE
36
Towards this goal, we create a mapping from non-bipartite graphs to bipartite graphs.
Given a non-bipartite graph G = (V, E), we build a win-lose game, that is, a bipartite
directed graph G0 = (R ∪ C, E 0 ) as follows. We set R = C = V . Thus, for each vi ∈ V we
have a row vertex ri ∈ R and a column vertex ci ∈ C. Next, for each arc a = (vi , vj ) in G,
we create two arcs (ri , cj ) and (ci , rj ) in G0 . Finally, for each vi ∈ V we add an arc (ri , ci ).
This mapping is illustrated in Figure 5.1. In the figure, each arc in G is replaced by two
arcs (a red one and a blue one) in G0 . For each vi ∈ V , we have an arc (ri , ci ) in G0 , shown
in black.
R
r1
c1
r2
c2
r3
c3
v1
v2
v3
(a) Directed Non-Bipartite graph
(b) Directed Bipartite Graph
Fig. 5.1: A Mapping from Non-Bipartite Graphs to Bipartite Graphs
Now let’s understand what this mapping does to cycle and uncovered sets. First, suppose G contain a cycle of length `. Then, observe that G0 contains a cycle of length ` if `
is even and of length ` + 1 if ` is odd. On the other hand, suppose the minimum length
cycle in G0 is ` + 1. This cycle will use at most one arc of type (ri , ci ), otherwise we can
find a shorter cycle. Thus, G contains a cycle of length ` or ` + 1.
Second, consider an uncovered set S ⊆ V of size ` in G. Then S ⊆ C is uncovered in
G0 . (Note S ⊆ R may be covered because we added arcs of the form (ri , ci ) to G0 ). On the
other hand if S is uncovered in G0 (either in R or C) then S is also uncovered in G.
For example, consider again Figure 5.1. The 2-cycle C = {v2 , v3 } in G corresponds to
a 2-cycle C 0 = {c2 , r3 } in G0 . In contrast the 3-cycle C1 = {v1 , v3 , v2 } corresponds to a
4-cycle C10 = {c1 , r3 , c2 , r1 } in G0 that uses one black arc. Furthermore, the set S = {v1 , v3 }
is uncovered in G and so S 0 = {c1 , c3 } is uncovered in G0 . However, the set S1 = {v1 , v2 }
5 A Characterization of Win-lose Games with Small Support -WSNE
37
is covered by v3 in G, and so S10 = {c1 , c2 } is covered by r3 in G0 .
Applying this mapping in conjunction with the characterization of Theorem 5.5 we
have:
Theorem 5.7. if Conjecture 5.6 is false then, for any δ > 0, there exist win-lose games
for which every (1 − δ)-WSNE has supports of super-constant cardinality.
Therefore, from a game-theoretic viewpoint, a negative resolution of Conjecture 5.6
would be much more interesting than a positive resolution. The reason being that it
would illustrate a fundamental structural distinction between -WSNE and -NE. This
structural distinction would have important implications, in particular, with regards to
popular equilibria search algorithms that focus upon small supports.
39
Chapter 6
Conclusion
We examined the structure of -WSNE via a directed bipartite graph representation of
win-lose bimatrix games. We proved that to obtain an approximation guarantee better
than 23 we must allow for supports of polylogarithmic cardinality.
The main open question in this area is whether or not a (1 − δ)-WSNE can always be
obtained using supports of constant cardinality, for some constant δ > 0. We proved that
resolving this conjecture is equivalent to determining whether or not there is some constant
k such that any directed graph always contains either an uncovered set or a directed cycle
or cardinality k.
40
References
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Nash equilibrium”, SIAM Journal on Computing, 39(1), pp195-259, 2009.
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[10] W. Hoeffding, “Probability inequalities for sums of bounded random variables”, Journal of the American Statistical Association, Vol 58, 13-30, 1963.
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42
Appendix A
The Approximation Lemma
In this appendix, we present a proof of Althöfer’s Approximation Lemma by Althöfer [2].
To do this, we will need to use Hoeffding’s Inequality [10].
Lemma A.1. (Hoeffding’s Inequality [10])
Let Y1 , .., Yn be independent random variables where each Yi is distributed over the interval
P
[0, 1]. If Y = ni=1 Yi then, for all > 0, P[|Y − E[Y ]| ≥ n] ≤ 2 · exp(−2n2 ).
Lemma 3.2. (Approximation Lemma [2])
Let A ∈ Rm×n with entries in [0, 1]. For any m-probability vector, p ∈ ∆m and any
> 0,
log(2n)
there exists another k-uniform probability vector p̂ ∈ ∆m (k) with |sup(p̂)| ≤ k ≡
22
and sup(p̂) ⊆ sup(p), such that, |pT Aej − p̂T Aej | ≤ , ∀j ∈ [n].
Proof. Given p, define k independent and identically distributed random variables X1 , .., Xk
with values in [m] = {1, .., m} where P[X1 = i] = pi for each i ∈ [m]. (Recall m is the
number of rows in A.) To define p̂, we set p̂i = kki where
ki = #{t : Xt = i, 1 ≤ t ≤ k}
Observe that, by construction, p̂ is a k-uniform strategy.
Now fix some column j of A. Let Yt be the Xt th entry of column j. That is Yt = aXt ,j .
P
By definition of A, the variables Yt satisfy 0 ≤ Yt ≤ 1. Furthermore, E(Yt ) = m
i=1 pi aij
A The Approximation Lemma
43
for all t. We now apply Hoeffding’s Inequality to obtain that
P[|Y − E[Y ]| ≥ k] = P[|
k
X
a Xt j − k ·
t=1
−22 k
m
X
pi aij | > k]
i=1
< 2e
Applying the union bound over all n columns gives
k
m
X
1X
2
a Xt j −
pi aij | > for at least one column j] < 2ne−2 k
P[|
k t=1
i=1
Setting k ≥ log(2n)
gives that the RHS is at most 1. Thence, the random construction
22
gives deviations at most for every column with positive probability. By the probabilistic
method, this implies that there exists a probability vector with the desired property.