An Empirical Study of Finding Approximate Equilibria in Bimatrix Games

While there have been a number of studies about the efficacy of methods to find exact Nash equilibria in bimatrix games, there has been little empirical work on finding approximate Nash equilibria. Here we provide such a study that compares a number of approximation methods and exact methods. In particular, we explore the trade-off between the quality of approximate equilibrium and the required running time to find one. We found that the existing library GAMUT, which has been the de facto standard that has been used to test exact methods, is insufficient as a test bed for approximation methods since many of its games have pure equilibria or other easy-to-find good approximate equilibria. We extend the breadth and depth of our study by including new interesting families of bimatrix games, and studying bimatrix games upto size $2000 \times 2000$. Finally, we provide new close-to-worst-case examples for the best-performing algorithms for finding approximate Nash equilibria

Distributed Methods for Computing Approximate Equilibria

We present a new, distributed method to compute approximate Nash equilibria in bimatrix games. In contrast to previous approaches that analyze the two payoff matrices at the same time (for example, by solving a single LP that combines the two players payoffs), our algorithm first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then compute approximate Nash equilibria using only limited communication between the players. Our method has several applications for improved bounds for efficient computations of approximate Nash equilibria in bimatrix games. First, it yields a best polynomial-time algorithm for computing \emph{approximate well-supported Nash equilibria (WSNE)}, which guarantees to find a 0.6528-WSNE in polynomial time. Furthermore, since our algorithm solves the two LPs separately, it can be used to improve upon the best known algorithms in the limited communication setting: the algorithm can be implemented to obtain a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE. The algorithm can also be carried out to beat the best known bound in the query complexity setting, requiring $O(n \log n)$ payoff queries to compute a 0.6528-WSNE. Finally, our approach can also be adapted to provide the best known communication efficient algorithm for computing \emph{approximate Nash equilibria}: it uses poly-logarithmic communication to find a 0.382-approximate Nash equilibrium

New algorithms for approximate Nash equilibria in bimatrix games

We consider the problem of computing additively approximate Nash equilibria in non-cooperative two-player games. We provide a new polynomial time algorithm that achieves an approximation guarantee of 0.36392. Our work improves the previously best known (0.38197¿+¿e)-approximation algorithm of Daskalakis, Mehta and Papadimitriou [6]. First, we provide a simpler algorithm, which also achieves 0.38197. This algorithm is then tuned, improving the approximation error to 0.36392. Our method is relatively fast, as it requires solving only one linear program and it is based on using the solution of an auxiliary zero-sum game as a starting point. The first author was supported by NWO. The second and third author were supported by the EU Marie Curie Research Training Network, contract numbers MRTN-CT-2003-504438-ADONET and MRTN-CT-2004-504438-ADONET respectively

New Algorithms for Approximate Nash Equilibria in Bimatrix Games

We consider the problem of computing additively approximate Nash equilibria in noncooperative two-player games. We provide a new polynomial time algorithm that achieves an approximation guarantee of 0.36392. We first provide a simpler algorithm, that achieves a 0.38197-approximation, which is exactly the same factor as the algorithm of Daskalakis, Mehta and Papadimitriou.This algorithm is then tuned, improving the approximation error to 0.36392. Our method is relatively fast and simple, as it requires solving only one linear program and it is based on using the solution of an auxiliary zero-sum game as a starting point. Finally we also exhibit a simple reduction that allows us to compute approximate equilibria for multi-player games by using algorithms for two-player games

Computing Approximate Nash Equilibria in Polymatrix Games

In an $\epsilon$-Nash equilibrium, a player can gain at most $\epsilon$ by unilaterally changing his behaviour. For two-player (bimatrix) games with payoffs in $[0,1]$, the best-known$\epsilon$ achievable in polynomial time is 0.3393. In general, for $n$-player games an $\epsilon$-Nash equilibrium can be computed in polynomial time for an $\epsilon$ that is an increasing function of $n$ but does not depend on the number of strategies of the players. For three-player and four-player games the corresponding values of $\epsilon$ are 0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general $n$-player games where a player's payoff is the sum of payoffs from a number of bimatrix games. There exists a very small but constant $\epsilon$ such that computing an $\epsilon$-Nash equilibrium of a polymatrix game is \PPAD-hard. Our main result is that a $(0.5+\delta)$-Nash equilibrium of an $n$-player polymatrix game can be computed in time polynomial in the input size and $\frac{1}{\delta}$. Inspired by the algorithm of Tsaknakis and Spirakis, our algorithm uses gradient descent on the maximum regret of the players. We also show that this algorithm can be applied to efficiently find a $(0.5+\delta)$-Nash equilibrium in a two-player Bayesian game

Polylogarithmic Supports are required for Approximate Well-Supported Nash Equilibria below 2/3

In an epsilon-approximate Nash equilibrium, a player can gain at most epsilon in expectation by unilateral deviation. An epsilon well-supported approximate Nash equilibrium has the stronger requirement that every pure strategy used with positive probability must have payoff within epsilon of the best response payoff. Daskalakis, Mehta and Papadimitriou conjectured that every win-lose bimatrix game has a 2/3-well-supported Nash equilibrium that uses supports of cardinality at most three. Indeed, they showed that such an equilibrium will exist subject to the correctness of a graph-theoretic conjecture. Regardless of the correctness of this conjecture, we show that the barrier of a 2/3 payoff guarantee cannot be broken with constant size supports; we construct win-lose games that require supports of cardinality at least Omega((log n)^(1/3)) in any epsilon-well supported equilibrium with epsilon < 2/3. The key tool in showing the validity of the construction is a proof of a bipartite digraph variant of the well-known Caccetta-Haggkvist conjecture. A probabilistic argument shows that there exist epsilon-well-supported equilibria with supports of cardinality O(log n/(epsilon^2)), for any epsilon> 0; thus, the polylogarithmic cardinality bound presented cannot be greatly improved. We also show that for any delta > 0, there exist win-lose games for which no pair of strategies with support sizes at most two is a (1-delta)-well-supported Nash equilibrium. In contrast, every bimatrix game with payoffs in [0,1] has a 1/2-approximate Nash equilibrium where the supports of the players have cardinality at most two.Comment: Added details on related work (footnote 7 expanded

Parameterized Two-Player Nash Equilibrium

We study the computation of Nash equilibria in a two-player normal form game from the perspective of parameterized complexity. Recent results proved hardness for a number of variants, when parameterized by the support size. We complement those results, by identifying three cases in which the problem becomes fixed-parameter tractable. These cases occur in the previously studied settings of sparse games and unbalanced games as well as in the newly considered case of locally bounded treewidth games that generalizes both these two cases