Computing Approximate Nash Equilibria in Polymatrix Games

Abstract

In an $$\epsilon $$∈-Nash equilibrium, a player can gain at most $$\epsilon $$∈ by unilaterally changing his behavior. For two-player (bimatrix) games with payoffs in [0, 1], the best-known $$\epsilon $$∈ achievable in polynomial time is 0.3393 (Tsaknakis and Spirakis in Internet Math 5(4):365---382, 2008). 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 $$\mathtt {PPAD}$$PPAD-hard. Our main result is that a $$(0.5+\delta )$$(0.5+ź)-Nash equilibrium of an n-player polymatrix game can be computed in time polynomial in the input size and $$\frac{1}{\delta }$$1ź. Inspired by the algorithm of Tsaknakis and Spirakis [28], our algorithm uses gradient descent style approach on the maximum regret of the players. We also show that this algorithm can be applied to efficiently find a $$(0.5+\delta )$$(0.5+ź)-Nash equilibrium in a two-player Bayesian game.