Computing Approximate Nash Equilibria in Polymatrix Games

Argyrios Deligkas, John Fearnley, Rahul Savani, Paul G. Spirakis

Abstract: In an $$\epsilon $$ ϵ -Nash equilibrium, a player can gain at most $$\epsilon $$ ϵ by unilaterally changing his behavior. For two-player (bimatrix) games with payoffs in [0, 1], the best-known $$\epsilon $$ ϵ 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 $$\epsilon $$ ϵ -Nash equilibrium can be computed in polynomial time for an $$\epsilon $$ ϵ 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 $$\epsilon $$ ϵ 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 $$\epsilon $$ ϵ such that computing an $$\epsilon $$ ϵ -Nash equilibrium of a polymatrix game is $$\mathtt {PPAD}$$ PPAD -hard. Our main result is that a $$(0.5+\delta )$$ ( 0.5 + δ ) -Nash equilibrium of an n-player polymatrix game can be computed in time polynomial in the input size and $$\frac{1}{\delta }$$ 1 δ . 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 + δ ) -Nash equilibrium in a two-player Bayesian game.