In an -Nash equilibrium, a player can gain at most by unilaterally changing his behavior. For two-player (bimatrix) games with payoffs in [0, 1], the bestknown achievable in polynomial time is 0.3393 (Tsaknakis and Spirakis in Internet Math 5(4): [365][366][367][368][369][370][371][372][373][374][375][376][377][378][379][380][381][382] 2008). In general, for n-player games an -Nash equilibrium can be computed in polynomial time for an 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 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 such that computing an -Nash equilibrium of a polymatrix game is PPAD-hard. Our main result is that a (0.5 + δ)-Nash equilibrium of an n-player polymatrix game can be computed in time polynomial in the input size and 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 + δ)-Nash equilibrium in a two-player Bayesian game.
On a positive note, REF presented an algorithm for computing a (0.5 + δ)-MSNE of an n-player polymatrix game.
Computing Approximate Nash Equilibria in Polymatrix Games
Algorithmica
Algorithmica
Mathematics Computer Science