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 approximate wellsupported 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 approximate Nash equilibria: it uses poly-logarithmic communication to find a 0.382-approximate Nash equilibrium.Communication complexity of equilibria in games has been studied in previous works [3,13]. The recent paper of Goldberg and Pastink [11] initiated the study of communication complexity in the bimatrix game setting. There they showed Θ(n 2 ) communication is required to find an exact Nash equilibrium of an n × n bimatrix game. Since these games have Θ(n 2 ) payoffs in total, this implies that there is no communication efficient protocol for finding exact Nash equilibria in bimatrix games. For approximate equilibria, they showed that one can find a 3 4 -Nash equilibrium without any communication, and that in the nocommunication setting, finding an 1 2 -Nash equilibrium is impossible. Motivated by these positive and negative results, they focused on the most interesting setting, which allows only a polylogarithmic (in n) amount of communication (number of bits) between the players. They demonstrated that one can compute 0.438-Nash equilibria and 0.732-well-supported Nash equilibria in this setting.

The ε-well-supported Nash equilibrium is a strong notion of approximation of a Nash equilibrium, where no player has an incentive greater than ε to deviate from any of the pure strategies that she uses in her mixed strategy. The smallest constant ε currently known for which there is a polynomial-time algorithm that computes an ε-well-supported Nash equilibrium in bimatrix games is slightly below 2/3. In this paper we study this problem for symmetric bimatrix games and we provide a polynomial-time algorithm that gives a (1/2 + δ)-well-supported Nash equilibrium, for an arbitrarily small positive constant δ.