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 computes an approximate Nash equilibrium using only limited communication between the players. Our method gives improved bounds on the complexity of computing approximate Nash equilibria in a number of different settings. Firstly, it gives a polynomial-time algorithm for computing approximate well supported Nash equilibria (WSNE) that always finds a 0.6528-WSNE, beating the previous best guarantee of 0.6608. Secondly, since our algorithm solves the two LPs separately, it can be applied to give an improved bound in the limited communication setting, giving a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and finds a 0.6528-WSNE, which beats the previous best known guarantee of 0.732. It can also be applied to the case of approximate Nash equilibria, where we obtain a randomized expected-polynomial-time algorithm that uses poly-logarithmic communication and always finds a 0.382-approximate Nash equilibrium, which improves the previous best guarantee of 0.438. Finally, the method can also be applied in the query complexity setting to give an algorithm that makes $$O(n \log n)$$O(nlogn) payoff queries and always finds a 0.6528-WSNE, which improves the previous best known guarantee of 2/3.