An Efficient PTAS for Two-Strategy Anonymous Games

Abstract: We present a novel polynomial time approximation scheme for two-strategy anonymous games, in which the players' utility functions, although potentially different, do not differentiate among the identities of the other players. Our algorithm computes an $eps$-approximate Nash equilibrium of an $n$-player 2-strategy anonymous game in time $poly(n) (1/eps)<sup>{O(1/eps<sup>2)}$, which significantly improves upon the running time $n<sup>{O(1/eps<sup>2)}$ required by the algorithm of Daskalakis & Papadimitriou, 2007. The improved running time is based on a new structural understanding of approximate Nash equilibria: We show that, for any $eps$, there exists an $eps$-approximate Nash equilibrium in which either only $O(1/eps<sup>3)$ players randomize, or all players who randomize use the same mixed strategy. To show this result we employ tools from the literature on Stein's Method.