Playing Anonymous Games using Simple Strategies

Abstract

We investigate the complexity of computing approximate Nash equilibria in anonymous games. Our main algorithmic result is the following: For any n-player anonymous game with a bounded number of strategies and any constant δ > 0, an O(1/n1−δ)-approximate Nash equilibrium can be computed in polynomial time. Complementing this positive result, we show that if there exists any constant δ > 0 such that an O(1/n1+δ)-approximate equilibrium can be computed in polynomial time, then there is a fully polynomial-time approximation scheme (FPTAS) for this problem.We also present a faster algorithm that, for any n-player k-strategy anonymous game, runs in time O((n + k)knk) and computes an O(n−1/3k11/3)-approximate equilibrium. This algorithm follows from the existence of simple approximate equilibria of anonymous games, where each player plays one strategy with probability 1 − δ, for some small δ, and plays uniformly at random with probability δ.Our approach exploits the connection between Nash equilibria in anonymous games and Poisson multinomial distributions (PMDs). Specifically, we prove a new probabilistic lemma establishing the following: Two PMDs, with large variance in each direction, whose first few moments are approximately matching are close in total variation distance. Our structural result strengthens previous work by providing a smooth tradeoff between the variance bound and the number of matching moments.