An Efficient PTAS for Two-Strategy Anonymous Games

TL;DR

This paper introduces a faster polynomial time approximation scheme for finding approximate Nash equilibria in two-strategy anonymous games, leveraging new structural insights and Stein's Method.

Contribution

It provides a significantly improved algorithm for approximate Nash equilibria in anonymous games, with a novel structural understanding of equilibria.

Findings

The algorithm computes an $eps$-approximate Nash equilibrium in polynomial time with respect to $n$ and $1/eps$.

Existence of structured approximate equilibria where only a few players randomize or all randomizing players use the same strategy.

The new structural results lead to a more efficient approximation algorithm compared to previous methods.

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 $poly(n) (1/eps)^{O(1/eps^2)}$, which significantly improves upon the running time $n^{O(1/eps^2)}$ 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 $O(1/eps^3)$ 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.