Logarithmic Query Complexity for Approximate Nash Computation in Large Games

TL;DR

This paper presents a randomized algorithm with logarithmic query complexity for approximating Nash equilibria in large, Lipschitz-type games, achieving near 1/8 accuracy in a fully uncoupled setting with minimal communication.

Contribution

It introduces a novel logarithmic-query algorithm for approximate Nash computation in large games, including extensions to multi-strategy and different largeness parameters.

Findings

Achieves  approximation with O(log n) queries in 2-strategy games

Allows slight improvements with minimal communication

Extends results to multi-strategy large games

Abstract

We investigate the problem of equilibrium computation for "large" $n$-player games. Large games have a Lipschitz-type property that no single player's utility is greatly affected by any other individual player's actions. In this paper, we mostly focus on the case where any change of strategy by a player causes other players' payoffs to change by at most $\frac{1}{n}$. We study algorithms having query access to the game's payoff function, aiming to find $\epsilon$-Nash equilibria. We seek algorithms that obtain $\epsilon$ as small as possible, in time polynomial in $n$. Our main result is a randomised algorithm that achieves $\epsilon$ approaching $\frac{1}{8}$ for 2-strategy games in a {\em completely uncoupled} setting, where each player observes her own payoff to a query, and adjusts her behaviour independently of other players' payoffs/actions. $O(\log n)$ rounds/queries are required. We also show how to obtain a slight improvement over $\frac{1}{8}$, by introducing a small amount of communication between the players. Finally, we give extension of our results to large games with more than two strategies per player, and alternative largeness parameters.