Nash Equilibria in Perturbation Resilient Games

TL;DR

This paper studies equilibrium computation in perturbation-stable bimatrix games, showing that such games allow more efficient approximation algorithms and are computationally easier than general games, especially under uniform stability.

Contribution

It introduces the concept of perturbation resilience in bimatrix games and demonstrates improved algorithms for approximate Nash equilibria in these stable games.

Findings

Efficient approximation algorithms for stable games.

Approximate equilibria are close to true equilibria even with perturbed data.

Stable games are computationally easier than general bimatrix games.

Abstract

Motivated by the fact that in many game-theoretic settings, the game analyzed is only an approximation to the game being played, in this work we analyze equilibrium computation for the broad and natural class of bimatrix games that are stable to perturbations. We specifically focus on games with the property that small changes in the payoff matrices do not cause the Nash equilibria of the game to fluctuate wildly. For such games we show how one can compute approximate Nash equilibria more efficiently than the general result of Lipton et al. \cite{LMM03}, by an amount that depends on the degree of stability of the game and that reduces to their bound in the worst case. Furthermore, we show that for stable games the approximate equilibria found will be close in variation distance to true equilibria, and moreover this holds even if we are given as input only a perturbation of the actual underlying stable game. For uniformly-stable games, where the equilibria fluctuate at most quasi-linearly in the extent of the perturbation, we get a particularly dramatic improvement. Here, we achieve a fully quasi-polynomial-time approximation scheme: that is, we can find $1/\poly(n)$-approximate equilibria in quasi-polynomial time. This is in marked contrast to the general class of bimatrix games for which finding such approximate equilibria is PPAD-hard. In particular, under the (widely believed) assumption that PPAD is not contained in quasi-polynomial time, our results imply that such uniformly stable games are inherently easier for computation of approximate equilibria than general bimatrix games.