An Empirical Study of Finding Approximate Equilibria in Bimatrix Games

TL;DR

This paper empirically compares various methods for finding approximate Nash equilibria in bimatrix games, analyzing their trade-offs in quality and computational time, and introduces new game families and challenging examples.

Contribution

It provides the first comprehensive empirical evaluation of approximation methods, extends testing to larger games, and introduces new challenging instances.

Findings

Existing test libraries are insufficient for approximation methods.

Some algorithms perform well on certain game families but poorly on others.

New worst-case examples challenge current approximation algorithms.

Abstract

While there have been a number of studies about the efficacy of methods to find exact Nash equilibria in bimatrix games, there has been little empirical work on finding approximate Nash equilibria. Here we provide such a study that compares a number of approximation methods and exact methods. In particular, we explore the trade-off between the quality of approximate equilibrium and the required running time to find one. We found that the existing library GAMUT, which has been the de facto standard that has been used to test exact methods, is insufficient as a test bed for approximation methods since many of its games have pure equilibria or other easy-to-find good approximate equilibria. We extend the breadth and depth of our study by including new interesting families of bimatrix games, and studying bimatrix games upto size $2000 \times 2000$. Finally, we provide new close-to-worst-case examples for the best-performing algorithms for finding approximate Nash equilibria.