An Empirical Study of Finding Approximate Equilibria in Bimatrix Games

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

On Tightness of the Tsaknakis-Spirakis Algorithm for Approximate Nash Equilibrium

Finding the minimum approximate ratio for Nash equilibrium of bi-matrix games has derived a series of studies, started with 3/4, followed by 1/2, 0.38 and 0.36, finally the best approximate ratio of 0.3393 by Tsaknakis and Spirakis (TS algorithm for short). Efforts to improve the results remain not successful in the past 14 years. This work makes the first progress to show that the bound of 0.3393 is indeed tight for the TS algorithm. Next, we characterize all possible tight game instances for the TS algorithm. It allows us to conduct extensive experiments to study the nature of the TS algorithm and to compare it with other algorithms. We find that this lower bound is not smoothed for the TS algorithm in that any perturbation on the initial point may deviate away from this tight bound approximate solution. Other approximate algorithms such as Fictitious Play and Regret Matching also find better approximate solutions. However, the new distributed algorithm for approximate Nash equilibrium by Czumaj et al. performs consistently at the same bound of 0.3393. This proves our lower bound instances generated against the TS algorithm can serve as a benchmark in design and analysis of approximate Nash equilibrium algorithms