On Tightness of Tsaknakis-Spirakis Descent Methods for Approximate Nash Equilibria

TL;DR

This paper proves the tightness of the Tsaknakis-Spirakis method's approximation ratio for Nash equilibria in bi-matrix games and introduces a generator for challenging benchmark instances.

Contribution

It establishes the tightness of the 0.3393 approximation bound for the TS algorithm and provides a method to generate instances for benchmarking.

Findings

The 0.3393 bound is tight for the TS algorithm.

Most generated instances are unstable, but stable ones exist.

Other algorithms outperform TS on these instances.

Abstract

This article explores the minimum approximation ratio for Nash equilibrium in bi-matrix games, focusing on the Tsaknakis and Spirakis (TS) methods. The previous SOTA, TS algorithm, achieved an approximation ratio of 0.3393, but efforts to improve the analysis of the TS algorithm have been unsuccessful. This work demonstrates that the bound of 0.3393 is tight for the TS algorithm and presents a theoretical worst-case analysis. A condition for identifying tight instances is provided, along with a generator. While most generated instances are unstable, indicating potential improvements, stable instances exist where perturbations cannot enhance the 0.3393 bound. Other approximate algorithms, such as regret-matching and fictitious play, achieve better ratios on these instances. The generated instances can serve as benchmarks for approximate Nash equilibrium algorithms. The article also mentions progress in the TS algorithm, achieving an approximation ratio of 1/3, which can be further studied using the presented techniques.