The Search-and-Mix Paradigm in Approximate Nash Equilibrium Algorithms

TL;DR

This paper introduces an automatic method for analyzing approximation bounds of algorithms for computing Nash equilibria, using a search-and-mix paradigm that automates the analysis process based on LP-relaxation structures.

Contribution

It presents a novel automated approach to analyze approximation algorithms for Nash equilibria, eliminating manual proof efforts and leveraging LP-relaxation structures.

Findings

Successfully automates approximation bound analysis for existing algorithms

Reproduces known bounds without manual proofs

Potentially extends to other algorithms using LP relaxation

Abstract

AI in Math deals with mathematics in a constructive manner so that reasoning becomes automated, less laborious, and less error-prone. For algorithms, the question becomes how to automate analyses for specific problems. For the first time, this work provides an automatic method for approximation analysis on a well-studied problem in theoretical computer science: computing approximate Nash equilibria in two-player games. We observe that such algorithms can be reformulated into a search-and-mix paradigm, which involves a search phase followed by a mixing phase. By doing so, we are able to fully automate the procedure of designing and analyzing the mixing phase. For example, we illustrate how to perform our method with a program to analyze the approximation bounds of all the algorithms in the literature. Same approximation bounds are computed without any hand-written proof. Our automatic method heavily relies on the LP-relaxation structure in approximate Nash equilibria. Since many approximation algorithms and online algorithms adopt the LP relaxation, our approach may be extended to automate the analysis of other algorithms.