Discovering Expert-Level Nash Equilibrium Algorithms with Large Language Models

TL;DR

This paper introduces LegoNE, a framework that automates the discovery and proof of algorithms for computing approximate Nash equilibria, leveraging large language models to outperform human-designed algorithms.

Contribution

LegoNE integrates algorithm design and formal analysis into an automated process, enabling AI to rediscover and innovate Nash equilibrium algorithms efficiently.

Findings

AI rediscovered the state-of-the-art algorithm for two-player games in hours.

AI discovered a novel algorithm for three-player games surpassing existing ones.

LegoNE demonstrates a new collaborative paradigm for theoretical algorithm discovery.

Abstract

Algorithm design and analysis is a cornerstone of computer science, but it confronts a major challenge. Proving an algorithm's performance guarantee across all inputs has traditionally required extensive and often error-prone human effort. While AI has shown great success in finding solutions to specific problem instances, automating the discovery of general algorithms with such provable guarantees has remained a significant barrier. This challenge stems from the difficulty of integrating the creative process of algorithm design with the rigorous process of formal analysis. To address this gap, we propose LegoNE, a framework that tightly fuses these two processes for the fundamental and notoriously difficult problem of computing approximate Nash equilibria. LegoNE automatically translates any algorithm written by a simple Python-like language into a constrained optimization problem. Solving this problem derives and proves the algorithm's approximation bound. Using LegoNE, a state-of-the-art large language model rediscovered the state-of-the-art algorithm for two-player games within hours, a feat that had taken human researchers 15 years to achieve. For three-player games, the model discovered a novel algorithm surpassing all existing human-designed ones. This work demonstrates a new human-machine collaborative paradigm for theoretical science: humans reason at a higher-abstract level, using symbols to compress the search space, and AI explores within it, achieving what neither could alone.