Discovering Expert-Level Nash Equilibrium Algorithms with Large Language Models

Designing polynomial-time algorithms for approximate Nash equilibria (ANE) with provable worst-case guarantees is a fundamental open problem in algorithmic game theory. While large language models (LLMs) can generate candidate algorithms at scale, certifying worst-case guarantees requires formal analysis over all game instances -- a task for which no automated system previously existed. Here, we present LegoNE, a framework encoding expert proof strategies into a symbolic language that automatically compiles any candidate algorithm into a finite optimization problem certifying its worst-case guarantee. Integrating LegoNE with a reasoning LLM, we rediscovered an algorithm matching the best polynomial-time guarantee for two-player games, and discovered a three-player algorithm improving the best guarantee from $0.6+\delta$ to $0.5+\delta$ -- provably beyond the reach of the extension technique, the only previously known multi-player ANE design paradigm. These results show that encoding domain-specific proof strategies into a machine-tractable language can support LLM-driven discovery of algorithms outside known human design paradigms.