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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-play","title":"Discovering Expert-Level Nash Equilibrium Algorithms with Large Language Models","url":"https://arxiv.org/abs/2508.11874","vendor":"arxiv_cs_ai"},"summary":"arXiv:2508.11874v2 Announce Type: replace-cross \nAbstract: 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. 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