Over the past few decades, algorithm design and analysis have become increasingly complex, often resulting in errors and hindering researchers' ability to verify results or share ideas across disciplines. Developing machine-assisted approaches is crucial to simplify these processes and enhance reliability.

We introduce LegoNE, a framework automating the design and analysis of polynomial-time algorithms for approximate Nash equilibrium (ANE), a fundamental and cutting-edge problem in algorithmic game theory. It enables researchers to design ANE algorithms using high-level code descriptions while automatically deriving and proving approximation bounds. LegoNE successfully replicates approximation bounds for all existing polynomial-time ANE algorithms in the literature.

By integrating large language models (LLMs) with LegoNE's automated analysis, we further automate the algorithm design process. DeepSeek-R1, a state-of-the-art LLM, rediscovers the best two-player ANE algorithm in two rounds. For three-player ANE, within two rounds, it discovers a new algorithm whose approximation bound outperforms all existing algorithms designed by human experts. This work highlights the efficacy of human-machine collaboration: humans define high-level problem structures, while machines invent algorithms and verify their theoretical guarantees efficiently.