Computing Nash Equilibria: Approximation and Smoothed Complexity

Xi Chen; Xiaotie Deng; Shang-Hua Teng

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Computing Nash Equilibria: Approximation and Smoothed Complexity

Xi Chen·Xiaotie Deng·Shang-Hua Teng

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Abstract

We show that the BIMATRIX game does not have a fully polynomial-time approximation scheme, unless PPAD is in P. In other words, no algorithm with time polynomial in n and 1/εcan compute an ε-approximate Nash equilibrium of an n by nbimatrix game, unless PPAD is in P. Instrumental to our proof, we introduce a new discrete fixed-point problem on a high-dimensional cube with a constant side-length, such as on an n-dimensional cube with side-length 7, and show that they are PPAD-complete. Furthermore, we prove, unless PPAD is in RP, that the smoothed complexity of the Lemke-Howson algorithm or any algorithm for computing a Nash equilibrium of a bimatrix game is polynomial in n and 1/σunder perturbations with magnitude σ. Our result answers a major open question in the smoothed analysis of algorithms and the approximation of Nash equilibria.

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Xi Chen, Xiaotie Deng, Shang-Hua Teng. 2006-02-22. Computing Nash Equilibria: Approximation and Smoothed Complexity. https://arxiv.org/abs/cs/0602043

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