Computing Nash Equilibria: Approximation and Smoothed Complexity
<|reference_start|>Computing Nash Equilibria: Approximation and Smoothed Complexity: 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/\epsilon can compute an \epsilon-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/\sigma under perturbations with magnitude \sigma. Our result answers a major open question in the smoothed analysis of algorithms and the approximation of Nash equilibria.<|reference_end|>
@article{chen2006computing,
    title={Computing Nash Equilibria: Approximation and Smoothed Complexity},
    author={Xi Chen, Xiaotie Deng and Shang-Hua Teng},
    journal={arXiv preprint arXiv:cs/0602043},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0602043},
    primaryClass={cs.CC cs.GT}
}