Abstract

By proving that the problem of computing a 1/n Θ(1)-approximate Nash equilibrium remains PPAD-complete, we show that the BIMATRIX game is not likely to have a fully polynomial-time approximation scheme. In other words, no algorithm with time polynomial in n and 1/ǫ can compute an ǫ-approximate Nash equilibrium of an n×n bimatrix game, unless PPAD ⊆ 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 that it is unlikely, unless PPAD ⊆ 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/σ under perturbations with magnitude σ. Our result answers a major open question in the smoothed analysis of algorithms and the approximation of Nash equilibria.

Citations