Computing Nash Equilibria: Approximation and Smoothed Complexity

TL;DR

This paper proves the computational hardness of approximating Nash equilibria in bimatrix games and establishes the smoothed complexity bounds for existing algorithms, resolving key open questions in game theory and algorithm analysis.

Contribution

It introduces a new PPAD-complete fixed-point problem and shows the limitations of polynomial-time approximation schemes for Nash equilibria.

Findings

No polynomial-time -approximation scheme exists unless PPAD in P.

The Lemke-Howson algorithm has polynomial smoothed complexity under perturbations.

The paper resolves major open questions in the complexity and smoothed analysis of Nash equilibria.

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/\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.