Smooth Nash Equilibria: Algorithms and Complexity

A fundamental shortcoming of the concept of Nash equilibrium is its computational intractability: approximating Nash equilibria in normal-form games is PPAD-hard. In this paper, inspired by the ideas of smoothed analysis, we introduce a relaxed variant of Nash equilibrium called $\sigma$-smooth Nash equilibrium, for a smoothness parameter $\sigma$. In a $\sigma$-smooth Nash equilibrium, players only need to achieve utility at least as high as their best deviation to a $\sigma$-smooth strategy, which is a distribution that does not put too much mass (as parametrized by $\sigma$) on any fixed action. We distinguish two variants of $\sigma$-smooth Nash equilibria: strong $\sigma$-smooth Nash equilibria, in which players are required to play $\sigma$-smooth strategies under equilibrium play, and weak $\sigma$-smooth Nash equilibria, where there is no such requirement. We show that both weak and strong $\sigma$-smooth Nash equilibria have superior computational properties to Nash equilibria: when $\sigma$ as well as an approximation parameter $\epsilon$ and the number of players are all constants, there is a constant-time randomized algorithm to find a weak $\epsilon$-approximate $\sigma$-smooth Nash equilibrium in normal-form games. In the same parameter regime, there is a polynomial-time deterministic algorithm to find a strong $\epsilon$-approximate $\sigma$-smooth Nash equilibrium in a normal-form game. These results stand in contrast to the optimal algorithm for computing $\epsilon$-approximate Nash equilibria, which cannot run in faster than quasipolynomial-time. We complement our upper bounds by showing that when either $\sigma$ or $\epsilon$ is an inverse polynomial, finding a weak $\epsilon$-approximate $\sigma$-smooth Nash equilibria becomes computationally intractable.