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.

翻译：纳什均衡概念的一个根本缺陷在于其计算难解性：在标准式博弈中逼近纳什均衡是PPAD-hard的。受平滑分析思想的启发，本文引入了一种称为σ-平滑纳什均衡的放松变体，其中σ为平滑参数。在σ-平滑纳什均衡中，参与者只需获得至少不低于其偏离至σ-平滑策略（一种不会在任何固定动作上分配过大质量（由σ参数化）的分布）所能达到的效用。我们区分了两种σ-平滑纳什均衡变体：强σ-平滑纳什均衡（要求参与者在均衡状态下采用σ-平滑策略）和弱σ-平滑纳什均衡（无此要求）。我们证明弱和强σ-平滑纳什均衡均比纳什均衡具有更优越的计算性质：当σ、近似参数ε以及参与者数量均为常数时，存在常数时间随机算法可在标准式博弈中找到弱ε-近似σ-平滑纳什均衡。在相同的参数设定下，存在多项式时间确定性算法可在标准式博弈中找到强ε-近似σ-平滑纳什均衡。这些结果与计算ε-近似纳什均衡的最优算法形成对比——后者无法以快于拟多项式时间运行。作为上界的补充，我们证明当σ或ε为逆多项式时，寻找弱ε-近似σ-平滑纳什均衡将变得计算上难解。