Smooth Nash Equilibria: Algorithms and Complexity
Constantinos Daskalakis, Noah Golowich, Nika Haghtalab, Abhishek Shetty
TL;DR
The work introduces $σ$-smooth Nash equilibria, a smoothed-analysis-based relaxation of Nash equilibria, to address the computational intractability of finding exact equilibria in normal-form games.Two variants are studied: weak, which allows deviations to $σ$-smooth distributions, and strong, which imposes $σ$-smoothness on the equilibrium strategies themselves; these notions enable tractable computation under favorable parameter regimes.The authors provide both algorithmic and hardness results: constant-time randomized algorithms for weak equilibria and polynomial-time deterministic algorithms for strong equilibria when $m$, $ε$, and $σ$ are constants, alongside PPAD- and ETH-based hardness when $ε$ or $σ$ are inverse-polynomial.Structural results show the existence of sparse, sample-based representations (k-uniform profiles) that underpin efficient algorithms; connections to quantal response, polyhedral games, and regularization are discussed, with implications for multi-agent learning.
Abstract
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 $σ$-smooth Nash equilibrium, for a smoothness parameter $σ$. In a $σ$-smooth Nash equilibrium, players only need to achieve utility at least as high as their best deviation to a $σ$-smooth strategy, which is a distribution that does not put too much mass (as parametrized by $σ$) on any fixed action. We distinguish two variants of $σ$-smooth Nash equilibria: strong $σ$-smooth Nash equilibria, in which players are required to play $σ$-smooth strategies under equilibrium play, and weak $σ$-smooth Nash equilibria, where there is no such requirement. We show that both weak and strong $σ$-smooth Nash equilibria have superior computational properties to Nash equilibria: when $σ$ as well as an approximation parameter $ε$ and the number of players are all constants, there is a constant-time randomized algorithm to find a weak $ε$-approximate $σ$-smooth Nash equilibrium in normal-form games. In the same parameter regime, there is a polynomial-time deterministic algorithm to find a strong $ε$-approximate $σ$-smooth Nash equilibrium in a normal-form game. These results stand in contrast to the optimal algorithm for computing $ε$-approximate Nash equilibria, which cannot run in faster than quasipolynomial-time. We complement our upper bounds by showing that when either $σ$ or $ε$ is an inverse polynomial, finding a weak $ε$-approximate $σ$-smooth Nash equilibria becomes computationally intractable.