Inapproximability of Nash Equilibrium

TL;DR

This paper establishes that computing an ε-approximate Nash equilibrium remains PPAD-complete even in simple game classes, highlighting fundamental computational hardness in game theory and economic equilibria.

Contribution

It proves PPAD-completeness for ε-approximate Nash equilibria in simple polymatrix and graphical games, extending inapproximability results to various economic and game-theoretic models.

Findings

PPAD-completeness for ε-approximate Nash in simple games

Inapproximability results for Bayesian and market equilibria

Hardness extends to generalized circuits and indivisible goods markets

Abstract

We prove that finding an $\epsilon$-approximate Nash equilibrium is PPAD-complete for constant $\epsilon$ and a particularly simple class of games: polymatrix, degree 3 graphical games, in which each player has only two actions. As corollaries, we also prove similar inapproximability results for Bayesian Nash equilibrium in a two-player incomplete information game with a constant number of actions, for relative $\epsilon$-Well Supported Nash Equilibrium in a two-player game, for market equilibrium in a non-monotone market, for the generalized circuit problem defined by Chen, Deng, and Teng [CDT'09], and for approximate competitive equilibrium from equal incomes with indivisible goods.