Inapproximability of Nash Equilibrium

Abstract

We prove that finding an $\epsilon$-approximate Nash equilibrium is $\mathsf{PPAD}$--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 nonmonotone market, for the generalized circuit problem defined by Chen, Deng, and Teng [J. ACM, 56 (2009)], and for approximate competitive equilibrium from equal incomes with indivisible goods.