Inapproximability of Nash Equilibrium

We prove that finding an ε-approximate Nash equilibrium is PPAD-complete for constant ε 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 ε-Nash equilibrium in a two-player game, for market equilibrium in a non-monotone market, for the generalized circuit problem defined by Chen et al. [4], and for approximate competitive equilibrium from equal incomes with indivisible goods.