Settling the complexity of computing approximate two-player Nash equilibria

We prove that there exists a constant epsilon > 0 such that, assuming the Exponential Time Hypothesis for PPAD, computing an epsilon-approximate Nash equilibrium in a two-player nxn game requires time n(log1-o(1) n). This matches (up to the o (1) term) the algorithm of Lipton, Markakis, and Mehta [54]. Our proof relies on a variety of techniques from the study of probabilistically checkable proofs (PCP); this is the first time that such ideas are used for a reduction between problems inside PPAD. En route, we also prove new hardness results for computing Nash equilibria in games with many players. In particular, we show that computing an epsilon-approximate Nash equilibrium in a game with n players requires 2(Omega(n)) oracle queries to the payoff tensors. This resolves an open problem posed by Hart and Nisan [43], Babichenko [13], and Chen et al. [28]. In fact, our results for n-player games are stronger: they hold with respect to the (epsilon, delta)-WeakNash relaxation recently introduced by Babichenko et al. [15].