Settling the Complexity of Computing Approximate Two-Player Nash Equilibria

We prove that there exists a constant ε > 0 such that, assuming the Exponential Time Hypothesis for PPAD, computing an ε-approximate Nash equilibrium in a two-player (n × n) game requires quasi-polynomial time, nlog1-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 ε-approximate Nash equilibrium in a game with n players requires 2Ω(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 (ε,δ)-WeakNash relaxation recently introduced by Babichenko et al. [15].