Well-Supported versus Approximate Nash Equilibria: Query Complexity of Large Games

We study the randomized query complexity of approximate Nash equilibria (ANE) in large games. We prove that, for some constant $\epsilon>0$, any randomized oracle algorithm that computes an $\epsilon$-ANE in a binary-action, $n$-player game must make $2^{{\Omega(n/\log} n)}$ payoff queries. For the stronger solution concept of well-supported Nash equilibria (WSNE), Babichenko previously gave an exponential $2^{\Omega(n)}$ lower bound for the randomized query complexity of $\epsilon$-WSNE, for some constant $\epsilon>0$; the same lower bound was shown to hold for $\epsilon$-ANE, but only when $\epsilon=O(1/n)$. Our result answers an open problem posed by Hart and Nisan and Babichenko and is very close to the trivial upper bound of $2^n$. Our proof relies on a generic reduction from the problem of finding an $\epsilon$-WSNE to the problem of finding an $\epsilon/(4\alpha)$-ANE, in large games with $\alpha$ actions, which might be of independent interest.