Well-Supported vs. 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 ǫ > 0, any randomized oracle algorithm that computes an ǫ-ANE in a binary-action, n-player game must make 2 logn) payoff queries. For the stronger solution concept of well-supported Nash equilibria (WSNE), Babichenko [Bab14] previously gave an exponential 2 lower bound for the randomized query complexity of ǫ-WSNE, for some constant ǫ > 0; the same lower bound was shown to hold for ǫ-ANE, but only when ǫ = O(1/n). Our result answers an open problem posed by Hart and Nisan in [HN13] and by Babichenko in [Bab14], and is very close to the trivial upper bound of 2. Our proof relies on a generic reduction from the problem of finding an ǫ-WSNE to the problem of finding an ǫ/(4α)-ANE, in large games with α actions, which might be of independent interest. Email: [email protected]. Supported in part by NSF grants CCF-1149257 and CCF-1423100. Email: [email protected]. Supported in part by NSF grant CCF-1111270 and Shang-Hua Teng’s Simons Investigator Award. Email: [email protected]. Supported in part by ERC grant 321171. This work was done in part while the authors were visiting the Simons Institute.