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 Ω(n/ log n) payoff queries. For the stronger solution concept of well-supported Nash equilibria (WSNE), Babichenko [Bab14] previously gave an exponential 2 Ω(n) 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). , 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 ǫ-WSNE to the problem of finding an ǫ/(4α)-ANE, in large games with α actions, which might be of independent interest. *
Finally, REF showed (for small enough positive ǫ) an exponential lower bound for ǫ-Nash equilibria of binary-action games.
Well-Supported versus Approximate Nash Equilibria: Query Complexity of Large Games
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Mathematics Computer Science