We study the deterministic and randomized query complexity of finding approximate equilibria in a k × k bimatrix game. We show that the deterministic query complexity of finding an -Nash equilibrium when < 1 2 is (k 2 ), even in zero-one constant-sum games. In combination with previous results [Fearnley et al. 2013 ], this provides a complete characterization of the deterministic query complexity of approximate Nash equilibria. We also study randomized querying algorithms. We give a randomized algorithm for finding a ( 2 ) payoff queries, which shows that the 1 2 barrier for deterministic algorithms can be broken by randomization. For well-supported Nash equilibria (WSNE), we first give a randomized algorithm for finding an -WSNE of a zero-sum bimatrix game using O( k·log k 4 ) payoff queries, and we then use this to obtain a randomized algorithm for finding a ( 2 3 + )-WSNE in a general bimatrix game using O( k·log k 4 ) payoff queries. Finally, we initiate the study of lower bounds against randomized algorithms in the context of bimatrix games, by showing that randomized algorithms require (k 2 ) payoff queries in order to find an -Nash equilibrium with < 1 4k , even in zero-one constant-sum games. In particular, this rules out query-efficient randomized algorithms for finding exact Nash equilibria.
In the other direction, REF showed a randomized query algorithm that finds a 0.382-approximate Nash equilibrium by making O(N log N ) queries.
Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries
TEAC
Mathematics Computer Science