Simple strategies for large zero-sum games with applications to complexity theory

Abstract

Von Neumann's Min-Max Theorem guarantees that each player of a zero-sum matrix game hss an optimal mixed strategy. We show that each player has a near-optimal mixed strategy that chooses uniformly from a multiset of pure strategies of size logarithmic in the number of pure strategies available to the opponent. Thus, for exponentially large games, for which even representing an optimal mixed strategy can require exponential space, there are nearoptimal, linear-size strategies. These strategies are eaay to play and serve as small witnesses to the approximate value of the game. Because of the fundamental role of games, we expect this theorem to have many applications in complexity theory and cryptography. We use it to strengthen the connection established by Yao between randomized and distributional complexity and to obtain the following results: (1) Every language has anti-checkers - small hard multisets of inputs certifying that small circuits can't decide the language. (2) Circuits of a given size can generate random instances that are hard for all circuits of linearly smaller size. (3) Given an oracle M for any exponentially large game, the approximate value of the game and near-optimal strategies for it can be computed in ΣP2(M). (4) For any NP-complete language L, the problems of (a) computing a hard distribution of instances of L and (b) estimating the circuit complexity of L are both in ΣP2.