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

Von Neumann's Min-Max Theorem guarantees that each player of a zero-sum matrix game has an optimal mixed strategy. This paper gives an elementary proof that each player has a near-optimal mixed strategy that chooses uniformly at random from a multiset of pure strategies of size logarithmic in the number of pure strategies available to the opponent. For exponentially large games, for which even representing an optimal mixed strategy can require exponential space, it follows that there are near-optimal, linear-size strategies. These strategies are easy to play and serve as small witnesses to the approximate value of the game. As a corollary, it follows that every language has small ``hard'' multisets of inputs certifying that small circuits can't decide the language. For example, if SAT does not have polynomial-size circuits, then, for each n and c, there is a set of n^(O(c)) Boolean formulae of size n such that no circuit of size n^c (or algorithm running in time n^c) classifies more than two-thirds of the formulae succesfully.