neal young / Lipton94Simple

Von Neumann's MinMax Theorem guarantees that each player of a zerosum matrix game has an optimal mixed strategy. This paper gives an elementary proof that each player has a nearoptimal 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 nearoptimal, linearsize 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 polynomialsize 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 twothirds of the formulae successfully.
The "simple strategies" observation was generalized to Nash equilibrium by Lipton et al. in their paper Playing large games using simple strategies.
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