### Playing large games using simple strategies

We prove the existence of ε-Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payoffs to all players in any (exact) Nash equilibrium can be ε-approximated by the payoffs to the players in some such logarithmic support ε-Nash equilibrium. These strategies are also uniform on a multiset of logarithmic size and therefore this leads to a quasi-polynomial algorithm for computing an ε-Nash equilibrium. To our knowledge this is the first subexponential algorithm for finding an ε-Nash equilibrium. Our results hold for any multiple-player game as long as the number of players is a constant (i.e., it is independent of the number of pure strategies). A similar argument also proves that for a fixed number of players m, the payoffs to all players in any m-tuple of mixed strategies can be ε-approximated by the payoffs in some m-tuple of constant support strategies.We also prove that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support. This implies that if the payoff matrices can be well approximated by low rank matrices, the game has an ε-equilibrium with small support. It also implies that if the payoff matrices have constant rank we can compute an exact Nash equilibrium in polynomial time.

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