Repeated matching pennies with limited randomness

We consider a repeated Matching Pennies game in which players have limited access to randomness. Playing the (unique) Nash equilibrium in this n-stage game requires n random bits. Can there be Nash equilibria (or epsilon-Nash equilibria) that use less than n random coins? Our main results are as follows: - We give a full characterization of approximate equilibria, showing that, for any gamma in [0,1], the game has a gamma-Nash equilibrium if and only if both players have (1 - gamma)n random coins. - When players are bound to run in polynomial time with nδ bits of randomness, approximate Nash equilibria can exist if and only if one-way functions exist. - It is possible to trade-off randomness for running time. In particular, under reasonable assumptions, if we give one player only O(log n) random coins but allow him to run in arbitrary polynomial time with nδ bits of randomness and we restrict his opponent to run in time nk, for some fixed k, then we can sustain an epsilon-Nash equilibrium. - When the game is played for an infinite amount of rounds with time discounted utilities, under reasonable assumptions, we can reduce the amount of randomness required to achieve a epsilon-Nash equilibrium to nδ, where n is the number of random coins necessary to achieve an approximate Nash equilibrium in the general case.