  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 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 e in [0, 1], the game has a
e-Nash equilibrium if and only if both players have (1 - e)n random coins. When
players are bound to run in polynomial time, 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 and we restrict his opponent to run in time n^k, for some fixed k, then we
can sustain an 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 e-Nash equilibrium to n,
where n is the number of random coins necessary to achieve an approximate Nash
equilibrium in the general case.
