  We study the performance of Fictitious Play, when used as a heuristic for
finding an approximate Nash equilibrium of a 2-player game. We exhibit a class
of 2-player games having payoffs in the range [0,1] that show that Fictitious
Play fails to find a solution having an additive approximation guarantee
significantly better than 1/2. Our construction shows that for n times n games,
in the worst case both players may perpetually have mixed strategies whose
payoffs fall short of the best response by an additive quantity 1/2 -
O(1/n^(1-delta)) for arbitrarily small delta. We also show an essentially
matching upper bound of 1/2 - O(1/n).
