We study the performance of Fictitious Play (FP), when used as a heuristic for finding an approximate Nash equilibrium of a two-player game. We exhibit a class of two-player games having payoffs in the range <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$[0,1]$$</EquationSource> </InlineEquation> that show that FP fails to find a solution having an additive approximation...</equationsource></inlineequation>