On the Rate of Convergence of Fictitious Play

Fictitious play is a simple learning algorithm for strategic games that proceeds in rounds. In each round, the players play a best response to a mixed strategy that is given by the empirical frequencies of actions played in previous rounds. There is a close relationship between fictitious play and the Nash equilibria of a game: if the empirical frequencies of fictitious play converge to a strategy profile, this strategy profile is a Nash equilibrium. While fictitious play does not converge in general, it is known to do so for certain restricted classes of games, such as constant-sum games, non-degenerate 2×n games, and potential games. We study the rate of convergence of fictitious play and show that, in all the classes of games mentioned above, fictitious play may require an exponential number of rounds (in the size of the representation of the game) before some equilibrium action is eventually played. In particular, we show the above statement for symmetric constant-sum win-lose-tie games.