 Fictitious play: George Brown (1951) introduced fictitious play as “pre-play” reasoning, in which a player mentally simulates repeated trials of a game against an imaginary opponent so as to predict her real opponent’s actions. The phrase “fictitious play” has become a misnomer, because researchers now typically apply it to models in which players learn based upon their actual experience of repeated play. In paradigmatic fictitious play models, each player plays a “best reply” to the observed historical frequency of her opponents’ past actions. This policy is rational if: the player assumes that each opponent plays some stationary (either pure or mixed) strategy; the player employs Bayesian updating to determine the probability that each opponent will perform a given action in the next round; and the player seeks to maximize her expected payoff for that round based upon her current probability distribution over her opponents’ actions. It is easy to show that, if players engaged in fictitious play enter into a strict Nash equilibrium, then they stay in it forever. Moreover, there are some circumstances (e.g., zero-sum two-person games) in which fictitious play converges to Nash equilibrium behavior. However, as Lloyd Shapley (1964) first showed, there are games in which fictitious play does not always converge to equilibrium behavior.