5.3 Fictitious-Play-Based Algorithms 129 5.3 FICTITIOUS-PLAY-BASED ALGORITHMS Aside from the Cournot t atonnement, one of the oldest iterative algorithms for com- ^ puting equilibria is given by the fictitious play (FP) procedure (Brown, 1949, 1951; Robinson, 1951). Fictitious play and its variants (stochastic fictitious play, smooth fictitious play, weakened generalized fictitious play, etc.) are simple models of learn- ing which are extensively used in the literature of game theory. One of the reasons for considering FP is that its structure allows one to better understand reinforcement-type learning algorithms. 5.3.1 Brown's Fictitious Play Model The standard fictitious play algorithm introduced by Brown (1949, 1951) and Robin- son (1951) assumes that each player knows his actions and his own utility function, but also the last actions played by the other players. In it, each player presumes that the other players are playing stationary (possibly mixed) strategies. At each game instance/stage, each player thus best responds to the empirical frequency of play of his opponent (this presumes also that he/she observes the last actions chosen by the other