.1 Discrete-time Fictitious Play

In this lecture we define several variants of the fictitious play dynamic, and study some of the properties of fictitious play. Throughout the section we consider a finite N -player game, where each player i has a finite pure action set Ai. We let ai denote a pure action for player i, and let si ∈ ∆(Ai) denote a mixed action for player i. We will typically view si as a vector in Ri , with si(ai) equal to the probability that player i places on ai. We let Πi(a) denote the payoff to player i when the composite pure action vector is a, and by an abuse of notation also let Πi(s) denote the expected payoff to player i when the composite mixed action vector is s. We let BRi(s−i) denote the best response mapping of player i; here s−i is the composite mixed action vector of players other than i. We will typically restrict attention to the case where N = 2.