Lecture 9: “Adaptive learning”

In this lecture, we develop the notion of “adaptive learning” as proposed by Milgrom and Roberts [1]. Although the learning definition they give is of interest in its own right, it primarily derives power in the case of dominance solvable games, or for games where there is a straightforward characterization of the set of strategies surviving iterated strict dominance (hereafter ISD). Throughout the lecture we consider a finite N -player game, where each player i has a finite pure action set Ai; let A = ∏ i 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 need some additional notation involving ISD. Given T ⊂ ∏ i Ai, we define Ui(T ) as follows: