Econ 618 Mixed, Behavioral, Distributional Strategies and Existence of Equilibrium

1.1 Definitions The individual strategy set Ai is finite with mi elements, {ai . . .a mi i }, indexed a ki i . A mixed strategy for player i, αi is a probability distribution over Ai, with αi(ai i ) as the probability of choosing a ki i . Thus ∑i ki=1 αi(a ki i ) = 1. The space of all mixed strategies for player i is denoted ∆Ai. Let a ∈ A = (a1 1 ,a k2 2 , . . .a kN N ) be a profile of pure strategies in the joint strategy space. Under the mixed strategy profile α = {αi}i=1 = (α1,α2 . . .αN), the probability of the outcome a is α1(a1 1 ).α2(a k2 2 ). . . .αN(a kN N ) = ∏i∈N αi(a ki i ). Note that, ∑a∈A ∏i∈N αi(a ki i ) = 1. That is, a given mixed strategy profile α, induces a probability distribution over the joint strategy set A. Another way of saying this is that the given mixed strategy profile α yields a well defined probability for each of the joint outcome a in A. A mixed strategy profile is a lottery over A. Denote by Ui(α), the expected payoff to player i,