Eco514-Game Theory Lecture 2:(Iterated) Best Response Operators Marciano siniscalchi September 21. 1999 Introduction This lecture continues the analysis of normal-form games. We analyze general, non-zeros ames, emphasizing the informalequation Rational Behavior Assumptions about Beliefs= Solution Concepts Before we tackle the new material. let us review what we have learned about zerosum games in light of this "equation". Rational behavior in the context of normal-form games (zerosum or otherwise) means that players choose strategies which maximize their expected payoff, given their beliefs about their opponents'choices. Today we shall examine a few technical aspects related to the existence and characteristics of the best reply correspondence but the basics should be familiar to you from decision theory The minmax theory makes the following assumption about beliefs: each player expects her opponent to(1)understand that she will best-respond to his strategy, and(2) play so as to minimize her expected payoff. That is, each player reasons as if she were to observe the (possibly random) choice of her opponent and best-respond to it, and as if her opponent anticipating this, chose a strategy in order to make her payoff as low as possible Formally, the problem min max ui(a,a;) a∈△(4)a2∈△(A) defines the belief a; of Player i about Player j Best Replies Let us begin with a formal definition of best replies in the case of finite gamesEco514—Game Theory Lecture 2: (Iterated) Best Response Operators Marciano Siniscalchi September 21, 1999 Introduction This lecture continues the analysis of normal-form games. We analyze general, non-zerosum games, emphasizing the informal “equation”: Rational Behavior + Assumptions about Beliefs = Solution Concepts Before we tackle the new material, let us review what we have learned about zerosum games in light of this “equation”. Rational behavior in the context of normal-form games (zerosum or otherwise) means that players choose strategies which maximize their expected payoff, given their beliefs about their opponents’ choices. Today we shall examine a few technical aspects related to the existence and characteristics of the best reply correspondence, but the basics should be familiar to you from decision theory. The minmax theory makes the following assumption about beliefs: each player expects her opponent to (1) understand that she will best-respond to his strategy, and (2) play so as to minimize her expected payoff. That is, each player reasons as if she were to observe the (possibly random) choice of her opponent and best-respond to it, and as if her opponent, anticipating this, chose a strategy in order to make her payoff as low as possible. Formally, the problem min α i j∈∆(Aj ) max α i i∈∆(Ai) ui(α i i , αi j ) defines the belief α i j of Player i about Player j. Best Replies Let us begin with a formal definition of best replies in the case of finite games. 1