Eco514|Game Theory Problem Set 2: Due Thursday, October 14 Recall the following de nitions: in any model M = ( ;(Ti;ai;pi)i2N), Ri is the event \Player i is rational"; R = Ti2N Ri. Also, Bi(E) is the event \Player i is certain that E is true" and B(E) = Ti2N Bi(E). This is as in Lecture 7. Let me introduce the following notation for iterated mutual certainty: B(0)(E) = E, B(k)(E) = B(B(k 1)(E)). Then the de nition of Bk in Lecture 7 can be rewritten as Bk = B(k)(R) for k 0. 1. A characterization of Correlated Rationalizability Consider a game G = (N;(Ai;ui)i2N). Construct a ( nite) model M = ( ;(Ti;ai;pi)i2N) for G such that, for any k 0 and for any pro le of actions a2A: a2Ak+1 if and only if there exists a state !2 such that (i) ai(!) = ai for all i2N, and (ii) !2Sk‘=0Bk. A comment: the \if" part follows from Part (1) of Proposition 5 in Lecture 7; however, notice that Part (2) therein asserts the existence of a model that \works" for a single ratio- nalizable action pro le. The \only if" part of the claim I am asking you to prove instead applies to all such pro les. Also, [hint! hint!] an elegant proof of the claim does not rely on Proposition 5 in Lecture 7 for the \if" part. Also, I specify that I am looking for a nite model because obviously the complicated \universal" model constructed in the notes for Lecture 6 will \work". 2. Mutual and Common Certainty Consider the normal-form game with payo uncertainty in Fig. 1: A N A , 1,0 N 0, 1 0,0 Figure 1: 2f12; 32g; Pr( = 32) = . 1 (i) Begin by assuming that the probability that = 32 is commonly known among the players. Formalize this assumption by representing the situation as a game with payo uncertainty G = (N; ;(Ai;ui;Ti)i2N) as de ned in Lecture 4, and compute all its Bayesian Nash equilibria (again as de ned in Lectures 4 and 5). (ii) Now x in the region where (N,N) and (A,A) are both Bayesian Nash equilibria of the game in which is common knowledge. Consider the following (k + 1) statements: E = B(0)(E) = Every player believes that = 32 with probability ; B(1)(E) = Every player is certain that E is true; ::: B(k)(E) = Every player is certain that B(k 1)(E) is true. Show that, for any k = 0;1;:::, you can provide an alternative formulation of the game Gk = (N; k;(Ai;uki;Tki )i2N) (with the same action sets) and a pair of priors pki 2 ( k), i = 1;2, such that: (i) there exists some state !2 k for which !2Tk‘=0B(‘)(E) (where of course E and certainty operators are de ned in k and with respect to the priors pki ); (ii) yet, there is a unique Bayesian Nash equilibrium of the game in which the players’ priors are given by pki , i = 1;2, and in this equilibrium both players choose A at every state, i.e. regardless of their respective types (in the notation of Lecture 4, i(ti) = A for i = 1;2 and all ti2Tki ). In words, to support (N,N) as an equilibrium, the value of must be common certainty; mutual certainty of arbitrarily high (but nite) order is not enough. 4. From OR: 19.1, 35.2, 42.1 (important!) 2