Eco514-Game Theory Problem Set 2: Due Thursday, October 14 Recall the following definitions: in any model M=(Q, (Ti, ai, piie), Ri is the event Player i is rational";R=nieN Ri. Also, Bi(E) is the event "Player i is certain that E is true" and B(E)=neN Bi(E). This is as in Lecture 7. Let me introduce the following notation for iterated mutual certainty: B()(E)=E B()(E)=B(B-I)(E)). Then the definition of Bk in Lecture 7 can be rewritten as Bk B((R)fork≥0 1. A characterization of Correlated Rationalizability Consider a game G=(N, (Ai, ui Construct a(finite)model M=(Q, (Ti, ai, piie for G such that, for any k>0 and for any profile of actions a E A: a E A+ if and only if there exists a state w∈ such that(i)a(u)= a, for all i∈N,and(i)u∈∪=0B 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 profile. The "only if"part of the claim I am asking you to prove instead applies to all such profiles. 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 finite 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 payoff uncertainty in Fig. 1 A6.66-1,0 N|0,-10,0 1:B∈{,3Pr(0=2)Eco514—Game Theory Problem Set 2: Due Thursday, October 14 Recall the following definitions: in any model M = (Ω,(Ti , ai , pi)i∈N ), Ri is the event “Player i is rational”; R = T i∈N Ri . Also, Bi(E) is the event “Player i is certain that E is true” and B(E) = T i∈N 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 definition 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)i∈N ). Construct a (finite) model M = (Ω,(Ti , ai , pi)i∈N ) for G such that, for any k ≥ 0 and for any profile of actions a ∈ A: a ∈ Ak+1 if and only if there exists a state ω ∈ Ω such that (i) ai(ω) = ai for all i ∈ N, and (ii) ω ∈ Sk `=0 Bk . 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 profile. The “only if” part of the claim I am asking you to prove instead applies to all such profiles. 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 finite 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 payoff uncertainty in Fig. 1: A N A θ,θ θ − 1,0 N 0,θ − 1 0,0 Figure 1: θ ∈ { 1 2 , 3 2 }; Pr(θ = 3 2 ) = π. 1