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 rationalizable 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
(i) Begin by assuming that the probability T that 0 =3 is commonly known among the players. Formalize this assumption by representing the situation as a game with payoff uncertainty=(N, Q, (Ai, ui, Tien as defined in Lecture 4, and compute all its Bayesian Nash equilibria(again as defined in Lectures 4 and 5) (ii) Now fix T in the region where(N, N)and(A, A)are both Bayesian Nash equilibria of the game in which T is common knowledge. Consider the following(k+1)statements E=B(O(E)= Every player believes that 0=3 with probability T; BO(E)=Every player is certain that E is true c)(E)= Every player is certain that B(-1)(E)is true Show that, for any k=0, 1, .., you can provide an alternative formulation of the game G=(N,。92,(A,t,T)∈N)( with the same action sets) and a pair of priors p∈△(92) i=1.2. such that () there exists some state∈ Qk for which w∈∩0B((E)( where of course e and certainty operators are defined in Q and with respect to the priors Pa) (ii) yet, there is a unique Bayesian Nash equilibrium of the game in which the players priors are given by pi, 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, ai(ti)= dA for i= 1, 2 and all t∈T) In words, to support (N, N)as an equilibrium, the value of T must be common certainty mutual certainty of arbitrarily high(but finite) order is not enough 4. From OR: 19.1, 35.2, 42.1(important
(i) Begin by assuming that the probability π that θ = 3 2 is commonly known among the players. Formalize this assumption by representing the situation as a game with payoff uncertainty G = (N, Ω,(Ai , ui , Ti)i∈N ) as defined in Lecture 4, and compute all its Bayesian Nash equilibria (again as defined in Lectures 4 and 5). (ii) Now fix π in the region where (N,N) and (A,A) are both Bayesian Nash equilibria of the game in which π is common knowledge. Consider the following (k + 1) statements: E = B(0)(E) = Every player believes that θ = 3 2 with probability π; 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, Ω k ,(Ai , uk i , Tk i )i∈N ) (with the same action sets) and a pair of priors p k i ∈ ∆(Ωk ), i = 1, 2, such that: (i) there exists some state ω ∈ Ω k for which ω ∈ Tk `=0 B(`) (E) (where of course E and certainty operators are defined in Ω k and with respect to the priors p k i ); (ii) yet, there is a unique Bayesian Nash equilibrium of the game in which the players’ priors are given by p k i , 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, αi(ti) = δA for i = 1, 2 and all ti ∈ T k i ). In words, to support (N,N) as an equilibrium, the value of π must be common certainty; mutual certainty of arbitrarily high (but finite) order is not enough. 4. From OR: 19.1, 35.2, 42.1 (important!) 2
