(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