Game Theory

4. Two individuals each receives a ticket on which one of the four possible prizes $10, $20, $30 and $40 is written. The probability of each individual receiving one of the tickets is 1/4. Each individual observes the prize on her own ticket, but not the other's. Both individuals can simultaneously decide whether to exchange their tickets. If both wish to exchange, then two tickets are exchanged; otherwise each individual keeps her own ticket and receives the prize accordingly. In this static Bayesian game, each individual has two actions: \"exchange\" (E) and \"does not exchange\" (N). Each individual has four types and sixteen strategies. For example, (E, N, E, N ) is one strategy, which means \"exchange $10, does not exchange $20, exchange $30, and does not exchange $40\". Please answer the following questions. (a) Is both individuals choosing (E, E, E, E) a Bayesian Nash equilibrium? How about individual 1 choosing (E, E, E, E) and individual 2 choosing (N, N, N, N )? Please justify your answer. (b) Please show that both individuals choosing ( N, N, N, N) is a Bayesian Nash equi librium. (c) Are there any other purestrategy Bayesian Nash equilibria? Please nd all of them. (d) Suppose there are to possible prizes of each ticket, which are drawn from a set V 2 {UL . . . ,vn}, where U1 0 and Billing = 1. Please nd all purestrategy Bayesian Nash equilibria. Can you conclude when exchange will take place in general