3. (40 Minutes - 20 Points) Harvard and MIT are both considering whether to admit a particular student to their economics Ph.D. programs. Assume that MIT has read the student's application carefully And knows the quality of the student. Assume that Harvard faculty members are to busy to read applications carefully. Instead they must base their decisions on their prio about the student's ability. Harvard's prior is that q may be 1, 2 or 3 and that each o these values is equally likely. Assume that each school must make one of two decisions on the student: admit with financial aid or reject (the student has no source of support and could not attend graduate school without financial aid). The schools make these decisions simultaneously. Assume that each school's payoff in the game is 0 if they do not offer the student admission, -1 if the student is offered admission and turns them down (this is costly both because the school loses prestige and because the slot could have been given to another student), and q - 1.5 if the student is offered admission and decides to come. Assume that if the student is admitted to both schools she chooses to come to MIT with probability 0.65 and to go to Harvard with probability 0.35. In the following questions treat this as a two player game between Harvard (player 1 and MIT (player 2). (a) What type spaces, and , would you use to represent this situation as a static game of incomplete information? How many elements are in each set? Write down the val ues of the utility functions ui(Q1, 22:01,02) for a couple values of i, 41, 42,61,62 to illustrate how to compute them. How many pure Bayesian strategies does each player have? (b) What actions are strictly conditionally) dominated for each possible type of each player? (c) Find the Bayesian Nash equilibrium of this game. (d) Would Harvard be any better off if it could observe MIT's admission decision before making its decision? 3. (40 Minutes - 20 Points) Harvard and MIT are both considering whether to admit a particular student to their economics Ph.D. programs. Assume that MIT has read the student's application carefully And knows the quality of the student. Assume that Harvard faculty members are to busy to read applications carefully. Instead they must base their decisions on their prio about the student's ability. Harvard's prior is that q may be 1, 2 or 3 and that each o these values is equally likely. Assume that each school must make one of two decisions on the student: admit with financial aid or reject (the student has no source of support and could not attend graduate school without financial aid). The schools make these decisions simultaneously. Assume that each school's payoff in the game is 0 if they do not offer the student admission, -1 if the student is offered admission and turns them down (this is costly both because the school loses prestige and because the slot could have been given to another student), and q - 1.5 if the student is offered admission and decides to come. Assume that if the student is admitted to both schools she chooses to come to MIT with probability 0.65 and to go to Harvard with probability 0.35. In the following questions treat this as a two player game between Harvard (player 1 and MIT (player 2). (a) What type spaces, and , would you use to represent this situation as a static game of incomplete information? How many elements are in each set? Write down the val ues of the utility functions ui(Q1, 22:01,02) for a couple values of i, 41, 42,61,62 to illustrate how to compute them. How many pure Bayesian strategies does each player have? (b) What actions are strictly conditionally) dominated for each possible type of each player? (c) Find the Bayesian Nash equilibrium of this game. (d) Would Harvard be any better off if it could observe MIT's admission decision before making its decision