2. (20 points) A student, player 1, has to hand in a problem set at the other end of (a large) campus but needs to rush into an exam. They have two options. They can deliver the problem set after the exam (call this option H for "hand-deliver") and incur a late penalty. Alternatively she can give the problem set to player 2, a random student who happens to be next to player 1 (call this option G for "give"). Player 2 can then either deliver the problem set on time (call this option D for "deliver") or throw it away in the nearest trash can (call this option T for "throw"). For player 1 the payoff is 1 if the problem set is delivered on time, -1 if it is thrown away, and 0 if it is delivered late. The payoffs for player 2 are x if they delivers and y if they throw it away. (a) Draw the game tree that represents this game. Using some equilibrium consideration, under what conditions on x and y can we justify player 1 trusting player 2 to deliver the problem set ? (b) Now assume that a proportion p of the students one might meet on campus are "altruistic" (A) : they like to help and have payoffs given by x = 1 and y = 0. The remaining proportion 1 - p is "mean" (M) and have payoffs x = 0, y = 1. You can think of the game now as Nature first drawing the type of player 2 that player 1 will meet (but player 1 does not observe that type). Draw the new game tree. (c) Now assume p = 3/4. What are the pure-strategy Bayesian Nash equilibria of this game ? (d) Are there BNE that are not weak perfect bayesian equilibria ? For which values of p are all the BNE also wPBE