A student, player 1, has to hand in a problem set at the other end of campus but needs to rush into a midterm exam. She has two options. She can deliver the problem set after the exam (call this L) 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 S). Player 2 can either deliver the problem set on time (call this D) or throw it away in the nearest compost bin (call this T). 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 he delivers and y if he throws it away.
(a) Draw the game tree for this game.
(b) What conditions do you have to place on x and y in order for player 1 to trust that player 2 will deliver the problem set in an equilibrium?
(c) Now assume that some proportion of students are "nice guys" (N) for which x = 1 and y = 0, while a proportion 1 - p are "jerks" (J) for which x = 0 and y = 1. Modify the game to allow Nature to choose what type player 2 is before the game begins. Only player 2 knows his type. Draw the new game tree.
(d) Assume p = 3/4. What are teh pure strategy perfect Bayesian equilibria of this game?