Question: Consider the two-player game illustrated in figure. In the first period, players I and 2 simultaneously choose U1 or DI (player 1) and L I or R 1 (player 2); these choices are revealed at the end of period 1 with payoffs as in the left-hand matrix. In period 2, players choose U2 or 1)2 and L2 or D2. with payoff as in the matrix on the right. Each player's objective is to maximize the average of his per-period payoff.
(a) Find the subgame-perfect equilibria of this game, and compute the convex hull of the associated payoffs.
(b) Now suppose that the players can jointly observe the outcome y1 of a public randomizing device before choosing their first-period actions, where
y1 has a uniform distribution on the unit interval. Find the set of subgame-perfect equilibria, and compare the resulting payoffs against the answer to part a of this exercise.
(c) Suppose that the players jointly observe y, at the beginning of period 1 and y, at the beginning of period 2, with y1 and y2 being in-dependent draws from a uniform distribution on the unit interval. Again, find the subgame-perfect equilibrium payoffs.
(d) Relate your answers to parts a-c to the role of public randomizations in the proof of the Folk Theorem.
