Consider the same setting as that in Exercise 7, except now consider that both players have private information. Player 1's type is either high or low as well. The players' types are independently selected by nature with the probability p. (Thus, p2 is the probability that both players are the high type, p(1 - p) is the probability that player 1's type is H and player 2's type is L, and so on.) Suppose that if the firm is not formed, both players receive 0.
If the firm is formed, then the payoffs are as follows: If both players have high productivity, then they each receive 10. If both have low productivity, then they each receive 0. If one of the players has high productivity and the other has low productivity, then the high-type player gets -4 and the low-type player gets 5.
(a) Consider the game in which the players simultaneously and independently select Y or O. The firm is formed if they both select Y. Otherwise, the firm is not formed. Note that a strategy for each player specifies what the player should do conditional on being the high type and conditional on being the low type. Demonstrate that (O, O; O, O) is a Bayesian Nash equilibrium in this game, regardless of p.
(b) Under what values of p is (Y, Y; Y, Y) an equilibrium? (In this equilibrium, both types of both players choose Y.)
(c) Now suppose that before the players choose between Y and N, they have an opportunity to give each other gifts. The players simultaneously and independently choose between giving (G) and not giving (N) gifts, with the cost and benefit of gifts as specified in Exercise 7. Under what conditions is there a perfect Bayesian equilibrium in which the high types give gifts and the firm is formed if and only if both players give gifts? Specify the equilibrium as best you can.