Suppose that two people (person 1 and person 2) are considering whether to form a partnership firm. Person 2's productivity (type) is unknown to person 1 at the time at which these people must decide whether to create a firm, but person 1 knows that, with probability p, person 2's productivity is high (H) and, with probability 1 - p, person 2's productivity is low (L). Person 2 knows her own productivity. If these two people do not form a firm, then they each receive a payoff of 0. If they form a firm, then their payoffs are as follows: If person 2's type is H, then each person receives 10. If person 2's type is L, then person 2 receives 5 and person 1 receives -4 (which has to do with person 1 having to work very hard in the future to salvage the firm).
(a) Consider the game in which person 1 chooses between forming the firm (F) or not forming the firm (O). Draw the extensive form of this game (using a move of nature at the beginning to select person 2's type). Note that only one person has a move in this game. What is the Bayesian Nash equilibrium of the game? (It depends on p.)
(b) Now suppose that before person 1 decides whether to form the firm, person 2 chooses whether or not to give person 1 a gift (such as a dinner). Player 1 observes person 2's choice between G and N (gift or no gift) before selecting F or O. The gift entails a cost of g units of utility for person 2; this cost is subtracted from person 2's payoff designated earlier. The gift, if given, adds w to person 1's utility. If person 2 does not give the gift, then it costs her nothing. Assume that w and g are positive numbers. Draw the extensive form of this new game.
(c) Under what conditions (values of g and w) does a (separating) perfect Bayesian equilibrium exist in which the low type of person 2 does not give the gift, the high type gives the gift, and person 1 forms the firm if and only if a gift is given? Completely specify such an equilibrium.
(d) Is there a pooling equilibrium in this game? Fully describe it and note how it depends on p.