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Econ 171 Fall 2008 Problem Set 3 Solutions to two-star problems
In some cases I’ve provided more explanation than was asked of you.
** Problem 1 Watson 19.2 (Postponed from PS2)
Solution:
(a) Because δ is so small, the home owner is very impatient (as are you) and would be willing to accept a low offer on the first day. Specifically, because δ < 1 /2, the responder on the first day prefers accepting less than half of the surplus to rejecting and getting all of the surplus on the second day. This means that whoever offers on day 1 will get over half of the surplus, which means you should make the first offer. (b) Now, you are very patient and would be willing to wait even until the last day rather than accept a small amount at the beginning. On the last day, you could keep the entire surplus because the owner would accept any offer because otherwise she would get nothing. The discounted value of the entire surplus on the last day is δ^9 > 1 /2, which is greater than the surplus you could get on the first day. This means that it is better for you to make the second offer.
** Problem 2
(a) Watson 22.2b You can do a) and c), but you only need to hand in b).
Solution: To support cooperation by Player 1, δ needs to be at least 1/2. For Player 2 to be willing to cooperate, we need δ ≥ 3 /5. Thus, we need δ ≥ 3 /5. (b) When the stage game in Watson 22.2b is repeated infinitely, what is the set of discounted average payoffs for Nash equilibria when δ is close to 1. It may be easiest for you to show your answer graphically.
Solution: For any feasible payoff profile in which each player’s payoff exceeds her minmax payoff, there is a Nash equilibrium in which that is the payoff profile. In this case, the set of feasible average discounted payoff profiles is the area bounded by the four payoff profiles of the stage game: (3, 4), (5, 0), (0, 7), and (1, 2),. The minmax payoffs are 1 for Player 1 and 2 for Player 2. This means that for δ sufficiently close to 1, the set of discounted average payoffs is as depicted in the shaded area of Figure 1. (c) What about when the stage game is Battle of the Sexes?
Solution: See Figure 2.
Figure 1: The set of feasible average discounted payoff profiles is bounded by the quadrilateral and the shaded area shows those profiles for which each player’s average discounted payoff exceeds her minmax payoff.
Figure 2: The set of feasible average discounted payoff profiles and the profiles that are possible in equilibrium with sufficiently large discount factor for the infinitely-repeated BoS.
** Problem 3 Consider a two-player Bayesian game where both players are not sure whether they are playing the game X or game Y , and they both think that the two games are equally likely.
(a) This game has a unique Bayesian Nash equilibrium, which involves only pure strategies. What is it? (Hint: start by looking for Player 2’s best response to each of Player 1’s actions.)
Solution: The unique BNE is (B, L), yielding each player a payoff of 2. Player 1’s payoffs do not depend upon which version of the game is actually being played. Her best response to L is to play B and T is a best response to M or R. If 1 plays T , then both M and R give Player 2 an expected utility of .15, so her best
which implies that b = p−^1 /α. This is decreasing in p, so the more likely the project is to succeed, the lower the bonus in equilibrium. This suggests that higher-powered incentives are required for riskier projects.
** Problem 5 Firm 1 is considering taking over Firm 2. It does not know Firm 2’s current value, but believes that is equally likely to be any dollar amount from 0 to 100. If Firm 1 takes over firm 2, it will be worth 50% more than its current value, which Firm 2 knows to be x. Firm 1 can bid any amount y to take over Firm 2 and Firm 2 can accept or reject this offer. If 2 accepts 1’s offer, 1’s payoff is 32 x − y and 2’s payoff is y. If 2 rejects 1’s offer, 1’s payoff is 0 and 2’s payoff is x. Find a Nash equilibrium of this game. What does this situation have to do with dating and shopping for used cars?
Solution: Firm 1 will bid zero and Firm 2 will accept any offer greater than or equal to x. Firm 2’s simply accepts offers that are higher than the firm’s own value. Firm 1 knows that the value of a firm that accepts an offer of y is anywhere from 0 to y. Thus, the expected value of a firm that accepts is y/2, which means that Firm 1’s expected payoff as a function of it’s bid is 32 (y/2) − y = −^14 y. In other words, it expects to lose money on any positive bid it makes. It’s best response, then is to bid zero. Just like in dating and the used-car market, this market is plagued by adverse selection, which in this case leads the market to unravel completely.
