Assignment 5
1. In experimental tests of the ultimatum game, the offerer rarely offer a near-zero share of the surplus. Furthermore, responders sometimes reject positive offers.
Many scholars conclude that the payoffs specified in the basic model do not represent the actual preferences of the people who participate in the game. In reality, people care about more than their own monetary rewards. For example, people also care about fairness. Suppose that in the ultimatum game the offerer (O) can make any offer m between 0 and 100 to the responder (R). If the offer is accepted, O’s payoff is 100 – m and R’s is m + c(m – (100-m)) where c is some positive real number. If the offer is rejected, both their payoffs are zero.
a. (5 pts) Find the sub game perfect Nash equilibrium.
b. (5 pts) Describe how the SPNE depends on c and explain why.
2. Suppose the CEO of a company and a union leader will bargain about the worker’s
salaries. There are a set of outcomes that are acceptable to both sides. The CEO makes a take-it-or-leave-it offer between 0 and 1 where 0 represents the CEO’s most preferred outcome and 1 represents the union’s most preferred outcome. If the offer is accepted, it is executed. If it rejected, they both receive a payoff of zero.
a. (5 pts) Find the subgame perfect Nash equilibrium.
b. (10 pts) Suppose that, before the union leader hears the CEO’s offer, she makes a statement to the members of the union that she will not accept an offer less than some number z < 1. If the union leader accepts an offer less than z, she will lose her position and also feel bad. The total cost, in terms of utility, is C < 1. Find the SPNE of this new game.
c. (5 pts) Given your answer to part (b), what statement should the union leader make to the members of the union? That is, what is her optimal choice of z?
3. (10 pts) There are two players bargaining over a pie worth 100 utils. In period 1, Player 1 proposes a split. If it is rejected, then in period 2, Player 1 again proposes a split. If it is rejected, then in period 3, Player 2 proposes a split. If it is rejected, then both players receive zero. The discount factor is δ . What is the subgame perfect Nash equilibrium?
4. (10 pts) Ashely and her perspective employer, the YMCA, are negotiating her contract, which will stipulate her title (tennis instructor or swim instructor) and salary (s). If she teaches tennis, her payoff is s – 10,000 and the YMCA’s payoff is 60,000 – s. If she teaches swimming, her payoff is s – 20,000 and the YMCA’s payoff is 65,000 – s. If they don’t reach an agreement, her payoff is 20,000 and the YMCA’s payoff is 10,000.
Suppose their bargaining weights are ¾ and ¼. What is the Nash bargaining solution?
5. (10 pts) For what values of the discount factor, δ, is “cooperation” (i.e. (C, C) is played every period) possible?
6. Consider the normal-form game below.
a. (5 pts) What are the pure-strategy Nash equilibria?
b. (10 pts) If this game is repeated twice and there is no time discounting (i.e. δ = 1), is there a SPNE in which (A, X) is played in the first period? If so, fully describe it. If not, explain why not.
7. A seller and a buyer will interact in an infinite number of periods. In each period, the
seller chooses a quality level, q from [0,5] at a personal cost of q, and the buyer chooses whether or not to accept the seller’s product. If the buyer accepts, the seller’s payoff is 6 – q and the buyer’s payoff is 2q - 6. If the buyer rejects, the seller’s payoff is -q and the buyer’s payoff is 0.
a. (5 pts) What is the most efficient outcome, in a period, in terms of total utility?
b. (10 pts) If the discount factor, δ, is sufficiently big, is there an SPNE in which the most efficient outcome occurs every period?