1. Use the standard bargaining solution to find the outcomes of the following bargaining problems. Player 1's payoff is u1 = v1(z) + t and player 2's payoff is u2 = v2(z) - t. In each case report the chosen z and t as well as players' individual payoffs.
(a) z ∈ {5, 10, 15}, v1(z) = 1/2 . z, v2(z) = z, d1 = 1, d2 = 0, Π1 = and Π2 = 3 .
(b) z ∈{0, 2, 10, 20}, v1(z) = -√z- v2(z) = 2 . z, d1 = 0, d2 = 0, Π1 = 1/2, and Π2 = 1/2.
(c) z ∈ {1, 3, 5, 7, 9}, v1(z) = z2, v2(z) = 3 . z - 3, d1 = 2, d2 = 4, Π1 = 7, and Π2 = 3/7
(d) z ∈ [1, 15], v1(z) = 1/2 . z, v2(z) = z1/3, d1 = 1, d2 = 0, Π1 = 4, and Π2 = 1/4.
2. A contract is being negotiated between a firm (player F) and a worker (player W). The contract specifies two things: Job description and salary t. Regarding the job description, W can be either a production supervisor or a maintenance supervisor. If W works as a production supervisor the payoff to W is t - 5, 000 and the payoff to F is 55, 000 - t. If W works as a maintenance supervisor, his payoff is t- 15, 000 and the payoff to F is 70, 000 - t. If W and F fail to reach an agreement, W has an outside job opportunity worth 15, 000 and F has an alternative job candidate whose services are worth 5, 000 to F.
(a) Let ΠF and Πw denote the bargaining weights of W and F respectively. Solve this bargaining problem using the standard bargaining solution. Note: The solution needs to specify a job description and the salary t.
(b) Suppose ΠF = 2/3 and Πw = 1/3. What is the salary t predicted by the standard bargaining solution?
(c) Suppose ΠF = Πw = What is the salary t predicted by the standard bargaining solution?
3. Consider a three-player bargaining game, where players are negotiating over how to split a surplus of $1. The game begins with player 1 proposing a three-way split of the surplus. Then player 2 must decide whether to accept the proposal or to substitute player l's proposal with his own alternative proposal. Finally, player 3 must decide whether to accept or reject the current proposal (which would be player l's original proposal if it was accepted by player 2, OR player 2's alternative proposal if player 2 decided to replace the original proposal made by player 1). If player 3 accepts, then the players obtain the specified shares of the surplus. If player 3 rejects, then each player gets 0.
(a) Draw the extensive form of game.
(b) Determine the subgame perfect equilibria.
4. Suppose that you are attempting to buy a property, and that you are bargaining with the current owner of the property over the sale price. The property is worth 200, 000 to you and 100, 000 to the current owner. Assume that bargaining takes place with alternating offers and that each stage of bargaining takes a full day to complete. Suppose that if an agreement is not reached after ten days of bargaining, then the sale with definitely not take place and no further bargaining can occur. Suppose that you and the current owner have the same discount factor of (5 per day. The real estate agent has allowed you to decide whether you will make the first offer.
(a) Suppose δ < 1/4 (both you and the owner are relatively "impatient"). Should you make the first offer or should you let the current owner make the first offer? Why?
(b) Suppose instead that S is closer to one, so both you and the owner are relatively "patient". Specifically suppose δ > (1/2)1/9* (note, this implies S > 0.925874712). Should you make the first offer or should you let the current owner make the first offer? Why?