There is a seller who has a single object to sell (the seller's reservation value is zero). There are two potential buyers, and they each value the object at 1. The seller and the two buyers share the same discounting factor de (0, 1). If the seller and buyer i agree to trade at price p in period t , then the seller receives a payoff of 81-\p , buyer i a payoff of 5+-(1 p), and buyer j E i a payoff of zero. Consider alternating offer bargaining, with the seller choosing a buyer to make an offer to (name a price to). If the buyer accepts, the game is over, if the buyer rejects, then play proceeds to the next period, when the buyer who received the offer in the proceeding period makes a counter- offer, If the offer is accepted, it is implemented. If the offer is rejected, then the seller makes a new proposal to either the same buyer or to the other buyer. Thus, the seller is free to switch the identity of the buyer he is negotiating with after every rejected offer from the buyers. Suppose that the bargaining lasts for 4 periods and that the seller makes an offer to buyer 1 in period 1. a. In each subgame that starts in period t = 3 , describe the subgame perfect equilibria. [5 points] b. Describe the subgame perfect equilibria. [10 points) Now consider the following alternative. Suppose that the bargaining lasts for at most 5 periods and that the seller makes an offer to buyer 1 in period 1. Suppose that if the seller rejects the offer from buyer 1 in period 2, he can either wait one period to make a counteroffer to buyer 1, or he can immediately make an offer to buyer 2. The bargaining ends if the seller rejects buyer 1's offer in period 5, or he rejects buyer 2's offer in period 4. c. Describe the subgame perfect equilibria. [5 points] There is a seller who has a single object to sell (the seller's reservation value is zero). There are two potential buyers, and they each value the object at 1. The seller and the two buyers share the same discounting factor de (0, 1). If the seller and buyer i agree to trade at price p in period t , then the seller receives a payoff of 81-\p , buyer i a payoff of 5+-(1 p), and buyer j E i a payoff of zero. Consider alternating offer bargaining, with the seller choosing a buyer to make an offer to (name a price to). If the buyer accepts, the game is over, if the buyer rejects, then play proceeds to the next period, when the buyer who received the offer in the proceeding period makes a counter- offer, If the offer is accepted, it is implemented. If the offer is rejected, then the seller makes a new proposal to either the same buyer or to the other buyer. Thus, the seller is free to switch the identity of the buyer he is negotiating with after every rejected offer from the buyers. Suppose that the bargaining lasts for 4 periods and that the seller makes an offer to buyer 1 in period 1. a. In each subgame that starts in period t = 3 , describe the subgame perfect equilibria. [5 points] b. Describe the subgame perfect equilibria. [10 points) Now consider the following alternative. Suppose that the bargaining lasts for at most 5 periods and that the seller makes an offer to buyer 1 in period 1. Suppose that if the seller rejects the offer from buyer 1 in period 2, he can either wait one period to make a counteroffer to buyer 1, or he can immediately make an offer to buyer 2. The bargaining ends if the seller rejects buyer 1's offer in period 5, or he rejects buyer 2's offer in period 4. c. Describe the subgame perfect equilibria. [5 points]
