A seller is bargaining to sell a good to a buyer. The good is worth $0 to the seller and $1 to the buyer. The bargaining is in two periods. In period 1, one party is chosen with probability 1/2 to propose a sale price. The other party then accepts or rejects. Acceptance results in sale with the proposed price. Rejection moves the game to the second period. In the second period, one party is again chosen with probability 1/2 to propose a sale price, and the other party accepts or rejects. Acceptance results in sale at the proposed price. Rejection results in no sale, in which case each player receives zero payoff. Both players discount period 2 payoffs with a discount factor de (0,1): if a sale occurs at price pı in period 1, then the seller and the buyer collect p1 and 1-P1, respectively; but if a sale occurs at price p2 in period 2, the parties collect ∂p2 and ∂(1 – P2), respectively. Required:
a. For any pe [0, 1], show that there is a Nash equilibrium in which a sale occurs at price p in period 1.
b. Find all subgame perfect Nash equilibria (SPNE). (You must describe the associated strategy profile, and prove that it is a SPNE.)