1. Consider the following variant of alternating offer game. In period 1, player 1 makes a proposal, and player 2 decides either to accept or reject the proposal. If player 2 accepts and the game ends and the dollar is divided according to the proposal. If player 2 rejects, then the game moves onto period 2. In the beginning of period 2, nature determines who makes a proposal. Specifically, player 1 is selected as a proposer with probability a and player 2 is selected as a proposer with probability 1- a. Then the selected proposer makes a proposal, and the other player decides either to accept or to reject, and the game ends. If the proposal is rejected, then both players get payoff 0, and if accepted, the dollar is divided according to the proposal, but the payoff is discounted by δ.

a. Solve the game for sub-game perfect equilibrium.

b. Calculate the equilibrium payoffs for both players. Is it true that player 1' s payoff is increasing in a and player 2's payoff is decreasing in a?
2. Now consider an infinite horizon alternating offer game where, in the beginning of each period, nature chooses the proposer: in every period, player 1 is selected as a proposer with probability a and player 2 is selected with probability 1 — a.

a. Find a sub-game perfect equilibrium.

b. Calculate the equilibrium payoffs for both players. Is it true that player 1's payoff is increasing in a and player 2's payoff is decreasing in a?