Consider a bargaining game among three political parties-the Conservative Party, the Liberal Democratic Party, and the Labor Party (each party is a player). There are T periods of bargaining, numbered t = 1, c, T. In each period, one party is the "proposer," one is the "responder," and the third is the "bystander." The proposer makes an offer to the responder, who may then accept or reject. If the responder accepts, then the game ends and the offer is implemented.
Otherwise, the game continues to the following period unless the period is T, in which case the game ends. If the game ends without an offer being accepted, all parties get zero. Suppose that the Liberal Democratic Party is always the responder. In odd-numbered periods the Conservative Party is the proposer, and in evennumbered periods the Labor Party is the proposer. The total amount of value to be divided among the parties is normalized to 1. A proposal specifies a vector (x, y), where x is the amount that the proposer will receive, and y is an amount that the responder will receive. (The bystander is left out.) The players discount the future according to the common discount factor d.
(a) Suppose that T = 1. Prove that in the subgame perfect equilibrium, the Conservative Party offers x = 1 and y = 0, and the Liberal Democratic Party accepts any offer with y = 0.
(b) Suppose that T is odd. Determine the unique subgame perfect equilibrium and describe what offer, if any, is accepted and in which period.
(c) Suppose that T = 
. Describe a subgame perfect equilibrium in which an offer is accepted in the first period. What is the first-period offer?