Consider a bargaining game among three political parties; the Conservative party, the Liberal party and the Labor party (each party is a player). There are T periods of bargaining, numbered t= = 1,2,...,T .
In each period one party is the proposer; one is the responder and 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 even numbered periods the Labor party is the proposer. The total amount of value to be divided among parties is normalized to 1.
A proposal specifies a vector (x,y), where x is the amount 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 a common discount factor d.
a) Suppose T=1. Prove that in 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=8. Describe a subgame perfect equilibrium in which an offer is accepted in the first period. What is the first period offer?