Consider a variant of the alternative offers bargaining game. Two players are bargaining over the split of a pie of total size/value 1; each player likes as big a share as he can get. In each round, a player can be a proposer or a proposee (i.e. someone to whom a proposal is made). Every round, player 1 gets to be the proposer with probability π, and correspondingly, 2 gets to be the proposer with probability 1 − π. The proposer offers a split of the pie. If the proposee accepts the offer then the pie is split as per the offer and the game ends. If the offer is not accepted the game proceeds to the next round. Each player has a discount factor δ ∈ (0, 1).
Using the one shot deviation principle, find the sub-game perfect Nash equilibrium of the game. Restrict your attention to stationery strategies, wherein in each round a player offers the same split if he is the proposer, and uses the same accept/reject rule if he is the proposee.
