Alternating offers game Barack and Joe can together implement a project that will jointly yield them a profit $100. How should they divide this sum of money between them?
They decide to implement the following mechanism: Barack will offer Joe a split of (x, 100 - x), where x is a number in the interval [0, 100], signifying the amount of money that Barack will receive under this offer. Joe may accept or reject this offer. If he accepts, this will be the final split.
If he rejects the offer, the next day he proposes a counteroffer (y, 100 - y), where y is a number in the interval [0, 100], signifying the amount of money that Barack will receive, under this offer. Barack may accept or reject this offer. If he accepts, this will be the final split. If he rejects the offer, the next day he proposes a counteroffer, and so on.
Every delay in implementing the project reduces the profit they will receive: if the two of them agree on a division of the money (x, 100 - x) on day n, Barack's payoff is βn-1 × x, and Joe's payoff is βn-1 × (100 - x), where β ∈ (0, 1) is the discount factor in the game (in other words, 100( 1/β - 1) is the daily interest rate in the game).
Depict this situation as an extensive-form game, and find all the subgame perfect equilibria of the game.