Alternating Offers game Debby and Barack are jointly conducting a project that will pay them a total payoff of $100. Every delay in implementing the project reduces payment for completing the project. How should they divide this money between them?
The two decide to implement the following procedure: Debby starts by offering a division (xD, 100 - xD), where xD is a number in [0, 100] representing the amount of money that Debby receives under the terms of this offer, while Barack receives 100 - xD. Barack may accept or reject Debby's offer.
If he rejects the offer, he may propose a counteroffer (yD, 99 - yD) where yD is a number in [0, 99] representing the amount of money that Debby receives under the terms of this offer, while Barack receives 99 - yD. Barack's offer can only divide $99 between the two players, because the delay caused by his rejection of Debby's offer has reduced the payment for completing the project by $1. Debby may accept or reject Barack's offer.
If she rejects the offer, she may then propose yet another counteroffer, and so on. Each additional round of offers, however, reduces the amount of money available by $1: if the two players come to an agreement on a division after the kth offer has been passed between them, then they can divide only (101 - k) dollars between them. If the two players cannot come to any agreement, after 100 rounds of alternating offers, they drop plans to conduct the project jointly, and each receives 0.
Describe this situation as an extensive-form game, and find all of its subgame perfect equilibria.