5. Hungry Bargaining (5 marks total) Alice and Bob are trying to decide how to divide a delicious cherry pie between them. They both agree to the following rules: e On the first day. Alice makes a proposal about how to divide the pie. Alice can propose any a; [0,1]. Here, a, represents the fraction of the pie that Alice will give to Bob, leaving a fraction 1 ay of the pie for herself. Bob can choose to accept or reject Alice's offer. If Bob accepts, then the game is over; Alice gets 1 a; of the pie and Bob gets a1 of the pie. If Bob rejects, then nobody gets any pie on day 1, and we move to day 2. e On day 2, the roles reverse. Bob proposes by [0,1] to Alice. Here, by represents the fraction of the pie that Bob will give to Alice, leaving 1 by of the pie for himself. Alice can accept or reject Bob's offer. If Alice accepts, then the game is over; Alice gets b2 of the pie, and Bob gets 1 bz of the pie. If Alice rejects, then nobody gets any pie on day 2, and we move to day 3. e On day 3, the roles reverse again. Alice makes an offer az; Bob can accept or reject. If Bob accepts, he gets ag of the pie, and Alice gets 1 a3 of the pie. If Bob rejects the offer, then we go to day 4. e This same pattern continues indefinitely until an offer is accepted. We assume that Bob and Alice are both hungry: they would rather eat the pie today than tomorrow, and similarly would rather eat it tomorrow than in two days' time. More specifically, Alice values an entire pie today at 1, the same pie tomorrow at d,4, in two days at (4.4)2, and so on. Bob values the pie today at 1, tomorrow at dg, in two days at (3)%, and so on. Note that 0