Question

Two individuals, Susan and Bill, are involved in a bitter dispute over splitting a small prize of £1000. The prize donator, who acts as an independent arbitrator, rules that if an agreement cannot be reached between the two individuals within two months then they will each receive one third of the prize with the remainder going to charity. They are allowed two chances to settle: Susan can make an offer to Bill immediately which he can either accept or reject. If Bill rejects this offer, he can make an alternative offer to Susan one month from now. Bill is more impatient than Susan and so their monthly discount factors are different, og for Susan and 88 for Bill (0 <0B < s < 1). (i) Determine the minimum offer Susan can propose now that Bill will certainly accept. (ii) If ds = 9/10 for what range of discount values OB would result in Bill agreeing to accept less than half the prize money immediately? (iii) Given different discount rates OB and ds, what would the out- come be for the infinite horizon game where the alternating offers continue forever?

Answer 1

Solution Consider the game in the following notation we have three periods t = 0, 1,2 At tão since Sussan offers a to bill of accepted by bill, the payoffs [us, UB) = (1000 -2,2) if rejected by bill, Bill offers y to susan at til If aceopted by susan then the payoffs are (Us, Ub) = (ssy, SB(1000-4)). If rejected by susan at t=2, the payoffs are (us, ub) = (SP 1000, s 1900) ♡ Coming to the analysis, Susan accepts y at tal if and only it Ssy = $ 100 which is same as yz 10008. Since Bill wants to maximize his payoff, he will actually propose y* = roods, and get Up = dp (1000 - gt) for himself. Therefore,

the minimum offer that will would accept X 1000 We get that a = 400&B we want 100<p < 500 026 <5/7 l het (x, y) and (y , g) denote the future payott's for susan and bill when susan and bill are making an often respectively. That is, when susan is making an offer y, she knows that bill goes gets dy by rejecting i she will offer x=ly and she will get 1000-12: similarly, bill would offer y = y, and get 1000-y for himself. Thus, we need that SB (1000 -4, )=x2 and Ss (1000 - 12) =Y

Solving these a equations in a variables, we get x2 = 1000 s fl-ds 11-8 Sp and Y = 1000 Sg (1-56 ) 11-Sg Sb) Also, we have that = 1000 CI- Sb 1- ES Sb and you = 22186 = 1000 / ross LT-6 Sb