Question

g 2. 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, δS for Susan and δB for Bill (0 < δB < δS < 1). (i) Determine the minimum offer Susan can propose now that Bill will certainly accept. 1 (ii) If δS = 9/10 for what range of discount values δB would result in Bill agreeing to accept less than half the prize money immediately? (iii) Given different discount rates δB and δS, what would the outcome be for the infinite horizon game where the alternating offers continue forever?

Answer 1

Answer:

(i) The minimum offer Susan can propose now that Bill will certainly accept is x* = (1000 / 3) * δ$_{B}$ * (3 - δ$_{S}$).

(ii) The range of discount values   that would result in Bill agreeing to accept less than half the prize money immediately is  0 < δ$_{B}$ < 5/7.

(iii) Given different discount rates  and  , the outcome for the infinite horizon game where the alternating offers continue forever would be as follows:

$x_{1}$ = (1000(1 – δ$_{B}$)) / (1 +  δ

$x_{2}$ = (1000δ$_{B}$(1 – δ$_{S}$)) / (1 +  δ

$y_{1}$ = (1000δ$_{S}$(1 – δ$_{B}$)) / (1 +  δ

$y_{2}$ = (1000(1 – δ$_{S}$)) / (1 +  δ$_{B}$ δ

Explanation:

Note: See the attached Microsoft word file for the full explanations.

Note: Some of the signs in the questions are not stated properly. They are therefore restated before answering the question. See the restated question in the attached file for these.

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