Four offers, accept or walk

Question

You will receive up to four offers, one at a time, each an independent uniform random amount between $0 and $100. You must accept or reject each the moment it arrives, with no going back. Reject the first three and you are stuck with the fourth.

What is the optimal strategy, and what is the expected amount you walk away with?

Accepted Answer

Backward induction, one stage at a time. For a uniform draw on [0,100] with acceptance bar t, the value of a stage is E[\max(x, t)]: accept when the draw beats t (a value known to exceed t averages 	frac{t+100}{2}), otherwise fall through to t. Fourth offer (last). Value = 50. Third offer. Bar = 50: value = 	frac12(50) + 	frac12(75) = 62.5. Second offer. Bar = 62.5: value = 0.625(62.5) + 0.375(81.25) = 69.53. First offer. Bar = 69.53. With acceptance chance 	frac{100 - 69.53}{100} = 0.3047 and a kept average of 	frac{69.53 + 100}{2} = 84.77: V1 = 0.6953(69.53) + 0.3047(84.77) \approx 74.…

Four offers, accept or walk

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