Three offers, accept or walk

Question

You will receive up to three offers, one at a time. Each offer is an independent uniform random amount between $0 and $100. You must accept or reject each offer the moment it arrives, with no going back to a rejected one. If you reject the first two, you are stuck taking the third.

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

Accepted Answer

Work backwards from the last offer. At each stage you compare the offer in hand against the value of continuing. Third offer (last). No choice: you take it. Its expected value is the middle of the range, \50. Second offer. You should accept it only if it beats the continuation value of \50. So if the second offer x exceeds \50 you keep it, otherwise you gamble on the third. The value of reaching the second offer is E[\max(x, 50)] = \underbrace{	frac12 \cdot 50}{x < 50,\ 	ext{take third}} + \underbrace{	frac12 \cdot 75}{x 50,\ 	ext{keep it}} = 62.5, using that a uniform value known to exce…

Three offers, accept or walk

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