Two offers only

Question

You will receive up to two offers, one at a time, each an independent uniform random amount between $0 and $100. You must decide on each the moment it arrives, with no going back. Reject the first and you are stuck with the second.

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

Accepted Answer

Backward induction with only two stages. Second offer (last). You must take it. Its expected value is the middle of the range, \50. First offer. Accept it only if it beats the continuation value of \50. So keep the first offer x when x 50, otherwise gamble on the second. The value of the game is E[\max(x, 50)] = \underbrace{	frac12 \cdot 50}{x < 50,\ 	ext{take second}} + \underbrace{	frac12 \cdot 75}{x 50,\ 	ext{keep it}} = 62.5, using that a uniform value known to exceed 50 averages 75. Strategy: accept the first offer if it beats \50; otherwise take the second. Expected take = 62.50. Th…

Two offers only

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