Three offers, whole dollars one to ten

Question

You will receive up to three offers, one at a time, each an independent uniform whole-dollar amount from $1 to $10. You must accept or reject each on the spot, with no going back. Reject the first two and you take the third.

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

Accepted Answer

Backward induction with a discrete range. Third offer (last). You must take it, expected value 	frac{1+10}{2} = 5.5. Second offer. Accept if it beats the \5.50 continuation, that is, keep 6,7,8,9,10 and reject 1 through 5. Half the time you keep an average of 8, half the time you fall through to \5.50: V2 = 	frac12 \cdot 8 + 	frac12 \cdot 5.5 = 6.75. First offer. Now the bar is \6.75, so accept only 7,8,9,10 (average 8.5), a 	frac{4}{10} chance, and otherwise continue to \6.75: V1 = 	frac{4}{10} \cdot 8.5 + 	frac{6}{10} \cdot 6.75 = 3.4 + 4.05 = 7.45. Strategy: accept the first offer if…

Three offers, whole dollars one to ten

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