Re-roll the die, but it costs you a dollar

Question

You roll a fair six-sided die and are paid its face value in dollars. You may keep it, or pay a fee of $\$ 1 $to re-roll once, in which case you are paid the second face minus the$ $1$ fee.

What is the optimal strategy, and what is the fair value of the game?

Accepted Answer

The continuation value of re-rolling is the expected second face minus the fee: E[	ext{re-roll}] = 3.5 - 1 = 2.5. So keep the first face if and only if it beats 2.5: keep 3, 4, 5, 6; re-roll 1, 2. The fee shifts the threshold up from 3.5 (in the free-re-roll game) toward the lower faces, because paying to re-roll is only worth it when your current face is quite bad. Value of the game: V = 	frac{4}{6} \cdot 	frac{3+4+5+6}{4} + 	frac{2}{6} \cdot 2.5 = 	frac23(4.5) + 	frac13(2.5) = 3 + 	frac{5}{6} = 	frac{23}{6} \approx 3.83. So the game is worth about \3.83, which is still above the pla…

Re-roll the die, but it costs you a dollar

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