A reroll that costs a dollar

Question

You roll a fair six-sided die and are paid its face value in dollars. Before being paid, you may reroll once, but choosing to reroll costs you a $1 fee (subtracted from your payout). If you reroll, you must accept the second roll.

What is the fair value of this game under optimal play?

Accepted Answer

If you reroll, you get a fresh die worth 3.5 but pay the \1 fee, so the continuation value is 3.5 - 1 = 2.5. Keep any first roll worth more than 2.5, that is 3, 4, 5, 6, and reroll only on 1 or 2. Value the game at the start: - With probability 	frac{4}{6}, the first roll is 3 to 6, averaging \dfrac{3+4+5+6}{4} = 4.5. - With probability 	frac{2}{6}, you reroll, worth 2.5 (already net of the fee). E = 	frac{4}{6}(4.5) + 	frac{2}{6}(2.5) = 3.0 + 0.8333 = 3.833. The game is worth \dfrac{23}{6} \approx 	extbf{\3.83}. The fee lowered the reroll's value from 3.5 to 2.5, so the keep threshold d…

A reroll that costs a dollar

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