One re-roll, but the re-roll costs you

Question

You roll a fair six-sided die and get paid the face value in dollars. After the first roll you may re-roll once and keep that second roll, but each re-roll costs you a fixed fee of $\$ 0.50$, deducted from your payout.

Find the optimal strategy and the fair value of the game.

Accepted Answer

The re-roll delivers a fresh face worth E[X] = 3.5 on average, minus the \0.50 fee paid regardless of the outcome, so its net continuation value is 3.5 - 0.5 = 3.0. Decision on the first roll: keep it if the face beats 3.0. Faces 4, 5, 6 clearly beat it; face 3 ties (keeping gives 3, re-rolling gives 3.0), so you are indifferent; faces 1, 2 are worse, so re-roll. Take the keep set as \{4, 5, 6\}: V = 	frac{3}{6} \cdot \frac{4 + 5 + 6}{3} + 	frac{3}{6} \cdot 3.0 = 	frac{1}{2}(5) + 	frac{1}{2}(3.0) = 2.5 + 1.5 = 4.0. (Keeping the tied 3 as well gives 	frac{4}{6}\cdot 4.5 + 	frac{2}{6}\cdo…

One re-roll, but the re-roll costs you

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