Unlimited rerolls, but each one costs you

Question

You roll a fair six-sided die and are paid its face value. You may reroll as many times as you want, but each reroll costs $\$ 0.50$ (you replace your current value with the new roll).

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

Accepted Answer

With unlimited rerolls the game no longer has a fixed number of stages, so instead of counting down you solve for a fixed point. Let C be the value of choosing to reroll from any position. Rerolling costs 0.50, and afterward you again keep the new roll Y or continue, so C = -0.5 + E\big[\max(Y, C)\big]. The optimal rule is "keep iff the roll is at least C." Guess C lands so the keep set is \{4,5,6\} (faces below C get rerolled). Then E[\max(Y, C)] = \frac{3C + (4+5+6)}{6} = \frac{3C + 15}{6} = 0.5C + 2.5, so C = -0.5 + 0.5C + 2.5, giving 0.5C = 2 and C = 4. This is consistent: with C = 4 you …

Unlimited rerolls, but each one costs you

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