Roll a die, keep it or pay to re-roll, what's it worth?

Question

You roll a fair six-sided die and will be paid its face value in dollars. After any roll you may keep the current face value and stop, or pay $\$ 1 $to re-roll (replacing the current value). You may re-roll as many times as you like, and the$ $1$ costs are subtracted from your winnings.

What is your stopping rule, and what is the game worth?

Accepted Answer

Let G be the game's value, the expected net payoff when you hold a roll and play optimally. After a roll of x, you keep x or re-roll, and re-rolling is worth G - 1 (a fresh roll worth G, minus the \1 cost). So the value of holding x is \max(x,\ G - 1), and G = E\big[\max(X,\ G - 1)\big], \qquad X \sim 	ext{Uniform}\{1,\dots,6\}. Let R = G - 1 be the re-roll threshold: you keep x iff x \ge R. Guess R \in [3,4), so you keep \{3,4,5,6\} and re-roll \{1,2\}. Then G = \frac{1}{6}\Big[\underbrace{R + R}{x=1,2 	o R} + 3 + 4 + 5 + 6\Big] = \frac{2R + 18}{6}. With R = G - 1: G = \frac{2(G-1) + 18}{6…

Roll a die, keep it or pay to re-roll, what's it worth?

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