A die game with two rerolls

Question

You roll a fair six-sided die and are paid its face value in dollars. Before being paid, you may reroll, and after that roll you may reroll one more time; you keep whatever the final roll shows.

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

Accepted Answer

Solve by backward induction, stage by stage. One reroll left. This is the standard single-reroll game. You keep a roll only if it beats the value of a fresh roll, 3.5, so keep 4, 5, 6. Its value is V1 = 	frac12 \cdot \frac{4+5+6}{3} + 	frac12 \cdot 3.5 = 	frac12(5) + 	frac12(3.5) = 4.25. Two rerolls left. Now the continuation value if you reroll is V1 = 4.25, so keep the current roll only if it beats 4.25, meaning keep 5 or 6 and reroll on 1, 2, 3, 4: V2 = \frac{2}{6}\cdot\frac{5+6}{2} + \frac{4}{6}\cdot 4.25 = \frac{2}{6}(5.5) + \frac{4}{6}(4.25) = \frac{11}{6} + \frac{17}{6} = \frac{28}…

A die game with two rerolls

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