Die game with two rerolls

Question

You roll a fair six-sided die and get paid its face value in dollars. After the first or second roll you may choose to reroll; you must accept the third roll if you take it. (Up to $3$ rolls total.)

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

Accepted Answer

Backward induction, one stage at a time. Last roll (no options left): value E1 = 3.5. One reroll left: keep the current roll iff it beats 3.5, i.e. keep \{4,5,6\}: E2 = \frac{3}{6}\cdot\frac{4+5+6}{3} + \frac{3}{6}\cdot 3.5 = \frac12(5) + \frac12(3.5) = 4.25. Two rerolls left (first roll): the continuation is now worth 4.25, so keep only \{5, 6\}: E3 = \frac{2}{6}\cdot\frac{5+6}{2} + \frac{4}{6}\cdot 4.25 = \frac13(5.5) + \frac23\left(\frac{17}{4}\right) = \frac{11}{6} + \frac{17}{6} = \frac{28}{6} = \frac{14}{3} \approx 4.67. The game is worth \frac{14}{3} \approx \4.67. Notice the diminishi…

Die game with two rerolls

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