Eight-sided die with two rerolls

Question

You roll a fair eight-sided die (faces 1 through 8) and are paid its face value. After the first or second roll you may 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

Work backward one stage at a time, always using the rule "keep the current roll only if it beats the value of playing on." Last roll (no options left): the average of 1 through 8 is E1 = 4.5. One reroll left: keep the current roll iff it beats 4.5, i.e. keep \{5,6,7,8\}: E2 = \frac{4}{8}\cdot\frac{5+6+7+8}{4} + \frac{4}{8}\cdot 4.5 = 	frac12(6.5) + 	frac12(4.5) = 5.5. Two rerolls left (first roll): the continuation is now worth 5.5, so keep only \{6,7,8\}: E3 = \frac{3}{8}\cdot\frac{6+7+8}{3} + \frac{5}{8}\cdot 5.5 = \frac{3}{8}(7) + \frac{5}{8}(5.5) = \frac{21}{8} + \frac{55}{16} = \frac{9…

Eight-sided die with two rerolls

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