Ten-sided die with one reroll

Question

You roll a fair ten-sided die (faces 1 through 10) and are paid its face value. After the first roll you may reroll once; if you reroll, you must accept the second roll. What is the fair value of the game, and which first rolls should you keep?

Accepted Answer

With one reroll, there are just two stages. The reroll (no options left): the average of 1 through 10 is \frac{1 + 10}{2} = 5.5. So a forced final roll is worth E1 = 5.5. First roll (one reroll available): keep the current roll iff it beats the continuation value 5.5, i.e. keep \{6,7,8,9,10\} and reroll \{1,2,3,4,5\}: E2 = \frac{5}{10}\cdot\frac{6+7+8+9+10}{5} + \frac{5}{10}\cdot 5.5 = 	frac12(8) + 	frac12(5.5) = 4 + 2.75 = 6.75. The game is worth \6.75, and the optimal rule is keep any first roll of 6 or higher, reroll 5 or lower. The single reroll adds 6.75 - 5.5 = \1.25 over just taking …

Ten-sided die with one reroll

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