Die with up to two re-rolls, price the game

Question

You roll a fair six-sided die and get paid the face value in dollars. After the first roll you may re-roll; after the second roll you may re-roll again. You must keep the third roll. Each re-roll discards the previous result.

Find the optimal strategy and the fair value of the game.

Accepted Answer

Backward induction, stage by stage. Last roll (no options left): worth E[X] = 3.5. Second roll: keep if it beats the continuation value 3.5, keep 4,5,6, re-roll 1,2,3: V2 = 	frac{1}{2} \cdot \frac{4+5+6}{3} + 	frac{1}{2} \cdot 3.5 = 	frac{1}{2}(5) + 	frac{1}{2}(3.5) = 4.25. First roll: now the continuation value is V2 = 4.25, so keep only 5 or 6 (a kept 4 is worth 4 < 4.25): V3 = 	frac{2}{6} \cdot \frac{5+6}{2} + 	frac{4}{6} \cdot 4.25 = 	frac{1}{3}(5.5) + 	frac{2}{3}(4.25) = 	frac{11}{6} + 	frac{17}{6} = 	frac{28}{6} = 	frac{14}{3} \approx 4.67. Optimal strategy: first roll, keep…

Die with up to two re-rolls, price the game

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