Roll a die with two re-rolls allowed, price it

Question

You roll a fair six-sided die and are paid its face value in dollars. This time you may re-roll up to twice (so up to three rolls in total), and whenever you re-roll you must keep taking the later roll unless you still have a re-roll left.

What is the optimal strategy, and what is the fair value of the game?

Accepted Answer

Work backwards through the layers. Last roll: worth its expectation, 3.5. One re-roll left (the two-roll subgame): keep any face above 3.5, so keep 4, 5, 6 and otherwise re-roll for 3.5. Its value is V2 = 	frac12 \cdot 	frac{4+5+6}{3} + 	frac12 \cdot 3.5 = 	frac12(5) + 	frac12(3.5) = 4.25. Two re-rolls left (the first roll): the continuation value is now V2 = 4.25, so keep the first face only if it beats 4.25: keep 5, 6; re-roll 1, 2, 3, 4. The value is V3 = 	frac{2}{6} \cdot 	frac{5+6}{2} + 	frac{4}{6} \cdot 4.25 = 	frac13(5.5) + 	frac23(4.25) = 	frac{11}{6} + 	frac{17}{6} = 	fr…

Roll a die with two re-rolls allowed, price it

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