Roll a die with at most two free re-rolls

Question

You roll a fair six-sided die and are paid its face value. Re-rolls are free, but you get at most two of them, so you see at most three rolls in total. After each roll you decide to keep the current face or re-roll (if you have a re-roll left); on the final roll you must keep whatever comes up.

What is your strategy, and what is the game worth?

Accepted Answer

With a finite number of rolls, solve backward. Let Vk be the value when you have k rolls remaining including the current one. Last roll (V1). You must keep it: V1 = E[X] = 3.5. One re-roll left (V2). You hold a face x; keeping is worth x, re-rolling is worth V1 = 3.5. So keep x iff x \ge 3.5, i.e. keep \{4,5,6\}, re-roll \{1,2,3\}: V2 = 	frac{1}{6}(4+5+6) + 	frac{3}{6}(3.5) = 	frac{15}{6} + 	frac{10.5}{6} = 	frac{25.5}{6} = 4.25. Two re-rolls left (V3, the first roll). Keeping x is worth x; re-rolling is worth V2 = 4.25. Keep x iff x \ge 4.25, i.e. keep \{5,6\}, re-roll \{1,2,3,4\}: V3 =…

Roll a die with at most two free re-rolls

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