Reroll a die for fifty cents

Question

You roll a fair six-sided die and are paid its face value in dollars. You may pay $0.50 to reroll (discarding the current face), as many times as you like, and you keep whatever face you finally stop on.

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

Accepted Answer

Let V be the fair value of a fresh roll. Rerolling costs \0.50 and returns you to a fresh-roll situation worth V, so it is worth V - 0.5. After seeing x you keep it if x \ge V - 0.5. The fee is cheaper than before, so you can afford to be pickier. Guess you keep only 5, 6 and reroll 1, 2, 3, 4, then verify: V = \frac{1}{6}\Big[\underbrace{5 + 6}{	ext{kept}} + \underbrace{4(V - 0.5)}{	ext{reroll 1,2,3,4}}\Big] = \frac{1}{6}\big[11 + 4V - 2\big] = \frac{2V}{3} + 1.5. Solving V = 	frac{2V}{3} + 1.5 gives 	frac{V}{3} = 1.5, so V = 4.5. The threshold is V - 0.5 = 4.0, meaning you keep 5 or 6 a…

Reroll a die for fifty cents

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