Reroll a die for two dollars

Question

You roll a fair six-sided die and are paid its face value in dollars. You may pay $2 to reroll (discarding the current face), as many times as you like, keeping whatever 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 value of a fresh roll. A reroll costs \2 and returns you to a fresh roll worth V, so it is worth V - 2. You keep x when x \ge V - 2. The fee is high, so the reroll option is nearly worthless and you will reroll only the very worst face. Guess you reroll only a 1 (keep 2 through 6) and verify: V = \frac{1}{6}\Big[\underbrace{(V - 2)}{	ext{reroll the 1}} + \underbrace{2 + 3 + 4 + 5 + 6}{	ext{kept}}\Big] = \frac{1}{6}\big[V - 2 + 20\big] = \frac{V + 18}{6}. Solving 6V = V + 18 gives 5V = 18, so V = \frac{18}{5} = \3.60. The threshold is V - 2 = 1.6, so you keep anything 2 or above…

Reroll a die for two dollars

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