Reroll a die for a dollar each time

Question

You roll a fair six-sided die and get paid its face value in dollars. After seeing a roll you may pay a fee of $1 to reroll (discarding the current face). You may reroll as many times as you like, each reroll costs another $1, and you keep whatever you finally stop on.

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

Accepted Answer

Because rerolls are unlimited and identical, the situation looks exactly the same every time you are about to roll fresh. Call that fair value V. If you decide to reroll, you pay \1 and land back in a fresh-roll situation worth V, so rerolling is worth V - 1. After a roll shows x, you keep it (worth x) or reroll (worth V - 1). You should keep exactly when x \ge V - 1. So the whole strategy hinges on the single threshold V - 1. Guess that you keep 4, 5, 6 and reroll 1, 2, 3, then check it is consistent. Under that rule, V = \frac{1}{6}\Big[\underbrace{4 + 5 + 6}{	ext{kept}} + \underbrace{3(V …

Reroll a die for a dollar each time

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