You roll a fair six sided die. After seeing the result you may re-roll once and must then take the second number. What is the value of the game and what is the optimal policy?

You roll a fair six sided die. After seeing the result you may re-roll once and must then take the second number. What is the value of the game and what is the optimal policy?

Approach: Compare each first roll against the value of a fresh roll to set a threshold, then compute the expectation as the kept outcomes plus the re-roll branch.

17/4. A fresh roll of a fair die is worth 3.5, so the continuation value is 3.5 and you re-roll whenever the first roll falls below it. That means keeping 4, 5 and 6 and re-rolling 1, 2 and 3. The value is then (4 + 5 + 6)/6 + (3/6) * 3.5 = 2.5 + 1.75, which is 4.25 or 17/4. The threshold policy is the optimal rule because the continuation value does not depend on what you saw, so the decision reduces to a single comparison. With two re-rolls the same argument runs backwards from this answer: the one re-roll game is worth 4.25, so you keep only 5 and 6, giving 1.8333 + 2.8333, so (5 + 6)/6 + (4/6) * 4.25 = 14/3.

Follow-up: Each re-roll now costs $0.20. How many re-rolls do you buy, and what is the value of the game?

Key concepts: continuation value, threshold policy, optimal rule, fair die.