You roll a fair six sided die and may take the number shown in dollars or reroll, with at most three rolls in total. You are paid whatever you accept, and the third roll must be accepted. Playing optimally, what is the fair value of the game?

You roll a fair six sided die and may take the number shown in dollars or reroll, with at most three rolls in total. You are paid whatever you accept, and the third roll must be accepted. Playing optimally, what is the fair value of the game?

Approach: Value the game from the last roll backwards, accepting a roll exactly when it exceeds the value of continuing with the rolls that remain.

14/3. Use backward induction. With one roll left the value is 3.5, so with two rolls left you accept 4, 5 or 6 and reroll otherwise, giving (4 + 5 + 6)/6 + (3/6) * 3.5 = 2.5 + 1.75 = 4.25. With three rolls left the acceptance threshold rises: 4 is now below the continuation value of 4.25, so you keep only 5 or 6, giving (5 + 6)/6 + (4/6) * 4.25 = 11/6 + 17/6 = 28/6 = 14/3, about 4.667. The rule is to accept exactly when the roll beats the value of the option to reroll. With unlimited rolls the value rises to 6.

Follow-up: With a fee of $0.50 charged for each reroll, what is the value and what are the thresholds?

Key concepts: backward induction, acceptance threshold, option to reroll.