Up to three rolls
You roll a fair die and may either stop and receive the number shown, in dollars, or roll again. You may roll at most three times in total; if you reach the third roll you must take it.
With the best strategy, what is your expected payout?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Work backwards from the last roll. What is a roll worth when it is your final one?
With two rolls left, you should keep the first one only if it beats the value of rolling again.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Backward induction: the value of the game with rolls left tells you exactly which results to keep with rolls left.
One roll left. You must take it, so its value is .
Two rolls left. Keep the roll if it is larger than , that is if it is , or ; otherwise roll again and get on average:
Three rolls left. Keep the first roll only if it beats , that is if it is or :
The strategy. On the first roll keep or ; on the second roll keep , or ; take whatever the third roll gives. The threshold drops as you run out of rolls, which is the general pattern in optimal stopping.