A fair die is rolled some number of times. You can choose whether to stop after

Question

A fair die is rolled some number of times. You can choose whether to stop after 1, 2, or 3 rolls, and your decision can be based on the values that have appeared so far. You receive the value shown on the last roll of the die, in dollars. What is your optimal strategy (to maximize your expected winnings)? Find the expected winnings for this strategy. Hint: Start by considering a simpler version of this problem, where there are at most 2 rolls. For what values of the first roll should you continue for a second roll?

Explanation / Answer

Case 1: You are allowed one die roll. Your expected value is 3.5

Let us summarize this by saying E1=3.5.

Case 2: You are allowed up to two die rolls. If your first roll is 3 or less, you should reroll, since the expectation of your next roll is greater than 3. If your first roll is 4 or greater, you should take that value, since if you roll again, your expected value is less than what you have. So in this case half the time you are taking 4,5,6 (equally likely);

Thus your expected value here is

E2=(1/2)*5+(1/2)*3.5=4.25

Case 3: You are allowed up to three die rolls. If your first roll is greater than 4.25 (the expected value if you pass on your first roll and go into case 2), you should keep your first roll. Otherwise you should go on and roll again.

E3=(1/3)5.5+(2/3)4.25

E3=14/3

Regarding long term behavior, I would think you should be able to approach an expected value of 6, as the number of allowed die rolls increases.

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