Solution
Paid Rerolls
Show the problem again
You roll a fair six-sided die. After seeing the result you may either stop and receive the face value in dollars, or pay $1 and roll again. You may pay to reroll as many times as you like, and each reroll replaces the previous result. There is no entry fee. Playing to maximize your expected net amount (payout minus all reroll fees), what is your expected net amount? Give an exact number of dollars.
Worked solution
The answer is 4 dollars. The game looks the same before every roll, so its value V satisfies V = E[max(X, V − 1)]: after a roll of X you either take X or pay 1 to face the same game again. A stationary threshold strategy is optimal. Accepting only 4, 5 or 6 takes 2 rolls on average (one paid reroll) and pays 5 on average, net 4. Accepting 3 or more takes 3/2 rolls on average and pays 4.5, also net 4. Accepting only 5 or 6 nets 5.5 − 2 = 3.5, and accepting only 6 nets 6 − 5 = 1. Check: with V = 4, E[max(X, 3)] = (3 + 3 + 3 + 4 + 5 + 6)/6 = 4, so V = 4 is the fixed point. A roll of 3 is exactly the break-even point, which is why two thresholds tie.
Source: Standard optimal-stopping exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.