Keep rolling or bank it, the greedy pig game

Question

You roll a fair die repeatedly, adding each face value to a running score. But if you ever roll a $\mathbf{1}$ , you lose the entire accumulated score for the round and it ends. At any point you may instead choose to stop and bank your current score.

What is your stopping rule, and roughly what is the game worth?

Accepted Answer

One-step reasoning. Suppose your current banked-at-risk score is s. Roll once more: - With probability 1/6 you roll a 1 and lose s (change = -s). - With probability 1/6 each you roll 2,3,4,5,6, gaining that amount (average gain \frac{2+3+4+5+6}{6}). Expected change from one more roll: E[\Delta] = \frac{1}{6}(-s) + \frac{1}{6}(2+3+4+5+6) = \frac{-s + 20}{6} = \frac{20 - s}{6}. This is positive exactly when s < 20. So keep rolling while your score is below 20; stop once you reach 20 or more. (At s = 20 you're indifferent.) This one-step rule is genuinely optimal here because the game is a "burn…

Keep rolling or bank it, the greedy pig game

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