The ten-percent blow-up

Question

Each round of a game, with probability 0.1 a blow-up occurs and your whole pile is wiped to zero. Otherwise (probability 0.9) the round adds a payout averaging \4 to your pile. You may stop and bank at any point before the next round. At what pile size should you stop, to maximize the expected banked amount?

Accepted Answer

Weigh one more round at pile size t: - With probability 0.1 you blow up and lose t, a change of -t. - With probability 0.9 you gain \4 on average, a change of +4. Expected change of one more round: 0.1(-t) + 0.9(4) = -0.1\,t + 3.6. This is positive while -0.1\,t + 3.6 0, that is while t < 36. So you keep going below \36 and stop once your pile reaches \36 or more. Because the blow-up is rare (only 10\%), you can afford to let the pile grow quite large, all the way to \36, before the wipeout risk on that pile finally outweighs the modest \4 you expect to add each round. \boxed{	ext{Stop as so…

The ten-percent blow-up

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