Double or nothing, when do you walk away?

Question

You start with \1. Each round you flip a fair coin: heads doubles your entire bankroll, tails wipes it to zero and the game ends. Before any flip you may stop and bank what you have. When should you stop, and what is the game worth?

Accepted Answer

One-step reasoning, and why it stalls. With bankroll x, one more flip gives E[\Delta] = 	frac{1}{2}(+x) + 	frac{1}{2}(-x) = 0. The expected change is exactly zero at every bankroll. Unlike the classic pig game, there is no score at which continuing becomes a losing proposition, so the expected-value threshold rule gives no guidance: every stopping policy yields the same expected value. The expected value is flat. Each flip is a fair gamble, so the bankroll is a martingale. Whether you stop at \1, after one flip, or after ten, the expected bank is always your starting \1: with probability 2^…

Double or nothing, when do you walk away?

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