Why not just maximize expected wealth?

Question

A coin lands heads with probability 0.60. On heads you double whatever you stake; on tails you lose it. A colleague argues: "Each round, staking everything has expected value 0.6 	imes 2 = 1.2 times my money, the highest possible. So I should bet my whole bankroll every round to maximize expected wealth." Is the colleague right? What actually happens, and what fraction should be bet instead?

Accepted Answer

The colleague is technically correct that betting everything maximizes expected wealth, and disastrously wrong about what that means. Consider all-in every round for n rounds: - You survive only if every flip is heads: probability 0.6^n, which decays to zero. - If you ever see one tail (probability 1 - 0.6^n 	o 1), your bankroll is wiped to zero, permanently. Expected wealth is (0.6 \cdot 2)^n W0 = 1.2^n W0, growing without bound, but that average is carried by the vanishingly rare all-heads path. The typical outcome is ruin: with probability approaching 1 you go broke. Maximizing expected w…

Why not just maximize expected wealth?

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