For each dollar staked, a bet pays a profit of 2 dollars with probability 3/10, loses the dollar with probability 1/2, and returns it unchanged with probability 1/5. What fraction of your bankroll maximises the expected logarithm of wealth?

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Expected valueHard

For each dollar staked, a bet pays a profit of $2$ dollars with probability $\frac{3}{10}$ , loses the dollar with probability $\frac{1}{2}$ , and returns it unchanged with probability $\frac{1}{5}$ . What fraction of your bankroll maximises the expected logarithm of wealth?

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Hint

Set the derivative of the expected log growth to zero; the push contributes nothing.

Solution

Wealth is multiplied by $1 + 2 f$ , $1 - f$ or $1$ with probabilities $\frac{3}{10}$ , $\frac{1}{2}$ and $\frac{1}{5}$ , and the last outcome adds nothing to the derivative.
The condition is $(\frac{3}{10}) \times \frac{2}{1 + 2 f} = \frac{\frac{1}{2}}{1 - f}$ , which becomes $0.6 (1 - f) = 0.5 (1 + 2 f)$ .
Then $0.1 = 1.6 f$ , so $f = \frac{1}{16}$ .

Answer: $\frac{1}{16}$

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