Kelly sizing when the payout isn't even money

Question

You can repeatedly bet on an event with win probability $p = 0.5$ . When you win you receive net odds $b = 2$ (win $\$ 2 $for every$ $1$ staked); when you lose you forfeit the stake. You may bet any fraction of your bankroll each time.

What fraction should you bet, and what growth rate does it achieve?

Accepted Answer

Wealth compounds multiplicatively, so maximize the expected log growth of betting fraction f: g(f) = p\,\ln(1 + bf) + q\,\ln(1 - f), \qquad q = 1 - p. Differentiate and set to zero: g'(f) = \frac{pb}{1 + bf} - \frac{q}{1 - f} = 0 \;\implies\; pb(1 - f) = q(1 + bf). Expanding: pb - pbf = q + qbf \implies pb - q = bf(p + q) = bf, so \boxed{f^ = \frac{pb - q}{b} = p - \frac{q}{b}.} Plug in p = 0.5,\ q = 0.5,\ b = 2: f^ = \frac{0.5 \cdot 2 - 0.5}{2} = \frac{1 - 0.5}{2} = \frac{0.5}{2} = 0.25. Bet 25\% of bankroll. The numerator pb - q = 1 - 0.5 = 0.5 is your edge per dollar staked (expected profi…

Kelly sizing when the payout isn't even money

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