Kelly sizing a fair coin at 2-to-1 odds

Question

You have a \10{,}000 bankroll and can repeatedly bet on a fair coin (p = 0.5), but the payout is generous: a winning 1 stake returns 2 in profit (net odds b = 2), while a loss forfeits the 1 staked. You may bet any fraction of your bankroll, repeatedly. How much do you bet per flip, and why not more?

Accepted Answer

Because bankroll compounds multiplicatively, maximize the expected growth rate E[\log(	ext{wealth})]. Betting fraction f with net odds b: g(f) = 0.5\ln(1 + bf) + 0.5\ln(1 - f) = 0.5\ln(1 + 2f) + 0.5\ln(1 - f). Differentiate and set to zero: \frac{2 \cdot 0.5}{1 + 2f} - \frac{0.5}{1 - f} = 0 \Rightarrow \frac{1}{1+2f} = \frac{0.5}{1-f} \Rightarrow 2(1-f) = 1 + 2f \Rightarrow f = 0.25. Bet f^ = 25\% of bankroll, \2{,}500 on the first flip, then re-size to 25% each flip. This matches the general Kelly formula f^ = \frac{bp - q}{b} = \frac{2(0.5) - 0.5}{2} = \frac{0.5}{2} = 0.25, that is, your e…

Kelly sizing a fair coin at 2-to-1 odds

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