Optimal Betting & the Kelly Criterion

Positive edge is not enough

Suppose you can repeatedly bet on a coin that wins with probability $p = 0.6$ at even odds. Betting your whole bankroll maximises expected wealth each round — and goes broke the first time you lose, which happens eventually with probability 1.

The relevant quantity for repeated, compounding bets is the growth rate. If you bet a fraction $f$ each time, after $n$ bets with $k$ wins your wealth is $W_0(1+f)^k(1-f)^{n-k}$, so

$$\frac1n \ln\frac{W_n}{W_0} \;\longrightarrow\; g(f) = p\ln(1+f) + q\ln(1-f) .$$

The Kelly fraction

Maximising $g(f)$ gives, for a bet paying $b$-to-1,

$$f^* = \frac{bp - q}{b} = \frac{\text{edge}}{\text{odds}} .$$

At even odds ($b = 1$) this is $f^* = p - q$. Key properties:

• Kelly maximises long-run growth and never risks ruin (it never bets everything).
• Betting more than $f^*$ lowers growth; at roughly $2f^*$ the growth rate falls to about zero.
• Betting less (e.g. half Kelly) gives up a little growth for much less volatility — popular in practice because your estimate of $p$ is uncertain.