Kelly Criterion

Bet everything and ruin is certain; bet nothing and the edge is wasted — the log finds the fraction in between.

The idea

The Kelly criterion determines what fraction of wealth to stake, each round, on a favorable bet that can be repeated indefinitely.

Neither extreme works. Stake everything, and the first loss takes everything: wealth reaches zero and stays there, whatever run of wins came before. Stake nothing, and the favorable bet goes unused. The right stake is a fraction $f$ strictly in between.

Repeated betting multiplies wealth rather than adding to it. Staking the same fraction $f$ every round multiplies wealth by one factor on a win and by another on a loss, so after many rounds wealth is a long product of random factors. A logarithm turns that product into a sum, and sums obey the law of large numbers: over many rounds the average of $\ln(\text{one-round multiplier})$ settles at its expected value, so wealth after $n$ rounds behaves like $e^{ng}$, where

$g = \mathbb{E}\left[\ln(\text{one-round multiplier})\right]$

is the long-run growth rate per round.

Definition (Kelly criterion).

For a favorable bet repeated indefinitely, with the same fraction $f$ of wealth staked each round, the Kelly fraction $f^{}$ is the stake that maximizes the long-run growth rate $g(f) = \mathbb{E}\left[\ln(\text{one-round multiplier})\right]$. The Kelly criterion is the rule: stake $f^{}$.

The curve $g(f)$ rises from zero to its maximum at $f^{*}$, then falls, and turns negative once the stake is large enough, as the figure shows.

Ways to work on it

• Walkthrough. Maximize the expected log growth of a repeated even-money bet, and see the cost of overbetting.
• Practice. Compute Kelly fractions and the exact growth rates they achieve.
• Hardest. Find the best fraction to bet at general odds, and when to bet less than it.

Not sure where to start? Take the ten-question placement test.