In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet) is a formula for risk allocation with the sizing a sequence of bets by maximizing the long-term expected value of the logarithm of wealth, which is equivalent to maximizing the long-term expected geometric growth rate. John Larry Kelly Jr., a researcher at Bell Labs, described the criterion in 1956.

The practical use of the formula has been demonstrated for gambling, and the same idea was used to explain diversification in investment management. In the 2000s, Kelly-style analysis became a part of mainstream investment theory and the claim has been made that well-known, successful investors including Warren Buffett and Bill Gross use Kelly methods (also see intertemporal portfolio choice).^{[page needed]} It is also the standard replacement of statistical power in anytime-valid statistical tests and confidence intervals, based on e-values and e-processes.

Gamblers often state the size of their bets relative to the Kelly criterion. A full Kelly bet is a bet made at the Kelly Criterion. A half Kelly bet is half the size of a full Kelly bet. A quarter Kelly bet is a quarter of the size of a full Kelly. Gamblers would use less than full Kelly in order to reduce the chance of ruin, reduce volatility, and account for model error. Due to the high drawdowns, gamblers in practice find fractional Kellies much better emotionally than full Kelly. This reduced volatility is a tradeoff, as it increases the time to reach an intended wealth or decreases the wealth growth rate. It has been found that betting an amount larger than the Kelly amount increases the risk of ruin.

In a system where the return on an investment or a bet is binary, so an interested party either wins or loses a fixed percentage of their bet, the expected growth rate coefficient yields a very specific solution for an optimal betting percentage.

Where losing the bet involves losing the entire wager, the Kelly bet is:

where:

As an example, if a gamble has a 60% chance of winning (${\displaystyle p=0.6}$, ${\displaystyle q=0.4}$), and the gambler receives 1-to-1 odds on a winning bet (${\displaystyle b=1}$), then to maximize the long-run growth rate of the bankroll, the gambler should bet 20% of the bankroll at each opportunity (${\textstyle f^{*}=0.6-{\frac {0.4}{1}}=0.2}$).

If the gambler has zero edge (i.e., if ${\displaystyle b=q/p}$), then the criterion recommends the gambler bet nothing (see gambler's ruin).