What Is the Kelly Criterion?

The criterion, published by John Kelly in 1956 in the context of information transmission, answers a specific question: given repeated independent opportunities with a known probability and payoff, which constant fraction of current wealth maximises the long-run growth rate.

For a binary bet at decimal odds d with win probability p, the optimal fraction is:

f = (p × d − 1) / (d − 1)

The numerator is the expected value per unit staked, and the denominator is the net payoff per unit. So the fraction is edge divided by net odds. If the expected value is zero or negative, the formula returns zero or less, meaning no stake.

A worked example. With p equal to 0.55 and d equal to 2.00, the numerator is 0.55 × 2 − 1, which is 0.10, and the denominator is 1.00, so f is 0.10. The formula recommends staking ten percent of the bankroll.

Why logarithms. Maximising expected wealth alone recommends staking everything on any favourable bet, which guarantees eventual ruin. Maximising expected log wealth values proportional growth, which produces the property Kelly is known for: over a long enough run it produces higher terminal wealth than any other constant fraction, and it never risks total loss on a single bet.

Multiple simultaneous positions require solving jointly, accounting for correlation. Treating each independently and summing the fractions overstakes, sometimes severely.

It assumes the probability is exact. The formula takes p as given. In sports, p is an estimate from a model that is itself uncertain, and the formula has no term for that uncertainty.

Overestimation is asymmetric. Staking above the Kelly fraction reduces growth and increases variance, and staking at twice the optimal fraction produces an expected growth rate of zero even when the edge is real. Since most estimated edges are smaller than believed, systematic overestimation of p is the common failure.

Bets are not independent. Correlated positions behave like one larger position. Same-game legs, related markets, and outcomes driven by the same underlying event violate the independence assumption directly.

Bankroll is not continuous. Minimum stakes, maximum limits, and rounding all deviate from the model, particularly at small bankrolls.

Variance is high even when correct. Full Kelly produces large drawdowns as a matter of mathematics, not bad luck. Many people cannot tolerate them, and abandoning a strategy mid-drawdown is worse than having staked smaller throughout.

Fractional Kelly. Staking a fixed fraction of the Kelly amount, such as a half or a quarter, is the standard response. It reduces growth modestly while reducing variance substantially, and it provides protection against the overestimation of p that is nearly universal.

What this page is not

As a sensitivity tool. Recomputing the fraction across a plausible range of p shows how quickly the recommendation changes. If a small error in p moves the fraction dramatically, that is a statement about the fragility of the whole analysis.

As a comparison of opportunities. Because the formula is edge divided by net odds, it ranks opportunities by more than raw edge, which is informative even if the absolute numbers are not acted on.

As a check on calibration. Kelly is only valid with calibrated probabilities. Building the fraction into a workflow creates pressure to measure calibration, which is the useful discipline.

In simulation. Simulating staking strategies against a record of estimates and outcomes shows the drawdown distribution, which is far more informative than the growth rate alone.

Records. Whatever staking approach is used, the record should store the estimate, the price available, the fraction implied, and what was actually staked. Without the estimate, none of it can be evaluated later.

This page describes data and method and is not betting advice.

Why the fraction is usually reduced in practice