What is the Kelly criterion?
The Kelly criterion is a formula that determines the optimal fraction of a player's bankroll to wager on a bet, in order to maximize the long-term growth rate of the bankroll. It was developed by John L. Kelly Jr. in 1956 at Bell Labs, originally in the context of information theory, and has since become a fundamental tool in gambling, finance, and investment.
The formula, expressed as a fraction of the bankroll, is:
k: Kelly fraction
e: expected value (as a fraction of the wager)
p: prize (decimal odd minus 1)
k = e / p
Since the expected value is e = w · p − l, where w is the actual probability of winning and l = 1 − w is the actual probability of losing, the formula can also be written as:
k = (w · p − l) / p
The result k is the fraction of the bankroll that should be wagered. If k is negative, the bet has negative expected value and should not be placed at all.
Example
Suppose a player estimates that an event has a 55% actual probability of occurring, and the decimal odd offered is 2.10. The prize is:
p = 2.10 − 1 = 1.10
The expected value is:
e = .55 · 1.10 − .45 = .605 − .45 = .155 = 15.5%
The Kelly fraction is:
k = .155 / 1.10 ≈ .1409 ≈ 14.09%
This means the player should wager approximately 14.09% of his bankroll on this bet to maximize long-term growth. If the bankroll is ¤1000, the optimal wager would be approximately ¤141.
Why the Kelly criterion maximizes growth
The Kelly criterion maximizes the expected logarithmic growth rate of the bankroll. This is formally expressed as:
G: expected growth rate
f: fraction of bankroll wagered
G(f) = w · log(1 + f · p) + l · log(1 − f)
The Kelly fraction k is the value of f that maximizes G(f). Wagering more than k increases the variance without increasing the long-term growth rate — in fact, it decreases it. Wagering exactly double the Kelly fraction produces a growth rate of zero, equivalent to not betting at all. Wagering more than double the Kelly fraction produces negative growth, meaning the bankroll will shrink over time despite the bet having positive expected value.
This is a counterintuitive but mathematically certain result: a bet can have positive expected value, and yet a player can lose money in the long term by wagering too much on it.
Fractional Kelly
In practice, full Kelly betting is rarely used. The formula assumes that the player's estimate of the actual probability is perfectly accurate, but in reality, probability estimates carry uncertainty. If the estimated probability is even slightly wrong, the Kelly fraction may be too large, exposing the player to excessive risk.
For this reason, most practitioners use a fraction of the Kelly criterion, commonly between 25% and 50%. For instance, half-Kelly means wagering half of what the full Kelly formula indicates:
f = k / 2
In the example above, where k ≈ 14.09%, half-Kelly would be approximately 7.05%. This reduces the growth rate, but it also substantially reduces the variance and the risk of large drawdowns. The reduction in growth rate is modest compared to the reduction in risk: half-Kelly achieves approximately 75% of the growth rate of full Kelly, with significantly less volatility.
Relationship to expected value and edge
The Kelly criterion is directly proportional to the expected value. A bet with a higher expected value will produce a larger Kelly fraction, and a bet with zero or negative expected value will produce a Kelly fraction of zero or less, indicating that the bet should not be placed.
The edge, defined as the difference between the actual probability and the implied probability, is closely related:
edge = w − i
Where w is the estimated actual probability and i is the implied probability (1 / d). A positive edge means the player believes the event is more likely than the odd implies. The expected value and the edge are both measures of the player's advantage, expressed in different terms: the expected value is expressed as a fraction of the wager, while the edge is expressed as a difference in probabilities.
Limitations
The Kelly criterion is optimal under specific assumptions: the player knows the actual probability exactly, can place bets of any size, and is concerned exclusively with maximizing long-term growth. In practice, none of these assumptions hold perfectly.
Probability estimates are always uncertain. The Kelly fraction is sensitive to errors in these estimates: overestimating the probability leads to over-betting, which can produce catastrophic losses. This sensitivity is the primary reason why fractional Kelly strategies are preferred.
Furthermore, the Kelly criterion does not account for the player's risk tolerance or time horizon. A player who cannot withstand a 50% drawdown in his bankroll — which is statistically likely even under full Kelly betting — may need a more conservative strategy.
Despite these limitations, the Kelly criterion remains the most theoretically sound framework for bet sizing, and it serves as the reference point from which other staking strategies are derived or evaluated.