Kelly criterion explained: the staking formula & its limits
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How the Kelly formula sizes a bet from an edge & the odds, where it came from, & why small errors in the inputs matter so much.
What the Kelly criterion is
Wikipedia’s entry describes the Kelly criterion as a formula for risk allocation that sizes a sequence of bets by maximising the long-term expected value of the logarithm of wealth, which the entry says is equivalent to maximising the long-term expected geometric growth rate. Geometric growth is the compounded rate at which a bankroll grows across many bets.
In their paper Good & bad properties of the Kelly criterion, dated 1 January 2010, Leonard MacLean, Edward Thorp & William Ziemba write that its main advantage is that “it maximizes the limiting exponential growth rate of wealth”. They name the main disadvantage too: “its suggested wagers may be very large”, so the criterion “can be very risky in the short term”.
Wikipedia notes that gamblers often describe their bets relative to the criterion. A full Kelly bet is the amount the formula gives, “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.”
The formula
Karl Whelan of University College Dublin sets out the rule in a paper on parlay bets. A bet at decimal odds D returns a profit of (D − 1) times the stake if it wins & loses the stake if it does not. With p as the bettor’s estimated chance of winning, the optimal share of wealth to stake is (pD − 1) / (D − 1). Whelan writes: “This is the Kelly (1956) criterion rule.”
The top of the fraction, pD − 1, is the edge: the expected profit on each unit staked. The bottom, D − 1, is the net odds, often written b. Rewritten with b = D − 1 & q = 1 − p (the chance of losing), the same rule reads f* = (bp − q) / b, where f* is the fraction of the bankroll to stake.
MacLean, Thorp & Ziemba put it in words: “the optimal Kelly wager is the edge (expected return) divided by the odds”. Whelan adds that the stake “increases with the size of the perceived edge” & decreases with the odds, “so higher risk bets will typically be allocated lower stakes”.
If the edge is zero, the top of the fraction is zero & so is the stake. If the edge is negative, the formula gives a figure below zero. Whelan restricts his own analysis to cases where pD is greater than 1 & says he “will not examine negative stakes”.
Worked examples
Take a coin that lands heads 60% of the time, with even-money bets (decimal odds of 2.00, so b = 1). The formula gives (0.6 × 2 − 1) / 1 = 0.2. Wikipedia reaches the same answer: “the right approach would be to bet 20% of one’s bankroll on each toss of the coin”.
Wikipedia adds that this works out to “a 2.034% average gain each round”. That is a geometric mean, the compounded rate, & not the simple average of 4% per bet.
The same formula applies at a sports price. At decimal odds of 2.10 with an estimated 50% chance of winning, the full Kelly stake is (0.5 × 2.10 − 1) / (2.10 − 1) of the bankroll. Half Kelly would be half of that & quarter Kelly a quarter of it.
In both cases p is an input the bettor supplies. In the coin game it is known exactly; at a sports price it is an estimate.
Kelly’s 1956 paper
Wikipedia says John Larry Kelly Jr, “a researcher at Bell Labs, described the criterion in 1956”. His paper, A New Interpretation of Information Rate, appeared in the Bell System Technical Journal in July 1956; the manuscript was received on 21 March 1956.
Kelly framed it as a problem in communication theory. Where the symbols sent over a channel represent the outcomes of a chance event on which bets are available “at odds consistent with their probabilities (i.e., ‘fair’ odds)”, he wrote, “a gambler can use the knowledge given him by the received symbols to cause his money to grow exponentially”, & “The maximum exponential rate of growth of the gambler’s capital is equal to the rate of transmission of information over the channel.” He adds that “This result is generalized to include the case of arbitrary odds.”
His example is a gambler with a “private wire” carrying baseball results before they become common knowledge. If the wire is perfect, the gambler would bet everything. If it sometimes carries errors, betting everything each time maximises expected capital, but Kelly notes the gambler “would be broke with probability one if he continued indefinitely”. So the gambler bets a fraction of capital instead.
Edward Thorp’s use of Kelly
According to Wikipedia’s entry on Edward O. Thorp, Thorp “used the IBM 704 as a research tool in order to investigate the probabilities of winning while developing his blackjack game theory, which was based on the Kelly criterion, which he learned about from the 1956 paper by Kelly”. He learned the programming language Fortran to run the equations.
Thorp wrote Beat the Dealer in 1962. The entry says the book “mathematically proved that the house advantage in blackjack could be overcome by card counting”.
He later applied probability to financial markets. His first hedge fund was Princeton/Newport Partners, which ran from 1969 to 1989, & he wrote many articles on the Kelly criterion. He is also a co-author of the MacLean, Thorp & Ziemba paper cited in this guide.
Fractional Kelly & why bettors use it
Fractional Kelly means staking a set share of the full Kelly amount, such as half or a quarter. Wikipedia says gamblers use less than full Kelly “in order to reduce the chance of ruin, reduce volatility, and account for model error”, & that they find fractional Kellies “much better emotionally” because of the large drawdowns, the falls from a bankroll’s peak.
The cost is slower growth. Wikipedia says the reduced volatility “increases the time to reach an intended wealth or decreases the wealth growth rate”. MacLean, Thorp & Ziemba describe fractional Kelly as “simply a blend of Kelly and cash” & write that “growth is traded off for security with the use of fractional Kelly strategies”.
Their paper includes a blackjack table for a 2% advantage. At full Kelly, the chance of doubling a bankroll before halving it is 0.67; at half Kelly it is 0.89, with a growth rate 0.75 of the full Kelly rate.
The risk of overestimating your edge
Wikipedia warns that “the Kelly criterion is perfectly valid only for fully known outcome probabilities, which is almost never the case with investments”. It describes “garbage in, garbage out” problems when the inputs are only estimates, & says one way to counter unknown risk is to invest less than the Kelly amount.
MacLean, Thorp & Ziemba cite earlier asset allocation research finding that errors in means, variances & covariances “were about 20:2:1 in importance”. The mean is the expected return, the same quantity that forms the edge in the formula.
Overbetting is costly. The paper says “it never pays to bet more than the Kelly strategy because then risk increases (lower security) and growth decreases”. For processes close to continuous time, growth falls to zero plus the risk-free rate at exactly twice the Kelly stake. Wikipedia adds that betting more than the Kelly amount “increases the risk of ruin”.
An overestimate feeds straight into the stake. Because the stake “increases with the size of the perceived edge”, in Whelan’s words, a bettor who overstates p computes a larger stake than the true probability supports.
Behaviour adds risk even when the odds are known. In a study posted on arXiv, 61 “young, quantitatively trained men and women” were given $25 to bet for 30 minutes on a coin “biased to come up heads 60% of the time”. The authors report that “About 30% of the subjects actually went bust”.
Wikipedia’s account of the study puts the figure at 28%, with an average payout of $91. It says 18 of the 61 participants bet everything on one toss.
Why Kelly often gives a stake of zero
Whelan writes that “Sportsbooks build in a profit margin on their odds, so on average they come out ahead.” His figures for New Jersey show sportsbooks keeping 9.5% of all stakes in 2025, measured as gross gaming revenue (what the books keep) divided by handle (the total staked).
A margin shades the odds below fair value. For a bettor whose probability estimate is no better than the price, pD − 1 is zero or below, & the formula’s answer is to stake nothing.
A positive stake needs a belief that the price is wrong. Whelan notes that odds giving the bettor an edge imply the sportsbook “has implicitly underestimated” the probability, & that the amount staked “increases with the size of the perceived edge”. The word “perceived” matters: the formula sizes the stake on the bettor’s estimate, not on a known truth.
Questions
What is the Kelly criterion formula?
In Whelan’s notation the stake is (pD − 1) / (D − 1) of the bankroll, where p is the estimated chance of winning & D the decimal odds. With b = D − 1 & q = 1 − p, this is (bp − q) / b, which MacLean, Thorp & Ziemba describe as “the edge (expected return) divided by the odds”.
What is half Kelly?
Wikipedia says “A half Kelly bet is half the size of a full Kelly bet”. Gamblers use fractions of Kelly “to reduce the chance of ruin, reduce volatility, and account for model error”, at the cost of slower growth.
Who invented the Kelly criterion?
John Larry Kelly Jr, a researcher at Bell Labs, described it in “A New Interpretation of Information Rate”, published in the Bell System Technical Journal in July 1956.
What happens if I bet more than the Kelly amount?
MacLean, Thorp & Ziemba write that betting more than Kelly means risk increases & growth decreases. Wikipedia says it “increases the risk of ruin”.
Why does the Kelly formula sometimes say to bet nothing?
If the edge pD − 1 is zero or negative, the formula gives zero or less. Whelan notes that sportsbooks build a profit margin into their odds, so a positive stake requires believing the price underestimates the true probability.
Sources
Sources checked 30 October 2026. Information only. 18+.
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