Suppose you are offered a bet. A coin comes up heads with probability and tails with probability . If you bet one dollar, you gain one dollar on heads and lose one dollar on tails. The bet has positive expected value: on average, each bet gains you dollars per dollar wagered. So you should bet… how much?
The naive answer is “all your money”: after all, the expected value of betting your whole bankroll is the highest possible. But this strategy is catastrophic. With any positive probability of losing, betting everything bankrupts you the moment the coin comes up tails — and that happens almost certainly if you play long enough. The maximum-expected-value strategy is the maximum-probability-of-ruin strategy.
The opposite extreme — betting an arbitrarily small amount — avoids ruin but grows your bankroll too slowly to be worth the effort. Somewhere between is a unique optimal fraction. The formula for this fraction was derived by John L. Kelly Jr. at Bell Labs in 1956. The Kelly criterion specifies the bet size that maximises the long-run growth rate of your bankroll, balancing growth against ruin in a mathematically precise way. This article is about what the Kelly criterion says, why it works, and where it has been used.
The Kelly formula
For a binary bet with win probability , loss probability , and net payout (i.e., a one-dollar stake returns dollars on a win, or zero on a loss), the optimal fraction of your bankroll to bet on each round is
The numerator is the expected net gain per dollar wagered, called the edge. The denominator is the payout odds. The Kelly fraction is edge divided by odds, a memorable and easy-to-apply rule of thumb.
For the example above (, , ), . You should bet of your bankroll on each coin toss. Win or lose, your next bet is of the new bankroll, and so on. The strategy adjusts your bet size to your current wealth, so a string of losses automatically shrinks your bets and a string of wins automatically grows them.
The growth-rate landscape
Why is optimal? Define your long-run growth rate per bet as
This is the expected value of the logarithm of the multiplicative factor your wealth gains per bet. Maximising over (taking and solving) gives the Kelly formula. For our example, the function looks like:
The curve has a unique maximum at , where . This means your bankroll grows, on average, by a factor of about per bet — about per bet. Over bets, your bankroll should grow by a factor of . Over a year of daily bets (say bets), the expected multiplication is about .
Bet too little — say — and you grow at , less than half the optimal rate. Bet too much — say — and the growth rate is actually negative, . You are over-betting and slowly losing money on average, despite the positive edge.
Beyond in this example (for ), the growth rate drops below the “lose everything in one bet” floor: you are taking risks so large that the typical sequence of bets ruins you.
Why the logarithm?
The Kelly criterion can be derived in two equivalent ways.
First derivation: long-run growth. Imagine you play the bet times and your bankroll evolves multiplicatively: , where is the multiplicative factor on bet (either on a win or on a loss). The long-run growth rate is
By the law of large numbers, the actual rate concentrates almost surely around its expectation. Maximising therefore maximises the long-run growth rate, which is the natural objective for a gambler who plays many rounds.
Second derivation: expected log-wealth. Equivalently, is the expected one-period log-return of the strategy, and the Kelly criterion maximises the expected logarithm of wealth at any given horizon. This is the optimal strategy under logarithmic utility, the unique utility function with constant relative risk aversion equal to .
These two perspectives are mathematically equivalent: maximising expected log-utility per period is the same as maximising long-run geometric growth.
Why not just maximise expected value?
The crucial point is that expected value of wealth and expected value of log-wealth are different. Maximising expected wealth on a single bet recommends betting everything; maximising expected log-wealth gives a moderate fraction. Over many bets, the two strategies diverge dramatically: the expected-value-maximising strategy almost certainly bankrupts you, while the Kelly strategy almost certainly grows your wealth at the maximum sustainable rate.
This is the fundamental insight of the Kelly criterion: when the outcome of repeated bets is multiplicative, you should optimise the geometric mean (equivalently, the log) of returns, not the arithmetic mean. Logarithm converts multiplication to addition, and the law of large numbers does its work on the additive sum.
Caveats and fractional Kelly
The Kelly criterion has well-known limitations:
Estimation error is brutal. The formula assumes the true probabilities and odds are known. In practice, estimates of are noisy. Over-estimating your edge by leads to over-betting that costs much more in growth than the same absolute under-betting would.
Drawdowns are uncomfortable. Even the optimal Kelly strategy can experience large temporary drawdowns. The probability of your bankroll halving before doubling under full Kelly is — a sobering number for anyone deploying real money.
For both reasons, many practitioners use fractional Kelly: betting some fraction (often or even ) of the full Kelly amount. Half Kelly captures about of the growth rate while approximately halving the standard deviation of returns. Edward Thorp, the mathematician who used Kelly betting in both blackjack card counting and the stock market, recommends about of full Kelly as the practical sweet spot.
Where the criterion has been used
Blackjack card counting. Edward Thorp’s 1962 book Beat the Dealer showed how to count cards in blackjack to obtain a small probabilistic edge over the casino. The book also used the Kelly criterion to determine optimal bet sizes — varying with the current count to size bets to the current edge. Within a few years, casinos had banned card counting and introduced multiple-deck shoes specifically to defeat Thorp’s strategy. The mathematics had become a real-world economic force.
Sports betting. Professional sports bettors using statistical models often size positions with fractional Kelly. The same logic applies as in blackjack: positive expected value, repeated bets, and the desire to grow a bankroll without going bust.
Quantitative finance. Edward Thorp moved from blackjack to running a hedge fund, Princeton-Newport Partners, which traded statistical-arbitrage strategies in the 1970s and 1980s. The fund used Kelly-style sizing throughout its life. Thorp’s fund had a remarkable track record — averaging around annual returns for nearly two decades — and Kelly sizing was a central part of the methodology.
Information theory. Kelly’s original 1956 paper framed the betting problem in information theory terms: the optimal long-run growth rate of the bankroll equals the mutual information between the bettor’s information and the betting outcome. So a side channel that gives you a -bit advantage on each coin toss earns you exponential growth per bet — a precise connection between information and money.
A small formula with large consequences
The Kelly criterion is one of those mathematical results that sits at the intersection of probability, economics, and information theory. The formula itself is short, the derivation is clean, and the conclusions are precise enough to be acted on with real money.
It captures a deep insight: the right way to compound positive-expected-value bets is to maximise expected log-wealth, not expected wealth. The two are different, and the difference is the gap between long-run prosperity and bankruptcy.
For an individual investor, the Kelly criterion is a reminder that bet sizing is at least as important as bet selection. For a quantitative trader, it is a tool that has been used to compound real money over decades. For an information theorist, it is a clean illustration of how information can be converted into wealth at a precisely measurable rate. And for the general reader, it is a useful corrective to the everyday intuition that “you should bet bigger when you’re sure”—because the Kelly criterion shows, precisely, that beyond a certain point, betting bigger destroys long-run growth even when the bet is genuinely favourable.
Bet too little and you grow too slowly. Bet too much and you go broke. The right answer, as Kelly showed at Bell Labs in 1956, is to bet exactly over — and to keep doing it, with discipline, until the law of large numbers does its quiet work.
Frequently asked
Who invented the Kelly criterion?
John L. Kelly Jr. was a Bell Labs scientist who derived the criterion in 1956, while studying information-theoretic problems for AT&T. His paper, 'A New Interpretation of Information Rate', framed gambling as a communication channel and showed that the optimal long-run growth rate of a bankroll is bounded by the channel's information rate. The mathematician Edward O. Thorp picked up the result in the 1960s and applied it to blackjack card counting (his book Beat the Dealer became a bestseller) and later to the stock market through his hedge fund Princeton-Newport Partners, which used Kelly-style sizing for decades.
What does the Kelly fraction mean in practice?
It is the fraction of your bankroll you should wager on each bet to maximise the geometric mean of your wealth — equivalently, to maximise the long-run growth rate of your bankroll. For a binary bet with win probability p, loss probability q = 1 − p, and net payout odds b (i.e., a win of $1 returns $b plus the stake), the Kelly fraction is f* = (bp − q) / b. For a fair coin with edge — say p = 0.6 on an even-money bet — f* = 0.2, meaning you should bet 20% of your bankroll.
Why not just bet for maximum expected value?
Because maximising expected value of a single bet does not maximise long-run growth. If the bet has positive expected value, the expected-value-maximising strategy is to bet everything every time — but with any positive probability of losing, this strategy goes bankrupt almost surely. The Kelly criterion maximises the expectation of the logarithm of wealth, which is equivalent to maximising the geometric mean of returns, and avoids ruin by automatically scaling bet sizes down as the chance of catastrophic loss grows. This is the precise mathematical formulation of 'compound your wealth optimally'.
What is 'fractional Kelly' and why use it?
Many practitioners use 'half-Kelly' or even 'quarter-Kelly' — a fraction smaller than the full Kelly amount. The reason is that the Kelly fraction is computed assuming the true probabilities and odds are known exactly. In practice, estimates of p and b are imperfect, and an overestimate of edge leads to over-betting, which is much more costly than under-betting. Half-Kelly sacrifices only about a quarter of the growth rate but dramatically reduces drawdown and is more forgiving of estimation error. The famous mathematician Edward Thorp recommends about 0.4 of the full Kelly fraction for most practical use.