Where the formula comes from
The Kelly criterion is a result from probability theory about how large a fraction of a growing pool to risk on each of a long series of favourable bets, if the aim is to maximise long-run growth. It is named after John L. Kelly Jr., who published the underlying idea in 1956 while working at Bell Labs, in the context of information theory.
It is a theoretical tool with strict assumptions. It does not find an advantage, it does not make a losing game profitable, and it is not a recommendation to bet. This article explains the idea and, mostly, why using it for real money is fragile.
The formula
For a bet that pays b units of profit per unit staked if it wins, and where the chance of winning is p and the chance of losing is q = 1 − p, the Kelly fraction of the pool to stake is:
f* = (b × p − q) / b
The numerator, b × p − q, is the expected profit per unit staked. Dividing by b scales it by the payout. The formula returns a fraction of the current pool, so as the pool shrinks or grows, the stake shrinks or grows with it.
Here are two checked examples. They are illustrations, and games with such an advantage are rare.
- Even-money bet (b = 1) with p = 0.55: f* = (0.55 − 0.45) / 1 = 0.10, so 10% of the pool.
- A bet that pays 2 to 1 (b = 2) with p = 0.4: b × p = 0.8, q = 0.6, so f* = (0.8 − 0.6) / 2 = 0.10, also 10%.
If the expected profit is zero or negative, the formula gives zero or a negative number, which means do not bet. With p = 0.45 at even money, f* = (0.45 − 0.55) / 1 = −0.10. In games with a house edge, Kelly’s answer is not to play, so the criterion is of no help to someone looking for a way to make money from them.
What it is optimising
Kelly maximises the long-run average of the logarithm of the pool, which is equivalent to maximising the long-run compound growth rate. For a single bet staking a fraction f, the average log growth per bet is:
g(f) = p × ln(1 + b × f) + q × ln(1 − f)
Using the p = 0.55, even-money example, with natural logarithms rounded to four or five places:
- At f = 0.10: g = 0.55 × ln(1.1) + 0.45 × ln(0.9) = 0.55 × 0.09531 − 0.45 × 0.10536 = 0.05242 − 0.04741 = about +0.0050 per bet.
- At f = 0.05 (half Kelly): g = 0.55 × ln(1.05) + 0.45 × ln(0.95) = 0.02684 − 0.02308 = about +0.0038 per bet.
- At f = 0.20 (twice Kelly): g = 0.55 × ln(1.2) + 0.45 × ln(0.8) = 0.10028 − 0.10042 = about −0.0001 per bet, which is roughly zero.
So half the Kelly stake gives about three quarters of the maximum growth rate (0.0038 / 0.0050 is 0.76), while twice the Kelly stake gives roughly no growth at all. Stakes beyond that point have negative growth, even though each bet has a favourable average. This is one of the main reasons the idea is taken seriously: betting too much is worse than betting a little too little.
The assumptions behind it
Kelly’s result depends on conditions that rarely hold:
- The probabilities are known exactly. In real life the chance p is an estimate, not a fact.
- Bets are repeated many times. The criterion concerns long-run growth, not a handful of plays.
- Outcomes are independent and each bet is available again at the same odds.
- Any fraction can be staked, with no minimums, maximums or rounding.
- Winnings are reinvested, so the pool compounds.
- Only growth matters. The goal is long-run growth, not comfort, income or avoiding bad stretches.
- No fees or limits interfere.
Why real use is fragile
Errors in p. Because the formula depends on the size of the advantage, small mistakes matter. Suppose the true chance is 0.52 at even money, while the bettor believes it is 0.55 and stakes 10%. Then g = 0.52 × ln(1.1) + 0.48 × ln(0.9) = 0.04956 − 0.05057 = about −0.0010 per bet, which is negative. The bettor believes in growth and is actually shrinking the pool slowly. Estimating p from past results is hard, because a limited sample size leaves wide uncertainty. The range given by confidence intervals around an estimated probability can be as wide as the advantage being estimated, or wider.
Large swings. Even when p is known, full Kelly staking produces big fluctuations. Large drawdowns, including a halving of the pool at some stage, are not unusual in idealised models. Many people would find that unbearable, which is why a fraction of the Kelly stake, often called fractional Kelly, is sometimes discussed. That trades some growth for a gentler path.
No real advantage to size. The criterion assumes an advantage exists. In games with a house edge, or with a bookmaker margin built into the prices, the best the formula can say is that the right stake is zero. Claims of an edge need very strong evidence, and the idea of gaining an edge is easy to overstate.
Practical frictions. Betting limits, fees, delays in payment and non-independent outcomes all depart from the model.
How to read it
Kelly is best understood as a statement about the trade-off between growth and risk, illustrating that staking too much destroys growth even when the odds are favourable. Its main lesson is about humility. A formula that looks precise rests on inputs that are not.
None of this is advice to bet. Outcomes are uncertain, games with negative expectation lose money over time, and gambling can be harmful. If play stops being fun or starts to cause stress, a local problem gambling support service can help.