The Kelly criterion, and what it really answers

The "Kelly criterion" is often presented as a "betting strategy" for the casino, a formula telling the player how much of the "bankroll" (the money set aside for gambling) to stake on each bet so as to make it grow as fast as possible. It is a real formula with a real use, and on a game with a house edge it gives an answer that the pages promoting it rarely print.

Applied honestly to any bet at roulette, Kelly says "do not bet", and for an even-money bet the number it returns is the "house edge" with a minus sign. We consider it the strongest argument of the subject, since it can be checked in two lines and comes from the formula that system sellers like to quote.

What the formula computes

The formula was published in July 1956 by John Larry Kelly Jr., a scientist at Bell Labs, in a paper on information theory ("A New Interpretation of Information Rate", Bell System Technical Journal, volume 35, pages 917 to 926). Its purpose is precise: for a bet with a positive expectation, it gives the fraction of the bankroll that maximises the long-run "growth rate" of that bankroll (technically, the average of its logarithm).

In its simplest form the formula reads f* = (b·p − q) / b, where f* is the fraction to stake, b the "net odds" (1 for a bet paying even money, 35 for a single number at roulette), p the probability of winning and q the probability of losing. The "numerator" (the top of the fraction) is nothing other than the expected gain per dollar staked, so the sign of the answer is decided before any sizing begins, and a negative fraction has no meaning except "stay out".

The calculation on a European wheel

The arithmetic fits in two lines. Take an even-money bet on red, where b = 1, p = 18/37 (eighteen red numbers out of thirty-seven) and q = 19/37 (eighteen black plus the zero). The formula gives 18/37 − 19/37 = −1/37, that is minus 2.70 per cent, which is exactly the house edge of the European wheel with a minus sign in front of it!

The same happens on every other bet of the table. For a single number b = 35 and p = 1/37, the numerator is once again minus one thirty-seventh, and the fraction is that number divided by 35, about minus 0.08 per cent: smaller in size, because the odds are longer, but with the same sign. Kelly at the roulette table, in other words, does not recommend a "small" or "careful" stake; it recommends none.

There is a second way to read the result, which we find just as instructive. If you stake any positive fraction of the bankroll on red, spin after spin, the long-run growth rate turns negative (by our computation about minus 0.07 per cent per spin at a 2 per cent stake, and worse as the stake grows), so the bankroll shrinks over time whatever fraction is chosen.

Where Kelly does apply

Kelly applies, in our reading, where a bettor has a real positive expectation (a "value bet", whose odds pay more than the true chance deserves), and the examples that circulate come almost all from sports betting, where the bettor supposes that their own estimate of an outcome is better than the bookmaker's. A hypothetical bettor who judges a team's chance of winning at 50 per cent, at decimal odds of 2.10 (a net b of 1.10), finds a Kelly fraction of 4.55 per cent, and with a bankroll of $1,000 would stake about $45.

The whole calculation rests on that 50 per cent, which is the bettor's opinion and not a fact of the game. If the true chance is 48 per cent the fraction falls to 0.73 per cent ($7 on the same bankroll); below 47.6 per cent (the break-even point at those odds, one divided by 2.10) the formula turns negative and says what it says at the wheel.

When the estimate is too generous

The error of an optimistic estimate is not symmetrical, and this is the part of Kelly that we would underline for anyone tempted to use it. A bettor who believes in 50 per cent and stakes 4.55 per cent of the bankroll each time, when the true chance is 48 per cent, places bets with a small positive expectation (0.8 per cent per dollar). Yet the growth rate is negative, and the bankroll shrinks in the long run even though each bet is favourable!

The name for it is "overbetting", and it is the error that turns a real advantage into a shrinking bankroll. The reason is that the growth rate falls on both sides of the right fraction, and much faster on the side of excess. For our hypothetical bettor, with the chance really at 50 per cent, staking twice the Kelly fraction (9.1 per cent) brings the long-run growth to zero and staking more makes it negative, while staking half of it keeps three quarters of the growth.

Fractional Kelly

For this reason the advice usually given to bettors who use Kelly in earnest is to stake a fraction of the fraction, the "half Kelly" or the "quarter Kelly", accepting slower growth in exchange for smaller swings and a margin against their own optimism.

In the example above half Kelly means about $23 instead of $45 on the $1,000 bankroll. If the true chance is 49 per cent rather than 50, the half stake grows faster than the full one; if it is 48, only the quarter stake (about $11) still grows at all.

None of this, we should add, concerns the casino. Fractional Kelly is a refinement for bettors who have, or believe they have, a positive expectation, and a fraction of a negative number is still a negative number.

The error of the usual sources

The common presentation of Kelly as a "casino strategy" makes, in our view, a substitution that is worth naming. It borrows its examples from sports betting, where the bettor supposes an advantage, and applies the conclusion to roulette or blackjack, where the advantage belongs to the house and follows from the rules (the house edge page gives it game by game).

The honest use of the formula at a casino is the one it was built for: it tells the player whether a bet has a positive expectation, and at the wheel, as at every table where the house holds an edge, the answer is "no", bet after bet. If you want to play anyway, for entertainment, flat betting is a better instrument, which sets the price of the evening before it starts, and the page on betting systems shows that no other system changes that price either.

If you have arrived here hoping that Kelly would turn a run of losses into a "plan", we would put it plainly: the formula agrees with the house, and the stake it recommends is the one that leaves your money where it is.

Questions

By the Rand Casinos editorial team · Last updated October 2026 · Online since 2007