The Kelly criterion: the formula that tells most gamblers to bet nothing

A formula for a noisy telephone line

John Kelly Jr. was working at Bell Labs on information theory when he published the result in 1956. The problem he was formally solving was not gambling at all — it was how much of a channel’s capacity you could use given noise. The gambling framing was the illustration.

The question it answers is precise and narrow: given a repeated bet you have an edge on, what fraction of your bankroll should you stake each time to maximise the long-run growth rate of that bankroll? Not to maximise the chance of a profit, and not to minimise risk — to compound fastest.

It became foundational in professional betting and investing because the answer is not obvious. Betting too little wastes the edge. Betting too much destroys the bankroll even when the edge is real, because losses compound as brutally as gains.

The formula, and the term that kills it

For a simple bet paying b to 1, won with probability p, lost with probability q = 1 − p:

f* = (bp − q) ÷ bthe fraction of bankroll to stake

The whole result lives in the numerator. bp is what you expect to win; q is what you expect to lose. If bp is bigger than q you have an edge and the formula hands you a positive fraction. If they are equal — a genuinely fair bet — it returns zero.

And if bp is smaller than q, which is the definition of a house edge, the formula returns a negative number. Read literally, it is telling you to take the other side. Since you cannot, the actionable answer is zero.

The honest answer for a casino game

Every game in an honest casino is published with a return below 100%. That is the product: you are buying a session, and the edge is the price. It also means the bottom half of that table is where every casino bet sits.

So a casino publishing an article about Kelly arrives somewhere slightly absurd: the correct application of the most respected bet-sizing formula in existence, to our own games, says do not bet. We think saying so plainly is more useful than pretending otherwise, because the alternative is letting people believe a staking system might rescue a negative edge. None can. Not Kelly, not martingale, not anything.

What Kelly does establish, usefully, is the shape of the mistake. Overbetting relative to your edge is not merely riskier — past a threshold it drives long-run growth negative even with a real edge, because a bankroll that halves needs to double to recover. With no edge at all, every size is overbetting, and the only lever left is how fast you would like to arrive.

What to take from it anyway

The formula is not useless to a player. It reframes the decision in a way that survives contact with a negative edge.

1. Bet size should be a fraction of a bankroll, not a number that feels right. Kelly’s deepest idea is proportional staking — the stake moves with the stack.
2. Without an edge, the fraction is not a growth decision but a duration decision: it sets how long the budget lasts and how wild the ride is.
3. The correct frame is therefore a session budget you are willing to spend, divided into bets small enough to make the session last. That is the same discipline Kelly enforces, applied to the only variable actually available.

Which lands on the least surprising advice in gambling, arrived at from the most rigorous direction available: decide what the evening is worth to you, then bet in fractions of it. The mathematics of optimal growth and the mathematics of not hurting yourself turn out to agree.