How much to stake with the Kelly criterion

What the Kelly criterion is

The Kelly criterion is a formula that sets your stake as a percentage of your bankroll, using the price on offer and your own estimate of how often the bet wins.

It aims at one thing: growing that bankroll as fast as possible over the long run without wiping it out along the way. Mathematicians call this maximizing the growth rate.

It did not start in betting. John Kelly, a physicist at Bell Labs, published it in 1956 as a way of measuring how much information a noisy telephone line could carry. Gamblers took it over afterwards, Ed Thorp carried it from blackjack into investing, and it has been standard in both ever since.

It gives back one number and nothing else. It has no opinion on the bet: it will not tell you who wins, and it cannot tell whether your guess is any good. It only sizes the stake.

A word about that money. Your betting pot — bankroll, in the industry's own term, and the two mean the same thing everywhere below — is the amount set aside for betting. Not savings, not rent. Only what you are willing to lose.

The answer always comes as a share of it, never as cash. Ten percent of a 1000 bankroll is 100. Lose that bet and the bankroll is 900, so ten percent is now 90. The stake shrinks on its own after a bad run and grows again after a good one, and that quiet self-correction does a lot of the work.

Why not just pick an amount yourself

Because left alone, almost everybody stakes too much.

Staking big pays more on the bets you get right. It also empties the bankroll on the ones you get wrong, and an empty bankroll ends the run — there is nothing left to be right with next time. That ending has a name in staking theory: risk of ruin.

Kelly looks for the size in the middle. The one that grows the bankroll fastest across hundreds of bets while keeping it alive through the bad weeks.

A price, a guess, a stake

Take a price of 2.10, which is +110 in American odds.

Every price has a chance built into it. Divide 1 by 2.10 and you get 47.6 percent. That is what the bookmaker's number says about how often this happens, and it is the figure you have to beat.

Say you think the real answer is 50 percent — a straight coin flip. The gap between 47.6 and 50 is your edge, and it is worth 5 percent on every unit staked. That 5 percent is the expected value of the bet, EV for short.

Kelly turns that edge into a stake of 4.55 percent of the bankroll. On a bankroll of 1000, that is 45.50.

Most people would have put on more than 45.50. That is the whole point of the rule.

The formula

Take the decimal price and subtract 1. At 2.10 that leaves 1.10, which is the profit you make per 1 staked.

stake share = your chance − (1 − your chance) ÷ profit per 1

For the example: 0.5 minus 0.5 divided by 1.10 gives 0.0455, or 4.55 percent.

The textbook case runs the same way. A price of 2.00, or +100, on something you rate at 60 percent gives 0.6 minus 0.4 divided by 1, which is 20 percent of the bankroll.

If your prices come in American format, convert them before anything else. The standard −110 line is 1.91 in decimal, and 1 divided by 1.91 says you need 52.4 percent just to break even on it. The odds converter does the conversion in both directions.

The same edge at different prices

Your edge is the gap in percentage points between your own chance and the break-even chance inside the price. The table below shows the full Kelly stake for three sizes of that gap. Half Kelly is exactly half of each figure.

Read the rows before the columns. The same edge is worth far more at a short price than at a long one: ten points at 1.50 asks for 30 percent of the bankroll, the same ten points at 5.00 asks for 12.5.

The reason is the profit per unit. At 1.50 you win 0.50 per 1 staked, so the stake has to be large for the edge to be worth anything. At 5.00 you win 4.00, and a small stake carries the same weight.

That top-left corner is also where full Kelly becomes unusable. Nobody should be putting 30 percent of a bankroll on one football match, however good the number looks — which is what the next two sections are about.

Where your guess is supposed to come from

Kelly treats your probability as a fact and does no checking. Feed it a feeling about a team and it hands that feeling back to you as a stake with a decimal point on it. It looks precise. It is not.

There is a steadier way to get the number. Take the price at a bookmaker known for accurate pricing, strip out the built-in margin, and use the no-vig figure that is left. That is the closest thing to a fair price you can get without doing your own modelling, and the odds converter guide walks through it.

What happens when the guess is wrong

This is the part worth reading twice.

Say you rate that same 2.10 shot at 55 percent when the true chance is 50. Kelly now calls for 14.1 percent of the bankroll instead of 4.55.

The strange part is that the bet is still good. At a true 50 percent it carries the same 5 percent EV it carried before. Nothing about the price got worse. Only the stake did.

And the stake is now bigger than the edge can carry. At the correct 4.55 percent the bankroll grows by about 0.11 percent a bet. At 14.1 percent it shrinks by about 0.39 percent a bet. Staking past the number has its own name: overbetting.

Five points too high triples the stake. The edge is still real, and the bankroll shrinks anyway, because 14.1 percent is more than that edge can carry.

Nobody believes they are the one guessing too high. Everybody is, some of the time.

Why almost nobody stakes the full amount

The full Kelly number is correct math and hard to live with.

Follow it exactly and there is roughly a one in three chance the bankroll halves before it ever doubles. Stretch the window across the whole run rather than to the doubling point, and the chance of touching half at some stage sits nearer one in two.

Not lost, in either case. Halved, then slowly rebuilt. Very few people keep following a rule through that.

The swings come from the size. At 20 percent of the bankroll, four losses in a row take out more than half of it.

Fractional Kelly: half, quarter, and the multiplier

Cutting the Kelly number down is so common that it has its own name — fractional Kelly — and most calculators expose it as a Kelly multiplier you can set to 0.5 or 0.25.

Half Kelly is the usual setting, and it costs less than people expect.

Half Kelly gives up a quarter of the growth and removes about half of the swings. The risk figures move further than the growth does: against roughly one in three at full Kelly, the half Kelly bettor faces about one in nine of halving the bankroll before doubling it.

It also softens the damage when your estimate was a little too high, which is the more common problem. Bet half of a number built on a guess that was five points generous, and you land close to the stake the true edge deserved.

On the 2.10 example, half Kelly is 2.27 percent of the bankroll and quarter Kelly is 1.14 percent.

The ceiling above the number

Stake twice the Kelly amount and the growth is gone. On the 2.10 example that lands on exactly zero; at other prices it lands a fraction below. All of the risk, none of the reward.

Go past that and the bankroll shrinks, even when every single bet holds a genuine edge. That is what happened in the 14.1 percent example above — 14.1 is more than double the 4.55 the numbers supported.

Staying under the number costs a slice of the growth. Going over it costs all of the growth, and then starts working through the bankroll itself.

Kelly against flat staking

Flat staking is the alternative most bettors actually use: the same amount on every bet, or the same one or two percent of the bankroll every time, regardless of price or confidence.

Kelly does something flat staking cannot. It stakes more when the edge is bigger and less when the price is longer, and it rescales after every win and loss.

Flat staking has one real advantage in return: it does not care whether your probabilities are any good. A bad estimate produces a bad bet, but not an oversized one. Kelly turns the same bad estimate into a bad bet three times the size, which is exactly what the 14.1 percent example shows.

So the honest split is this. If your probabilities come from stripped-down sharp prices or a model you have tested, Kelly is the better tool and half Kelly is the sane setting. If they come from watching the games and forming a view, flat staking will treat you more gently.

When the rule says no

If your guess matches the chance already inside the price, the formula returns zero. No edge, no bet.

If your guess is lower, it returns a minus number. Textbooks read that as a signal to take the other side, and in a fair two-way market that is right. At a bookmaker it usually is not: the two prices add up to more than 100 percent, so the margin sits on both sides and both of them lose money over time.

Treat a minus as «leave this alone». Backing the opposite outcome is a separate calculation, and it only becomes sensible at a price your own numbers make favorable, which often means a different bookmaker.

A staking rule that keeps telling you to bet nothing is working properly. Most prices are not worth taking.

Three things the rule does not know about

The bankroll keeps moving. Every stake is a share of what you have right now, so it falls after losses and rises after wins. Recalculating each time is the whole idea, not an optional extra.

Some bets move together. Kelly assumes each one stands alone. Four picks from the same match rise and fall as a group, so staking the full amount on each one risks far more than the rule intends.

Bookmakers set limits. The stake the math calls correct may be more than your account is allowed to place. Once an account has been restricted, the bookmaker decides the size and the formula becomes theory.