The Kelly criterion explained

The Kelly criterion is a formula for deciding how much to stake. Instead of a fixed amount every time, it sizes each stake as a share of your bankroll (the money you have set aside for betting and can afford to lose), based on two things: the odds on offer and how likely you think the outcome is. Where you see an edge, it stakes more; where you see none, it stakes nothing.

John L. Kelly Jr., a researcher at Bell Labs, published it in 1956 as a problem about a gambler receiving advance information over a noisy communication channel. It has been used to size stakes and investments ever since.

The formula

With decimal odds d and your probability estimate p:

• b = d − 1, the profit on a stake of 1 if it wins
• q = 1 − p, the chance it loses
• Kelly fraction = (b × p − q) ÷ b

An equivalent way to write it uses your expected value:

• Edge = p × d − 1, the expected profit per unit staked
• Kelly fraction = edge ÷ b

Both give the same number: the share of your current bankroll to stake. If you use American or fractional odds, convert them to decimal first with the odds converter.

A worked example

Say a price is 2.20 (+120), and your own estimate is that the outcome happens 50% of the time.

• b = 2.20 − 1 = 1.20
• q = 1 − 0.50 = 0.50
• Kelly fraction = (1.20 × 0.50 − 0.50) ÷ 1.20 = 0.10 ÷ 1.20 = 8.33%

Using the second form: the edge is 0.50 × 2.20 − 1 = 0.10, or 10%, and 0.10 ÷ 1.20 = 8.33%. With a bankroll of 1,000 units, full Kelly stakes 83.33 units.

Now take a price of 1.91 with the same 50% estimate. The edge is 0.50 × 1.91 − 1 = −0.045, a negative expected value. The formula returns a negative fraction, and the answer is to stake nothing.

Your probability estimate is the input that matters most. A common starting point is a fair price built from many sportsbooks with the margin removed, which you can turn into a probability with the implied probability guide.

Fractional Kelly

Full Kelly is the stake that grows a bankroll fastest over the long run, but only if your probability is exactly right. In practice, many people stake a fixed share of it instead:

The numbers come from the 2.20 example above. In the standard model behind the formula, half Kelly keeps about three quarters of full Kelly's long-run growth rate while cutting the size of the swings roughly in half. That trade is why half and quarter Kelly are common choices.

The risks

Your estimate is probably wrong

The formula treats your probability as the truth. Suppose in the example the real chance was 46%, not 50%. Then:

• The correct Kelly fraction is (1.20 × 0.46 − 0.54) ÷ 1.20 = 0.012 ÷ 1.20 = 1%
• Staking 8.33% is more than eight times that

Staking more than about twice the correct Kelly amount gives a negative long-run growth rate: the bankroll tends to shrink over time, even though each stake here still has a small positive expected value (0.46 × 2.20 − 1 = 0.012). Overestimating your edge is the most common way Kelly goes wrong, and fractional Kelly is the usual protection against it.

The swings are large

Even when every estimate is right, full Kelly is a bumpy ride. In the standard model, there is about a 1 in 2 chance the bankroll falls to half its starting size at some point; at half Kelly it is about 1 in 8. Long losing runs are a normal part of it, not a sign that something has broken.

It assumes one outcome at a time

The basic formula sizes one stake, settled before the next. Several stakes running at once need more care: adding up separate full Kelly stakes can commit far more of the bankroll than the formula intends. A parlay counts as one stake at its combined price, but the probability you enter must be the chance that every leg wins together; multiplying the legs' chances is only right when the legs are unrelated.

It does not create an edge

Kelly only sizes a stake. No staking plan turns a negative expected value into a positive one, and if your probability is too high, Kelly will size stakes on an edge that is not really there. Tracking how your prices compare with the final prices, as explained in the closing line value guide, is one way to test whether your estimates hold up.

Betting always carries a risk of losing money, so if you bet, set a limit you are comfortable with and check that the sportsbook is licensed where you are; that guide also shows where to find free help if gambling stops being fun.