The Kelly criterion for sports betting, worked

The Kelly criterion says to stake f = (bp − q) ÷ b of your bankroll, where b is the net odds (decimal odds minus 1), p is the chance of winning and q = 1 − p. The top line is just your expected value per dollar, so Kelly is edge ÷ net odds. A 5% edge at +100 means betting 5% of your bankroll. Staked that way, a bankroll grows fastest in the long run, but only if p is right. Because it never is exactly, nearly everyone who uses Kelly bets a fraction of it, typically a quarter or less.

Why it is edge divided by odds

Bigger edges deserve bigger bets, and longer odds deserve smaller ones, because a longshot loses more often and the losses come in runs. Kelly balances the two so that the bankroll's expected logarithm grows as fast as possible, which in plain terms means it grows fastest over many bets while never staking everything. The staking guide covers the wider idea; this one works the numbers.

What fractional Kelly costs you

Betting a fraction c of the Kelly stake gives up surprisingly little growth. When edges are small, the long-run growth rate at c × Kelly is roughly c × (2 − c) of the maximum, while the swings in your bankroll shrink roughly in proportion to c. Half Kelly keeps about 75% of the growth with half the volatility. Quarter Kelly keeps about 44% with a quarter of it. Betting double Kelly gets you zero growth with twice the swings.

What happens when your edge is wrong

Kelly's promise depends on knowing p. You estimate it from a devigged fair price, and that estimate is noisy. Suppose you think a +100 bet is 55% (Kelly says 10%) when it is really 52.5% (true Kelly 5%). Betting 10% is double the true Kelly: in the long run the bankroll goes nowhere while swinging wildly. Overestimate a little more and it shrinks. Underestimate and you merely grow more slowly. The penalties are lopsided, so the sensible response to uncertainty is to bet less.

Simultaneous bets

The simple formula assumes one bet at a time, settled before the next. On a Sunday you may have ten open at once. For a few small, independent bets the correct joint stakes are barely below the single-bet figures. With many at once, the joint optimum falls further and total exposure becomes the thing to watch: at full Kelly on ten independent 10%-Kelly bets, the best stakes are about 9% each, nearly 90% of the bankroll in play at once. Fractional Kelly deals with most of this. Bets on the same game are not independent, so treat correlated bets as one position and size them together.

Doing it without the arithmetic

On the Positive EV board, each row's stake box is pre-filled with a fractional-Kelly stake worked from the bankroll and risk level in your Settings, scaled down by the engine's confidence in that fair price and capped at your own maximum. It is a suggestion you can overwrite. The walkthrough sizes one bet step by step.