The Kelly Criterion

The Kelly criterion is the mathematically optimal stake for maximising long-run bankroll growth, given a known edge. It is elegant, it is correct, and applying it literally to sports betting is one of the faster ways to destroy a bankroll — because the one thing it assumes you have is the one thing you do not.

The Formula

For a simple win-lose bet, the Kelly stake as a fraction of bankroll is:

f = (bp − q) / b

where b is the decimal profit per unit staked, p is your win probability and q is 1 − p.

Worked example. A bet at -110 (b = 0.909) that you rate at 55%:

f = (0.909 × 0.55 − 0.45) / 0.909 = (0.500 − 0.45) / 0.909 = 0.055

Full Kelly says stake 5.5% of your bankroll. On a 5,000 bankroll that is 275 on one football game, which should already feel wrong to anybody who has read bankroll management.

Why Full Kelly Is Dangerous Here

Kelly assumes p is known. In sports betting it is estimated, and the formula is extremely sensitive to error.

An estimate two points too high roughly doubles the recommended stake. Since most bettors overestimate their edge, Kelly applied honestly to dishonest inputs produces catastrophic sizing.

Even with perfect inputs, full Kelly drawdowns are severe. A full-Kelly bettor with a genuine edge should expect to be down 50% from a peak at some point, which almost nobody tolerates without abandoning the plan.

Fractional Kelly

The standard practical answer is to stake a fixed fraction of the Kelly recommendation — typically a quarter or an eighth.

Quarter Kelly on the 5.5% example gives 1.375% of bankroll, which is squarely in the range a simple unit scheme would have produced. That convergence is the useful insight: conventional 1-to-2% unit sizing is approximately quarter-Kelly for the edges realistically available in sports betting.

Fractional Kelly retains most of the growth — half Kelly captures about 75% of the growth rate at far lower variance — while being much more forgiving of estimation error. If your edge estimate is wrong by a factor of two, a quarter-Kelly stake is still survivable; a full-Kelly stake is not.

When Kelly Thinking Is Genuinely Useful

Not as a stake calculator, but as a way to reason about relative sizing.

Kelly correctly says that stake should scale with edge and inversely with odds. A 5% edge on a -110 bet warrants more than a 5% edge on a +400 bet, because the latter's variance is far higher. That relationship is real and most bettors get it backwards, sizing up on long shots because the payout is exciting.

It also formalises something worth internalising: at zero edge the optimal stake is zero. There is no Kelly stake for a bet you have no edge on, which is a more rigorous statement of why betting for action loses money. See expected value and variance and drawdowns.

Frequently Asked Questions