Kelly criterion for football betting

Kelly tells you how much of your bankroll to stake. Getting the fraction wrong in either direction hurts. Getting it wrong upward kills you. Getting it wrong downward just slows you down.

This piece covers the formula, a worked example using real Premier League odds, and the three separate arguments for why sharp bettors almost never use full Kelly in practice.

TL;DR

Formula, decimal odds: f* = (bp − q) / b, where b = odds − 1, p = your win probability, q = 1 − p.
The formula gives the growth-rate-optimal stake assuming you know p exactly.
You don't. Fractional Kelly (half, quarter, or less) is the rational response to estimation error, not timidity.
Overstaking is asymmetrically damaging. Betting twice the Kelly fraction produces zero expected growth; more than twice turns growth negative.

The formula

For a single bet at decimal odds, the Kelly fraction is:

f* = (bp − q) / b

Where b is the decimal odds minus one (your net profit per unit staked if the bet wins), p is your estimate of the win probability, and q = 1 − p. If f* comes out negative, the bet is not +EV and you don't place it. If positive, that's the fraction of your bankroll that maximises long-run exponential growth, given your inputs.

A quick textbook illustration. You think a coin lands heads with 60% probability, and you are offered even money (decimal odds 2.00). Here b = 1, p = 0.6, q = 0.4, so f* = (1 × 0.6 − 0.4) / 1 = 0.20. Kelly says stake 20% of your bankroll. Wikipedia's treatment of the same example is the standard form.

A real football example

Premier League, 13 April 2026. Manchester United hosted Leeds. Our model gave the match a 72.8% probability of the total corners going over 8.5. Paddy Power offered decimal odds of 1.50 on that line, which implies a 66.7% market probability.

Plug the numbers in:

f* = (0.50 × 0.728 − 0.272) / 0.50
   = (0.364 − 0.272) / 0.50
   = 0.092 / 0.50
   = 0.184

Full Kelly says stake 18.4% of your bankroll on a single corners over/under line. If you have £1,000 of money set aside for betting, that is £184 on a Tuesday night in April.

That is a lot. Kelly says it is the growth-rate-optimal amount, given those inputs. Quarter-Kelly would bring the stake to 4.6% of bankroll, or £46. Sixteenth-Kelly to 1.15%, or £11.50. The stake shrinks aggressively as the fraction shrinks, and for the reasons below, aggressive shrinking is usually the right call.

(We picked this fixture because the arithmetic is clean and the numbers are round. The question here is what Kelly says to stake, not whether the stake won or lost.)

Why fractional Kelly

The practical answer for sharp bettors is almost never "full Kelly." Fractional Kelly (half, quarter, or less) is standard. Three distinct arguments converge on the same conclusion, and all three apply to football.

Variance reduction

Half-Kelly retains most of full-Kelly's growth rate while cutting volatility substantially. MacLean, Thorp and Ziemba (2010) published a Kelly-fraction-versus-relative-growth table for blackjack with a 2% edge (p = 0.51, q = 0.49):

Kelly fraction	Relative growth rate
0.5	0.75
1.0 (full)	1.00
2.0	0.00

Half-Kelly retains 75% of full-Kelly's growth rate. You sacrifice a quarter of the theoretical upside to halve your volatility.

Buchdahl has computed the same trade-off for football bettors specifically. His simulations show the probability of seeing any profit after a long sequence of bets goes from 66% at full Kelly to 73% at half Kelly, while the most likely bankroll growth drops from 41% to 29%. A meaningful lift in "am I in the black?" for a modest cut in expected growth.

Model uncertainty

This is the argument that matters most for any real bettor, and the one the Kelly formula itself does not contain.

The formula f* = (bp − q) / b assumes you know p. You don't. You have an estimate of p, produced by some model, and that estimate has error.

Take the Manchester United example. Our model said 72.8%. Suppose the true probability was 70%, a 2.8 percentage-point miss that is well within normal model noise for corners markets. Full Kelly now says (0.50 × 0.70 − 0.30) / 0.50 = 0.10, or 10% of bankroll. Almost half of the 18.4% the model suggested. If the true probability was 65%, inside the bookmaker's implied 66.7%, the bet is not +EV at all and Kelly says do not bet.

The implication is sharp. Kelly-sized stakes are hyper-sensitive to errors in p, and errors in p are the norm, not the exception. MacLean, Thorp and Ziemba state the sensitivity directly: "Given the extreme sensitivity of E log calculations to errors in mean estimates, these estimates must be accurate and to be on the safe side, the size of the wagers should be reduced."

Buchdahl puts the same point in football terms: "if you overestimate the advantage that you think you hold over the bookmaker's odds, you can easily turn a winning system into an unprofitable one."

Fractional Kelly is the rational response to that fact. Half-Kelly, quarter-Kelly, or less is not paranoia. It is the mathematically appropriate haircut to apply when you know your probability estimates are noisy.

Drawdown tolerance

Full Kelly is brutal on the way to being right. A standard result under the log-optimal assumption: the probability that your bankroll drops to X% of its starting value at some point is approximately X%. Fifty-percent chance of a 50% drawdown. Twenty-percent chance of an 80% drawdown. These are not tail events. They are baseline outcomes for full Kelly over a long enough run.

Wikipedia's Kelly entry puts it plainly: "gamblers in practice find fractional Kellies much better emotionally than full Kelly."

Emotional tolerance is not window-dressing. A bettor who cannot sit through a 50% drawdown without abandoning the method has effectively blown up their strategy, whatever the long-run maths says. Fractional Kelly trades some theoretical growth for a drawdown profile you can actually sit through.

Choosing a fraction

There is no formula for picking the right fraction. It depends on how noisy you think your probability estimates are, how much drawdown you can tolerate, and how correlated your bets tend to be. A reasonable starting heuristic for football bettors:

Half-Kelly if your model has been calibrated against closing lines for at least a few hundred bets and the calibration is tight. You are still exposed to tail drawdowns but the profit-probability lift is meaningful.
Quarter-Kelly if you have a working model but limited live validation, or if your probability estimates are known to be noisier than the bookmaker's closing prices.
Eighth-Kelly or less if you are trialling a new model, running on sparse data, or betting derived markets (corners, cards, throw-ins) where the model uncertainty compounds.

The honest answer is that most sharp bettors tune their fraction down over time after seeing how their model actually performs live. Starting conservative and increasing the fraction as evidence accumulates is always safer than the reverse.

Overstaking is asymmetric

The three arguments above are mostly about making fractional Kelly feel rational. There is also a sharp mathematical reason the errors that come from overstaking are worse than the errors from understaking.

In continuous time with a geometric Wiener process, betting exactly twice the Kelly fraction produces zero expected growth plus the risk-free rate. The result is attributed to Thorp (1997), Stutzer (1998) and Janacek (1998), and the Markowitz-style proof appears in the appendix of MacLean, Thorp and Ziemba (2010). Betting more than twice Kelly produces negative expected growth. Your bankroll is expected to shrink, even though every bet you are placing was genuinely +EV at the moment of placing.

Half the full Kelly fraction gives you 75% of the growth and roughly half the variance. Double the full Kelly fraction gives you zero growth and roughly double the variance. Those two errors are not symmetric.

MacLean, Thorp and Ziemba state the asymmetry flatly: "As you exceed the Kelly bets more and more, risk increases and long term growth falls, eventually becoming more and more negative. Long Term Capital is one of many real world instances in which overbetting led to disaster."

Understaking just slows you down. Overstaking compounds against you.

How Spectral stakes

This is a methodology statement, not a performance claim. For the current track-record numbers, see /track-record.

Our scanner flags any pick with at least 5% edge over fair odds. For stake sizing, we do not use full Kelly on any tier:

Confidence tier	Kelly fraction	Multiplier on full Kelly
High	0.25	quarter
Medium	0.0625	sixteenth
Low	0	display only, no stake suggested

Tier assignment is not static. When a league's calibration drifts, or when our recent realised edge in that league drops to noise level, the confidence tier itself gets demoted upstream. A pick that would have carried high confidence during a well-calibrated window may be downgraded to medium, or demoted to low (where we publish the pick but suggest no stake), before it reaches the Kelly step. The stake shrinks because the tier shrinks, not because we overlay a second fractional adjustment.

It looks conservative. It is. Not because we are timid, but because of three things the formula itself ignores:

Our probability estimate for any given fixture has error we can measure. Medium-confidence picks have visibly wider calibration bands than high-confidence picks.
Corners markets carry more model uncertainty than headline 1X2 markets. We are pricing derived counts from possession, set-piece and tactical inputs, and the noise compounds.
The asymmetric overbetting result means a 2x Kelly error zeroes out long-run growth. A 0.25x Kelly error cuts it by 75% but leaves it clearly positive. The downside of conservatism is bounded; the downside of aggression is not.

Betting less than the formula recommends is the rational response to an estimate we know is imperfect. Betting exactly the formula recommends would assume the estimate is perfect. Betting twice the formula recommends would be the fastest way to turn a good model into a blown-up bankroll. The full derivation of these thresholds is on our /methodology page.

Common misconceptions

"Kelly maximises my returns." Kelly maximises the growth rate of your bankroll in the long run. It does not maximise any particular bettor's utility, and it does not make any single bet more likely to win. Plenty of bettors prefer a strategy with lower growth and much lower drawdown variance. Fractional Kelly is one way to express that preference.

"Kelly applies to parlays the same way." Not cleanly. Kelly assumes bets are independent. Placing five separate Kelly-sized bets simultaneously means your actual exposure is five times what Kelly intended, and those bets are often correlated (two picks on the same team, two picks on the same weekend's weather). Parlays violate Kelly's independence assumption structurally. The right way to size a parlay is to compute Kelly on the joint probability, which is much harder to estimate reliably.

"Kelly assumes I know p." Correct, and that is exactly the problem. Every real application of Kelly lives or dies on how well you have estimated p. If your p is unbiased but noisy, fractional Kelly corrects for the noise. If your p is biased upward (overconfident), fractional Kelly mitigates the damage but does not eliminate it. There is no substitute for validating p externally. Closing line value is one of the few credible tests of whether your probability estimates are actually sharper than the closing market.

"Simultaneous bets should each be Kelly-sized." No. If Kelly would have you staking 10% of bankroll on each of three picks at once, you are holding 30% of bankroll at risk simultaneously, which is not the growth-optimal exposure for the combined position. Multi-bet Kelly sizing requires jointly optimising the full portfolio, not each bet in isolation.