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Kelly Criterion: The Mathematics of Optimal Bet Sizing
Published: June 2026 · 6 min read
Having a positive-expected-value (EV) bet is only half the battle. The question every bettor must answer: how much should I stake? Too little and you leave profit on the table. Too much and you risk ruin. The Kelly criterion, developed by John L. Kelly Jr. at Bell Labs in 1956, provides the mathematically optimal answer.
1. The Core Insight: Maximize Growth, Not Wealth
Kelly's key insight: the goal isn't to maximize expected wealth (which would mean betting everything on any positive-EV opportunity). The goal is to maximize the expected logarithm of wealth — the geometric growth rate. This automatically prevents ruin because log(0) = −∞.
Why Log?
If you win 50% of your bankroll and lose 50% on alternating bets: starting with $100 → $150 (win) → $75 (lose) → $112.5 → $56.25 ... despite 50% win rate, you go broke. The arithmetic mean return is 0%, but the geometric mean is −13.4%. Kelly's log-wealth criterion accounts for this compounding effect.
2. The Kelly Formula for Sports Betting
For a bet with decimal odds b (the amount returned per unit staked, including the stake) and win probability p:
f* = fraction of bankroll to bet. If f* ≤ 0, do not bet.
Alternatively, expressed in terms of expected value (EV) where EV = p × b − 1:
This is the version FootballMoney uses in every AI report's Layer 1 recommendation.
3. Worked Examples
Example 1: Clear Value
EV = 0.55 × 2.10 − 1 = 1.155 − 1 = +0.155 (15.5%)
Kelly f* = 0.155 / (2.10 − 1) = 0.155 / 1.10 = 14.1% of bankroll
Half-Kelly = 7.05% of bankroll
Example 2: Marginal Edge
EV = 0.52 × 2.00 − 1 = +0.04 (4%)
Kelly f* = 0.04 / 1.00 = 4% of bankroll
Half-Kelly = 2% of bankroll
Example 3: Negative EV — Do Not Bet
EV = 0.52 × 1.80 − 1 = −0.064 (−6.4%)
Kelly f* = −0.064 / 0.80 = −8.0% → Do not bet
4. Why Half-Kelly?
Full Kelly assumes your probability estimates are perfect. They never are. Three practical reasons to use half-Kelly:
- Estimation error: Your model says 55% but the true probability might be 52%. Full Kelly at 14.1% becomes reckless if the edge is smaller than estimated. Half-Kelly at 7% provides a margin of safety.
- Psychological: Watching 14% of your bankroll disappear on a single loss is stressful. Half-Kelly reduces drawdowns by 25% while retaining 75% of the expected growth rate.
- Fat tails: Sports outcomes have fatter tails than the binomial model assumes. Unexpected events (red cards, VAR decisions, weather) create variance beyond what standard probability models capture.
The Growth-Variance Tradeoff
| Full Kelly | 100% growth rate | 100% variance |
| Half Kelly | ~75% growth rate | ~50% variance |
| Quarter Kelly | ~44% growth rate | ~25% variance |
5. Practical Guidelines
- Cap at 25%: Never bet more than 25% of bankroll on a single wager, regardless of Kelly.
- Rebalance: Recalculate your bankroll after each bet. Kelly assumes continuous rebalancing.
- No chasing: A loss doesn't change the next bet's EV. Don't increase stakes to "recover."
- Simultaneous bets: If betting on multiple matches simultaneously, reduce each individual Kelly fraction to account for correlated risk.
- Minimum edge threshold: We only recommend bets where half-Kelly ≥ 1% of bankroll. Below that, transaction costs and estimation noise dominate.
6. How FootballMoney Uses Kelly
Every AI report's Layer 1: Quick Recommendation includes the Kelly calculation:
Kelly f* = Edge / (Odds − 1)
Half-Kelly = f* / 2 (recommended)
When Edge is negative, Kelly is negative — our AI explicitly states "do not bet" or "negative EV, avoid."
The combination of ELO ratings for probability estimation, Bayesian Poisson for score distribution, and Kelly criterion for position sizing creates a complete quantitative betting framework — from data to probability to action.
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