The Kelly Criterion is a mathematical formula for sizing bets in proportion to your edge. It was developed by John L. Kelly Jr. at Bell Labs in 1956 for signal-noise optimisation in telecommunications, but it translates directly to any situation where you wager capital on probabilistic outcomes — including sports betting.
This guide explains the formula, works through practical examples using real prediction-market data, and covers the fractional Kelly adjustments that experienced bettors use to manage variance.
Why Bet Sizing Matters
Most bettors focus entirely on finding an edge — the right pick, the right price. But even a profitable strategy will blow up if you stake too much on any single bet. Conversely, staking too little leaves money on the table.
The Kelly Criterion solves this by answering a specific question: given a known edge and known odds, what fraction of your bankroll should you stake to maximise long-term growth?
It balances two forces:
- Bet too big and a losing streak wipes you out before the edge compounds.
- Bet too small and your bankroll grows slower than it could.
The Formula
The standard Kelly formula for a simple win/lose bet is:
f* = (p × b − q) / b
Where:
- f* = fraction of bankroll to wager
- p = your estimated probability of winning
- b = net odds received (decimal odds minus 1)
- q = probability of losing (1 − p)
If f* is zero or negative, the Kelly Criterion says don't bet — there's no edge.
Worked Example: Premier League Match
Suppose you're looking at a Premier League match where:
- bet365 decimal odds on Home Win: 2.50 (implied probability: 40%)
- Your estimated true probability of Home Win: 50%
Here, p = 0.50, q = 0.50, b = 2.50 − 1 = 1.50.
| Variable | Value |
|---|---|
| p (true probability) | 0.50 |
| q (1 − p) | 0.50 |
| b (net odds) | 1.50 |
| f* = (0.50 × 1.50 − 0.50) / 1.50 | 0.167 |
The Kelly stake is 16.7% of bankroll. On a £1,000 bankroll, that's £167.
That's a large single bet — which leads to the most important practical adjustment.
Fractional Kelly
Full Kelly maximises the theoretical long-run growth rate, but it assumes your probability estimate is perfectly accurate. In practice, no one's probability estimates are perfect. Overestimate your edge slightly and full Kelly can lead to severe drawdowns.
Most professional bettors and fund managers use fractional Kelly — typically quarter-Kelly or half-Kelly:
| Kelly Fraction | Stake (£1k bankroll) | Growth Rate | Max Drawdown Risk |
|---|---|---|---|
| Full (1.0×) | £167 | Highest theoretical | High variance |
| Half (0.5×) | £83 | ~75% of full Kelly | Significantly lower |
| Quarter (0.25×) | £42 | ~50% of full Kelly | Much lower |
Half-Kelly retains roughly 75% of the growth rate of full Kelly but cuts variance dramatically. Quarter-Kelly is the most conservative option — you give up half the theoretical growth rate but your bankroll curve becomes far smoother.
Our recommendation: Start with quarter-Kelly. Move to half-Kelly only once you have a track record demonstrating that your probability estimates are well-calibrated.
Using Kelly with MarketGap Data
The Kelly Criterion becomes especially useful when you have two independent price sources disagreeing on the same event — which is exactly what MarketGap tracks.
When bet365 and Polymarket show a pricing gap, you can use the Polymarket price as your probability estimate (p) and the bet365 odds as your payout (b). The logic: if one market is mispriced, there's a quantifiable edge.
Example: MarketGap Opportunity
Suppose MarketGap flags the following:
- Match: Sunderland vs Fulham — Home Win
- bet365 odds: 3.00 (implied: 33.3%)
- Polymarket price: 45¢ (implied: 45.0%)
- Gap: 11.7 percentage points
Using the Kelly formula with Polymarket as our probability estimate:
| Variable | Value |
|---|---|
| p (Polymarket probability) | 0.45 |
| q (1 − p) | 0.55 |
| b (bet365 net odds) | 2.00 |
| f* = (0.45 × 2.00 − 0.55) / 2.00 | 0.175 |
Full Kelly says 17.5% of bankroll. At quarter-Kelly, that's 4.4% of bankroll — a manageable position.
You can run these numbers instantly using our Arbitrage Calculator, which includes Kelly sizing, expected return, break-even price, and a visual gap bar.
Expected Return and Break-Even
Two other numbers matter beyond Kelly stake size:
Expected return per unit staked:
EV = (p × b) − q
In our Sunderland example: EV = (0.45 × 2.00) − 0.55 = +0.35 units per unit staked (35% edge). That's a strong theoretical edge, though it depends entirely on the accuracy of the 45% probability estimate.
Break-even probability:
Break-even = 1 / decimal odds
At decimal odds of 3.00, break-even is 33.3%. Any true probability above 33.3% makes the bet positive-EV. The gap between break-even (33.3%) and your estimated probability (45%) is your margin of safety.
When Kelly Says Don't Bet
If the Kelly fraction comes out at zero or below, there is no mathematical edge and the formula says pass.
This happens when:
- The implied probability from odds already matches or exceeds your probability estimate
- The MarketGap gap is too narrow to overcome the bookmaker's margin
- The Polymarket price is below the bet365 implied probability (the gap favours the other side)
Discipline here matters more than the formula. The Kelly Criterion's biggest practical value isn't telling you how much to bet — it's telling you when not to bet at all.
Limitations
The Kelly Criterion is a powerful framework, but it has real limitations:
- Probability estimation error — Kelly assumes you know the true probability. You don't. This is why fractional Kelly is essential.
- Binary outcomes only — The standard formula covers win/lose. Three-way markets (home/draw/away) require modified versions.
- Correlated bets — Kelly assumes independent events. If you have multiple bets from the same match or dependent markets, you need to adjust.
- Bankroll definition — Kelly optimises for a fixed bankroll. If your "bankroll" is really money you need for other things, full Kelly is inappropriate regardless of edge size.
- Bookmaker limits — In practice, if a bookmaker consistently sees you sizing correctly according to Kelly, they may restrict your account. This is a real-world constraint the formula doesn't address.
Putting It Into Practice
Here's a practical workflow using WagerBase tools:
- Find a gap — Check MarketGap for pricing disagreements of 8%+ between bet365 and Polymarket.
- Assess the probability — Use the Polymarket price as a starting point, but consider whether it makes sense given what you know about the match.
- Size the bet — Enter the numbers into the Arbitrage Calculator. Use quarter-Kelly or half-Kelly.
- Check the break-even — If break-even is close to your probability estimate, the margin of safety is thin. Consider passing.
- Track results — Over time, compare your estimated probabilities against actual outcomes. If you're consistently overestimating, reduce your Kelly fraction further.
Key Takeaways
| Principle | Detail |
|---|---|
| The Kelly formula | f* = (p × b − q) / b |
| When to bet | Only when f* > 0 (positive edge) |
| Practical sizing | Use quarter-Kelly (0.25×) or half-Kelly (0.5×) |
| Where to find edge | MarketGap gaps of 8%+ between bet365 and Polymarket |
| Calculator | WagerBase Arbitrage Calculator |
The Kelly Criterion is a mathematical framework, not financial advice. All betting involves risk. Never stake more than you can afford to lose. Please gamble responsibly — visit BeGambleAware.org or call 0808 8020 133 for support.