The Kelly Criterion: Optimal Bet Sizing for Sports Bettors

What Is the Kelly Criterion?

The Kelly Criterion is a formula that tells you what fraction of your bankroll to bet, given your edge. It maximizes the expected logarithmic growth of your bankroll — meaning it grows your money as fast as possible while managing risk of ruin.

The Formula

For a bet with decimal odds $d$ and true win probability $p$:

$$f^* = \frac{bp - q}{b}$$

where:

• $b = d - 1$ (the profit per unit bet)
• $p$ = probability of winning
• $q = 1 - p$ = probability of losing
• $f^*$ = fraction of bankroll to bet

Example

You find a bet at +150 (decimal 2.50) and estimate a 45% chance of winning:

$$b = 2.50 - 1 = 1.50$$ $$f^* = \frac{1.50 \times 0.45 - 0.55}{1.50} = \frac{0.675 - 0.55}{1.50} = \frac{0.125}{1.50} = 8.33\%$$

With a $10,000 bankroll, you'd bet $833.

Why Full Kelly Is Dangerous

Full Kelly maximizes growth rate but creates enormous variance. In practice:

• A 50% drawdown is almost certain over a long career
• The formula assumes you know the exact true probability — you don't
• Overestimating your edge by even a small amount leads to catastrophic overbetting

Fractional Kelly

Most sharp bettors use a fraction of Kelly:

Strategy	Fraction	Growth Rate	Variance
Full Kelly	100%	Maximum	Very high
3/4 Kelly	75%	~94% of max	Moderate
Half Kelly	50%	~75% of max	Low
Quarter Kelly	25%	~44% of max	Very low

Half Kelly is the most popular choice. You sacrifice 25% of optimal growth but reduce variance dramatically.

The Edge Requirement

Kelly only recommends a bet when you have positive expected value. If $f^* \leq 0$, don't bet:

$$f^* > 0 \iff bp > q \iff p > \frac{1}{d}$$

In other words, your estimated win probability must exceed the implied probability from the odds.

Practical Guidelines

1. Never bet full Kelly unless you have extremely accurate probability estimates
2. Half Kelly is a good default — it captures most of the growth with much less risk
3. Track your CLV (closing line value) to validate your probability estimates over time
4. Reduce Kelly fraction when uncertain — if you're unsure of your edge, bet smaller
5. Don't Kelly multiple correlated bets — the formula assumes independent events

The Connection to Expected Value

Kelly and EV are related but different:

$$\text{EV} = p \cdot b - q = b \cdot f^*$$

A positive EV bet always has positive Kelly. But Kelly also accounts for your bankroll — it tells you not just whether to bet, but how much.

A 1% edge on a -110 bet suggests a small Kelly fraction. A 10% edge on the same bet suggests a much larger one. EV alone doesn't capture this.