Kelly Criterion for Optimal Bet Sizing

The Kelly Criterion is a mathematical formula that calculates the optimal bet size to maximize bankroll growth over time. Developed by John Kelly in 1956, it's used by professional gamblers, investors, and traders worldwide. Kelly betting ensures you bet aggressively when you have a strong edge, conservatively when edge is small, and not at all when edge is negative. Understanding and applying Kelly separates professional bettors from everyone else.

The Kelly Formula

Kelly Percentage = (bp - q) / b

Where:
- b = decimal odds - 1 (so +200 American = 2.0 - 1 = 1)
- p = probability of winning
- q = probability of losing (1 - p)

Example:
Odds: +200 (3.0 decimal, so b = 2)
Your assessed win probability: 40% (p = 0.4, q = 0.6)

Kelly% = (2 × 0.4 - 0.6) / 2 = (0.8 - 0.6) / 2 = 0.2 / 2 = 0.1 = 10%

Kelly says bet 10% of your bankroll.

With $1,000 bankroll: Bet $100
With $10,000 bankroll: Bet $1,000

Why Kelly Works

Kelly maximizes the logarithm of wealth, which means:

1. Fastest bankroll growth possible without risk of ruin
2. Aggressive when edge is large, conservative when edge is small
3. Automatically adjusts bet size as bankroll changes
4. Never bets when expected value is zero or negative
5. Proportional to both edge AND odds

Key insight: Kelly accounts for variance. At +200 odds, you need less edge to bet big than at -200 odds because upside is much higher. The formula perfectly balances risk and reward.

Full Kelly vs Fractional Kelly

Full Kelly is mathematically optimal but extremely aggressive:

Problems with Full Kelly:
- Requires perfect probability assessment (impossible)
- Huge swings in bankroll (50%+ drawdowns possible)
- Psychological difficulty maintaining discipline
- Small errors in probability cause over-betting
- Assumes you only have one bet at a time

Fractional Kelly solves this:

Half Kelly (Most Popular):
- Bet 50% of Full Kelly recommendation
- Much smoother variance
- 75% of Full Kelly growth rate
- Tolerates probability errors better

Quarter Kelly (Conservative):
- Bet 25% of Full Kelly
- Very smooth variance
- ~50% of Full Kelly growth
- Suitable for risk-averse or newer bettors

Most professionals use Half Kelly or less.

Practical Kelly Application

Real-world Kelly betting:

Scenario 1: Strong edge
Odds: +150 (2.5 decimal, b = 1.5)
Your probability: 50%
Kelly = (1.5 × 0.5 - 0.5) / 1.5 = 0.25 / 1.5 = 16.7%
Half Kelly = 8.35% of bankroll

$5,000 bankroll = $417 bet

Scenario 2: Moderate edge
Odds: -110 (1.91 decimal, b = 0.91)
Your probability: 55%
Kelly = (0.91 × 0.55 - 0.45) / 0.91 = 0.0505 / 0.91 = 5.5%
Half Kelly = 2.75% of bankroll

$5,000 bankroll = $137 bet

Scenario 3: No edge
Odds: +100
Your probability: 50% (same as implied)
Kelly = (1 × 0.5 - 0.5) / 1 = 0 = 0%

Kelly correctly says don't bet.

Common Kelly Mistakes

Errors that destroy Kelly effectiveness:

• Overestimating win probability: Leads to over-betting and ruin
• Using Kelly for every bet: Only works if bets are uncorrelated
• Not adjusting for multiple simultaneous bets: Total Kelly exposure can exceed 100%
• Ignoring psychological tolerance: Full Kelly is mathematically optimal but mentally unbearable
• Failing to recalculate as bankroll changes: Kelly is dynamic, not static
• Applying to -EV bets: Kelly assumes +EV; using on -EV destroys bankroll
• Using Full Kelly with imperfect information: Always use fractional with estimated probabilities

Kelly with Multiple Bets

The Kelly formula assumes one bet at a time. With multiple simultaneous bets:

Approach 1: Reduced Kelly
- Use Quarter Kelly when making multiple bets
- Accounts for correlation and compounding risk
- More conservative but safer

Approach 2: Portfolio Kelly
- Calculate Kelly for entire portfolio
- Complex but optimal for correlated bets
- Requires advanced modeling

Approach 3: Cap Total Exposure
- Calculate Kelly for each bet individually
- Ensure total Kelly allocation never exceeds 25-50% of bankroll
- Simple and practical

Example: Three bets each showing 10% Kelly
- Don't bet 10% on each (30% total exposure)
- Bet 5% on each (15% total) or
- Prioritize the best one at 10%, others at 5%

Implementing Kelly Successfully

Start with Quarter Kelly until you build confidence in your probability assessments. Track every bet with your assessed probability and actual results to calibrate your skill. Adjust Kelly fraction based on your psychological tolerance - if you can't sleep, reduce the fraction. Remember that Kelly assumes you can accurately assess probabilities - if you can't, you're just guessing aggressively. Use Kelly as a guide, not a rule - round down, not up. The difference between 8% and 10% Kelly is significant, but 8% and 7% is negligible. Most importantly: Kelly prevents over-betting, which is the #1 way bettors go broke. Even if you never use the formula exactly, understanding Kelly principles will dramatically improve your bet sizing discipline and long-term profitability.