Kelly Criterion

Kelly Criterion
How mathematical bankroll management maximizes long-term betting growth

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📘 Definition

The Kelly Criterion is a mathematical formula used to determine the optimal size of a bet relative to your bankroll when you believe you have an edge. Developed by John L. Kelly Jr. in 1956 for telecommunications signal optimization, it was later adopted by gamblers and investors as a system for maximizing long-term capital growth while minimizing the risk of total ruin.

In sports betting, the Kelly Criterion helps bettors calculate exactly how much of their bankroll to wager on a given bet when their assessed probability of winning differs from the bookmaker’s implied probability. It is not about choosing who to bet on but about sizing bets correctly to balance risk and reward.

When applied consistently and correctly, Kelly outperforms flat staking or arbitrary bet sizing because it is grounded in probability theory and logarithmic wealth growth. However, it requires accurate probability estimates—if your edge is miscalculated, Kelly can magnify mistakes.

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🧮 Structure

The core Kelly formula for decimal odds is:

f∗=bp−qbf^* = \frac{bp – q}{b}f∗=bbp−q​

Where:

• f* = fraction of bankroll to stake

• b = decimal odds minus 1 (e.g., 2.50 odds → b = 1.5)

• p = your estimated probability of winning

• q = 1 – p (probability of losing)

Example calculation:

• Odds: 2.50 (implied probability 40%)

• Your estimate: 50% chance of winning

• b = 1.5, p = 0.50, q = 0.50

• Kelly fraction: (1.5 × 0.50 – 0.50) ÷ 1.5 = 0.25 ÷ 1.5 = 0.1667

• Recommendation: Bet 16.7% of bankroll.

This fraction scales with edge:

• Bigger edge → larger stake

• Smaller edge → smaller stake

• Negative edge → no bet

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🎯 In Practice

The Kelly Criterion is widely used in sports betting, poker, and financial trading. Professional bettors rely on it to size wagers in proportion to their perceived advantage.

Scenario 1: Small Edge

• Odds: 1.91 (implied 52.4%)

• Your model: 55% win chance

• Kelly fraction: (0.91 × 0.55 – 0.45) ÷ 0.91 ≈ 0.032

• Bet size: 3.2% of bankroll.

Scenario 2: Big Edge

• Odds: 3.00 (implied 33.3%)

• Your model: 45% win chance

• Kelly fraction ≈ 0.18 → Bet 18% of bankroll.

Kelly naturally adjusts bet size to the strength of your edge.

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🔢 Example Bet

Bankroll: €5,000
Bet: Tennis match, underdog priced at 3.20 (implied 31.25%).
Your analysis: 40% chance of winning.

• b = 2.20, p = 0.40, q = 0.60

• Kelly = (2.20 × 0.40 – 0.60) ÷ 2.20 = 0.28 ÷ 2.20 = 0.127

• Bet size = 12.7% of bankroll → €635

If the player wins, payout = €2,032 (profit €1,397).
If the player loses, bankroll drops by €635 but remains intact.

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💸 Pros and Cons

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💡 Strategy Tips

1. Use Fractional Kelly

   • Many bettors use half-Kelly (50%) or quarter-Kelly to reduce volatility while keeping growth.

2. Be conservative with probabilities

   • If your model is uncertain, scale down Kelly fractions to avoid oversized bets.

3. Track your edge

   • Only use Kelly when confident in your advantage. Avoid applying it blindly.

4. Apply consistently

   • Kelly only works long term if applied rigorously across many bets. Sporadic use removes its benefits.

5. Compare with flat staking

   • Flat stakes simplify risk but ignore edge size. Kelly ensures edges are weighted appropriately.

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📊 Best Use Cases

• Value betting: When you have strong models showing clear edges over implied probabilities.

• Futures markets: Longer odds bets benefit from correct bet sizing to avoid overexposure.

• Poker bankrolls: Kelly is widely applied to bankroll management in poker tournaments.

• Professional betting syndicates: Groups using data models rely on Kelly for sustainable growth.

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⚠️ Common Mistakes

• Overconfidence: Plugging in unrealistic win probabilities leads to massive over-bets.

• Misusing Kelly on small bankrolls: Volatility can wipe out casual bettors.

• Ignoring fractional Kelly: Full Kelly is mathematically optimal but psychologically brutal.

• Confusing Kelly with “staking plans”: It is not a progression system but a probability-based formula.

• Applying to coin-flip odds without edge: Kelly will show zero stake—forcing discipline not to bet.

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📌 Summary