The Kelly Criterion is a mathematical formula for calculating the optimal bet size when you have an edge. The formula is f* = (bp - q) / b, where f* is the fraction of your bankroll to wager, b is the decimal odds minus 1, p is your win probability, and q = 1 - p. Developed by John L. Kelly Jr. at Bell Labs in 1956 (original paper), it maximizes long-term bankroll growth by betting proportionally to your edge — more when the edge is large, less when it is small. For sports betting at standard -110 odds with a 55% win probability, Kelly recommends wagering approximately 5.5% of your bankroll. Most professional bettors use fractional Kelly (25-50% of the full recommendation) to reduce variance while preserving most of the growth rate.
What Is the Kelly Criterion?
The Kelly Criterion is a formula for determining the optimal size of a bet when you have an edge. It was developed in 1956 by John L. Kelly Jr., a researcher at Bell Labs, in a paper titled "A New Interpretation of Information Rate." Kelly was originally solving a problem about signal transmission over noisy telephone lines, but the mathematics turned out to be directly applicable to any situation involving repeated bets with a known edge: horse racing, blackjack, stock trading, and sports betting.
The core insight is deceptively simple: if you have an edge, you should bet more when your edge is larger and less when it is smaller. Bet too little and you leave money on the table. Bet too much and you risk catastrophic drawdowns that can wipe out your bankroll even when your underlying edge is real. The Kelly formula finds the exact sweet spot that maximizes the long-term geometric growth rate of your bankroll.
The formula was famously adopted by Edward Thorp, who used it alongside his card-counting system to beat blackjack in the 1960s (documented in his book "Beat the Dealer") and later applied it to the stock market with extraordinary success. Today, every quantitative hedge fund, professional poker player, and serious sports bettor uses some form of Kelly-based sizing.
The Kelly Formula
The Kelly formula for sports betting is:
f* = (bp - q) / b
Where:
- f* = the fraction of your bankroll to bet (Kelly percentage)
- b = the decimal odds minus 1 (net profit per dollar wagered)
- p = your estimated probability of winning the bet
- q = the probability of losing = 1 - p
Let us break this down with a clear example.
Worked Example: Standard -110 Bet
You are betting on a spread at standard -110 American odds, and your Monte Carlo model estimates a 55% win probability.
First, convert American odds to the net payout multiplier (b):
Step 1: Convert -110 to decimal odds
For negative American odds: decimal = 1 + (100 / |odds|) = 1 + (100 / 110) = 1.909
Step 2: Calculate b (net payout)
b = decimal - 1 = 1.909 - 1 = 0.909
Step 3: Define p and q
p = 0.55 (your estimated win probability)
q = 1 - 0.55 = 0.45
Step 4: Apply Kelly formula
f* = (0.909 × 0.55 - 0.45) / 0.909
f* = (0.500 - 0.45) / 0.909
f* = 0.050 / 0.909
f* = 0.055 = 5.5%
Kelly recommends wagering 5.5% of your bankroll on this bet. On a $10,000 bankroll, that is $550.
Notice what happens if your probability estimate changes. At 52% (barely above breakeven for -110 odds), Kelly recommends only 1.7%. At 60%, it recommends 11.0%. The formula automatically scales bet size to the strength of the edge, which is precisely the behavior you want.
When Kelly Is Negative
If the Kelly formula returns a negative number, it means you do not have an edge. Do not bet. For -110 odds, the breakeven probability is 52.4% (accounting for the vig). Any estimated probability below that produces a negative Kelly, correctly telling you to stay away.
Full Kelly vs. Fractional Kelly
Full Kelly maximizes the theoretical long-term growth rate of your bankroll. In practice, however, almost no professional uses full Kelly. Here is why:
- Probability estimates are uncertain. Your Monte Carlo model says 55%, but the true probability might be 53% or 57%. Full Kelly assumes your probability estimate is perfectly accurate. If you are even slightly overconfident, full Kelly oversizes your bets, and the compounding effect of oversizing is devastating over hundreds of bets.
- Drawdowns are severe. Full Kelly produces a theoretical maximum drawdown that approaches 100% of bankroll with enough bets. The median bettor using full Kelly will experience drawdowns of 50% or more, which is psychologically and financially unsustainable.
- Variance is extreme. The path to long-term profit under full Kelly is wild. You might be down 40% after 200 bets and up 300% after 500. Most people do not have the psychological resilience or bankroll depth to survive the variance.
The solution is fractional Kelly: multiply the Kelly recommendation by a fraction.
| Strategy | Multiplier | 5.5% Kelly Becomes | Drawdown Risk | Growth Rate |
|---|---|---|---|---|
| Full Kelly | 1.0x | 5.5% | Very High | Maximum (theoretical) |
| Half Kelly | 0.5x | 2.75% | Moderate | 75% of max |
| Quarter Kelly | 0.25x | 1.375% | Low | 50% of max |
The mathematical insight that makes fractional Kelly so attractive: half Kelly achieves 75% of the growth rate with dramatically less variance. You sacrifice only 25% of your theoretical growth to gain a much smoother, more survivable equity curve. Most professionals use between quarter Kelly and half Kelly.
Converting Kelly Percentage to Unit Sizing
Most sports bettors do not think in exact bankroll percentages. They think in "units," a standardized measure where 1 unit = a standard bet size (typically 1-2% of bankroll). The Kelly percentage can be mapped to a unit system for practical use:
| Kelly % | Unit Size | Confidence Level |
|---|---|---|
| 0 - 1% | 0.5 units | Minimum threshold |
| 1 - 3% | 1.0 unit | Standard play |
| 3 - 6% | 1.5 units | Above average edge |
| 6 - 10% | 2.0 units | Strong edge |
| 10 - 15% | 2.5 units | Very strong edge |
| 15%+ | 3.0 units | Maximum (capped) |
Note the cap at 3.0 units. Even when Kelly suggests a very large bet, professional-grade systems impose a hard maximum to protect against model error. If your model says there is a 20% Kelly edge, something might be wrong with your model rather than right with the bet.
League-Specific Caps and Why They Matter
Different sports have different levels of inherent predictability. NBA games, with 48 minutes of continuous play and many possessions, produce more stable outcomes than NHL games, where a single fortunate bounce can decide the result. This means the appropriate maximum bet size varies by league:
| League | Max Units | Rationale |
|---|---|---|
| NBA | 3.0u | High possession count, more predictable |
| NFL | 3.0u | Deep data, but single-game variance |
| CBB | 2.5u | Shorter games, larger talent disparities |
| NHL | 2.5u | Low-scoring, goaltending variance |
| Soccer | 2.0u | Very low scoring, draws common |
| MLB | 3.0u | 162-game season, deep data |
These caps exist because model accuracy is not uniform across sports. A 60% probability estimate in the NBA, derived from a possession-level Monte Carlo simulation with deep player data, is more reliable than a 60% estimate in soccer, where a single deflection can change the result. The caps reflect this epistemic humility.
Common Mistakes with Kelly Criterion
Using Raw Model Probabilities
This is the most dangerous mistake. If your Monte Carlo model says 65% and the true probability is 60%, full Kelly will oversize your bet by roughly 40%. Over hundreds of bets, this compounds into significant losses. The solution is Bayesian probability calibration before Kelly sizing, which corrects for systematic overconfidence in model outputs.
Ignoring the Vig
The sportsbook's vig (commission) is real and must be accounted for in the odds conversion. A bet at -110 does not pay even money. It pays $0.909 per dollar risked. Using the vig-adjusted decimal odds in the Kelly formula automatically accounts for this, but some bettors accidentally use the raw implied probability without removing the vig, which understates the true breakeven threshold.
Not Updating Bankroll
Kelly is designed for a dynamically adjusting bankroll. After a winning day, your bankroll is larger and your next bet should be larger in dollar terms (but the same percentage). After a losing day, the opposite. Some bettors calculate Kelly once on their starting bankroll and use that flat dollar amount forever, which defeats the purpose of Kelly's compounding advantage.
Betting Correlated Outcomes at Full Kelly
If you bet full Kelly on five NBA games in the same night, and all five share a correlated factor (e.g., all are unders on a night when referees are calling fewer fouls league-wide), your effective bet size is much larger than any single Kelly recommendation implied. The standard practice is to reduce each individual bet when you have multiple correlated bets in play.
Kelly Criterion vs. Flat Betting
Flat betting means wagering the same dollar amount on every bet, regardless of edge size. It is the simplest possible strategy. How does Kelly compare?
Flat Betting
Wager $100 on every bet. Simple. Easy to track. But mathematically suboptimal: you bet the same on a 52% edge as a 62% edge, wasting edge when it is large and over-risking when it is small.
Kelly Betting
Wager proportional to edge size. Mathematically optimal for long-term growth. Requires accurate probability estimates. More complex to implement. But over 500+ bets, Kelly-sized bettors accumulate significantly more profit than flat bettors with the same win rate.
A concrete comparison: assume a bettor with a consistent 3% edge at -110 odds over 1,000 bets starting with a $10,000 bankroll. A flat bettor wagering 2% per bet ($200 fixed) would expect to finish with approximately $11,800. A quarter-Kelly bettor would expect approximately $13,400. A half-Kelly bettor would expect approximately $17,100. The same edge, the same bets, but Kelly's proportional sizing compounds more effectively.
The caveat: Kelly's advantage assumes accurate probability estimates. If the "3% edge" is actually a 1% edge due to model overconfidence, the flat bettor loses less than the full Kelly bettor. This is another reason to use fractional Kelly and always calibrate your probabilities before sizing.
Bayesian Shrinkage Before Kelly
Raw model probabilities from Monte Carlo simulations are almost always overconfident. A model that predicts 70% win probability for a team will typically see that team win about 64-66% of the time. This overconfidence, even if it is only 4-6 percentage points, causes Kelly to systematically oversize bets.
The solution is Bayesian shrinkage: pulling the raw probability toward a prior (typically 50%, representing no edge) before feeding it to the Kelly formula. The shrinkage formula is:
Shrunk Probability = (model_prob × w) + (0.50 × (1 - w))
Where w is the weight given to the model (typically 0.85, meaning 15% shrinkage toward 50%).
Example: Model says 70%. Shrunk = 0.70 × 0.85 + 0.50 × 0.15 = 0.595 + 0.075 = 67.0%
This 3-percentage-point reduction might seem small, but its impact on Kelly sizing is substantial. At -110 odds, a raw 70% produces a Kelly of 19.2% of bankroll. A calibrated 67% produces 14.7%. That is a 24% reduction in bet size, which provides a critical buffer against model overconfidence.
Read the full Bayesian calibration guide for a deeper treatment of why models are overconfident and how to correct for it.
How Olympus Bets Implements Kelly
At Olympus Bets, Kelly Criterion is the final step in a multi-stage pipeline:
- Monte Carlo simulation produces raw win/cover/total probabilities for each game
- Bayesian calibration applies Platt scaling and ensemble stacking to correct overconfidence
- Edge detection compares calibrated probabilities to sportsbook implied probabilities
- Kelly sizing converts the edge into an optimal bet fraction, then maps to our unit system
- League caps enforce maximum unit sizes per sport
- Profitability zone gating blocks bets in historically unprofitable sub-niches regardless of Kelly size
This layered approach ensures that Kelly never sees a raw, overconfident probability estimate. By the time a probability reaches the Kelly formula, it has been calibrated, shrunk, and validated against historical performance patterns. The result is a unit recommendation that reflects genuine edge, not model hubris.
Further Reading
- Monte Carlo Simulation in Sports Betting — how the probability estimates that feed Kelly are generated
- Bayesian Probability Calibration — why raw model probabilities must be calibrated before Kelly sizing
- Kelly Criterion Calculator — interactive tool with full/half/quarter Kelly and unit mapping
- Our Methodology — full technical overview of the Olympus Bets platform