Kelly Criterion Calculator
The Kelly Criterion is a mathematical formula for optimal bet sizing that maximizes long-term bankroll growth while managing risk. Originally developed by John L. Kelly Jr. at Bell Labs in 1956 for information theory applications, it has become a cornerstone of bankroll management in poker, sports betting, and investment theory.
This calculator helps you determine the optimal percentage of your bankroll to wager based on your edge and the odds offered. Understanding Kelly is essential for serious players who want to maximize growth while avoiding the risk of ruin. For related concepts, see our guides on bankroll management and expected value calculation.
Kelly Criterion Calculator
Your estimated probability of winning
Odds offered (e.g., 2.0 = even money)
Your total available bankroll
Conservative betting uses fractional Kelly
Strategy Recommendation
With a 5% edge at even money, the full Kelly suggests betting 10% of your bankroll. However, using half Kelly (5%) is recommended due to edge uncertainty. This provides 75% of the growth rate with significantly reduced variance and risk of ruin.
Kelly Fraction Comparison
Different Kelly fractions offer trade-offs between growth rate and risk. The table below shows how reducing Kelly fraction affects your long-term results:
| Kelly Fraction | Bet Size | Growth Rate | Risk of 50% Drawdown | Recommended For |
|---|---|---|---|---|
| Full Kelly (100%) | $1,000 | 100% | Very High | Perfect edge knowledge |
| Three-Quarter (75%) | $750 | 94% | High | Confident edge estimates |
| Half Kelly (50%) | $500 | 75% | Moderate | Most bettors (recommended) |
| Quarter Kelly (25%) | $250 | 44% | Low | Uncertain edge estimates |
| Tenth Kelly (10%) | $100 | 19% | Very Low | Maximum safety |
Understanding the Kelly Criterion
The Kelly Criterion was developed by John Kelly while working at Bell Labs on signal noise issues in long-distance telephone calls. He realized the same mathematical principles could optimize betting strategies. According to research published in the American Mathematical Society Notices, the formula has since been adopted by legendary investors like Warren Buffett's partner Charlie Munger and hedge fund managers.
The Basic Formula
The Kelly formula for determining optimal bet size is:
b = decimal odds minus 1 (net odds received on a 1:1 bet)
p = probability of winning
q = probability of losing (1 - p)
For example, if you have a 55% chance of winning at even money (2.0 decimal odds), the calculation is:
- b = 2.0 - 1 = 1.0
- p = 0.55, q = 0.45
- f* = (1.0 × 0.55 - 0.45) / 1.0 = 0.10 = 10%
Why Fractional Kelly?
Full Kelly betting assumes you know your exact edge, which is rarely true in practice. Research from ScienceDirect's Journal of Economic Dynamics and Control demonstrates that fractional Kelly approaches outperform full Kelly when edge estimates contain uncertainty.
Benefits of fractional Kelly (typically 25-50%):
- Reduced variance: Smoother bankroll growth with fewer large swings
- Error buffer: Protects against overestimating your edge
- Lower risk of ruin: Much smaller probability of catastrophic losses
- Psychological comfort: Easier to stick with during downswings
Kelly Criterion in Poker
In poker, Kelly applies differently than in fixed-odds betting. Rather than calculating bet sizes for individual hands, Kelly helps determine appropriate stakes relative to your bankroll. For more on poker-specific bankroll management, see our bankroll calculator and variance simulator.
Cash Game Application
For cash games, a modified Kelly approach considers your win rate and standard deviation. The formula becomes:
For a player with 5 bb/100 win rate and 80 bb/100 standard deviation: (5)² / (2 × 80²) = 25 / 12,800 = 0.002, suggesting risking about 0.2% of bankroll per session, or having 500 buy-ins. In practice, most players use 20-40 buy-ins with fractional Kelly adjustments.
Tournament Application
Tournaments have higher variance than cash games, so Kelly suggests even more conservative bankroll requirements. A strong tournament player with 30% ROI might still need 100+ buy-ins using Kelly principles. See our ICM calculator and tournament strategy guide for related concepts.
Common Kelly Mistakes
| Mistake | Problem | Solution |
|---|---|---|
| Overestimating edge | Leads to over-betting and high ruin risk | Use conservative edge estimates; apply half Kelly |
| Ignoring bankroll changes | Fixed bet sizes don't adjust to wins/losses | Recalculate Kelly after significant bankroll changes |
| Betting beyond Kelly | Expected growth decreases; ruin risk soars | Never exceed full Kelly; prefer fractional |
| Applying to single events | Kelly requires repeated bets to work | Use Kelly for long-term strategies, not one-offs |
| Ignoring correlation | Multiple correlated bets increase risk | Reduce total exposure when bets are correlated |
Kelly and Risk of Ruin
One of Kelly's key properties is that it theoretically eliminates the risk of ruin since you never bet your entire bankroll. However, in practice, several factors make full Kelly risky:
- Edge uncertainty: You rarely know your true edge precisely
- Minimum bet sizes: Real bets have minimums that may exceed fractional Kelly
- Practical ruin: Losing 90% of your bankroll is functionally equivalent to ruin for most people
- Psychological factors: Large swings affect decision-making quality
According to research from JSTOR's Management Science journal, half Kelly provides approximately 75% of the growth rate of full Kelly while substantially reducing drawdown risk.
Related Tools
Combine the Kelly Criterion with these other calculators for comprehensive bankroll analysis:
- Bankroll Calculator - Determine proper bankroll requirements for different stakes
- Expected Value Calculator - Calculate the profitability of specific poker decisions
- Variance Simulator - Visualize how variance affects your results over time
- Pot Odds Calculator - Compare pot odds to determine profitable calls
- Session Tracker - Track your win rate over time for accurate Kelly inputs
- ICM Calculator - Understand tournament equity for Kelly-based stake selection
Frequently Asked Questions
What is the Kelly Criterion?
The Kelly Criterion is a formula developed by John L. Kelly Jr. at Bell Labs in 1956 for determining the optimal size of a series of bets. It maximizes the expected geometric growth rate of your bankroll while minimizing the risk of ruin. The basic formula is: f* = (bp - q) / b, where f* is the fraction of bankroll to bet, b is the decimal odds minus 1, p is the probability of winning, and q is the probability of losing (1 - p).
Why use fractional Kelly instead of full Kelly?
Full Kelly betting is extremely aggressive and assumes perfect knowledge of your edge, which is rarely the case in real-world scenarios. Fractional Kelly (typically 25-50% of full Kelly) reduces variance and risk of ruin while still capturing most of the long-term growth. It provides a buffer for estimation errors in your edge calculation and creates a smoother, less volatile growth curve.
Can the Kelly Criterion be applied to poker?
Yes, but with important caveats. In poker, Kelly applies more to bankroll management decisions (choosing stakes) and tournament buy-ins rather than individual hand decisions. Your edge in poker is determined by your long-term win rate, and Kelly helps determine what stakes you can sustainably play. For cash games, many use a modified approach considering their bb/100 win rate and standard deviation.
What happens if I bet more than Kelly suggests?
Betting more than the Kelly amount (over-betting) actually decreases your expected long-term growth rate and dramatically increases your risk of ruin. At exactly 2x Kelly, your expected growth becomes zero despite having a positive edge. Beyond 2x Kelly, you're mathematically expected to lose money over time even with a winning edge. This is why conservative approaches like half-Kelly are popular.
How accurate does my edge estimate need to be?
Edge estimation is the biggest challenge in applying Kelly. In poker, you need thousands of hands to reliably estimate your win rate. In sports betting, most bettors overestimate their edge. The Kelly formula is highly sensitive to edge estimates - a small error in your edge calculation leads to significantly different bet sizing. This uncertainty is why fractional Kelly is strongly recommended.
Responsible Gambling Note
The Kelly Criterion is a mathematical tool for theoretical optimal betting. Even with a positive edge and proper Kelly sizing, gambling involves risk and variance. Never bet more than you can afford to lose, and be aware that edge estimates may be incorrect. If you're concerned about your gambling behavior, resources are available at the National Council on Problem Gambling.