Kelly Criterion for Bettors

Introduction to the Kelly Criterion

The Kelly Criterion, also known as the Kelly formula or Kelly bet, is a mathematical formula used to determine the optimal size of a series of bets to maximize the long-term growth of a bankroll. Developed by John L. Kelly Jr., a researcher at Bell Labs, in 1956, the formula was not originally intended for gambling. Kelly’s work was focused on long-distance telephone signal noise, but its applications in the world of investment and betting were quickly recognized.

At its core, the Kelly Criterion is a staking strategy that tells you what percentage of your bankroll to risk on a bet when you have an “edge” – a situation where you believe the probability of an outcome is higher than the odds offered by the sportsbook suggest. Unlike flat betting, where you wager the same amount on every game, the Kelly Criterion provides a dynamic approach, adjusting your stake size based on the perceived value of the bet.

For bettors, the Kelly Criterion offers a powerful tool for bankroll management that, when used correctly, can theoretically lead to faster bankroll growth than any other staking method. However, it is a double-edged sword. Its aggressive nature and reliance on precise probability estimates make it a high-risk, high-reward strategy that is not suitable for everyone.

The Kelly Criterion Formula Explained

The most common version of the Kelly Criterion formula used in sports betting is as follows:

Kelly % = ( (Probability of Winning * Decimal Odds) - 1 ) / (Decimal Odds - 1)*

Let's break down the components:

• Kelly %: This is the output of the formula – the percentage of your current bankroll that you should wager on this specific bet.
• Probability of Winning (p): This is your personal assessment of the likelihood that your bet will win, expressed as a decimal. For example, a 55% chance of winning is expressed as 0.55. This is the most critical and most difficult variable to determine accurately.
• Decimal Odds (b): These are the odds offered by the sportsbook, converted to decimal format. American odds of -110 are equivalent to 1.91 in decimal odds. To convert positive American odds to decimal, you use the formula: (American Odds / 100) + 1. To convert negative American odds, you use: (100 / Absolute American Odds) + 1.

A Simple Example:

Let's say you are betting on an NFL game. The Kansas City Chiefs are playing the Denver Broncos, and the point spread is -3.5 for the Chiefs. The odds for this bet are -110, which is 1.91 in decimal odds.

You have done your research and believe that the Chiefs have a 55% chance of covering the spread. So:

• Probability of Winning (p) = 0.55
• Decimal Odds (b) = 1.91

Now, let's plug these numbers into the formula:

Kelly % = ( (0.55 * 1.91) - 1 ) / (1.91 - 1) Kelly % = ( 1.0505 - 1 ) / 0.91 Kelly % = 0.0505 / 0.91 Kelly % = 0.0555 or 5.55%*

According to the Kelly Criterion, you should wager 5.55% of your bankroll on this bet. If your bankroll is $1,000, your stake would be $55.50.

The Core Concept: Your “Edge”

The entire Kelly Criterion revolves around the concept of having an “edge.” An edge exists when your estimated probability of an event occurring is greater than the implied probability derived from the sportsbook’s odds. If you don’t have an edge, the Kelly formula will tell you not to bet.

Implied Probability = 1 / Decimal Odds

In our previous example, the implied probability of the Chiefs covering the spread, according to the sportsbook, is:

Implied Probability = 1 / 1.91 = 0.5236 or 52.36%

Since your personal assessment of the Chiefs' chances is 55%, you have an edge. The difference between your probability and the implied probability is the source of your expected profit.

The critical challenge of using the Kelly Criterion is accurately assessing your winning probability. This is far more art than science. It requires extensive knowledge of the sport, statistical analysis, and the ability to create your own odds that are more accurate than the market’s. If you overestimate your edge, the Kelly Criterion will recommend a dangerously large bet size, which can quickly lead to ruin.

A Practical Example in a Real-World Scenario

Let's consider a real-life scenario. It's Week 10 of the NFL season, and the Green Bay Packers are playing the Chicago Bears. The Packers are favored by 7 points, with odds of -110 on the spread.

• Bankroll: $2,000
• Bet: Green Bay Packers -7
• Odds: -110 (1.91 decimal)

Your analysis, which includes looking at team performance, injuries, and historical data, leads you to believe that the Packers have a 58% chance of winning by more than 7 points.

1. Calculate Your Edge:

   • Implied Probability from odds: 1 / 1.91 = 52.36%
   • Your Edge: 58% - 52.36% = 5.64%

2. Apply the Kelly Criterion Formula:

   • Kelly % = ( (0.58 * 1.91) - 1 ) / (1.91 - 1)
   • Kelly % = ( 1.1078 - 1 ) / 0.91
   • Kelly % = 0.1078 / 0.91
   • Kelly % = 0.1185 or 11.85%

3. Determine Your Stake:

   • Stake = 11.85% of $2,000
   • Stake = 0.1185 * 2000 = $237

The Kelly Criterion recommends a very aggressive bet of $237 on this game. This illustrates the high-risk nature of the formula. A single loss at this stake size would represent a significant portion of your bankroll.

The Dangers and Pitfalls of the Kelly Criterion

While the Kelly Criterion is mathematically sound, its practical application is fraught with danger.

• Overestimating Your Edge: This is the single biggest risk. Even professional bettors struggle to quantify their edge with precision. A small error in your probability assessment can lead to a drastically oversized bet.
• High Volatility and Drawdowns: The Kelly Criterion is designed for maximum growth, not for a smooth ride. It leads to a highly volatile bankroll trajectory with significant upswings and terrifying downswings (drawdowns). Many bettors find the psychological pressure of these drawdowns to be unbearable, leading them to abandon the strategy at the worst possible time.
• Risk of Ruin: While the formula is designed to prevent you from betting your entire bankroll on a single event, a series of losses on over-leveraged bets can quickly decimate your funds. If your probability estimates are consistently wrong, the Kelly Criterion will accelerate your path to ruin.

Fractional Kelly: A Safer Approach

Given the risks of the “full Kelly,” most experienced bettors who use this method opt for a “fractional Kelly” approach. This means they bet only a fraction of the amount recommended by the formula.

Common fractional Kelly strategies include:

• Half Kelly (50%): You bet half of the recommended Kelly stake.
• Quarter Kelly (25%): You bet a quarter of the recommended Kelly stake.

Let's revisit our Packers vs. Bears example with a Half Kelly approach:

• Full Kelly Recommendation: 11.85% of bankroll ($237)
• Half Kelly Stake: 0.5 * 11.85% = 5.925% of bankroll
• New Stake: 0.05925 * 2000 = $118.50

By using a fractional Kelly, you significantly reduce your risk and the volatility of your bankroll. While this will slow down your potential growth rate, it makes the strategy far more sustainable and psychologically manageable. For most bettors, a fractional Kelly is the only sensible way to implement this strategy.

Implementing the Kelly Criterion in Your Betting

If you are considering using the Kelly Criterion, follow these steps:

1. Develop a Proven Edge: Before you even think about using Kelly, you need a long track record of profitable betting. You must be able to prove that you can consistently identify value in the market.
2. Start with Paper Trading: Use the Kelly formula to calculate stakes but don't actually place the bets with real money. Track your hypothetical results over a large sample size (at least 500 bets) to see if your probability assessments are accurate.
3. Use a Fractional Kelly: If you decide to implement it with real money, start with a very small fraction, such as a Quarter Kelly or even a Tenth Kelly (10%).
4. Be Honest and Conservative: Be brutally honest in your assessment of your winning probability. When in doubt, be conservative. It is far better to under-bet on a winning opportunity than to over-bet and risk your bankroll.
5. Combine with Bankroll Management Fundamentals: The Kelly Criterion should be used as a staking tool within a broader framework of sound bankroll management. This includes having a dedicated bankroll and tracking all of your bets.

Conclusion

The Kelly Criterion is a powerful but dangerous tool. It offers the tantalizing promise of maximizing long-term growth, but it demands a level of precision and emotional discipline that few bettors possess. For the vast majority of sports bettors, a simple flat betting model is a much safer and more effective approach to bankroll management.

However, for the advanced bettor who has a demonstrable edge and a deep understanding of probability, the Kelly Criterion, particularly in its fractional form, can be a valuable method for optimizing bet sizing. The key is to approach it with extreme caution, respect its inherent risks, and never forget that the formula is only as good as the probability estimates you feed into it.