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The Kelly Criterion is a mathematical formula developed by John L. Kelly Jr. in 1956, designed to determine the optimal size of a series of bets.

In the context of gambling, it is used to calculate the portion of a gambler’s bankroll that should be put at stake in a bet to maximize the logarithm of wealth.

The main advantage of the Kelly Criterion is its ability to balance both the growth and the risk of financial ruin by tailoring wager sizes.

It has gained popularity among professional gamblers and investors alike for its straightforward approach to money management.

The formula takes into account both the probability of winning a bet and the odds received on the bet, aiming to maximize the exponential growth rate of the gambler’s bankroll over the long term.

By doing so, it helps to avoid the risk of losing the entire bankroll, as it never recommends betting more than is optimal for the given situation.

While the Kelly Criterion offers an enticing strategy for risk management, it requires an accurate estimation of one’s edge, which is the advantage over the odds. This precision in estimation is often difficult to achieve in practice, making the application of the criterion challenging.

Despite this, for gamblers and investors who can accurately assess their edge, the Kelly Criterion provides a methodical approach to bet sizing that can potentially lead to sustainable growth of their capital over time.

Concept of the Kelly Criterion

The Kelly Criterion is a betting strategy that determines the optimal size of a series of bets to maximize wealth over time.

Origins of the Kelly Criterion

Developed by John L. Kelly in 1956, the strategy originated from information theory and was originally intended for use in long distance telephone signal noise. It was soon adopted by gamblers for its efficiency in capital growth within games of chance.

Definition and Formula

The Kelly Criterion is defined as a formula that helps a bettor to decide how much of their bankroll to stake on a bet. The standard formula is:

f = (bp – q) / b*


  • f* is the fraction of the bankroll to wager;
  • b is the odds received on the bet (b to 1);
  • p is the probability of winning;
  • q is the probability of losing, which is 1 – p.

Fundamental Assumptions

The implementation of the Kelly Criterion rests on several assumptions: A bettor has access to consistent true probabilities of outcomes, betting odds are fixed, and reinvestment of earnings. All these create a context where this strategy can theoretically optimize bankroll growth over the long term.

Application in Gambling

The Kelly Criterion is employed in gambling to determine the optimal size of a series of bets. By balancing the trade-off between risk and reward, the formula helps gamblers maximize the growth of their bankroll over the long term.

Bet Sizing Strategies

Kelly Formula: The Kelly Criterion is represented by the formula: [ f^* = \frac{bp – q}{b} ] where:

  • ( f^* ) is the fraction of the current bankroll to wager,
  • ( b ) is the odds received on the bet (b to 1),
  • ( p ) is the probability of winning,
  • ( q ) is the probability of losing, which is ( 1 – p ).

Implementation: Gamblers should bet a fraction of their bankroll equal to their edge divided by the odds. In scenarios where the edge is negative, the Kelly Criterion recommends not betting at all.

Risk Management

Protection Against Ruin: The Kelly Criterion inherently protects against going broke by never wagering the entire bankroll. This is achieved by adjusting bet sizes according to the likelihood of winning and the available odds.

Stability: By advising gamblers to bet less in situations with higher uncertainty or lower odds, the strategy ensures that the bankroll is subject to lower volatility, which may lead to more consistent growth over time.

Expected Value Optimization

Growth Maximization: The Kelly Criterion aims to maximize the expected value of the logarithm of wealth, which leads to the highest possible long-term growth rate of the gambler’s bankroll.

Calculating the Optimal Bet: Gamblers calculate the size of their bets by determining the probability of a win and the offered odds. This calculation informs them of the optimal amount to bet to maximize the expected logarithmic growth of their bankroll.

Advantages of the Kelly Criterion

The Kelly Criterion offers strategic benefits for betters seeking optimal bet sizes on Bitcoin casinos and gambling establishments. This section examines its core advantages.

Maximizing Bankroll Growth

The Kelly Criterion is known for its ability to maximize bankroll growth effectively. By specifying the optimal amount to bet based on one’s bankroll and the perceived edge over the bookmaker, bettors can achieve the fastest growth without overexposure on single bets.

Long-Term Wealth Accumulation

Adopting the Kelly Criterion promotes long-term wealth accumulation as it dynamically adjusts the stake depending on the size of the bankroll and the odds. This means the amount to bet grows with the bankroll, helping to compound wealth consistently over time.

Reducing the Risk of Ruin

Crucially, the Kelly Criterion minimizes the risk of total ruin. In gambling, this term indicates the loss of the entire bankroll. Following the Kelly strategy ensures that bettors never wager more than what is calculated to be the most they can afford to lose while still maintaining a positive bankroll trajectory.

Criticism and Limitations

The Kelly Criterion, despite its theoretical appeal, faces various practical limitations and has been subject to criticism regarding its real-world application.

Practical Challenges

Execution: In practice, the Kelly Criterion requires bettors to constantly adjust their wager sizes, which can be logistically challenging. Larger bets may also move the market, potentially reducing the value of a bet.

Risk of Ruin: Although the Kelly Criterion aims to avoid it, there’s still a non-zero risk of ruin if outcomes are determined to be binary or if bettors overestimate their edge.

Parameter Estimation Issues

Edge Estimation: A key factor in the Kelly Criterion is the accurate estimation of one’s edge, which can be incredibly difficult in dynamic environments like sports or financial markets.

Volatility Misjudgment: Misjudging the odds’ volatility can lead to overbetting and significant drawdowns, as the formula is highly sensitive to changes in these parameters.

Alternatives to the Kelly Criterion

Fractional Kelly: Some bettors opt for a more conservative approach, using a fraction of the Kelly-recommended bet size to reduce volatility and risk.

Fixed Wager: Others may choose a fixed wager strategy, which ignores the variability in perceived edge and volatility, providing more consistent stake sizes.

Utility Theory: Decision models based on utility theory take into account personal risk preferences, which can lead to different bet sizing compared to the Kelly Criterion.

Case Studies

This section examines specific instances where the Kelly criterion has been applied successfully and compares its performance to fixed betting systems.

Historical Success Stories

In the 1950s, John Kelly, a researcher at Bell Labs, formulated what is now known as the Kelly criterion. This betting strategy was quickly adopted by gamblers seeking an edge in the stock market and other betting scenarios. Edward O. Thorp, a mathematician and hedge fund manager, exploited the Kelly criterion in blackjack and the stock market, significantly increasing his capital over time. Bill Gross, the co-founder of PIMCO, attributed some of his investment success to the use of the Kelly formula.

Comparative Analysis with Fixed Betting Systems

When comparing the Kelly criterion to fixed ratio betting systems, the differential becomes evident. Fixed systems do not adjust bet size based on the prevailing odds and the player’s current bankroll.

Aspect Kelly Criterion Fixed Betting System
Bet Size Adjustment Dynamic, based on odds and bankroll Static, does not vary
Risk of Ruin Minimized Potentially higher
Long-term Growth Optimized for maximum growth Could be sub-optimal

Investors using the Kelly criterion adapt their position size by calculating the optimal fraction of their bankroll to wager based on the probability of winning and the payoff odds.

This contrasts with a fixed betting system, where the investor adheres to an unchanging bet size regardless of these factors. This adaptive approach generally leads to a lower risk of ruin and ensures capital growth over the long term when applied correctly.

Kelly Criterion in the Real World

The Kelly Criterion, originally formulated for gambling, has found substantial application in the real-world areas of stock market investing and portfolio management. Investors use this strategy to optimize bet size and capital allocation to maximize long-term growth.

Adaptations in Stock Market Investing

In stock market investing, the Kelly Criterion is adapted to determine the proportion of a portfolio to allocate to each investment.

Calculation involves the probability of a stock’s success, the anticipated value of the return, and the potential loss if the investment fails. Successful investors gauge the accuracy of their predictions and the volatility of the stocks to apply the criterion most effectively.

Evidence of its application includes:

  • Portfolio Allocation: Traders apply the Kelly percentage to allocate their capital effectively among diverse investments.
  • Risk Management: By sticking to the prescribed Kelly fraction, investors limit their exposure in any single investment.

Frequent reassessment is crucial in the stock market as conditions change rapidly.

Implications for Portfolio Management

Portfolio managers leverage the Kelly Criterion to balance growth against risk. They use it to avoid overconcentration in high-risk assets while still capturing sufficient upside from potential winners. The Kelly Criterion has the following implications for portfolio management:

  • Optimized Growth: The strategy aims to achieve compounded growth over time rather than short-term gains.
  • Downside Protection: It inherently considers the worst-case scenario to protect against undue losses.

Diversification strategies often incorporate the Kelly Criterion to decide on the optimal betting size for each asset within a broader portfolio.

Mathematical Considerations

In the context of the Kelly criterion, mathematical considerations form the backbone of this betting strategy, incorporating an analysis of the probability of winning and the appropriate sizing of bets to maximize wealth over time without risking drawdowns that would take the bettor out of the game.

Probabilistic Analysis

The Kelly criterion operates on the premise that a bettor has an edge, which is a probabilistic advantage over the other side of the bet. The edge can be quantified as ( p \cdot b – q ), where ( p ) is the probability of winning, ( b ) represents the net odds received on the wager (odds received, minus 1), and ( q ) is the probability of losing (( q = 1 – p )). The formula for the Kelly stake ( f^* ) is given by:

[ f^* = \frac{pb – q}{b} ]


  • ( f^* ) is the fraction of the current bankroll to wager,
  • ( b ) is the decimal odds – 1,
  • ( p ) is the probability of winning,
  • ( q ) is the probability of losing.

If a bettor’s calculated edge is positive, the Kelly criterion suggests betting a fraction of the bankroll corresponding to this edge. If the edge is negative, it suggests not betting at all, as a negative edge means the odds are in favor of the house.

Leverage and Fractional Kelly Strategies

Leverage in betting refers to using borrowed capital to increase the size of one’s bet. However, when it comes to the Kelly criterion, leverage is typically avoided due to the increased risk of large losses that can exceed the bettor’s bankroll.

Fractional Kelly strategies involve betting a fixed fraction, such as half (1/2) or a quarter (1/4), of the calculated Kelly stake. This approach is often adopted to reduce risk and volatility. For instance, if the full Kelly stake is 20% of the bankroll, a bettor using a half-Kelly would only wager 10%.

Pros of Fractional Kelly:

  • Reduces the volatility of betting returns
  • Decreases the risk of significant drawdowns

Cons of Fractional Kelly:

  • Slower bankroll growth compared to full Kelly betting
  • Less optimal in the mathematically ideal sense, as it deviates from the Kelly optimal bet size

Advanced Topics

In the realm of gambling, advanced applications of the Kelly criterion explore adaptations and integrations that cater to dynamic betting environments and theoretical synergies.

Dynamic and Partial Kelly Strategies

Dynamic Kelly strategies allow bettors to adjust their bet size in response to a constantly changing probability distribution of outcomes. This flexibility can lead to more nuanced betting strategies where stake sizes are continuously updated to reflect new information or shifts in the market.

  • Adjusting for Real-Time Data: Bettors recalibrate bet sizes as fresh data comes in, potentially enhancing the edge over static models.
  • Reduced Risk: By scaling back bets during losing streaks, partial Kelly strategies aim to minimize drawdowns and protect capital.

Example: If a bettor starts with a full Kelly bet and then encounters information suggesting a reduction in edge, they might reduce the stake to a half Kelly bet to mitigate risk.

Integrating the Kelly Criterion with Other Theories

Integrating the Kelly criterion with other investment and betting theories can create hybrid strategies that potentially offer improved risk management and return profiles.

  1. Mean-Variance Analysis: Combines the Kelly criterion’s focus on maximizing bankroll growth rate with the variance minimization of the mean-variance portfolio theory.
    1. Aim: To strike a balance between growth and volatility.
  2. Game Theory: Applying Kelly in the context of game theory can inform strategic betting in situations involving multiple players with different strategies.
    1. Focus: Bet sizing in relation to the strategies and bet sizes of other players.

Italics indicate specific nuanced applications whereas bold highlights critical strategy aspects. Integrating these advanced topics requires a deep understanding of both the Kelly criterion and the additional theories being applied.

Tools and Resources

Before engaging in any betting system, it is crucial for individuals to utilize the appropriate tools and resources. These aids can significantly enhance one’s understanding and application of the Kelly Criterion in gambling.

Software and Calculators

Various software solutions and online calculators are readily available to assist gamblers in applying the Kelly Criterion to their betting strategies. Some notable examples include:

  • Kelly Calculator: A user-friendly online tool specifically designed for quickly determining bet sizes according to the Kelly Criterion.
  • BetSize Pro: This advanced software offers detailed analysis and helps users simulate different betting scenarios based on the Kelly formula.

Educational Material and Courses

For those seeking a deeper understanding of the Kelly Criterion, a range of educational materials and courses are available. They include:

  • Books:
    • Fortune’s Formula by William Poundstone provides a thorough exploration of the Kelly Criterion’s history and application in various fields.
  • Online Courses:
    • Sites like Coursera or Udemy often host finance or gambling-related courses where the Kelly Criterion is covered, providing both theoretical and practical insights.


In conclusion, the Kelly Criterion stands as a robust mathematical tool for optimizing bet sizes in gambling and has found applications in stock market investing.

Despite challenges in practical implementation and edge estimation, it offers a systematic approach to balancing growth and risk. The strategy’s impact extends to portfolio management, demonstrating its versatility

beyond gambling. While fractional Kelly strategies provide alternatives, the Kelly Criterion remains a valuable framework for individuals seeking optimized returns in their financial endeavors.