The Kelly Criterion, translated as the "Kelly Method" or "Kelly Strategy" in Canadian, is a mathematical formula introduced by John Larry Kelly, Jr. in 1956 in an essay published in the Bell System Technical Journal. This formula has gained significant attention in the field of finance and gambling due to its ability to determine the optimal amount of capital to allocate to a particular investment or bet. The Kelly Criterion takes into account the probability of success and the potential payoff, aiming to maximize long-term growth while minimizing the risk of ruin. By following the Kelly Criterion, investors and gamblers can make more informed decisions and manage their capital more effectively. It is important to note, however, that the Kelly Criterion is not without its limitations and should be used alongside other risk management strategies to achieve optimal results.

Wettexperten nutzen die Kelly-Formel als populäre Sportwetten-Strategie, um ihre Einsätze effektiv zu verwalten.

The formula, in fact, allows for the determination of the optimal wager for fixed odds sports betting, taking into account the uncertainty about the outcome of the game.

More specifically, the aim is to determine the optimal proportional amount for each individual bet from a total capital. The overarching goal is to maximize the betting capital in the long run using this approach.

Since it pertains to the allocation and growth of financial resources, the Kelly Criterion serves as both a money management technique and a meticulously followed strategy for sports betting, particularly by seasoned wagering experts. This approach is equally focused on organizing funds and advancing them, making it an essential tool for astute bettors in the realm of sports gambling.

This is how the Kelly formula works for sports betting.

In her discussion of the formula, Kelly draws upon the analogy of a gambler who has access to inside information about a baseball game (with two possible outcomes). This mythical "all-knowing player," who is aware of the outcome beforehand, cannot exist in reality, of course.

And Kelly also imposes restrictions on the player: Contrary to his fortune of a completely risk-free bet, there is the fact that the game result is transmitted to him via the telephone network, whose interference may cause the message to not be correctly received. Standing in opposition to the player's desire for an absolutely safe wager, Kelly introduces limitations. These constraints arise from the manner in which the outcome of the game is relayed to the player through the telephone network, which unfortunately experiences disruptive noises that may impede the accurate delivery of the message.

Given the provided conditions, how should the player proceed? The likelihood of a correct transmission is p, while the likelihood of an erroneous transmission is q.


The modern player is faced with the dilemma of which team to bet on. However, their uncertainty in betting does not stem from the probability of transmission errors, but rather from entirely different probabilities.

The uncertainty experienced by players in Kelly is not a trivial matter, as it is not easily explained from a contemporary standpoint. The fact remains that there is no omnipotent source of information unless there are arrangements between the parties involved that resemble acts of deception.

Understanding Kelly's choice, however, primarily relies on taking into account the context and the circumstances in which the essay was created.

Kelly is currently conducting research at Bell Laboratories on minimizing noise interference in telecommunications, thereby generating the idea for the formula. In order to contextualize this within his research and ensure the publication of the formula in the in-house journal remains uncompromised, the original text necessarily incorporates discussions on telephone lines and noise interference.

However, Kelly himself already declares the formula's openness to other contexts.

The investor's arsenal of knowledge, whether acquired through a tangible communication channel or existing as a reservoir of privileged information, embodies what the theory refers to as the 'channel'. Kelly (1956: 926) posits that this channel serves as a conduit for the investor to tap into a wealth of insights, guiding their decision-making process. Whether this channel manifests as a tangible medium of communication or as a comprehensive pool of exclusive knowledge, it plays a pivotal role in shaping the investor's perspective and ultimately influencing their investment choices.

The Kelly formula gained later fame in its utilization by the financial industry: as an investment theory, in the depiction by Edward O. Thorp on the one hand, and in the practical implementation on financial markets by Warren Buffet on the other hand.

It is actually a logical consequence that an algorithm aiming to maximize investments would pique the interest of other disciplines.

Kelly had already anticipated this development by mentioning it in the original manuscript.

While the inspiration for this model is rooted in the realm of gambling, its potential applicability to various economic scenarios cannot be overlooked. It is plausible that this framework may hold relevance beyond the realm of games of chance, extending its reach to encompass certain other economic contexts. This notion, as proposed by Kelly in 1956 (p. 926), suggests that the model's underlying principles could potentially transcend the boundaries of gambling, offering valuable insights into other economic dynamics. By acknowledging the broader spectrum of its potential applications, we open ourselves to a deeper understanding of the intricate interplay between risk, reward, and decision-making within the realm of economics.

Sports betting according to Kelly: Formula, explanation, and calculation example.

As previously stated, Kelly's objective is to optimize high-risk investments in order to maximize capital in the long run. Her primary focus lies in enhancing the performance of risky investments to ensure long-term capital growth.

According to Kelly, when it comes to investments that may not necessarily lead to success, the focus should not be on maximizing the increase in wealth, otherwise the entire budget would have to be allocated. This is because only then can the maximum possible yield be ensured, but it also means that a maximum loss is possible. The key is to strike a balance between potential gains and the risk of potential losses. By carefully managing the budget and making informed investment decisions, one can aim for optimal outcomes while minimizing the potential for substantial financial setbacks.

Rather, the focus should be on maximizing the anticipated increase in assets ("maximize the expected logarithm of wealth"). Consequently, Kelly introduces a utility function into his equation.

There is much more to be said about the background and understanding of the formula at this point, as the original text by Kelly or the formulas therein are difficult to access without a deeper mathematical understanding that is provided by a degree, but not solely by school education.

Kelly's complex calculations can, however, be distilled down to their essence, rendering the formula applicable for sports bettors.

Kelly Formel: f* = (bp-q) / b = [p (b + 1) – 1] / b

f* represents the wager (percentage of total capital) that needs to be determined using the subsequent formula, b denotes the odds offered by the bookmaker, p is, to put it in a more general term, the likelihood of success, and q is analogous to the likelihood of loss (in percentage). The objective is to calculate the wager f* by considering the odds b, the probability of winning p, and the probability of loss q.

Choosing between the two calculation methods presented above is purely a matter of personal preference. To provide an explanation for those who may not be as familiar with formulas, here is a breakdown of the two different calculation approaches:

The probability of winning and losing combined equals 100% (= 1), which is why q is logically 1 - p (see second calculation method). Consequently, the percentages of the probabilities of winning and losing must be expressed in decimal numbers (e.g.: 40% is 0.40).

The scientific discourse surrounding the Kelly Formula is primarily confined to the Anglo-American sphere. This also applies to its implementation within the sports community. It is important to consider that the formula, as described above, operates within the American odds system.

The formula will be demonstrated in its practical application according to the American quota system through the subsequent calculation examples, followed by a gradual adaptation for "European use". The main aim is to showcase the formula's versatility across different contexts and regions.

Assumptions for the calculation example
The European quote for b is 2.5, while the American quote is +150.

The American odds need to be converted for Formula 1: In order to calculate the potential net winnings, the payout ratio of 150:100 (meaning that for every €100 wagered, €150 can be won) should be used. This equates to odds of 1.5 (150 divided by 100).

p = probability of winning = 60% = 0.60
q = probability of loss = 40% = 0.40
 
Calculation of bet amount in % using American odds (rounded result)
[0,60 x (1,5 + 1) – 1] / 1,5 = 0,33 = 33%

As the detour via the American odds is cumbersome, the formula is presented below, demonstrating its applicability to decimal odds calculations.

f* = (bp – 1) / (b – 1)
Calculation of bet amount in % using European odds (result rounded)
(2,5 x 0,60 – 1) / (2,5 – 1) = 0,33 = 33%

If you want to calculate the total amount for your wager instead of the percentage, simply include the betting capital (V) in the formula.

f* = [V (bp – 1)] / (b – 1)
Calculating the betting amount in euros for V = 1,000 euros (result rounded)
[1.000 x (2,5 x 0,60 – 1)] / (2,5 – 1) = 333 $

Two elements are left to the sports bettor when it comes to calculating the wager: the betting capital and the probability of winning. In addition to the bookmaker's assessment (= odds) of the outcome of the weather event, another probability comes into play.

Kelly calculates the probabilities of whether the bet will succeed or not based on the error-proneness of information transmission. However, this is not useful for actual or modern betting transactions. Instead, the sports betting community has established two alternative approaches in place of this.

  • The probability of the personal prediction winning is determined based on the individual betting history (percentage of those tips from all previous bets that were won/lost).
  • The likelihood of a specific tip winning is estimated based on experience and knowledge.

This list can still be supplemented with the statistically determined probability.

The second approach is also utilized to determine a value wager. And indeed: when comparing the Kelly formula to calculate a value bet, the resemblance becomes apparent. The formula component above the fraction line (bp - 1) corresponds to the formula for calculating the "valuable wager." Hence, the Kelly formula is also characterized as "f* = edge / odds" ("bet size equals advantage/expected winnings divided by odds").

The practical application of the Kelly formula: Conditions for its applicability.

If the practical implementation of the Kelly Criterion is intended, the initial requirement prior to its application is to determine a sum for personal betting capital. This presents the first challenge: simulations have demonstrated the effectiveness of the Kelly Formula, but it necessitates a significant amount of both time and initial capital.

In practical terms, it is possible to calculate a percentage share of any amount, even if the initial capital were to be reduced to just 1 $ at some point. However, the reality faced by bookmakers in their operations presents a challenge to this applicability, as they impose restrictions in the form of minimum bets. Ultimately, this clash between mathematical calculation and the constraints imposed by bookmakers highlights the inherent limitations in practice.

From the necessity of a substantial capital arises, however, a psychological dilemma, as the capital increases, so does the individual bet amount. If such a large sum is lost repeatedly, one's nerves could gradually unravel. Yet, a sizable capital is essential for successful ventures. The key lies in maintaining a delicate balance between risk and reward, ensuring the preservation of both one's financial resources and mental well-being.

Another issue arises from the proneness to errors in self-assessment. Thorp's simulation reveals that "employing an excessively large f* and engaging in overbetting is significantly more heavily penalized than employing an excessively small f* and engaging in underbetting" (Thorp 1997: 18).

Consequently, the general advice is to always go for "less than Kelly" - or even just "half Kelly," which means betting only half of f* - and never exceed 20-25% of the initial capital. Opting for "half Kelly" typically yields approximately 75% of the growth rate achieved by "full Kelly."

Sources cited:

In the prestigious Bell System Technical Journal, published in 1956, a groundbreaking article by J.L. Kelly presents a fresh perspective on the concept of information rate. Kelly's work, titled "A New Interpretation of Information Rate," delves into the intricacies of this fundamental notion. Spanning from page 917 to 926, the article explores uncharted territory and challenges existing paradigms. With meticulous analysis and profound insights, Kelly's research revolutionizes our understanding of information rate and paves the way for future advancements in the field. Dive into this seminal work to unlock a wealth of knowledge and gain a deeper appreciation for the intricacies of information theory.

Edward O. Thorp's paper titled "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market" was presented at the 10th International Conference on Gambling and Risk Taking in Montreal, back in June 1997. The paper, which has gained significant recognition in the field, explores the application of the Kelly Criterion in various gambling and investment scenarios. Thorp's insights and findings garnered further attention, leading to corrections being added to the paper on April 20, 2005. This seminal work remains a valuable resource for those interested in the intersections of mathematics, probability, and financial decision-making.

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