**The Kelly Criterion Formula** stands as a well-established, mathematically grounded strategy within the realm of sports betting. Delve into the article below to gain insights into the Kelly Criterion, its practical application, and the mechanics that underpin its effectiveness.

In the realm of gambling, optimal play relies on mathematical principles that, when put into practice, offer players a strategic advantage over the long haul. This advantage can manifest in various forms, whether it’s through increased winning frequency, optimized profit and loss dynamics, enhanced profit rates, or accumulating greater overall profits. However, it’s important to note that not every game of chance lends itself to optimal play.

One of the most renowned forms of optimal play is rooted in expectation-based strategies. The primary objective here is to minimize cumulative losses and maximize cumulative profits over an extended period. This approach involves making decisions at each juncture that yield the highest expected value for your wagers. This strategy finds particular prominence in games like blackjack and extends to other games such as poker.

In the realm of sports betting, where skills revolve around predicting outcomes based on external information, the concept of optimal play may not seem directly applicable. However, there exists a mathematical principle that theoretically optimizes a financial metric known as the “expected growth rate of the bankroll.” This principle is famously known as the Kelly criterion, or Kelly strategy.

In this article, we will delve into the intricacies of the Kelly criterion, explore its relevance in the context of sports betting, and dissect its strengths and limitations.

## Kelly Criterion formula

The Kelly criterion can be succinctly expressed through a straightforward formula, which holds true for general betting scenarios:

f = (bp – q)/b, where the terms are defined as follows:

- f represents the optimal size of your bet, expressed as a percentage of your bankroll.
- b denotes the net payout rate of the bet.
- p signifies the probability of winning the bet.
- q stands for the probability of losing the bet.

The net payout rate, b, quantifies the profit you would make if you win the bet based on your stake. In the context of sports betting, b is often calculated as the decimal odds of the bet minus 1. For instance, if you have a bet with decimal odds of 2.5, the corresponding b would be 2.5 – 1 = 1.5. In the realm of casino games, payout odds like 3:1 translate to b = 3 – 1 = 2.

This formula can also be expressed as f = p – q/b. Since q can be represented as 1 – p, we can further simplify it to f = (bp + p – 1)/b or even f = [p(b + 1) – 1]/b, where b + 1 corresponds to the gross payout rate, often denoted as o (from “odds”). Consequently, the formula can be alternatively written as f = (po – 1)/b.

The Kelly formula furnishes you with the ideal bet size (f) given the other variables at play (net payout rate and probabilities of winning and losing). But what does “optimal” entail in this context? It’s important to note that the calculated bet size isn’t an absolute amount but rather a proportion relative to your existing bankroll. Thus, there are no specific profit or loss figures or profit rates subject to optimization; instead, the focus is on your bankroll itself. To be more precise, the criterion seeks to maximize the growth rate of your bankroll under conditions of a positive expected value. In simpler terms, it applies to strategies aimed at ensuring the most significant possible growth rate for your bankroll or capital. To gain a clearer understanding of this interpretation, let’s delve into the mathematical context of the Kelly formula and where it finds applicability.

## Mathematical context and applications

The Kelly formula originates from the realm of probability theory and serves as a mathematical framework for determining the appropriate bet size that yields a positive expected value. Its primary objective is to satisfy the following criterion: maximizing the expected value of the natural logarithm of wealth, essentially optimizing the expected geometric growth rate. This formula assumes that the returns on the bet are known, even if a bet might result in a positive return when lost, albeit less than the stake. Additionally, it presupposes that the probability of winning is quantifiable.

Within these stipulated conditions, the Kelly criterion surpasses all other strategies in the long run, consistently yielding higher wealth. In practical terms, this means that as the number of bets tends towards infinity, the theoretical overall return is maximized.

In its general representation, the Kelly formula takes the form of **f = (p/a) – (q/b)**, where ‘a’ signifies the loss rate associated with the bet.

In the realm of gambling, lost bets typically result in a complete loss of the stake, equating to a loss rate of 100%, or simply 1. Under this context, the formula reverts to its initial form, which we originally introduced.

However, it’s worth noting that there are domains where the concept of a loss in a bet, defined as an action involving uncertainty, is not absolute. One such domain is investment management, which includes activities like stock market trading. In these contexts, gambling and investment intersect, and the Kelly criterion emerges as a valuable tool for determining the optimal bet or investment size concerning the growth rate of one’s bankroll or capital.

## Is the Kelly criterion applicable in sports betting?

**It’s essential to understand that the Kelly criterion is specifically designed for bets with positive expected value**, making it a strategic tool for enhancing the profitability of already favorable wagers and maximizing your bankroll growth.

In contrast, **you won’t encounter the Kelly strategy applied in casino games** because most casino bets inherently come with negative expected values, meaning they are structured in a way that favors the house over the long run.

Take, for instance, a column bet in European roulette. Applying the parameters to the Kelly formula **(b = 2; p = 12/37; q = 25/37)**, we find that f = 12/37 – 25/(37 x 2) = – 1/74, resulting in a negative value. This negative value signifies that placing such a bet would be unwise since the payout odds are lower than the true odds, essentially indicating a negative expectation and an overall loss in the long term. Therefore, continuously wagering on this type of bet would not lead to bankroll growth.

However, sports betting presents a different scenario. Unlike casino games, precise probabilities for the outcomes of sporting events are challenging to determine due to the numerous deterministic factors involved. Sportsbooks establish payout odds based on various factors, including statistical analysis of bettors’ intentions. Informed bettors, on the other hand, assess the likelihood of outcomes based on available information, creating a subjective probability.

In sports betting, handicapping comes into play as a process of estimating true odds based on external information and analysis. Here, the true odds replace the role of the probability of winning in the Kelly formula. As long as these true odds are higher than the payout odds offered by the bookmaker, the Kelly criterion can be applied effectively. This condition is often met when bettors select wagers as a result of their handicapping efforts. In essence, sports betting involves identifying “profitable” bets that deviate from the bookmaker’s assessments, with bettors relying on their own analysis and estimations to make informed decisions.

It’s worth noting that implied probability, which represents the inverse of payout odds, doesn’t serve as true odds. Only the bettor’s subjective estimation of true odds can be used as the probability ‘p’ in the Kelly formula when applying it to sports bets.

## Concrete application of Kelly criterion in sports betting

As explained earlier, the Kelly criterion serves as a valuable tool for determining the optimal size of your bets when you’re dealing with bets that offer positive expectations, ultimately aiming to maximize the growth rate of your bankroll. Once you’ve identified a bet where you believe the probability of winning exceeds the payout odds, you can harness the power of the Kelly criterion to calculate the optimal stake, denoted as ‘f’.

When working with parameter ‘b’, it’s most convenient to use the decimal format of the payout odds. If you encounter odds presented in American or British formats, it’s essential to convert them into decimal format before plugging them into the formula.

While implied probability isn’t a direct substitute for true odds, it can be a handy tool for gauging whether your subjective probability estimate surpasses the payout odds. This involves comparing two percentages to help you make more informed betting decisions.

Now, let’s delve into some practical examples to illustrate the application of the Kelly criterion.

**Example 1:** Imagine a European match between Team A and Team B, where Team B is offered at decimal odds of 3.8 for victory. You’ve assessed a subjective probability of 35% for Team B to win.

First, we calculate the implied probability: 100 divided by 380 equals 0.2631, which translates to 26.31%. Since your estimated probability (35%) surpasses the implied probability (26.31%), this bet holds a positive expectation.

With values of b = 2.8, p = 35%, and q = 65%, we apply the Kelly formula: f = p – q/b = 35% – 23.21% = 11.79%. This result indicates that you should wager 11.79% of your bankroll on this event.

**Example 2:** Now consider a match between Team A and Team B in the UK, with fractional odds of 7/5 for Team B’s victory. Your subjective probability estimate for Team B winning is 45%.

To work with the Kelly criterion, convert the fractional odds to decimal format: 12 divided by 5 equals 2.4.

The implied probability is then found as 5 divided by 12, resulting in 0.4166 or 41.66%. Since your estimated probability (45%) is higher than the implied probability (41.66%), this bet has a positive expectation.

With parameters b = 1.8, p = 45%, and q = 55%, we apply the Kelly formula: f = p – q/b = 45% – 30.55% = 14.45%. Here, the calculation suggests that you should bet 14.45% of your bankroll on this event.

**Example 3:** In the US, a match between Team A and Team B provides +250 (moneyline) odds for Team B’s victory. Your subjective probability assessment for Team B’s win stands at 35%.

To work with the Kelly criterion, convert the moneyline odds to decimal format: 350 divided by 100 equals 3.5.

The implied probability is determined as 100 divided by 350, resulting in 0.2857 or 28.57%. Since your estimated probability (35%) exceeds the implied probability (28.57%), this bet presents a positive expectation.

Utilizing values b = 2.5, p = 35%, and q = 65%, we apply the Kelly formula: f = p – q/b = 35% – 26% = 9%. This calculation advises that you should stake 9% of your bankroll on this event.

These examples demonstrate the practical application of the Kelly criterion in assessing optimal bet sizing for favorable opportunities.

Should you find the Kelly criterion’s suggested percentage to be too high and perceive it as a risk to your bankroll, you have the option to mitigate this risk by reducing the value of ‘f.’ Commonly, bettors opt for half or a quarter of ‘f.’ Employing these modified versions of the criterion allows for a more cautious betting strategy. It’s essential to note that this adjustment comes at the expense of not fully optimizing your bankroll’s long-term growth rate, even though it still maintains a positive trajectory.

## Effectiveness of the Kelly strategy

When assessing the effectiveness of the **Kelly strategy** in practical betting scenarios, it’s essential to keep in mind that it operates on an expectation-based model. As such, it functions as a statistical average over the long term. This means that while using the Kelly strategy, your bankroll won’t consistently experience optimal growth. It will have periods of faster growth, slower growth, and occasional decreases due to inevitable losses. The Kelly strategy aims for theoretically optimal long-term growth, contingent on adhering to its criteria and maintaining the patience for sustained, similar betting endeavors.

It’s crucial to recognize that the Kelly strategy isn’t designed for those seeking surprises, high odds, or quick, substantial wins. Instead, it serves as a relatively safe and defensive approach, considerably safer than aggressive martingale-style progressive betting. It best suits bettors who prioritize gradually building their bankrolls through prudent and conservative play. Nevertheless, it comes with its own set of risks. Since wagers are specified in terms of bankroll percentages rather than fixed amounts, an extended losing streak can erode a bankroll. This risk can be mitigated by selecting bets with higher win probabilities, even if they offer smaller payouts.

Handicapping is a fundamental component of the Kelly strategy’s success. Its application and outcomes hinge on your ability to accurately assess the true odds. Therefore, your handicapping proficiency plays a pivotal role in determining whether the Kelly strategy yields success in your betting endeavors.

In summary, the Kelly criterion leverages the relationship between your bet’s winning likelihood and payout odds to ascertain an optimal wager size for maximizing your bankroll’s growth rate. While it is mathematically proven to be sound, its real-world efficacy is subject to both the inherent uncertainty of betting and human factors, such as subjective estimations and betting behavior.