Sports Betting: Understanding the Kelly Criterion

In this series, I would like to explore sports betting and the different strategies that are used in the industry by individuals and bookies. In particular, I am interested in examining the strategies through building Python models to simulate their application.

In this first article, I would like to provide an overview of Kelly Betting before building a python model that will implement the strategy.

Kelly betting is a money management strategy used in gambling and investing, which aims to maximise the long-term growth rate of one’s bankroll or investment portfolio. The strategy was first introduced by John L. Kelly Jr. in 1956 in his paper A New Interpretation of Information Rate.

The basic formula for the Kelly Criterion is given by:

-> f^* is the optimal fraction of the bankroll to bet;
-> b is the net odds received on the wager (i.e., the amount of money you will win for each dollar bet, not including your initial bet);
-> p is the probability of winning the bet; and
-> q = 1 — p is the probability of losing the bet.

The derivation of this formula comes from maximising the expected value of the logarithm of wealth after a bet. This is done by differentiating the expected value with respect to the fraction bet (f) and setting the result equal to zero.

Let W represent the wealth of a bettor. Suppose the bettor places a fraction, f, of their wealth on a bet with probability, p, of winning and probability q=1 − p of losing. If the bettor wins, their wealth increases by a factor of 1+bf, where b represents the net odds received on the wager. If they lose, their wealth is reduced by the fraction f. The new wealth after a single bet, W′, can be represented by:

The objective is to maximise the expected value of the logarithm of wealth after the bet, denoted as E[log(W′)]. We can calculate the expected value as follows:

To maximise the expected value with respect to the fraction bet (f), we differentiate E[log(W′)] with respect to f and set the result equal to zero:

Using the chain rule, we can differentiate the terms in the equation:

Now substitute these derivatives back into the above equation:

To solve for f, we rearrange the terms:

Example 1

Suppose you bet on the outcome of a fair coin toss. If the coin lands heads, you win double your bet; if it lands tails, you lose your bet. In this case, the net odds received (b) are equal to 1, the probability of winning (p) is 0.5, and the probability of losing (q) is also 0.5.

Using the Kelly formula:

The optimal fraction of your bankroll to bet is 0, meaning you should not bet at all in this situation. This is because there’s no advantage or positive expected return from betting on a fair coin toss (EV = (0.5 × $1) — (0.5 × $1) = $0.5 — $0.5 = $0), and thus, it does not provide any opportunity for long term bankroll growth.

Example 2

Consider an investment opportunity where the probability of the stock increasing in value by 50% is 0.6, and the probability of it decreasing in value by 40% is0.4. In this case, b=0.5, p=0.6, and q=0.4.

Applying the Kelly formula:

The optimal fraction of your investment portfolio to allocate to this stock is 20%.

Example 3

Assume you are betting on a football match with the following decimal odds: Team A wins at 2.1, and Team B wins at 3.2. After analysing the teams’ performances, you estimate that the probability of Team A winning is 0.55, and the probability of Team B winning is 0.45.

For TeamA, b=2.1−1=1.1, p=0.55, and q=1−0.55=0.45. Applying the Kelly formula for Team A:

The optimal fraction of your bankroll to bet on Team A winning is approximately 13.64%.

For Team B, b = 3.2−1 = 2.2, p = 0.45, and q = 1−0.45 = 0.55. Applying the Kelly formula for Team B:

The optimal fraction of your bankroll to bet on Team B winning is approximately 4.55%.

Limitations

One of the key challenges in implementing the Kelly Betting strategy is the accurate estimation of probabilities for each outcome. In practice, obtaining precise probabilities can be difficult, especially for complex events or sports games. When probabilities are misestimated, this can lead to suboptimal bet sizes and reduced performance. As a result, bettors must carefully consider the quality of the information and data they use to calculate probabilities in order to optimise their betting strategy.

Another concern with the Kelly Criterion is the high volatility it can introduce to a bettor’s bankroll. Although the strategy maximises the expected growth rate in the long run, there might be significant short-term ups and downs in the bankroll. These fluctuations can be emotionally challenging for some bettors to handle, as they require a strong tolerance for risk and a long-term perspective on their betting activities. It is essential for bettors to be aware of this aspect of the Kelly Criterion and to manage their emotions and expectations accordingly.

In addition to the aforementioned challenges, applying the Kelly Criterion becomes more complex when multiple simultaneous bets with correlated outcomes are involved. In such cases, determining optimal bet sizes requires more intricate calculations that consider the relationships between the various bets. Bettors need to be prepared to invest additional time and effort into understanding these correlations and adapting their betting strategy accordingly, particularly when faced with a large number of concurrent wagers.

Lastly, the Kelly Criterion can sometimes suggest large bet sizes that may not be practical due to betting limits imposed by bookmakers or the need for bettors to maintain liquidity in their bankroll. In these situations, bettors must exercise discretion and consider alternative strategies or modifications to the Kelly Criterion, such as using a fraction of the suggested bet size. By doing so, they can strike a balance between adhering to the principles of the Kelly Criterion and accommodating the practical constraints of their specific betting environment.

After conducting a theoretical examination of the Kelly Betting strategy and analysing three applications, our next objective is to develop a series of Kelly Betting models using Python. By creating these models, we aim enhance our understanding of the strategy, explore its practical implications, and optimise its performance in various betting scenarios.