# Kelly criterion in sports betting

21.10.2021

The Kelly Criterion is a professional financial management system for advanced bettors in bookmakers. To use the Kelly criterion correctly, you need to have a very deep understanding of the mechanics of value betting, as well as an excellent understanding of the chosen sport or league.

## The essence of the Kelly criterion

This method involves changing the size of the bet depending on two main factors:

1. The value of the coefficient;
2. The volume of value in the market.

The larger the volume of value, the more it is proposed to bet on the outcome. This is reasonable, because it is value that is the bettor's profit at a distance. Also, another divisor has been added to the formula, which creates a dependence on the coefficient. At high odds, the size of the bet decreases, and at low odds, with a greater probability of passing, it increases.

## Kelly criterion bet sizing formulas

The classic formula looks like this:

(p x k – 1) / (k – 1) x B

The variable k in this formula is the coefficient of the outcome, p is the probability of the outcome according to the player's opinion. And variable B is the size of the bankroll. Thus, we identify the value of the bet, because the dividend is the formula for calculating the value of the outcome. Divide it by the divisor “coefficient minus one”, and multiply it by the size of the bankroll. The resulting number is the recommended Kelly bet size in money.

## Examples of rate calculation by Kelly criterion

It will be easier to understand with an example. ## Example №1

Let's say we estimate the probability of victory for the Slovak national team at 35%, that is, this is a very valuable outcome in our opinion. Our bankroll is 100,000 rubles. Let's substitute the data into the formula.

(4.35 x 35% - 1) / (4.35 - 1) x 100,000 = 15597 rubles for a bet on P2

Despite the high coefficient, the Kelly criterion recommends bet a pretty impressive amount of 15% of your bankroll. If the value was less, then the size would be much more modest:

(4.35 x 25% - 1) / (4.35 - 1) x 100,000 = 2,612 rubles

And if the volume value was negative, then the bet amount would also be negative.

## Example №2

Now let's calculate the amount of the bet on TM (2.5) goals when estimating the probability of 62% and 75%:

(1.65 x 62% - 1) / (1.65 - 1) x 100,000 = 3538 rubles

(1.65 x 75% - 1) / (1.65 - 1) x 100,000 = 36,538 rubles

The difference is more than 10 times! In the first case, the outcome contains minimum value, so the bet on it is small, 3.5% of the pot. In the second case, there is a lot of value, so the size of the bet is huge - 36.5% of the bankroll.

## Example №3

Now let's consider the dependence on the odds for the same amount of bet value. Let's take the odds for the victory of Slovakia with a probability of 25% and quotes on TM (2.5) goals with a probability of 66%. The value is approximately the same, about 9%.

(4.35 x 25% - 1) / (4.35 - 1) x 100,000 = 2,612 rubles for a w2 bet

(1.65 x 66% - 1) / (1.65 - 1) x 100,000 = 13692 rubles for a bet on TM (2.5) goals

Thus, we see that with an equal volume of value, less money is put on high odds, and more money on low odds. This allows you to reduce risks and not to bet large amounts on outcomes with a low probability of passing.

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A very important nuance. All of the above mechanics only work if the bettor ideally determines the probabilities of the outcomes. But this is a priori impossible, because the exact probabilities in sports are unknown to anyone. Therefore, you need to be extremely careful with the Kelly criterion, and use its modification, which will be discussed later.

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## Fractional Kelly criterion

The fractional Kelly criterion bet is calculated using the same formula. The difference is that the counting result is then divided several more times. The denomination of this divisor is determined by each bettor for himself, it can be 2, 3, 4, 5 or even 10 - it depends on the personal desire to take risks.

The point of additional division is to keep the dependence of the bet size on the determined value in the end, but at the same time reduce the risks to an acceptable value of 1-5% of the bank for one outcome.

That is, we take our most powerful example:

(1.65 x 75% - 1) / (1.65 - 1) x 100,000 = 36,538 rubles

And divide the result additionally, say, by 5. Total, we get a more or less sane rate of 7307 rubles, that is, 7.3% of the bank. At the same time, we still have a high probability of passing in 75% (3 out of 4) and increased sizing due to the increased volume of value.

Even if the bettor made a decent mistake in determining the probability, it will still be quite high at this ratio. According to the bookmaker, it is about 60%. Losing 7.3% will not be critical for your bankroll, as opposed to losing 36.5% on one outcome.

## Conclusion

The Kelly Criterion is an excellent financial tool for determining the sizing of subsequent bets. It takes into account the amount of value in the outcome and varies the size of the bet depending on it. This approach can literally explode a game bankroll, building it up very quickly if used correctly.

Only professional players who are very good at determining the probabilities of outcomes in sports events can use the Kelly criterion. But even for them, we recommend using the fractional Kelly criterion, that is, dividing the resulting number several more times. With time and experience, the size of this divider can be reduced, but it is still better to end up leaving it at the level of at least 2-3.