# Elo Rating for Sports Betting

## Statistics, rankings, and predictive probabilities of sports results

**A complete guide: from statistics to optimal bets**

Unlike games of pure chance (lotto, roulette, etc.), and even if they contain an undeniable element of chance and unpredictability, sports betting can be identified and studied using mathematical tools: statistics and probabilities.

Bookmakers are the best demonstration of this: with a good level of expertise and mathematical calculations, sports betting can be profitable.

Good, but how? Filling in tables of past results, calculating averages and expected values, or other standard deviations, of course, but how to transform this data into effective bets?

For example, how does knowing that a player who has won 80% of his last matches help me to know if it is interesting to bet in a future match with odds of 1.4?

The Elo raing system is a ranking system that everyone can implement, judiciously using the previous statistics noted. Then, once the ranking has been established, the probability of victory can be directly and simply calculated, and therefore be compared with the bookmaker's odds, and finally judge mathematically whether it is interesting to bet (value bet situation or not).

Finally, the Kelly criterion can be used to accurately and optimally size the amount of the bet.

This entire mathematical method is described, detailed and discussed on this page.

## Establish your own statistical model and Elo rating

The first part of the method consists of exploiting statistical data with both expertise and some mathematical calculations.At this stage, you must have already collected data, statistics, over the last few years (the last two for example).

### Specialize your data

First of all, you choose to specialize, according to your expertise in the sport. For example, for tennis sports betting, I can decide to specialize my data, to categorize them by type of surface. I then separate my data into 4 categories: match results on clay, on grass, on hard surfaces, and a final overall category for all surfaces combined.For sports betting on football matches, I can distinguish championship, cup, etc.

I thus specialize my data according to my categories, my knowledge.

### Making your own Elo Ratings

The next step will make it possible to use this statistical data by category. We calculate an Elo ranking from these past results as follows:- To begin with, we give an identical ranking to all players, or all teams, of 1500 ELo points.
- We then take the history of the results, our statistical data, in chronological order. At each match, from the result, we calculate the new Elo ranking of each opponent using the mathematical formula or using the following calculator:

We can initially leave the parameter*K*= 20. This parameter will be discussed later, and also in the "validating and/or adjusting the overall model" section. - We start the previous step again, calculating the new rankings, after each match in the chronological order of our database.

## Calculating probabilities and odds from Elo rating

The whole point of the Elo type ranking is precisely to allow the direct calculation of the opponents' probability of victory. For a ranking gap*d*, the probability of victory is given by the mathematic formula

^{−d / 400}

You can also use the following calculator which does these mathematical calculations automatically:

For example, if A ranked 1900 meets B ranked 1750, the ranking difference is

*d*= 150 and the probability for A to win is

^{−150 / 400}≃ 70%

## Spotting interesting bets - Detecting a value bet

We now arrive at the central stage: to bet or not!We therefore compare our previous result with the odds of bookmakers looking for a value bet : if a bookmaker offers a higher odds, for example 1.5 with the previous example, I bet! otherwise, I will refrain from investing in this bet. .

## How much to bet? Calculate the optimal stake to bet

In case we choose to bet. We can clearly see that the greater the difference between our estimated odds and those of the bookmaker, the more it is in our interest to bet a lot. But how much exactly?The mathematical formula for the Kelly criterion gives the mathematical answer to this question exactly.

With the previous example, probability of victory which I estimate at 70% therefore an odds of 1.4, if a bookmaker offers an odds of 1.5, then using the Kelly calculator I find that I must invest 10% of my bankroll.

My Elo ranking therefore allows to automatically detect favorable situations and how much to bet optimally.

## Simulating, testing, evaluating and validating the mathematical model

The preceding procedure (an algorithm, actually) makes it possible to exploit a statistical database until the exact and optimal bet to be invested. Like any mathematical model, it is important to go through a testing phase and possible adjustment of the model. Of course, free and without risk!**How to test the model and procedure without risk?**

We moreover recall that the model includes a parameter ( K for updating the Elo rankings) whose value can still be validated or adapted, see also the following paragraph.

The complete mathematical model must therefore be adjusted and validated. Simulations allow this, simulation for example over a full month.

To do this, we systematically use our model on all matches falling within one of our specialized categories for which we have established an Elo ranking. We apply the previous complete algorithm: we therefore calculate whether we should bet or not, in each bet offered by our bookmaker. But, in this testing period:

**we don't really bet : we only simulate**. We note what we would have bet, and the gain or loss.

At the end of the test period, free therefore because we have never bet, we can evaluate the quality of our mathematical model.

You can use this same test period as many times as you want, by varying parameters (value of parameter

*K*, amount of bets, etc.) in order to obtain the maximum gain. This gain is then both theoretical: nothing has really been gained, and practical because it comes from real data.

Then, we can either do the same the following month, another month of testing, with the previous adjusted and optimized model, or launch into real bets if the model suits us.

Setting up a model takes time, from collecting past data and statistics to mathematical calculations of rankings, probabilities, odds, and finally stakes, then the testing/adjustment/validation phase.

We then just finally recall the leitmotif: Living off your earnings?! yes, it's possible, then it's called a job!

## Back to parameter *K*

The *K*parameter used in the calculation of new rankings after the result of a match can be adjusted in different ways. The greater the value of

*K*, the greater the variation in rankings will be. In the Elo model, this is a parameter of stability/volatility of the rankings: if

*K*is small (close to 0), the rankings vary shortly after a new result while for larger values the rankings vary much more quickly.

The Elo rating was originally designed for chess players, where the following rule is used:

*K*= 40 for a player new to the rating list until 30 matchs,*K*= 20 for players who have always been rated under 2400*K*= 10 for players with any published rating of at least 2400 and at least 30 games played in previous events. Thereafter it remains permanently at 10.

For sports betting on football matches, in addition to the result of the match, victory/loss, we can take into account the difference in level during the match using this parameter. This difference can be measured by the difference in the number of goals and its impact on the parameter:

*K*= 10 for 1 goal difference*K*= 20 for 2 goal difference*K*= 30 for 3 goal difference*K*= 40 for more than 4 goal difference

See also: