# Football Club Elo Ratings

## Articles - Rating System

### Elo Formula

The Elo system is based on probabilities, each club has an Elo value based on its past performance. The difference between two clubs' Elo values represent the probability of one team winning against the other team. This probability can also be called result expectancy. Results and result expectancies can be expressed as percentages or fractions of 1. A victory is 1 (100%), a draw is 0.5 (50%) and a defeat is 0 (0%).
If two teams play each other, the result expectancy can be calculated according to the Elo formula:
E = 1 / (10(-dr/400) + 1),
where dr is the Elo point difference of the 2 clubs.

The result expectancy will be somewhere between 1 and 0. A team winning will always exceed the result expectancy and steal points from the losing team. The number of points stolen is determined by the following formula:
P = W * F * (R-E),
where R is the actual result and E is the result expectancy, W is a weight index and F is an additional factor depending on the margin of victory. High victories can be seen as more significant, so the number of stolen points increases.
The Elo system is a zero-sum game, points that are won by one team are lost by its opponent and vice versa. If a team exceeds expectations, it wins Elo points, if it stays behind expectations, it loses Elo points.

Created: 23 Mar 2013 - Modified: 20 May 2012

As football is a game of probabilities, where a weaker team can beat a stronger team occasionally, narrow victories are not as significant as landslide victories.
In these Elo ratings, the number of points exchanged will be multiplied with the square-root of the goal difference if there is a winner. For a draw, the factor F stays 1. The square-root-formula is inspired by WSASU.

Created: 23 Mar 2013 - Modified: 20 May 2012

### Weight Index

The weight index determines how many points are exchanged per game. A low weight index (10 or lower) would lead to a long-term-ranking where results from the distant past still have an effect on todays Elo values. A very high value instead would be a good indicator on current shape of clubs and recent results.
This ranking uses the weight index that allows the best prediction for the next match. Simulations for every weight index have been performed and compared. At a weight index of 20, the difference between the actual results and the result expectations are at a minimum (least-squares).

These results have been obtained by all games on record.

Created: 23 Mar 2013 - Modified: 20 May 2012

In football as in other sports, the home field advantage has a measurable effect that is taken into account in these Elo ratings as well.
If the Elo formulas were applied without any home field advantage adjustment, home teams would win a lot more Elo points than away teams. To counter this effect, home teams have to be rated artificially higher until the expectation to win or lose Elo points is balanced between home and away sides. The following graph shows how many points the home team would win on average depening on how big the adjustment is.

Empirical analysis shows that the home advantage is around 90 Elo points for modern domestic games. For modern European Cup games it is 120. [When I analysed the whole database, I discovered that in the past - especially in the 1970s - the home advatage was much more significant than it is nowadays. Click here for details] When team A plays team B at home, with team B having 90 Elo points more than team A, the probability for either team to win are the same.

Created: 23 Mar 2013 - Modified: 14 Jul 2012

Teams from different leagues play against each other in European competitions and exchange Elo points. This means that leagues exchange Elo points as well. However, when one observes the leagues' Elo differences over time, it becomes clear that leagues do not exchange enough Elo points just by the few European matches each year. The Elo values will then be very inert and it takes decades for formerly weak leagues to catch up with formerly strong leagues.
So these ranking will apply an additional measure: Every time a club exchanges Elo points with a club from another league, on top of the exchanged points from the clubs, the leagues will exchange points as well. A certain amount of points will be added to the league's clubs, uniformly distributed. To find out how many points are ideal, I tested what factor is required to provide the best predictability for European Cup games. The following table shows how many points are exchanged per team and per league in a European Cup game:

Elo points factor: Team + LeagueNLOP1L2L
Team's league not covered2+02+02+02+0
Only national playoff matches covered1½+½1½+1½1½+1½1½+4½
One League from team's assossiation covered1+11+21+51+5
Two Leagues from team's assossiation covered1+11+51+51+11

A team that plays in a league that is not covered will gain or lose twice as many points as it would normally do to compensate for the effect that the team has very few games. Simliar if there are very few domestic matches from an era available, then the factor is 1½. If a team plays regularly in a league then a European Cup game counts as a normal game for this team.
If a team does not play in a league, then there is no Inter-league adjustment. If a team plays regularly against national opposition, then the Elo points that one team wins or loses at international level are multiplied by 6 (minus the Elo points of the playing team itself) and uniformly added to the league's teams. If both team's assossiacions have two leagues this factor is doubled.
This leads to the unelegant consequence that a club can start a game with a different Elo value than its last post-game-value.
However, this method reacts quickly to new balances of power in European football. These Football Club Elo rankings do always want to represent the exact winning probabilities of a potential or actual encounter between two teams based entirely on results.

Created: 23 Mar 2013 - Modified: 22 May 2012

### Promotion and Relegation

Clubs that are newly introduced in this system need an initial Elo value to start from.
Being relegated is defined here as not playing for one year because of real relagation or - if the league matches are not available - non-qualification for the European Cups. Teams count as relegated if there is already clear that they will not play a match in the covered competitions in the next 12 months.
A club that has not played for at least 12 months or enters the system for the first time is a promoted team. If a team is promoted it needs a new Elo value, which is set to the average Elo value of all relegated teams of its league. The club that was inactive for the longest time disappears from the system and the remaining relegated clubs are set to the same average value. In this way, no Elo points are won or lost, the league does not change its average Elo value just by promotion/relegation.
If there are more promoted clubs than relegated clubs, the last Elo value that has been given to a promoted team will be given to the new team as well.

Created: 23 Mar 2013 - Modified: 20 May 2012

### New Formula for 2-Leg-Games

The current method of calculating two-leg games takes into account the aggregate result, increases its weighting and subtracts the result of the first leg. Please see here for details.
However, there are some issues that I have with that method:

• Once the second leg is played, the first becomes irrelevant
• The workaround for two single victories seems (and is) constructed
• The expectancy of losing or winning Elo points in the second leg is not 0
The way it should be in my eyes is rather:
• Winning or losing Elo points based on one game only
• The expectancy of losing or winning Elo points should be 0 in every match
The first leg should be considered as a standard game, I do not see a reason in changing that.
The second leg however depends very much on the first leg. Winning, drawing or losing the second leg only is irrelevant, but what matters is to qualify for the next round. Qualifying should earn you points, not qualifying should make you lose points, based on the probability of qualifying. Just what are the probabilities for that?
We will for this purpose refer to the probabilities from the Poisson/Elo-Model. By using this model, it can be quite accurately calculated what the probability of qualifying for the next round is.
This probability replaces the one from the Elo Formula in order to determine how many points will be exchanged per game. Practically, this means that if you carry a large lead from the first leg, then qualifying does not earn you many Elo points as this is the expected result. Losing however will cost you a lot of points. The expectancy of losing or winning Elo points is zero.
This new method will be applied to all games and will take the away goals rule into account in games where it applied. Penalty shootouts will count as draws.

Created: 23 Mar 2013 - Modified: 20 Jan 2013