# Ranking Methods 2: Elo Ratings for women's football

This is the second part of the series "Ranking Methods in football". The series introduces a variety of methods to determine the strongest teams on various levels of football. This article explains how Elo ratings (originating from chess) can be used to rank continental football teams. Like PageRank, the procedure comes with the desired feature of adjusting for the strength of schedule. That is, winning against stronger teams is more beneficial than winning against weaker teams. Instead of our dataset on male football, we here use a newly gathered dataset comprising 50 European leagues of women’s football.

2018-02-28

## What is Elo?

The Elo rating system, introduced by Arpad Elo, is a widely used method to measure relative strength. Originally, it has been invented as an improved chess raiting, but since then has also been used as a rating system in a variety of other sports, even eSports. Clubelo is probably the best known version for football on European level.

The beauty of the Elo rating system is its simplicity. When teams or players compete, the winner is not awarded a fixed number of points, but both competitors exchange points. The competition is thus viewed as a zero-sum game. The number of points exchanged, depends on the rating difference between the two competitors before the match. This difference is used as a predictor of the match. Say team A has a rating that is 100 points greater than the opponent, team B. Then, team A is expected to score 64% (or 0.64). Of course the exact percentage depends on the chosen probability distribution. The interested reader may refer to the wikipedia article for details.

The exchanged points in a match A vs. B can then be calculated with the following formula. $$points = K\cdot(R-E)$$ R is the actual result (1 for home win, 0.5 for draw and 0 for away win) and E the expected score, as explained above, for team A. The factor K is used to scale the number of points. The basic setting is $$K=20$$. The points are calculated from the perspective of team A. For team B, we simply take -points (remember, it's a zero-sum game).

Considering again two teams A and B with a rating difference of 100 points (So E=0.64). If team A wins, they are awarded $$20\cdot(1-0.64) = 7.2$$ points and team B looses $$7.2$$ points. If the match ends in a draw, team A gets $$20\cdot(0.5-0.64) = -2.8$$ and team B gets $$2.8$$. So also the game ended in a draw, which usually awards the same amount of points for each team, team B gains and team A looses points. This is because the initial rating of team A was higher than the one of B and A was expected to win against B.

Notice that the rating after a match always depends on the rating before the match. So we run into problems at the beginning of the ranking period since no previous ratings exist. This problem can be solved by assigning all teams the same value (like 1500) at the beginning and using the first few games as a training period, until the rating has converged to a reasonable value.

Many extensions to this basic Elo scheme are possible when ranking football teams. The factor K, for instance, could be scaled with the goal difference of the match. The higher a team wins, the more points it gets. Also, a home-field advantage factor could be introduced, by inflating the home teams rating by a fixed factor, to increase the expected outcome for the home team. Clubelo uses an interesting dynamic approach for home-field advantage factor.

## Elo ratings of women's football teams

A few months ago, I started gathering results of European leagues in women's football and came to the realization that there does not seem to exist a (official or inofficial) ranking for teams in Europe (or the world for that matter). Since the official FIFA ranking of national teams is also based on the Elo rating system, I decided to create a ranking of European teams based on the described Elo rating system. So far, it is very close to the basic version and it may be adjusted during the upcoming months.

The Elo rating only differs in two aspects from the basic version. The factor K for domestic league matches is set according to the UEFA women's Champions League Association Coefficient Rankings. For top-tier countries (Germany, France, Sweden and England) K=30, for second-tier countries (Spain, Russia, Italy and Denmark) K=25 and for third-tier countries (Czech-Republic, Austria, Scotland, Norway and Switzerland) K=20. For the remaining countries, the factor is set to 10. For Champion's League matches, K is set to 30 for the qualification round and 60 for the main stage.

The initial value of the ratings is set to 1500 (clubs from top-tier leagues), 1400 (second-tier), 1350 (third-tier) and 1250 (remaining teams). Both the K factor and initial ratings are differentiated between sets of leagues is due to the still existing discrepancy between women's football in different countries and should help to not overestimate the rating of lower-tier teams.

The below table shows the top ten as of 2018-02-28 according to the described Elo ratings.

Rank Team Country Rating
1 Olympique Lyonnais Féminin France 2171
2 VFL Wolfsburg Germany 2079
3 Paris Saint Germain FC France 2023
4 Manchester City LFC England 2070
5 FC Bayern Muenchen Germany 1941
6 FC Barcelona Spain 1926
7 Montpellier HSC France 1923
8 Chelsea LFC England 1918