# FIFA Women’s World Ranking and goal difference in Elo ratings.

The FIFA rankings for women’s national teams use a quite different methodology than the one used in ranking men’s national teams. The Women’s World Ranking (WWR) is based on the Elo rating system I wrote about in the previous post. The details for the men’s ranking can be found here, and the details the women’s ranking can be found here.

One thing that makes the WWR interesting is how goal differences are accounted for in the ratings. This is not something found in the ordinary chess-based Elo ratings. The method used in WWR is to let the ‘winning percentage’ change depending on two things: Goal difference and the number of goals the loosing team scored. This is in contrast to the original Elo ratings where the winning team wins 100% and the loosing team wins 0% (a draw is 50%). In WWR a team can never win 100%, the most it can win is 99%. This is the case when the goal difference is more than 6 and the loosing team haven’t scored any goals. The table below is from the pdf file linked to above and shows how many percent the loosing team win.

One strange thing about this table is the column for goal difference 0, implying a draw. I am guessing this is an error since it means that the winning percentage for the loosing team will be greater than the winning team. In the paper I mentioned in my last post where a number of different rating methods were compared (The predictive power of ranking systems in association football by J. Laset et. al), it was assumed that draw would yield both teams 50%, as in the original Elo-ratings. That paper also showed that the Women’s World Ranking was among the rating systems with best prediction.

Here is a plot showing the win percentage when the loosing team has scored 0 and 5 goals. We can see that there is not much gained for the loosing team to score one extra goal (assuming the goal difference stays the same, which of course is dubious), and most of the gain in winning percentage is when a team scores a goal such that the stance goes from a draw to a win.

The table above only goes up to 5 goals for the loosing team, but for the sake of implementation it is easy to generalize the rule about how much the loosing team gains by scoring an additional goal (with the goal difference is the same). For example will the loosing team gain 0.9 extra winning percentage points when the goal difference is 2. Similarly the gain is 0.6 percentage points when the goal difference is 5.

Below is a R function to compute the win percentage for football matches. It takes as input two vectors with the number of goals scored by the two opponents and returns a vector of win percentages (a number between 0 and 1) for the first team.

```winPercentageWWR <- function(team1Goals, team2Goals){
#calculates the win percentage for team 1.

stopifnot(length(team1Goals) == length(team2Goals))

perc <- c(0.01, 0.02, 0.03, 0.04, 0.08, 0.15, 0.50, 0.85, 0.92, 0.96, 0.97, 0.98, 0.99)
add <- c(0, 0.01, 0.009, 0.008, 0.007, 0.006, 0.005)

goalDifferences <- (team1Goals - team2Goals)
goalDifferences[goalDifferences < -6] <- -6
goalDifferences[goalDifferences > 6] <- 6

team1WinPercentage <- numeric(length=length(goalDifferences))

for (idx in 1:length(goalDifferences)){

team1WinPercentage[idx] <- perc[goalDifferences[idx]+7] -