Introduction
My past couple of articles have focused on Elo ratings and how they can be applied to football teams to rank them against each other and to estimate win probabilities.
Problems with Elo Ratings
On the whole the Elo system works okay but it was not designed with football in mind and so there are some issues with it, for example it can only handle two distinct outcomes – winning and losing.
Elo ratings try to get around this problem by considering each draw to be half a win and half a loss. However, this means that the win probabilities calculated using the Elo equation are actually the probability of winning or drawing versus the probability of losing or drawing, which isn’t particularly useful.
For a game such as chess, which Elo ratings were originally developed for, this may not be too much of an issue as tied matches are comparatively rare but in football draws are a common occurrence so we really need to be able to model three outcomes – win, loss and draw.
The Eastwood Index
So instead of combining draws with wins and losses, we need to be able to calculate their probabilities individually. To do this I have been developing my own ranking system, which for want of a better name I am currently calling the Eastwood Index, or EI for short (it feels rather pretentious to be naming it after myself so if anyone has any better names for it then feel free to let me know!)
The Eastwood Index allows football teams to be ranked using a mathematical rating system that evaluates relative strength based on previous performances weighted so that more recent matches have a greater impact on a team’s ranking.
Methodology
Teams EI ratings are scaled so that the average rating is 2000. The higher the rating the better a team is compared with the rest of the league.
EI ratings increase when a team wins a match or draws against superior opposition. Conversely, EI ratings decrease when teams lose matches or draw against weaker opposition. The size of this increase or decrease in ratings is linked to the quality of the opposition. For example, beating a superior team is worth more than winning against a lower ranked team.
The change in EI rating is also weighted by the score line so that the greater the difference in goals scored or conceded then the greater the change in ratings. Home advantage is also included in the calculations so that the home team is considered to perform better at home compared with away.
So far this all sounds similar to an Elo rating. However, the Eastwood Index has a major advantage over Elo in that it is multinomial, meaning it can function with multiple outcomes. This makes it possible to accurately calculate the probabilities of teams winning, drawing or losing matches.
A further advantage of the Eastwood Index is that it is does not rely on the Logistic distribution the same way Elo ratings do. The use of the Logistic distribution in Elo ratings originates from chess where it was considered to predict chess outcomes reasonably well. Football and chess are different games with different outcomes so instead the Eastwood Index uses custom curves developed using football data. This means that predictions for football should be more accurate using the EI compared with Elo ratings.
Example
The underlying mathematics for the EI is completely different to how an Elo rating is calculated but rather than wade through a list of equations it is simpler to show how it works using the recent Liverpool versus Swansea match.
Prior to the game Liverpool had an EI rating of 2151 compared with Swansea’s rating of 1891. Team performances are considered to be normally distributed around their rating so on any given day a team may play above or below their true skill level. By comparing the distribution curves for the two teams we can then calculate the probabilities of each outcome of the match before it is played.
Although both teams have similar ratings Liverpool has the home advantage giving them overall a 52% chance of a win compared with a 25% chance of Swansea winning and a 23% chance of a draw (Figure 1).
Figure 1: Predicting Liverpool Versus Swansea City
We can also use these probabilities to calculate the expected points from the match. If these two teams were to play the same match repeatedly then on average Liverpool would be expected to earn (0.52 * 3) + (0.23 * 1) = 1.79 points while Swansea would be expected to earn (0.25 * 3) + (0.23 * 1) = 0.98 points.
Once we know the actual result we can then update the EI for each team based on their current ratings and the score line, which was Liverpool 5 – 0 Swansea. Since Liverpool already had a higher EI rating and had beaten somewhat lesser opposition they would expect only a small rise in their EI but taking into account their high score in the match Liverpool’s rating moves up to 2183 while Swansea’s falls to 1859.
Conclusions
The EI Index offers a potentially superior way of rating football teams compared with other ranking systems, with the advantage that it can predict wins, losses and draws, and uses mathematics specifically designed to accurately model football data.
I will be discussing the EI in more detail in future posts and showing how it can be used to analyse and predict football matches.
As ever, get in touch if you have any comments of questions!