In the previous posts on the weighted goalscoring metric, I used a logistic function to describe the weight that I apply to goals scored against opponents based on their current rank in the league table. The idea is that league position is a proxy for opponent strength, which isn't perfect, of course, but is something to get us started on the subject.
I would like to examine whether the weighted goalscorer list changes if we use a much simpler weighting function. The following function is as simple as it gets:
It is a simple rational function, the quotient of two polynomials which is itself nonlinear. (The inverse is x, which is a simple linear function that Goran used in his goalkeeper metric.) The difference between this weighting function and the previous one is that this function attaches very little importance to goals scored against opponents outside of the top three. A goal against the league leader counts as a full goal, while a goal scored against the second placed team counts as one-half and a goal against the third-placed team as one-third, and so on. Players who score lots of goals will end up at the top of the list, but those who score consistently against league opponents in the upper half will also rank very highly.
So here are the top twenty players on the weighted goalscorers list, now applying the simple rational function as the weighting:
|Cristiano Ronaldo||Real Madrid||33||5.343|
|Xabi Prieto||Real Sociedad||7||1.695|
So CR7, Messi, and David Villa top the list once more, but there are a lot of changes as you proceed down the table. David Trézéguet and Manu move way up into the top six, while Agüero and Rossi move down toward the second ten. There are more top twenty goalscorers with fewer total goals, but with more goals scored against the top sides.
The weighting function does make a big difference in this metric, so it has to be one that is defensible. I don't know if such a simple rational function is a credible description of opponent strength, but I'm open to other opinions on the subject. And the idea of marginal utility in goalscoring needs to be considered further. But right now, I need to get some sleep!