# Marginal utility and goalscoring

Over the last few days I have presented a goalscoring metric that makes the value of the goal a function of the quality of the opponent, for which incoming league position acts as a proxy.  The main concept is that a goal scored against a top team should count for more than three or four scored against a bottom-placed team.  The top players on the raw goalscoring list more often than not end up at the top of the weighted goalscoring list, but the position of the remaining players depends rather strongly on the weighting function used.

I would like to place the matter of which weighting function to use aside for a moment and consider an extension to the weighted goalscoring metric.  To the current metric I add the concept that a goal scored to place a team in the lead is much more important than a goal that wraps up a rout, or one that is a consolation after being down 0-4.  This is the concept of marginal utility.  (There are exceptions, of course, but I am considering matches in isolation from each other, at least for now.)

Marginal utility is a concept from microeconomics that is the measure of the increase (or decrease) of utility due to an increase (or decrease) in consumption of a good or service.  Utility means different things to different people in different situations, but essentially it is a measure of relative satisfaction.

How does this relate to soccer?  If your team scores the opening goal in a match, there is quite a bit of satisfaction among the players, the coaches, and the supporters.  For the opposing team, not so much.  But what if it's 2-0 and your team scores a third goal to put the outcome beyond any doubt, and then scores another just because?  Most supporters would be delighted, of course, especially if it's a derby match.  But how much more useful were those additional two goals to the desired outcome, which was to win the match?  Not much I would argue; the result is well in hand.  This situation is an illustration of diminishing marginal utility.  If that opposing team grabs a very late consolation to make the final score 4-1, there is some satisfaction from that team for finally getting on the scoreboard, but tempered by the fact that the match is lost.  In this situation, the marginal utility is also very small.

So to extend this weighted goalscoring model, I make the metric a function of two quantities:

• the league position of the opponent at the beginning of the matchday, and
• the goal difference Δ between the two teams at the time of the goal.

I expressed weighting function for opposing team strength as either a logistic function or a simple rational function, both in terms of league position at the start of the matchday.  I do something similar to express goal utility, with a logistic function that increases sharply for minimum goal differentials and ties, and then saturates for lopsided goal differentials.  Such a function is expressed here

and plotted below:

What I am really interested in is the marginal utility function, and I obtain that by taking the derivative of the utility function:

I plot the marginal utility function.  Notice where it peaks:

The marginal utility peaks at goal differential Δ = 0, which means that for this model, a goal has maximum utility in a tied match.  The steepness of the slope can be adjusted by placing a multiplier or a divisor in the exponential, and I can scale the marginal utility by 2 so that the maximum utility is unity for zero goal differential.

This post is getting long, and the content is quite heavy already, so I'll stop here and come back tomorrow with the revised formulation of the weighted goalscoring metric and some results from the Champions League group stage.