# Weighted goalscoring metric, updated with marginal utility

Last night I wrote a post on the concept of marginal utility and how it might apply to the goalscoring metric I developed early in the week.  Chris Anderson let me know about his post on team leverage and how it related to my notion of marginal utility, which I elaborated upon in a post earlier today.  Now it's time to put it all together in a revised weighted goalscoring expression.

Below is the weighted goalscoring metric from last time:

In addition to revising the above equation, I want to clean up the notation as well.  I define M to be the total number of matchdays in the tournament and i the specific (ith) matchday.  I will let gi be the number of goals scored by the player on the ith matchday, and j the jth goal scored by the player (1,…,gi).  Let Δj be the goal differential between the two teams (player's team – opponent) just before the jth goal was scored.

The revised weighted goalscoring metric accounts for the strength of the opponent, represented by the current league position, and the marginal utility of the goals scored by the player in the game.  The new expression is presented below:

The equation might look scary, but it really isn't.  For starters, the opponent weighting function phi can be moved outside the inner summation, so what you're doing first is summing up the marginal utilities of the goals scored by a player in a single match.  Then you multiply the result by the weighting function (the opponent's league position).  Repeat for all of the matchdays of the competition and you are done.

Now is the time to show some results, so I return to this season's Group D of the UEFA Champions League.  I illustrate the new metric with this competition because there is less work involved in collecting the goal data and figuring out the goal differential at the time of the 24 goals scored in the group.  (I disregard own-goals in the calculation, so Djibril Cissé is not included in the list.)  Here are the results for all of the goalscorers in the group:

 Player Team Total Lionel Messi Barcelona 3.270 Christian Noboa Rubin Kazan 1.500 Claudemir Copenhagen 0.786 David Villa Barcelona 0.762 Dame N'Doye Copenhagen 0.750 Pedro Barcelona 0.565 Sidney Govou Panathinaikos 0.500 Martin Vingaard Copenhagen 0.447 Andreu Fontàs Barcelona 0.333 Víctor Vázquez Barcelona 0.262 Jesper Gronkjaer Copenhagen 0.197 Dani Alvés Barcelona 0.090 Cédric Kanté Panathinaikos 0.090

And to compare, here are the weighted goalscorers without marginal utility included:

 Player Team Total Lionel Messi Barcelona 3.750 Christian Noboa Rubin Kazan 1.500 Pedro Barcelona 1.000 David Villa Barcelona 0.833 Dame N'Doye Copenhagen 0.750 Claudemir Copenhagen 1.000 Sidney Govou Panathinaikos 0.500 Cédric Kanté Panathinaikos 0.500 Daniel Alves Barcelona 0.500 Martin Vingaard Copenhagen 0.500 Andreu Fontás Barcelona 0.333 Víctor Vázquez Barcelona 0.333 Jesper Gronkjaer Copenhagen 0.250

The weighted function is the simple rational function (y = 1/x).  Once again Messi and Noboa are the top goalscorers, but the inclusion of marginal utility causes significant movement among the other goalscorers in the table.  There does appear to be a reward for scoring winning goals in a match (which Dame N'Doye did on two occasions), but an even greater one for scoring the equalizer against the top team in the group, which Claudemir did against Barcelona.  Dani Alvés and Cédric Kanté scored meaningless goals in matches that were routs, and that fact is reflected in their weighted goal score.

There are two further extensions to be made: an inclusion of home/away effects, and the revision of the utility function with match result data, like Chris Anderson's leverage function for the various European leagues.