Even the best teams have a weak link. In “The Numbers Game”, Chris Anderson and David Sally suggest that a soccer team is only as good as its worst player. I’m not quite ready to sign on to their “O-ring” theory yet, but I do have a tool for finding the players who drag down their teams or simply don’t fit.
Shapley values are a sort of sophisticated plus-minus or with-or-without-you metric. I use them to create a series of statistical hypotheticals that answer the question, “If I formed this team in every possible order, what would the average contribution of each player be, at the moment he joined, to the team’s overall results?” The most important thing to keep in mind about Shapley values is that they’re contextual, just like every other metric taken from match data. This affects their interpretation.
In particular, a player’s Shapley value may depend on the gap in skill between him and the player next to him in the depth chart. An example I cited recently on Twitter is Mathieu Flamini. Because there is a significant gulf between him and Francis Coquelin at defensive midfield, Arsenal’s results suffer markedly when Flamini plays. Flamini therefore has a low Shapley value, and Coquelin has a high one. But if Flamini were on another team where the alternatives to him were of equal quality, his Shapley value might be higher.
A low Shapley value may also indicate a poor fit. Again at Arsenal, Alex Oxlade-Chamberlain is a prime example. He consistently posts low Shapley values despite good underlying numbers in his mechanistic metrics. He’s clearly a talented player, but Arsenal doesn’t work as well when he’s on the field.
This is exactly the point of Shapley values – they help us to ask the right questions. Why doesn’t Ox fit? Where would he fit better? Can Arsenal find a better option than Flamini to back up Coquelin? (They may have.)
During this transfer window, there’s just enough data for Shapley values to start asking some of these questions. Of course, Shapley values can be even more revealing at the end of a season. Here are the players with the lowest Shapley values – by some distance – for a few of the top teams in Europe after the 2014-15 season (all with 900+ league minutes):
- Barcelona: Jérémy Mathieu
- Real Madrid: Luka Modric
- Chelsea: Loïc Rémy, Filipe Luís, Didier Drogba
- Manchester City: Edin Dzeko
- Bayern Munich: Pierre-Emile Höjbjerg
- VfL Wolfsburg: Ivica Olic, Sebastian Jung
A couple of these players stand out to me. Luka Modric has been a regular at Madrid this season after playing roughly 1,300 league minutes in 2014-15. Did Carlo Ancelotti use him incorrectly, or have this year’s managers been making a mistake by using the 30-year-old midfielder? Well, Modric’s Shapley value this season is the highest at Real Madrid among players with 900+ minutes, so it seems like the former. Perhaps if Ancelotti had known about and investigated this problem earlier in his tenure, results might have been different.
Sebastian Jung is another interesting case. Despite his reputation – Arsenal reportedly wanted him in 2014 – Wolfsburg performed worse when he was in the squad. Did he really represent a big step down from Vierinha, or was something else going on? His underlying figures suggested some good defending but overall fairly mediocre play. If the club still had faith in his ability, it could have used match video, interviews with teammates and coaches, and other tools to figure out what was wrong. If not, the club could have tried to move him on. Perhaps neither has happened; he’s stayed at Wolfsburg and has had just two starts in the league this season.
Of the rest of the players, the majority left their clubs: Dzeko, Olic (in midseason), Filipe Luís, and Höjbjerg (on loan). Rémy seems to be perpetually in the shop window, and Mathieu, at 32, has been benched several times and injured twice already this season. The good news is that Filipe Luís and Dzeko have been doing well at their new homes. Höjbjerg is only 20 and may still have much to offer; in his 400+ league minutes this season, he’s performed pretty well for a player of his age.
Clearly, Shapley values aren’t a measure of ability alone. We shouldn’t expect them to be consistent from year to year when a player is being used in different ways or with different teammates. But in conjunction with more consistent metrics, they can help us to find situations where a talented player will thrive – and to avoid those where he won’t.