Shapley Values

Who really matters on a soccer team? At NYA we've developed several bottom-up ways of assessing performance based on actions on the field, but we know that these metrics can't capture all the ways players contribute to results. That's why we also calculate Shapley values.

The Shapley value is a concept invented by the Nobel laureate economist and mathematician Lloyd Shapley as the solution to a bargaining problem involving a team of workers. In Shapley's scenario, the workers can create a certain amount of output together, but they have to decide how the proceeds of selling that output should be split between them. The Shapley value uses a series of hypotheticals to arrive at a distribution of proceeds that all the workers will agree is fair.

Viewed in another light, Shapley values measure who is pivotal in a team. The most pivotal workers - or players - make the biggest contributions to the team's output and have the highest Shapley values. Because of how they are calculated, Shapley values are agnostic to how a team member contributes; they simply estimate how much output might be expected to rise when the member joins the team.

In soccer, Shapley values can help to identify players with the "X" factor. These are the players who account for unexpectedly high shares of their teams' production on the field. If a player who isn't a well known star has a high Shapley value, it's probably worth trying to figure out how he manages to make such a big contribution, perhaps by watching video and talking to his teammates.

Of course, players can simply be lucky enough to take the field at the right time. For that reason, NYA's Shapley values are based on expected goals - the number of goals a team would be expected to score given the locations and conditions of its shots - calculated using data from Opta. This is a more granular and repeatable measure of a team's performance. But as a result, these Shapley values don't encompass exceptional skill in shooting or saving.

We think Shapley values are most effective at figuring out which players are pivotal in a team's tactical setup and which players are being used in ways that fail to exploit their talents. The correlation between Shapley values from season to season rises for players that stay on the same team, then rises again for players who also have the same manager, and again for players who also play the same position. In other words, a player's Shapley value is least likely to change if he's being used in the same way. Tactics make a difference as well as talent.

Below are the Shapley values for all players in the English Premier League who played roughly a third or more of their teams' minutes in the 2012-13 season. (The data are more idiosyncratic for players with fewer minutes.) This version of the formula measures players' production as the expected goals they generate per minute of play, compared with a predicted value based on the strength of their team and the opposition. A player with a Shapley value of zero is not expected to add or subtract from his team's predicted production.

Shapley values can also be created for different measures of a team's production, groups of players rather than individuals, shorter runs of games, and specific matchups. All of these variations - as well as Shapley values for any league covered by Opta - are available from NYA. Click here to read an application of Shapley values, and please get in touch to find out more.

North Yard Analytics LLC is a limited liability company formed in the State of New York.