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A Model-Based Restricted Shapley Value to Measure the Players' Contribution to Shot Actions in Football

Mattia Cefis, Rodolfo Metulini, Maurizio Carpita

TL;DR

A novel framework to assess individual player contributions in football, explicitly accounting for the cooperative nature of shot-ending offensive actions is proposed, and the Player's Restricted Shapley (PRS) statistic is introduced, a contribution metric based on restricted coalition structures derived from observed passing interactions, where xGA is adopted to compute the cohesion function.

Abstract

This paper proposes a novel framework to assess individual player contributions in football, explicitly accounting for the cooperative nature of shot-ending offensive actions. By incorporating team interaction into player evaluation, it also supports economically sustainable decision-making, with practical implications for performance analysis and player scouting. Extending the expected Goal (xG) paradigm, we propose the expected Goal Action (xGA), a measure of shot quality that incorporates build-up play and passing networks. Furthermore, we adapt cooperative game theory and introduce the Player's Restricted Shapley (PRS) statistic, a contribution metric based on restricted coalition structures derived from observed passing interactions, where xGA is adopted to compute the cohesion function. Unlike traditional Shapley approaches, the PRS one restricts coalitions to tactically admissible player subsets, offering action-specific, interpretable measures of marginal contribution in a cooperative context. We apply the framework to 8,421 shot-actions from the Italian League Serie A season 2022/23, and the case studies of AC Milan and SSC Napoli reveal some heterogeneity in contributions within teams. Furthermore, combining the PRS statistic with a final efficiency metric highlights the discrepancies between cooperative engagement and goal conversion.

A Model-Based Restricted Shapley Value to Measure the Players' Contribution to Shot Actions in Football

TL;DR

A novel framework to assess individual player contributions in football, explicitly accounting for the cooperative nature of shot-ending offensive actions is proposed, and the Player's Restricted Shapley (PRS) statistic is introduced, a contribution metric based on restricted coalition structures derived from observed passing interactions, where xGA is adopted to compute the cohesion function.

Abstract

This paper proposes a novel framework to assess individual player contributions in football, explicitly accounting for the cooperative nature of shot-ending offensive actions. By incorporating team interaction into player evaluation, it also supports economically sustainable decision-making, with practical implications for performance analysis and player scouting. Extending the expected Goal (xG) paradigm, we propose the expected Goal Action (xGA), a measure of shot quality that incorporates build-up play and passing networks. Furthermore, we adapt cooperative game theory and introduce the Player's Restricted Shapley (PRS) statistic, a contribution metric based on restricted coalition structures derived from observed passing interactions, where xGA is adopted to compute the cohesion function. Unlike traditional Shapley approaches, the PRS one restricts coalitions to tactically admissible player subsets, offering action-specific, interpretable measures of marginal contribution in a cooperative context. We apply the framework to 8,421 shot-actions from the Italian League Serie A season 2022/23, and the case studies of AC Milan and SSC Napoli reveal some heterogeneity in contributions within teams. Furthermore, combining the PRS statistic with a final efficiency metric highlights the discrepancies between cooperative engagement and goal conversion.
Paper Structure (8 sections, 12 equations, 2 figures, 4 tables)

This paper contains 8 sections, 12 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The Player's Restricted Shapley (PRS) statistic
  3. The xGA model
  4. The actions dataset
  5. Empirical analysis
  6. The xGA model results
  7. The PRS statistic results for AC Milan and SSC Napoli
  8. Concluding remarks

Figures (2)

  • Figure 1: Features importance according to the two classifiers. (a) XGBoost, (b) BR cloglog. Original sample estimates (horizontal bars) and 90% bootstrap percentiles intervals (B = 1,000 bootstrap replications).
  • Figure 2: Scatterplot of the $PRS$ statistics (B = 1,000 bootstrap replications) (x-axis) and the score efficiency $(G90 - xG90)$ (y-axis).