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

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.