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arxiv: 2603.11016 · v2 · submitted 2026-03-11 · 📊 stat.AP

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

Mattia Cefis , Rodolfo Metulini , Maurizio Carpita This is my paper

Pith reviewed 2026-05-15 12:36 UTC · model grok-4.3

classification 📊 stat.AP
keywords football analyticsShapley valueplayer contributionxGApassing networkscooperative gamessports performanceSerie A
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The pith

A restricted Shapley value computed only over observed passing coalitions and valued by xGA measures each football player's marginal contribution to shot actions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper adapts cooperative game theory to football by defining coalitions strictly from real passing interactions rather than all possible player combinations. It extends the expected goals idea into xGA to score the full build-up sequence leading to a shot. The resulting Player's Restricted Shapley statistic then assigns credit only within those data-derived groups, producing action-level numbers that reflect actual team play. A reader would care because standard individual stats ignore the cooperative setup of shots and therefore misattribute value to scorers versus creators.

Core claim

By restricting Shapley-value coalitions to subsets of players connected through observed passes and using xGA as the coalition value function, the PRS statistic yields interpretable, action-specific marginal contributions that differ across teammates within the same club, as shown in the 2022/23 Serie A data for AC Milan and SSC Napoli.

What carries the argument

The Player's Restricted Shapley (PRS) statistic, which calculates marginal contributions over only the tactically admissible player subsets defined by the season's passing network and values each subset with the xGA cohesion function.

If this is right

  • PRS supplies action-specific numbers that distinguish players who create shots through build-up from those who only finish them.
  • Pairing PRS with a final efficiency score exposes gaps between a player's cooperative involvement and actual goal conversion.
  • Team-level analysis of 8,421 shots shows measurable differences in contribution patterns inside the same squad.
  • The framework supplies a quantitative input for scouting and squad-building decisions that already incorporate passing networks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Transfer-market valuations could add a PRS term to better price players whose main value lies in facilitating shots rather than scoring them.
  • The same restricted-coalition logic could be tested on defensive actions or set pieces once comparable interaction data exist.
  • Coaches might experiment with line-ups that maximize high-PRS pairings identified from recent matches.

Load-bearing premise

Observed passing interactions in the data correctly define the tactically admissible coalitions whose xGA values reflect true individual impact without distortion from opponent pressure or missing tracking information.

What would settle it

Recomputing PRS rankings after replacing the passing-based coalitions with random subsets of the same size and checking whether the new rankings lose all correlation with independent video-coded impact ratings or future goal-creation rates.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript proposes extending the expected goals (xG) framework with expected Goal Action (xGA), which incorporates build-up play and passing networks to evaluate shot quality. It introduces the Player's Restricted Shapley (PRS) statistic, adapting cooperative game theory by restricting coalitions to subsets observed in passing interactions and using xGA as the cohesion function to quantify individual marginal contributions to shot-ending actions. The approach is applied to 8,421 shot-actions from Serie A 2022/23, with case studies on AC Milan and SSC Napoli illustrating within-team heterogeneity in contributions and discrepancies between cooperative engagement and goal conversion.

Significance. If the framework is rigorously validated, PRS could advance player evaluation in football analytics by explicitly modeling cooperative interactions, offering more interpretable and action-specific metrics than standard xG or traditional Shapley values. This has potential value for scouting and performance analysis, particularly in distinguishing individual skill from team passing patterns, provided the restrictions and value function are shown to be unbiased.

major comments (3)
  1. [Abstract and framework description] Framework description (abstract and methods): No equations or algorithmic details are supplied for computing xGA from passing networks or for evaluating xGA on restricted coalitions; without these, it is impossible to verify how marginal contributions are obtained or whether re-simulation of build-ups is performed.
  2. [PRS definition and coalition restriction] PRS construction (methods): The restriction of coalitions to observed passing sequences is presented without justification or sensitivity analysis; the manuscript does not demonstrate that unobserved but tactically feasible combinations are correctly excluded rather than merely missing due to tracking gaps or opponent pressure, which directly affects the validity of the marginal contribution claims.
  3. [Application to Serie A data and case studies] Empirical application (results): The analysis of 8,421 actions reports case-study heterogeneity but supplies no validation of xGA against realized goals, no calibration metrics, and no error analysis; this leaves the reported PRS values and efficiency discrepancies without external grounding.
minor comments (1)
  1. [Notation and definitions] Notation: The relationship between standard xG and the proposed xGA would benefit from an explicit comparison table or equation highlighting the additional network terms.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: Framework description (abstract and methods): No equations or algorithmic details are supplied for computing xGA from passing networks or for evaluating xGA on restricted coalitions; without these, it is impossible to verify how marginal contributions are obtained or whether re-simulation of build-ups is performed.

    Authors: We agree that the current version lacks explicit equations. In the revised manuscript we will insert the full mathematical definition of xGA as a function of the passing network (including the precise cohesion value derived from build-up sequences) and the algorithm for evaluating it on the restricted coalitions. We will also state explicitly that no re-simulation of unobserved build-ups is performed; xGA is computed directly from the observed sequences only. revision: yes

  2. Referee: PRS construction (methods): The restriction of coalitions to observed passing sequences is presented without justification or sensitivity analysis; the manuscript does not demonstrate that unobserved but tactically feasible combinations are correctly excluded rather than merely missing due to tracking gaps or opponent pressure, which directly affects the validity of the marginal contribution claims.

    Authors: The restriction is intentional and data-driven: only coalitions that actually occurred in the recorded action are admitted, thereby reflecting the realized cooperative structure. We will add a dedicated paragraph in the methods section justifying this choice on substantive grounds (tactical admissibility within the specific action) and will include a sensitivity analysis that perturbs the observed coalitions (e.g., by adding or removing low-probability links) to quantify robustness of the PRS values. revision: yes

  3. Referee: Empirical application (results): The analysis of 8,421 actions reports case-study heterogeneity but supplies no validation of xGA against realized goals, no calibration metrics, and no error analysis; this leaves the reported PRS values and efficiency discrepancies without external grounding.

    Authors: We acknowledge that the present results section focuses on illustrative case studies rather than formal validation. In the revision we will add calibration plots of xGA versus realized goal outcomes, Brier scores, and a brief error analysis for the xGA model. These additions will provide the external grounding requested while preserving the paper’s primary emphasis on the PRS metric. revision: yes

Circularity Check

1 steps flagged

PRS marginal contributions reduce to xGA values on empirically observed coalitions by construction

specific steps
  1. fitted input called prediction [Abstract (PRS definition) and methods section on coalition restriction]
    "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"

    The paper first constructs xGA from the identical set of 8,421 shot-actions and their passing networks, then defines PRS by restricting coalitions to exactly the subsets that appear in those observed sequences and valuing them with the same xGA. The resulting marginal contributions are therefore computed directly from the empirical interaction graph and the xGA fit on the same data, reducing the 'prediction' of individual impact to a re-expression of the input quantities.

full rationale

The central derivation defines xGA from the same passing networks and shot-actions, then restricts Shapley coalitions exactly to the observed subsets of those actions and values them with the same xGA function. This makes the reported player contributions a direct algebraic function of the input data's interaction graph and xGA estimates, with no external benchmark or independent model for admissible coalitions. The paper's own description confirms the restriction is taken from observed sequences and xGA is adopted as the cohesion function, satisfying the fitted-input-called-prediction pattern at the core of the PRS statistic.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 2 invented entities

The framework rests on two new constructs (xGA and PRS) plus the domain assumption that passing data directly encodes admissible coalitions; no independent evidence is supplied for either construct.

free parameters (1)
  • xGA model parameters
    xGA extends xG and must incorporate fitted coefficients for build-up and passing features; these are not enumerated but are required to compute the cohesion function.
axioms (1)
  • domain assumption Observed passing interactions define tactically admissible coalitions
    The restriction step in PRS assumes that the passing network observed in each action accurately reflects which player subsets are allowed to cooperate.
invented entities (2)
  • xGA no independent evidence
    purpose: Shot-quality measure that incorporates build-up play and passing networks
    New quantity introduced to serve as the value function for the cooperative game.
  • PRS statistic no independent evidence
    purpose: Player contribution metric based on restricted Shapley values
    New statistic defined on top of xGA and the restricted coalitions.

pith-pipeline@v0.9.0 · 5506 in / 1421 out tokens · 31382 ms · 2026-05-15T12:36:01.872351+00:00 · methodology

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