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

This paper introduces a novel framework combining the expected Goal Action (xGA) metric with a Player's Restricted Shapley (PRS) statistic to quantify individual player contributions in football shot actions by modeling team interactions through tactically admissible passing coalitions, as demonstrated in an analysis of the 2022/23 Serie A season.

The Gist

Imagine football not just as a game of kicking a ball into a net, but as a giant, high-speed puzzle where every piece (player) has to fit together perfectly to create a picture (a goal).

For a long time, when we tried to figure out who was the "best" player, we mostly looked at the final picture: Who scored the goal? or Who made the final pass?

This paper introduces a new way to look at the game. It argues that judging a player only by the final shot is like judging a chef only by the taste of the dessert, ignoring the fact that they might have chopped the vegetables, seasoned the sauce, or built the fire that made the whole meal possible.

Here is the breakdown of their new system, explained simply:

1. The Problem: The "Solo Star" Trap

Traditional stats (like "Expected Goals" or xG) are great at telling us: "If a player shoots from this spot, how likely is it to go in?"
But they miss the teamwork.

• The Old Way: If Player A passes to Player B, who passes to Player C, who scores, the stats mostly credit Player C. Players A and B are invisible.
• The Reality: Football is a cooperative game. You can't score without the team moving the ball, creating space, and setting up the play. The paper says we need a way to measure how much each person added to the value of that specific play.

2. The Solution: The "Recipe" (xGA)

The authors created a new metric called xGA (Expected Goal Action).

• Think of it like a recipe: Traditional xG looks at the final ingredient (the shot). xGA looks at the entire recipe.
• It asks: How many people touched the ball? How far did they pass? How fast was the movement? Was the team in a good position?
• By analyzing the whole "build-up," xGA gives a score to the entire sequence, not just the final kick. It tells us, "This specific play was a high-quality opportunity because of how the team moved together."

3. The Math Magic: The "Restricted Shapley Value" (PRS)

Now, how do we split that "high-quality play" score among the 5 or 6 players involved?
They used a concept from game theory called the Shapley Value.

• The Analogy: Imagine a group of friends building a sandcastle.
  • If you add a bucket of sand to an empty beach, it's not much.
  • If you add a bucket of sand to a half-built castle, it's very valuable.
  • The Shapley Value calculates the average extra value a person brings to every possible group they could join.
• The Twist (Restricted): In football, you can't just put any 11 players together randomly. The coach has a specific plan. A goalkeeper doesn't usually pass to a striker in the middle of a play.
  • The authors created a "Restricted" version. They only looked at the groups of players that actually played together in real games (the passing networks).
  • This creates the PRS (Player's Restricted Shapley) score. It tells you: "Based on the specific plays this player was actually part of, how much did they increase the chance of a goal?"

4. The Results: Who is the Real MVP?

The authors tested this on the Italian Serie A (2022/23 season), looking at teams like AC Milan and Napoli.

• The Surprise: The stats showed that some players who don't score many goals are actually superstars of teamwork.
  • Example: A defender might have a high PRS score because they consistently start the attacks that lead to goals, even if they never touch the ball in the penalty box.
  • Example: A striker might have a low PRS score if they only get the ball when the team is already in a bad position, even if they are a great finisher.
• The "Two-Dimensional" Map: They plotted players on a graph:
  • X-Axis (PRS): How good are you at building the play? (Teamwork)
  • Y-Axis (Finishing): How good are you at scoring the goal? (Individual Skill)
  • The Stars: Players in the top-right corner are the "Unicorns"—great at building and great at scoring (like Rafael Leão for Milan).
  • The Unsung Heroes: Players in the top-left are "Architects." They build amazing plays but don't score much (like some midfielders).
  • The "One-Trick Ponies": Players in the bottom-right score well but don't help build the play.

Why Does This Matter?

This isn't just for nerds with spreadsheets. It helps clubs make smarter money decisions.

• Scouting: If a team needs someone to organize the attack, they shouldn't just look at who scores the most. They should look for high PRS players (the Architects) who might be cheaper because traditional stats ignore them.
• Fairness: It helps players get paid what they are actually worth to the team's success, not just what their goal count says.

In short: This paper gives us a new lens to see football. It stops us from only cheering for the person who kicks the ball into the net and starts appreciating the whole orchestra that made the music possible.

Technical Summary

Here is a detailed technical summary of the paper "A Model-Based Restricted Shapley Value to Measure the Players' Contribution to Shot Actions in Football."

1. Problem Statement

Traditional football performance metrics, such as goals, assists, and Expected Goals (xG), suffer from significant limitations in evaluating player contributions:

• Lack of Context: Standard xG models focus solely on the shot itself (location, angle) and ignore the build-up play, passing sequences, and teamwork required to create the opportunity.
• Cooperative Blindness: Football is inherently a cooperative game, but existing metrics often fail to quantify individual marginal contributions within a coalition of players.
• Inapplicability of Standard Game Theory: The classical Shapley value assumes that all possible coalitions of players are admissible and observable. In football, this is false; only specific subsets of players (those involved in a specific passing network) contribute to a specific shot, and coaches restrict formations based on tactics. Standard Shapley calculations overestimate the sample space by including unobserved or tactically impossible coalitions.

The paper aims to develop a framework that quantifies individual player contributions to shot-ending actions by explicitly modeling the cooperative nature of offensive sequences and accounting for restricted coalition structures.

2. Methodology

The proposed framework consists of three main components: a new predictive model for shot quality, a restricted cooperative game theory approach, and a statistical inference mechanism.

A. The Expected Goal Action (xGA) Model

To serve as the "worth" or "cohesion" function ($\upsilon$) for the game theory model, the authors introduce xGA (Expected Goal Action).

• Concept: Unlike standard xG, which predicts the probability of a goal based only on shot characteristics, xGA predicts the probability of a goal based on the entire offensive sequence leading to the shot.
• Features: The model incorporates:
  • Shot features: Location ($X, Y$), angle.
  • Action features: Number of passes, number of players involved, average pass distance, coordinates of the first pass.
  • Contextual features: Match situation (open play, free kick, etc.), home/away status.
  • Performance features: A composite offensive performance index of the involved teammates (derived from Sofifa data).
• Algorithms: Two models were compared: Binary Regression with a complementary log-log link (BR cloglog) and XGBoost. XGBoost was selected as the final estimator due to superior sensitivity and performance metrics on the test set.

B. The Player's Restricted Shapley (PRS) Statistic

The authors adapt the Shapley value to the specific constraints of football:

• Restricted Coalitions: Instead of considering all $2^n$ possible subsets of players, the framework restricts coalitions to those tactically admissible and empirically observed in the passing network of a specific shot action.
• The Formula: The PRS for player $i$ is calculated as a weighted sum of marginal contributions over the restricted set of coalitions $C_i$:
  $$\phi^R_i(\upsilon) = \sum_{C \in C_i} w_i(C) [\upsilon(C \cup \{i\}) - \upsilon(C)]$$
  Where weights $w_i(C)$ are normalized probabilities conditional on the observed coalition structures.
• Worth Function: The value $\upsilon(S)$ of a coalition $S$ is the sum of the predicted xGA probabilities for all actions where the players in $S$ participated. For unobserved but compatible coalitions, out-of-sample xGA predictions are used.

C. Statistical Inference and Normalization

Since the worth function is estimated via a statistical model and the data is finite, the raw Shapley values have sampling uncertainty.

• Bootstrap: The authors use a bootstrap procedure ($B=1,000$ replications) to estimate the standard error ($SE$) of the PRS estimator.
• PRS Statistic: To allow for meaningful comparison across players with different participation levels and uncertainty, they define the Player's Restricted Shapley (PRS) statistic as a Signal-to-Noise Ratio:
  $$PRS_i = \frac{\hat{\phi}^R_i}{SE(\hat{\phi}^R_i)}$$
  This dimensionless metric indicates how large a player's contribution is relative to its estimation noise.

3. Key Contributions

1. Novel Metric (xGA): Extends the xG paradigm to include build-up play and passing networks, providing a more holistic measure of shot quality.
2. Theoretical Adaptation: Successfully adapts cooperative game theory (Shapley value) to a domain with restricted, non-random coalition structures, moving beyond the assumption that all player combinations are possible.
3. Inferential Framework: Introduces a bootstrap-based approach to quantify the uncertainty of player contributions, allowing for statistically robust comparisons (via the PRS statistic) rather than just point estimates.
4. Action-Specific Attribution: Provides a method to attribute value to specific actions based on the actual passing network, rather than averaging over a full season or match without context.

4. Empirical Results

The framework was applied to 8,421 shot actions from the Italian Serie A 2022/23 season, with case studies on AC Milan and SSC Napoli.

• Model Performance: The xGA model (using XGBoost) significantly outperformed the traditional xG benchmark. Key improvements included:

  • Matthews Correlation Coefficient (MCC): +10.5%
  • Precision: +7.1%
  • F1 Score: +5.7%
  • Specificity: +6.2%
  • This confirms that incorporating build-up context improves the prediction of scoring events.

• Player Contributions (PRS):

  • Heterogeneity: There was significant variation in contributions within and between teams.
  • Top Performers: Victor Osimhen (Napoli) had the highest PRS (4.84), reflecting his central role in Napoli's attack. Olivier Giroud (3.81) and Rafael Leão (3.44) were the top contributors for AC Milan.
  • Defensive Involvement: Defenders like Fikayo Tomori (AC Milan) and Min-jae Kim (Napoli) showed positive PRS values, highlighting their role in ball progression and build-up, not just defense.
  • Negative Contributions: Some players had negative PRS values, indicating that their involvement in passing networks was associated with lower-quality shot opportunities compared to the average.

• Integration with Finishing Efficiency:

  • By plotting PRS (cooperative contribution) against $G90 - xG90$ (finishing overperformance), the authors identified four distinct player archetypes:
    1. High Contribution / High Finishing: (e.g., Leão) – Elite all-round attackers.
    2. Low Contribution / High Finishing: (e.g., Messias) – Players who finish well but are less involved in the build-up.
    3. High Contribution / Low Finishing: (e.g., Giroud, De Ketelaere) – Players who create high-quality chances but may underperform in converting them personally.
    4. Low Contribution / Low Finishing: Players with limited offensive impact.

5. Significance and Implications

• Decision Support: The PRS statistic offers clubs a tool for "economically sustainable" decision-making. It helps identify players whose collective impact is undervalued by traditional stats (e.g., a midfielder who creates high-quality chances but doesn't score) and those whose goal-scoring masks a lack of team involvement.
• Scouting and Recruitment: The framework allows scouts to evaluate players based on their specific tactical fit and ability to integrate into cooperative structures, rather than just raw output.
• Tactical Analysis: Coaches can use the metric to understand how different players affect the quality of offensive sequences, aiding in lineup selection and tactical adjustments.
• Future Directions: The authors suggest extending the framework to non-shot possession sequences, incorporating tracking data for off-ball movement, and applying the model to multiple seasons and leagues to enhance generalizability.

In conclusion, this paper provides a rigorous, mathematically grounded, and empirically validated method to deconstruct the cooperative nature of football, offering a more nuanced and actionable view of player performance than existing metrics.