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Fair feature attribution for multi-output prediction: a Shapley-based perspective

Umberto Biccari, Alain Ibáñez de Opakua, José María Mato, Óscar Millet, Roberto Morales, Enrique Zuazua

Abstract

In this article, we provide an axiomatic characterization of feature attribution for multi-output predictors within the Shapley framework. While SHAP explanations are routinely computed independently for each output coordinate, the theoretical necessity of this practice has remained unclear. By extending the classical Shapley axioms to vector-valued cooperative games, we establish a rigidity theorem showing that any attribution rule satisfying efficiency, symmetry, dummy player, and additivity must necessarily decompose component-wise across outputs. Consequently, any joint-output attribution rule must relax at least one of the classical Shapley axioms. This result identifies a previously unformalized structural constraint in Shapley-based interpretability, clarifying the precise scope of fairness-consistent explanations in multi-output learning. Numerical experiments on a biomedical benchmark illustrate that multi-output models can yield computational savings in training and deployment, while producing SHAP explanations that remain fully consistent with the component-wise structure imposed by the Shapley axioms.

Fair feature attribution for multi-output prediction: a Shapley-based perspective

Abstract

In this article, we provide an axiomatic characterization of feature attribution for multi-output predictors within the Shapley framework. While SHAP explanations are routinely computed independently for each output coordinate, the theoretical necessity of this practice has remained unclear. By extending the classical Shapley axioms to vector-valued cooperative games, we establish a rigidity theorem showing that any attribution rule satisfying efficiency, symmetry, dummy player, and additivity must necessarily decompose component-wise across outputs. Consequently, any joint-output attribution rule must relax at least one of the classical Shapley axioms. This result identifies a previously unformalized structural constraint in Shapley-based interpretability, clarifying the precise scope of fairness-consistent explanations in multi-output learning. Numerical experiments on a biomedical benchmark illustrate that multi-output models can yield computational savings in training and deployment, while producing SHAP explanations that remain fully consistent with the component-wise structure imposed by the Shapley axioms.
Paper Structure (24 sections, 12 theorems, 65 equations, 2 figures, 4 tables)

This paper contains 24 sections, 12 theorems, 65 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction and motivations
  2. Mathematical origins of SHAP analysis and the axiomatic approach to interpretability
  3. Biomedical motivation for multi-output modeling and interpretability
  4. SHAP in context: comparison with related interpretability methods
  5. Paper's contributions
  6. Paper's organization
  7. SHAP values for model interpretability
  8. Scalar cooperative games and the Shapley value
  9. Shapley operators for vector-valued games
  10. Shapley axioms in the vector-valued setting
  11. On axiomatic choices in the vector-valued setting
  12. SHAP values for ML models
  13. Numerical experiments
  14. Experimental setup and dataset description
  15. Model architectures and training
  16. ...and 9 more sections

Key Result

Theorem 2.4

There exists a unique value operator $\Phi:\mathcal{G}_{[n]} \to{\mathbb R}^n$ satisfying the axioms of efficiency, symmetry, dummy, and additivity. For each game $v\in\mathcal{G}_{[n]}$ and player $i\in[n]$, this operator is given by where $\mathfrak{S}_{[n]}$ is the set of all possible permutations of the players $[n]$ and $P_i(\pi)$ denotes the set of players preceding $i$ in permutation $\pi\

Figures (2)

  • Figure 1: Cumulative explained variance obtained from PCA on the standardized input features. The shape of the curve indicates that the dataset does not admit a strong low-dimensional linear representation.
  • Figure 2: SHAP beeswarm plots comparing multi-output and single-output models for the three prediction tasks considered. From top to bottom: age, MetSCORE, and sex. In each row, the left panel corresponds to the multi-output model, while the right panel shows the associated single-output model trained independently for the same target.

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Shapley operator
  • Theorem 2.4: shapley1953value
  • Theorem 2.5: roth1988expected
  • Definition 2.6: Vector-valued value operator
  • Definition 2.7: Vector-valued Shapley operator
  • Theorem 2.8: Rigidity of vector-valued Shapley operators
  • proof : Proof sketch.
  • Theorem 2.9
  • ...and 17 more