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

Biccari U, Ibáñez de Opakua A, Mato JM, Millet O, Morales R, Zuazua Iriondo E (2026)

Publication Language: English

Publication Status: Submitted

Publication Type: Unpublished / Preprint

Future Publication Type: Journal article

Publication year: 2026

Open Access Link: https://doi.org/10.48550/arXiv.2602.22882

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.

Authors with CRIS profile

Enrique Zuazua Iriondo Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

CIC bioGUNE ES Spain (ES) University of Deusto / Universidad de Deusto ES Spain (ES) Deutsches Zentrum für Neurodegenerative Erkrankungen (DZNE) DE Germany (DE)

How to cite

APA:

Biccari, U., Ibáñez de Opakua, A., Mato, J.M., Millet, O., Morales, R., & Zuazua Iriondo, E. (2026). Fair feature attribution for multi-output prediction: A Shapley-based perspective. (Unpublished, Submitted).

MLA:

Biccari, Umberto, et al. Fair feature attribution for multi-output prediction: A Shapley-based perspective. Unpublished, Submitted. 2026.

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