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

TL;DR

This paper characterizes the theoretical constraints of fair feature attribution in multi-output models using Shapley axioms, revealing that explanations must decompose component-wise unless axioms are relaxed, with practical implications demonstrated on biomedical data.

Contribution

It extends Shapley axioms to multi-output settings, establishing a rigidity theorem that clarifies the structure of fair explanations in such models.

Findings

Component-wise attribution is necessary under classical axioms.

Multi-output models can be computationally efficient while maintaining explanation consistency.

Theoretical insights are supported by experiments on biomedical benchmarks.

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.