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

In this paper, we provide a theoretically grounded and computationally efficient alternative to SHAP. To this end, we study feature attribution through the lens of cooperative game theory by formulating a class of XAI--TU games. Building on this formulation, we investigate equal-surplus-type and proportional-allocation-type attribution rules and propose a low-cost attribution rule, ESENSC_rev2, constructed by combining two polynomial-time closed-form rules while ensuring the null-player property in the XAI--TU domain. Extensive experiments on tabular prediction tasks demonstrate that ESENSC_rev2 closely approximates exact SHAP while substantially improving scalability as the number of features increases. These empirical results indicate that equal-surplus-type attribution rules can achieve favorable trade-offs between computational cost and approximation accuracy in high-dimensional explainability settings. To provide theoretical foundations for these findings, we establish an axiomatic characterization showing that ESENSC_rev2 is uniquely determined by efficiency, the null-player axiom, a restricted differential marginality principle, an intermediate inessential-game property, and axioms that reduce computational requirements. Our results suggest that axiomatically justified and computationally efficient attribution rules can serve as practical and theoretically principled substitutes for SHAP-based approximations in modern explainability pipelines.

Machine learning pipelines commonly flatten relational data into single-table representations, discarding structural constraints. Widely used Shapley value-based feature attributions then rely on feature independence, evaluating the model on combinations that could never arise in the underlying data, producing misleading explanations. We propose RelShap, a framework that incorporates relational constraints and data provenance into Shapley value computation, restricting both background data and coalition evaluation to relationally valid configurations. The framework is estimator-agnostic and composes with Kernel SHAP, Monte Carlo, and Leverage SHAP without altering their sampling or weighting properties. Functional dependencies further induce equivalence classes over feature coalitions, which RelShap exploits to reduce runtime without changing Shapley values; we provide a combinatorial characterization of the expected speedup. Experiments across multiple datasets, models, and estimators show that RelShap produces explanations that are more faithful to the data-generating process, correctly identifying the dominant feature in controlled settings where existing methods, including Conditional SHAP and ManifoldShap, do not. Our code is available at: https://github.com/duneag2/relshap.

Originating in game theory, Shapley values are widely used for explaining a machine learning model's prediction by quantifying the contribution of each feature's value to the prediction. This requires a scalar prediction as in binary classification, whereas a multiclass probabilistic prediction is a discrete probability distribution, living on a multidimensional simplex. In such a multiclass setting the Shapley values are typically computed separately on each class in a one-vs-rest manner, ignoring the compositional nature of the output distribution. In this paper, we introduce Shapley compositions as a well-founded way to properly explain a multiclass probabilistic prediction, using the Aitchison geometry from compositional data analysis. We prove that the Shapley composition is the unique quantity satisfying linearity, symmetry and efficiency on the Aitchison simplex, extending the corresponding axiomatic properties of the standard Shapley value. We demonstrate this proper multiclass treatment in a range of scenarios.