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.
Figures & tables
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.
Model
Output
R2
RMSE
MAE
M1 Multi-output
Age
0.5614
8.9306
6.9946
MetSCORE
0.728266
0.1343
0.0963
M2 Multi-output
Age
0.6206
8.3061
6.3454
MetSCORE
0.8123
0.1116
0.0764
M2 Single-output
Age
0.6529
7.9438
6.0959
MetSCORE
0.8173
0.1101
0.0745
Table 1. Test-set regression performance for the continuous outputs (Age and MetSCORE). Results are reported for the linear multi-output baseline (M1), the nonlinear multi-output model (M2), and the corresponding single-output models based on the same architecture.
Model
ACC
AUC
F1-score
M1 Multi-output
0.9373
0.9814
0.9462
M2 Multi-output
0.9485
0.9869
0.9559
M2 Single-output
0.9498
0.9849
0.9568
Table 2. Test-set classification performance for sex prediction. Results are shown for the multi-output models M1 and M2, as well as for the single-output model derived from the M2 architecture.
Time (s)
RAM (MB)
Age
81.71
605.96
MetSCORE
84.60
603.93
Sex
83.13
603.11
Total M2 single-output
249.44
605.96
Total M2 multi-output
89.3678
599.15
Table 3. Training time and peak RAM usage for the M2 architecture in the multi-output and single-output settings. Single-output models are trained independently for each target (Age, MetSCORE, and Sex), and their total training time is reported as the sum over tasks, while the RAM value corresponds to the maximum peak memory observed among the single-output runs.
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.
Output
Cosine similarity
Spearman correlation
Age
0.9867
0.9460
MetSCORE
0.9949
0.9003
Sex
0.9740
0.8639
Table 4. Cosine similarity and Spearman rank correlation between feature-importance ranking, computed over the full set of input features for each output variable.