Model-Agnostic Decomposition of the Difference in Mean Response Between Two Groups via Game-Theoretic Attribution with Minimum-Switching-Distortion Coupling

Abstract

The difference in mean response between two groups can arise from differences in covariate-response relationships, covariate distributions, or both. We propose a model-agnostic decomposition (MAD) of this difference into structure and composition effects. We formulate a cooperative switching game with a model-switch player and individual covariate-switch players, organized into model and covariate unions. MAD employs the Owen value, whose quotient game yields the aggregate structure and composition effects and whose internal game allocates the aggregate composition effect across covariates, and thereby inherits its fair allocation properties. In the internal game, partial covariate switches create hybrid covariate vectors whose joint distributions are not determined by the group-specific covariate distributions alone. To resolve this indeterminacy, we propose a minimum-switching-distortion (MSD) coupling, formulated as an optimal-transport problem. The MSD coupling satisfies admissibility conditions of identity at coincidence, transposition equivariance, and coordinate equivariance, and its sparse support reduces allocation cost. Because MAD imposes no particular functional form, it accommodates machine-learning models that adapt to complex nonlinear and interactive relationships, thereby yielding more accurate decompositions. Simulation studies demonstrate the effectiveness of MAD and the MSD coupling in terms of decomposition accuracy, coupling admissibility, and computational cost. As an illustration, we decompose the difference in mean body mass index between UK and US males aged 30-70 years using microbiome data, at the aggregate level and across microbial genera.