Better than Just Average: Computing and Using Shapley Variance

Shapley values summarize a player’s marginal contributions across coalitions by their mean, potentially hiding large, inconsistent effects behind a constant attribution. Shapley variance complements this mean by measuring the squared variation of marginal contributions around it. Despite this natural interpretation, Shapley variance remains rarely used in explainable AI, in part because tools for computing it are limited. We develop exact and efficient algorithms for games represented by sparse polynomials, tree ensembles, and Gaussian processes with product-Hamming kernels. For general games, we build on recent surrogate-based approaches to Shapley value estimation, using our exact algorithms on learned surrogates to efficiently estimate Shapley variance. On local explanation games, for example, our polynomial estimator achieves 16-fold lower median squared error than the prior Monte Carlo estimator. Beyond enabling computation, our work demonstrates how Shapley variance also provides a new lens on Shapley value estimation itself. We show that complementary paired sampling reduces the error of Monte Carlo estimation exactly when the game’s odd component has less total Shapley variance than its even component, and derive a corresponding residual decomposition for Kernel SHAP and Leverage SHAP. Together, these results make Shapley variance practical to compute and reveal how variation in marginal contributions governs the behavior of widely used Shapley value estimators. We hope our work enables explanations that characterize not only the mean marginal contribution but also more accurately summarizes their distribution.