Fairness Analysis with Shapley-Owen Effects

Abstract: We argue that relative importance and its equitable attribution in terms of Shapley-Owen effects is an appropriate one, and, if we accept a small number of reasonable imperatives for equitable attribution, the only way to measure fairness. On the other hand, the computation of Shapley-Owen effects can be very demanding. Our main technical result is a spectral decomposition of the Shapley-Owen effects, which decomposes the computation of these indices into a model-specific and a model-independent part. The model-independent part is precomputed once and for all, and the model-specific computation of Shapley-Owen effects is expressed analytically in terms of the coefficients of the model's \emph{polynomial chaos expansion} (PCE), which can now be reused to compute different Shapley-Owen effects. We also propose an algorithm for computing precise and sparse truncations of the PCE of the model and the spectral decomposition of the Shapley-Owen effects, together with upper bounds on the accumulated approximation errors. The approximations of both the PCE and the Shapley-Owen effects converge to their true values.