A note on the Sobol' indices and interactive criteria

TL;DR

This paper explores the use of Sobol' indices to measure interaction in multicriteria decision making, linking them to Fourier transforms and existing interaction measures like Banzhaf, providing new insights and computations.

Contribution

It introduces a novel approach to defining interaction using Sobol' indices, connecting them to Fourier transforms and extending analysis to the 2-additive Choquet integral.

Findings

Sobol' indices of the multilinear extension equal the square of the Fourier transform.

Established relationships between Sobol' indices, Fourier transforms, and Banzhaf interaction.

Computed Sobol' indices specifically for the 2-additive Choquet integral.

Abstract

The Choquet integral and the Owen extension (or multilinear extension) are the most popular tools in multicriteria decision making to take into account the interaction between criteria. It is known that the interaction transform and the Banzhaf interaction transform arise as the average total variation of the Choquet integral and multilinear extension respectively. We consider in this note another approach to define interaction, by using the Sobol' indices which are related to the analysis of variance of a multivariate model. We prove that the Sobol' indices of the multilinear extension gives the square of the Fourier transform, a well-known concept in computer sciences. We also relate the latter to the Banzhaf interaction transform and compute the Sobol' indices for the 2-additive Choquet integral.