Sobol’ indices and Shapley value

Let f be a finite variance function of d independent input variables. Sobol’ indices are used to measure the importance of input variables and subsets of them. They are based on a variance decomposition. A similar problem arises in economics, when the value produced through the joint efforts of a team is to be attributed to individual members of that team. The Shapley value is widely used to solve the attribution problem. If we take the joint benefit of a set of variables to be the variance that they explain and then compute the Shapley value we get something different from the Sobol’ indices. The resulting measure splits each component of variance equally among the variables it contains. Sobol’ indices are in fact Shapley values for variance derived quantities, but those quantities are not the variance explained.