Sobol' Indices and Shapley Value

Global sensitivity analysis measures the importance of some input variables to a function f by looking at the impact on f of making large random perturbations to subsets of those variables. Using measures like those of Sobol' we can attribute importance to input variables based on the extent to which they help predict the target function f. There is a longstanding literature in economics and game theory that considers how to attribute the value of a team effort to individual members of that team. The primary result, known as the Shapley value, is the unique method satisfying some intuitively necessary criteria. In this paper we find the Shapley value of individual variables when we take "variance explained" as their combined value. The result does not match either of the usual Sobol' indices. It is instead bracketed between them for variance explained or indeed any totally monotone game. Because those indices are comparatively easy to compute, Sobol' indices provide effectively computable bounds for the Shapley value.