Axiomatic characterization of pointwise Shapley decompositions

A common problem in various applications is the additive decomposition of the output of a function with respect to its input variables. Functions with binary arguments can be axiomatically decomposed by the famous Shapley value. For the decomposition of functions with real arguments, a popular method is the pointwise application of the Shapley value on the domain. However, this pointwise application largely ignores the overall structure of functions. In this paper, axioms are developed which fully preserve functional structures and lead to unique decompositions for all Borel measurable functions.