A note on representations of the Choquet integral via bases and transforms

Abstract

In this note, we discuss the validity of formula (4.54) in Grabisch's book from 2016. This equation expresses the Choquet integral of nonnegative functions w.r.t. a game in the form of linear invertible transform of set functions. However, the formula (erroneously) excludes the empty set, which has consequences for deriving Choquet integral representations in the literature using this approach. We point out that results for several transforms (e.g. Möbius, or co-Möbius transform) are not affected, but it is not the case of others (e.g. Fourier transform). In this spirit, we also provide a new formula for the Choquet integral representation via the Shapley interaction transform.