Substitution Decomposition of Multilinear Functions with Applications to Utility and Game Theory

Abstract

A theory of decomposition “by substitution” for multi-linear (i.e. multi-affine) functions is presented. A representation theorem for such functions is shown to be given by a Moebius inversion formula. The concept of autonomous sets of variables (a “linear separability” of some kind, also known as “generalized utility independence”) captures the decomposition possibilities of a multi-linear function. Their entirety can be hierarchically represented by a so-called composition tree. Distinguished, strong forms of decompositions are shown to be given by multiplicative or additive functions. Important applications to the theories of multi-attribute expected-utilitv functions, switching circuits and cooperative n-person games are outlined.