On Order Invariant Aggregation Functionals

This paper deals with aggregation functionals defined on arbitrary sets of ordinal numerical values in the framework of representational measurement theory. Our basic assumption is that such a functional represents a meaningful relation between variables; i.e., it is invariant under actions from an appropriate automorphism group. We prove that an aggregation functional is continuous, idempotent, and invariant if and only if it can be represented in the form of the Choquet integral with respect to a monotonic {0, 1}- valued set function. We also establish a polynomial representation for these functionals.