Parallel axiomatizations of the concave and convex integrals

Abstract

We provide parallel axiomatic foundations for the concave and convex integrals, which represent optimistic and pessimistic rules for aggregating interacting attributes. For the concave integral, we show that the existing characterizations hold once accordance for additive capacities is required for every finite attribute set rather than only when the attribute set is a singleton. Without this strengthening, the original axioms admit a family of distinct functionals. For the convex integral, we derive a representation as the maximum of ordinary integrals over additive measures dominated by the capacity and establish parallel axiomatic characterizations, with the relevant inequalities reversed relative to their concave counterparts.