On the existence of aggregation functions with given super-additive and sub-additive transformations

TL;DR

This paper investigates the conditions under which super-additive and sub-additive transformations of aggregation functions are compatible, revealing that certain convexity properties force the functions to be linear or identical, thus limiting their coexistence.

Contribution

It establishes new restrictions on super-additive and sub-additive transformations, showing when they must coincide or be linear, and demonstrates the non-existence of certain aggregation functions under mild conditions.

Findings

If $A^*$ is strictly directionally convex, then $A=A^*$ and $A_*$ is linear.

If $A_*$ is strictly directionally concave, then $A=A_*$ and $A^*$ is linear.

There exist pairs of sub-additive and super-additive functions with no corresponding aggregation function.

Abstract

In this note we study restrictions on the recently introduced super-additive and sub-additive transformations, $A\mapsto A^*$ and $A\mapsto A_*$, of an aggregation function $A$. We prove that if $A^*$ has a slightly stronger property of being strictly directionally convex, then $A=A^*$ and $A_*$ is linear; dually, if $A_*$ is strictly directionally concave, then $A=A_*$ and $A^*$ is linear. This implies, for example, the existence of pairs of functions $f\le g$ sub-additive and super-additive on $[0,\infty[^n$, respectively, with zero value at the origin and satisfying relatively mild extra conditions, for which there exists no aggregation function $A$ on $[0,\infty[^n$ such that $A_*=f$ and $A^*=g$.