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

Abstract

In this note we study restrictions on the recently introduced super-additive and sub-additive transformations, $A\mapsto A^{\ast }$ and $A\mapsto A_{\ast }$, of an aggregation function $A$. We prove that if $A^{\ast }$ has a slightly stronger property of being strictly directionally convex, then $A=A^{\ast }$ and $A_{\ast }$ is linear; dually, if $A_{\ast }$ is strictly directionally concave, then $A=A_{\ast }$ and $A^{\ast }$ 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 $A_{\ast }=f$ and $A^{\ast }=g$.

12 pages, 1 figure