Aggregation functions with given super-additive and sub-additive transformations

Abstract

Aggregation functions and their transformations have found numerous applications in various kinds of systems as well as in economics and social science. Every aggregation function is known to be bounded above and below by its super-additive and sub-additive transformations. We are interested in the 'inverse' problem of whether or not every pair consisting of a super-additive function dominating a sub-additive function comes from some aggregation function in the above sense. Our main results provide a negative answer under mild extra conditions on the super-and sub-additive pair. We also show that our results are, in a sense, best possible.

Remarks on Super-Additive and Sub-Additive Transformations of Aggregation Functions

“…Again, the terminology reflects the fact that A * is always a super-additive and A * is always a sub-additive aggregation function. The properties of these transformations have been extensively studied in the literature, see, e.g., [3]- [7], [10]- [12]. Also note that A * might assume the value of ∞; if this happens at some point then we will say that A * escapes locally.…”

“…It builds on two basic notions of super-and sub-additive transformations of aggregation functions introduced recently in [3] and studies their further properties and variations. We remark that transformations of aggregation functions as defined in [3] have been extensively studied as well, especially in connection with existence of aggregation functions with given super-and sub-additive transformations; see, e.g., [4]- [7], [10]- [12].…”

Sub-Additive Aggregation Functions and Their Applications in Construction of Coherent Upper Previsions

“…Some properties of revenue transformations are proven; in particular, it is proven that the revenue transformation transforms any aggregation function to some other aggregation function that is sub-additive and is bounded belowed by the transformed function. There already exists a sub-additive transformation of aggregation functions introduced in [7] and is heavily studied by researchers [8][9][10]. This sub-additive transformation always exists and is bounded above by the aggregation function that is being transformed, whereas examples are given in this paper of aggregation functions that do not have a bounded revenue transformation.…”

Coherent Upper Conditional Previsions Defined through Conditional Aggregation Operators

“…In [12,13], construction methods of coherent lower previsions based on collection integrals were proposed; in particular, a new type of integral, named the super-additive integral, was introduced. In [14], coherent upper previsions were constructed by aggregation functions [15][16][17][18] which are shift-invariant, positively homogeneous, and sub-additive [19][20][21][22]. In this paper, the construction of coherent upper conditional previsions with conditional aggregation operators is investigated.…”