An Abstract Result on Projective Aggregation Functions

Juan Candeal

doi:10.3390/axioms7010017

Abstract

A general characterization result of projective aggregation functions is shown, the proof of which makes use of the celebrated Arrow’s theorem, thus providing a link between aggregation functions theory and social choice theory. The result can be viewed as a generalization of a theorem obtained by Kim (1990) for real-valued aggregation functions defined on the n-dimensional Euclidean space in the context of measurement theory. In addition, two applications of the core theorem of the article are shown. The first is a simple extension of the main result to the context of multi-valued aggregation functions. The second offers a new characterization of projective bijection aggregators, thus connecting aggregation operators theory with social choice.

Highlights

This short note aims to further contribute to the study of the interplay existing among aggregation functions, aggregation operators and social choice
This paper provides a general characterization result of projective aggregation functions defined on an abstract set by using a purely social choice approach; to wit, a suitable version of Arrow’s theorem
The variant of Arrow’s theorem used in this situation states that a partial social welfare function G : (W≤ )n → R that satisfies the condition of independence of irrelevant alternatives and the Pareto condition is projective.)

Summary

Introduction

This short note aims to further contribute to the study of the interplay existing among aggregation functions, aggregation operators and social choice. Generalizations of the concept of an aggregation function appear when considering, for example, aggregation with ordinal and nominal scales, aggregation on posets and lattices or aggregation of preferences (see, e.g., [1,2,3,4,5]) All these developments have been closely linked to an increasing number of applications to many different topics such as image processing, decision making, pattern recognition, fuzzy control and social sciences, just to mention a few of them. Aggregation functions play an important role in measurement theory, as well (see [13] for a detailed account of this discipline) In this context, a key feature has to do with the preservation of particular numerical scales used to measure both the input and the output variables, which are, in turn, real variables. Not developed in the article, these two consequences could be applied to the context of aggregation of preferences and aggregation of classifications, as seen, for example, in [22]

Preliminaries

The Main Result

Some Consequences of the Main Result

Findings

Conclusions