Open Access
Representation and aggregation of crisp and fuzzy ordered structures: applications to social choice
(2021) Raventós Pujol, Armajac; Induráin Eraso, Esteban; Campión Arrastia, María Jesús; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
The present memory is structured as follows: after the Introduction, in the Chapter 2 of preliminaries, we will pay attention to the three areas which sustain the development of this thesis. These are, binary relations, Social Choice and Fuzzy sets. Chapter 3 is devoted to the study of fuzzy Arrovian models. First, it is introduced the concept of a fuzzy preference. Next, we define fuzzy aggregation rules and all of the restrictions of common sense, which are inspired by the restrictions that come from the classic Arrovian model. Next, different models are defined in the fuzzy setting. Their definitions depend on the particular nuances and features of a preference (choosing a transitivity type and a connectedness type) and the restrictions on an aggregation function (choosing an independence of irrelevant alternatives property,an unanimity property, etc). Different possibility and impossibility theorems have been proved depending on the set of definition and restrictions. In Chapter 4 it is studied the problem of the decomposition of fuzzy binary relations. There, it is defined clearly the problem of setting suitable decomposition rules. That is, we analyze how to obtain a strict preference and an indifference from the weak preference in a fuzzy approach. In this chapter, the existence and the uniqueness of certain kind of decomposition rules associated to fuzzy unions are characterized. In Chapter 5, the decomposition rules studied in Chapter 4 are used to achieve a new impossibility result. It is important to point out that in the proof of the main result in this chapter it is introduced a new technique. In this proof, fuzzy preferences are framed through an auxiliary tuple of five crisp binary relations, that we name a pseudofuzzy preference. An aggregation model à la Arrow of pseudofuzzy preferences is also studied,but the main result is about the aggregation of fuzzy preferences that come from decompositions.Chapters 3, 4 and 5 constitute the main body of this memory. Then a section of conclusions is included. It contains suggestions for further studies, open problems and several final comments. Finally, an Appendix has been added in order to give an account of the work done within these three years, that can not be included in the body of the present memory.
Open Access
Why using topological and analytical methods in aggregation of fuzzy preferences?
(2020) Campión Arrastia, María Jesús; Induráin Eraso, Esteban; Raventós Pujol, Armajac; Estatistika, Informatika eta Matematika; Institute for Advanced Research in Business and Economics - INARBE; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
The Arrow’s Impossibility Theorem states that there is no function fusing individual preferences into a social one satisfying certain properties of 'common sense'. On the contrary, in some of the fuzzy extensions of the Arrovian model, possibility arises. We have developed a technique which has been able to prove new impossibility results in the fuzzy approach. In this poster, we will explain the fundaments of this technique and in which models we can apply it. This technique, is based on controlling the aggregation of fuzzy preferences through some aggregation functions of dichotomic preferences. For each fuzzy aggregation function, we get a family of dichotomic aggregation functions. Studying this family, we obtain information about the initial aggregation function. We will discuss why the fuzzy Arrovian models in which we can apply this technique are, in some sense, less fuzzy. Moreover, we will expose why we should use topological and analytical methods in the fuzzy models out of the scope of our technique.
Open Access
Unexpected thresholds from independence of irrelevant alternatives in fuzzy arrow theorems
(2019) Raventós Pujol, Armajac; Estatistika, Informatika eta Matematika; Institute for Advanced Research in Business and Economics - INARBE; Estadística, Informática y Matemáticas
It is well known Arrow Theorem and its impact into Social Choice. It states that under an apparently mild set of conditions no rule fusing individual preferences into a social one is possible. In order to solve this situation, a possibility is to skip from dichotomic preferences to fuzzy ones. All conditions imposed to aggregation rules should be adapted to the fuzzy setting and due to the existence of different generalizations for each condition, depending on the chosen combination, a possibility or an impossibility result arises. In addition, in case we find a reasonable fuzzy aggregation rule, in most situations dichotomic decisions have to be taken at the end of the day, so the use of thresholds over fuzzy preferences is compulsory to make any decision. Surprisingly, independence of irrelevant alternatives axioms induce different thresholds which, besides they can be used on discrete and dichotomic decision making, transform fuzzy spaces of preferences and its aggregation functions into discrete ones allowing the application of new techniques to their study.

Raventós Pujol, Armajac

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