Decomposition and Arrow-Like Aggregation of Fuzzy Preferences

Abstract

We analyze the concept of a fuzzy preference on a set of alternatives, and how it can be decomposed in a triplet of new fuzzy binary relations that represent strict preference, weak preference and indifference. In this setting, we analyze the problem of aggregation of individual fuzzy preferences in a society into a global one that represents the whole society and accomplishes a shortlist of common-sense properties in the spirit of the Arrovian model for crisp preferences. We introduce a new technique that allows us to control a fuzzy preference by means of five crisp binary relations. This leads to an Arrovian impossibility theorem in this particular fuzzy setting.

Fuzzy Arrovian Theorems when preferences are complete

“…In this work, we have described a new fuzzy model by means of crisp binary relations similarly as it has been done in other works with the same spirit (e.g. [4,6,22]). We decided to focus on complete preferences instead of general S-connected preferences as a starting point of a more general study.…”

“…Notice that other models in the literature use qualitative formulations of the independence of irrelevant alternatives property (see [7,19]). In these models, it is quite natural to build associated preorders compatible with the IIA property and use them to describe the aggregation function (as we made in [22]). However, in this paper, the qualitative IIA is a consequence of the quantitative IIA and the completeness.…”

“…A fuzzy preference R plays the role of a binary relation but in the fuzzy setting, we need the equivalent of the strict preference ≻ in the fuzzy setting. There are many definitions of the strict preference P R derived from R (see [15,20,22]). However, in this article we do not require those amount of analysis and precision.…”

“…First, we have been able to implement the technique we started in [21] and used in [22]. This new technique is based on controlling fuzzy preferences by means of crisp preferences.…”

Decomposition of Fuzzy Relations: An Application to the Definition, Construction and Analysis of Fuzzy Preferences

“…But, without this condition, if X = {x, y} and a fuzzy binary relation R on X is defined as R(x, y) = R(y, x) = R(x, x) = R(y, y) = 1, it decomposes to (P, I) with I(x, y) = I(x, x) = I(y, y) = 1 and P(x, y) = 0.5 and 0 otherwise. Finally, in a previous paper [23] the authors proved some necessary conditions for these kinds of decompositions. We will include them here for the sake of completeness.…”