Fuzzy Weak Preference Relation Research Articles

Transitive full covers of incomplete preference relations

When a decision maker is asked to compare a set of alternatives, it may happen that the information provided is incomplete because she has no time to compare all the options or is unable to compare some alternatives against others. This contribution departs from an incomplete fuzzy weak preference relation by completing it on a consistent way with the known information. Herein the original notion of fuzzy transitivity, min-transitivity, is considered. If the decision maker is assumed to be coherent, i.e. under the assumption that the known preferences satisfy transitivity, a complete transitive preference relation that preserves the information given by the coherent decision maker is derived. In the case of the given preference values violate transitivity, a degree of transitivity is defined, and an algorithm is presented to provide the most coherent preference relation that preserves the original information according to that degree of transitivity.

On the Rationality of Some Crisp Choice Functions Based on Strongly Complete Fuzzy Pre-Orders

Barrett, Pattanaik and Salles [Fuzzy Sets and Systems 34 (1990) 197–212] introduced nine alternative rules for generating exact choice sets from fuzzy weak preference relations (FWPR) and four rationality properties. They showed that, when preferences are fuzzy pre-orders, most of these alternative rules (preference-based choice functions or PCFs) violate at least two rationality properties. Following in the same direction, Fotso and Fono [New Mathematics and Natural Computation 8 (2012) 257–272] characterized these PCFs and analyzed their consistency in the cases of strongly complete fuzzy pre-orders and crisp complete pre-orders. In this paper, based on results of the two previous papers, we determine to what extend the PCFs are rational with respect to the structure of the underlying relation. More specifically, for each of the nine alternative rules violating a given rationality property, we determine respectively crisp pre-orders and strongly complete fuzzy pre-orders for which the PCF satisfies the property.

Generalized manipulability of fuzzy social choice functions

In many social decision making contexts, a manipulator attempts to change the social choice in his favor by misrepresenting his preferences. This paper deals with the strategic manipulation problem of social choice functions aggregating fuzzy individual preferences. It defines how the strategic misrepresentation of fuzzy preferences can be profitable for an individual with a fuzzy weak preference relation. The case of max-$\top$-transitive fuzzy preference relations is considered where $\top$ is a t-norm. Then, the impossibility of building a non-manipulable fuzzy social choice function except the dictatorial one is established, generalizing thus the well-known Gibbard-Satterthwaite's result. The obtained results generalizes also the one of Ben Abdelaziz et al. for max-min transitive fuzzy preference relations.

FUZZY STRICT PREFERENCE RELATIONS COMPATIBLE WITH FUZZY ORDERINGS

One of the most important issues in the field of fuzzy preference modelling is the construction of a fuzzy strict preference relation and a fuzzy indifference relation from a fuzzy weak preference relation. Here, we focus on a particular class of fuzzy weak preference relations, the so-called fuzzy orderings. The definition of a fuzzy ordering involves a fuzzy equivalence relation and, in this paper, the latter will be considered as the corresponding fuzzy indifference relation. We search for fuzzy strict preference relations compatible with a given fuzzy ordering and its fuzzy indifference relation. In many situations, depending on the t-norm and t-conorm used, this quest results in a unique fuzzy strict preference relation. Our aim is to characterize these fuzzy strict preference relations and to study their transitivity.

On the manipulability of the fuzzy social choice functions

In many social decision-making contexts, a manipulator has incentives to change the social choice in his favor by strategically misrepresenting his preference. Gibbard [Manipulation of voting schemes: a general result, Econometrica 41(4) (1973) 587–601] and Satterthwaite [Strategy-proofness and Arrow's conditions: existence and correspondence theorems for voting procedures and social welfare functions. J. Econom. Theory 10 (1975) 187–217] have shown that any non-dictatorial voting choice procedure is vulnerable to strategic manipulation. This paper extends their result to the case of fuzzy weak preference relations. For this purpose, the best alternative set is defined in three ways and consequently three generalizations of the Gibbard–Satterthwaite theorem to the fuzzy context are provided.

Factorization of fuzzy preferences

In the ordinary framework, the factorization of a weak preference relation into a strict preference relation and an indifference relation is unique. However, in fuzzy set theory, the intersection and the union of fuzzy sets can be represented different ways. Furthermore, some equivalent properties in the ordinary case have generalizations in the fuzzy framework that may be not equivalent. For these reasons there exist in the literature several factorizations of a fuzzy weak preference relation. In this paper we obtain and characterize different factorizations of fuzzy weak preference relations by means of two courses of action which are equivalent in the ordinary framework: axioms and definitions of strict preference and indifference.

The Paretian liberal with intuitionistic fuzzy preferences: A result

In the present paper we consider a situation in which the individual preferences and the social preference relation are intuitionistic fuzzy and study the compatibility between the Pareto principle and Sen’s minimal liberalism condition. A possible factorization of the intuitionistic fuzzy weak preference relation allows us to prove a possibility result for the case of max-min transitive social preference.

Fuzzy preferences and Arrow-type problems in social choice

There are alternative ways of decomposing a given fuzzy weak preference relation into its antisymmetric and symmetric components. In this paper I have provided support to one among these alternative specifications. It is shown that on this specification the fuzzy analogue of the General Possibility Theorem is valid even when the transitivity restrictions on the individual and the social preference relations are relatively weak. In the special case where the individual preference relations are exact but the social preference relation is permitted to be fuzzy it is possible to distinguish between different degrees of power of the dictator. This power increases with the strength of the transitivity requirement.

On choosing rationally when preferences are fuzzy

In real life, exact choices are induced by fuzzy preferences. These choices eventually satisfy certain plausible rationality properties. Several alternative rules for generating exact choices from fuzzy weak preference relations are introduced and the extent to which exact choice sets generated by these rules satisfy fairly weak rationality conditions is studied. Max-min transitivity is shown to be crucial for obtaining a positive result.