Weak Preference Relation Research Articles

Small amendment arguments: how they work and what they do and do not show

AbstractThe small improvement argument has been said to establish that the standard weak preference or value relation can be incomplete. We first show that the argument is one of three possible ‘small amendment arguments’, each of which would yield the same conclusion. Generalizing the analysis thus, we subsequently present a strong and a weak version of small amendment arguments and derive the exact rationality conditions under which they reveal incompleteness. The results show that the arguments (in any of their variants) need not reveal a problem for the possibility of rational choice. In fact, it can be argued that they only reveal such a problem if the underlying relation is complete rather than incomplete.

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.

A characterization of reciprocal fuzzy preference structures and its compatibility with standard fuzzy preference structures

Fuzzy relations R:A2→[0,1] are widely used in decision-making in order to represent degrees of preference between alternatives in A. Examining fuzzy relations, this paper first reflects on an inherent incompatibility inside the domain of reciprocal fuzzy preference relations and preference structures. Aiming at clarifying the semantics of reciprocal fuzzy relations by means of that of weak fuzzy relations, a general frame is then proposed for studying the transformation between them, based on the notion of reciprocal functions. Next, we show that reciprocal relations can be naturally endowed with a preference structure of their own, containing two relations respectively showing a semantics of strict preference and lack of preference. Finally, we study the compatibility between these reciprocal preference structures and the standard preference structures of weak fuzzy preference relations by means of an axiomatic approach, leading to a system of functional equations that is shown to admit different solutions, allowing to explicitly assign one of the available semantics for reciprocal preference relations.

Superset-robust collective choice rules

A new property of collective choice rules that we refer to as superset robustness is introduced, and we employ it in several characterization results. The axiom requires that if all individual preference orderings expand weakly (in the sense of set inclusion), then the corresponding social preference relation must also expand weakly. In other words, if a given profile is changed by adding instances of weak preference to some individual relations, then the social weak preference relation for the expanded profile must contain the social weak preference relation for the original—that is, the social relation cannot contract in response to the addition of pairs to the individual relations. We begin by examining social welfare functions (that is, collective choice rules such that the resulting social preferences are orderings) and then move on to rules that generate transitive (but not necessarily complete) social rankings. The remaining results of the paper focus on Suzumura-consistent collective choice rules. In all of these cases, it turns out that the property of superset robustness is closely related to classes of agreement-based collective choice rules. These are rules such that the social relation is determined by collecting the pairs on whose relative rankings the members of the decisive sets agree.

PROMETHEE-MP: a generalisation of PROMETHEE for multi-period evaluations under uncertainty

Sustainability is a major concern and decisions have to be made based on the triple bottom line while simultaneously evaluating the economic, social and environmental impacts. In this context, decisions generally have a planning horizon of several years or even decades and consequently need to be evaluated in the short, medium and long term under uncertainty. This paper intends to tackle this complexity and proposes a multi-period generalisation of PROMETHEE under uncertainty, named PROMETHEE-MP, which is based on a double aggregation (a multi-criteria aggregation and a temporal aggregation), followed by an exploitation phase. The multi-criteria aggregation step uses a generalisation of PROMETHEE III in a situation of random uncertainty with intervals generated by Monte Carlo simulation. For temporal aggregation, we use a measure of distance between pre-orders that captures indifference, strict preference, weak preference and incomparability relations. Finally, we illustrate the proposed PROMETHEE-MP in the context of sustainable forest management.

(1705-3711) UNCERTAINTY DATA CREATING INTERVAL-VALUED FUZZY RELATION IN DECISION MAKING MODEL WITH GENERAL PREFERENCE STRUCTURE

The paper introduces a new approach to preference structure, where from a weak preference relation derive the following relations:strict preference, indifference and incomparability, which by aggregations and negations are created and examined. We decomposing a preference relation into a strict preference, anindifference, and an incomparability relation.This approach allows one to quantify different types of uncertainty in selecting alternatives.In presented preference structure we use interval-valued fuzzy relations, which can be interpreted as a tool that may help to model in a better way imperfect information, especially under imperfectlydefined facts and imprecise knowledge.Preference structures are of great interest nowadays because of their applications, so we propose at the end the algorithm of decision making by use new preference structure.

Rational Choice with Intransitive Preferences

Traditional rational choice theory assumes that the weak preference relation of an agent is an ordering that is it satisfies reflexivity, completeness and transitivity. It is also well known that the ordering property is essential to build the traditional ordinal utility analysis of consumer behaviour. However, there can be many situations when the weak preference relation of an agent may violate transitivity property, and hence, is not an ordering. In such situations traditional ordinal utility analysis breaks down. This paper develops a framework and discusses all the important results of rational choice theory when preferences are intransitive. It looks at weaker rationality properties such as quasi-transitivity and acyclicity and based on that it introduces weaker concepts of rationality such as quasi-transitive rationality and acyclic rationality and characterizes them. It also brings in the congruence axioms and property of path independence, and establishes the link with rationality. Finally, it analyzes how the results will change if we bring in restricted domain assumption of the choice function. JEL Classification: D01, D10, D11

Fuzzy preorders: conditional extensions, extensions and their representations

The crisp literature provides characterizations of the preorders that admit a total preorder extension when some pairwise order comparisons are imposed on the extended relation. It is also known that every preorder is the intersection of a collection of total preorders. In this contribution we generalize both approaches to the fuzzy case. We appeal to a construction for deriving the strict preference and the indifference relations from a weak preference relation, that allows to obtain full characterizations in the conditional extension problem. This improves the performance of the construction via generators studied earlier.

Robust ordinal regression for decision under risk and uncertainty

We apply the Robust Ordinal Regression (ROR) approach to decision under risk and uncertainty. ROR is a methodology proposed within multiple criteria decision aiding (MCDA) with the aim of taking into account the whole set of instances of a given preference model, for example instances of a value function, which are compatible with preference information supplied by the Decision Maker (DM) in terms of some holistic preference comparisons of alternatives. ROR results in two preference relations, necessary and possible; the necessary weak preference relation holds if an alternative is at least as good as another one for all instances compatible with the DM’s preference information, while the possible weak preference relation holds if an alternative is at least as good as another one for at least one compatible instance. To apply ROR to decision under risk and uncertainty we have to reformulate such a problem in terms of MCDA. This is obtained by considering as criteria a set of quantiles of the outcome distribution, which are meaningful for the DM. We illustrate our approach in a didactic example based on the celebrated newsvendor problem.

A new criterion of choice between generalized triangular fuzzy numbers

In this paper we consider a new criterion of choice between generalized (or rounded) triangular fuzzy numbers based on the concept of the weighted possibilistic mean introduced by Fullér and Majlender. First we introduce a preference relation which establishes a partial order on the family of generalized triangular fuzzy numbers. Second, we present a weak preference relation which leads to a total order on this family. Then we consider the special case of triangular fuzzy numbers and rephrase the preceding results in terms of selected parameters characterizing a triangular fuzzy number. Finally we compare our results with a number of new approaches recently discussed in the literature.

Multiple criteria ranking and choice with all compatible minimal cover sets of decision rules

We introduce a new multiple criteria ranking/choice method that applies Dominance-based Rough Set Approach (DRSA) and represents the Decision Maker’s (DM’s) preferences with decision rules. The DM provides a set of pairwise comparisons indicating whether an outranking (weak preference) relation should hold for some pairs of reference alternatives. This preference information is structured using the lower and upper approximations of outranking (S) and non-outranking (Sc) relations. Then, all minimal-cover (MC) sets of decision rules being compatible with this preference information are induced. Each of these sets is supported by some positive examples (pairs of reference alternatives from the lower approximation of a preference relation) and it does not cover any negative example (pair of alternatives from the upper approximation of an opposite preference relation). The recommendations obtained by all MC sets of rules are analyzed to describe pairwise outranking and non-outranking relations, using probabilistic indices (estimates of probabilities that one alternative outranks or does not outrank the other). Furthermore, given the preference relations obtained in result of application of each MC set of rules on a considered set of alternatives, we exploit them using some scoring procedures. From this, we derive the distribution of ranks attained by the alternatives. We also extend the basic approach in several ways. The practical usefulness of the method is demonstrated on a problem of ranking Polish cities according to their innovativeness.

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.

A new distance measure including the weak preference relation: Application to the multiple criteria aggregation procedure for mixed evaluations

We introduce a new distance measure between two preorders that captures indifference, strict preference, weak preference and incomparability relations. This measure is the first to capture weak preference relations. We illustrate how this distance measure affords decision makers greater modeling power to capture their preferences, or uncertainty and ambiguity around them, by using our proposed distance measure in a multiple criteria aggregation procedure for mixed evaluations.

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.

Processing additional information in a multiple criteria context with mixed evaluations: including the weak preference relation

In a multiple criteria decision context facing inevitable and various kinds of information imperfections (uncertainty, imprecision, ambiguity...), it is rather natural to seek additional information. In this paper, we propose a unified procedure for processing additional information in such a context. It is based on the Bayesian model and the adaptation of information revision rules from the uniattribute to the multiattribute context. It leads to prior and posterior global preference relations including the strict preference, the indifference, the incomparability and the weak preference relations. The impact of additional information can then be assessed through the comparison of prior and posterior results.

On continuity of incomplete preferences

A weak (strict) preference relation is continuous if it has a closed (open) graph; it is hemicontinuous if its upper and lower contour sets are closed (open). If preferences are complete these four conditions are equivalent. Without completeness continuity in each case is stronger than hemicontinuity. This paper provides general characterizations of continuity in terms of hemicontinuity for weak preferences that are modeled as (possibly incomplete) preorders and for strict preferences that are modeled as strict partial orders. Some behavioral implications associated with the two approaches are also discussed.

Continuity, completeness and the definition of weak preferences

This note explores the connections between continuity and completeness under alternative conceptions of preference relations. For non-trivial preorders, it shows that, unlike the standard definitions, the weak preference relation defined in Galaabaatar and Karni (2010) allows for incomplete preferences while maintaining all the continuity properties of complete preference relations. It also makes it possible to distinguish indifference between alternatives from non-comparability of alternatives. If the preference relations are complete, this definition agrees with the customary definitions.

On the Ferrers property of valued interval orders

We study the relationship between the Ferrers property and the notion of interval order in the context of valued relations. Given a crisp preference structure without incomparability, the strict preference relation satisfies the Ferrers property if and only if the associated weak preference relation does. These conditions characterize a total interval order. For valued relations the Ferrers property can be written in two different and non-equivalent ways. In this work, we compare these properties by finding the kind of completeness they imply. Moreover, we study whether they still characterize a valued total interval orders.

Non-additive robust ordinal regression: A multiple criteria decision model based on the Choquet integral

Abstract Within the multicriteria aggregation–disaggregation framework, ordinal regression aims at inducing the parameters of a decision model, for example those of a utility function, which have to represent some holistic preference comparisons of a Decision Maker (DM). Usually, among the many utility functions representing the DM’s preference information, only one utility function is selected. Since such a choice is arbitrary to some extent, recently robust ordinal regression has been proposed with the purpose of taking into account all the sets of parameters compatible with the DM’s preference information. Until now, robust ordinal regression has been implemented to additive utility functions under the assumption of criteria independence. In this paper we propose a non-additive robust ordinal regression on a set of alternatives A, whose utility is evaluated in terms of the Choquet integral which permits to represent the interaction among criteria, modelled by the fuzzy measures, parameterizing our approach. In our methodology, besides holistic pairwise preference comparisons of alternatives from a subset of reference alternatives A ′ ⊆ A , the DM is also requested to express the intensity of preference on pairs of alternatives from A ′ , and to supply pairwise comparisons on the importance of criteria, and the sign and intensity of interaction among pairs of criteria. The output of our approach defines a set of fuzzy measures (capacities) such that the corresponding Choquet integral is compatible with the DM’s preference information. Moreover, using linear programming, our decision model establishes two preference relations for any a , b ∈ A : the necessary weak preference relation, if for all compatible fuzzy measures the utility of a is not smaller than the utility of b, and the possible weak preference relation, if for at least one compatible fuzzy measure the utility of a is not smaller than the utility of b.

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.