Preference relations, transitivity and the reciprocal property (original) (raw)

Utility Theory and reciprocal pairwise comparisons: The Eigenvector Method

Socio-Economic Planning Sciences, 1986

The criticisms of Utility Theory focus on either its axioms or the construction of utility functions. Here we present a method which avoids the problems of uniqueness encountered in the construction of utility functions when using either the certainty equivalence method or the probability equivalence method. The method is based on the construction of ratio scale value functions from reciprocal pairwise comparisons and Saaty's Eigenvector Method. We show that under the assumption of cardinal consistency utility functions are a particular case of these ratio scales. Reciprocal pairwise comparisons allow decision makers to relax the transitivity assumption and help to derive a unique scaling of preferences.

Formal Properties of the Preference Relation with Comparability

In practical decision-making, it seems clear that if we hope to make an optimal or at least defensible decision, we must weigh our alternatives against each other and come to a principled judgment between them. In the formal literature of classical decision theory, it is taken as an indispensable axiom that cardinal rankings of alternatives be defined for all possible alternatives over which we might have to decide. Whether there are any items " beyond compare " is thus a crucial question for decision theorists to consider when constructing a formal framework. At the very least, it seems problematic to presuppose that no such incommensurability is possible on the grounds that it would make formalizing axioms for decision-making more difficult, or even intractable. With this in mind, I plan to argue in this paper that a formal notion of comparability can be introduced to the classical understanding of preference relations such that the question of comparability between alternatives can be taken non-trivially. Building on the work of Richard Bradley and Ruth Chang, I argue that the comparability relation should be understood to be transitive but not complete. I contend that this understanding of comparability within decision theory can explain both why we believe that some alternatives may be incommensurable, yet we are still able to make justified decisions despite incomplete preference relations. In Section I, I lay the groundwork for understanding the conceptual relationship between comparability and commensurability with respect to decision-making. In Section II, I will argue that Bradley's definition of the preference relation with comparability leads to absurdity and contradiction due to a small oversight, which I propose to remedy. Then,

Consistency of Reciprocal Preference Relations

2007 Ieee International Fuzzy Systems Conference, 2007

The consistency of reciprocal preference relations is studied. Consistency is related with rationality, which is associated with the transitivity property. For fuzzy preference relations many properties have been suggested to model transitivity and, consequently, consistency may be measured according to which of these different properties is required to be satisfied. However, we will show that many of them are not appropriate for reciprocal preference relations. We put forward a functional equation to model consistency of reciprocal preference relations, and show that self-dual uninorms operators are the solutions to it. In particular, Tanino's multiplicative transitivity property being an example of such type of uninorms seems to be an appropriate consistency property for fuzzy reciprocal preferences.

Underlying Criteria in Valued Preference Relations

Translation of Classical Dimension Theory into a valued context should allow a comprehensible view of alternatives, by means of an informative representation, being this representation still manageable by decision makers. In fact, there is an absolute need for this kind of representations, since being able to comprehend a valued preference relation is most of the time the very first difficulty decision makers afford, even when dealing with a small number of alternatives. Moreover, we should be expecting deep computational problems, already present in classical crisp Dimension Theory. A natural approach could be to analyze dimension of every α-cut of a given valued preference relation. But due to complexity in dealing with valued preference relations, imposing max-min transitivity to decision makers in order to assure that every α-cut defines a crisp partial order set seems quite unrealistic. In this paper we propose an alternative definition of crisp dimension, based upon a general representation result, that may allow the possibility of skipping some of those computational problems.

Majority-consistent preference orderings

Social Choice and Welfare, 1996

This paper considers the construction of sets of preferences that give consistent outcomes under majority voting. Fishburn [7] shows that by combining the concepts of single-peaked and single-troughed preferences (which are themselves examples of value restriction) it is possible to provide a simple description of the extent of agreement between individuals that allows the construction of sets that are as large as those previously known (for fewer than 7 alternatives) and larger than those previously known (for 7 or more alternatives). This paper gives a characterisation of the preferences generated through these agreements and makes observations on the relation between the sizes of such sets as the number of alternatives increases.

A note on the representation of preferences

Mathematical Social Sciences, 1995

We consider a class of relations which includes irreflexive preference relations and interdependent preferences. For this class, we obtain necessary and sufficient conditions for representation of the relation by two numerical functions in the sense of a ~ x if and only if u(a) < vex).

On the Preference Relations with Negatively Transitive Asymmetric Part. I

Comptes rendus de l'Académie bulgare des sciences: sciences mathématiques et naturelles

Given a linearly ordered set I, every surjective map p : A → I endows the set A with a structure of set of preferences by "replacing" the elements ι ∈ I with their inverse images p −1 (ι) considered as "balloons" (sets endowed with an equivalence relation), lifting the linear order on A, and "agglutinating" this structure with the balloons. Every ballooning A of a structure of linearly ordered set I is a set of preferences A whose preference relation (not necessarily complete) is negatively transitive and every such structure on a given set A can be obtained by ballooning of certain structure of a linearly ordered set I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of balloons. As a consequence of this characterization, under certain natural topological conditions on the set of preferences A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.

Some algebraic characterizations of preference structures

Journal of Interdisciplinary Mathematics, 2004

In decision theory several preference structures are used for modeling coherence and rational behavior. In this paper we establish, from an algebraic approach, characterizations of some general properties involving preference and indifference relations, as well as the more common preference structures used in the literature.