A Characterization of Strict Concordance Relations

2006

The purpose of this note is to sharpen the results in Bouyssou and Pirlot (2005) giving an axiomatic characterization of concordance relations. We show how the conditions used in that paper can be weakened so as to become independent from the conditions needed to characterize a general conjoint measurement model tolerating intransitive and/or incomplete relations. This leads to a clearer characterization of concordance relations within this general model.

European Journal of Operational Research, 2009

Outranking methods propose an original way to build a preference relation between alternatives evaluated on several attributes that has a definite ordinal flavor. Indeed, most of them appeal the concordance / non-discordance principle that leads to declaring that an alternative is "superior" to another, if the coalition of attributes supporting this proposition is "sufficiently important" (concordance condition) and if there is no attribute that "strongly rejects" it (non-discordance condition). Such a way of comparing alternatives is rather natural. However, it is well known that it may produce binary relations that do not possess any remarkable property of transitivity or completeness. This explains why the axiomatic foundations of outranking methods have not been much investigated, which is often seen as one of their important weaknesses. This paper uses conjoint measurement techniques to obtain an axiomatic characterization of preference relations that can be obtained on the basis of the concordance / non-discordance principle. It emphasizes their main distinctive feature, i.e., their very crude way to distinguish various levels of preference differences on each attribute. We focus on outranking methods, such as ELECTRE I, that produce a reflexive relation, interpreted as an "at least as good as" preference relation. The results in this paper may be seen as an attempt to give such outranking methods a sound axiomatic foundation based on conjoint measurement.

Journal of Mathematical Psychology, 2004

This paper analyzes conjoint measurement models allowing for intransitive and/or incomplete preferences. This analysis is based on the study of marginal traces induced on coordinates by the preference relation and uses conditions guaranteeing that these marginal traces are complete.

European Journal of Operational Research, 2007

The purpose of this note is to sharpen the results in giving an axiomatic characterization of concordance relations. We show how the conditions used in that paper can be weakened so as to become independent from the conditions needed to characterize a general conjoint measurement model tolerating intransitive and/or incomplete relations. This leads to a clearer characterization of concordance relations within this general model.

2002

This paper analyzes conjoint measurement models allowing for intransitive and/or incomplete preferences. This analysis is based on the study of marginal traces induced on coordinates by the preference relation and uses conditions guaranteeing that these marginal traces are complete.

2013

Outranking relations such as produced by the Electre I or II or the Tactic methods are based on a concordance and non-discordance principle that leads to declaring that an alternative is "superior" to another, if the coalition of attributes supporting this proposition is "sufficiently important" (concordance condition) and if there is no attribute that "strongly rejects" it (non-discordance condition). Such a way of comparing alternatives is rather natural and does not require a detailed analysis of tradeoffs between the various attributes. However, it is well known that it may produce binary relations that do not possess any remarkable property of transitivity or completeness. The axiomatic foundations of outranking relations have recently received attention. Within a conjoint measurement framework, characterizations of reflexive concordance-discordance relations have been obtained. These relations encompass those generated by the Electre I and II methods, which are non-strict (reflexive) relations. A different characterization has been provided for strict (asymmetric) preference relations such as produced by Tactic. The goal of this paper is to analyze the relationships between reflexive and asymmetric outranking relations. Co-duality plays an essential rôle in our analysis. It allows to understand the correspondence between the previous characterizations. Making a step further, we provide a common axiomatic characterization for both types of relations. Applying the co-duality operator to concordance-discordance relations also yields a new and interesting type of preference relation that we call concordance relation with bonus. The axiomatic characterization of such relations results directly from co-duality arguments.

2009

Abstract Consistency of preferences is related to rationality, which is associated with the transitivity property. Many properties suggested to model transitivity of preferences are inappropriate for reciprocal preference relations. In this paper, a functional equation is put forward to model the ldquocardinal consistency in the strength of preferencesrdquo of reciprocal preference relations.

2022

We propose a class of semimetrics for preference relations any one of which is an alternative to the classical Kemeny-Snell-Bogart metric. (We take a fairly general viewpoint about what constitutes a preference relation, allowing for any acyclic order to act as one.) These semimetrics are based solely on the implications of preferences for choice behavior, and thus appear more suitable in economic contexts and choice experiments. In our main result, we obtain a fairly simple axiomatic characterization for the class we propose. The apparently most important member of this class (at least in the case of finite alternative spaces), which we dub the top-difference semimetric, is characterized separately. We also obtain alternative formulae for it, and relative to this metric, compute the diameter of the space of complete preferences, as well as the best transitive extension of a given acyclic preference relation. Finally, we prove that our preference metric spaces cannot be isometically embedded in a Euclidean space.

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,

Decision Analysis

This paper characterizes lexicographic preferences over alternatives that are identified by a finite number of attributes. Our characterization is based on two key concepts: a weaker notion of continuity called “mild continuity” (strict preference order between any two alternatives that are different with respect to every attribute is preserved around their small neighborhoods) and an “unhappy set” (any alternative outside such a set is preferred to all alternatives inside). Three key aspects of our characterization are as follows: (i) we use continuity arguments; (ii) we use the stepwise approach of looking at two attributes at a time; and (iii) in contrast with the previous literature, we do not impose noncompensation on the preference and consider an alternative weaker condition.