An axiomatic analysis of concordance–discordance relations (original) (raw)
Related papers
A consolidated approach to the axiomatization of outranking relations: a survey and new results
Annals of Operations Research, 2015
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. In this paper we briefly review the various kinds of axiomatizations of outranking relations proposed so far in the literature. Then we analyze the relationships between reflexive and asymmetric outranking relations in a conjoint measurement framework, consolidating our previous work. Co-duality plays an essential rôle in our analysis. It allows us 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.
On the relationship between strict and non-strict outranking relations
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.
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,
A characterization of concordance relations (2002, révisé juin 2003)
The notion of concordance is central to many multiple criteria techniques relying on ordinal information, e.g. outranking methods. It leads to compare alternatives by pairs on the basis of a comparison of coalitions of attributes in terms of "importance". This note proposes a characterization of the binary relations that can be obtained using such comparisons, within a general framework for conjoint measurement that allows for intransitive preferences. We show that such relations are mainly characterized by the very rough differentiation of preference differences that they induce on each attribute.
A characterization of concordance relations
European Journal of Operational Research, 2005
The notion of concordance is central to many multiple criteria techniques relying on ordinal information, e.g. outranking methods. It leads to compare alternatives by pairs on the basis of a comparison of coalitions of attributes in terms of "importance". This paper proposes a characterization of the binary relations that can be obtained using such comparisons, within a general framework for conjoint measurement that allows for intransitive preferences. We show that such relations are mainly characterized by the very rough differentiation of preference differences that they induce on each attribute.
HAL (Le Centre pour la Communication Scientifique Directe), 2005
This paper studies strict preference relations on product sets induced by "ordinal aggregation methods". Such methods are interpreted here as performing paired comparisons of alternatives based on the "importance" of attributes favoring each element of the pair: alternative x will be preferred to alternative y if the attributes for which x is better than y are "more important" than the attributes for which y is better than x. Based on a general framework for conjoint measurement that allows for intransitive preferences, we propose a characterization of such preference relations. This characterization shows that the originality of these relations lies in their very crude way to distinguish various levels of "preference differences" on each attribute when compared to the preference relations usually studied in conjoint measurement. The relation between such preference relations and P. C. Fishburn's noncompensatory preferences is investigated.
Electre I choice procedures: An axiomatization approach
RAIRO - Operations Research, 2019
This paper discusses choice procedures that select the set of best alternatives taking into account reflexive binary relations (called pseudo tournaments in the paper), such as those that can be obtained when constructing an outranking relation `a' la Electre. The paper contains interesting results which link together the second exploitation step in the Electre I outranking method with two choice procedures (Gocha and Getcha choice procedures also known in the literature as Schwartz set and Smith set respectively). A set of results that characterize some properties of the two outranking methods (ElectI and ElectIP choice procedures) is also presented.
Handling effects of reinforced preference and counter-veto in credibility of outranking
European Journal of Operational Research, 2008
The aim of this paper is to generalize the way of computing the credibility of outranking in a multiple criteria aggregation procedure, in view of taking into account two new effects called reinforced preference and counter-veto. These effects concern only those criteria for which, as soon as action a is ''judged very strongly preferred'' to action b, one wishes that the credibility of outranking of a over b is greater than that for the case where (all things equal elsewhere) the preference is not ''judged very strong''. To achieve this goal, we propose two complementary ways. The first one involves a reinforced preference threshold which affects the concordance degree, and the second one involves a counter-veto threshold which affects the insertion of discordance degree in the calculation of the credibility of outranking. The introduction of these two new effects remains compatible with the handling of ordinal preference scales. The resulting new index of the credibility of outranking can be used, in particular, in ELECTRE methods.
The ordinal consistency of an incomplete reciprocal preference relation
Fuzzy Sets and Systems, 2014
The ordinal consistency is the usual weak transitivity condition that a logical and consistent person should use if he/she does not want to express inconsistent opinions, and therefore becomes the minimum requirement condition that a consistent reciprocal preference relation should verify. In this paper, we define and study the ordinal consistency of an incomplete reciprocal preference relation. We then develop an algorithm to judge whether an incomplete reciprocal preference relation is ordinally consistent. This proposed algorithm can also find all cycles of length 3 to n in the incomplete digraph of the incomplete reciprocal preference relation. Based on this proposed algorithm and two rules, we develop another algorithm to repair an inconsistent incomplete reciprocal preference relation and to convert it to one with ordinal consistency. Our algorithm eliminates the cycles of length 3 to n in the digraph of an incomplete reciprocal preference relation most effectively. Our proposed method can preserve the initial preference information as much as possible. Furthermore, the proposed method can be used for an incomplete reciprocal preference relation with strict comparison and non-strict comparison information. Finally, the effectiveness and validity of the proposed method are illustrated with examples.