An axiomatic analysis of concordance–discordance relations

Abstract

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.

An Overview of ELECTRE Methods and their Recent Extensions

“…Finally, Bouyssou and Pirlot () consider also the veto condition, proposing the following result . Theorem The aforementioned Theorem 2 holds also when Proposition 1 of Theorem 2 is replaced by the following one :for each x j , y j , w j , z j ∈ X j , a − j , b − j , c − j , d − j , e − j , f − j ∈ X − j , j = 1, …, n , and aforementioned conditions ( RC1 ), ( RC2 ) and ( M1 ) hold.…”

A note on the asymmetric part of an outranking relation

“…). This is standard definition of already used for the study of CDR‐AT in Bouyssou and Pirlot (). We have the following lemma.…”

“…The strategy of Bouyssou and Pirlot (2002b, 2005a, 2009, first presented in Bouyssou et al (1997) and Bouyssou and Pirlot (1999), to characterize outranking relations is to introduce conditions limiting the number of equivalence classes of * i and * * i . Indeed, it is easy to see that an outranking relation generates a relation * i that has at most four equivalence classes.…”

“…We have studied, from an axiomatic point of view, both reflexive outranking relations (Bouyssou and Pirlot, , , ) and asymmetric outranking relations (Bouyssou and Pirlot, , ). We have also recently unified and generalized the analysis of both cases in Bouyssou and Pirlot ().…”

“…The strategy of Bouyssou and Pirlot (, , , ), first presented in Bouyssou et al. () and Bouyssou and Pirlot (), to characterize outranking relations is to introduce conditions limiting the number of equivalence classes of and .…”