On some properties of outranking relations based on a concordance-discordance principle

The purpose of this paper is to study some properties of outranking relations based on a concordance-discordance principle. We show that, whenever the structure of the set of alternatives is sufficiently rich, imposing "nice" transitivity properties on such outranking relations always leads to a somewhat unappealing distribution of "power" among the various attributes. These results directly apply to methods, such as TACTIC, that produce a crisp asymmetric outranking relation. We explore the links between these results and classical ones in the field of Social Choice Theory and show their relevance for users of outranking methods. IIntroduction A classical problem in the field of MCDM is to build a preference relation on a set of multiattributed alternatives on the basis of preferences expressed on each attribute and "inter-attribute" information such as weights and/or tradeoffs. A common way (see Keeney and Raiffa (1976)) to do so is to attach a number v(x) to each alternative x and to declare that x is preferred to y if and only if v(x) > v(y). Usually, the number v(x) depends on the evaluations x1, x2, ..., xn of x on the n attributes and we have v(x) = V(x1, x2, ..., xn). When one uses such a method, the preference relation that is built has "nice" transitivity properties. However, the definition of the aggregation function V may not always be an easy task (see, e.g., Roy and Bouyssou (1987)). Starting with ELECTRE I (see Roy (1968) or, for a presentation in English, Goicoechea et al. (1982)), a number of MCDM techniques, the so-called outranking methods, have been proposed that use an alternative way to build a preference relation based on a concordance-discordance principle (see, e.g., Roy and Bertier (1973), Vansnick (1986) and the bibliography of Siskos et al. (1983)). In these methods, the preference relation, which is often called an outranking relation, is built through a series of pairwise comparisons. Such pairwise comparisons can be done in many ways. The idea of concordance-discordance consists in declaring that an alternative x is preferred to an alternative y if a "majority" of the attributes supports this assertion (concordance condition) and if the opposition of the other attributes is not "too strong" (non-discordance condition). In this paper we will restrict our attention to methods aiming at building a crisp (i.e., nonfuzzy) and, for reasons to be explained in section 4, asymmetric preference relation1. 1 A (crisp) binary relation S on a set K is a subset of K2. Throughout the paper we will classically write a S b instead of (a, b) ∈ S. We say that a binary relation S on a set K is (for all a, b, c ∈ K): complete if a S b or b S a, asymmetric if a S b implies Not (b S a), transitive if a S b and b S c imply a S c, negatively transitive if Not(a S b) and Not(b S c) imply Not(a S c), It is without circuit if for all k≥1 and all a1, a2,..., ak ∈ K, a1 S a2, a2 S a3, ..., ak-1 S ak imply Not ak S a1.