A characterization of the min ranking method for valued preference relations

This paper deals with the problem of ranking several alternatives on the basis of a valued preference relation. We present a system of axioms which is shown to characterize a ranking method based on the Min operator that has been introduced in the literature. IIntroduction In order to compare a number of alternatives taking into account several criteria, many aggregation methods (e.g. ELECTRE III, Roy (1978) or PROMETHEE, Brans et al. (1984)) associate with each ordered pair (a, b) of alternatives a number indicating the strength or the credibility of a proposition such as: "a is preferred to b", e.g. the sum of the weights of the criteria for which a is preferred to b. At least since Condorcet, we know that, when the different criteria taken into account are conflictual, it may not be easy to compare the alternatives on the basis of these numbers. Many methods can be envisaged to rank alternatives on the basis of such information. In order to compare these methods we may study their behavior with regards to a number of "desirable" properties (see, e.g., Vincke (1991)). Alternatively, we may try to find a set of properties that would characterize a particular method, i.e. a set of properties that is satisfied by a unique method. Following Bouyssou (1991), this is the route followed in this paper in which we study and characterize the Min Method which ranks alternatives on the basis of the minimum credibility to which an alternative is preferred to all the others. After having introduced our definitions and notations in section 2, we present, in section 3, a number of "reasonable" properties of ranking methods and show in section 4 that the Min Method is the only ranking method satisfying all of them. An alternative characterization of the Min Method has been obtained by Pirlot (1991). II-Definitions and Notations Let A be a finite set of objects called "alternatives" such that |A| = n ≥ 2.We define a valued (binary) relation on A as a function R associating with each ordered pair of alternatives (a, b) ∈ A2 with a ≠ b an element of [0, 1]. Let R(A) be the set of all valued relations on A. A crisp (binary) relation S on A is a subset of A2. We will write a S b instead of (a, b) ∈ S. A crisp relation S on A is complete if for all a, b ∈ A either a S b or b S a. It is transitive if for all a, b, c ∈ A, a S b and b S c imply a S c. A complete and transitive crisp relation will be called a ranking. We define a ranking method ≥ as a function associating a ranking ≥(R) on A to any valued relation R on A.