Ranking methods based on valued preference relations: A characterization of the net flow method1

This paper deals with the problem of ranking several alternatives on the basis of a valued preference relation. A system of three independent axioms is shown to characterize a ranking method based on "net flows" which contains as particular cases the rules of Copeland and Borda and is used in one of the PROMETHEE methods. IIntroduction Suppose that you want to compare a number of alternatives taking into account different points of view, e.g. several criteria or the opinion of several voters. A common practice in such situations is to associate with each ordered pair (a, b) of alternatives a number indicating the strength or the credibility of the proposition "a is at least as good as b", e.g. the sum of the weights of the criteria favouring a or the percentage of voters declaring that a is preferred or indifferent to b. Since Condorcet, we know that, when the different points of view taken into account are conflictual, it may not be easy to compare the alternatives on the basis of these numbers. In this paper we study a particular method allowing to build a ranking, i.e. a complete and transitive binary (crisp) relation2, on a set of alternatives given such information. In a similar context, Bouyssou and Perny (1990) envisage more general methods building a partial ranking ranking, i.e. a reflexive and transitive binary relation, on a set of alternatives. Let A be a finite set of objects called "alternatives" with at least two elements.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]. From a technical point of view, the condition a ≠ b could be omitted from this definition at the cost of a minor modification of our third axiom. However, since it is clear that the values R(a, a) are immaterial in order to rank the alternatives, we will use this definition throughout the paper. A ranking method ≥ is a function assigning a ranking ≥(R) on A to any valued relation R on A. An obvious way to obtain a ranking method is to associate a score S(a, R) with each alternative a and to rank the alternatives according to their scores, i.e. a ≥(R) b iff S(a, R) ≥ S(b, R) (1) 1 I wish to thank Patrice Perny, Marc Pirlot and Philippe Vincke for their extremely helpful comments on earlier drafts of this text. 2 A (crisp) binary 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. It is connected if for all a, b ∈ A with a ≠ b, either a S b or b S a. It is asymmetric if for all a, b ∈ A, a S b implies Not b S a. It is reflexive if a S a for all a ∈ A.