A continuous rating method for preferential voting: the complete case

Abstract

A method is given for quantitatively rating the social acceptance of different options which are the matter of a complete preferential vote. Completeness means that every voter expresses a comparison (a preference or a tie) about each pair of options. The proposed method is proved to have certain desirable properties, which include: the continuity of the rates with respect to the data, a decomposition property that characterizes certain situations opposite to a tie, the Condorcet-Smith principle, and clone consistency. One can view this rating method as a complement for the ranking method introduced in 1997 by Markus Schulze. It is also related to certain methods of cluster analysis and one-dimensional scaling.The outcome of a vote is commonly expected to specify not only a winner and an ordering of the candidates, but also a quantitative estimate of the social acceptance of each of them. Such a quantification is expected even when the individual votes give only qualitative information.The simplest voting methods are clearly based upon such a quantification. This is indeed the case of the plurality count as well as that of the Borda count. However, it is well known that these methods do not comply with basic majority principles nor with other desirable conditions. In order to satisfy certain combinations of such principles and conditions, one must resort to other more elaborate methods, such as the celebrated rule of Condorcet, Kemény and Young (see Monjardet 1990 and Tideman 2006, pp. 182-190), the method of ranked pairs (Tideman 1987;Zavist and Tideman 1989; Tideman 2006, pp. 219-223), or what we will refer to as the method of paths, which was introduced by Schulze in 1997 (posted in a mailing list about election methods; R. Camps · X. Mora (B) · L. Saumell

A general method for deciding about logically constrained issues

“…On the other hand, when the doctrine corresponds to the notion of a total order, then the proposed method corresponds essentially to the one that was introduced in 1997 by Markus Schulze (posted in a mailing list about election methods; see [24,25], [27, pp. 228-232] and [4,5]). …”

Fraction-like ratings from preferential voting

“…Notice that, contrarily to ρ x , lower mean ranks correspond to a higher acceptance. The ratings R x that were considered in [1,2] are nothing else than the mean ranks that are obtained after transforming the Llull matrix by means of the CLC projection. The mean preference scores ρ x can certainly be rescaled to add up to 1.…”

“…Section 2 is devoted to Zermelo's method by itself, with some new results, especially in connection with the continuous dependence of the ratings on the data in the reducible case. Section 3 looks at certain properties of the paired-comparison matrices that arise from the CLC projection of [1,2]. Section 4 combines the previous results to show that the concatenation of the CLC projection and Zermelo's method achieves the desired properties.…”

Social choice rules driven by propositional logic

“…In this case, our general method corresponds essentially to the method introduced in 1997 by Markus Schulze (posted in a mailing list about election methods; see [22,23], [25, p. 228-232] and [6,7]), sometimes called the method of paths (in the incomplete case, however, it does not coincide with any of the variants given in [23]). In the way that we have introduced it, it is clearly a method for ranking all the candidates.…”

“…8.1]). Another interesting property of the method of paths is clone consistency, also known as independence of clones, which refers to the effect of replacing a single option c by a set C of several options similar to c; for more details we refer the reader to [ Moreover, it has also been shown [6,7] that this method can be extended to a continuous rating method that allows to sense the closeness of two candidates at the same time that it allows to recognise certain situations that are quite opposite to a tie.…”