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 a property of 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 one-dimensional scaling or cluster analysis.
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 [12 ; 20 : p. 182-190], the method of ranked pairs [19 , 21 ; 20 : p. 219-223], or the method introduced in 1997 by Markus Schulze [15 , 16 ; 20 : p. 228-232], which we will refer to as the method of paths. Now, as they stand, these methods rank the candidates in a purely ordinal way, without properly quantifying the social acceptance of each of them. So, it is natural to ask for a method that combines the above-mentioned principles and conditions with a quantitative rating of the candidates.
A quantitative rating should allow to sense the closeness between two candidates, such as the winner and the runner-up. For that purpose, it is essential that the rates vary in a continuous way, especially through situations where ties or multiple orders occur.
On the other hand, it should also allow to recognise certain situations that are opposite to a tie. For instance, a candidate should get the best possible rate if and only if it has been placed first by all voters. This is a particular case of a more general condition that we will call decomposition. This condition, that will be made precise later on, places sharp constraints on the rates that should be obtained when the candidates are partitioned in two classes X and Y such that each member of X is unanimously preferred to every member of Y .
In this article we will produce a rating method that combines such a quantitative character with other desirable properties of a qualitative nature. Among them we will be especially interested in the following extension of the Condorcet principle introduced in 1973 by John H. Smith [17]: Assume that the set of candidates is partitioned in two classes X and Y such that for each member of X and every member of Y there are more than half of the individual votes where the former is preferred to the latter; in that case, the social ranking should also prefer each member of X to any member of Y . This principle is quite pertinent when one is interested not only in choosing a winner but also in ranking all the alternatives (or in rating them).
To our knowledge, the existing literature does not offer any other rating method that combines this principle with the above-mentioned quantitative properties of continuity and decomposition. We will refer to our method as the CLC rating method, where the capital letters stand for “Continuous Llull Condorcet”. The reader interested to try it can use the CLC calculator which has been made available at [13].
Of course, any rating automatically implies a ranking. In this connection, it should be noticed that the CLC rating method is built upon Schulze’s method of paths as underlying ranking method. As we will remark in the concluding section, we doubt that any of the other ranking methods mentioned above could be extended to a rating method with the properties of continuity and decomposition. Having said that, it should be clear that the present work is not aimed at saying anything new about ranking methods as such. Whatever we might say about them will always be in reference to the rating issue.
By reasons of space, this article is restricted to the complete case and to a particular class of rates that we call rank-like rates. We are in the complete case when every individual expresses a comparison (a preference or a tie) about each pair of options. The incomplete case requires some additional developments that are dealt with in a separate article [3]. In another separate article we deal with another class of rates that have a fraction-like character [4].
The present article is organized as follows: Section 1 gives a more precise statement of the problem and finishes with a general remark. Section 2 presents an heuristic outline of our proposal, ending with a summary of the procedure and an illustrative example. Section 3 introduces some mathematical language. Sections 4-10 give detailed mathematical proofs of the claimed properties. Sections 11-12 are devoted to other interesting properties of the concomitant social ranking, namely clone consistency and two weak forms of monotonicity. Finally, section 13 makes some concluding remarks and poses a few open questions.
1 Statement of the problem and a general remark 1.1 Let us consider a set of N opti
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