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Recursive Monotonicity of the Scorix: Borda Meets Condorcet

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Abstract

Back in the eighteenth century, the works of Borda and Condorcet laid the foundations of social choice theory. On the one hand, Borda proposed to exploit the positions at which each candidate is ranked. On the other hand, Condorcet proposed to exploit the relative positions of each pair of candidates. Both proposals have equally called the attention of the scientific community, leading to two diametrically opposed points of view of the theory of social choice. Here, we introduce the intuitive property of recursive monotonicity of the scorix, which will be proven to be a natural condition for Borda and Condorcet to agree. Furthermore, we propose a ranking rule that focuses on the search for recursive monotonicity of the scorix.

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Notes

  1. This vector of scores is unique up to a positive multiplicative constant.

  2. A weak order relation is a complete and transitive relation.

  3. As a ranking can be seen as a profile containing a single ranking, the restriction of a ranking to a subset of the set of candidates follows from Definition 6.

  4. It is fair to mention that two similar ranking rules, such as the methods of Dodgson (1876) and Kemeny (1959), are proved to be NP-hard problems (Bartholdi et al. 1989).

  5. \(S'\) actually is strictly recursively monotone w.r.t. \(a\succ b\succ c\succ d\).

  6. As the winning ranking according to the method of Kemeny (1959) for this profile of rankings is the ranking \(a\succ b\succ c\succ d\), the independence of the search for recursive monotonicity of the scorix w.r.t. the method of Kemeny is also proven.

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Authors and Affiliations

  1. KERMIT, Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Ghent, Belgium

    Raúl Pérez-Fernández & Bernard De Baets

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  1. Raúl Pérez-Fernández
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  2. Bernard De Baets
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Correspondence to Raúl Pérez-Fernández.

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Pérez-Fernández, R., De Baets, B. Recursive Monotonicity of the Scorix: Borda Meets Condorcet. Group Decis Negot 26, 793–813 (2017). https://doi.org/10.1007/s10726-017-9525-y

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