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Metrizable preferences over preferences

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Abstract

A hyper-preference is a weak order over all linear orders defined over a finite set A of alternatives. An extension rule associates with each linear order p over A a hyper-preference. The well-known Kemeny extension rule ranks all linear orders over A according to their Kemeny distance to p. More generally, an extension rule is metrizable iff it extends p to a hyper-preference consistent with a distance criterion. We characterize the class of metrizable extension rules by means of two properties, namely self-consistency and acyclicity across orders. Moreover, we provide a characterization of neutral and metrizable extension rules, based on a simpler formulation of acyclicity across orders. Furthermore, we establish the logical incompatibility between neutrality, metrizability and strictness. However, we show that these three conditions are pairwise logically compatible.

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Notes

  1. Such as Bossert and Storcken (1992), Bossert and Sprumont (2014), Athanasoglou (2016), Doğan and Lainé (2016), who analyze the manipulation of social welfare functions, or Haeringer and Halaburda (2016) who introduce the concept of monotone strategy-proofness for social choice functions.

  2. The reader may refer to Jeffrey (1974), van der Veen (1981), Igersheim (2007), and Priddat (2014) for further development of the role hyper-preferences can have in questioning the relationship between morality and rationality.

  3. Axiomatic characterizations of the Kemeny metric are provided in Kemeny and Snell (1962), Bogart (1973), and Barthélémy (1979). The introduction of Can and Storcken (2013) gives an account of the usages of the swap distance which predates Kemeny.

  4. There are also rank-weighted versions of the Kemeny metric (giving a semi-metric) considered in García-Lapresta and Pérez-Román (2011) (see also Can (2014) and Can and Storcken (2015) for a generalization). In fact, both in the statistics literature and in the growing field of studying consensus, one can find beyond the Kemeny distance many measures of concordance or discordance between two linear or weak orders. In particular, different rank correlation indices, such as Spearman’s rho (Spearman 1904), Kendall’s tau (Kendall 1962) and Gini’s cograduation index (Gini 1954) have been considered for assigning grades of agreement between two rankings. These indices have been extended to the analysis of concordance within a set of rankings (see Eckert and Klamler (2011), García-Lapresta and Pérez-Román (2011), Alcalde-Unzu and Vorsatz (2011) and Borroni and Zenga (2007)). One can also note the alternative (semi-)metrics considered in Blin (1976), Cook and Seiford (1978), Armstrong et al. (1982), Cook and Kress (1985), and Monjardet (1997, 1998).

  5. Let \(A=\{a,b,c,d\}\). Consider, without loss of generality, the bijection \( f(\{a,b\})=1,\)\(f(\{b,c\})=2,f(\{b,d\})=3,f(\{c,d\})=4,f(\{a,c\})=5\) and \( f(\{a,d\})=6\). Hence \(w_{ab}^{f}=\frac{1}{2}\), \(w_{bc}^{f}=\frac{1}{4}\), \( w_{bd}^{f}=\frac{1}{8}\), \(w_{cd}^{f}=\frac{1}{16}\), \(w_{ac}^{f}=\frac{1}{32}\) , and \(w_{ad}^{f}=\frac{1}{64}\). For \(p=abcd\), \(q=bacd\), and \(q^{\prime }=adcb\), we have q\(\varepsilon _{K}^{*}(p)\)\(q^{\prime }\) for the Kemeny rule and \(q^{\prime }\varepsilon _{d_{w}}^{*}(p)\)\(q^{\prime }\) for the weigthed-pair rule.

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Authors and Affiliations

  1. Murat Sertel Center for Advanced Economic Studies, Istanbul Bilgi University, Istanbul, Turkey

    Gilbert Laffond & Jean Lainé

  2. Lirsa, Cnam, Paris, France

    Jean Lainé

  3. Université Paris-Dauphine, Université PSL, CNRS, LAMSADE, 75016, Paris, France

    M. Remzi Sanver

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  1. Gilbert Laffond
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Correspondence to Jean Lainé.

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This project has been supported by the ANR-14-CE24-0007-01 (CoCoRICoCoDEC), the project IDEX ANR-10-IDEX-0001-02 PSL MIFID and the PICS CNRS exchange programme. We thank Goksel Asan, Onur Dogan and Dominik Peters for useful discussions.

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Laffond, G., Lainé, J. & Sanver, M.R. Metrizable preferences over preferences. Soc Choice Welf 55, 177–191 (2020). https://doi.org/10.1007/s00355-019-01235-0

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