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When are committees of Condorcet winners Condorcet winning committees?

A Correction to this article was published on 12 October 2021

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Abstract

We consider seat-posted (or designated-seat) committee elections, where disjoint sets of candidates compete for each seat. We assume that each voter has a collection of seat-wise strict rankings of candidates, which are extended to a strict ranking of committees by means of a preference extension. We investigate conditions upon preference extensions for which seat-wise Condorcet candidates, whenever all exist, form the Condorcet winner among committees. We characterize the domain of neutral preference extensions for which the committee of seat-wise winners is the Condorcet winning committee, first assuming the latter exists (Theorem 1) and then relaxing this assumption (Theorem 2). Neutrality means that preference extensions are not sensitive to the names of candidates. Moreover, we show that these two characterizations can be stated regardless of which preference level is considered as a premise.

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Notes

  1. 1.

    A good real-life example of the difficulties brought by the presence of preferential dependencies is provided in Ratliff (2006). See also Hodge and Schwallier (2006) for a detailed analysis of preferential dependencies in multiple referenda.

  2. 2.

    The reader may refer to Lang and Xia (2016) for a review of methods that partially overcome the trade-off between expressivity and costs.

  3. 3.

    A preference over committees is separable if there exists a set of seat-wise rankings such that for any two committees disagreeing on a single seat, the preferred one is comprising the more preferred candidate for that seat.

  4. 4.

    This result extends previous ones obtained for multiple referenda. In particular, (Özkal-Sanver and Sanver 2006) show that if preferences over committees are separable, every anonymous seat-based procedure is Pareto inefficient. For the same setting, Cuhadaroglu and Lainé (2012) show that the committee formed by seat-wise plurality winners is Pareto-optimal at any profile of (separable) preferences over committees if and only if these preferences emanate from the Hamming distance criterion.

  5. 5.

    A generalization of the Ostrogorski paradox is analyzed in Laffond and Lainé (2009). Furthermore, Laffond and Lainé (2012) show that the committee of seat-wise plurality winners may also fail at implementing a compromise solution.

  6. 6.

    If a voter’s ideal committee is A, the Hamming distance between A and another committee B is the number of seats with different appointed candidates.

  7. 7.

    A detailed argument is provided in Sect. 6.

  8. 8.

    A CCG is exposed to several paradoxes (see Ratliff 2003; Diss and Mahajne 2020). Studies of voting methods selecting a CCG can be found in Barberà and Coelho (2008), Kamwa and Merlin (2015), Diss and Doghmi (2016), Aziz et al. (2017), Kamwa (2017a, 2017b), Diss et al. (2020) and Bubboloni et al. (2020).

  9. 9.

    In this context, a preference extension maps every ranking of candidates to a ranking of sets of candidates. Kamwa and Merlin (2018) consider the Leximin and the Leximax preference extensions.

  10. 10.

    A preliminary step towards the answer is provided in Kamwa and Merlin (2018), who prove that if preferences over committees are separable, a CCG coincides with the CCF if the latter exists.

  11. 11.

    The reader may refer to Ratliff and Saari (2014), who analyze at-large committee selection methods that address diversity constraints such as gender balance.

  12. 12.

    Another example of a lexicographic preference extension is given by the choice problem defined above Definition 3. In this example, \(\delta ^{i}\) is q-lexicographic with 1 q 2.

  13. 13.

    Under neutrality, a preference extension generates a unique committee preference up to a reshuffling of candidates’ names.

  14. 14.

    This simple remark already appears in Benoît and Kornhauser (1991) and is discussed in Kamwa and Merlin (2018) for at-large committees.

  15. 15.

    By Lemma 3 a responsive preference extension induces a separable committee preference. The fact that a profile of separable preference generates a separable majority preference and its consequences are emphasized in particular by Kadane (1972), Koehler (1975), Miller (1975), Hollard and Le Breton (1996) and Vidu (2002).

  16. 16.

    As this relabelling does not interact with the one made above, it does not imply any loss of generality.

  17. 17.

    We write the projection of \(C_{k}\) to \(\{k^{*},\ldots ,k\}\) as \( C_{k}\mid _{\{k^{*},\dots ,k\}}\)instead of \(C_{k\{k^{*},\dots ,k\}}\) in order to avoid a too heavy notation.

  18. 18.

    Intuitively, \(q_{\delta }\) defines a priority ordering over seats. Observe that the definition of \(\delta |_{\Omega }\) induces the inverse of this priority order. This expresses the fact that if seat s is given lower priority than seat \(s^{\prime }\), and when starting from the best rank vector \(\overrightarrow{1}\), losing one rank in s should be less harmful than losing one rank in \(s^{\prime }\).

  19. 19.

    Observe that one cannot have \(s^{*}=S\), since \(\delta \) is responsive.

  20. 20.

    Note that if \(s^{*}=1\), the designated columns for \(1\le s<s^{*}-1\) in preferences are redundant. And also straightforward changes are required in the notation, e.g., \((\overset{s^{*}-1}{\overbrace{1,...,1}},\overset{ i}{\overbrace{2,...,2}},1,\overset{S-s^{*}-i}{\overbrace{2,...,2}})\) becomes \((\overset{i}{\overbrace{2,...,2}},1,\overset{S-s^{*}-i}{ \overbrace{2,...,2}})\).

  21. 21.

    Observe that one may have \(\forall {\mathbf {P}}\in {\mathcal {L}}_{{\mathcal {C}}}\), \(\varepsilon ({\mathbf {P}})\in {\mathbb {L}}\). For instance, denote by \(A^{*}( {\mathbf {P}})\) the first-best committee according to \({\mathbf {P}}\), and define \( P_{s}^{i}\) by \(\forall s\in {\mathcal {S}}, \forall a_{s},b_{s}\in {\mathcal {C}} _{s}\), \(a_{s} \varepsilon _{s}({\mathbf {P}}) b_{s} \Leftrightarrow [a_{s},A^{*}({\mathbf {P}})_{-s}] {\mathbf {P}}^{i} [b_{s},A^{*}({\mathbf {P}})_{-s}]\). With words, for any seat s, we elicit voter i ’s preferences for two different candidates \(a_{s}\) and \(b_{s}\) by considering all committees which overlap the first-best committee for all but seat s. If \({\mathbf {P}}\) ranks the committee comprising \(a_{s}\) above the one comprising \(b_{s}\), we say that i prefers \(a_{s}\) to \(b_{s}\). This elicitation method generalizes the one above by allowing to pick different reference committees for different seats. However, for the elicitation maps characterized in our results, some committee preferences are mapped to \( {\triangle }\).

  22. 22.

    The reader may refer to Gehrlein and Lepelley (2017) for an overview of the main probabilistic approaches.

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Acknowledgements

Funding was provided by Istanbul Bilgi University (Grant No. BILGI Research Development Innovation Programme, POlarization viewed from SOcial choice Perspective (POSOP)) National Research Development and Innovation Office (Grant No. TKP2020, National Challenges Program, BME NC TKP2020).

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Correspondence to Jean Lainé.

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Authors are grateful to two reviewers for their valuable comments and suggestions. This research has been partially funded by the BILGI Research Development Innovation Programme, POlarization viewed from SOcial choice Perspective (POSOP), and the “TKP2020, National Challenges Program” of the National Research Development and Innovation Office (BME NC TKP2020).

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Aslan, F., Dindar, H. & Lainé, J. When are committees of Condorcet winners Condorcet winning committees?. Rev Econ Design (2021). https://doi.org/10.1007/s10058-021-00260-9

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Keywords

  • Committee selection
  • Condorcet choice rules
  • Separability
  • Preference extensions
  • Lexicographic property

JEL Classification

  • D71