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Distance rationalization of anonymous and homogeneous voting rules

Author

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  • Benjamin Hadjibeyli

    (ENS Lyon)

  • Mark C. Wilson

    (University of Auckland)

Abstract

The concept of distance rationalizability of voting rules has been explored in recent years by several authors. Roughly speaking, we first choose a consensus set of elections (defined via preferences of voters over candidates) for which the result is specified a priori (intuitively, these are elections on which all voters can easily agree on the result). We also choose a measure of distance between elections. The result of an election outside the consensus set is defined to be the result of the closest consensual election under the distance measure. Most previous work has dealt with a definition in terms of preference profiles. However, most voting rules in common use are anonymous and homogeneous. In this case there is a much more succinct representation (using the voting simplex) of the inputs to the rule. This representation has been widely used in the voting literature, but rarely in the context of distance rationalizability. We show exactly how to connect distance rationalizability on profiles for anonymous and homogeneous rules to geometry in the simplex. We develop the connection for the important special case of votewise distances, recently introduced and studied by Elkind, Faliszewski and Slinko in several papers. This yields a direct interpretation in terms of well-developed mathematical concepts not seen before in the voting literature, namely Kantorovich (also called Wasserstein) distances and the geometry of Minkowski spaces. As an application of this approach, we prove some positive and some negative results about the decisiveness of distance rationalizable anonymous and homogeneous rules. The positive results connect with the recent theory of hyperplane rules, while the negative ones deal with distances that are not metrics, controversial notions of consensus, and the fact that the $$\ell ^1$$ ℓ 1 -norm is not strictly convex. We expect that the above novel geometric interpretation will aid the analysis of rules defined by votewise distances, and the discovery of new rules with desirable properties.

Suggested Citation

  • Benjamin Hadjibeyli & Mark C. Wilson, 2019. "Distance rationalization of anonymous and homogeneous voting rules," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 52(3), pages 559-583, March.
    • Handle: RePEc:spr:sochwe:v:52:y:2019:i:3:d:10.1007_s00355-018-1156-y
    DOI: 10.1007/s00355-018-1156-y
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    References listed on IDEAS

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    1. Edith Elkind & Piotr Faliszewski & Arkadii Slinko, 2015. "Distance rationalization of voting rules," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 45(2), pages 345-377, September.
    2. Edith Elkind & Piotr Faliszewski & Arkadii Slinko, 2012. "Rationalizations of Condorcet-consistent rules via distances of hamming type," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(4), pages 891-905, October.
    3. Tommi Meskanen & Hannu Nurmi, 2008. "Closeness Counts in Social Choice," Springer Books, in: Matthew Braham & Frank Steffen (ed.), Power, Freedom, and Voting, chapter 15, pages 289-306, Springer.
    4. Donald G. Saari & Vincent R. Merlin, 2000. "Changes that cause changes," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 17(4), pages 691-705.
    5. Peyton Young, 1995. "Optimal Voting Rules," Journal of Economic Perspectives, American Economic Association, vol. 9(1), pages 51-64, Winter.
    6. Lerer, Ehud & Nitzan, Shmuel, 1985. "Some general results on the metric rationalization for social decision rules," Journal of Economic Theory, Elsevier, vol. 37(1), pages 191-201, October.
    7. Cervone, Davide P. & Dai, Ronghua & Gnoutcheff, Daniel & Lanterman, Grant & Mackenzie, Andrew & Morse, Ari & Srivastava, Nikhil & Zwicker, William S., 2012. "Voting with rubber bands, weights, and strings," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 11-27.
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