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Isomorphic Distances Among Elections

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Computer Science – Theory and Applications (CSR 2020)

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Abstract

This paper is an invitation to study the problem of measuring distances between elections, for the case where both the particular names of the candidates and the voters are irrelevant. In other words, we say that two elections are at distance zero (or, that they are isomorphic) if it is possible to make them identical by renaming their candidates and voters, and we are interested in measuring how far are two given elections from being isomorphic. The study of such distances has just begun and in this paper we outline why we believe that it is interesting and what are the natural research directions.

Piotr Faliszewski was supported by the funds of the Polish Ministry of Science and Higher Education assigned to AGH University. Piotr Skowron was supported by the Foundation for Polish Science (Homing programme). Arkadii Slinko was supported by Marsden Fund grant 3706352 of The Royal Society of New Zealand. Stanisław Szufa was supported by NCN project 2018/29/N/- ST6/01303. Nimrod Talmon was supported by the Israel Science Foundation (grant No. 630/19).

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Notes

  1. 1.

    The references here are just examples giving pointers to both classic and new works.

  2. 2.

    Thus we are effectively interested in pseudodistances over elections. We will, however, typically omit the “pseudo” prefix to maintain simplicity of our writing.

  3. 3.

    One can verify that, indeed, Definition 3 provides functions \({d\hbox {-}\mathrm {ID}}\) that satisfy the requirements of (pseudo)distances. Further, one can also verify that the Diaconis-Graham inequality also holds for \({d_{{\mathrm {swap}}}\hbox {-}\mathrm {ID}}\) and \({d_{{\mathrm {Spear}}}\hbox {-}\mathrm {ID}}\).

  4. 4.

    We thank Gerhard Woeginger for suggesting this idea.

  5. 5.

    This follows from the fact that being able to compute \({d_{{\mathrm {swap}}}\hbox {-}\mathrm {ID}}\) implies the ability to compute Kemeny rankings, and this problem is \({{\mathrm {NP}}}\)-hard already for four voters [11]; see also the results of Bartholdi et al. [4] and Hemaspaandra et al. [26].

  6. 6.

    These approximation algorithms, however, would have to have approximation ratios very close to 1 if the computed distances were to be meaningful.

  7. 7.

    We thank Tuomas W. Sandholm for suggesting this idea.

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Acknowledgments

Some of the discussions related to the ideas presented in this paper happened during Dagstuhl Seminar 19381.

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Authors and Affiliations

  1. AGH University, Kraków, Poland

    Piotr Faliszewski

  2. University of Warsaw, Warsaw, Poland

    Piotr Skowron

  3. University of Auckland, Auckland, New Zealand

    Arkadii Slinko

  4. Jagiellonian Univeristy, Kraków, Poland

    Stanisław Szufa

  5. Ben-Gurion University, Be’er Sheva, Israel

    Nimrod Talmon

Authors
  1. Piotr Faliszewski
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  4. Stanisław Szufa
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Correspondence to Piotr Faliszewski .

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Editors and Affiliations

  1. University of Trier, Trier, Germany

    Henning Fernau

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Faliszewski, P., Skowron, P., Slinko, A., Szufa, S., Talmon, N. (2020). Isomorphic Distances Among Elections. In: Fernau, H. (eds) Computer Science – Theory and Applications. CSR 2020. Lecture Notes in Computer Science(), vol 12159. Springer, Cham. https://doi.org/10.1007/978-3-030-50026-9_5

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  • DOI: https://doi.org/10.1007/978-3-030-50026-9_5

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