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Measuring Nearly Single-Peakedness of an Electorate: Some New Insights

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Algorithmic Decision Theory (ADT 2021)

Abstract

After introducing a new distance measure of a preference profile to single-peakedness, directly derived from the very definition of single-peaked preferences by Black [4], we undertake a brief comparison with other popular distance measures to single-peakedness. We then tackle the computational aspects of the optimization problem raised by the proposed measure, namely we show that the problem is NP-hard and we propose an integer programming formulation. Finally, we carry out numerical tests on real and synthetic voting data. The obtained results show the interest of the proposed measure, but also shed new light on the advantages and drawbacks of some popular distance measures.

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Notes

  1. 1.

    As in any axiomatic approach, the specific situation considered here does not need to often happen in practice: it is a thought experiment in which one considers a hypothetical situation and examines whether the measure would behave well in such a case.

  2. 2.

    Tests performed on an Intel Core i7 (1.3 GHz base, 3.9 GHz turbo) with 8 GB RAM.

  3. 3.

    http://www.cs.toronto.edu/~lex/code/asprgen.html.

  4. 4.

    https://www.preflib.org/data/index.php.

  5. 5.

    https://zenodo.org/record/1199545.

  6. 6.

    https://cran.r-project.org/package=PerMallows.

  7. 7.

    For a weak axis A, each preference relation in the profile is generated in two steps: (1) an axis \(A'\) compatible with A is generated uniformly at random; (2) a candidate \(c^*\) is drawn uniformly at random in \(\mathcal {C}\), then a preference relation is generated uniformly at random among preferences of peak \(c^*\) single-peaked w.r.t. \(A'\).

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Correspondence to Olivier Spanjaard .

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Escoffier, B., Spanjaard, O., Tydrichová, M. (2021). Measuring Nearly Single-Peakedness of an Electorate: Some New Insights. In: Fotakis, D., Ríos Insua, D. (eds) Algorithmic Decision Theory. ADT 2021. Lecture Notes in Computer Science(), vol 13023. Springer, Cham. https://doi.org/10.1007/978-3-030-87756-9_2

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  • DOI: https://doi.org/10.1007/978-3-030-87756-9_2

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