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Divisor-Based Biproportional Apportionment in Electoral Systems: A Real-Life Benchmark Study

Author

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  • Sebastian Maier

    (Institute of Mathematics, University of Augsburg, D-86135 Augsburg, Germany)

  • Petur Zachariassen

    (University of the Faroe Islands, FO-100 Tórshavn, Faroe Islands)

  • Martin Zachariasen

    (Department of Computer Science, University of Copenhagen, DK-2100 Copenhagen, Denmark)

Abstract

Biproportional apportionment methods provide two-way proportionality in electoral systems where the electoral region is subdivided into electoral districts. The problem is to assign integral values to the elements of a matrix that are proportional to a given input matrix, and such that a set of row- and column-sum requirements are fulfilled. In a divisor-based method for biproportional apportionment, the problem is solved by computing appropriate row and column divisors, and by rounding the quotients. We present a comprehensive experimental evaluation of divisor-based biproportional apportionment in an electoral system context. By performing experiments on real-life benchmark instances (election data with multimember districts), we evaluate the general quality of divisor-based apportionments with respect to, e.g., deviation from quota, reversal orderings, and occurrences of ties. For example, we studied the frequency in which a party with a higher vote count in a district ended up with fewer seats in that district.

Suggested Citation

  • Sebastian Maier & Petur Zachariassen & Martin Zachariasen, 2010. "Divisor-Based Biproportional Apportionment in Electoral Systems: A Real-Life Benchmark Study," Management Science, INFORMS, vol. 56(2), pages 373-387, February.
    • Handle: RePEc:inm:ormnsc:v:56:y:2010:i:2:p:373-387
    DOI: 10.1287/mnsc.1090.1118
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    References listed on IDEAS

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    1. Michel L. Balinski & Gabrielle Demange, 1989. "Algorithm for Proportional Matrices in Reals and Integers," Post-Print halshs-00585327, HAL.
    2. Gassner, Marjorie, 1988. "Two-dimensional rounding for a quasi-proportional representation," European Journal of Political Economy, Elsevier, vol. 4(4), pages 529-538.
    3. Friedrich Pukelsheim, 2006. "Current Issues of Apportionment Methods," Studies in Choice and Welfare, in: Bruno Simeone & Friedrich Pukelsheim (ed.), Mathematics and Democracy, pages 167-176, Springer.
    4. Vito Fragnelli & Guido Ortona, 2006. "Comparison of Electoral Systems: Simulative and Game Theoretic Approaches," Studies in Choice and Welfare, in: Bruno Simeone & Friedrich Pukelsheim (ed.), Mathematics and Democracy, pages 65-81, Springer.
    5. Lawrence R. Ernst, 1994. "Appointment Methods for the House of Representatives and the Court Challenges," Management Science, INFORMS, vol. 40(10), pages 1207-1227, October.
    6. Sebastian Maier, 2006. "Algorithms for Biproportional Apportionment," Studies in Choice and Welfare, in: Bruno Simeone & Friedrich Pukelsheim (ed.), Mathematics and Democracy, pages 105-116, Springer.
    7. Petur Zachariassen & Martin Zachariassen, 2006. "A Comparison of Electoral Formulae for the Faroese Parliament," Studies in Choice and Welfare, in: Bruno Simeone & Friedrich Pukelsheim (ed.), Mathematics and Democracy, pages 235-251, Springer.
    8. Marjorie B. Gassner, 1991. "Biproportional Delegations," Journal of Theoretical Politics, , vol. 3(3), pages 321-342, July.
    9. Gisèle De Meur & Marjorie Gassner & Xavier Hubaut, 1985. "A Mathematical Model for Political Bipolarization," ULB Institutional Repository 2013/232145, ULB -- Universite Libre de Bruxelles.
    10. Gaffke, Norbert & Pukelsheim, Friedrich, 2008. "Divisor methods for proportional representation systems: An optimization approach to vector and matrix apportionment problems," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 166-184, September.
    11. Benoit, Kenneth, 2000. "Which Electoral Formula Is the Most Proportional? A New Look with New Evidence," Political Analysis, Cambridge University Press, vol. 8(4), pages 381-388, July.
    12. Aline Pennisi, 2006. "The Italian Bug: A Flawed Procedure for Bi-Proportional Seat Allocation," Studies in Choice and Welfare, in: Bruno Simeone & Friedrich Pukelsheim (ed.), Mathematics and Democracy, pages 151-165, Springer.
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    Cited by:

    1. Demange, Gabrielle, 2012. "On party-proportional representation under district distortions," Mathematical Social Sciences, Elsevier, vol. 63(2), pages 181-191.
    2. Gabrielle Demange, 2013. "On Allocating Seats To Parties And Districts: Apportionments," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 15(03), pages 1-14.
    3. Oelbermann, Kai-Friederike, 2016. "Alternate Scaling algorithm for biproportional divisor methods," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 25-32.
    4. Javier Cembrano & Jos'e Correa & Gonzalo D'iaz & Victor Verdugo, 2024. "Proportionality in Multiple Dimensions to Design Electoral Systems," Papers 2410.03304, arXiv.org.
    5. Zhaonan Qu & Alfred Galichon & Johan Ugander, 2023. "On Sinkhorn's Algorithm and Choice Modeling," Papers 2310.00260, arXiv.org.
    6. Haydar Evren & Manshu Khanna, 2021. "Affirmative Action's Cumulative Fractional Assignments," Papers 2111.11963, arXiv.org, revised Feb 2024.
    7. Victoriano Ramírez-González & Blanca Delgado-Márquez & Antonio Palomares & Adolfo López-Carmona, 2014. "Evaluation and possible improvements of the Swedish electoral system," Annals of Operations Research, Springer, vol. 215(1), pages 285-307, April.
    8. Kerem Akartunalı & Philip A. Knight, 2017. "Network models and biproportional rounding for fair seat allocations in the UK elections," Annals of Operations Research, Springer, vol. 253(1), pages 1-19, June.
    9. Friedrich Pukelsheim, 2014. "Biproportional scaling of matrices and the iterative proportional fitting procedure," Annals of Operations Research, Springer, vol. 215(1), pages 269-283, April.

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