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


  • 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)


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

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    References listed on IDEAS

    1. 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.
    2. Michel Balinski & Gabrielle Demange, 1989. "Algorithm for Proportional Matrices in Reals and Integers," Post-Print halshs-00585327, HAL.
    3. Gassner, Marjorie, 1988. "Two-dimensional rounding for a quasi-proportional representation," European Journal of Political Economy, Elsevier, vol. 4(4), pages 529-538.
    4. Marjorie B. Gassner, 1991. "Biproportional Delegations," Journal of Theoretical Politics, , vol. 3(3), pages 321-342, July.
    5. Gisèle De Meur & Marjorie Gassner & Xavier Hubaut, 1985. "A Mathematical Model for Political Bipolarization," ULB Institutional Repository 2013/232145, ULB -- Universite Libre de Bruxelles.
    6. 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.
    7. Benoit, Kenneth, 2000. "Which Electoral Formula Is the Most Proportional? A New Look with New Evidence," Political Analysis, Cambridge University Press, vol. 8(04), pages 381-388, July.
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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. repec:hal:wpaper:halshs-00623031 is not listed on IDEAS
    3. Oelbermann, Kai-Friederike, 2016. "Alternate Scaling algorithm for biproportional divisor methods," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 25-32.
    4. repec:hal:wpaper:halshs-00644439 is not listed on IDEAS


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