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A majorization comparison of apportionment methods in proportional representation


  • Friedrich Pukelsheim

    () (Institut für Mathematik, Universität Augsburg, D-86315 Augsburg, Germany)

  • Albert W. Marshall

    (Statistics Department, University of British Columbia, Vancouver BG, V6T 1Z2, Canada)

  • Ingram Olkin

    (Department of Statistics, Stanford University, Stanford CA, 94305-4065, USA)


From the inception of the proportional representation movement it has been an issue whether larger parties are favored at the expense of smaller parties in one apportionment of seats as compared to another apportionment. A number of methods have been proposed and are used in countries with a proportional representation system. These apportionment methods exhibit a regularity of order, as discussed in the present paper, that captures the preferential treatment of larger versus smaller parties. This order, namely majorization, permits the comparison of seat allocations in two apportionments. For divisor methods, we show that one method is majorized by another method if and only if their signpost ratios are increasing. This criterion is satisfied for the divisor methods with power-mean rounding, and for the divisor methods with stationary rounding. Majorization places the five traditional apportionment methods in the order as they are known to favor larger parties over smaller parties: Adams, Dean, Hill, Webster, and Jefferson.

Suggested Citation

  • Friedrich Pukelsheim & Albert W. Marshall & Ingram Olkin, 2002. "A majorization comparison of apportionment methods in proportional representation," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(4), pages 885-900.
  • Handle: RePEc:spr:sochwe:v:19:y:2002:i:4:p:885-900
    Note: Received: 5 August 2000/Accepted: 24 October 2001

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    Cited by:

    1. Heinrich Lothar & Pukelsheim Friedrich & Schwingenschlögl Udo, 2005. "On stationary multiplier methods for the rounding of probabilities and the limiting law of the Sainte-Laguë divergence," Statistics & Risk Modeling, De Gruyter, vol. 23(2/2005), pages 117-129, February.
    2. Bittó, Virág, 2017. "Az Imperiali és Macau politikai választókörzet-kiosztási módszerek empirikus vizsgálata
      [Empirical Analysis of the Imperiali and Macau Apportionment Methods]
      ," MPRA Paper 79554, University Library of Munich, Germany.
    3. Grimmett, G.R. & Oelbermann, K.-F. & Pukelsheim, F., 2012. "A power-weighted variant of the EU27 Cambridge Compromise," Mathematical Social Sciences, Elsevier, vol. 63(2), pages 136-140.
    4. Heinrich Lothar & Pukelsheim Friedrich & Schwingenschlögl Udo, 2004. "Sainte-Laguë’s chi-square divergence for the rounding of probabilities and its convergence to a stable law," Statistics & Risk Modeling, De Gruyter, vol. 22(1/2004), pages 43-60, January.
    5. Jones, Michael A. & Wilson, Jennifer M., 2010. "Evaluation of thresholds for power mean-based and other divisor methods of apportionment," Mathematical Social Sciences, Elsevier, vol. 59(3), pages 343-348, May.
    6. Brams, S.J. & Kaplan, T.R., 2002. "Dividing the Indivisible: Procedures for Allocating Cabinet Ministries to Political Parties in a Parliamentary System," Working Papers 02-06, C.V. Starr Center for Applied Economics, New York University.
    7. Steven J Brams & D Marc Kilgour, 2012. "Narrowing the field in elections: The Next-Two rule," Journal of Theoretical Politics, , vol. 24(4), pages 507-525, October.
    8. Laszlo A. Koczy & Peter Biro & Balazs Sziklai, 2017. "US vs. European Apportionment Practices: The Conflict between Monotonicity and Proportionality," IEHAS Discussion Papers 1716, Institute of Economics, Centre for Economic and Regional Studies, Hungarian Academy of Sciences.
    9. Ulrich Kohler & Janina Zeh, 2012. "Apportionment methods," Stata Journal, StataCorp LP, vol. 12(3), pages 375-392, September.

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