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Error minimization methods in biproportional apportionment


  • Federica Ricca


  • Andrea Scozzari


  • Paolo Serafini


  • Bruno Simeone


One of the most active research lines in the area of electoral systems to date deals with the Biproportional Apportionment Problem, which arises in those proportional systems where seats must be allocated to parties within territorial constituencies. A matrix of the vote counts of the parties within the constituencies is given, and one has to convert the vote matrix into an integer matrix of seats “as proportional as possible” to it, subject to the constraints that each constituency be granted its pre-specified number of seats, each party be allotted the total number of seats it is entitled to on the basis of its national vote count, and a zero-vote zero-seat condition be satisfied. The matrix of seats must simultaneously meet the integrality and the proportionality requirement, and this not infrequently gives rise to self-contradictory procedures in the electoral laws of some countries. Here we discuss a class of methods for Biproportional Apportionment characterized by an “error minimization” approach. If the integrality requirement is relaxed, fractional seat allocations (target shares) can be obtained so as to achieve proportionality at least in theory. In order to restore integrality, one then looks for integral apportionments that are as close as possible to the ideal ones in a suitable metric. This leads to the formulation of constrained optimization problems called “best approximation problems” which are solvable in polynomial time through the use of network flow techniques. These error minimization methods can be viewed as an alternative to the classical axiomatic approach introduced by Balinski and Demange (in Math Oper Res 14:700–719, 1989a ; Math Program 45:193–210, 1989b ). We provide an empirical comparison between these two approaches with a real example from the Italian Elections and a theoretical discussion about the axioms that are not necessarily satisfied by the error minimization methods. Copyright Sociedad de Estadística e Investigación Operativa 2012

Suggested Citation

  • Federica Ricca & Andrea Scozzari & Paolo Serafini & Bruno Simeone, 2012. "Error minimization methods in biproportional apportionment," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(3), pages 547-577, October.
  • Handle: RePEc:spr:topjnl:v:20:y:2012:i:3:p:547-577
    DOI: 10.1007/s11750-012-0252-x

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    References listed on IDEAS

    1. repec:spr:stchwe:978-3-642-20441-8 is not listed on IDEAS
    2. M. L. Balinski & G. Demange, 1989. "An Axiomatic Approach to Proportionality Between Matrices," Mathematics of Operations Research, INFORMS, vol. 14(4), pages 700-719, November.
    3. Minoux, M., 1984. "A polynomial algorithm for minimum quadratic cost flow problems," European Journal of Operational Research, Elsevier, vol. 18(3), pages 377-387, December.
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    Cited by:

    1. repec:spr:annopr:v:253:y:2017:i:1:d:10.1007_s10479-016-2323-0 is not listed on IDEAS
    2. Paolo Serafini, 2015. "Certificates of optimality for minimum norm biproportional apportionments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 44(1), pages 1-12, January.


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