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Certificates of optimality for minimum norm biproportional apportionments

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  • Paolo Serafini

Abstract

Computing a biproportional apportionment that satisfies some given properties may require a high degree of mathematical expertise, that very few voters can share. It seems therefore that the voters have to accept the electoral outcome without any possibility of checking the validity of the stated properties. However, it is possible in some cases to attach to the computed apportionment a certificate which can guarantee the voters of the validity of the apportionment. This type of investigation has been first proposed in Serafini and Simeone (Soc Choice Welf 38:247–268, 2012 . In this paper we pursue the same line of approach and show that a certificate can be produced and easily checked by a layman for apportionments that minimize either an $$L_1$$ L 1 - or an $$L_2$$ L 2 -norm deviation from given quotas. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • 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.
    • Handle: RePEc:spr:sochwe:v:44:y:2015:i:1:p:1-12
    DOI: 10.1007/s00355-014-0821-z
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    References listed on IDEAS

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    1. Dan S. Felsenthal & Moshé Machover (ed.), 2012. "Electoral Systems," Studies in Choice and Welfare, Springer, number 978-3-642-20441-8, December.
    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. 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.
    4. N. Gaffke & F. Pukelsheim, 2008. "Vector and matrix apportionment problems and separable convex integer optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(1), pages 133-159, February.
    5. 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.
    6. 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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