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Bidimensional allocation of seats via zero-one matrices with given line sums

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  • Isabella Lari
  • Federica Ricca
  • Andrea Scozzari

Abstract

In some proportional electoral systems with more than one constituency the number of seats allotted to each constituency is pre-specified, as well as, the number of seats that each party has to receive at a national level. “Bidimensional allocation” of seats to parties within constituencies consists of converting the vote matrix V into an integer matrix of seats “as proportional as possible” to V, satisfying constituency and party totals and an additional “zero-vote zero-seat” condition. In the current Italian electoral law this Bidimensional Allocation Problem (or Biproportional Apportionment Problem—BAP) is ruled by an erroneous procedure that may produce an infeasible allocation, actually one that is not able to satisfy all the above conditions simultaneously. In this paper we focus on the feasibility aspect of BAP and, basing on the theory of (0,1)-matrices with given line sums, we formulate it for the first time as a “Matrix Feasibility Problem”. Starting from some previous results provided by Gale and Ryser in the 60’s, we consider the additional constraint that some cells of the output matrix must be equal to zero and extend the results by Gale and Ryser to this case. For specific configurations of zeros in the vote matrix we show that a modified version of the Ryser procedure works well, and we also state necessary and sufficient conditions for the existence of a feasible solution. Since our analysis concerns only special cases, its application to the electoral problem is still limited. In spite of this, in the paper we provide new results in the area of combinatorial matrix theory for (0,1)-matrices with fixed zeros which have also a practical application in some problems related to graphs. Copyright Springer Science+Business Media New York 2014

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  • Isabella Lari & Federica Ricca & Andrea Scozzari, 2014. "Bidimensional allocation of seats via zero-one matrices with given line sums," Annals of Operations Research, Springer, vol. 215(1), pages 165-181, April.
    • Handle: RePEc:spr:annopr:v:215:y:2014:i:1:p:165-181:10.1007/s10479-013-1440-2
    DOI: 10.1007/s10479-013-1440-2
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    References listed on IDEAS

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    1. Gabrielle Demange & Michel L. Balinski, 1989. "An Axiomatic Approach to Proportionality between Matrices," Post-Print halshs-00670952, 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. 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.
    4. 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.
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    Cited by:

    1. Haydar Evren & Manshu Khanna, 2021. "Affirmative Action's Cumulative Fractional Assignments," Papers 2111.11963, arXiv.org, revised Feb 2024.

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