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Algorithm for Proportional Matrices in Reals and Integers

Author

Listed:
  • Michel L. Balinski

    (X-DEP-ECO - Département d'Économie de l'École Polytechnique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris)

  • Gabrielle Demange

    (CECO - Laboratoire d'économétrie de l'École polytechnique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

Abstract

Let R be the set of nonnegative matrices whose row and column sums fall between specific limits and whose entries sum to some fixed h > 0. Closely related axiomatic approaches have been developed to ascribe meanings to the statements: the real matrix fe R and the integer matrix a ~ R are "proportional to" a given matrix p ~> 0. These approaches are described, conditions under which proportional solutions exist are characterized, and algorithms are given for finding proportional solutions in each case.

Suggested Citation

  • Michel L. Balinski & Gabrielle Demange, 1989. "Algorithm for Proportional Matrices in Reals and Integers," Post-Print halshs-00585327, HAL.
    • Handle: RePEc:hal:journl:halshs-00585327
    DOI: 10.1007/BF01589103
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00585327
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    References listed on IDEAS

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    1. 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.
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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. Ricca, Federica & Scozzari, Andrea & Simeone, Bruno, 2011. "The give-up problem for blocked regional lists with multi-winners," Mathematical Social Sciences, Elsevier, vol. 62(1), pages 14-24, July.
    3. Gabrielle Demange, 2021. "On the resolution of cross-liabilities," Working Papers halshs-03151128, HAL.
    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. Michel Balinski, 2007. "Equitable representation and recruitment," Annals of Operations Research, Springer, vol. 149(1), pages 27-36, February.
    6. Oelbermann, Kai-Friederike, 2016. "Alternate Scaling algorithm for biproportional divisor methods," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 25-32.
    7. Gabrielle Demange, 2020. "Resolution rules in a system of financially linked firms," Working Papers hal-02502413, HAL.
    8. 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.
    9. 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.

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    Keywords

    algorithm; proportional matrices;

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