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An Axiomatic Approach to Proportionality between Matrices

Author

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  • Michel Balinski

    (Department of Economics, Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique)

  • Gabrielle Demange

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

Abstract

Given a matrix p ≥ 0 what does it mean to say that a matrix f (of the same dimension), whose row and column sums must fall between specific limits, is "proportional to" p? This paper gives an axiomatic solution to this question in two distinct contexts. First, for any real "allocation" matrix f. Second, for any integer constrained "apportionment" matrix f. In the case of f real the solution turns out to coincide with what has been variously called biproportional scaling and diagonal equivalence and has been much used in econometrics and statistics. In the case of f integer the problem arises in the simultaneous apportionment of seats to regions and to parties and also in the rounding of tables of census data.

Suggested Citation

  • Michel Balinski & Gabrielle Demange, 1989. "An Axiomatic Approach to Proportionality between Matrices," Post-Print hal-00686748, HAL.
  • Handle: RePEc:hal:journl:hal-00686748
    Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-00686748
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    References listed on IDEAS

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    1. Aumann, Robert J. & Maschler, Michael, 1985. "Game theoretic analysis of a bankruptcy problem from the Talmud," Journal of Economic Theory, Elsevier, vol. 36(2), pages 195-213, August.
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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. Michel Balinski & Gabrielle Demange, 1989. "Algorithm for Proportional Matrices in Reals and Integers," Post-Print halshs-00585327, HAL.
    3. 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.
    4. Demange, Gabrielle, 2017. "Mutual rankings," Mathematical Social Sciences, Elsevier, vol. 90(C), pages 35-42.
    5. repec:eee:matsoc:v:90:y:2017:i:c:p:43-55 is not listed on IDEAS
    6. Gabrielle Demange, 2018. "Mechanisms in a digitalized world," Post-Print hal-01715951, HAL.
    7. 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.
    8. Attila Tasnádi, 2008. "The extent of the population paradox in the Hungarian electoral system," Public Choice, Springer, vol. 134(3), pages 293-305, March.
    9. Oelbermann, Kai-Friederike, 2016. "Alternate Scaling algorithm for biproportional divisor methods," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 25-32.
    10. Paolo Serafini & Bruno Simeone, 2012. "Certificates of optimality: the third way to biproportional apportionment," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(2), pages 247-268, February.
    11. Demange, Gabrielle, 2014. "A ranking method based on handicaps," Theoretical Economics, Econometric Society, vol. 9(3), September.
    12. MESNARD, Louis de, 1999. "Interpretation of the RAS method : absorption and fabrication effects are incorrect," LATEC - Document de travail - Economie (1991-2003) 9907, LATEC, Laboratoire d'Analyse et des Techniques EConomiques, CNRS UMR 5118, Université de Bourgogne.
    13. Gabrielle Demange, 2013. "A ranking method based on handicaps," Working Papers halshs-00687180, HAL.
    14. Gabrielle Demange, 2016. "Mutual rankings," Working Papers halshs-01353825, HAL.
    15. 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.
    16. Moulin, Hervé, 2016. "Entropy, desegregation, and proportional rationing," Journal of Economic Theory, Elsevier, vol. 162(C), pages 1-20.
    17. 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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