IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-00686748.html
   My bibliography  Save this paper

An Axiomatic Approach to Proportionality between Matrices

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

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 L. 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.science/hal-00686748
    as

    Download full text from publisher

    File URL: https://hal.science/hal-00686748/document
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. H. Peyton Young, 1987. "On Dividing an Amount According to Individual Claims or Liabilities," Mathematics of Operations Research, INFORMS, vol. 12(3), pages 398-414, August.
    2. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Demange, Gabrielle, 2012. "On party-proportional representation under district distortions," Mathematical Social Sciences, Elsevier, vol. 63(2), pages 181-191.
    2. Gabrielle Demange, 2018. "New electoral systems and old referendums," PSE Working Papers hal-01852206, HAL.
    3. Marjorie B. Gassner, 1991. "Biproportional Delegations," Journal of Theoretical Politics, , vol. 3(3), pages 321-342, July.
    4. Demange, Gabrielle, 2017. "Mutual rankings," Mathematical Social Sciences, Elsevier, vol. 90(C), pages 35-42.
    5. Gabrielle Demange, 2018. "Mechanisms in a Digitalized World," CESifo Working Paper Series 6984, CESifo.
    6. Gabrielle Demange, 2013. "On Allocating Seats To Parties And Districts: Apportionments," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 15(03), pages 1-14.
    7. Attila Tasnádi, 2008. "The extent of the population paradox in the Hungarian electoral system," Public Choice, Springer, vol. 134(3), pages 293-305, March.
    8. Oelbermann, Kai-Friederike, 2016. "Alternate Scaling algorithm for biproportional divisor methods," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 25-32.
    9. 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.
    10. ,, 2014. "A ranking method based on handicaps," Theoretical Economics, Econometric Society, vol. 9(3), September.
    11. Gabrielle Demange, 2021. "On the resolution of cross-liabilities," Working Papers halshs-03151128, HAL.
    12. Michel L. Balinski & Gabrielle Demange, 1989. "Algorithm for Proportional Matrices in Reals and Integers," Post-Print halshs-00585327, HAL.
    13. 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.
    14. Moulin, Herve, 2017. "Consistent bilateral assignment," Mathematical Social Sciences, Elsevier, vol. 90(C), pages 43-55.
    15. 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.
    16. Michel Balinski, 2007. "Equitable representation and recruitment," Annals of Operations Research, Springer, vol. 149(1), pages 27-36, February.
    17. Gabrielle Demange, 2020. "Resolution rules in a system of financially linked firms," Working Papers hal-02502413, HAL.
    18. Victoriano Ramírez-González & Blanca Delgado-Márquez & Antonio Palomares & Adolfo López-Carmona, 2014. "Evaluation and possible improvements of the Swedish electoral system," Annals of Operations Research, Springer, vol. 215(1), pages 285-307, April.
    19. 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.
    20. 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.
    21. 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.
    22. Moulin, Hervé, 2016. "Entropy, desegregation, and proportional rationing," Journal of Economic Theory, Elsevier, vol. 162(C), pages 1-20.
    23. Friedrich Pukelsheim, 2014. "Biproportional scaling of matrices and the iterative proportional fitting procedure," Annals of Operations Research, Springer, vol. 215(1), pages 269-283, April.
    24. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. William Thomson, 2007. "On the existence of consistent rules to adjudicate conflicting claims: a constructive geometric approach," Review of Economic Design, Springer;Society for Economic Design, vol. 11(3), pages 225-251, November.
    2. Juan Moreno-Ternero & Antonio Villar, 2006. "The TAL-Family of Rules for Bankruptcy Problems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 27(2), pages 231-249, October.
    3. Erik Ansink & Hans-Peter Weikard, 2012. "Sequential sharing rules for river sharing problems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(2), pages 187-210, February.
    4. Jingyi Xue, 2018. "Fair division with uncertain needs," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 51(1), pages 105-136, June.
    5. Lahiri, Somdeb, 2001. "Axiomatic characterizations of the CEA solution for rationing problems," European Journal of Operational Research, Elsevier, vol. 131(1), pages 162-170, May.
    6. Bas Dietzenbacher & Yuki Tamura & William Thomson, 2024. "Partial-implementation invariance and claims problems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 63(1), pages 203-229, August.
    7. Sanchez-Soriano, Joaquin, 2021. "Families of sequential priority rules and random arrival rules with withdrawal limits," Mathematical Social Sciences, Elsevier, vol. 113(C), pages 136-148.
    8. Peris, Josep E. & Jiménez-Gómez, José M., 2012. "A Proportional Approach to Bankruptcy Problems with a guaranteed minimum," QM&ET Working Papers 12-7, University of Alicante, D. Quantitative Methods and Economic Theory.
    9. Hougaard, Jens Leth & Moreno-Ternero, Juan D. & Østerdal, Lars Peter, 2012. "A unifying framework for the problem of adjudicating conflicting claims," Journal of Mathematical Economics, Elsevier, vol. 48(2), pages 107-114.
    10. William Thomson, 2011. "Consistency and its converse: an introduction," Review of Economic Design, Springer;Society for Economic Design, vol. 15(4), pages 257-291, December.
    11. William Thomson, 2014. "Compromising between the proportional and constrained equal awards rules," RCER Working Papers 584, University of Rochester - Center for Economic Research (RCER).
    12. repec:ebl:ecbull:v:3:y:2008:i:56:p:1-10 is not listed on IDEAS
    13. Miguel Ángel Mirás Calvo & Carmen Quinteiro Sandomingo & Estela Sánchez-Rodríguez, 2022. "The average-of-awards rule for claims problems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 59(4), pages 863-888, November.
    14. Herrero, Carmen & Maschler, Michael & Villar, Antonio, 1999. "Individual rights and collective responsibility: the rights-egalitarian solution," Mathematical Social Sciences, Elsevier, vol. 37(1), pages 59-77, January.
    15. Chambers, Christopher P., 2006. "Asymmetric rules for claims problems without homogeneity," Games and Economic Behavior, Elsevier, vol. 54(2), pages 241-260, February.
    16. Jaume García-Segarra & Miguel Ginés-Vilar, 2023. "Additive adjudication of conflicting claims," International Journal of Game Theory, Springer;Game Theory Society, vol. 52(1), pages 93-116, March.
    17. Rodica Branzei & Sirma Zeynep Alparslan Gok, 2008. "Bankruptcy problems with interval uncertainty," Economics Bulletin, AccessEcon, vol. 3(56), pages 1-10.
    18. Hervé Moulin & Jay Sethuraman, 2013. "The Bipartite Rationing Problem," Operations Research, INFORMS, vol. 61(5), pages 1087-1100, October.
    19. Gallo, Oihane & Inarra, Elena, 2018. "Rationing rules and stable coalition structures," Theoretical Economics, Econometric Society, vol. 13(3), September.
    20. Patrick Harless, 2017. "Endowment additivity and the weighted proportional rules for adjudicating conflicting claims," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(3), pages 755-781, March.
    21. Moreno-Ternero, Juan D. & Villar, Antonio, 2004. "The Talmud rule and the securement of agents' awards," Mathematical Social Sciences, Elsevier, vol. 47(2), pages 245-257, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:journl:hal-00686748. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.