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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-00686748v1
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    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. Robert J. Aumann, 2025. "Game-Theoretic Analysis of a Bankruptcy Problem from the Talmud," World Scientific Book Chapters, in: SELECTED CONTRIBUTIONS TO GAME THEORY, chapter 9, pages 219-242, World Scientific Publishing Co. Pte. Ltd..
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