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An Axiomatic Approach to Proportionality between Matrices

Listed author(s):
  • Michel Balinski

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

  • Gabrielle Demange

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

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.

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File URL: https://hal.archives-ouvertes.fr/hal-00686748/document
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Paper provided by HAL in its series Post-Print with number hal-00686748.

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Date of creation: 1989
Publication status: Published in Mathematics of Operations Research, INFORMS, 1989, 14 (4), pp.700-719
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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  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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