An Axiomatic Approach to Proportionality Between Matrices

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&...

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Bibliographic Details
Published in:Mathematics of operations research 1989-11, Vol.14 (4), p.700-719
Main Authors: Balinski, M. L, Demange, G
Format: Article
Language:English
Subjects:
allocation
apportionment
axiomatics
Axioms
contingency
Economics and Finance
fairness
Humanities and Social Sciences
Integers
Mathematical models
Mathematical vectors
Mathematics
Matrices
Operations research
optimality
Political parties
Problems
rounding
scaling
Solutions
Theory
Towns
Uniformity
Uniqueness
Voting
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Summary: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.
ISSN:0364-765X
1526-5471
DOI:10.1287/moor.14.4.700