Majorization comparison of closed list electoral systems through a matrix theorem

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• 作者：José Manuel Gutiérrez
• 关键词：Majorization ; Apportionment ; Electoral systems
• 刊名：Annals of Operations Research
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：235
• 期：1
• 页码：807-814
• 全文大小：374 KB
• 参考文献：Balinski, M. L., & Young, H. P. (2001). Fair representation: Meeting the ideal of one man, one vote (2nd ed.). Washington: Brookings Institution Press. (1st edition 1982).
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  Marshall, A. W., & Olkin, I. (1983). Inequalities via majorization: An introduction. General Inequalities 3 International Series of Numerical Mathematics, 64, 165-87.CrossRef
  Marshall, A. W., Olkin, I., & Arnold, B. C. (2011). Inequalities: Theory of majorization and Its applications (2nd ed.). Dordrecht: Springer. (1st edition 1979).
  Marshall, A. W., Olkin, I., & Pukelsheim, F. (2002). A majorization comparison of apportionment methods in proportional representation. Social Choice and Welfare, 19, 885-00.CrossRef
  Pukelsheim, F. (2014). Proportional representation: Apportionment methods and their applications. Heidelberg: Springer.CrossRef
  
• 作者单位：José Manuel Gutiérrez (1)
  
  1. Facultad de Economía y Empresa, Universidad de Salamanca, 37007, Salamanca, Spain
  
• 刊物类别：Business and Economics
• 刊物主题：Economics
  Operation Research and Decision Theory
  Combinatorics
  Theory of Computation
  
• 出版者：Springer Netherlands
• ISSN：1572-9338

文摘

Let \({\mathcal {M}}\) be the space of all the \(\tau \times n\) matrices with pairwise distinct entries and with both rows and columns sorted in descending order. If \(X=(x_{ij})\in {\mathcal {M}}\) and \(X_{n}\) is the set of the \(n\) greatest entries of \(X\), we denote by \(\psi _{j}\) the number of elements of \(X_{n}\) in the column \(j\) of \(X\) and by \(\psi ^{i}\) the number of elements of \(X_{n}\) in the row \(i\) of \(X\). If a new matrix \(X^{\prime }=(x_{ij}^{\prime })\in {\mathcal {M}}\) is obtained from \(X\) in such a way that \(X^{\prime }\) yields to \(X\) (as defined in the paper), then there is a relation of majorization between \((\psi ^{1},\psi ^{2},\ldots ,\psi ^{\tau })\) and the corresponding \((\psi ^{\prime 1},\psi ^{\prime 2},\ldots ,\psi ^{\prime \tau })\) of \(X^{\prime }\), and between \((\psi _{1}^{\prime },\psi _{2}^{\prime },\ldots ,\psi _{n}^{\prime })\) of \(X^{\prime }\) and \((\psi _{1},\psi _{2},\ldots ,\psi _{n})\). This result can be applied to the comparison of closed list electoral systems, providing a unified proof of the standard hierarchy of these electoral systems according to whether they are more or less favourable to larger parties. Keywords Majorization Apportionment Electoral systems