Majorization comparison of closed list electoral systems through a matrix theorem

Abstract

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.