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.