Majorization comparison of closed list electoral systems through a matrix theorem

Let $${\mathcal {M}}$$M be the space of all the $$\tau \times n$$τ×n matrices with pairwise distinct entries and with both rows and columns sorted in descending order. If $$X=(x_{ij})\in {\mathcal {M}}$$X=(xij)∈M and $$X_{n}$$Xn is the set of the $$n$$n greatest entries of $$X$$X, we denote by $$\psi _{j}$$ψj the number of elements of $$X_{n}$$Xn in the column $$j$$j of $$X$$X and by $$\psi ^{i}$$ψi the number of elements of $$X_{n}$$Xn in the row $$i$$i of $$X$$X. If a new matrix $$X^{\prime }=(x_{ij}^{\prime })\in {\mathcal {M}}$$X′=(xij′)∈M is obtained from $$X$$X in such a way that $$X^{\prime }$$X′yields to$$X$$X (as defined in the paper), then there is a relation of majorization between $$(\psi ^{1},\psi ^{2},\ldots ,\psi ^{\tau })$$(ψ1,ψ2,…,ψτ) and the corresponding $$(\psi ^{\prime 1},\psi ^{\prime 2},\ldots ,\psi ^{\prime \tau })$$(ψ′1,ψ′2,…,ψ′τ) of $$X^{\prime }$$X′, and between $$(\psi _{1}^{\prime },\psi _{2}^{\prime },\ldots ,\psi _{n}^{\prime })$$(ψ1′,ψ2′,…,ψn′) of $$X^{\prime }$$X′ and $$(\psi _{1},\psi _{2},\ldots ,\psi _{n})$$(ψ1,ψ2,…,ψn). 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.