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 = ( x i j ) ∈ M and $$X_{n}$$ X n is the set of the $$n$$ n greatest entries of $$X$$ X , we denote by $$\psi _{j}$$ ψ j the number of elements of $$X_{n}$$ X n in the column $$j$$ j of $$X$$ X and by $$\psi ^{i}$$ ψ i the number of elements of $$X_{n}$$ X n in the row $$i$$ i of $$X$$ X . If a new matrix $$X^{\prime }=(x_{ij}^{\prime })\in {\mathcal {M}}$$ X ′ = ( x i j ′ ) ∈ 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 $$(\psi ^{1},\psi ^{2},\ldots ,\psi ^{\tau })$$ ( ψ 1 , ψ 2 , … , ψ τ ) 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 $$(\psi _{1}^{\prime },\psi _{2}^{\prime },\ldots ,\psi _{n}^{\prime })$$ ( ψ 1 ′ , ψ 2 ′ , … , ψ n ′ ) of $$X^{\prime }$$ X ′ and $$(\psi _{1},\psi _{2},\ldots ,\psi _{n})$$ ( ψ 1 , ψ 2 , … , ψ 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. Copyright Springer Science+Business Media New York 2015