Proportional Representation by Friedrich Pukelsheim

Author:Friedrich Pukelsheim
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

b.(ℓ ≥ 3) Systems with three or more parties are such that for every seat vector x ∈ A(h; v) there exists a seat vector y ∈ A(H; v) satisfying x ≤ y, and that for every seat vector y ∈ A(H; v) there exists a seat vector x ∈ A(h; v) satisfying y ≥ x.

Property a offers a symmetric handling of the two situations when stepping up from a small house size h to a larger house size H, and when going down from a larger house size H to a smaller house size h. Either way all seat vectors (x 1, x 2) for house size h and ( y 1, y 2) for house size H satisfy (x 1, x 2) ≤ ( y 1, y 2); that is, ( y 1, y 2) ≥ (x 1, x 2).

Property b handles the two situations separately because in the first situation the seat vector y depends on x, while in the second x depends on y. A symmetric dependence is too much to ask for. It is not hard to see why. The example in Table 4.​2 displays ten tied seat vectors x for house size 18. A vector y that dominates all of them must satisfy y ≥ (6, 5, 4, 3, 2) and have component sum y + ≥ 20. House size 19 would be missed out. For this reason property b is more modest: For every seat vector x ∈ A(18; v) there exists a solution y ∈ A(19; v) such that x ≤ y, and for every seat vector y ∈ A(19; v) there exists a solution x ∈ A(18; v) satisfying y ≥ x.

All divisor methods are house size monotone. In fact, the monotone progression from house size h with seat vector x to house size h + 1 with seat vector y ≥ x is part of the argument that divisor methods are well-defined; see Sect. 4.​5.

In the presence of coherence house size monotonicity comes for free.

Theorem

Every coherent apportionment method is house size monotone.

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