Comparisons: Sainte-Laguë ~ Proportionality 1

Disproportionality Indices

For party-list CHPV, the Gallagher or Least Squares Index (LSI) is employed as a measure of the discrepancy between the party shares of the tallies (t) and the party shares of the seats (s); where 0 ≤ LSI ≤ 1. If necessary, please refer back to the Evaluations: Proportionality of CHPV section. For convenience, the definition of the least squares index for S parties is repeated here; see opposite.

To facilitate a fair comparison with CHPV, the proportionality of Sainte-Laguë is approached in an identical manner. Two LSIs are again employed to assess disproportionality; one for the seat resolution and the other for the voting system itself. The resolution LSI is based on the discrepancies between the centres and the corners or extremes of the domains. These domains relate to those of an optimally proportional voting (OPV) system. For a given number of parties (S) and seats (W), the number and average size of the domains are identical irrespective of the voting system used. Hence, the resolution LSI is almost identical for OPV, CHPV and Sainte-Laguë.

By separating the common discrepancies due to resolution from the unique ones introduced by the voting system itself, a clearer comparison between two rival systems can be achieved. The voting system least squares index (LSI_VS) is also repeated here; see opposite. For a given election (S and W), there are k domains in total and each domain j requires its own LSI to be determined.

The centre of an OPV domain is where perfect party proportionality is achieved between tallies and seats. Like CHPV, the centre of the corresponding Sainte-Laguë domain will potentially be offset from this ideal point. The least squares index for domain j (LSIj) hence measures the disproportionality introduced by the Sainte-Laguë algorithm in offsetting its centre. The overall LSI_VS simply combines these individual domain LSIs into one value within, as always, the range 0 ≤ LSI ≤ 1.

Sainte-Laguë Domain Centres

In order to determine the LSI_VS for any particular Sainte-Laguë Party-List election, the position of each domain centre must first be identified. In the example two-party map opposite, all the five domains and their centres for a four-winner election are illustrated. The domain boundaries are shown as vertical event markers and the domain centres as dots.

Notice that there are three full-length domains with a half-length one at each end. The length of a full domain is 1/W; or 1/4 here. Each inner domain is simply bisected to locate its centre. These centres are uniformly spaced one domain length of 1/W apart. The centre of each end domain is also one domain length from its adjacent centre and also one half-length from the boundary that separates the two adjacent domains. These two requirements place the end domain centres at the end points of the unidimensional map.

All the domain centres are hence uniformly spaced along the map at intervals of one full-length domain. This is exactly how the seat share dots (points of perfect proportionality) are also distributed along the map; see the earlier Sainte-Laguë Method ~ Maps 1 page. There is therefore no displacement of the Sainte-Laguë domain centres (or boundaries) relative to the optimally proportional voting (OPV) ones. The LSI_VS for this and indeed any other two-party map is zero as no disproportionality is introduced. This is no surprise as two-party Sainte-Laguë elections always produce optimally proportional outcomes.

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