Comparisons: Droop Quota ~ Maps 1

Two-Party Droop Quota Party-List Elections

Two-party maps contain all the possible election outcomes for a given number of winners in a two-party election. The maps for two parties A and B contesting up to six vacancies in closed Droop Quota Party-List elections are shown below. Recall that the actual per-unit tally shares for A and B are represented by the appropriate point along the line. The domain boundaries of an optimally proportional voting (OPV) system are indicated by the dotted markers underneath the line. These optimal boundaries are necessarily midway between the dots that represent perfect proportionality between the outcome seat share ratio (as stated underneath the dot) and the tally share ratio (point on the line) as determined by the voters.

The boundaries for the Droop Quota domains are derived from the various two-way critical ties that may occur in elections with W winners. For the specific details of how bar charts are used to deduce the position of these boundaries, please refer to the Map Construction appendix for the Two-Party Droop Quota Maps page.

For all points along the linear map that are in the same Droop Quota and OPV domains the outcome is optimally proportional. For all points in dissimilar domains the outcome is not optimal. For Droop Quota domains, notice that there are W+1 of them and that they are all of equal length; namely 1/(W+1). For OPV domains, the two end ones are each half the length of the other domains. Counting these two half-length domains as just one full-length one, there would then be W domains of full-length 1/W.

It is this difference in domain length that accounts for the mismatch between the Droop Quota and OPV domain boundaries. Two corresponding boundaries are at their closest in the centre of the map but steadily diverge as they approach either end of the map. At the very end of a map, the two corresponding boundaries are however necessarily equal as the domains terminate here. Therefore, the Droop Quota method is more likely to produce an optimally proportional outcome when the two parties have comparable rather than dissimilar levels of support. This feature is largely independent of the number of seats available.

As the number of winners increases, extra domains and boundaries appear on the map. With more boundaries, there are more portions along the map where points relate to dissimilar domains. However, with more domains, these portions are smaller in length. The net effect of these two conflicting features is that the optimality of two-party Droop Quota Party-List elections deteriorates slowly as the number of winners increase; see next section.

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