Proportionality for solid coalitions (PSC) is a criterion of proportionality for ranked voting systems. It is an adaptation of the quota rule to voting systems in which there are no official party lists, and voters directly support candidates. The criterion was first proposed by the British philosopher and logician Michael Dummett.[1] [2]
PSC is a weak definition of proportionality, which only guarantees proportional representation for cloned candidates, also called solid coalitions. In other words, voters must rank all candidates within the same party first (before candidates of other parties) to ensure it is adequately represented. On the other hand, PSC does not guarantee proportional representation if voters rank candidates of different parties together (as they will no longer form a solid coalition). As a result, PSC systems like the single transferable vote[3] can become disproportional if there are substantial cross-cutting cleavages.[4] [5] [6]
See main article: Solid coalition. In party-list systems, proportional representation guarantees each party a number of representatives proportional to its number of votes. In systems without parties, the natural analogue of a "party" is a solid coalition. A solid coalition is a group of voters who prefer any candidate within a certain set of candidates over any candidate not in the set. A set of voters
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When a voter is part of a solid coalition that prefers some set of candidates, they are said to be "solidly supporting" or "solidly committed to" that set of candidates.[7] [8] Any voter who ranks a single candidate as their 1st choice solidly supports that candidate.
Note that a solid coalition may be "nested" within another solid coalition; for example, there may be a faction of voters that can further be split into subfactions.
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See also: Electoral quota and quota rule.
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In the single-winner case, it is equivalent to the unanimity criterion, as a Hare quota in the single-winner case includes all voters.
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It is a generalization of the majority criterion in the sense that it relates to groups of supported candidates (solid coalitions) instead of just one candidate, and there may be more than one seat to be filled. An advantage of Droop proportionality is that any solid coalition with a majority will always be able to elect at least half of seats; however, the use of the Droop quota tends to create substantial seat biases in favor of larger parties.
Examples of quota-proportional include the expanding approvals rule, the method of equal shares, and the single transferable vote.
Aziz and Lee define a property called generalized PSC, and another property, called inclusion PSC, that apply also to weak rankings (rankings with indifferences). Their expanding approvals rule satisfies these generalizations of PSC.[9]
Brill and Peters define a fairness property called Rank-PJR+, which also applies to weak rankings, but makes positive guarantees also to coalitions that are only partially solid. Rank-PJR+ is attained by the expanding approvals rule, but violated by the single transferable vote. It can be decided in polynomial time whether a given committee satisfies Rank-PJR+.