Proportionality for solid coalitions

Proportionality for solid coalitions (PSC) is a fairness criterion for ranked voting systems. It is an adaptation of the proportional representation criterion to voting systems in which there are no parties, the voters can vote directly for candidates, and can rank candidates in any way they want. This criterion was proposed by the British philosopher and logician Michael Dummett.

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 of their own faction first (before candidates of other parties) to ensure their faction 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 can become disproportional if there are substantial cross-cutting cleavages.

Solid coalitions
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. Informally, 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 $$V$$ is a solid coalition for a set of candidates $$C$$, if every voter in $$V$$ ranks every candidate in $$C$$ ahead of every candidate that is not in $$C$$.

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. 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.

In the following let $$n$$ be the number of voters, $$k$$ be the number of seats to be filled and $$j$$ be some positive integer.

k–PSC
$$k$$–PSC is defined with respect to the Hare quota $$n/k$$. If $$V$$ is a solid coalition for $$C$$ and the number of Voters in $$V$$ is at least $$j$$ Hare quotas, then at least $$j$$ candidates from $$C$$ must be elected (if $$C$$ has less than $$j$$ candidates at all, then all of them have to be elected). This criterion was proposed by Michael Dummett.

In the single-winner case, k-PSC is equivalent to the unanimity criterion, as a Hare quota there would comprise all voters.

The most prominent example of a rule that satisfies k-PSC is single transferable vote.

k + 1–PSC
$$k+1$$–PSC is defined like $$k$$–PSC, but with respect to the Hagenbach-Bischoff quota $$n/(k+1)$$ instead of the Hare quota: the number of voters in $$V$$ must exceed $$j$$ Hagenbach-Bischoff quotas. (The reason it is "exceed" rather than "at least" here is because there can be more HB quotas than seats.)

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. Because some authors call the fraction $$n/(k+1)$$ Droop quota, $$k+1$$–PSC is also known as Droop proportionality criterion.

One major implication of Droop proportionality is that a majority solid coalition will always be able to elect at least half of the seats. This is because a majority is always over n/2 voters, which is equivalent to a number of voters exceeding half of the Hagenbach-Bischoff quotas (There are (k+1) Hagenbach-Bischoff quotas in an election, since (n/(k+1)) * (k+1) = n, so (k+1)/2, which is half of the quotas * n/(k+1), which is the quota, = n/2).

Generalizations
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 generalized PSC.

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+.