Proportionality for solid coalitions

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 can directly support candidates. The criterion was first proposed by the British philosopher and logician Michael Dummett.

PSC is a relatively minimal definition of proportionality. To be guaranteed representation, a coalition of voters must rank all candidates within the same party first before candidates of other parties. And PSC does not guarantee proportional representation if voters rank candidates of different parties together (as they will no longer form a 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 ${\displaystyle V}$ is a solid coalition for a set of candidates ${\displaystyle C}$, if every voter in ${\displaystyle V}$ ranks every candidate in ${\displaystyle C}$ ahead of every candidate that is not in ${\displaystyle C}$.

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 candidate as their first-preference solidly supports that candidate.

Note that a solid coalition may be "nested" within another solid coalition, so there may be some faction of voters that can further be split into subfactions. However, solid coalitions cannot cut across different factions. For example, say voters are organized along a political spectrum, with factions on the far-left, center-left, center, center-right, and far-right. Then, the three moderate groups will not form a solid coalition, because some members of the center-right may not rank the center-left candidate above the far-right candidate.

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

${\displaystyle k}$–PSC or Hare-PSC is defined with respect to the Hare quota ${\displaystyle n/k}$. It says that if there is a solid coalition for a set of candidates with at least ${\displaystyle j}$ Hare quotas, then at least ${\displaystyle j}$ candidates from this set must be elected. (If ${\displaystyle C}$ has less than ${\displaystyle j}$ candidates, all of them must be elected). This criterion was proposed by Michael Dummett.

In the single-winner case, it is equivalent to the unanimity criterion, as a Hare quota in the single-winner case includes all voters.