Proportionality for solid coalitions

Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results 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.

[1][2] PSC is a relatively minimal definition of proportionality.

And PSC does not guarantee proportional representation if voters rank candidates of different parties together (as they will no longer form a solid coalition).

For example, say voters are organized along a political spectrum, with factions on the far-left, center-left, center, center-right, and far-right.

–PSC or Hare-PSC is defined with respect to the Hare quota

Droop quotas entitle a solid coalition to

[5] 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.

[2] An advantage of Droop proportionality is that any solid coalition with a majority will always be able to elect at least half of seats.

[citation needed] However, this comes at the cost of a substantial seat bias in favor of larger parties.

Examples of quota-proportional methods include the expanding approvals rule, the method of equal shares, and the single transferable vote.

[7] 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.

[8] 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+.