Proportionality for solid coalitions
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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]
Solid coalitions
[edit]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 is a solid coalition for a set of candidates , if every voter in ranks every candidate in ahead of every candidate that is not in .
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.
In the following let be the number of voters, be the number of seats to be filled and be some positive integer.
Quotas
[edit]–PSC or Hare-PSC is defined with respect to the Hare quota . It says that if there is a solid coalition for a set of candidates with at least Hare quotas, then at least candidates from this set must be elected. (If has less than candidates, all of them must be elected).[5] This criterion was proposed by Michael Dummett.[1]
In the single-winner case, it is equivalent to the unanimity criterion, as a Hare quota in the single-winner case includes all voters.
–PSC, also called Droop-PSC, is defined like –PSC but using the Droop quota instead of the Hare quota, i.e. Droop quotas entitle a solid coalition to candidates.[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; 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.[3]
Generalizations
[edit]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+.[4]
See also
[edit]- Justified representation – properties analogous to proportional representation for electoral systems using approval ballots.
References
[edit]- ^ a b Dummett, M.: Voting procedures. Oxford Clarendon Press (1984).
- ^ a b D. R. Woodall: Monotonicity of single-seat preferential election rules. Discrete Applied Mathematics 77 (1997), p. 83–84.
- ^ a b Tideman, Nicolaus (1995-03-01). "The Single Transferable Vote". Journal of Economic Perspectives. 9 (1): 27–38. doi:10.1257/jep.9.1.27. ISSN 0895-3309.
- ^ a b Brill, Markus; Peters, Jannik (2023). "Robust and Verifiable Proportionality Axioms for Multiwinner Voting". arXiv:2302.01989 [cs.GT].
- ^ a b c Tideman N.: Collective Decisions and Voting. Ashgate Publishing Ltd, Aldershot, 2006, p. 268–269.
- ^ Aziz, Haris; Lee, Barton (2018-06-04). "The Expanding Approvals Rule: Improving Proportional Representation and Monotonicity". arXiv:1708.07580 [cs.GT].
- ^ Aziz, Haris; Lee, Barton E. (2020). "A characterization of proportionally representative committees". arXiv:2002.09598 [cs.GT].
- ^ Aziz, Haris; Lee, Barton (2017). "The Expanding Approvals Rule: Improving Proportional Representation and Monotonicity". arXiv:1708.07580 [cs.GT].
- ^ Aziz, Haris; Lee, Barton E. (2019-08-09). "The expanding approvals rule: improving proportional representation and monotonicity". Social Choice and Welfare. 54 (1). Springer Science and Business Media LLC: 8. arXiv:1708.07580. doi:10.1007/s00355-019-01208-3. ISSN 0176-1714. S2CID 46926459.