Proportionality for Solid Coalitions: Difference between revisions
Proportionality for Solid Coalitions (view source)
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'''Proportionality for Solid Coalitions''' ('''PSC''') is a criterion for proportional methods requiring that sufficiently-sized groups of voters (solid coalitions) always elect a proportional number of candidates from their set of mutually most-preferred candidates. In general, any time any group of voters prefers any set of candidates over all others, a certain minimum number of candidates from that set must win to pass the criterion, and the same must hold if the preferred set of candidates for a group can be shrunk or enlargened. It is the main conceptualization of Proportional Representation generally used throughout the world ([[Party List]] and [[STV]] pass versions of it.)
The two main types of PSC are k-PSC (aka. Hare-PSC, a condition requiring a solid coalition of c candidates supported by k Hare quotas to be always elect at least <math>\min(c, k)</math> most-preferred candidates) and k+1-PSC (aka. Droop-PSC, which is the same as Hare-PSC but holding for Droop quotas instead). The Droop-PSC criterion is also called the '''Droop proportionality criterion'''.
Any voting method that collects enough information to distinguish solid coalitions (generally scored or ranked methods, since preferences can be inferred from their ballots) can be forced to be PSC-compliant by first electing the proportionally correct number of candidates from each solid coalition before doing anything else.
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== Generalised solid coalitions ==
The [[Expanding Approvals Rule]] passes a stricter PR axiom than PSC:
<blockquote>'''Definition 5 (Generalised solid coalition)''' A set of voters ''N''′ is a ''generalised solid coalition'' for a set of candidates ''C''′ if every voter in ''N''′ weakly prefers every candidate in ''C''′ at least as high as every candidate in C\C′. That is, for all i ∈ N′ and for any c′ ∈ C′
: <math>\forall c \in C \setminus C^{\prime}\quad c^{\prime} \succsim_{i} c</math>.
We note that under strict preferences, a generalised solid coalition is equivalent to solid coalition. Let c(i, j) denotes voter i’s j-th most preferred candidate. In case the voter’s preference has indifferences, we use lexicographic tie-breaking to identify the candidate in the j-th position.
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exists a set C′′ ⊆ W with size at least min{ℓ, |C′|} such that for all c′′ ∈ C′′
: ∃i ∈ N′ : c′′ i c(i,|C′ |).<ref name="Aziz Lee p. 8">{{cite journal | last=Aziz | first=Haris | last2=Lee | first2=Barton E. | title=The expanding approvals rule: improving proportional representation and monotonicity | journal=Social Choice and Welfare | publisher=Springer Science and Business Media LLC | volume=54 | issue=1 | date=2019-08-09 | issn=0176-1714 | doi=10.1007/s00355-019-01208-3 | page=8 | url=https://arxiv.org/abs/1708.07580
By analogy to [[Descending Acquiescing Coalitions]], the generalized PSC could also be called proportionality for acquiescing coalitions.
== Notes ==
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Arguably there is some kind of coalition of 45 voters backing candidates A through J here, and since the largest opposing coalition is 8 voters, D'Hondt would say that the 45-voter coalition ought to win all 5 seats. At that point, one could eliminate all candidates outside the 45-voter coalition (K and L) at which point A through E all are a Hare quota's 1st choice and must all win. This sort of thinking is generally what Condorcet PR methods such as Schulze STV do.
== References ==
[[Category:Voting theory]]
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