Proportionality for Solid Coalitions: Difference between revisions
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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.) |
'''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.) |
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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'''. |
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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. |
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 == |
== Generalised solid coalitions == |
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The Expanding Approvals Rule passes a stricter PR axiom than PSC: |
The [[Expanding Approvals Rule]] passes a stricter PR axiom than PSC: |
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<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′ |
<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′ |
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: <math>\forall c \in C \setminus C^{\prime}\quad c^{\prime} \succsim_{i} c</math>. |
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: ∀c ∈ C\C′ c′ i c. |
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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. |
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′′ |
exists a set C′′ ⊆ W with size at least min{ℓ, |C′|} such that for all c′′ ∈ C′′ |
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: ∃i ∈ N′ : c′′ i c(i,|C′ |).<ref>https://arxiv.org/abs/1708.07580 |
: ∃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}}</ref></blockquote> |
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By analogy to [[Descending Acquiescing Coalitions]], the generalized PSC could also be called proportionality for acquiescing coalitions. |
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== Notes == |
== 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. |
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. |
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== References == |
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[[Category:Voting theory]] |