Proportionality for Solid Coalitions: Difference between revisions

 

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.

 

=== Weak forms of PSC ===

 

PSC is a requirement that holds for honest voters. Many voting methods pass weaker requirements that hold only for strategic voters, with the difficulty of the strategy depending on the method. In general, any method that passes such weaker versions of PSC is considered to be at least semi-proportional. Note that PSC implies all of these weaker forms of PSC. Here are some of these weaker requirements (note that the requirements vary slightly depending on whether you're using the Hare quota, HB quota, or other quota):

* with rated, approval, and choose-one ballots, all of the candidates in the set are given the maximum score or are marked.)

<blockquote>If a solid coalition of k quotas evenly distributes its support among k of their preferred candidates such that each of the k candidates receives maximal support from a quota of voters, with each quota of voters giving no support to any other candidates, and the solid coalition as a whole giving no support to any candidates other one of the k candidates, then at least k of their preferred candidates must win.</blockquote>[[SNTV]] and [[Cumulative voting|cumulative voting]] pass this with Droop quotas.<blockquote>If a solid coalition of k quotas gives maximal support to k of their preferred candidates, and no support to all other candidates, then at least k of their preferred candidates must win.</blockquote>Most cardinal PR methods pass this for Hare quotas.<blockquote>If a solid coalition of k quotas gives maximal support to at least k of their preferred candidates, and less-than-maximal support to all other candidates, at least k of those preferred candidates must win.</blockquote>[[Sequential Monroe voting]] passes this, making it the best cardinal PR method from the perspective of PSC. This is probably the strongest PSC-like requirement that a natural voting method can pass without actually passing PSC.

 

== Examples ==

<blockquote>So, the Hare quota here is 20. A1 and A2 are immediately elected, but post-transfer A3 only has 11 votes, and is thus eliminated first. B1, B2, B3 take the remaining 3 seats.<ref name="reddit 2011">{{cite web | title=Can Ranked-Choice Voting Save American Democracy? : EndFPTP | website=reddit | date=2011-01-26 | url=https://www.reddit.com/r/EndFPTP/comments/ermb1s/comment/ff7a7f8 | access-date=2020-02-19}}</ref></blockquote>

 

There can be quota overlaps when assigning PSC claims; suppose a group constituting 80% of a quota of voters vote A>B>C=D, another group of 80% of a quota vote B>A>C=D, and another group of 50% of a quota vote C>A=B=D. Then, 2 candidates must be elected from the set (A, B, C, D), since in total 2.1 quotas mutually most prefer that set, but a further constraint is that 1 candidate must win from within (A, B), since 1.6 quotas mutually most prefer them. It would not satisfy PSC if the final winner set had neither A or B in it in other words, even if it had C and D.

 

 

By analogy to [[Descending Acquiescing Coalitions]], the generalized PSC could also be called proportionality for acquiescing coalitions.

 

== Notes ==

With rated ballots, it is possible for a voter to express less-than-full support for any candidate. Because of this, one way to apply PSC to rated ballots and thus cardinal PR methods would be to require that every voter in each solid coalition give at least one candidate in their coalition the maximum score. An even stricter requirement could be to require them to give every candidate in the coalition a perfect score, though this could instead be thought of as a genuinely slightly weaker form of PSC; it is already satisfied by [[SMV]]. This is modeled off of the [[Majority criterion for rated ballots|majority criterion for rated ballots]].

 

Droop-PSC implies Hare-PSC, since a Hare quota is simply a large Droop quota, but the same doesn't hold the other way around. Hare-PSC is equivalent to the unanimity criterion and Droop-PSC to the mutual majority criterion in the single-winner case. Note that this means cardinal PR methods can only pass Hare-PSC and not Droop-PSC in order to reduce to cardinal methods that fail the mutual majority criterion in the single-winner case, which is most of them.

 

Though Droop-PSC guarantees a majority half of the seats, it doesn't guarantee a plurality group half the seats when it could take them using [[vote management]]. 5-winner example:

 

The HB quota is (110/(5)+1)=~18.333, so A is guaranteed only (51/18.333 rounded down) = 2 out of the 5 seats, less than half. With vote management, Party A could split into 3 groups of 17 votes each, and most PR methods would then give them 3 seats. So it may be worth considering a stronger type of Droop-PSC based on giving solid coalitions at least as many seats as they'd get in [[D'Hondt]]. [[Schulze STV]] is an example of a method that does so.

 

Note that PSC doesn't hold if some voters in a coalition back out-of-coalition candidates i.e. 1-winner example with Droop quota of 51:

 

 

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.

 

== See Also ==