Difference between revisions of "Proportionality for Solid Coalitions"

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=== Droop-PSC ===
'''k+1-PSC''' or '''Droop-PSC''' is the same as Hare-PSC but holds for [[Hagenbach-Bischoff quota|Hagenbach-Bischoff quotas]] instead (many authors call the [[Hagenbach-Bischoff quota]]/Hb quota a Droop quota), and requires the solid coalition's preferred candidates to be supported by ''more'' than k HB quotas, rather than at least that amount. The Droop-PSC criterion is also called the '''Droop proportionality criterion'''. Note that Droop proportionality implies the [[mutual majority criterion]], and more generally guarantees that a majority will always win at least half of the seats.
The main difference between Hare-PSC and Droop-PSC can be seen with an example: Suppose you can buy two boxes of pizza, with over 2/3rds of voters wanting Cheese pizza, and under 1/3rds of the voters wanting Pepperoni pizza. Hare-PSC would say that you should buy at least one box of Cheese pizza, but has no opinion on what you should buy for the second box, whereas Droop-PSC would say that you should buy two boxes of Cheese pizza. This can be explained as happening partially because if the 2/3rds group of cheese-preferring voters split themselves into two equally sized groups of over 1/3rd of voters each, then these "two" groups that want Cheese would each outnumber the group of under 1/3rds of voters that want Pepperoni.
=== Weak forms of PSC ===
Most methods that pass weak forms of PSC (i.e. any method that passes a weak form of Droop-PSC) allow a [[majority]] to strategically vote to get at least half of the seats.
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):
Looking at the top 5 lines, 50 voters, a Hare quota, mutually most prefer the set of candidates (A1-5) over all other candidates, so Hare-PSC requires at least one of (A1-5) must win. (Note that Sequential Monroe voting fails Hare-PSC in this example. However, one could forcibly make SMV do so by declaring that the candidate with the highest Monroe score within the set (A1-5) must win the first seat, for example.) <ref name="The Center for Election Science 2020">{{cite web | title=An example of maximal divergence between SMV and Hare-PSC | website=The Center for Election Science | date=2020-01-31 | url=https://forum.electionscience.org/t/an-example-of-maximal-divergence-between-smv-and-hare-psc/586 | access-date=2020-02-19}}</ref>
Generally,The reason Droop-PSC makes it more likely thatguarantees a majority will winwins at least half of the seats than only Hare-PSC. The reason for this is that majority solid coalitions almost always constitute enoughmore Droopvoters quotasthan (andthe alwaysnumber constitute enoughof Hagenbach-Bischoff quotas) corresponding to alwayshalf win at least halfof the seats, while with Hare quotas they can only guarantee they will win just under half the seats and have over half a Hare quota left to win the additional seat required to get at least half the seats. 5-winner example using STV with Hare quotas:
{| class="wikitable"
Note that 51 voters, a majority, prefer (A1-3) over all other candidates, and thus electing all 3 of them would mean 3 out of 5 seats, a majority, would belong to the majority. Also, the DroopHB quota here is 17~16.667, and thus by Droop-PSC the majority has (51/1716.667 rounded down to the nearest number)=3 seat or PR guarantees, which would give them a majority of seats. However, by Hare-PSC, they only have (51/20 rounded down) = 2 PR guarantees, and thus bySTV using Hare quotas, which passes Hare-STVPSC, yields:
<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.
== Complying methods ==
[[STV]], [[Party list]], and [[Expanding Approvals Rule]] pass forms of PSC. Specifically, the choice of quota in those methods determines which type of PSC they pass.
== Generalised solid coalitions ==
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