Difference between revisions of "Proportionality for Solid Coalitions"

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== Types of PSC ==
For the purposes of PSC, maximal support to a candidate or set of candidates is generally determined as:
 
- with ranked ballots, all of the candidates are ranked above all candidates not in the set.
 
- with rated, approval, and choose-one ballots, all of the candidates in the set are given the maximum score or are marked.
 
=== Hare-PSC ===
'''k-PSC''' or '''Hare-PSC''' is 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 (i.e. k candidates whenever c is not less than k.)
 
=== Droop-PSC ===
'''k+1-PSC''' or '''Droop-PSC (since many authors call the [[Hagenbach-Bischoff quota]] a Droop quota)''' is the same as Hare-PSC but holds for [[Hagenbach-Bischoff quota|Hagenbach-Bischoff quotas]] instead (many authors call the [[Hagenbach-Bischoff 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]].
 
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):
 
(For the purposes of PSC, maximal support to a candidate or set of candidates is generally determined as:
 
-* with [[Ranked ballot|ranked ballots]], all of the candidates in the set are ranked above all candidates not in the set.
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):<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 at least a quota of voters,then at least k of their preferred candidates must win.</blockquote>[[SNTV]] passes 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.
-* with rated, approval, and choose-one ballots, all of the candidates in the set are given the maximum score or are marked.)
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== Examples ==
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.
 
Methods that reduce to [[D'Hondt]] in the [[Party list case|party list case]]<nowiki/>tend to pass at least a weak form of Droop-PSC; this is because D'Hondt guarantees every party will get at least the number of HB quotas it has rounded down.
 
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:
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