Difference between revisions of "Proportionality for Solid Coalitions"

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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.)
 
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.
 
== Types of PSC ==
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 ===
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.
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.
 
Cardinal PR methods generally don't pass PSC, though they pass weaker, related versions relating to Hare quotas of voters being able to force the proportionally correct number of their most-preferred candidates to win through strategic voting. In general, any method that passes such weaker versions of PSC is considered to be at least semi-proportional.
 
== Examples ==
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