Cardinal proportional representation: Difference between revisions
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'''Cardinal proportional representation''' (or "'''Cardinal PR'''") is a class containing [[cardinal voting systems]] used for [[proportional representation]] in multi-seat elections.
It should be noted that these methods follow different types and philosophies of proportionality than most other PR methods; they all fail [[PSC]] (though [[Sequential Monroe voting]] comes closest).▼
Because of the nature of [[Ratings ballot|rated ballots]], it is possible to make assumptions that allow us to examine many different variations of what it means to "represent" voters in the multi-winner context, and to observe to what degree they are all represented.
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Also see the following section for some categories.
When investigating cardinal PR, it is often categorized into optimal PR methods, which generally work by assigning every possible [[winner set]] a score based on how good it is, and picking the best winner set out of all possible winner sets, and sequential PR methods, which elect one candidate at a time. Optimal PR has the issue of being non-hand-countable and very computationally expensive and complex (in fact, with large committees, they may be both completely impossible to compute and very, very vulnerable to strategic voting<ref>{{Cite web|url=https://www.rangevoting.org/QualityMulti.html|title="Optimal proportional representation" multiwinner voting systems I: methods, algorithms, advantages, and inherent flaws|last=Smith|first=Warren D.|date=October 2015|website=rangevoting.org|url-status=live|archive-url=|archive-date=|access-date=}}</ref>), so in practice, many sequential cardinal PR methods are designed to approximate certain optimal PR methods. When simulating the quality of various cardinal PR methods, sometimes it's common to use optimal PR methods more as "benchmarks" of how good the winner set chosen by the sequential method is, rather than as an actual way of running an election.
The [[KP transform]] can be very useful in allowing '''Approval PR''' methods ([[Approval voting]]-based cardinal PR methods) to work with rated ballots with more than two allowed scores.
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*
*Under the [[Phragmén's Method| Phragmén interpretation]], voting is a distribution problem where the representation weight of candidates must be fairly spread across the different voters to produce the most equitable representation possible. The winner set composed of candidates which best distribute the candidates representation is the most proportional.
*Under the [[Monroe's method | Monroe
*Under the Thiele interpretation, voters have vote weight which should be distributed across candidates. The proportion of ballot weight assigned to each winner is the amount which that candidate supports their election. Under this interpretation, the more an outcome maximizes the sum of all score when reweighted by ballot weight, the more proportional it is.
**Thiele's [[party list case]] is the [[Highest averages method]]<nowiki/>s.
*Under the [[Vote unitarity | Unitary interpretation]] interpretation of each voter has an fixed amount of utility to be spent on candidates. When a candidate is elected their power to elect subsequent candidates is lower directly proportionally to the amount of utility previously spend on prior candidates. This interpretation can be thought of as an additional constraint on the [[Monroe's method | Monroe interpretation]] but since the philosophy is about voters spending points on candidates rather than voters themselves being assigned to candidates it is a distinct interpretation of proportional representation. The [[Vote unitarity | Unitary interpretation]] is in some way the inverse interpretation of the [[Phragmén's Method| Phragmén interpretation]]. In the former each '''voter''' has a conserved amount of vote weight to spend on candidates and in the latter the each '''candidate''' has a conserved amount of representation weight to distribute over the voters.
*Under the [[COWPEA]] interpretation, the weight received by a candidate approved on a particular ballot would not be equal to the other candidates also approved on that ballot, but in a proportional manner according to the rest of the electorate.
===Comparison===▼
To compare, [[PSC]] can be thought of to some extent as a separate philosophy to Monroe because rather than trying to look at utility, it requires coherent groups to have a certain number of seats. PSC and Monroe can be made to conflict with examples where a solid coalition has some differences within itself, while another, smaller group is more unified; see [[PSC#Examples]]. ▼
▲To compare, [[
[[Proportionality for Solid Coalitions]] is praised for ensuring that groups of voters get what would intuitively be considered an at least somewhat proportional outcome, even if they disagree about what candidate best represents their faction. Most methods passing this criterion behave like Monroe and don't use preferences outside the faction to decide who will be a faction's best representative, but exceptions like [[STV-CLE]] exist.
===Example
{| class="wikitable"
|-
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|[[Sequential Monroe voting]]||[[Monroe's method | Monroe interpretation]]||-
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|[[Method of Equal Shares]]||[[Monroe's method | Monroe interpretation]]||-
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|[[Sequentially Spent Score]]||[[Vote unitarity | Unitary interpretation]]||-
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|[[Sequentially Shrinking Quota]]||[[Vote unitarity | Unitary interpretation]]||May not be strictly Unitary but follows from the theory
|-
|[[Sequential proportional approval voting]]||[[Proportional approval voting | Thiele Interpretation]]||Approval ballots only
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|[[Reweighted Range Voting]]||[[Proportional approval voting | Thiele Interpretation]]||May not be strictly Thiele but follows from the theory▼
|-
|[[Single distributed vote]]||[[Proportional approval voting | Thiele Interpretation]]||A more Thiele implementation of [[Reweighted Range Voting]]▼
|-
|[[Sequential proportional score voting]]||[[Proportional approval voting | Thiele Interpretation]]||
▲|[[Reweighted Range Voting]]||Thiele Interpretation||May not be strictly Thiele but follows from the theory
|-
|[[Harmonic Voting]]||[[Proportional approval voting | Thiele Interpretation]]||
▲|[[Single distributed vote]]||Thiele Interpretation||A more Thiele implementation of [[Reweighted Range Voting]]
|-
|[[Sequential Phragmen]]||[[Phragmén's Method| Phragmén interpretation]]||
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|[[Sequential Ebert]]||[[Phragmén's Method| Phragmén interpretation]]||
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|[[PAMSAC]]||[[Phragmén's Method | Phragmén interpretation]]||
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|[[COWPEA Lottery]]||[[COWPEA| COWPEA interpretation]]||
|}
▲===Comparison===
===The backstory===
Thiele, a Danish statistician, and Phragmen, a mathematician have been debating these two philosophies in Sweden. Thiele originally proposed [[
The Monroe interpretation named after the first first person to formalize the concept, Burt Monroe.<ref name="Monroe 1995 pp. 925–940">{{cite journal | last=Monroe | first=Burt L. | title=Fully Proportional Representation | journal=American Political Science Review | publisher=Cambridge University Press (CUP) | volume=89 | issue=4 | year=1995 | issn=0003-0554 | doi=10.2307/2082518 | pages=925–940|url=https://www.cambridge.org/core/journals/american-political-science-review/article/fully-proportional-representation/ACD79636D5CF12D1E56D43EF7AB7AFE2 | access-date=2020-02-09}}</ref> [[Single transferable vote]] is a Monroe type system which predates this formalization so it is clear that the core idea had existed for some time.
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The trouble with this is, politicians are not like tap water and oranges. That reasoning would make sense if politicians were “wholly owned” by the Blues, just as Peter wholly-eats an apple. But even the most partisan politicians in Canada do a lot of work to help Joe Average constituent whose political leanings they do not even know. At least, so I am told.
===Incompatibility of the philosophies===
Pick your poison: it seems that all proportional voting methods must fail one of two closely related properties:
# If a group of voters gives all the candidates the same score, that cannot affect the election results (ex: if you gave every candidate a max score, your vote shouldn’t change who is and isn’t a winner any more so than you would change the results by just not voting).
# If some of the winners are given the same score by all voters, that cannot affect the proportionality of the election results among the remaining winners (ex: if you removed a candidate that is given a max score by all voters, and ran the election again such that you were electing 1 less winner, the only difference between that election result and the original election result should be that it does not contain the universally liked candidate).▼
▲If some of the winners are given the same score by all voters, that cannot affect the proportionality of the election results among the remaining winners (ex: if you removed a candidate that is given a max score by all voters, and ran the election again such that you were electing 1 less winner, the only difference between that election result and the original election result should be that it does not contain the universally liked candidate).
Phragmen/Monroe-type methods fail 1. and Thiele-type methods fail 2. and as of this point, it doesn’t seem possible to have them both without giving up PR.
Peters and [[Piotr Skowron | Skowron]] determined other properties that Phragmén passes but no Thiele-type method can pass, further indicating an incompatibility between the Phragmén and Theile philosophies.<ref name="Peters Skowron 2019">{{cite arXiv | last=Peters | first=Dominik | last2=Skowron | first2=Piotr | title=Proportionality and the Limits of Welfarism | date=2019-11-26 | eprint=1911.11747 | class=cs.GT}}</ref>
== Notes ==
Because rated voting methods allow a voter to give no candidate the highest score, it is possible to give some voters less power to their ballots if they choose it. See [[normalization]] for discussion on this.
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It is possible to use a cardinal PR method to fill all but one of the seats in an election, and then use either [[STAR voting]] or a [[:Category:Condorcet-cardinal hybrid methods|Condorcet-cardinal hybrid method]] to fill the final seat. For example, SPAV could be used to fill the first four of five seats, and then with the ballots in their reweighted forms, the [[Smith//Approval]] winner could be elected to the final seat. This can be done with ranked or rated ballots using [[Approval threshold|approval thresholds]] to find out both approvals and [[Head-to-head matchup|head-to-head matchups]]. One advantage this holds over using [[Single transferable vote#Deciding the election of the final seat|STV#Deciding the election of the final seat]] or any ranked PR method with a Condorcet method for the final seat is that there appear to be no simple, hand-countable ranked PR methods that reduce to [[D'Hondt]] in their [[party list case]], whereas SPAV and other cardinal PR methods do.
▲It should be noted that these methods follow different types and philosophies of proportionality than most other
== See also ==
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