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==Non-Partisan Definitions==
In the case of non-partisan voting, the definition of proportional Representation is undefined. Metrics like Gallagher index can no longer be defined. For non-partisan multi-member systems, for ranked methods, there is generally one minimum requirement for proportionality, [[Proportionality for Solid Coalitions]] (though see the [[Monroe's method#Fully Proportional Representation with Ordinal Balloting|Monroe's method]] article for an alternative idea), while for cardinal PR methods, there are four main competing philosophies between what is and is not proportional: [[Phragmén's Method | Phragmén]], [[Monroe's method | Monroe]], Thiele and [[Vote unitarity | Unitary]]. See the [[cardinal PR]] article for more information on these.
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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 interpretation]], voting is an attribution problem where every candidate has a [[Quota | quota]] of voters to be filled with specific voters. The winner set composed of candidates which maximizes the sum of score for the voters in that candidate’s quota is the most proportional. The voting method is impartial to how anybody outside of that candidate’s quota rates them.
* 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.
* 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.
[[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]].
===Example Systems===
| [[Sequential Ebert]] || [[Phragmén's Method | Phragmén interpretation]] ||
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===The backstory===
Thiele, a Danish statistician, and Phragmen, a mathematician have been debating these two philosophies in Sweden. Thiele originally proposed [[Sequential Proportional Approval Voting]] in 1900 and it was adopted in Sweden in 1909 before Sweden switched to [[Party List]] voting afterward. Phragmen believed there were flaws in Thiele’s method, and came up with his own sequential method to correct these flaws, and that started [https://rangevoting.org/NonlinQuality.html#debate a debate about what was the ideal metric of proportionality]. Thiele also came up with the approval ballot version of [[harmonic voting]], however during that time the harmonic method was too computationally exhaustive to be used in a governmental election. Both his [[sequential proportional approval voting]] and his approval ballot version of the harmonic method was lost to history until about a century later when they were independently rediscovered.
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.
[[Keith Edmonds]] saw a unification of [[Proportional representation|Proportional Representation]] and the concept of one person one vote which was maintained throughout winner the winner selection method. He coined the term "vote unitarity" for the second concept<ref>https://groups.google.com/forum/#!topic/electionscience/Tzt_z6pBt8A</ref> and designed a score reweighting system which satisfied both Hare Quota Criterion and Vote Unitarity. As such it would preserve the amount of score used through sequential rounds while attributing representation in a partitioned way similar to Monroe. It would assign Hare Quotas of score to winners which allowed for a voters influence to be spread over multiple winners as opposed to Monroe which assigns a whole ballot with no spreading. Since score is a conserved quantity which is spent like money there is a natural analogy to [https://rangevoting.org/MarketBasedVoting.html Market based voting]. This concept was heavily influence by economic theory not the Monroe interpretation even though the resultant mathematical formulation is quite similar.
===Comparison===
Many of the properties of these systems can be derived from their party list simplifications. The [[Balinski–Young theorem]] implies that not all desirable properties are possible in the same system. Theile type systems reduce to [[Highest averages method|divisor methods]] which means that adding voters or winners will not change results in undesirable ways. The other three reduce to [[Largest remainder methods]] which obey Quota Rules but adding voters or winners may change outcomes in undesirable ways. One such way is failure of [[Participation criterion]]. It is not clear which is a fundamentally better choice since Quota Rules are inanimately tied with some definitions of proportionality.
Phragmen and Monroe share many desirable and undesirable properties. Most importantly a lack of convexity, the ability for votes that give every candidate the same score to affect the outcome. There are also election scenarios where both philosophies pick what is clearly the wrong winner. Further details can be found in the “Pereira’s Complaints about Monroe” section of [https://rangevoting.org/MonroeMW.html Monroe’s method] or the “Major defect pointed out by Toby Pereira” section of this [http://scorevoting.net/PRintLinprog.html Phragmen-Type method])
However neither {{clarify}} fail the [[Universally liked candidate criterion]] which is a criterion that Thiele type methods fail.
'''Benefits of the Phragmen/Monroe/Unitary measure of proportionality:'''
Passes the ULC criteria. For Thiele-type methods, because they fail ULC, every time a candidate that every voter gave a max rating to wins, the distribution of the remaining winners becomes more majoritarian/utilitarian.
'''Benefits of the Thiele measure of proportionality:'''
Adding ballots that give every candidate the same score can’t change which outcome is considered the best.
Convexity.
Warren's multi-winner participation criteria.
'''Criticisms of the Phragmen metric:'''
Taken to its limits, Phragmen-thinking would say, once the 50% Reds elected a red MP, and the 50% Blues elected a blue MP, there was no benefit whatever to replacing the red MP by somebody approved by the entire populace.
'''Criticisms of the Thiele metric:'''
The [[Universally liked candidate criterion]] can be exemplified with the following example. Three people share a house and two prefer apples and one prefers oranges. One of the apple-preferrers does the shopping and buys three pieces of fruit. But instead of buying two apples and an orange, he buys three apples. Why? Because they all have tap water available to them already and he took this into account in the proportional calculations. And his reasoning was that the larger faction (of two) should have twice as much as the smaller faction (of one) when everything is taken into account, not just the variables. Taken to its logical conclusion, Thiele-thinking would always award the largest faction everything because there is so much that we all share – air, water, public areas, etc!
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.
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).
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.
Some common criticisms of [[STV]] (which would likely hold for many other nonpartisan PR methods) are that it is too complex in terms of filling out the ballot and tabulation, that it takes too long to count compared to partisan PR methods (many of which are [[Precinct-summable]] due to being based on [[FPTP]]), and that it can even make representatives parochialist and focused on representing their multi-member districts rather than the state or nation as a whole. Note that this last criticism is inapplicable when nonpartisan PR methods are proposed for a single national/statewide district, though this is usually not proposed or done (with the exception of some 21-seaters in Australia).
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