Proportional representation: Difference between revisions
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'''Proportional Representation''' ('''PR''') is a measure of the outcome of an election where there are multiple parties and multiple members are elected. It is one of many [[types of representation]] in a [[W:Representative government|representative government]].
In practice, the implementation involves ensuring that [[W:Political party|political parties]] in parliament or legislative assemblies receive a number of seats (approximately) proportional to the percentage of the vote they received by making use of a [[Partisan systems|partisan system]]. One system which achieves high levels of proportional representation is [[Party-list proportional representation|party-list proportional representation]]. Another kind of electoral system
== Measures ==
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: <math>\mathrm{LSq} = \sqrt{ \frac{1}{2} \sum_{i=1}^n ( V_i-S_i ) ^2}</math> {{sfn|Gallagher|1991|p=40}}
The index weighs the deviations by their own value, creating a responsive index, ranging from 0 to 100. The larger the differences between the percentage of the votes and the percentage of seats summed over all parties, the larger the Gallagher index. The larger the index value, the larger the disproportionality, and vice versa. Michael Gallagher included "other" parties as a whole category, and [[Arend Lijphart]] modified it, excluding those parties. Unlike the well-known [[Loosemore–Hanby index]], the Gallagher index is less sensitive to small discrepancies.
▲The while the [[Gallagher index]] is considered the standard measure for [[Proportional Representation]], Gallagher himself considered the [[Sainte-Laguë method]] "probably the soundest of all the measures." This is closely related to Pearson's chi-squared test which has better statistical underpinning.
:<math>\mathrm{SLI} = \sum {(S-V)^2 \over V}</math>
The failing of all such measures is the assumption that each vote is cast for one political party. This means that the only system which can be used
== Proportional Representation Criteria==
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===Proportional (Ideological) Representation Criterion===
Whenever a group of voters gives max support to their favoured candidates and min support to every other candidate, at least one seat less than the portion of seats in that district corresponding to the portion of seats that that group makes up{{clarify}} is expected to be won by those candidates.
One of the effects of this property is that if all voters vote solely on party lines (max support to everyone in your party and min support to everyone outside of it), then the proportion of popular vote for candidates associated to parties is roughly equal to the proportion of members elected for each party. This is identical to “Partisan Proportionality” in the case that all groups large enough to expect a winning candidate have a party which they identify with and their candidate belongs to.
===Partisan Proportionality Criterion===
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# There is a clear relationship between the vote and the endorsement for a single party
This means that only [[Partisan Systems]] can be exactly proportional. Conversely, no system has no{{clarify}} Proportional Representation since metrics like [[Gallagher index]] never reach the maximum values. The criteria above are often used to define proportionality for modern systems like [[Reweighted Range Voting]] or [[Sequential proportional approval voting]]. The most common being Hare Quota Criterion. These are normally implemented as a number of multi-member districts that together form a parliament. Each district produces results guaranteed to pass the Hare Quota Criterion.
The district magnitude of a system (i.e. the number of seats in a constituency) plays a vital role in determining how proportional an electoral system can be. When using such systems, the greater the number of seats in a district or constituency, the more Proportional Representation it will achieve.
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However, multiple-member districts do not need to use a system that passes any of these proportionality criteria. For example, a [[bloc vote]] would not pass any of the criteria.
An interesting quirk for implementation is that many [[Partisan Systems]] are altered in order to remove representation from groups. For example, in a [[Party List]] system it is common to
===Semi-proportional Systems===
A "semi-proportional" system is made of several [[Regional Systems | regional]] [[Multi-Member Districts]] with each
Semi-Proportional systems can be constructed from any multi-winner system. However, they are typically done with sequential non-partisan systems, such as the [[single transferable vote]] and [[Reweighted score voting]].
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==Non-Partisan Definitions==
There are three main competing philosophies between what is and is not proportional: Phragmen, Monroe and Thiele.
There are three main competing philosophies between what is and is not proportional, Phragmen, Monroe and Thiele. Under the most Phragmen interpretation, voting is a balancing problem where the weights of candidates must be balanced between the different voters and the outcomes composed of candidates that best balance these weights are the most proportional. Under the most Monroe interpretation, every candidate has a quota, and the more an outcome maximizes the scores voters in that candidate’s quota gives them, the more proportional the voting method is regardless of how anybody outside of that candidate’s quota rates them. Under the most Thiele interpretation, every voter has an honest utility of each candidate, and even if you completely resent a candidate, it is statistically impossible for your honest utility of any individual candidate to equal 0 exactly. Under this interpretation, the more an outcome maximizes the sum among all voters: ln( the sum of utilities that voter gave to each winner ), the more proportional it is. Since candidates can’t choose their honest utilities, they can choose the scores they give to candidates which means that it is much more likely that a candidate will give a set of candidates all zero scores which will blow up the natural log function (see footnote), so to counter-act this, the most Thiele voting methods instead use the partial sums of the harmonic function, which are closely related to the natural log (The natural log is the integral of 1/t from t=1 to t=x and the partial sums of the harmonic series are the summation of 1/n from n=1 to n=x).▼
* Under the Phragmen interpretation, voting is a balancing problem where the weights of candidates must be balanced between the different voters and the outcomes composed of candidates that best balance these weights are the most proportional.
* Under the Monroe interpretation, every candidate has a quota, and the more an outcome maximizes the scores voters in that candidate’s quota gives them,{{clarify}} the more proportional the voting method is regardless of how anybody outside of that candidate’s quota rates them.
▲
===The backstory===
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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 not{{clarify}} fail the [http://scorevoting.net/QualityMulti.html#faildesid universally liked candidate criterion] which is a criterion that Thiele type methods fail.
'''Benefits of the Phragmen/Monroe measure of proportionality:'''
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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
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).
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Footnote:
In addition, maximizing the natural log favors small parties a little too much to pass proportional criteria and when a voter’s satisfaction is zero is just the most extreme example of that. The partial sums of the harmonic series equation does however pass the proportional criteria that a maximization of the natural log can’t. I{{who}} personally think that the partial sums of the harmonic series are better for determining the winners of an election, but the natural log of summed utilities is a better tool for measuring proportionality in computer simulations even if those simulations are skewed to representing small parties too much (which may or may not be a bad thing).
== Alternatives ==
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