Proportional representation

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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 representative government.

In practice, the implementation involves ensuring that 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 system. One system which achieves high levels of proportional representation is party-list proportional representation. Another kind of electoral system strives to achieve proportional representation, but without relying on the existence of political parties. A common example of this is the single transferable vote (STV).


There are several metrics that are used to define Proportional Representation explicitly. A well-accepted form is the Gallagher index, which measures the difference between the percentage of votes each party gets and the percentage of seats each party gets in the resulting legislature, and aggregates across all parties to give a total measure in any one given election result. This measure attributes a specific level or Proportional Representation to a given election which can then be used in comparing various levels of proportionality among various elections from various Voting systems.

Michael Gallagher, who created the index, referred to it as a "least squares index", inspired by the residual sum of squares used in the method of least squares. The index is therefore commonly abbreviated as "LSq" even though the measured allocation is not necessarily a least squares fit. The Gallagher index is computed by taking the square root of half the sum of the squares of the difference between percent of votes () and percent of seats () for each of the political parties ().


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.

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.

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 is a Partisan system. Under the assumption that a plurality vote for a candidate represents a vote for their party, these measures can be applied to plurality voting systems like Single Member Plurality and Mixed Member Proportional. The consequence of this limitation is that Proportional Representation is not defined for systems without vote splitting.

Proportional Representation Criteria

Since the standard definitions of Proportional Representation do not apply to nearly all modern systems it has become common to define proportional representation in terms of passing some sort of criteria. There is no consensus on which criteria need to be passed for a parliament to be said to be proportional.

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 upTemplate: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

How similar are the proportion of the voters who support a party to the proportion of the parliament when voters deploy the strategy that maximizes the number of seats their preferred party gets (in most methods, this strategy is voting solely on party lines, i.e. max support to everyone in your party and min support to everyone outside of it)? This is a calculation for a specific outcome of a specific election. There are multiple different methods to be used but the most common is the Gallagher index. Specific systems can be judged under such metrics by the average expected value. This metric is nearly an exact restatement of the concept of Proportional Representation and as such, it cannot be defined in many cases.

Hare Quota Criterion

Whenever more than a Hare Quota of the voters gives max support to a single candidate and min support to every other candidate, that candidate is guaranteed to win regardless of how any of the other voters vote.

Any method that passes the Proportional Representation Criterion also passes the Hare Quota Criterion.

Winner Independent Proportionality Criterion

If at least n quotas of ballots approve the same set of candidates, but there is partial disagreement on m elected candidates outside of that set, then at least n-m candidates in the set must be elected. (If 2 quotas approve ABCD, 2 quotas approve ABCDE, and E is elected, the standard PR criterion would require 2 of ABCD to be elected, whereas this criterion would require 3 of ABCD to be elected.)

Combined Independent Proportionality Criterion

The winner set must be proportional even if some losing candidates were disqualified, scores for some losing candidates were reduced, and/or the scores for some winning candidates were increased. That is, if at least n quotas of ballots approve the same set of candidates, but there is partial disagreement on some candidates outside of that set, m of whom were elected, then at least n-m candidates in the set must be elected. (If 2 quotas approve ABCD, 2 quotas approve ABCDE, the standard PR criterion would require 2 of ABCD to be elected, whereas this criterion would require 4 of ABCDE to be elected.)

Proportional Systems

No system can be defined as giving exact proportional results unless a number of assumptions are made

  1. The metric for proportionality must be defined and the winner selection defined under those terms
  2. 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 noTemplate: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.

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 add a threshold, that a party needs some percent of votes to receive any seats. The effect of this is that the major parties receive relatively equitable results but the fringe parties receive none.

Semi-proportional Systems

A "semi-proportional" system is made of several regional Multi-Member Districts with each passing some measure of Proportional Representation. While each district is in itself going to produce results with High Proportional Representation, the assembly as a whole will not. For larger parties, the results will tend to be fairly high in proportional representation because the variation from each district is averaged out over the group. For smaller parties, there is a threshold for entry so they may receive no seats. This is normally viewed as a positive feature since partisan systems often impose such a threshold to keep out small extremist groups.

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.


Proportional representation is unfamiliar to many citizens of the United States. The dominant system in former British colonies was single member district plurality (SMDP), but mixed-member proportional representation (MMP) and single transferable vote (STV) replaced it in a number of such places.

Proportional representation does have some history in the United States. Many cities, including New York, once used it for their city councils as a way to break up the Democratic Party monopolies on elective office. In Cincinnati, Ohio, proportional representation was adopted in 1925 to get rid of a Republican party machine (the Republicans successfully overturned proportional representation in 1957).

Some electoral systems incorporate additional features to ensure more explicitly proportional representation, based on gender or minority status (like ethnicity). Note that features such as this are not strictly required for a system to be called "proportional representation". Many proportional representation advocates argue that, given their preferred system, voters will already be justly represented without demographic rules (and usually in a demographically proportional manner).

Non-Partisan Definitions

There are three main competing philosophies between what is and is not proportional: Phragmen, Monroe and Thiele.

  • 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,Template:Clarify the more proportional the voting method is regardless of how anybody outside of that candidate’s quota rates them.
  • Under the 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).

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 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.


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 Monroe’s method or the “Major defect pointed out by Toby Pereira” section of this Phragmen-Type method)

However neither notTemplate:Clarify fail the universally liked candidate criterion which is a criterion that Thiele type methods fail.

Benefits of the Phragmen/Monroe 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:

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.


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. ITemplate: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).


Due to the ambiguity and difficulty in the definition of Proportional Representation academic work often uses another more robust metric. This is the concept of a Stable Winner Set. The requirement that a system always produces a stable winner set when there exists one is definable in all possible systems. This makes it more useful than the concept of Proportional Representation which is typically tied to Partisan voting and as such cannot be defined for all systems. This concept evolved out of the economics field of Participatory Budgeting but can be equally suitable in Social Choice Theory. A less strict and more practical version of this is Justified representation.

See Also

Further reading

  • John Hickman and Chris Little. "Seat/Vote Proportionality in Romanian and Spanish Parliamentary Elections" Journal of Southern Europe and the Balkans Vol. 2, No. 2, November 2000safd
  • See the Proportional Representation Library (created by Professor Douglas J. Amy, Mount Holyoke College and now maintained by FairVote):
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