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 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 that strives to achieve proportional representation but which does not rely on the existence of political parties is the single transferable vote (STV). Some electoral systems, such as the single non-transferable vote and Reweighted score voting are sometimes categorized as "semi-proportional". A "semi-proportional" system is made of several regional districts with each of which passing some measure of Proportional Representation.


There are several metrics which 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.

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 the 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 in Partisan systems. Under the assumption that a plurality vote for a candidate represents a vote for their party, these meausres 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 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 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 relation 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 Proportional Representation since metrics like Gallagher index never reach they 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 implements as a number of multi-member districts which 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 which 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 put a cap that a party needs some percent of votes to receive any seats. The effect of this is that the major parties receive relatively relatively equitable results but the fringe parties receive none.


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 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 chose their honest utilities, they can chose 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 afterwards. Phragmen believed there was 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 were 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 (i.e. weak monotonicity, the ability for votes that give every candidate the same score to effect 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 not 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 (Defined as Warren’s weak monotonicity criteria on the document). Weak participation (also defined on the document).

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 “wholy owned” by the Blues, just as Peter wholy-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 effect 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 then you would change the results by just not voting)

If some of the winners are given the same score by all voters, that cannot effect 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. I 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).

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
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