Proportional representation

Proportional Representation (PR) is a measure of the outcome of an election where there are multiple parties and multiple member are election. It is one of many types of representation in a Representative Government.

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Proportional representation

In practice this usually involves ensuring that political parties in parliament or legislative assemblies receive a number of seats (approximately) proportional to the percentage of vote they received through a Partisan System. The simplest system which achieves high levels of Proportional Representation is the 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 cumulative voting are sometimes categorized as "semi-proportional".

Measures

There are several metrics which are used to define Proportional Representation explicitly. The most well accepted is the Gallagher index. As such, it measures the difference between the percentage of votes each party gets and the percentage of seats each party gets in the resulting legislature, and it also measures this disproportionality from all parties collectively in any one given election. That collective disproportionality from the election is given a precise score, which can then be used in comparing various levels of proportionality among various elections from various electoral 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 ( $V_{i}$ ) and percent of seats ( $S_{i}$ ) for each of the political parties ( $i=1,\ldots ,n$ ).

\mathrm {LSq} ={\sqrt {{\frac {1}{2}}\sum _{i=1}^{n}(V_{i}-S_{i})^{2}}}

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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.

\mathrm {SLI} =\sum {(S-V)^{2} \over V}

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

The metric for proportionality must be defined and the winner selection defined under those terms
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.

Advocacy

Proportional representation is unfamiliar to most citizens of the United States. There are many organizations who campaign for Proportional Representation but they often use the term loosly and use it to refer to systems. Such campaigns have use advocacy for the poorly defined term "proportional system" in order to gain support. The dominant system in former British colonies was [[Single Member Plurality] but mixed member system and [Single Transferable Vote]] 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 absolutely accurate or more comprehensive representation, based on gender or minority status (like ethnicity). Note that features such as this are not strictly part of proportional representation; depending on what kind of PR is used, people tend to be already represented proportionally according to these standards without such additional rules.

See Two-Party System: Arguments For and Against for a list of perceived advantages of proportional representation.