# 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).

## Measures

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 (${\displaystyle V_i}$) and percent of seats (${\displaystyle S_i}$) for each of the political parties (${\displaystyle i=1,\ldots,n}$).

${\displaystyle \mathrm{LSq} = \sqrt{ \frac{1}{2} \sum_{i=1}^n ( V_i-S_i ) ^2}}$  Template:Sfn

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.

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

### Lack of nonpartisan measures

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 Plurality Voting 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. In addition, if it is assumed that when some voters rank every candidate in a party ahead of all other candidates, that they prefer that party, then PSC and PSC-compliant voting methods can be used to measure how well ranked and rated PR methods satisfy partisan proportionality. The consequence of this limitation is that Proportional Representation is not defined for systems without vote splitting.

The reliance of the standard definition of Proportional representation on the system being Partisan is clearly limiting on the usefulness of such a definition. A Partisan system itself has long been considered a flaw which undermines the Ideal Representation of the individual. [1][2] If Proportional Representation cannot be robustly defined in a non-partisan system then it is of little use.

## 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, though most can agree that a voting method that passes one of the weak forms of PSC (several of which are listed here) is at least semi-proportional. It is worth noting that because there are disagreements on how best to conceptualize of PR, some measures look at how much each voter likes their favorite candidate i.e. the one meant to "represent them" (such as Monroe's method) while others look at how satisfied each voter is with all of the elected representatives.

### Proportionality for Solid Coalitions Criterion

PSC: If a sufficiently-sized group (generally at least a Droop or Hare quota) prefer a set of candidates above all others, do at least a proportional number (being the number of quotas the group comprises rounded down to the nearest integer) of candidates from that set (supposing there are enough of them) get elected?

### 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[clarification needed] 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 a weak form of PSC 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. This is a very weak form of PSC.

### 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.) These last two criteria are related to PSC.

## 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, all systems have some level of 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 Sequentially Spent Score 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. The most common criticism of such systems have to do with inequalities that arise from the difference in population densities. Having a 5 member district in a sparsely populated rural area would imply that the district be much larger than similar districts in cities. To avoid this it is sometimes proposed that rural areas have single member district while cities have multi-member districts. This then results in another inequality relating to the partisan allocation of funds do to some seats being simpler to win with different systems. A good example of such failures which ultimately resulted in returning to the original system is provincial Canada.[3]

An alternative, more common definition of semi-proportional is that a voting method must pass some weak form of Proportionality for Solid Coalitions e.g. allowing voters to get PSC-like outcomes through strategic voting. Something like SNTV would classify as semi-proportional under this definition.

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

Systems designed to have high levels of Proportional representation do have some history in the United States. Many cities, including New York, once used such systems for their city councils as a way to break up the Democratic Party monopolies on elective office. In Cincinnati, Ohio, a system was adopted in 1925 to get rid of Republican party dominance but was successfully overturned in 1957.

Some electoral systems incorporate additional constraints on winner selection to ensure quotas based on based on gender or minority status (like ethnicity). Note that features such as this are not typically associated with "proportional representation" although the goal of such systems is to ensure that elected member representation is proportional to such population percentages. Many proportional representation advocates argue that, voters will already be justly represented without these demographic rules since the particular immutable characteristics are independent of partisan allegiance, ideology or ability as a politician.

## 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 article for an alternative idea), while for cardinal PR methods, there are four main competing philosophies between what is and is not proportional: Phragmén, Monroe, Thiele and Unitary. See the cardinal PR article for more information on these.

### Example Systems

System Philosophy Comment
Single transferable vote PSC or Monroe interpretation Ordinal ballots
Sequential Monroe voting Monroe interpretation -
Sequentially Spent Score Unitary interpretation -
Sequentially Shrinking Quota Unitary interpretation May not be strictly Unitary but follows from the theory
Sequential proportional approval voting Thiele Interpretation Approval ballots only
Reweighted Range Voting Thiele Interpretation May not be strictly Thiele but follows from the theory
Single distributed vote Thiele Interpretation A more Thiele implementation of Reweighted Range Voting
Sequential Phragmen Phragmén interpretation
Sequential Ebert Phragmén interpretation

### Comparison

Proportionality for Solid Coalitions is praised for ensuring that voters get what would intuitively be considered an at least somewhat proportional outcome, but is criticized for focusing too much on giving a voter one "best" representative, rather than letting that voter have influence in electing several representatives.

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 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.

### Criticisms

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).

## Alternatives

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.

## Definitions

Ballot weight: The amount of power a voter's ballot has. It starts out at 1 vote, and can go down all the way until 0 i.e. if a candidate gets elected with a voter's support, then their ballot weight is reduced to allow other voters to elect someone they prefer.

## Notes

It may be desirable in some circumstances for a voting method to produce only semi-proportional outcomes. For example, the list of movie nominees for the Oscars may be improved with some diversity, but movies broadly recognized as excellent should still take priority over movies that are more polarizing. Similarly, if doing a primary election to decide which candidates should advance to the general election, it may be desirable for there to be some PR, in part to ensure more choices for the voters and to thwart large factions' attempts to pack the general election with only their side's candidates, but by and large the result should resemble a Bloc voting election. In order to do this, the best approach is often sequential: use some single-winner method to pick a best winner, then reduce the power of the voters who supported that winner and repeat until all seats are filled. For example, one could create a voting method in between RRV and Score voting by reducing the amount of ballot weight RRV takes from each voter in each round. See Condorcet PR for ideas on this with Condorcet methods.

### Party list case

The party list case of a proportional voting method is what type of Party list allocation method it becomes equivalent to when voters vote in a "Party list"-like manner (i.e. they give maximal support to some candidates and no support to all others, as if voting on party lines). Generally, the party list case of a PR method will either be a divisor method, such as D'Hondt, or a Largest remainder method, such as Hamilton. PR methods can generally be split into two categories: sequential (one winner is elected at a time) and optimal (every possible winner set is compared to each other and the best one is chosen).

Almost all sequential PR methods can have a single-winner method done to elect the final seat; this is because at that point there is only one seat left to elect. See ￼￼Single transferable vote#Deciding the election of the final seat for an example. Condorcet methods and STAR voting can be made to work with PR methods in this way.