Proportional representation

Proportional representation (PR) characterizes electoral systems in which divisions in an electorate are reflected proportionately in the elected body. The most widely used families of "proportional representation" electoral systems are party-list PR, the single transferable vote (STV), and mixed-member proportional representation (MMP).

Voting theorists frequently debate which systems can be called "proportional representation", and consider the levels of proportionality achieved by various systems from "low proportional" to "high proportional". The concept of "proportional representation" can be quantified as a measure of the outcome of an election where there are multiple parties and multiple members are elected, and the representatives are demographically similar to the voting population. It is one of many types of representation in a representative government.

The partisan definition for "proportional representation" is that the candidates are partitioned into disjoint parties, and each voter approves all candidates in a single party. For example, suppose we need to elect a committee of size 10. Suppose that exactly 50% of the voters approve all candidates in party A, exactly 30% approve all candidates in party B, and exactly 20% approve all candidates in party C. Then, proportional representation requires that the committee contains exactly 5 candidates from party A, exactly 3 candidates from party B, and exactly 2 candidates from party C. If the fractions are not exact, then some rounding method should be used, and this can be done by various apportionment methods. However, in approval voting there is a different challenge: the voters' approval sets might not be disjoint. For example, a voter might approve one candidate from party A, two candidates from B, and five from C. This raises the question of how proportional representation should be defined. The concepts of "justified representation" one approach to solving this problem.

Usage
Typically in the countries that use "high proportional" systems, there are more than two large political parties, and representatives will be chosen from many different parties. When the majority of voters favor one party, that party usually forms the government in parliamentary systems. Otherwise the government that is formed will usually have members of more than one party.

Countries which use systems designed to achieve high proportional representation include:
 * Africa: South Africa
 * Asia: Israel, Russia, South Korea,  Sri Lanka,  and Turkey.
 * Europe: Austria, Belgium, Bulgaria, Czech Republic, Denmark, Estonia, Finland, Greece, Iceland, Ireland, Netherlands, Norway, Poland, Portugal, Romania, Spain, Sweden, and Switzerland
 * South America: Argentina, Brazil,

Countries which have semi-proportional representation systems include: Australia, Germany, Hungary, India, Republic of Ireland, Italy, Japan, Mexico, New Zealand, Thailand and Wales. India is one of the most successful examples of a country with proportional representation.

Similar principles apply to sub-regions, who may have their own parliament or assembly. For example, in Scotland, after the passage of the Local Governance (Scotland) Act 2004, the Scottish portion of the United Kingdom uses single transferable vote to select many of their local representatives.

Measures
In practice, the implementation of "proportional representation" 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 proportionality of election methods explicitly.

Gallagher index
One well-accepted measure of proportionality 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 ($$V_i$$) and percent of seats ($$S_i$$) for each of the political parties ($$i=1,\ldots,n$$).
 * $$\mathrm{LSq} = \sqrt{ \frac{1}{2} \sum_{i=1}^n ( V_i-S_i ) ^2}$$

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.


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

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. 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
Proportionality for Solid Coalitions is often abbreviated as "PSC". The definition: 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 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. This is explicitly formalized for approval ballots as Proportional justified representation.

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.

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.

Party list PR
Party list proportional representation is an electoral system in which seats are first allocated to parties based on vote share, and then assigned to party-affiliated candidates on the parties' electoral lists. This system is used in many countries, including Finland (open list), Latvia (open list), Sweden (open list), Israel (national closed list), Brazil (open list), Nepal (closed list) as adopted in 2008 in first CA election, the Netherlands (open list), Russia (closed list), South Africa (closed list), Democratic Republic of the Congo (open list), and Ukraine (open list). For elections to the European Parliament, most member states use open lists; but most large EU countries use closed lists, so that the majority of EP seats are distributed by those. Local lists were used to elect the Italian Senate during the second half of the 20th century.

Closed list PR
In closed list systems, each party lists its candidates according to the party's candidate selection process. This sets the order of candidates on the list and thus, in effect, their probability of being elected. The first candidate on a list, for example, will get the first seat that party wins. Each voter casts a vote for a list of candidates. Voters, therefore, do not have the option to express their preferences at the ballot as to which of a party's candidates are elected into office. A party is allocated seats in proportion to the number of votes it receives.

There is an intermediate system in Uruguay ("Ley de Lemas"), where each party presents several closed lists, each representing a faction. Seats are distributed between parties according to the number of votes, and then between the factions within each party.

Open list PR
In an open list, voters may vote, depending on the model, for one person, or for two, or indicate their order of preference within the list. These votes sometimes rearrange the order of names on the party's list and thus which of its candidates are elected.

Local list PR
In a local list system, parties divide their candidates in single member-like constituencies, which are ranked inside each general party list depending by their percentages. This method allows electors to judge every single candidate as in a FPTP system.

Two-tier party list systems
Some party list proportional systems with open lists use a two-tier compensatory system, as in Denmark, Norway, and Sweden. In Denmark, for example, the country is divided into ten multiple-member voting districts arranged in three regions, electing 135 representatives. In addition, 40 compensatory seats are elected. Voters have one vote which can be cast for an individual candidate or for a party list on the district ballot. To determine district winners, candidates are apportioned their share of their party's district list vote plus their individual votes. The compensatory seats are apportioned to the regions according to the party votes aggregated nationally, and then to the districts where the compensatory representatives are determined. In the 2007 general election, the district magnitudes, including compensatory representatives, varied between 14 and 28. The basic design of the system has remained unchanged since its introduction in 1920.

Single transferable vote
The single transferable vote (STV), also called choice voting, is a ranked system: voters rank candidates in order of preference. Voting districts usually elect three to seven representatives. The count is cyclic, electing or eliminating candidates and transferring votes until all seats are filled. A candidate is elected whose tally reaches a quota, the minimum vote that guarantees election. The candidate's surplus votes (those in excess of the quota) are transferred to other candidates at a fraction of their value proportionate to the surplus, according to the voters' preferences. If no candidates reach the quota, the candidate with the fewest votes is eliminated, those votes being transferred to their next preference at full value, and the count continues. There are many methods for transferring votes. Some early, manual, methods transferred surplus votes according to a randomly selected sample, or transferred only a "batch" of the surplus, other more recent methods transfer all votes at a fraction of their value (the surplus divided by the candidate's tally) but may need the use of a computer. Some methods may not produce exactly the same result when the count is repeated. There are also different ways of treating transfers to already elected or eliminated candidates, and these, too, can require a computer.

In effect, the method produces groups of voters of equal size that reflect the diversity of the electorate, each group having a representative the group voted for. Some 90% of voters have a representative to whom they gave their first preference. Voters can choose candidates using any criteria they wish, the proportionality is implicit. Political parties are not necessary; all other prominent PR electoral systems presume that parties reflect voters wishes, which many believe gives power to parties. STV satisfies the electoral system criterion proportionality for solid coalitions – a solid coalition for a set of candidates is the group of voters that rank all those candidates above all others – and is therefore considered a system of proportional representation. However, the small district magnitude used in STV elections has been criticized as impairing proportionality, especially when more parties compete than there are seats available, and STV has, for this reason, sometimes been labelled "quasi proportional". While this may be true when considering districts in isolation, results are proportional. In Ireland, with particularly small magnitudes, results are "highly proportional". In the 1997 Irish general election, the average magnitude was 4.0 but eight parties gained representation, four of them with less than 3% of first preference votes nationally. Six independent candidates also won election. STV has also been described as the proportional system. The system tends to handicap extreme candidates because, to gain preferences and so improve their chance of election, candidates need to canvass voters beyond their own circle of supporters, and so need to moderate their views. Conversely, widely respected candidates can win election with relatively few first preferences by benefitting from strong subordinate preference support.

Australian Senate STV
The term STV in Australia refers to the Senate electoral system, a variant of Hare-Clark characterized by the "above the line" group voting ticket, a party list option. It is used in the Australian upper house, the Australian Senate, most state upper houses, the Tasmanian lower house and the Capital Territory assembly. Due to the number of preferences that are compulsory if a vote for candidates (below-the-line) is to be valid – for the Senate a minimum of 90% of candidates must be scored, in 2013 in New South Wales that meant writing 99 preferences on the ballot – 95% and more of voters use the above-the-line option, making the system, in all but name, a party list system. Parties determine the order in which candidates are elected and also control transfers to other lists and this has led to anomalies: preference deals between parties, and "micro parties" which rely entirely on these deals. Additionally, independent candidates are unelectable unless they form, or join, a group above-the-line. Concerning the development of STV in Australia researchers have observed: "... we see real evidence of the extent to which Australian politicians, particularly at national levels, are prone to fiddle with the electoral system".

As a result of a parliamentary commission investigating the 2013 election, from 2016 the system has been considerably reformed (see 2016 Australian federal election), with group voting tickets (GVTs) abolished and voters no longer required to fill all boxes.

Mixed compensatory systems
A mixed compensatory system is an electoral system that is mixed, meaning that it combines a plurality/majority formula with a proportional formula, and that uses the proportional component to compensate for disproportionality caused by the plurality/majority component. For example, suppose that a party wins 10 seats based on plurality, but requires 15 seats in total to obtain its proportional share of an elected body. A fully proportional mixed compensatory system would award this party 5 compensatory (PR) seats, raising the party's seat count from 10 to 15. The most prominent mixed compensatory system is mixed member proportional representation (MMP), used in Germany since 1949. In MMP, the seats won by plurality are associated with single-member districts.

Mixed member proportional representation
Mixed member proportional representation (MMP) is a two-tier system that combines a single-district vote, usually first-past-the-post, with a compensatory regional or nationwide party list proportional vote. The system aims to combine the local district representation of FPTP and the proportionality of a national party list system. MMP has the potential to produce proportional or moderately proportional election outcomes, depending on a number of factors such as the ratio of FPTP seats to PR seats, the existence or nonexistence of extra compensatory seats to make up for overhang seats, and electoral thresholds. It was invented for the German Bundestag after the Second World War and has spread to Lesotho, Bolivia and New Zealand. The system is also used for the Welsh and Scottish assemblies where it is called the additional member system.

Voters typically have two votes, one for their district representative and one for the party list. The list vote usually determines how many seats are allocated to each party in parliament. After the district winners have been determined, sufficient candidates from each party list are elected to "top-up" each party to the overall number of parliamentary seats due to it according to the party's overall list vote. Before apportioning list seats, all list votes for parties which failed to reach the threshold are discarded. If eliminated parties lose seats in this manner, then the seat counts for parties that achieved the threshold improve. Also, any direct seats won by independent candidates are subtracted from the parliamentary total used to apportion list seats.

The system has the potential to produce proportional results, but proportionality can be compromised if the ratio of list to district seats is too low, it may then not be possible to completely compensate district seat disproportionality. Another factor can be how overhang seats are handled, district seats that a party wins in excess of the number due to it under the list vote. To achieve proportionality, other parties require "balance seats", increasing the size of parliament by twice the number of overhang seats, but this is not always done. Until recently, Germany increased the size of parliament by the number of overhang seats but did not use the increased size for apportioning list seats. This was changed for the 2013 national election after the constitutional court rejected the previous law, not compensating for overhang seats had resulted in a negative vote weight effect. Lesotho, Scotland and Wales do not increase the size of parliament at all, and, in 2012, a New Zealand parliamentary commission also proposed abandoning compensation for overhang seats, and so fixing the size of parliament. At the same time, it would abolish the single-seat threshold – any such seats would then be overhang seats and would otherwise have increased the size of parliament further – and reduce the electoral threshold from 5% to 4%. Proportionality would not suffer.

Dual member proportional representation
Dual member proportional representation (DMP) is a single-vote system that elects two representatives in every district. The first seat in each district is awarded to the candidate who wins a plurality of the votes, similar to first-past-the-post voting. The remaining seats are awarded in a compensatory manner to achieve proportionality across a larger region. DMP employs a formula similar to the "best near-winner" variant of MMP used in the German state of Baden-Württemberg. In Baden-Württemberg, compensatory seats are awarded to candidates who receive high levels of support at the district level compared with other candidates of the same party. DMP differs in that at most one candidate per district is permitted to obtain a compensatory seat. If multiple candidates contesting the same district are slated to receive one of their parties' compensatory seats, the candidate with the highest vote share is elected and the others are eliminated. DMP is similar to STV in that all elected representatives, including those who receive compensatory seats, serve their local districts. Invented in 2013 in the Canadian province of Alberta, DMP received attention on Prince Edward Island where it appeared on a 2016 plebiscite as a potential replacement for FPTP, but was eliminated on the third round. It was also one of three proportional voting system options on a 2018 referendum in British Columbia.

Biproportional apportionment
Biproportional apportionment applies a mathematical method (iterative proportional fitting) for the modification of an election result to achieve proportionality. It was proposed for elections by the mathematician Michel Balinski in 1989, and first used by the city of Zurich for its council elections in February 2006, in a modified form called "new Zurich apportionment" (Neue Zürcher Zuteilungsverfahren). Zurich had had to modify its party list PR system after the Swiss Federal Court ruled that its smallest wards, as a result of population changes over many years, unconstitutionally disadvantaged smaller political parties. With biproportional apportionment, the use of open party lists hasn't changed, but the way winning candidates are determined has. The proportion of seats due to each party is calculated according to their overall citywide vote, and then the district winners are adjusted to conform to these proportions. This means that some candidates, who would otherwise have been successful, can be denied seats in favor of initially unsuccessful candidates, in order to improve the relative proportions of their respective parties overall. This peculiarity is accepted by the Zurich electorate because the resulting city council is proportional and all votes, regardless of district magnitude, now have equal weight. The system has since been adopted by other Swiss cities and cantons.

Fair majority voting
Balinski has proposed another variant called fair majority voting (FMV) to replace single-winner plurality/majoritarian electoral systems, in particular the system used for the US House of Representatives. FMV introduces proportionality without changing the method of voting, the number of seats, or the –possibly gerrymandered –district boundaries. Seats would be apportioned to parties in a proportional manner at the state level. In a related proposal for the UK parliament, whose elections are contested by many more parties, the authors note that parameters can be tuned to adopt any degree of proportionality deemed acceptable to the electorate. In order to elect smaller parties, a number of constituencies would be awarded to candidates placed fourth or even fifth in the constituency – unlikely to be acceptable to the electorate, the authors concede – but this effect could be substantially reduced by incorporating a third, regional, apportionment tier, or by specifying minimum thresholds.

Other proportional systems
Generally, these differ from ranked choice voting by voters assigning a score instead of rank to each candidate. Each score is turned into a proportion by dividing by the sum of scores over candidates, for each position and voter (roughly similar, in effect, to each voter getting 100 percent to assign among candidates for each position).

Reweighted range voting
Reweighted range voting (RRV) is a multi-winner voting system similar to STV in that voters can express support for multiple candidates, but different in that candidates are graded instead of ranked. That is, a voter assigns a score to each candidate. The higher a candidate's scores, the greater the chance they will be among the winners.

Similar to STV, the vote counting procedure occurs in rounds. The first round of RRV is identical to range voting. All ballots are added with equal weight, and the candidate with the highest overall score is elected. In all subsequent rounds, ballots that support candidates who have already been elected are added with a reduced weight. Thus voters who support none of the winners in the early rounds are increasingly likely to elect one of their preferred candidates in a later round. The procedure has been shown to yield proportional outcomes if voters are loyal to distinct groups of candidates (e.g. political parties).

RRV was used for the nominations in the Visual Effects category for recent Academy Award Oscars from 2013 through 2017.

Proportional approval voting
Systems can be devised that aim at proportional representation but are based on approval votes on individual candidates (not parties). Such is the idea of Proportional approval voting (PAV). When there are a lot of seats to be filled, as in a legislature, counting ballots under PAV may not be feasible, so sequential variants have been proposed, such as Sequential proportional approval voting (SPAV). This method is similar to reweighted range voting in that several winners are elected using a multi-round counting procedure in which ballots supporting already elected candidates are given reduced weights. Under SPAV, however, a voter can only choose to approve or disapprove of each candidate, as in approval voting. SPAV was used briefly in Sweden during the early 1900s.

Asset voting
In asset voting, the voters vote for candidates and then the candidates negotiate amongst each other and reallocate votes amongst themselves. Asset voting was proposed by Lewis Carroll in 1884 and has been more recently independently rediscovered and extended by Warren D. Smith and Forest Simmons. As such, this method substitutes candidates' collective preferences for those of the voters.

Evaluative Proportional Representation (EPR)
Similar to Majority Judgment voting that elects single winners, Evaluative Proportional Representation (EPR) elects all the members of a legislative body. Both systems remove the qualitative wasting of votes. Each citizen grades the fitness for office of as many of the candidates as they wish as either Excellent (ideal), Very Good, Good, Acceptable, Poor, or Reject (entirely unsuitable). Multiple candidates may be given the same grade by a voter. Using EPR, each citizen elects their representative at-large for a city council. For a large and diverse state legislature, each citizen chooses to vote through any of the districts or official electoral associations in the country. Each voter grades any number of candidates in the whole country. Each elected representative has a different voting power (a different number of weighted votes) in the legislative body. This number is equal to the total number of votes given exclusively to each member from all citizens. Each member's weighted vote results from receiving one of the following from each voter: their highest grade, highest remaining grade, or proxy vote. No citizen's vote is "wasted" Unlike all the other proportional representation systems, each EPR voter, and each self-identifying minority or majority is quantitatively represented with exact proportionality. Also, like Majority Judgment, EPR reduces by almost half both the incentives and possibilities for voters to use Tactical Voting.

Advocacy
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.

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 intimately 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.

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 for an example. Condorcet methods and STAR voting can be made to work with PR methods in this way.

See the combinatorics article for more information.

Attribution

 * Portions copied from https://simple.wikipedia.org/w/index.php?title=Proportional_representation&oldid=7261369