There are many different electoral systems that have been used or proposed to achieve proportional representation. Most can be classified as party-list PR, the single transferable vote, or mixed-member PR.
Party-list PR methods Party-list PR is the most commonly used version of proportional representation. Each voter casts a vote for a single party and each party is allocated seats based on its share of the vote. The seats are assigned to party-affiliated candidates on the parties'
electoral lists. The mechanism that allocates seats to the parties or lists is how these systems achieve proportionality. A variety of list PR systems are in use today. Just a few party-list PR systems use overall country-wide vote counts. The Netherlands and Israel are the two main examples. Others count vote shares in separate districts and allocate seats in each part according to the party's vote count in the district. Denmark and some others use both, as a form of mixed member proportional. Some common types of electoral lists are: •
Closed list systems, where 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. •
Ley de lemas, an intermediate system formerly used in Uruguay, where each party (Lema) presents several closed lists (sublemas), each representing a faction or specific platform. Seats are allocated to parties according to the parties' shares of votes, then to each sublema proportionally, by the order of the names on the list. •
Open list systems, where voters may vote, depending on the model, for one person, or for two or more, or vote for a party list but indicate their order of preference within the list. The relative popularity of individual candidates are used to allocate the seats, apart from the list. Votes determine which of the party's candidates are elected. Nevertheless, the number of candidates elected from each list is determined by the number of votes that the list receives or that the candidates on the list receive overall. •
Localized list systems, where 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
first-past-the-post (FPTP) system. • Two-tier party list systems, as in Denmark, Norway, and Sweden. These operate similarly to mixed-member proportional systems or additional member systems. For example,
Denmark 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. It is cast for an individual candidate or for a party list on the district ballot. To determine district winners, parties are allocated district seats based on their district vote shares. Candidates in the district are apportioned their share of their party's district list vote plus their individual votes, and the most popular are elected to fill their party's seats. Compensatory seats are apportioned to regions according to the party vote share 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. • The simplest list PR system is the schedule plan where multiple members are elected in a contest, and each voter casts a single vote, which is counted both for a specific candidate and also for a party list. The seats are filled with the most popular candidates of the most popular parties. If quota is used, surplus votes of early winners can be put in the party pool of votes for determination of later seats. List PR systems are used in Finland (open list), Latvia (open list), Sweden (open list), Israel (national closed list), Brazil (open list), Kazakhstan (closed 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.
Example An example election where the assembly has 200 seats to be filled is presented below. Every voter casts their vote for the list created by their favourite party and the results of the election are as follows (popular vote). Under party-list PR, every party gets a number of seats proportional to their share of the popular vote. This is done by a proportional formula or method: for example, the
Sainte-Laguë methodthese are the same methods that may be used to allocate seats for geographic proportional representation (for example, how many seats each state gets in the US House of Representatives). Votes and seats often cannot be mathematically perfectly allocated, so some amount of rounding has to be done. The various methods deal with this in different ways, although the difference is reduced if there are many seatsfor example, if the whole country is one district. In practice, party-list PR is also more complicated than in the example, as list PR used by countries often use more than one district, two or three tiers (e.g. local, regional and national),
open lists and
electoral thresholds. Final seat allocations are frequently not proportional to the parties' vote share.
Single transferable vote (STV) The single transferable vote is an older method than party-list PR, and it does not need to involve parties. Instead of the process used in list PR, where parties put forward ordered lists of candidates from which winners are drawn in some order, under STV voters vote directly for candidates, who run by name. Instead of each voter only marking their first preference, as in FPTP and list PR, under STV a voter has opportunity to rank two or more candidates by preference, with only one marked preference used to place the vote. Votes cast for candidates determine the winners by relative popularity either by achieving a quota or by relative plurality at the end of the vote count process. STV uses
preferential ballots. The ranking is used to instruct election officials as to how the vote should be transferred in case the first preference is marked for an unelectable candidate or for an elected candidate who has an excess of votes needed to guarantee election. Each voter casts one vote. The district used elects multiple members (more than one, often 3 to 7, with 37 being the current maximum use in a government election in the world). Because parties play no role in the vote count, STV may be used for nonpartisan elections, as with the city council of
Cambridge, Massachusetts. A large proportion of the votes cast are used to actually elect someone, so the result is mixed and balanced, with no one voting block taking much more than its due share of the seats. Where party labels are indicated, proportionality party-wise is noticeable.
Counting votes under STV is more complicated than under
first-past-the-post voting, but the example belows shows how the vote count is performed and how proportionality is achieved in a district with 3 seats. In reality, districts usually elect more members than that in order to achieve more proportional results. A risk is that if the number of seats is larger than, for example, 10 seats, the ballot will be so large as to be inconvenient and voters may find it difficult to rank the many candidates, although 21 are elected through STV in some elections. In many STV systems, voters are not required to mark more choices than desired. Even if all voters marked only one preference, the resulting representation would be more balanced than under single-winner FPTP, due to each voter having just one vote and districts electing multiple members under STV. Under STV, the
quota, the share of the vote that guarantees election, is determined beforehand. The
Droop quota is commonly used. In a three-seat district, any candidate who earns more than 25 percent of the vote is declared elected. Note that it is only possible for three candidates to each achieve that quota. In Cambridge, under STV, 90 percent of voters see their vote help to elect a candidate, more than 65 percent of voters see their first-choice candidate elected, and more than 95 percent of voters see one of their top three choices win. Other reports claim that 90 percent 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. STV does not require political parties; party-list PR and MMP systems both presume that parties reflect voters wishes, which Nicolaus Tideman argues gives too much power to party officials. STV satisfies the
electoral system criterion proportionality for solid coalitionsa solid coalition for a set of candidates is the group of voters that rank all those candidates above all othersand is therefore considered a system of proportional representation. Even though Ireland has particularly small magnitudes (3 to 5 seats), results of STV elections are "highly proportional". This argument is made from the high natural threshold STV provides with low district magnitude. Conversely, widely respected candidates can win election even if they receive relatively few first preferences. They do this by benefiting from strong subordinate preference support. Of course, they must have enough initial support so that they are not in the bottom rung of popularity or they will be eliminated when the field of candidate is thinned. He believed STV to be fundamentally flawed, particularly regarding the allocation of "surplus" votes. His novel solution was to let the candidates themselves caucus and "club" votes together through the process of a negotiated consensus. As he stated: May I, in conclusion, point out that the method advocated in my pamphlet (where each elector names one candidate only, and the candidates themselves can, after the numbers are announced, club their votes, so as to bring in others besides those already announced as returned) would be at once perfectly simple and perfectly equitable in its result? However, his entreaties to
Lord Salisbury, leader of the United Kingdom's Conservative Party and future prime minister, to adopt "clubbing" were rejected in 1884 as "too sweeping a change". Subsequently, he joined with
Thomas Hare and several Conservative and Liberal members of Parliament to found the Proportional Representation Society (later the
Electoral Reform Society) and to pursue STV.
Example In the below example, five candidates from two parties run in a three-seat district. In the first count, the first preference (favourite candidate) marked on each of the ballots is counted. Candidates whose vote tally equals or exceeds the quota are declared elected as shown in the example below. The table below shows the initial count, or first round or stage, of the vote count process. Quota is 25 percent plus 1 (Droop quota). Jane Doe and Fred Rubble are elected in the first round. Next, surplus votes belonging to those already elected, votes the candidates received above the quota (votes that they did not need to be elected), are transferred to the next preference marked by the voters who voted for them. Continuing the example, suppose that all voters who marked first preference for Jane Doe marked John Citizen as their second choice. Based on this, Jane Doe's surplus votes are transferred to John Citizen. John Citizen achieves the quota and so is declared elected to the third and last seat that had to be filled. Even if all of Fred Rubble's surplus had gone to Joe Smith, the vote transfer plus Smith's original votes would not add up to quota. Party B did not have two quotas of votes so was not due two seats, while Party Awith 67 percent of the votewas. It is possible, in realistic STV elections, for a candidate to win without quota if they are still in the running when the field of candidates has thinned to the number of remaining open seats. In this example, the district result is balanced party-wise. No one party took all the seats, as frequently happens under FPTP or other non-proportional voting systems. The result is fairthe most popular party took two seats; the less popular party took just one. As well, the most popular candidates in each party won the party's seats. 81 percent of the voters saw their first choice elected. At least 15 percent of them (the Doe first, Citizen second voters) saw both their first and second choices electedthey were likely more than 15 percent if some "Citizen first" votes gave their second preference to Doe. Every voter had the satisfaction of seeing someone of the party they support elected in the district. Quota is 25 percent plus 1 Under STV, to make up a 200-seat legislature as large as in the examples that follow, about 67 three-seat districts would be used. Districts with more seats would provide more proportional resultsone form of STV in Australia uses a district with 21 members being elected at once. With a larger district magnitude, it is more likely that more than two parties will have some of their candidates elected. With a lower district magnitude, it is more likely that only two parties will have their candidates elected. For example, in
Malta, where STV is used with 5-member districts, it is common for successful candidates to receive 16.6 percent of the vote in the district. This produces a high effective threshold in the districts, and the country maintains a very strong two-party system. However, about 4000 voters in a district would be enough to elect a third-party candidate if voters desired, but this seldom happens. In this election, about 1/22nd of the vote in the state is enough to take a seat, and seven or eight parties take at least that many votes, demonstrating a different voting pattern than Malta exhibits.
Mixed-member proportional representation (MMP) Some election systems combine district results with more general proportionality through the use of levelling seats. The most prominent mixed compensatory system is mixed-member proportional representation (MMP). MMP combines election of individual district members with election of some members due to their party's vote share. In rare cases (such as New Zealand, Lesotho and a few other places), MMP systems use single-member districts to elect district members. These
mixed electoral systems combine a plurality/majority formula with a proportional formula or use the proportional component to compensate for disproportionality caused by the plurality/majority component. More commonly (in Denmark, Iceland, Sweden, and South Africa), multi-member districts are used in the countries' MMP electoral systems. After district results are known, additional compensatory members are then elected from party lists to achieve an overall party share proportional to the vote (according to an allocation method much like in party-list PR). Voters sometimes have two votes, one for the election of their district representative and one fused to allocate party list top-up seats. In some system such as Denmark's, each voter casts just one vote. The main idea behind MMP is that the levelling seats act as
compensation. The list-PR seat allocation is dependent on votes cast (in some systems, on the party list votes cast, separately from the district votes) and on the results of the district-level election contests. Single-member districts cannot be proportional as they are inherently winner-takes-all, so the disproportionalities are compensated by the party-list top-up seats as much as possible. MMP has the potential to produce proportional or moderately proportional election outcomes, depending on several 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, the use or not of fair voting in multi-member districts, and electoral thresholds. MMP was invented for the German
Bundestag after the Second World War, and its use has spread to Lesotho, Bolivia, New Zealand and Thailand. The system is also used for the
Scottish Parliament, where it is called the
additional member system. The proportionality of MMP can be compromised if the ratio of list to district seats is too low, as 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 (the additional seats were awarded to under-represented parties) 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, ruling that not compensating for the added overhang compensation seats had resulted in a
negative vote weight effect. Lesotho, Scotland and Wales do not increase the size of parliament at all, even if there are overhang seats. In 2012, a New Zealand parliamentary commission proposed abandoning compensation for overhang seats, and so fixing the size of parliament. At the same time, it proposed abolishing the single-seat threshold (the go-around past the electoral threshold used by some small parties to get their due share of seats). It was expected that such seats would be overhang seats. If that had been done without abolishing overhang compensation, it would have increased the size of parliament further through the overhang compensation. The commission also proposed reducing the electoral threshold from 5 percent to 4 percent. It was expected that proportionality would not suffer from these changes. A simple, yet common, version of MMP has as many list-PR seats as there are single-member districts. In the example it can be seen, as is often the case in reality, that the results of the district elections are highly disproportional: large parties typically win more seats than they should proportionally, but there is also randomnessa party that receives more votes than another party might not win more seats than the other. Any such disproportionality produced by the district elections is addressed, where possible, by the allocation of the compensatory additional members. A variant of MMP is
mixed single vote (MSV), in which voters only have one vote that functions for both district members and compensatory members. MSV may use a positive vote transfer system, where unused votes are transferred from the lower tier to the compensatory tier, where only these are used in the proportional formula. Alternatively, the MMP (seat linkage) algorithm can be used with a mixed single vote to "top-up" to a proportional result. With MSV, the similar requirements as in MMP apply to guarantee an overall proportional result.
Parallel voting systems use proportional formulas to allocate seats on a proportional tier separately from other tiers. Certain systems, like
scorporo, use a proportional formula after combining results of a parallel list vote with transferred votes from lower tiers (using negative or positive vote transfer).
Differences from mixed-member majoritarian system Compare the MMP example to a
mixed-member majoritarian system, where the party-list PR seat allocation is independent of the district results (this is also called parallel voting). Under a mixed-member majoritarian system, there is no compensation (no regard to how the district seats were filled) when allocating party-list seats so as to produce a proportional allocation of seats overall. The popular vote, the number of district seats won by each party, and the number of district and party-list PR seats are the same as in the MMP example above, yet the parties' seat tallies are different. Parallel voting (using non-compensatory party seats): The overall results are not proportional, although they are more balanced and fair than most single-winner first-past-the-post elections. Parallel voting is mostly
semi-proportional.
Mixed systems are the most proportional if the additional members are allocated in a compensatory way.
Dual-member mixed proportional (DMP) Another MMP type system is
dual-member mixed proportional (DMP). It 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
FPTP 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 aims to achieve proportionality in two dimensions, for example: proportionality by region and proportionality by party. There are several mathematical methods to attain biproportionality. One method is called
iterative proportional fitting (IPF). It was proposed for elections by the mathematician
Michel Balinski in 1989, and first used by the city of
Zürich for its council elections in February 2006, in a modified form called "new Zürich apportionment" (
Neue Zürcher Zuteilungsverfahren). Zürich 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 has not 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 favour of initially unsuccessful candidates, in order to improve the relative proportions of their respective parties overall. This peculiarity is accepted by the Zürich 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. Balinski has proposed another variant called
fair majority voting (FMV) to replace single-winner plurality or 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 thepossibly gerrymandereddistrict boundaries. Seats would be apportioned to parties in a proportional manner at the
state level.
Proportional approval voting Proportional approval voting (PAV) is like STV in that voters vote for candidates and not for parties. Rather than ranking candidates, each voter casts
approval votes for any number of candidates. It satisfies an adaptation of PR called
extended justified representation (EJR). When there are many seats to be filled, as in a legislature, counting ballots under PAV may not be feasible, so sequential variants have been used, such as
sequential proportional approval voting (SPAV). SPAV was used briefly in Sweden during the early 1900s. The vote counting procedure occurs in rounds. The first round of SPAV is identical to
approval 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 especially when voters are loyal to distinct groups of candidates (e.g. political parties). Reweighted range voting (RRV) uses the same method as
sequential proportional approval voting but uses a
score ballot. Reweighted range voting was used for the nominations in the Visual Effects category for recent Academy Award Oscars from 2013 through 2017, and is used in the city of
Berkeley, California, for sorting the priorities of the city council. == Districting under proportional representation ==