Proportional representation

Party list PR, the most commonly used version of proportional representation, involves parties in the election process. Voters do not primarily vote for candidates (persons), but for electoral lists (or party lists), which are lists of candidates that parties put forward. The mechanism that allocates seats to the parties/lists is how these systems achieve proportionality. Once this is done, the candidates who take the seats are based on the order in which they appear on the list. This is the basic, closed list version of list PR.

An example election where the assembly has 200 seats to be filled is presented below. Every voter votes 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.

Party		Popular vote	Party-list PR - D'Hondt (Jefferson) method
			Number of seats	Seats %
	Party A	43.91%	88	44%
	Party B	39.94%	80	40%
	Party C	9.98%	20	10%
	Party D	6.03%	12	6%
TOTAL		100%	200	100%

This is done by a proportional formula/method, for example the D'Hondt method (also called the Jefferson method). 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 seats ― for example, if the whole country is one district. Party list PR is also more complicated in reality than in the example, as countries often use more than one district, multiple tiers (e.g. local, regional and national), open lists or an electoral threshold.

The single transferable vote is an older method than party-list PR, and it does not need to formally involve parties in the election process. Instead of parties putting forward ordered lists of candidates, it is the voters themselves who rank the candidates. This is done using a preferential ballot. STV can be used for nonpartisan elections such as the city council of Cambridge, Massachusetts.[9]

Counting votes under STV in a realistic election scenario is complicated, but a simplified example can show how proportionality is achieved in a district with 4 seats. In reality, districts need to be larger to truly achieve proportionality, but not so large that there are too many candidates for voters to rank. In this example each candidate belongs to a party and for a candidate to be elected, they need to earn more than 25% of the votes (this is called a quota, more specifically the Droop quota. Note that it is impossible for 4 candidates to all get more than 25% of the votes. In the first step, the first preferences (favourite candidates) of all the voters are counted and all candidates who pass the quota are elected.

Simplified example of an STV ballot

Next, the votes the candidates received above the quota (surplus votes that they did not need to get elected) are transferred to the next preferences of the voters who voted for them. For the example we suppose that all voters of Joe Smith prefer John Citizen as their second choice (as he is also from Party A) and all voters of Fred Rubble prefer Mary Hill as their second choice (as she is also from Party B). Based on this we reallocate the votes and find that John Citizen has passed the quota and so got elected to the 3rd and last seat that we had to fill.

The mixed-member proportional system combines single member plurality voting (SMP), also known as first-past-the-post (FPTP), with party-list PR in a way that the overall result of the election is supposed to be proportional. The voter may vote for a district candidate as well as a party. The main idea behind MMP is compensation, meaning that the list-PR seat allocation is not independent of the results of the district level voting. First-past-the-post is a single winner system and cannot be proportional (winner-takes-all), so these disproportionalities are compensated by the party-list component.

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 districts elections are highly disproportional: large parties typically win more seats than they would under a simple list-PR system, but a party that won more votes than another party might not win more seats.

Party			Popular vote	FPTP seats (Number of districts won)	Compensatory seats under MMP (party-list PR seats)	Total number of seats under MMP	Seats % under MMP
		Party A	43.91%	64	24	88	44%
		Party B	39.94%	33	47	80	40%
		Party C	9.98%	0	20	20	10%
		Party D	6.03%	3	9	12	6%
TOTAL			100%	100	100	200	100%

MMP gives only as many compensatory seats to a party as they need to have the number of seats of each party be proportional. Another way to say this is that MMP focuses on making the final outcome proportional.

Compare this to parallel voting, which is a mixed-member majoritarian system: here the party-list PR seat allocation is independent of the district results, meaning that there is no compensation. The popular vote, the number of districts won by each party, and the number of districts and party-list PR seats are all the same, yet the total number of seats is different.

Party			Popular vote	FPTP seats (Number of districts won)	Party-list PR seats under parallel voting	Total number of seats under parallel voting	Seats % under parallel voting
		Party A	43.91%	64	44	108	54%
		Party B	39.94%	33	40	73	36.5%
		Party C	9.98%	0	10	10	5%
		Party D	6.03%	3	6	9	4.5%
TOTAL			100%	100	100	200	100%

The overall results are not proportional, but semi-proportional. This shows that for a mixed system to be proportional, it has to be compensatory. There are many versions of MMP in use. Some use only a single vote, some allocate compensatory seats to best losers, and some use levelling seats to compensate for potential overhang seats.

PR tries to resolve the unfairness of majoritarian and plurality voting systems where the largest parties receive an "unfair" seat bonus and smaller parties are disadvantaged, always under-represented, and on occasion win no representation at all (Duverger's law).[13][14][15]^: 6–7 An established party in UK elections can win majority control of the House of Commons with as little as 35% of votes (2005 UK general election). In certain Canadian elections, majority governments have been formed by parties with the support of under 40% of votes cast (2011 Canadian election, 2015 Canadian election). If turnout levels in the electorate are less than 60%, such outcomes allow a party to form a majority government by convincing as few as one quarter of the electorate to vote for it. In the 2005 UK election, for example, the Labour Party under Tony Blair won a comfortable parliamentary majority with the votes of only 21.6% of the total electorate.[16]^: 3 Such misrepresentation has been criticized as "no longer a question of 'fairness' but of elementary rights of citizens".[17]^: 22 However, intermediate PR systems with a high electoral threshold, or other features that reduce proportionality, are not necessarily much fairer: in the 2002 Turkish general election, using an open list system with a 10% threshold, 46% of votes were wasted.[2]^: 83

Plurality/majoritarian systems also benefit regional parties that win many seats in the region where they have a strong following but have little support nationally, while other parties with national support that is not concentrated in specific districts, like the Greens, win few or no seats. An example is the Bloc Québécois in Canada that won 52 seats in the 1993 federal election, all in Quebec, on 13.5% of the national vote, while the Progressive Conservatives collapsed to two seats on 16% spread nationally. The Conservative party although strong nationally had had very strong regional support in the West but in this election its supporters in the West turned to the Reform party, which won most of its seats west of Saskatchewan and none east of Manitoba.[15][18] Similarly, in the 2015 UK General Election, the Scottish National Party gained 56 seats, all in Scotland, with a 4.7% share of the national vote while the UK Independence Party, with 12.6%, gained only a single seat.[19]

The use of multiple-member districts enables a greater variety of candidates to be elected. Having more representatives per district and lowering the percentage of votes required for election allows more minor parties to gain representation. It has been argued that in emerging democracies, inclusion of minorities in the legislature can be essential for social stability and to consolidate the democratic process.[2]^: 58

Critics, on the other hand, claim this can give extreme parties a foothold in parliament, sometimes cited as a cause for the collapse of the Weimar government. With very low thresholds, very small parties can act as "king-makers",[20] holding larger parties to ransom during coalition discussions. The example of Israel is often quoted,[2]^: 59 but these problems can be limited, as in the modern German Bundestag, by the introduction of higher threshold limits for a party to gain parliamentary representation (which in turn increases the number of wasted votes).

Another criticism is that the dominant parties in plurality/majoritarian systems, often looked on as "coalitions" or as "broad churches",[21] can fragment under PR as the election of candidates from smaller groups becomes possible. Israel, Brazil, and Italy (until 1993) are examples.[2]^: 59, 89 However, research shows, in general, there is only a small increase in the number of parties in parliament (although small parties have larger representation) under PR.[22]

Open list systems and STV, the only prominent PR system which does not require political parties,[23] enable independent candidates to be elected. In Ireland, on average, about six independent candidates have been elected each parliament.[24] This can lead to a situation where forming a Parliamentary majority requires support of one or more of these independent representatives. In some cases these independents have positions that are closely aligned with the governing party and it hardly matters. The Irish Government formed after the 2016 election even included independent representatives in the cabinet of a minority government. In others, the electoral platform is entirely local and addressing this is a price for support.

The election of smaller parties gives rise to one of the principal objections to PR systems, that they almost always result in coalition governments.[2]^: 59[10]

Supporters of PR see coalitions as an advantage, forcing compromise between parties to form a coalition at the centre of the political spectrum, and so leading to continuity and stability. Opponents counter that with many policies compromise is not possible. Neither can many policies be easily positioned on the left-right spectrum (for example, the environment). So policies are horse-traded during coalition formation, with the consequence that voters have no way of knowing which policies will be pursued by the government they elect; voters have less influence on governments. Also, coalitions do not necessarily form at the centre, and small parties can have excessive influence, supplying a coalition with a majority only on condition that a policy or policies favoured by few voters is/are adopted. Most importantly, the ability of voters to vote a party in disfavour out of power is curtailed.[10]

Countries with PR do not appear to have more elections.

All these disadvantages, the PR opponents contend, are avoided by two-party plurality systems. Coalitions are rare; the two dominant parties necessarily compete at the centre for votes, so that governments are more reliably moderate; the strong opposition necessary for proper scrutiny of government is assured; and governments remain sensitive to public sentiment because they can be, and are, regularly voted out of power.[10] However, this is not necessarily so; a two-party system can result in a "drift to extremes", hollowing out the centre,[25] or, at least, in one party drifting to an extreme.[26] The opponents of PR also contend that coalition governments created under PR are less stable, and elections are more frequent. Italy is an often-cited example with many governments composed of many different coalition partners. However, Italy is unusual in that both its houses can make a government fall, whereas other PR nations have either just one house or have one of their two houses be the core body supporting a government. Italy's current parallel voting system is not PR, so Italy is not an appropriate candidate for measuring the stability of PR.

Plurality systems usually result in single-party-majority government because generally fewer parties are elected in large numbers under FPTP compared to PR, and FPTP compresses politics to little more than two-party contests. Relatively few votes in a few of the most finely balanced districts, the "swing seats", are able to swing majority control in the house. Incumbents in less evenly divided districts are invulnerable to slight swings of political mood. In the UK, for example, about half the constituencies have always elected the same party since 1945;[27] in the 2012 US House elections 45 districts (10% of all districts) were uncontested by one of the two dominant parties.[28] Voters who know their preferred candidate will not win have little incentive to vote, and even if they do their votes have no effect, although they are still counted in the popular vote calculation.[2]^: 10

With PR, there are no "swing seats". Most votes contribute to the election of a candidate, so parties need to campaign in all districts, not just those where their support is strongest or where they perceive most advantage. This fact in turn encourages parties to be more responsive to voters, producing a more "balanced" ticket by nominating more women and minority candidates.[15] On average about 8% more women are elected.[22]

Since most votes count, there are fewer "wasted votes", so voters, aware that their vote can make a difference, are more likely to make the effort to vote, and less likely to vote tactically. Compared to countries with plurality electoral systems, voter turnout improves and the population is more involved in the political process.[2][15][22] However, some experts argue that transitioning from plurality to PR only increases voter turnout in geographical areas associated with safe seats under the plurality system; turnout may decrease in areas formerly associated with swing seats.[29]

To ensure approximately equal representation, plurality systems are dependent on the drawing of boundaries of their single-member districts, a process vulnerable to political interference (gerrymandering). To compound the problem, boundaries have to be periodically re-drawn to accommodate population changes. Even apolitically drawn boundaries can unintentionally produce the effect of gerrymandering, reflecting naturally occurring concentrations.[30]^: 65

PR systems with their multiple-member districts are less prone to this – research suggests five-seat districts or larger are immune to gerrymandering.[30]^: 66

Equality of size of multiple-member districts is not important (the number of seats can vary) so districts can be aligned with historical territories of varying sizes such as cities, counties, states, or provinces. Later population changes can be accommodated by simply adjusting the number of representatives elected. For example, Professor Mollison in his 2010 plan for STV for the UK divided the country into 143 districts and then allocated a different number of seats to each district (to equal the existing total of 650) depending on the number of voters in each but with wide ranges (his five-seat districts include one with 327,000 voters and another with 382,000 voters). His district boundaries follow historical county and local authority boundaries, yet he achieves more uniform representation than does the Boundary Commission, the body responsible for balancing the UK's first-past-the-post constituency sizes.[27][31]

Mixed member systems are susceptible to gerrymandering for the local seats that remain a part of such systems. Under parallel voting, a semi-proportional system, there is no compensation for the effects that such gerrymandering might have. Under MMP, the use of compensatory list seats makes gerrymandering less of an issue. However, its effectiveness in this regard depends upon the features of the system, including the size of the regional districts, the relative share of list seats in the total, and opportunities for collusion that might exist. A striking example of how the compensatory mechanism can be undermined can be seen in the 2014 Hungarian parliamentary election, where the leading party, Fidesz, combined gerrymandering and decoy lists, which resulted in a two-thirds parliamentary majority from a 45% vote.[32][33] This illustrates how certain implementations of mixed systems (if non-compensatory or insufficiently compensatory) can produce moderately proportional outcomes, similar to parallel voting.

It is generally accepted that a particular advantage of plurality electoral systems such as first past the post, or majoritarian electoral systems such as the alternative vote, is the geographic link between representatives and their constituents.[2]^: 36[34]^: 65[17]^: 21 A notable disadvantage of PR is that, as its multiple-member districts are made larger, this link is weakened.[2]^: 82 In party list PR systems without delineated districts, such as the Netherlands and Israel, the geographic link between representatives and their constituents is considered weak, but has shown to play a role for some parties. Yet with relatively small multiple-member districts, in particular with STV, there are counter-arguments: about 90% of voters can consult a representative they voted for, someone whom they might think more sympathetic to their problem. In such cases it is sometimes argued that constituents and representatives have a closer link;[27][30]^: 212 constituents have a choice of representative so they can consult one with particular expertise in the topic at issue.[30]^: 212[35] With multiple-member districts, prominent candidates have more opportunity to be elected in their home constituencies, which they know and can represent authentically. There is less likely to be a strong incentive to parachute them into constituencies in which they are strangers and thus less than ideal representatives.[36]^{: 248–250} Mixed-member PR systems incorporate single-member districts to preserve the link between constituents and representatives.[2]^: 95 However, because up to half the parliamentary seats are list rather than district seats, the districts are necessarily up to twice as large as with a plurality/majoritarian system where all representatives serve single-member districts.[17]^: 32

An interesting case occurred in the Netherlands, when "out of the blue" a party for the elderly, the General Elderly Alliance gained six seats in the 1994 election. The other parties had not paid attention, but this made them aware. With the next election, the Party of the Elderly was gone, because the established parties had started to listen to the elderly. Today, a party for older folks, 50PLUS, has established itself in the Netherlands, albeit never with the same high number of seats. This can be seen as an example how geography in itself may not be a good enough reason to establish voting results around it and overturn all other particulars of the voting population. In a sense, voting in districts restricts the voters to a specific geography. Proportional voting follows the exact outcome of all the votes.[37]

In a presidential system, the president is chosen independently from the parliament. As a consequence, it is possible to have a divided government where a parliament and president have opposing views and may want to balance each other's influence. However, the proportional system favors government of coalitions of many smaller parties that require compromising and negotiating topics. As a consequence, these coalitions might have difficulties presenting a united front to counter presidential influence, leading to a lack of balance between these two powers. With a proportionally elected House, a President may strong-arm certain political issues

This issue does not happen in a parliamentary system, where the prime-minister is elected indirectly by the parliament itself. As a consequence, a divided government is impossible. Even if the political views change with time and the prime minister loses their support from parliament, they can be replaced with a motion of no confidence. Effectively, both measures make it impossible to create a divided government.

Academics agree that the most important influence on proportionality is an electoral district's magnitude, the number of representatives elected from the district. As magnitude increases, proportionality improves.[2] Some scholars recommend for STV voting districts of roughly four to eight seats,[38] which are considered small relative to PR systems in general, which frequently have district magnitudes in the hundreds.

At one extreme, the binomial electoral system used in Chile between 1989 and 2013,[39] a nominally proportional open-list system, features two-member districts. As this system can be expected to result in the election of one candidate from each of the two dominant political blocks in most districts, it is not generally considered proportional.[2]^: 79

At the other extreme, where the district encompasses the entire country (and with a low minimum threshold, highly proportionate representation of political parties can result), parties gain by broadening their appeal by nominating more minority and women candidates.[2]^: 83

After the introduction of STV in Ireland in 1921, district magnitudes slowly diminished as more and more three-member constituencies were defined, benefiting the dominant Fianna Fáil, until 1979 when an independent boundary commission was established reversing the trend.[40] In 2010, a parliamentary constitutional committee recommended a minimum magnitude of four.[41] Despite relatively low magnitudes, Ireland has generally experienced highly proportional results.[2]^: 73

In the FairVote plan for STV (which FairVote calls choice voting) for the US House of Representatives, three- to five-member super-districts are proposed.[42]

In Professor Mollison's plan for STV in the UK, four- and five-member districts are mostly used, with three and six seat districts used as necessary to fit existing boundaries, and even two and single member districts used where geography dictates.[27]

The electoral threshold is the minimum vote required to win one seat. The lower the threshold, the higher the proportion of votes contributing to the election of representatives and the lower the proportion of votes wasted.[2]

All electoral systems have electoral thresholds, either formally defined or the natural threshold, which is the mathematical consequence of the district magnitude and election parameters.[2]^: 83

A formal threshold usually requires parties to win a certain percentage of the vote in order to be awarded seats from the party lists. In Germany and New Zealand (both MMP), the threshold is 5% of the national vote but the threshold is not applied to parties that win a minimum number of constituency seats (three in Germany, one in New Zealand). Turkey defines a threshold of 10%, the Netherlands 0.67%.[2] Israel has raised its threshold from 1% (before 1992) to 1.5% (up to 2004), 2% (in 2006) and 3.25% in 2014.[43] South Africa has no explicit electoral threshold, only a natural threshold ~0.2%. A list of electoral thresholds by country shows a typical electoral threshold for party-list PR is 3-5%.

In STV elections, winning the quota (ballots/(seats+1)) of first preference votes assures election. However, well regarded candidates who attract good second (and third, etc.) preference support can hope to win election with only half the quota of first preference votes. Thus, in a six-seat district the effective threshold would be 7.14% of first preference votes (100/(6+1)/2).[27] For a district magnitude of 3, the STV electoral threshold is 12.5 %, significantly higher than typical party-list PR. The need to attract second preferences tends to promote consensus and disadvantage extremes. The electoral threshold has different effects on STV than on Party-list PR. For STV the votes for candidates below natural threshold are not wasted, but transferred to the next-indicated choice. For party-list PR a vote for a party below electoral threshold is an unrepresented vote, unless the spare vote system is applied.

Party magnitude is the number of candidates elected from one party in one district. As party magnitude increases a more balanced ticket will be more successful encouraging parties to nominate women and minority candidates for election.[44]

But under STV, nominating too many candidates can be counter-productive, splitting the first-preference votes and allowing the candidates to be eliminated before receiving transferred votes from other parties. An example of this was identified in a ward in the 2007 Scottish local elections where Labour, putting up three candidates, won only one seat while they might have won two had one of their voters' preferred candidates not stood.[27] The same effect may have contributed to the collapse of Fianna Fáil in the 2011 Irish general election.[45]

Other aspects of PR can influence proportionality such as the size of the elected body, the choice of open or closed lists, ballot design, and vote counting methods.

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.[54] Local lists were used to elect the Italian Senate during the second half of the 20th century. 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.[55][56] A party is allocated seats in proportion to the number of votes it receives.[57]
• Ley de Lemas - an intermediate system used in Uruguay, 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 systems, where 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. Nevertheless, the number of candidates elected from the list is determined by the number of votes the list receives.
• 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 FPTP system.
• Two-tier party list systems - 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.[58][59][60]

There are mixed electoral systems combining a plurality/majority formula with a proportional formula[61] or using the proportional component to compensate for disproportionality caused by the plurality/majority component.[62][63]

The most prominent mixed compensatory system is mixed member proportional representation (MMP). It combines a single-district vote, usually first-past-the-post, with a compensatory regional or nationwide party list proportional vote. 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. 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.[64][65][66] 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 Assembly and Scottish Parliament where it is called the additional member system.[67][4]

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. Any direct seats won by independent candidates are subtracted from the parliamentary total used to apportion list seats.[68]

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 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.[69] 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.[2][70]

Similarly to MMP, mixed single vote systems (MSV) use a proportional formula for allocating seats on the compensatory tier, but voters only have one vote that functions on both levels. MSV may use a positive vote transfer system, where unused votes are transferred from the lower tier to the upper, 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 (MMM) 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).

Another mixed system is dual-member proportional representation (DMP). It is a single-vote system that elects two representatives in every district.[71] 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.[72] 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,[73] but was eliminated on the third round.[74][75] It was also one of three proportional voting system options on a 2018 referendum in British Columbia.[76][77][78]

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 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 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 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.[79][80]

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.[80] 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.[81]

The single transferable vote (STV), also called ranked choice voting,[82][9] 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, while 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.[83][84]

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.[27] Political parties are not necessary; all other prominent PR electoral systems presume that parties reflect voters wishes, which many believe gives power to parties.[83] 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.[83] 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,[10]^: 50 and STV has, for this reason, sometimes been labelled "quasi proportional".[85]^: 83 While this may be true when considering districts in isolation, results overall are proportional. In Ireland, with particularly small magnitudes, results are "highly proportional".[2]^: 73[5] In 1997, 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.[40] STV has also been described as the most proportional system as it elects candidates without the need for parties which can distort proportionality.[85]^: 83 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.[86][87] Conversely, widely respected candidates can win election with relatively few first preferences by benefitting from strong subordinate preference support.[27]

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).[88] 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 used, such as sequential proportional approval voting (SPAV).

Sequential proportional approval voting (SPAV) 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.[89] That is, a voter approves or disapproves of each candidate. SPAV was used briefly in Sweden during the early 1900s.[90]

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).[91][92]

Reweighted Range Voting (RRV) uses the same method as sequential proportional approval voting but uses a score ballot.[93][94] Reweighted Range Voting was used for the nominations in the Visual Effects category for recent Academy Award Oscars from 2013 through 2017,[95][96] and is used in the city of Berkeley, California, for sorting the priorities of the city council.[97]

In asset voting,[89][98] 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[99] and has been more recently independently rediscovered and extended by Warren D. Smith and Forest Simmons.[100] As such, this method substitutes candidates' collective preferences for those of the voters.

Similar to Majority Judgment voting that elects single winners, Evaluative Proportional Representation (EPR) elects all the members of a legislative body. Both systems minimize the quantitative and qualitative wasting of votes.[101] 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 non-geographically defined 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"[102] because each citizen's vote equally adds to the voting power in the legislative body of the elected candidate they see as likely to represent their hopes and concerns most accurately. Unlike all the other proportional representation systems, each EPR voter, and each self-identifying minority or majority, is proportionally represented exactly. Also, like Majority Judgment, EPR reduces by almost half both the incentives and possibilities for voters to use Tactical Voting.