Editing Proportional representation (section)

=== Party-based systems ===

==== Party list PR ====
{{Main|Party-list proportional representation}}

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 list]]s. This system is used in many countries, including [[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 of the European Union|member states]] use open lists; but most large EU countries use closed lists, so that the majority of EP seats are distributed by those.<ref>As counted from the table in {{cite web |url=http://www.wahlrecht.de/ausland/europa.htm |title=Wahlsysteme in den EU-Mitgliedstaaten |language=de |website=Europawahlrecht |access-date=May 5, 2025}}</ref> 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.<ref>{{cite web |title=Party List PR |url=http://www.electoral-reform.org.uk/party-list |publisher=Electoral Reform Society |access-date=23 May 2016}}</ref><ref>{{cite book |title=Fixing Canadian Democracy |first=Gordon |last=Gibson |url=https://books.google.com/books?id=y7hFz5GawkcC&q=closed+list+PR&pg=PA58 |publisher=The Fraser Institute |page=76 |date=2003 |isbn=9780889752016}}</ref> A party is allocated seats in proportion to the number of votes it receives.<ref>{{cite book |last1=Gallagher |first1=Michael |last2=Mitchell |first2=Paul |date=2005 |title=The Politics of Electoral Systems |location=Oxford, New York |publisher=Oxford University Press |page=11 |isbn=978-0-19-925756-0}}</ref>
* [[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.{{citation needed|date=May 2015}}
* [[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. 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 each list is determined by the number of votes that the list receives or that the candidates on the list receive.<ref name="auto3">{{cite journal |first=Michal |last=Smrek |title=Mavericks or Loyalists? Popular Ballot Jumpers and Party Discipline in the Flexible-List PR Context |journal=[[Political Research Quarterly]]  |date=2023 |volume=76 |issue=1 |pages=323–336 |doi=10.1177/10659129221087961 }}</ref>
* [[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. These operate similarly to [[Mixed-member proportional representation|mixed-member proportional]] systems or additional member systems. For example, [[Folketing#Constitutional requirements|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 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.<ref name="denmark">{{cite web |title=The Parliamentary Electoral System in Denmark |url=http://www.thedanishparliament.dk/Publications/The%20Parliamentary%20Electoral%20System%20in%20DK.aspx |publisher=Ministry of the Interior and Health |location=Copenhagen |access-date=1 September 2014 |date=2011 |archive-date=23 November 2017 |archive-url=https://web.archive.org/web/20171123004726/http://www.thedanishparliament.dk/Publications/The%20Parliamentary%20Electoral%20System%20in%20DK.aspx}}</ref><ref>{{cite web |title=The main features of the Norwegian electoral system |url=http://www.regjeringen.no/en/dep/kmd/information-campaigns/election_portal/the-electoral-system/the-norwegian-electoral-system.html?id=456636 |publisher=Ministry of Local Government and Modernisation |location=Oslo |access-date=1 September 2014 |date=6 July 2017}}</ref><ref>{{cite web |title=The Swedish electoral system |url=http://www.val.se/sprak/engelska/general_information/index.html |publisher=Election Authority |location=Stockholm |access-date=1 September 2014 |date=2011 |archive-url=https://web.archive.org/web/20140818233057/http://www.val.se/sprak/engelska/general_information/index.html |archive-date=18 August 2014}}</ref>

==== Mixed systems ====
There are [[mixed electoral system]]s combining a plurality/majority formula with a proportional formula<ref name="ACEMixSys">{{cite web |last1=ACE Project Electoral Knowledge Network |title=Mixed Systems |url=https://aceproject.org/ace-en/topics/es/esd/esd03/default |access-date=29 June 2016}}</ref> or using the proportional component to compensate for disproportionality caused by the plurality/majority component.<ref name="MassicotteSearch">{{cite report |first1=Louis |last1=Massicotte |title=In Search of Compensatory Mixed Electoral System for Québec |url=https://www.institutions-democratiques.gouv.qc.ca/publications/mode_scrutin_rapport_en.pdf |date=2004 |access-date=29 December 2023 |archive-date=24 December 2019 |archive-url=https://web.archive.org/web/20191224070656/https://www.institutions-democratiques.gouv.qc.ca/publications/mode_scrutin_rapport_en.pdf}}</ref><ref name="BochslerTerritory">{{cite book |first1=Daniel |last1=Bochsler |title=Territory and Electoral Rules in Post-Communist Democracies |chapter=Chapter 5, How Party Systems Develop in Mixed Electoral Systems |chapter-url=https://books.google.com/books?id=qkCBDAAAQBAJ |date=13 May 2010 |publisher=Palgrave Macmillan |isbn=9780230281424}}</ref>

The most prominent mixed compensatory system is [[mixed-member proportional representation]] (MMP). It combines a single-district vote, usually [[First-past-the-post voting|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 voting|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 seat]]s, and electoral thresholds.<ref name="IfesElecSys">{{cite web |title=Electoral Systems and the Delimitation of Constituencies |url=http://www.ifes.org/publications/electoral-systems-and-delimitation-constituencies |date=2 July 2009 |publisher=[[International Foundation for Electoral Systems]]}}</ref><ref name="MoserMixedElec">{{cite journal |title=Mixed electoral systems and electoral system effects: controlled comparison and cross-national analysis |date=December 2004 |volume=23 |issue=4 |pages=575–599 |last1=Moser |first1=Robert G. |doi=10.1016/S0261-3794(03)00056-8 |journal=Electoral Studies}}</ref><ref name="MassicotteMixedElecSys">{{cite journal |title=Mixed electoral systems: a conceptual and empirical survey |date=September 1999 |volume=18 |issue=3 |pages=341–366 |last1=Massicotte |first1=Louis |doi=10.1016/S0261-3794(98)00063-8 |journal=Electoral Studies}}</ref> It was invented for the German [[Bundestag]] after the Second World War, and 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]].<ref name="ersAMS">{{cite web |title=Additional Member System |url=http://www.electoral-reform.org.uk/additional-member-system |access-date=16 October 2015 |publisher=[[Electoral Reform Society]] |location=London}}</ref>

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.<ref>{{cite web |title=MMP Voting System |url=http://www.elections.org.nz/voting-system/mmp-voting-system |publisher=[[Electoral Commission (New Zealand)|Electoral Commission New Zealand]] |location=Wellington |access-date=10 August 2014 |date=2011}}</ref>

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 seat]]s 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.<ref>{{cite news |title=Deutschland hat ein neues Wahlrecht |url=http://www.zeit.de/politik/deutschland/2013-02/bundestag-wahlrecht-beschluss |publisher=[[Zeit Online]] |language=de |date=22 February 2013}}</ref> 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 was 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.<ref name="ideaEsd" /><ref name="NZ2012EC">{{cite web |title=Report of the Electoral Commission on the Review of the MMP Voting System |url=http://www.elections.org.nz/events/past-events-0/2012-mmp-review/results-mmp-review |publisher=[[Electoral Commission (New Zealand)|Electoral Commission New Zealand]] |location=Wellington |access-date=10 August 2014 |date=2011 |archive-url=https://web.archive.org/web/20140707151142/http://www.elections.org.nz/events/past-events-0/2012-mmp-review/results-mmp-review |archive-date=7 July 2014}}</ref>

Similarly to MMP, [[Mixed single vote|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.<ref>{{Cite thesis |last=Graham |first=Sean |date=2016 |title=Dual-Member Mixed Proportional: A New Electoral System for Canada |publisher=University of Alberta |url=https://era.library.ualberta.ca/items/e3ab993a-d264-4d30-b819-290ab5fd6b62 |access-date=10 August 2022 |doi=10.7939/R3-QPPP-B676}}</ref> 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|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 [[mixed-member proportional representation|MMP]] used in the German state of [[Baden-Württemberg]].<ref>{{cite news |first=Antony |last=Hodgson |title=Why a referendum on electoral reform would be undemocratic |url=https://thetyee.ca/Opinion/2016/01/21/Why-Referendum-on-Electoral-Reform-Undemocratic/ |work=[[The Tyee]] |date=21 January 2016}}</ref> 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 [[Single transferable vote|STV]] in that all elected representatives, including those who receive compensatory seats, serve their local districts. Invented in 2013 in the [[Provinces and Territories of Canada|Canadian province]] of [[Alberta]], DMP received attention on [[Prince Edward Island]] where it appeared on a [[2016 Prince Edward Island electoral reform referendum|2016 plebiscite]] as a potential replacement for FPTP,<ref>{{cite news |first=Kerry |last=Campbell |title=P.E.I. electoral reform committee proposes ranked ballot |url=http://www.cbc.ca/news/canada/prince-edward-island/pei-electoral-reform-1.3537355 |work=[[CBC News]] |date=15 April 2016}}</ref> but was eliminated on the third round.<ref>{{cite web |author=Elections PEI |title=Plebiscite Results |url=http://www.electionspei.ca/plebisciteresults |date=7 November 2016 |access-date=26 October 2017 |archive-url=https://web.archive.org/web/20161108134958/http://www.electionspei.ca/plebisciteresults |archive-date=8 November 2016}}</ref><ref>{{cite news |first=Susan |last=Bradley |title=P.E.I. plebiscite favours mixed member proportional representation |url=http://www.cbc.ca/news/canada/prince-edward-island/mixed-member-proportional-representation-in-plebiscite-1.3840172 |work=[[CBC News]] |date=7 November 2016}}</ref> It was also one of three proportional voting system options on a [[2018 British Columbia electoral reform referendum|2018 referendum]] in [[British Columbia]].<ref>{{cite web |first=David |last=Eby |title=How We Vote: 2018 Electoral Reform Referendum Report and Recommendations of the Attorney General |url=https://engage.gov.bc.ca/app/uploads/sites/271/2018/05/How-We-Vote-2018-Electoral-Reform-Referendum-Report-and-Recommendations-of-the-Attorney-General.pdf |date=30 May 2018 |access-date=9 June 2018 |archive-url=https://web.archive.org/web/20180831060310/https://engage.gov.bc.ca/app/uploads/sites/271/2018/05/How-We-Vote-2018-Electoral-Reform-Referendum-Report-and-Recommendations-of-the-Attorney-General.pdf |archive-date=31 August 2018}}</ref><ref>{{cite news |first=Justin |last=McElroy |title=Know your voting systems: three types of electoral reform on B.C.'s ballot |url=http://www.cbc.ca/news/canada/british-columbia/pr-electoral-reform-questions-1.4688604?cmp=rss |work=[[CBC News]] |date=2 June 2018}}</ref><ref>{{cite web |publisher=[[Elections BC]] |title=2018 Referendum on Electoral Reform: Voting Results Available |date=20 December 2018 |url=https://elections.bc.ca/news/2018-referendum-on-electoral-reform-voting-results-available/ |access-date=1 November 2020}}</ref>

==== Biproportional apportionment ====
{{Main|Biproportional apportionment}}

Biproportional [[apportionment (politics)|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 [[Ward (electoral subdivision)|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 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 of Switzerland|cantons]].<ref name="pukZna">{{Cite journal |last=Pukelsheim |first=Friedrich |date=September 2009 |title=Zurich's New Apportionment |url=http://www.uni-augsburg.de/pukelsheim/2008e-en.pdf |journal=German Research |volume=31 |issue=2 |pages=10–12 |doi=10.1002/germ.200990024 |access-date=10 August 2014 |ref=ZurichNA}}</ref><ref name="balinskiFMV">{{Cite journal |last=Balinski |first=Michel |author-link=Michel Balinski |date=February 2008 |title=Fair Majority Voting (or How to Eliminate Gerrymandering) |url=http://www.maa.org/programs/maa-awards/writing-awards/fair-majority-voting-or-how-to-eliminate-gerrymandering |journal=The American Mathematical Monthly |volume=115 |issue=2 |pages=97–113 |doi=10.1080/00029890.2008.11920503 |access-date=10 August 2014 |ref=FMV |s2cid=1139441}}</ref>

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{{snd}}possibly gerrymandered{{snd}}district boundaries. Seats would be apportioned to parties in a proportional manner at the [[U.S. state|state]] level.<ref name="balinskiFMV" /> 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{{snd}} unlikely to be acceptable to the electorate, the authors concede{{snd}} but this effect could be substantially reduced by incorporating a third, regional, apportionment tier, or by specifying minimum thresholds.<ref>{{Cite journal |last1=Akartunalı |first1=Kerem |last2=Knight |first2=Philip A. |date=June 2017 |title=Network models and biproportional rounding for fair seat allocations in the UK elections |journal=Annals of Operations Research |publisher=[[University of Strathclyde]] |volume=253 |issue=1 |pages=1–19 |doi=10.1007/s10479-016-2323-0 |s2cid=30623821 |issn=0254-5330 |doi-access=free }}</ref>