Editing Proportional representation (section)

== Basics ==
''Proportional representation'' refers to the general principle found in any electoral system in which the popularly chosen subgroups (parties) of an electorate are reflected proportionately in the elected body.<ref name="JSM7" /> To achieve that intended effect, proportional electoral systems need to either have more than one seat in each district (e.g. [[single transferable vote]]), or have some form of compensatory seats (e.g. [[Mixed-member proportional representation#Apportionment methods|mixed-member proportional representation apportionment methods]]). A legislative body (e.g. assembly, parliament) may be elected proportionally, whereas there is no need for a single office (e.g. a president, or mayor) to be elected proportionately if no votes are for parties (subgroups).

In the [[European Parliament]], for instance, each member state has a number of seats that is (roughly) proportional to its population, enabling geographical and national proportional representation. For these elections, all European Union (EU) countries also must use a proportional electoral system (enabling political proportional representation): When ''n''% of the electorate support a particular [[political party]] or set of candidates as their favourite, then roughly ''n''% of seats are allotted to that party or those candidates.<ref name="ACEpr">{{cite web |title=Proportional Representation (PR) |url=http://aceproject.org/ace-en/topics/es/esd/esd02/default |access-date=9 April 2014 |publisher=[[ACE Electoral Knowledge Network]]}}</ref> All PR systems aim to provide some form of equal representation for votes but may differ in their approaches on how they achieve this.

=== How party-list PR works ===
{{Main|Party-list proportional representation}}

Party-list PR is the most commonly used version of proportional representation. Voters cast votes for parties and each party is allocated seats based on its party share.

Some party-list PR systems use overall country-wide vote counts; others count vote shares in separate parts of the country and allocate seats in each part according to that specific vote count. Some use both.

List PR involves parties in the election process. Voters do not primarily vote for candidates (persons), but for ''[[electoral list]]s'' (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 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.

{| class="wikitable"
! colspan="2" rowspan="2" |Party
! rowspan="2" |Popular vote
! colspan="3" |Party-list PR{{snd}}Sainte-Laguë method
|-
!Number of seats
!Seats %
| rowspan="6" |[[File:Party list pr-example total seats.svg|frameless|upright=0.75]]
|-
| style="background:#D10000" |
|Party A
|43.91%
|88
|44%
|-
| style="background:#0008A5" |
|Party B
|39.94%
|80
|40%
|-
| style="background:#03AA00" |
|Party C
|9.98%
|20
|10%
|-
| style="background:#820084" |
|Party D
|6.03%
|12
|6%
|-
| colspan="2" |''TOTAL''
|''99.86%''
|''200''
|''100%''
|}
This is done by a proportional formula or method; for example, the [[Sainte-Laguë method]]{{snd}}these are the same methods that may be used to allocate seats for geographic proportional representation (for example, how many seats each states 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 seats{{snd}}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 list]]s or an [[electoral threshold]]. This can mean that final seat allocations are frequently not proportional to the parties' vote share.

=== How the single transferable vote (STV) works ===
{{Main|Single transferable vote}}

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 from which winners are drawn in some order, candidates run by name, each voter casts one vote by marking preferences for candidates, with only one marked preference used to place the vote. Votes cast for the candidates determine the winners. 

STV uses [[Ranked voting|preferential ballots]]. The ranking is used to instruct election officials 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, and the district used elects multiple members (more than one, usually 3 to 7). Because parties play no role in the vote count, STV may be used for nonpartisan elections, as with the city council of [[Cambridge, Massachusetts]].{{Citation needed|date=May 2025}} 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 single transferable votes|Counting votes under STV]] is more complicated than under [[first-past-the-post voting]], but the following example 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 with no great difficulty.<ref>"Results: NSW Election 2019". ABC Elections. Archived from the original on 24 March 2019. Retrieved 23 March 2019</ref> 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'', an amount that guarantees election, is determined. In the example below, the [[Droop quota]] is 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 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. 
[[File:Preferential_ballot.svg|thumb|Simplified example of an STV ballot]]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.
{| class="wikitable"
!Candidate
! colspan="2" |Party
!Popular vote<br>(first preferences)
!Elected?
!If elected: surplus votes
|-
|Jane Doe
| style="background:#D10000" |
|Party A
|40%
|'''Yes'''
|15%
|-
|Fred Rubble
|
|Party B
|30%
|'''Yes'''
|5%
|-
|Joe Smith
| style="background:#D10000" |
|Party A
|16%
|
|
|-
|John Citizen
|
|Party A
|11%
|
|
|-
|Mary Hill
| style="background:#0008A5" |
|Party B
|3%
|
|
|-
| colspan="3" |''TOTAL''
|''100%''
|
|
|}
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 A{{Snd}}with 67 percent of the vote{{Snd}}was. 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 fair{{Snd}}the 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 elected{{Snd}}they 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. 

==== Final-round vote tallies and party satisfaction break-down ====
Quota is 25 percent plus 1 
{| class="wikitable"
!Candidate
! colspan="2" |Party
!Current vote total
!Elected?
!Party
!First-preference votes<br>for candidates of party
!Number of<br>seats
!Party seats %<br>under STV
|-
|Jane Doe
| style="background:#D10000" |
|Party A
|Already elected (25%+1 vote)
|'''Yes'''
| rowspan="3" |Party A
| rowspan="3" |67%
| rowspan="3" |2
| rowspan="3" |67%
|-
|John Citizen
| style="background:#D10000" |
|Party A
|11% + 15% = 26%
|'''Yes''' 
|-
|Joe Smith
| style="background:#D10000" |
|Party A
|16%
|
|-
|Fred Rubble
| style="background:#0008A5" |
|Party B
|Already elected (30%)<br>(surplus votes not transferred)
|'''Yes'''
| rowspan="2" |Party B
| rowspan="2" |33%
| rowspan="2" |1
| rowspan="2" |33%
|-
|Mary Hill
| style="background:#0008A5" |
|Party B
|3%
|
|-
| colspan="3" |''TOTAL''
|''100%''
|''3''
|
|''100%''
|''3''
|''100%''
|}

Under STV, to make up the 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 results{{Snd}}one 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.<ref name="auto1">{{cite web | url=https://electoral.gov.mt/ElectionResults/General | title=Electoral Commission of Malta }}</ref> However, about 4000 voters in a district would be enough to elect a third-party candidate if voters desired, but this seldom happens.<ref name="auto1" />{{Original research inline|date=May 2025}}

Conversely, New South Wales, which uses STV to elect its state legislative council in 21-seat contests, sees election of representatives of seven or eight different parties each time.<ref>{{cite web | url=https://elections.nsw.gov.au/elections/past-results/state-election-results/2023-nsw-state-election-results#legislativecouncilresults | title=2023 NSW State election results }}</ref> 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.

=== How mixed-member proportional representation (MMP) works ===
{{Main|Mixed-member proportional representation}}

Mixed-member proportional representation combines election of district members with election of additional members as compensatory top-up. Often MMP systems use single-member districts (SMDs) to elect district members. (Denmark, Iceland and Sweden use multi-member districts in their MMP systems.) MMP with SMDs is described here.

The mixed-member proportional system combines first-past-the-post voting 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 district elections are highly disproportional: large parties typically win more seats than they should proportionally, but there is also randomness{{snd}}a party that receives more votes than another party might not win more seats than the other. Any such dis-proportionality produced by the district elections is addressed, where possible, by the allocation of the compensatory additional members.
{| class="wikitable sortable"
|+Results under mixed-member proportional representation
! colspan="4" |
|[[File:Mixed compensatory example fptp.svg|frameless|upright=0.75]]
|[[File:Mixed-compensatory-example compensatory seats.svg|frameless|upright=0.75]]
|[[File:Mixed-compensatory-example total seats.svg|frameless|upright=0.75]]
| colspan="2" |[[File:Party list pr-example total seats.svg|center|frameless|upright=0.75]]
|-
! colspan="3" |Party
!Popular vote
!FPTP seats<br>(Number of districts won)
!Compensatory seats<br>(party-list PR seats)
! colspan="2" |Total number of seats
!Seats %
|-
| style="background:#D10000" |
| style="background:#ff2c2c" |
|Party A
|43.91%
|64
|24
| colspan="2" |88
|44%
|-
| style="background:#0008A5" |
| style="background:#000cff" |
|Party B
|39.94%
|33
|47
| colspan="2" |80
|40%
|-
| style="background:#03AA00" |
| style="background:#05ff00" |
|Party C
|9.98%
|0
|20
| colspan="2" |20
|10%
|-
| style="background:#820084" |
| style="background:#fb00ff" |
|Party D
|6.03%
|3
|9
| colspan="2" |12
|6%
|-
| colspan="3" |''TOTAL''
|''100%''
|''100''
|''100''
| colspan="2" |''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 outcome proportional.

==== Differences from mixed-member majoritarian system ====
Compare the MMP example to a [[Mixed-member majoritarian representation|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.{{Original research inline|date=May 2025}}

'''Parallel voting (using non-compensatory party seats)'''
{| class="wikitable"
|+Results under parallel voting
! colspan="4" |
|[[File:Mixed compensatory example fptp.svg|frameless|upright=0.75]]
|[[File:Mixed compensatory example non compensatory parallel seats.svg|frameless|upright=0.75]]
|[[File:Mixed compensatory example non compensatory total seats.svg|frameless|upright=0.75]]
| colspan="2" |[[File:Parallel voting mmm-example total seats.svg|center|frameless|upright=0.75]]
|-
! colspan="3" |Party
!Popular vote
!FPTP seats<br>(Number of districts won)
!Party-list PR seats
! colspan="2" |Total number of seats
!Seats %
|-
| style="background:#D10000" |
| style="background:#ff2c2c" |
|Party A
|43.91%
|64
|44
| colspan="2" |108
|54.0%
|-
| style="background:#0008A5" |
| style="background:#000cff" |
|Party B
|39.94%
|33
|40
| colspan="2" |73
|36.5%
|-
| style="background:#03AA00" |
| style="background:#05ff00" |
|Party C
|9.98%
|0
|10
| colspan="2" |10
|5.0%
|-
| style="background:#820084" |
| style="background:#fb00ff" |
|Party D
|6.03%
|3
|6
| colspan="2" |9
|4.5%
|-
| colspan="3" |''TOTAL''
|''100%''
|''100''
|''100''
| colspan="2" |''200''
|''100%''
|}
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 representation|semi-proportional]]. [[Mixed electoral system|Mixed system]]s are the most proportional if the additional members are allocated in a compensatory way.

There are many versions of MMP in use. Some use only a [[Mixed single vote|single vote]]; in some, voters cast two votes, one for a local candidate and one for a party. Some allocate compensatory seats to best losers; others allocate according to party lists. Some use levelling seats to compensate for potential [[overhang seat]]s; others don't. Most impose an electoral threshold in order for a party to be eligible for any additional seats; some allow parties that elect one or more district seats to be eligible for additional seats even if its party share is below the threshold. Any barrier to access to the additional seats may produce wasted votes and dis-proportionality in the final result.{{Original research inline|date=May 2025}}

As well, there is [[single non-transferable vote]], which is also semi-proportional. It has the advantage that parties play no direct role in elections and voters do not need to mark ranked votes. Each voter casts one vote for a candidate and as many candidates win by plurality as the number of seats in the district. Due to each voter casting just one vote, as in STV, and each district electing multiple members, as in STV, mixed representation is produced in each district and overall rough proportionality more or less. Without transferable votes, more votes are wasted than under STV.{{Original research inline|date=May 2025}}