Party-list proportional representation

Party-list proportional representation is a system of proportional representation based on preregistered political parties, with each party being allocated a certain number of seats roughly proportional to their share of the vote.
In these systems, parties provide lists of candidates to be elected, or candidates may declare their affiliation with a political party. Seats are distributed by election authorities to each party, in proportion to the number of votes the party receives. Voters may cast votes for parties, as in Spain, Turkey, and Israel ; or for candidates whose vote totals are pooled together to determine the share of representation of their respective parties, as in Finland, Brazil, and the Netherlands.

Voting

In most party list systems, a voter will only support one party. Open list systems may allow voters to support more than one candidate within a party list. Some open-list systems allow voters to support different candidates across multiple lists, which is called free list or panachage.

Selection of party candidates

The order in which a party's list candidates get elected may be pre-determined by some method internal to the party or the candidates or it may be determined by the voters at large or by districts.

Closed list

In a closed list system, each political party has pre-decided who will receive the seats allocated to that party in the elections, so that the candidates positioned highest on this list will always get a seat in the parliament while the candidates positioned very low on the closed list will not. Voters vote only for the party, not for individual candidates.

Open list

An open list describes any variant of a party-list where voters have at least some influence on the order in which a party's candidates are elected. Open lists can be anywhere from relatively closed, where a candidate can move up a predetermined list only with a certain number of votes, to completely open, where the order of the list completely depends on the number of votes each individual candidate gets.

Apportionment of party seats

Within party-list PR systems, there are a variety of different methods that can be used to determine how many seats are allocated to each party for a given vote breakdown. The method used to allocate seats within party-list proportional representation vary. Some apportionment methods may favor small parties; others may favor large parties:

D'Hondt method

The D'Hondt method is a highest averages method that allocates seats by dividing each party's total votes by a series of divisors.

Example :

Round	A quotient	B quotient	C quotient	Seat to
1	1000	800	400	A
2	500	800	400	B
3	500	400	400	A
4	333.33	400	400	B
5	333.33	266.67	400	C

Result: A = 2, B = 2, C = 1

Sainte-Laguë method

The Sainte-Laguë method is a highest averages method using odd-numbered divisors to promote more equal distribution.

Same vote totals as above.

Round	A quotient	B quotient	C quotient	Seat to
1	1000	800	400	A
2	333.33	800	400	B
3	333.33	266.67	400	C
4	333.33	266.67	133.33	A
5	200	266.67	133.33	B

Result: A = 2, B = 2, C = 1

Modified Sainte-Laguë method

Same as Sainte-Laguë but first divisor is 1.4 to favour larger parties.

Divisors: 1.4, 3, 5,...

Round	A quotient	B quotient	C quotient	Seat to
1	714.29	571.43	285.71	A
2	333.33	571.43	285.71	B
3	333.33	266.67	285.71	A
4	200	266.67	285.71	C
5	200	266.67	133.33	B

Result: A = 2, B = 2, C = 1

Hare quota (largest remainder)

The Hare quota uses a quota to allocate seats, then gives remaining seats to the parties with the largest remainders.

Quota = Total votes / Seats = 2200 / 5 = 440

Party	Votes	Initial seats	Remainder
A	1000	2	120
B	800	1	360
C	400	0	400

Remaining 2 seats to C and B
Result: A = 2, B = 2, C = 1

Imperiali quota (largest remainder)

The Imperiali quota is rarely used; favors large parties more.

Quota = Total votes / = 2200 / 7 = ~314.29

Party	Votes	Initial seats	Remainder
A	1000	3	57.14
B	800	2	171.42
C	400	1	85.71

1 leftover seat to B
Result: A = 3, B = 2, C = 1

Huntington–Hill method

The Huntington–Hill method is used for US congressional apportionment, based on geometric mean. Too specialized for vote quotas, usually used with population.
The apportionment methods can be classified into two categories:

The highest averages method, including the D'Hondt method is used in Armenia, Austria, Brazil, Bulgaria, Cambodia, Croatia, Estonia, Finland, Poland, and Spain; and the Sainte-Laguë method is used in Indonesia, New Zealand, Norway, and Sweden.
The largest remainder (LR) methods, including the Hamilton method and Droop method.

While the allocation formula is important, equally important is the district magnitude. The higher the district magnitude, the more proportional an electoral system becomes, with the most proportional results being when there is no division into constituencies at all and the entire country is treated as a single constituency. In some countries the electoral system works on two levels: at-large for parties, and in constituencies for candidates, with local party-lists seen as fractions of general, national lists. In this case, magnitude of local constituencies is irrelevant, seat apportionment being calculated at national level.
List proportional representation may also be combined with other apportionment methods in various mixed systems, using either additional member systems or parallel voting.

Example

Below it can be seen how different apportionment methods yield different results when apportioning 100 seats.
Webster's method yields the same result. Otherwise, all other methods give a different number of seats to the parties.
Notice how the D'Hondt method breaks the quota rule and favors the largest party by "rounding" an ideal apportionment of 35.91 up to 37.
Adams' method greatly favors smaller parties, giving 2 seats to the smallest party, and would give at least 1 seat to every party receiving at least one vote.

List of countries using party-list proportional representation

The table below lists countries that use a proportional electoral system to fill a nationally elected legislative body. Detailed information on electoral systems applying to the first chamber of the legislature is maintained by the ACE Electoral Knowledge Network. Countries using PR as part of a parallel voting or other mixed system are not included.

Country	Legislative body	List type	Variation of open lists	Apportionment method	Electoral threshold	Constituencies	Governmental system	Notes
Albania	Parliament	Open list	?	d'Hondt method	4% nationally or 2.5% in a district	Counties	Parliamentary republic
Algeria	People's National Assembly	Open list	?	Hare quota	5% of votes in respective district.	Provinces	Semi-presidential republic
Angola	Assembly (Angola)\|National Assembly]	Closed list	—	d'Hondt method		5 member districts and nationwide	Parliamentary republic with an executive presidency	Double simultaneous vote use to elect the President and the National Assembly at the same election.
Argentina	Chamber of Deputies	Closed list	—	d'Hondt method	3% of registered voters in the constituency	Provinces	Presidential republic
Armenia	National Assembly	Open list	?	D'Hondt method	5%, 7%	None	Parliamentary republic	Party lists run-off, but only if necessary to ensure stable majority of 54% if it is not achieved either immediately or through building a coalition. If a party would win more than 2/3 seats, at least 1/3 seats are distributed to the other parties.
Armenia	National Assembly	Closed list	—	D'Hondt method	5%, 7%	None	Parliamentary republic	Party lists run-off, but only if necessary to ensure stable majority of 54% if it is not achieved either immediately or through building a coalition. If a party would win more than 2/3 seats, at least 1/3 seats are distributed to the other parties.
Aruba	Parliament	Open list		D'Hondt method		None
Austria	National Council	Open list	More open: 14% on the district level	Hare quota	4% nationally	Single-member districts within federal states	Semi-presidential republic, de facto parliamentary
Austria	National Council	Open list	More open: 10% on the regional level	Hare quota	4% nationally	Federal states	Semi-presidential republic, de facto parliamentary
Austria	National Council	Open list	More open: 7% of the on the federal level	d'Hondt method	4% nationally	Single federal constituency	Semi-presidential republic, de facto parliamentary
Belgium	Chamber of Representative	Open list		D'Hondt method	5%	Electoral districts	Federal parliamentary constitutional monarchy
Bénin	National Assembly	Closed list	—	Largest remainder method	10%	Departments	Presidential republic
Bolivia	Chamber of Senators	Closed list	—	d'Hondt method		Departments	Unitary presidential republic	Ballots use the double simultaneous vote: voters cast a single vote for a presidential candidate and their party's list and local candidates at the same time
Bosnia and Herzegovina	House of Representative	Open list		Sainte-Laguë method		Electoral districts	Federal parliamentary directorial republic
Brazil	Chamber of Deputies	Open list	Fully open	D'Hondt method	2% distributed in at least 9 Federation Units with at least 1% of the valid votes in each one of them	States and Federal District	Presidential republic
Bulgaria	National Assembly	Open list		Hare quota and largest remainders	4%	Electoral districts	Unitary parliamentary republic
Cape Verde	National Assembly	Closed list	—	D'Hondt method		Electoral districts	Semi-presidential republic
Chile	Chamber of Deputies	Open list		D'Hondt method	No de jure threshold	Electoral districts	Presidential republic
Chile	Senate	Open list		D'Hondt method	No de jure threshold	Electoral districts	Presidential republic
Colombia	Chamber of Representatives	Closed list	—	D'Hondt method	3%	Departments	Unitary presidential republic
Colombia	Senate	Closed list	—	D'Hondt method	3%	None	Unitary presidential republic
Costa Rica	Legislative Assembly	Closed list	—	Hare quota and largest remainders	Subquota	Departments	Presidential republic
Croatia	Sabor	Open list		D'Hondt method	5%	Constituencies	Parliamentary republic
Cyprus	House of Representatives	Open list		Hare quota and largest remainders	No de jure threshold	Electoral districts	Presidential republic
Czech Republic	Chamber of Deputies	Open list	Relatively open: 5% on the district level	Imperiali quota with national remnant in the first scrutiny	5% nationally for single party lists, 7% for coalitions of two, 11% for coalitions of more than 2.	Regions and capital	Parliamentary republic
Czech Republic	Chamber of Deputies	Open list	Relatively open: 5% on the district level	Hagenbach-Bischoff quota + largest remainders	5% nationally for single party lists, 7% for coalitions of two, 11% for coalitions of more than 2.	National remnant seats redistributed to districts	Parliamentary republic
Denmark	Folketing	Open list		Modified Sainte-Laguë method for leveling seats	2%	Electoral districts	Parliamentary republic	135 constituency seats, 40 leveling seats
Dominican Republic	Chamber of Deputies	Closed list	—	D'Hondt method		Departments	Presidential republic
Ecuador	National Congress	Closed list	—	Sainte-Laguë method		Provinces	Presidential republic
El Salvador	Legislative Assembly	Open list		D'Hondt method	10%	Departments	Presidential republic
Estonia	Riigikogu	Open list		D'Hondt method	5%	Electoral districts	Parliamentary republic
Faroe Islands	Løgting	Open list		D'Hondt method		None	Parliamentary republic
Fiji	Parilament	Open list		D'Hondt method	5%	None	Parliamentary republic
Finland	Parliament	Open list	Fully open	D'Hondt method	No de jure threshold	Electoral districts	Parliamentary republic
Germany	Bundestag	Localized list	Separate vote for candidates	Only first place candidate may win seat	5% or 3 constituencies, first place for independents	Constituencies	Federal parliamentary republic	The system was recently modified to an essentially closed list proportional system with a local constituency vote to eliminate the need for overhang seats. In the new system, the number of seats a party can win is capped, if they "won" more seats by plurality, not all of their winners will be elected.
Germany	Bundestag	Closed list	—	Sainte-Laguë method	5% or 3 constituencies, first place for independents	Federal states	Federal parliamentary republic	The system was recently modified to an essentially closed list proportional system with a local constituency vote to eliminate the need for overhang seats. In the new system, the number of seats a party can win is capped, if they "won" more seats by plurality, not all of their winners will be elected.
Greenland	Inatsisartut	Open list		D'Hondt method		None	Parliamentary republic
Guatemala	Congress of the Republic	Closed list	—	D'Hondt method		Departments	Presidential republic
Guyana	National Assembly	Closed list	—	Hare quota + largest remainders	No de jure threshold	Regional constituencies	Parliamentary republic with an executive president	The president is elected by a first-past-the-post double simultaneous vote system, whereby each list nominates a presidential candidate and the presidential election itself is won by the candidate of the list having a plurality.
Guyana	National Assembly	Closed list	—	Hare quota + largest remainders	No de jure threshold	National constituency	Parliamentary republic with an executive president	The president is elected by a first-past-the-post double simultaneous vote system, whereby each list nominates a presidential candidate and the presidential election itself is won by the candidate of the list having a plurality.
Honduras	National Congress	Open list	Fully open with panachage			Departments	Presidential republic
Iceland	Althing	Open list				Regions	Parliamentary republic
Indonesia	House of Representative	Open list		Sainte-Laguë method	4%	3 to 10 members constituencies	Presidential system
Israel	Knesset	Closed list	—		3.25%	None	Parliamentary system
Kosovo	Assembly of the Republic	Open list		Sainte-Laguë method		None
Latvia	Saeima	Open list		Sainte-Laguë method	5%	Regions	Parliamentary republic
Lebanon	Parliament	Open list				Regions
Liechtenstein	Landtag	Open list			8%	Regions
Luxembourg	Chamber of Deputies	Open list	Panachage	d'Hondt method	No de jure threshold	Four multi-member constituencies, ranging from 7 to 23 members	Parliamentary system
Macedonia	Assembly	Closed list	—			Regions	Parliamentary system
Moldova	Parliament	Closed list	—	d'Hondt method	5%, 7%, 2%	None	Unitary parliamentary republic
Montenegro	Parilament	Closed list	—	d'Hondt method	3%	None	Parliamentary system
Namibia	National Assembly	Closed list	—			None	Presidential republic
Netherlands	House of Representatives	Open list	More open	d'Hondt method	No de jure threshold, but an effective threshold of 0.67% for a seat	None	Parliamentary system
Norway	Parliament (Storting)	Closed list	—	modified Sainte-Laguë method	No de jure threshold	19 multi-member constituencies	Parliamentary system	First divisor is 1,4 instead of 1.
Norway	Parliament (Storting)	Closed list	4% for leveling seats	modified Sainte-Laguë method	One seat in each constituency is used for nationwide leveling	-	Parliamentary system	First divisor is 1,4 instead of 1.
Paraguay	Chamber of Deputies	Closed list	—			Departments	Presidential republic
Peru	Congress of the Republic	Closed list	—	d'Hondt method	5%	Departments	Presidential republic
Poland	Sejm	Open list	Fully open	d'Hondt method	5% or 8% nationally, no threshold for national minority organisations	41 multi-member constituencies, ranging from 7 to 20 members	Semi-presidential republic	-
Portugal	Assembly of the Republic	Closed list	—	d'Hondt method	No de jure threshold	Districts	Semi-presidential republic
Romania	Chamber of Deputies	Closed list	—			Counties	Presidential republic
San Marino	Grand and General Council	Open list			3.5%	None		If needed to ensure a stable majority, the two best-placed parties participate in a run-off vote to receive a majority bonus.
São Tomé and Príncipe	National Assembly	Closed list	—			Constituencies
Serbia	National Assembly	Closed list	—		3%	None	Parliamentary republic
Sierra Leone	Parliament of Sierra Leone		—			Districts	Presidential republic
Sint Maarten	Parilament	Open list				None	Parliamentary republic
Slovakia	National Council	Open list		Droop quota and largest remainders	5%	None	Parliamentary republic
Slovenia	National Assembly	Open list	Fully open	Droop quota	4%	Districts	Parliamentary republic
Slovenia	National Assembly	Open list	Fully open	d'Hondt method	4%	Districts	Parliamentary republic
South Africa	National Assembly of South Africa	Closed list	—	Droop quota	No de jure threshold	Provinces of South Africa	Parliamentary republic with an executive president
Spain	Congress of Deputies	Closed list	—	d'Hondt method	3%	Provinces of Spain	Parliamentary system
Sri Lanka	Parliament	Open list	Panachage	d'Hondt method	5%	Constituencies	Semi-presidential system
Sri Lanka	Parliament	Closed list	—	?	No de jure threshold	None	Semi-presidential system
Suriname	National Assembly	Open list	Fully open	d'Hondt method	No de jure threshold	Districts of Suriname	Assembly-independent republic
Sweden	Riksdag	Open list	More open	Sainte-Laguë method	4% nationally or 12% in a given constituency	Counties of Sweden	Parliamentary system
Switzerland	National Council	Open list	Fully open with panachage	Hagenbach-Bischoff system	No de jure threshold	Cantons of Switzerland	Semi-direct democracy under an assembly-independent directorial republic
Timor-Leste	National Parliament	Open list		d'Hondt method		None
Tunisia	Assembly of the Representatives of the People	Closed list	—	Largest remainder method	No de jure threshold	Constituencies	Semi-presidential system
Turkey	Grand National Assembly	Closed list	—	d'Hondt method	7%. No threshold for independent candidates.	Provinces of Turkey	Presidential system
Uruguay	Chamber of Representatives	Closed list	—	d'Hondt method	No de jure threshold	Departments of Uruguay	Presidential system	Ballots use the double simultaneous vote, the same ballot is used for electing the president and the two chambers
Uruguay	Chamber of Senators	Closed list	—	d'Hondt method	No de jure threshold	None	Presidential system	Ballots use the double simultaneous vote, the same ballot is used for electing the president and the two chambers