Party-list proportional representation

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 (a closed list system) or it may be determined by the voters at large (an open list system) or by districts (a local list system).

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. There are various apportionment methods used to allocate seats within party-list proportional representation; they can be classified into two categories:

• The highest averages method (or divisor method), including the D'Hondt method (also known as the Jefferson method) is used in Armenia, Austria, Brazil, Bulgaria, Cambodia, Croatia, Estonia, Finland, Poland, and Spain; and the Sainte-Laguë method (also known as the Webster method or the Schepers method) is used in Indonesia, New Zealand, Norway, and Sweden.
• The largest remainder (LR) methods, including the Hare-Niemeyer (Hamilton) method and the Droop method.

While the allocation formula is important, equally important is the district magnitude (number of seats in a constituency). 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.

Some apportionment methods may favor small parties; others may favor large parties:

• D'Hondt method (biased towards large parties)
• Sainte-Laguë method (generally considered closest to proportional but does not ensure that a party receiving more than half the votes will win at least half the seats)
• Huntington–Hill method (biased towards small parties, automatically gives every party at least one seat)
• Adams' method (heavily biased towards small parties, automatically gives every party at least one seat)
• Hare-Niemeyer method (roughly unbiased, some bias towards smaller parties)
• Droop method (slightly biased towards 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 (1, 2, 3, ...).

• Example (5 seats):

• Result: A = 2, B = 2, C = 1

Sainte-Laguë method

The Sainte-Laguë (Webster, Schepers) method is a highest averages method using odd-numbered divisors (1, 3, 5, ...) to promote more equal distribution.

• Same vote totals as above.

• 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, ...

• 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

• Remaining 2 seats to C and B (highest remainders)

• Result: A = 2, B = 2, C = 1

Imperiali quota (largest remainder)

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

• Quota = Total votes / (seats + 2) = 2200 / 7 = ~314.29

• 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. It is too specialized for elections, as it requires every party to have at least one seat.

Example

Below it can be seen how different apportionment methods yield different results with the same number of seats and votes (100 and 2832 in this example).

As there are 100 seats, the percentage values for every party's share of the vote is equal to the party's vote count divided by the Hare quota (which is the ratio of vote and seat totals), and in this case the share of seats under the largest remainder method using this quota happens to be the same as the percentage values rounded to the nearest integer (because exactly 3 party's results has to be rounded up, same as there are 3 seats to assign with the largest remainder method after 97 seats are assigned based on the integer part of the vote share divided by the Hare quota). The Webster/Sainte-Laguë method yields the same result (but this is not always the case), otherwise all other methods give a different number of seats to the parties.

Notable is how the D'Hondt method breaks the quota rule (shown in red text) and favors the largest party with 37% of seats, even though it only got 35.91% of the vote (the quota rule would allow either 35 or 36 seats in this case = rounding up or rounding down, but no jump to 34 or 37). Also, the Adams and Huntington-Hill methods, which (without a threshold) greatly favour smaller parties gave 2 seats to the smallest party and would have both given at least 1 seat to every party if it got even just 1 vote from 2832. Of the highest average methods, modified versions of the formulas may not be strictly proportional. For example, the Imperiali method (not to be confused with the Imperiali quota) can be seen as a modified version of the D'Hondt (or Adams) method, and it is technically not proportional (e.g., if a party received 500/1000 votes, and there were 100 seats to be apportioned, it may sometimes not only get 50 - the clearly proportional number of seats - but could also get 51). The Macanese modification of the d'Hondt method, whose quotients are based on the formula ${\displaystyle \frac{V}{2^s}}$, is clearly disproportional; the great variations between the parties' vote shares are effectively reduced, and each party has a roughly equal number of seats, resulting in either completely equal or degressively proportional representation if the district magnitude is sufficient.

Electoral threshold

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 (mixed-member majoritarian) or other mixed system (e.g. MMP) are not included.

Authoritarian regimes