How Does Proportional Representation Work? Lists, Quotas and Divisors

Proportional representation (PR) works by electing several representatives from each constituency, or from the whole country, and sharing those seats among parties roughly in line with the votes they receive. A party with 30 per cent of the vote should end up with about 30 per cent of the seats. The most common form is party-list PR: parties put forward ranked lists of candidates, voters back a party (and sometimes a candidate), and a mathematical formula decides how many seats each party wins.

The formulas fall into two families, divisors and quotas, and a small example shows how they differ. For how PR compares with the UK's Westminster system, see first past the post vs proportional representation.

Why PR needs multi-member constituencies

One seat cannot be shared. Proportionality needs several seats in each area, and the number of seats per constituency, known as the district magnitude, matters a great deal: the more seats there are, the more proportional the result can be. The most proportional arrangement treats the whole country as a single constituency. Where constituencies are small, larger parties tend to gain. In Spain, where seats are divided among regional constituencies, the largest party won an absolute majority in the Congress of Deputies in 2011 with about 44 per cent of the national vote.

Divisor methods: D'Hondt and Sainte-Laguë

Divisor methods hand out seats one at a time, always to the party with the highest current score.

D'Hondt

Each party's vote total is divided by 1, then 2, then 3 and so on. All the results (called quotients) are ranked, and the seats go to the highest quotients until they run out. Once a party wins a seat, its next quotient is smaller, which gives other parties their turn. The method is also known as the Jefferson method, because Thomas Jefferson described it in 1792 before the Belgian lawyer Victor D'Hondt devised it again in 1878. It tends to favour larger parties more than Sainte-Laguë does.

D'Hondt is used in many countries, including Austria, Belgium, Finland, Poland, Portugal and Spain. In the UK it allocates the top-up seats in the Scottish Parliament and the London Assembly, and it divides ministerial posts among parties in the Northern Ireland Executive.

Sainte-Laguë

This works the same way but uses the odd numbers 1, 3, 5, 7 and so on as divisors. Because the gaps between divisors are bigger, a party's score falls faster after each seat, and smaller parties do comparatively better. Countries using it include Germany, Latvia and New Zealand. Some, including Norway and Sweden, use a modified version in which the first divisor is raised above 1 (Sweden moved to 1.2 for the 2018 general election), making it harder for a very small party to win a single seat.

Quota methods: Hare and Droop

Quota methods first work out how many votes "buy" a seat.

• The Hare quota is the total number of votes divided by the number of seats.
• The Droop quota is the total number of votes divided by the number of seats plus one.

Each party's vote is divided by the quota. Every party gets one seat for each whole quota, and any seats left over go, one each, to the parties with the largest remainders. This is called the largest remainder method. Because the Droop quota is smaller, more seats are filled in the first stage, which tends to help larger parties; the Hare quota is more generous to smaller ones.

A worked example

Imagine a constituency with six seats and 100,000 valid votes cast for four parties. The "entitlement" column shows each party's exact proportional share of the six seats.

Party	Votes	Entitlement	D'Hondt	Sainte-Laguë	Hare largest remainder	Droop largest remainder
A	42,000	2.52	3	2	2	3
B	30,000	1.80	2	2	2	2
C	17,000	1.02	1	1	1	1
D	11,000	0.66	0	1	1	0

How the columns were worked out:

• D'Hondt. The six highest quotients are 42,000 (A), 30,000 (B), 21,000 (A ÷ 2), 17,000 (C), 15,000 (B ÷ 2) and 14,000 (A ÷ 3). Party D's 11,000 would only have won the seventh seat.
• Sainte-Laguë. The six highest are 42,000 (A), 30,000 (B), 17,000 (C), 14,000 (A ÷ 3), 11,000 (D) and 10,000 (B ÷ 3).
• Hare. The quota is 100,000 ÷ 6, about 16,667. A, B and C each win their whole quotas (2, 1 and 1 seats), and the two remaining seats go to the largest remainders: B (0.80) and D (0.66).
• Droop. The quota is 100,000 ÷ 7, about 14,286. A, B and C win 2, 2 and 1 whole quotas, and the one remaining seat goes to the largest remainder, A (0.94), ahead of D (0.77).

The same votes give two different results. D'Hondt and the Droop quota give the largest party a third seat; Sainte-Laguë and the Hare quota give the smallest party its first. Neither answer is "wrong": each formula makes a different trade-off between rewarding larger parties and representing smaller ones.

Thresholds

Many list systems set a legal threshold, a minimum share of the vote a party needs before it can win any seats. Examples include 3.25 per cent in Israel, 5 per cent in Germany (where a party winning three constituency seats is exempt) and 7 per cent in Turkey. The Netherlands has no legal threshold, but with 150 seats elected nationally a party needs about 0.67 per cent of the vote to win one. Supporters of thresholds say they stop a parliament from splintering into many tiny parties and make governments easier to form. Critics say they leave the voters of small parties unrepresented and can distort the overall result.

Closed lists and open lists

The formula decides how many seats a party wins. The type of list decides which of its candidates fill them.

• Closed lists. The party fixes the order of its candidates in advance, and voters choose only a party. If a party wins three seats, the top three names are elected. Spain, Israel and Turkey use closed lists.
• Open lists. Voters can also vote for individual candidates, which can move them up the party's list. How much influence voters have varies, from systems where personal votes rarely change the order to ones where they decide it completely. Countries using open lists include Austria, Belgium, Brazil, Finland, Poland and Sweden.
• Free lists. In a few systems, such as Switzerland's, voters can spread their support across candidates from different parties, a practice called panachage.

Closed lists give parties more control and are simple for voters. Open lists give voters more say over which individuals are elected, at the cost of competition between candidates of the same party.

Where party-list PR is used

List PR, in one form or another, is used in many countries, from Norway and Portugal to Argentina and Angola. In South Africa, the 400 members of the National Assembly are elected by party-list PR, and since 2024 independent candidates have also been able to stand. In Wales, the Senedd Cymru (Members and Elections) Act 2024 replaced the previous mixed system: from the 2026 election, 16 constituencies each elect six members from closed party lists using D'Hondt.

Other proportional systems do not use lists at all. The single transferable vote lets voters rank individual candidates and achieves rough proportionality through transfers rather than a list formula.