Droop quota

Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results In the study of electoral systems, the Droop quota (sometimes called the Hagenbach-Bischoff, Britton, or Newland-Britton quota[1][a]) is the minimum number of votes a party or candidate needs to receive in a district to guarantee they will win at least one seat.

Just as any candidate with more than half of all votes is guaranteed to be declared the winner in single-seat election, any candidate with more than a Droop quota's worth of votes is guaranteed to win a seat in a multiwinner election.

In proportional quota-based systems such as STV or expanding approvals, these excess votes can be transferred to other candidates to preventing them from being wasted.

[6] It is also used in South Africa to allocate seats by the largest remainder method.

[7][8] Although common, the quota's use in proportional representation has been criticized both for its bias toward large parties[9] and for its ability to create no-show paradoxes, situations where a candidate or party loses a seat as a result of having won too many votes.

In the case of a single-winner election, this reduces to the familiar simple majority rule.

[1] A candidate who, at any point, holds strictly more than one Droop quota's worth of votes is therefore guaranteed to win a seat.

Modern variants of STV use fractional transfers of ballots to eliminate uncertainty.

However, some older implementations of STV with whole vote reassignment cannot handle fractional quotas, and so instead will either round up, or add one and truncate:[4]

Thus, even if there were only one unelected candidate who held all the remaining votes, they would not be able to defeat any of the Droop winners.

There are 4 candidates: George Washington, Alexander Hamilton, Thomas Jefferson, and Aaron Burr.

Thanks to Hamilton's support, Jefferson receives 30 votes to Burr's 20 and is elected.

If all of Hamilton's supporters had instead backed Burr, the election for the last seat would have been exactly tied, requiring a tiebreaker; generally, ties are broken by looking at who had the most first-preference votes.

There is a great deal of confusion among legislators and political observers about the correct form of the Droop quota.

[20] At least six different versions appear in various legal codes or definitions of the quota, all varying by one vote.

[20] The ERS handbook on STV has advised against such variants since at least 1976, as they can cause problems with proportionality in small elections.

[1][19] In addition, it means that vote totals cannot be summarized into percentages, because the winning candidate may depend on the choice of unit or total number of ballots (not just their distribution across candidates).

The two variants in the first line come from Droop's discussion in the context of Hare's STV proposal.

Hare assumed that to calculate election results, physical ballots would be reshuffled across piles, and did not consider the possibility of fractional votes.

[20] However, as Newland and Britton noted in 1974, this is not a problem: if the last two winners both receive a Droop quota of votes, rules can be applied to break the tie, and ties can occur regardless of which quota is used.

The confusion between the two quotas originates from a fencepost error, caused by forgetting unelected candidates can also have votes at the end of the counting process.

[4] The Hare quota gives more proportional outcomes on average because it is statistically unbiased.

[9] As a result, the Droop quota is the quota most likely to produce minority rule by a plurality party, where a party representing less than half of the voters may take majority of seats in a constituency.