In the study of electoral systems, the Droop quota (sometimes called the Hagenbach-Bischoff, Britton, or Newland-Britton quota) 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.
The Droop quota is used to extend the concept of a majority to multiwinner elections, taking the place of the 50% bar in single-winner elections. 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.
The Droop quota is used in almost all STV elections, including those in Australia, the Republic of Ireland, Northern Ireland, and Malta. It is also used in South Africa to allocate seats by the largest remainder method. Switzerland uses the Droop quota, calling it the Hagenbach-Bischof quota.
Although common, the quota's use in proportional representation has been criticized both for its bias toward large parties 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. However, this situation can occur regardless of whether the quota is used with largest remainders or STV. Charges of no-show paradoxes are based on having knowledge of how a vote would be transferred if a candidate were eliminated when that candidate may not have been in real life. It is clear that any system that uses ranked votes produces different results if candidates are in different order, which is partly determined by how votes are split and therefore that charge can apply to any ranked voting system no matter what quota is used. Some analysis states that no-show paradoxes are extremely rare in real-world elections. For one thing, transfers have little effect in general on who is elected, the winners usually being among the front runners in the first round of counting anyway.
Definition
The value of the exact Droop quota for a <math>k</math>-winner election is given by the expression:
<math>\frac{\text{total votes{k+1} </math>
In the case of a single-winner election, this reduces to the familiar simple majority rule. Under such a rule, a candidate can be declared elected as soon as they have more than 50% of the vote, i.e. their vote total exceeds <math display="inline">\frac{\text{total votes{2}</math>.
Sometimes, the Droop quota is written as a share of all votes, in which case it has value .
Original Droop quota
The original Droop as devised by Henry Droop was one more than the exact Droop:
<math>\frac{\text{total votes{k+1}+1 </math>
Modern variants of STV use fractional transfers of ballots to eliminate uncertainty and therefore do not need to use the original whole-vote Droop quota. The original Droop quota is not necessary in elections that allow fractional transfers of ballots.
However, some older implementations of STV with whole vote reassignment did not use handle fractional votes and so instead used either round up or add one and truncate: However, it is the most commonly used definition in legislative codes worldwide.
Derivation of the original Droop quota
The Droop quota was derived by considering what would happen if candidates (here called "Droop winners") have achieved the Droop quota. Could too many achieve quota? The goal is to identify whether an additional candidate could defeat any of the candidates who have quota. If each quota winner's share of the vote equals , all unelected candidates' share of the vote, taken together, is at most votes.
Thus, even if there were only one unelected candidate who held all the remaining votes, their vote tally would not exceed any of those with Droop quota. At least six different versions appear in various legal codes or definitions of the quota, all varying by one vote.
By contrast, the Droop quota is biased towards large parties.
See also
- List of democracy and elections-related topics