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. Besides establishing winners, the Droop quota is used to define the number of excess votes, i.e. votes not needed by a candidate who has been declared elected. In proportional quota–based systems such as STV or expanding approvals, these excess votes can be transferred to other candidates to prevent them from being wasted. (As well, the equivalent of one quota is the number of votes not used to elect someone in many STV contests.) The Droop quota was first suggested by the English lawyer and mathematician Henry Richmond Droop (1831–1884) as an alternative to the Hare quota and later by Swiss physicist Eduard Hagenbach-Bischof in the context of STV and not for the largest remainder method. 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 k {\displaystyle k} -winner election is given by the expression:
total votes k + 1 {\displaystyle {\frac {\text{total votes}}{k+1}}}
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 total votes 2 {\textstyle {\frac {\text{total votes}}{2}}} . A candidate who, at any point, holds strictly more than one Droop quota's worth of votes is therefore guaranteed to win a seat. Sometimes, the Droop quota is written as a share of all votes, in which case it has value 1⁄k+1.
Original Droop quota The original Droop as devised by Henry Droop was one more than the exact Droop:
total votes k + 1 + 1 {\displaystyle {\frac {\text{total votes}}{k+1}}+1}
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 fractional votes and so instead either rounded up or added one and truncated:
⌈ total votes k + 1 ⌉ ≈ ⌊ total votes k + 1 + 1 ⌋ {\displaystyle \left\lceil {\frac {\text{total votes}}{k+1}}\right\rceil \approx \left\lfloor {\frac {\text{total votes}}{k+1}}+1\right\rfloor }
This whole-vote variant of the quota is not necessary in the context of modern elections with fractional votes, and it can cause problems in small elections (see § Variety of Droop quotas). 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 k candidates (here called "Droop winners") have achieved the Droop quota: could too many achieve quota? The goal was 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 1⁄k+1, all unelected candidates' share of the vote, taken together, is at most 1⁄k+1 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.
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