Droop quota

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.^[3][4] It is commonly used in single transferable voting election contests.

The Droop quota is used to extend the concept of a majority to multiwinner elections, taking the place of the 50% plus 1 majority 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 a Droop quota's worth of votes is guaranteed to win a seat in a multiwinner election.^[4]

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.^[4]

The Droop quota was first suggested by the English lawyer and mathematician Henry Richmond Droop (1831–1884) as an alternative to the Hare quota.^[4]

Today, the Droop quota is used in almost all STV elections, including those in Australia,^[5] the Republic of Ireland, Northern Ireland, and Malta.^[6] It is also used in South Africa to allocate seats by the largest remainder method.^[7][8]

Although almost universally applied in STV government elections today, some say the Droop quota has a bias in favor of large parties (popular parties in a district)^[9] and the ability to create no-show paradoxes, situations where a candidate or party loses a seat as a result of receiving more votes. This is done when either that increase leaves fewer votes to another candidate, or enlarges the total number of votes cast and thus shifts the quota. Such changes alter the rank order of candidates, thus the order in which elections and eliminations take place, and often produce changes in who wins.^[10] Such paradoxes are said to occur regardless of whether the quota is used with largest remainders^[11] or STV.^[12] But such charges of no-show paradox is based on having knowledge of how a vote would be transferred if a candidate is eliminated, who may not have been in real life. Also it is clear that any system that uses ranked votes produces different results if candidates are in different order, which is partly set 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. ^[13] For one thing, transfers have little effect in general on whom is elected, the winners usually being among the front runners in the first round of counting anyway.^[14]

Definition

The value of the exact Droop quota for a ${\displaystyle k}$-winner election is given by the expression:^{[1][15][16][17][18][19][20][excessive citations]}

${\displaystyle {\frac {\text{total votes}}{k+1}}}$

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 ${\textstyle {\frac {\text{total votes}}{2}}}$.^[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.^[21][b]

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, given above. This was rounded off as old STV systems used whole votes. Some older implementations of STV used whole vote reassignment and cannot handle fractional quotas, and so instead will either round up, or add one and truncate:^[4]

${\displaystyle \left\lceil {\frac {\text{total votes}}{k+1}}\right\rceil \ or\left\lfloor {\frac {\text{total votes}}{k+1}}+1\right\rfloor }$

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 for fractional votes. It can cause problems in small elections (see below).^[1][22] However, it is the variant of Droop that is used in legislative elections worldwide today.^[23]

Derivation

The Droop quota can be derived by considering what would happen if k candidates (here called "Droop winners") exceed the exact Droop quota or at least match the exact Droop quota. The goal is to see whether an outside candidate could defeat any of these candidates. 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, they would not be able to defeat any of the Droop winners.^[4]

Example in STV

The following election has 3 seats to be filled by single transferable vote. There are 4 candidates: George Washington, Alexander Hamilton, Thomas Jefferson, and Aaron Burr. There are 102 voters, but two of the votes are spoiled.

The total number of valid votes is 100, and there are 3 seats. The Droop quota (if the original Droop quota is used) is therefore ${\textstyle {\frac {100}{3+1}}+1=26}$.^[18] These votes are as follows:

	45 voters	25 voters	20 voters	10 voters
1	Washington	Jefferson	Burr	Hamilton
2	Hamilton	Burr	Jefferson	Washington
3	Jefferson	Washington	Washington	Jefferson

First preferences for each candidate are tallied:

• Washington: 45 Y
• Jefferson: 25
• Burr: 20
• Hamilton: 10

Washington has 26 or more votes, so he is immediately declared elected. Washington has 19 excess votes that can be transferred to the candidate marked as the second ranking, Hamilton. The tallies therefore become:

• Washington: 26 Y
• Jefferson: 25
• Burr: 20
• Hamilton: 29Y

Hamilton is elected, so his excess votes are redistributed. Thanks to Hamilton's support, Jefferson receives 29 votes to Burr's 20 and is elected.

If it happens that candidates are tied, generally ties are broken by looking at who had more first preference votes, or taking the limit of the results as the quota approaches the Droop quota, or some random method such as a coin toss.

Variety of Droop quotas

Legislators and political observers show variation about what they mean when they use the term Droop quota.^[24] At least six different versions appear in various legal codes or definitions of the quota, all varying by one vote.^[24] 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][22] 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).^[1][22] Common variants of the Droop quota include:

${\displaystyle {\begin{array}{rlrl}{\text{Historical:}}&&\left\lceil {\frac {\text{votes}}{{\text{seats}}+1}}\right\rceil &&{\Bigl \lfloor }{\frac {\text{votes}}{{\text{seats}}+1}}+1{\Bigr \rfloor }&&{\Bigl \lfloor }{\frac {\text{votes}}{{\text{seats}}+1}}{\Bigr \rfloor }+1\\{\text{Accidental:}}&&{\phantom {\Bigl \lfloor }}{\frac {{\text{votes}}+1}{{\text{seats}}+1}}{\phantom {\Bigr \rfloor }}&&{\phantom {\Bigl \lfloor }}{\frac {\text{votes}}{{\text{seats}}+1}}+1{\phantom {\Bigr \rfloor }}\\{\text{Unusual:}}&&\left\lfloor {\frac {\text{votes}}{{\text{seats}}+1}}\right\rfloor &&\left\lfloor {\frac {\text{votes}}{{\text{seats}}+1}}+{\frac {1}{2}}\right\rfloor \end{array}}}$

A quota may be thought to be inadmissable if it allows more to achieve quota than the number of open seats. However preventing such an occurrence is not necessary. The original Droop quota (votes/seats+1, plus 1, rounded down) was traditionally seen as needed in the context of modern fractional transfer systems, and it was believed that any smaller portion of the votes, such as exact Droop, would not work because it would allow one more candidate than there are open seats to reach the quota.^[24] s 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.^[1][22] Due to this misunderstanding, Ireland, Malta and Australia have used Droop's original quota - votes/seats+1, plus 1 - for the last hundred years.

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. In such a situation, rounding the number of votes up (or, alternatively, adding one and rounding down^[c]) introduces as little error as possible, while maintaining the admissibility of the quota, by ensuring that no more can achieve quota than just the number of seats available.^[24][4]

Confusion with the Hare quota

The Droop quota is sometimes confused with the more intuitive Hare quota. While the Droop quota gives the number of votes needed to mathematically guarantee a candidate's election, the Hare quota gives the number of voters represented by each winner in an exactly-proportional system (i.e. one where each vote is represented equally and every vote is used). Unfortunately frequently one or more votes are found to be exhausted and there is no way for the last elected candidate to be elected with Hare.

The confusion between the two quotas may be caused by forgetting candidates who are neither elected nor eliminated may have votes at the end of the counting process.

Comparison with Hare

The Hare quota gives more proportional outcomes on average because it is statistically unbiased.^[9] By contrast, the Droop quota is more biased towards large parties than any other admissible quota.^[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.^[9] However, the Droop quota has the advantage that any party receiving more than half the votes will receive at least half of all seats.

See also

• List of democracy and elections-related topics

Notes

1. Some authors use the terms "Newland-Britton quota" or "exact Droop quota" to refer to the quantity described in this article, and reserve the term "Droop quota" for the archaic or rounded form of the Droop quota (the original found in the works of Henry Droop).^[2]
2. By abuse of notation, mathematicians may write the quota as votes⁄k+1 + 𝜖, where ${\displaystyle \epsilon >0}$ is taken arbitrarily close to 0 (i.e. as a limit), which allows breaking some ties for the last seat.
3. The two are only different when the quotient produced by the number of votes divided by one more than the number of seats is exactly a whole number.