# Droop quota

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 preventing them from being wasted.
The Droop quota was first suggested by the English lawyer and mathematician Henry Richmond Droop (1831–1884) as an alternative to the Hare quota.
Today, 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.
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. This occurs regardless of whether the quota is used with largest remainders or STV.

## Definition

The exact value of the Droop quota for a $k$-winner election is given by the expression:
${\frac {\text{total votes}}{k+1}}$
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

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:
$\left\lceil {\frac {\text{total votes}}{k+1}}\right\rceil \approx \left\lfloor {\frac {\text{total votes}}{k+1}}+1\right\rfloor$
This variant of the quota is generally not recommended in the context of modern elections that allow for fractional votes, where it can cause problems in small elections (see below). However, it is the most commonly-used definition in legislative codes worldwide.

### Derivation

The Droop quota can be derived by considering what would happen if k candidates (here called "Droop winners") have exceeded the Droop quota. The goal is to identify whether an outside candidate could defeat any of these candidates. In this situation, if each quota winner's share of the vote equals 1⁄k+1, while 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.

## 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 is therefore 100/(3+1)=25{\textstyle {\frac {100}{3+1}}=25}. These votes are as follows:
| | 45 voters | 20 voters | 25 voters | 10 voters |
| 1 | Washington | Burr | Jefferson | Hamilton |
| 2 | Hamilton | Jefferson | Burr | Washington |
| 3 | Jefferson | Washington | Washington | Jefferson |
First preferences for each candidate are tallied:
* Washington: 45 Y
* Hamilton: 10
* Burr: 20
* Jefferson: 25
Only Washington has strictly more than 25 votes. As a result, he is immediately elected. Washington has 20 excess votes that can be transferred to their second choice, Hamilton. The tallies therefore become:
* Washington: 25 Y
* Hamilton: 30Y
* Burr: 20
* Jefferson: 25
Hamilton is elected, so his excess votes are redistributed. 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 taking the limit of the results as the quota approaches the exact Droop quota.

## Common errors

There is a great deal of confusion among legislators and political observers about the correct form of the Droop quota. At least six different versions appear in various legal codes or definitions of the quota, all varying by one vote. The ERS handbook on STV has advised against such variants since at least 1976, as they can cause problems with proportionality in small elections. 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). Common variants of the Droop quota include:
${\begin{array}{rlrl}{\text{Historical:}}&&\left\lceil {\frac {\text{votes}}{{\text{seats}}+1}}\right\rceil &&{\Bigl \lfloor }{\frac {\text{votes}}{{\text{seats}}+1}}+1{\Bigr \rfloor }&&{\Bigl \lfloor }{\frac {\text{votes}}{{\text{seats}}+1}}{\Bigr \rfloor }+1\\{\text{Accidental:}}&&{\phantom {\Bigl \lfloor }}{\frac {{\text{votes}}+1}{{\text{seats}}+1}}{\phantom {\Bigr \rfloor }}&&{\phantom {\Bigl \lfloor }}{\frac {\text{votes}}{{\text{seats}}+1}}+1{\phantom {\Bigr \rfloor }}\\{\text{Inadmissible:}}&&\left\lfloor {\frac {\text{votes}}{{\text{seats}}+1}}\right\rfloor &&\left\lfloor {\frac {\text{votes}}{{\text{seats}}+1}}+{\frac {1}{2}}\right\rfloor \end{array}}$
A quota being "inadmissable" refers to the possibility that more could achieve quota than the number of open seats. However preventing such an occurrence is not necessary. The archaic rounded-off form of the 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 be possible for one more candidate than there are winners to reach the quota. 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. 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) 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.

### Confusion with the Hare quota

The Droop quota is often confused with the more intuitive Hare quota. While the Droop quota gives the number of voters 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 voter is represented equally).
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. In the case of a single-winner election, misapplying the Hare quota would lead to the incorrect conclusion that a candidate must receive 100% of the vote to be certain of victory; in reality, any votes exceeding a bare majority are excess votes.

## Comparison with Hare

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