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 needed for a party or candidate to guarantee they will win at least one seat in a legislature.

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 who holds 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, 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 used 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.

Standard Formula
The exact 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 $\frac{\text{total votes}}{2}$.

Sometimes, the Droop quota is written as a share of all votes, in which case it has value $1/k+1$. A candidate who, at any point, holds more than one Droop quota's worth of votes is therefore guaranteed to win a seat.

Archaic Droop quota
Modern variants of STV use fractional transfers of ballots to eliminate uncertainty. However, STV elections with whole vote reassignment cannot handle fractional quotas, and so instead will round up:

$$\left\lceil \frac{\text{total votes}}{k+1} \right\rceil $$

This variant of the quota should not be used in the context of modern elections that allow for fractional votes, where it can cause problems in small elections (see below).

Derivation
The Droop quota can be derived by considering what would happen if $k$ candidates (who we call "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. Newland and Britton noted that while a tie for the last seat is possible, such a situation can occur no matter which quota is used.

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 $ \frac{100}{3+1} = 25 $. These votes are as follows:

First preferences for each candidate are tallied:
 * Washington: 45 ✅
 * 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 ✅
 * Hamilton: 30✅
 * 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.

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. Such versions have been recognized as incorrect by the ERS handbook since 1976, as they can easily cause a failure of proportionality in small elections. Common variants include:

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

The first variant in the top-left arose from Droop's discussion of the quota in the context of Hare's original proposal for STV, which assumed a whole number of ballots would be transferred at random. In such a situation, a fractional quota would be physically impossible, leading Droop to describe the next-best value as "the whole number next greater than the quotient obtained by dividing $$m V$$, the number of votes, by $$n+1$$" (where n is the number of seats). In such a situation, rounding the number of ballots upwards introduces as little error as possible, while maintaining the admissibility of the quota.

There is a common misconception that the archaic form of the Droop quota is still needed in the context of modern fractional transfer systems, because otherwise it is possible to "elect" one more candidate than there are winners. However, as Newland and Britton noted in 1974, this is not the case: the situation where the last two candidates elected both receive a Droop quota of votes is simply a tie between the last two winners which must be broken, and this situation can occur regardless of which quota is used.

Spoiled ballots should not be included when calculating the Droop quota. However, some jurisdictions fail to correctly specify this in their election administration laws.

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 ideally-proportional system, i.e. one where every voter is treated equally. As a result, the Hare quota gives more proportional outcomes, while the Droop quota is more biased towards large parties than any other admissible quota.

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.

The Droop quota is today the most popular quota for STV elections.