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 a full seat in a legislature.

The Droop quota generalizes the concept of a majority to multiwinner elections. Just as a candidate with a majority (more than half of all votes) is guaranteed to be declared winner in a one-on-one election, a candidate who holds more than one Droop quota's worth of votes at any point 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, votes not needed by a candidate who has been declared elected. In proportional quota-rule systems such as STV and CPO-STV, these excess votes can be transferred to other candidates, preventing them from being wasted. The Droop quota was first devised 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 the Republic of Ireland, Northern Ireland, Malta, and Australia. It is also used in South Africa to allocate seats by the largest remainder method.

Standard Formula

The exact form of the Droop quota for a $k$-winner election is given by the formula:

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

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 strictly more than 50% of the vote, i.e. ${\textstyle {\frac {\text{total votes}}{2}}}$.

Sometimes, the Droop quota is written as a share (i.e. percentage) of the total votes, in which case it has value 1⁄k+1. A candidate holding a full Droop quota's worth of votes is guaranteed to win a seat.

Derivation

The Droop quota can be derived by considering what would happen if k candidates (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, each quota winner's share of the vote exceeds 1⁄k+1, while all unelected candidates' share of the vote, taken together, is less than 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 ${\textstyle {\frac {100}{3+1}}=25}$. 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, instead of a clear win for Jefferson, requiring a tiebreaker.

Nonstandard variants

There is a great deal of confusion among legislators and political observers about the exact definition of the Droop quota. At least six different versions appear in various legal codes or definitions of the quota, all varying by roughly one vote.

Such variants are typically considered problematic because they violate scale-invariance (i.e. the election depends on the total number of ballots). They have been discouraged by the ERS handbook since 1976. These variants are given by:

${\begin{array}{rlrl}{\text{Rounded up:}}&&{\Bigl \lfloor }{\frac {\text{votes}}{{\text{seats}}+1}}+1{\Bigr \rfloor }&&\left\lceil {\frac {\text{votes}}{{\text{seats}}+1}}\right\rceil \\{\text{Nonstandard:}}&&{\phantom {\Bigl \lfloor }}{\frac {\text{votes}}{{\text{seats}}+1}}+1{\phantom {\Bigr \rfloor }}&&{\phantom {\Bigl \lfloor }}{\frac {{\text{votes}}+1}{{\text{seats}}+1}}{\phantom {\Bigr \rfloor }}\\{\text{Inadmissible:}}&&\left\lfloor {\frac {\text{votes}}{{\text{seats}}+1}}+{\frac {1}{2}}\right\rfloor &&\left\lfloor {\frac {\text{votes}}{{\text{seats}}+1}}\right\rfloor \end{array}}$

The first two variants are from Droop himself, who was working with whole numbers of ballots (as Hare's proposal involved drawing ballots at random from a pile, rather than transferring a weighted fraction as in modern methods). As a result, Droop's writings used a rounded-up version of his quota (sometimes called the rounded Droop quota). Rounding to an integer is not necessary when working with non-random variants, and doing so can cause substantial headaches in small elections.

The middle two variants are nonstandard and arose, although they are still admissible quotas, i.e. they will never assign too many or too few seats.

The last two variants are simply too small and as a result can apportion more seats than there are available, making them disproportional.

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

Confusion with the Hare quota

The Droop quota is often confused with the more intuitive Hare quota. The Droop quota gives the number of voters needed to mathematically guarantee a candidate's election. By contrast, the Hare quota gives the number of voters who are represented by each winner in an ideally-proportional system, i.e. one where every voter is represented equally.

The confusion between the two quotas originates from a fencepost error, caused by forgetting that unelected candidates can also have votes at the end of the counting process. In the case of a single-winner election, using 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.

See also

• List of democracy and elections-related topics

Further reading

• Droop, Henry Richmond (1869). On the Political and Social Effects of Different Methods of Electing Representatives. London.{{cite book}}: CS1 maint: location missing publisher (link)
• Droop, Henry Richmond (1881). "On methods of electing representatives" (PDF). Journal of the Statistical Society of London. 44 (2): 141–196 [Discussion, 197–202]. doi:10.2307/2339223. JSTOR 2339223. Reprinted in Voting matters Issue 24 (October 2007) pp. 7–46.