Droop quota

In the study of electoral systems, the Droop quota (sometimes called the Hagenbach-Bischoff, Britton, or Newland-Britton quota ^{[1] [lower-alpha 1]} ) is the minimum number of votes needed for a party or candidate to guarantee they will win at least one seat in a legislature. ^{[3] [4]}

The Droop quota generalizes the concept of a majority to multiwinner elections. Just as a candidate with a majority (any number exceeding 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. ^[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-rule systems such as STV or expanding approvals, these excess votes can be transferred to other candidates, preventing 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 the Republic of Ireland, Northern Ireland, Malta, and Australia.^{[ citation needed ]} It is also used in South Africa to allocate seats by one of the largest remainder methods. ^{[5] [6]}

Standard Formula

The exact Droop quota for a ${\displaystyle k}$-winner election is given by the expression: ^{[1] [7] [8] [9] [10]}

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

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. ${\textstyle {\frac {\text{total votes}}{2}}}$. ^[1]

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. ^[11]

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 used: ^[4]

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

This variant on the quota should not be used in the context of modern STV elections, where it causes substantial problems. ^{[1] [12]}

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. ^[4] 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. ^{[1] [12]}

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 ${\textstyle {\frac {100}{3+1}}=25}$. ^[13] 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.

Common errors

There is a great deal of confusion among legislators and political observers about the correct form of the Droop quota. ^[14] At least six different versions appear in various legal codes or definitions of the quota, all varying by one vote. ^[14] 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. ^{[1] [12]} Common variants include:

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

The first variant, in the top-left, arises 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. ^[4] In such a situation, a fractional quota would be physically impossible, leading Droop to describe the next-best quota as "the whole number next greater than the quotient obtained by dividing ${\displaystyle mV}$, the number of votes, by ${\displaystyle n+1}$" (where n is the number of seats). ^[14] In such a situation, rounding the number of ballots upwards introduces the minimal error possible. Rounding down or rounding normally would instead create an infeasible quota. ^[14]

There is a common misconception that the archaic form of the Droop quota is still necessary in the context of modern fractional transfer systems, because otherwise it is possible to "elect" more candidates than winners in the exceptional case of a tied vote. ^[14] 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, which can occur under any system or quota. ^{[1] [12]}

Spoiled ballots should not be included when calculating the Droop quota. However, some jurisdictions fail to correctly specify this in their election administration laws.^{[ citation needed ]}

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 tends to give more proportional outcomes, ^[15] while the Droop quota is more biased towards large parties than any other admissible quota. ^[15]

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. ^[4]

The Droop quota is today the most popular quota for STV elections.^{[ citation needed ]}

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 rounded Droop quota (the original form in the works of Henry Droop). ^[2]

References

1. Lundell, Jonathan; Hill, ID (October 2007). "Notes on the Droop quota" (PDF). Voting Matters (24): 3–6.
2. Pukelsheim, Friedrich (2017). "Quota Methods of Apportionment: Divide and Rank". Proportional Representation. pp. 95–105. doi:10.1007/978-3-319-64707-4_5. ISBN   978-3-319-64706-7.
3. "Droop Quota", The Encyclopedia of Political Science, 2300 N Street, NW, Suite 800, Washington DC 20037 United States: CQ Press, 2011, doi:10.4135/9781608712434.n455, ISBN   978-1-933116-44-0 , retrieved 2024-05-03
4. Droop, Henry Richmond (1881). "On methods of electing representatives" (PDF). Journal of the Statistical Society of London . 44 (2): 141–196 [Discussion, 197–202] [33 (176)]. doi:10.2307/2339223. JSTOR   2339223. Reprinted in Voting matters Issue 24 (October 2007) pp. 7–46.
5. Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham ; New York : Springer. ISBN   978-3-319-03855-1.
6. "IFES Election Guide | Elections: South African National Assembly 2014 General". www.electionguide.org. Retrieved 2024-06-02.
7. Woodall, Douglass. "Properties of Preferential Election Rules". Voting Matters (3).
8. Lee, Kap-Yun (1999). "The Votes Mattered: Decreasing Party Support under the Two-Member-District SNTV in Korea (1973–1978)". In Grofman, Bernard; Lee, Sung-Chull; Winckler, Edwin; Woodall, Brian (eds.). Elections in Japan, Korea, and Taiwan Under the Single Non-Transferable Vote: The Comparative Study of an Embedded Institution. University of Michigan Press. ISBN   9780472109098.
9. Gallagher, Michael (October 1992). "Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities". British Journal of Political Science. 22 (4): 469–496. doi:10.1017/s0007123400006499.
10. Giannetti, Daniela; Grofman, Bernard (1 February 2011). "Appendix E: Glossary of Electoral System Terms". A Natural Experiment on Electoral Law Reform: Evaluating the Long Run Consequences of 1990s Electoral Reform in Italy and Japan (PDF). Springer Science & Business Media. ISBN   978-1-4419-7228-6.
11. Grofman, Bernard (23 November 1999). "SNTV, STV, and Single-Member-District Systems: Theoretical Comparisons and Contrasts". Elections in Japan, Korea, and Taiwan Under the Single Non-Transferable Vote: The Comparative Study of an Embedded Institution. University of Michigan Press. ISBN   978-0-472-10909-8.
12. Newland, Robert A. (June 1980). "Droop quota and D'Hondt rule". Representation. 20 (80): 21–22. doi:10.1080/00344898008459290. ISSN   0034-4893.
13. Gallagher, Michael (October 1992). "Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities". British Journal of Political Science. 22 (4): 469–496. doi:10.1017/s0007123400006499.
14. Dančišin, Vladimír (2013). "Misinterpretation of the Hagenbach-Bischoff quota". Annales Scientia Politica. 2 (1): 76.
15. Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Favoring Some at the Expense of Others: Seat Biases", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 127–147, doi:10.1007/978-3-319-64707-4_7, ISBN   978-3-319-64707-4 , retrieved 2024-05-10

Sources

• Robert, Henry M.; et al. (2011). Robert's Rules of Order Newly Revised (11th ed.). Philadelphia, Pennsylvania: Da Capo Press. p. 4. ISBN   978-0-306-82020-5.

Further reading

• Droop, Henry Richmond (1869). On the Political and Social Effects of Different Methods of Electing Representatives. London.