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 supporters a party or candidate needs to receive in a district to guarantee they will win at least one seat in a legislature. ^{[3] [4]}

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. ^[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, 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 Australia, ^[5] the Republic of Ireland, Northern Ireland, and Malta.^{[ citation needed ]} It is also used in South Africa to allocate seats by the largest remainder method. ^{[6] [7]}

Standard Formula

The exact Droop quota for a ${\displaystyle k}$-winner election is given by the expression: ^{[1] [8] [9] [10] [11] [12]}

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

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]

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. ^[13]

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: ^[4]

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

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). ^{[1] [14]}

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] [14]}

Example in STV

The following election has 3 seats to be filled by single transferable vote. The whole-vote version of STV, same as used in Ireland and Malta, is used here. Four candidates are in the running for the three seats: 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 ${\textstyle {\frac {100}{3+1}}=25}$. ^[15] These votes are as follows:

	45 voters	21 voters	24 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: 21
• Jefferson: 24

Only Washington has at least 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: 21
• Jefferson: 24

Hamilton is elected, so his surplus votes are transferred to Jefferson, the next usable preference marked on them. Jefferson then has 29 votes. Because this is more than Burr's 21, Jefferson is elected.

If all of Hamilton's supporters had instead backed Burr, Burr would have won the last seat, by being found to be the more generally acceptable candidate.

Common errors

There is a great deal of confusion among legislators and political observers about the correct form of the Droop quota. ^[16] At least six different versions appear in various legal codes or definitions of the quota, all varying by one vote. ^[16] Some say such versions are incorrect, and can cause a failure of proportionality in small elections. ^{[1] [14]} Common variants include:

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

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, and fractional votes would not be used. ^[4] 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 ${\displaystyle mV}$, the number of votes, by ${\displaystyle n+1}$" (where n is the number of seats). ^[16] In such a situation, rounding the number of votes upwards introduces as little error as possible, while maintaining the admissibility of the quota. ^[16]

In the same vein, any quota that is slightly larger than seats/votes plus 1 is workable in the way Droop envisioned its use. But an amount smaller than this archaic form of Droop can be used as long as rules are in place to deal with any problematic tie.

Some hold the misconception that the archaic form of the Droop quota (sometimes written (votes/seats plus 1) plus 1 is still needed in the context of modern fractional transfer systems, because if Droop as defined above in this Wikipedia article is used, it is possible for one more candidate than there are winners to achieve quota. ^[16] However, as 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] [14]} As well, if at any time in the vote count process, even one vote is found to be exhausted, then there is no possibility of more candidates achieving quota than the number of seats. (Even where a voter must mark full preferences, it is possible for a vote to become exhausted, as occurred in Australia.)

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 gives more proportional outcomes, ^[17] although sometimes under Hare a majority group will be denied the majority of seats. On the other hand the Droop quota is more biased towards large parties than any other admissible quota. ^[17]

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] (It is possible under certain circumstances to be elected under STV with less than quota.)

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 archaic or rounded form of the Droop quota (the original found 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. "Proportional Representation Voting Systems of Australia's Parliaments". Electoral Council of Australia & New Zealand. Archived from the original on 6 July 2024.
6. Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham ; New York : Springer. ISBN   978-3-319-03855-1.
7. "IFES Election Guide | Elections: South African National Assembly 2014 General". www.electionguide.org. Retrieved 2024-06-02.
8. Woodall, Douglass. "Properties of Preferential Election Rules". Voting Matters (3).
9. 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.
10. 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.
11. 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.
12. Graham-Squire, Adam; Jones, Matthew I.; McCune, David (2024-08-07), New fairness criteria for truncated ballots in multi-winner ranked-choice elections, arXiv: 2408.03926 , retrieved 2024-08-18
13. 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.
14. Newland, Robert A. (June 1980). "Droop quota and D'Hondt rule". Representation. 20 (80): 21–22. doi:10.1080/00344898008459290. ISSN   0034-4893.
15. 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.
16. Dančišin, Vladimír (2013). "Misinterpretation of the Hagenbach-Bischoff quota". Annales Scientia Politica. 2 (1): 76.
17. 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.