Droop quota

In the study of electoral systems, the Droop quota (sometimes called the Hagenbach-Bischoff) 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.^[1]^[2]

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 at least a Droop quota's worth of votes is guaranteed to win a seat in a multiwinner election. (Even if there is no official electoral threshold to that effect, there is no way for all the seats to be filled by others each having more votes than that.)^[2]

Besides establishing winners, the Droop quota is used in many electoral systems to establish how many votes remain with a successful candidate and thus how many of their votes are surplus votes and available to be transferred to other candidates in order to prevent them from being wasted. Such transfers are done in STV, quota-based proportional Largest remainder method and expanding approvals systems.^[2]

The Droop quota was first devised by the English lawyer and mathematician Henry Richmond Droop (1831–1884), as an alternative to the Hare quota.^[3] As Droop put it, "the whole number next greater than the quotient obtained by dividing mV , the number of votes, by n + 1, will be called the quota."^[4]

Later, Eduard Hagenbach-Bischoff (1833-1910) also wrote on the quota in his studies, Die Frage der Einführung einer Proportionalvertretung statt des absoluten Mehres (1888) and Die Verteilungsrechnung beim Basler Gesetz nach dem Grundsatz der Verhältniswahl (1905). Both Droop and Hagenbach-Bischoff gave their quota as some number just larger than votes/seats plus 1.^[5]

Today, the Droop quota is used in almost all STV elections, including those in Australia,^[6] the Republic of Ireland, Northern Ireland, and Malta.^[7] It is also used in South Africa to allocate seats by the largest remainder method.^[8]^[9]

Standard Formula

The Droop quota for a $k$ -winner election is:^[10]^[11]^[12]^[13]^[14]^[15]

${\frac {\text{total votes}}{k+1}}$ plus 1 or rounded up.

Sometimes, the Droop quota is written as a share of all votes, in which case it has value $.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1⁄k+1$ .

A candidate who, at any point, holds at least a Droop quota's worth of votes is guaranteed to win a seat.^[16]

Derivation

The Droop quota can be derived by considering what would happen if $k$ candidates (who we call "Droop winners") have achieved 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 $1 ⁄ k +1$ plus 1, while all unelected candidates' share of the vote, taken together, would be 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.^[2] 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.^[10]^[17]

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 104 voters, but two of the votes are spoiled.

The total number of valid votes is 102, and there are 3 seats. The Droop quota is therefore ${\textstyle {\frac {102}{3+1}}=25.5}$ . Rounded up, that is 26.^[18] These votes are as follows:

preferences marked	45 voters	23 voters	22 voters	10 voters
1	Washington	Jefferson	Burr	Hamilton
2	Hamilton	Burr	Jefferson	Washington
3	Jefferson	Washington	Washington	Jefferson

First preferences for each candidate are tallied:

Washington: 45 Y
Jefferson: 23
Burr: 22
Hamilton: 10

Only Washington has at least 26 votes. As a result, he is declared elected. Washington has 19 excess votes that are now transferred to their second choice, Hamilton. The tallies therefore become:

Washington: 26 Y
Jefferson: 23
Burr: 22
Hamilton: 29Y

Hamilton is elected. There is still one seat remaining to be filled so his excess votes are transferred. Thanks to the four vote transfer from Hamilton, Jefferson accumulates 27 votes to Burr's 22 and is declared elected. That fills the last empty seat.

If ties happen, pre-set rules deal with them, usually by reference to whom had the most first-preference votes.

Under plurality rules (such as block voting or SNTV), Burr would have been elected to a seat in the first round. But under STV he did not collect any transfers and Jefferson was proven to be the more generally supported candidate.

Burr, as a representative of a minority, would have been elected if his supporters numbered 26, but as they did not and as he did not receive any transfers from others, he was not elected and his voice was not heard in the chamber following the election.

Different versions of Droop

At least six different versions of the Droop quota appear in various legal codes or definitions of the quota.^[19] Some claim that, depending on which version is used, a failure of proportionality may arise in small elections.^[10]^[17] Common variants include:

${\begin{array}{rlrl}{\text{Droop:}}&&{\Bigl \lfloor }{\frac {\text{votes}}{{\text{seats}}+1}}+1{\Bigr \rfloor }\\{\text{Hagenbach-Bischoff:}}&&\left\lceil {\frac {\text{votes}}{{\text{seats}}+1}}\right\rceil {\text{or sometimes:}}{\Bigl \lfloor }{\frac {\text{votes}}{{\text{seats}}+1}}{\Bigr \rfloor }+1\\{\text{Unusual:}}&&\left\lfloor {\frac {\text{votes}}{{\text{seats}}+1}}\right\rfloor &&\left\lfloor {\frac {\text{votes}}{{\text{seats}}+1}}+{\frac {1}{2}}\right\rfloor \\{\text{Accidental:}}&&{\phantom {\Bigl \lfloor }}{\frac {{\text{votes}}+1}{{\text{seats}}+1}}{\phantom {\Bigr \rfloor }}\end{array}}$

Droop and Hagenbach-Bischoff derived new quota as a replacement for the Hare quota (votes/seats). Their quota was meant to produce more proportional results by having the quota as low as thought to be possible. Their quota was basically a number just larger than votes/seats plus 1.

This formula may yield a fraction, which was a problem as early STV systems did not use fractions. Droop went to votes/seats plus 1, plus 1, rounded down, as shown.^[2] Hagenbach-Bischoff went to votes/seats +1, rounded up or another formula, as shown. Hagenbach-Bischoff proposed a quota that is "the whole number next greater than the quotient obtained by dividing $mV$ , the number of votes, by $n+1$ " (where n is the number of seats).^[19]

Due to the use of fractions in many STV systems today, rounded-off variants of the Droop and Hagenbach-Bischoff quota may not be needed.

The Britton or Newland-Britton quota^[10]^[a], sometimes called the "exact Droop" quota, is not rounded off and is slightly smaller than the Droop quota as Droop originally proposed it. Its formula is vote/seats plus 1, with no rounding off or addition. Although it is smaller than Droop, which is billed as the lowest possible workable quota, N-B is said to be workable according to its originators, R.A. Newland and F.S. Britton.^[22]

It is un-necessary to ensure the quota is larger than vote/seats plus 1. When using the exact Droop quota (votes/seats plus 1) or any variant where the quota is slightly less than votes/seats plus 1, such as in an unusual formula shown above (votes/seats plus 1, rounded down), it is possible for one more candidate to reach the quota than there are seats to fill.^[19] 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, it would mean a tie. Rules are in place to break a tie, and ties can occur regardless of which quota is used.^[10]^[17] Even the Imperiali quota, a quota smaller than Droop, can work as long as rules indicate that relative plurality or some other method is to be used where more achieve quota than the number of empty seats.

Spoiled ballots should not be included when calculating the Droop quota. Some jurisdictions fail to specify in their election administration laws that valid votes should be the base for determining quota.^{[citation needed]}

Confusion with the Hare quota

The Droop quota is often confused with the 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 by exactly linear proportionality.

As a result, the Hare quota is said to give somewhat more proportional outcomes,^[23] by having large parties waste more votes and thus promoting representation of smaller parties. But sometimes under Hare a majority group will be denied the majority of seats, thus denying the principle of majority rule in such settings as a city council elected at-large. By contrast, the Droop quota is more biased towards large parties than any other admissible quota.^[23] The Droop quota sometimes allows a party representing less than half of the voters to take a majority of seats in a constituency.^[23]^[2]

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

Notes

^ 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).^[20]^[21]

References

^ "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{{citation}}: CS1 maint: location (link)
^ ^a ^b ^c ^d ^e ^f 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.
^ Droop, Henry Richmond, On methods of electing representatives. London, Macmillan and co., 1868
^ Henry R. Droop, "On Methods of Electing Representatives," Journal of the Statistical Society of London, Vol. 44, No. 2. (Jun., 1881), pp. 141–202 (Reprinted in Voting matters, No. 24 (Oct., 2007), pp. 7-46)
^ DANČIŠIN. MISINTERPRETATION OF THE HAGENBACH-BISCHOFF QUOTA.(online). accessed December 22, 2014
^ "Proportional Representation Voting Systems of Australia's Parliaments". Electoral Council of Australia & New Zealand. Archived from the original on 6 July 2024.
^ https://electoral.gov.mt/ElectionResults/General
^ Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham ; New York : Springer. ISBN 978-3-319-03855-1.
^ "IFES Election Guide | Elections: South African National Assembly 2014 General". www.electionguide.org. Retrieved 2024-06-02.
^ ^a ^b ^c ^d ^e Lundell, Jonathan; Hill, ID (October 2007). "Notes on the Droop quota" (PDF). Voting Matters (24): 3–6.
^ Woodall, Douglass. "Properties of Preferential Election Rules". Voting Matters (3).
^ 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.
^ 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.
^ 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.
^ 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
^ 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.
^ ^a ^b ^c Newland, Robert A. (June 1980). "Droop quota and D'Hondt rule". Representation. 20 (80): 21–22. doi:10.1080/00344898008459290. ISSN 0034-4893.
^ 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.
^ ^a ^b ^c Dančišin, Vladimír (2013). "Misinterpretation of the Hagenbach-Bischoff quota". Annales Scientia Politica. 2 (1): 76.
^ 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.
^ 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.
^ R A Newland and F S Britton (1976). How to conduct an election by the Single Transferable Vote (second edition). Electoral Reform Society of Great Britain and Ireland.
^ ^a ^b ^c 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.

v t e Types of majority and other electoral quotas
Single winner (majority)	Plurality Majority Absolute majority Supermajority Double majority
Electoral quotas	Hare quota Droop quota Imperiali quota (pseudoquota)
Other	Electoral threshold