Editing Droop quota

{{Short description|Quantity of votes in election studies}}
{{Electoral systems}}
In the study of [[Electoral system|electoral systems]], the '''Droop quota''' (sometimes called the [[Eduard Hagenbach-Bischoff|'''Hagenbach-Bischoff''']], '''Britton''', or '''Newland-Britton quota'''<ref name="mw-2007">{{Cite journal |last1=Lundell |first1=Jonathan |last2=Hill |first2=ID |title=Notes on the Droop quota |url=http://www.mcdougall.org.uk/voting-matters/ISSUE24/ISSUE24.pdf#page=7 |journal=Voting Matters |publication-date=October 2007 |issue=24 |pages=3–6}}</ref>{{efn|name="exact"|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).<ref name="pukelsheim-2017-quota">{{cite book |doi=10.1007/978-3-319-64707-4_5 |chapter=Quota Methods of Apportionment: Divide and Rank |title=Proportional Representation |date=2017 |last1=Pukelsheim |first1=Friedrich |pages=95–105 |isbn=978-3-319-64706-7 }}</ref>}}) is the [[Infimum|minimum]] number of votes a party or candidate needs to receive in a district to guarantee they will win at least one seat.<ref>{{Citation |title=Droop Quota |date=2011 |encyclopedia=The Encyclopedia of Political Science |url=http://dx.doi.org/10.4135/9781608712434.n455 |access-date=2024-05-03 |place=2300 N Street, NW, Suite 800, Washington DC 20037 United States |publisher=CQ Press|doi=10.4135/9781608712434.n455 |isbn=978-1-933116-44-0 |url-access=subscription }}</ref><ref name="droop-1881">{{cite journal |last=Droop |first=Henry Richmond |year=1881 |title=On methods of electing representatives |url=http://www.votingmatters.org.uk/ISSUE24/I24P3.pdf |journal=[[Journal of the Statistical Society of London]] |volume=44 |issue=2 |pages=141–196 [Discussion, 197–202] [33 (176)] |doi=10.2307/2339223 |jstor=2339223}} Reprinted in ''[[Voting matters]] Issue 24'' (October 2007) pp.&nbsp;7–46.</ref>

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 with more than a Droop quota's worth of votes is guaranteed to win a seat in a [[Multiwinner voting|multiwinner election]].<ref name="droop-1881"></ref>

Besides establishing winners, the Droop quota is used to define the number of [[excess vote]]s, i.e. votes not needed by a candidate who has been declared elected. In proportional [[electoral quota|quota]]-based systems such as [[Single transferable vote|STV]] or [[expanding approvals rule|expanding approvals]], these excess votes can be transferred to other candidates to prevent them from [[Wasted vote|being wasted]].<ref name="droop-1881"></ref>

The Droop quota was first suggested by the English lawyer and mathematician [[Henry Richmond Droop]] (1831–1884) as an alternative to the [[Hare quota]].<ref name="droop-1881"></ref>

Today, the Droop quota is used in almost all STV elections, including those in [[Australia]],<ref>{{Cite web |title=Proportional Representation Voting Systems of Australia's Parliaments |url=https://www.ecanz.gov.au/electoral-systems/proportional |url-status=live |archive-url=https://web.archive.org/web/20240706104711/https://www.ecanz.gov.au/electoral-systems/proportional |archive-date=6 July 2024 |website=Electoral Council of Australia & New Zealand}}</ref> the [[Republic of Ireland]], [[Northern Ireland]], and [[Malta]].<ref>{{Cite web|url=https://electoral.gov.mt/ElectionResults/General|title=Electoral Commission of Malta|website=electoral.gov.mt|accessdate=2025-01-20}}</ref> It is also used in [[South Africa]] to allocate seats by the [[largest remainder method]].<ref>{{Cite book |last=Pukelsheim |first=Friedrich |url=http://archive.org/details/proportionalrepr0000puke |title=Proportional representation : apportionment methods and their applications |date=2014 |publisher=Cham ; New York : Springer |others=Internet Archive |isbn=978-3-319-03855-1}}</ref><ref>{{Cite web |title=IFES Election Guide {{!}} Elections: South African National Assembly 2014 General |url=https://www.electionguide.org/elections/id/2721/ |access-date=2024-06-02 |website=www.electionguide.org}}</ref>

Although common, the quota's use in [[proportional representation]] has been criticized both for its bias toward large parties<ref name="pukelsheim-2017-bias" /> and for its ability to create [[no-show paradox|no-show paradoxes]], situations where a candidate or party loses a seat as a result of having won too ''many'' votes. This occurs regardless of whether the quota is used with [[Largest remainders method|largest remainders]]<ref>{{Cite journal |last=Dančišin |first=Vladimír |date=2017-01-01 |title=No-show paradox in Slovak party-list proportional system |url=https://www.degruyter.com/document/doi/10.1515/humaff-2017-0002/html |journal=Human Affairs |language=en |volume=27 |issue=1 |pages=15–21 |doi=10.1515/humaff-2017-0002 |issn=1337-401X|url-access=subscription }}</ref> or [[single transferable vote|STV]].<ref>{{Cite journal |last=Ray |first=Dipankar |date=1983-07-01 |title=Hare's voting scheme and negative responsiveness |url=https://www.sciencedirect.com/science/article/abs/pii/016548968390032X |journal=Mathematical Social Sciences |volume=4 |issue=3 |pages=301–303 |doi=10.1016/0165-4896(83)90032-X |issn=0165-4896|url-access=subscription }}</ref>

== Definition ==

The exact value of the Droop quota for a <math>k</math>-winner election is given by the expression:<ref name="mw-2007" /><ref>{{Cite book |last1=Delemazure |first1=Théo |last2=Peters |first2=Dominik |chapter=Generalizing Instant Runoff Voting to Allow Indifferences |date=2024-12-17 |title=Proceedings of the 25th ACM Conference on Economics and Computation |chapter-url=https://dl.acm.org/doi/10.1145/3670865.3673501 |series=EC '24 |location=New York, NY, USA |publisher=Association for Computing Machinery |at=Footnote 12 |doi=10.1145/3670865.3673501 |isbn=979-8-4007-0704-9|arxiv=2404.11407 }}</ref><ref>{{Cite journal |last=Woodall |first=Douglass |title=Properties of Preferential Election Rules |url=http://www.mcdougall.org.uk/VM/ISSUE3/P5.HTM |journal=Voting Matters |issue=3}}</ref><ref>{{cite book|chapter-url=https://books.google.com/books?id=ry26lbfP16sC&pg=PA252|last=Lee|first=Kap-Yun|chapter=The Votes Mattered: Decreasing Party Support under the Two-Member-District SNTV in Korea (1973–1978)|editor-first=Bernard|editor-last=Grofman|editor2-first=Sung-Chull|editor2-last=Lee|editor3-first=Edwin|editor3-last=Winckler|editor4-first=Brian|editor4-last=Woodall|title=Elections in Japan, Korea, and Taiwan Under the Single Non-Transferable Vote: The Comparative Study of an Embedded Institution|publisher=University of Michigan Press|year=1999|isbn=9780472109098}}</ref><ref name="gallagher-1992">{{cite journal |last1=Gallagher |first1=Michael |title=Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities |journal=British Journal of Political Science |date=October 1992 |volume=22 |issue=4 |pages=469–496 |doi=10.1017/s0007123400006499}}</ref><ref>{{cite book |last1=Giannetti |first1=Daniela |last2=Grofman |first2=Bernard |title=A Natural Experiment on Electoral Law Reform: Evaluating the Long Run Consequences of 1990s Electoral Reform in Italy and Japan |date=1 February 2011 |publisher=Springer Science & Business Media |isbn=978-1-4419-7228-6 |url=http://ndl.ethernet.edu.et/bitstream/123456789/77460/1/21.pdf#page=138 |language=en |chapter=Appendix E: Glossary of Electoral System Terms | page=134 |quote='''Droop quota''' of votes (for list PR systems, q.v., or single transferable vote, q.v.). This is equal to <math>E / (M + 1)</math>, where <math>E</math> is the size of the actual electorate and <math>M+1</math> is the number of seats to be filled.}}</ref><ref>{{Citation |last1=Graham-Squire |first1=Adam |title=New fairness criteria for truncated ballots in multi-winner ranked-choice elections |date=2024-08-07 |arxiv=2408.03926 |last2=Jones |first2=Matthew I. |last3=McCune |first3=David}}</ref>{{Excessive citations inline|date=January 2025}}

<math>\frac{\text{total votes}}{k+1} </math>

In the case of a single-winner election, this reduces to the familiar [[Simple majority voting|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 <math display="inline">\frac{\text{total votes}}{2}</math>.<ref name="mw-2007" /> A candidate who, at any point, holds strictly more than one Droop quota's worth of votes is therefore guaranteed to win a seat.<ref>{{cite book |last1=Grofman |first1=Bernard |title=Elections in Japan, Korea, and Taiwan Under the Single Non-Transferable Vote: The Comparative Study of an Embedded Institution |date=23 November 1999 |publisher=University of Michigan Press |isbn=978-0-472-10909-8 |chapter-url=https://books.google.com/books?id=Dkk_DwAAQBAJ&dq=SNTV,+STV,+and+Single-Member-District+Systems:+Theoretical+Comparisons+and+Constrasts&pg=PA317 |language=en |chapter=SNTV, STV, and Single-Member-District Systems: Theoretical Comparisons and Contrasts}}</ref>{{Efn|By [[abuse of notation]], mathematicians may write the quota as 
{{math|{{frac|votes|''k''+1}} + {{epsilon}}}}, where <math>\epsilon > 0</math> is taken arbitrarily close to 0 (i.e. as a limit), which allows breaking some ties for the last seat.|name=Abuse of notation}}

Sometimes, the Droop quota is written as a share of all votes, in which case it has value {{Math|{{frac|1|''k''+1}}}}. 

=== Original Droop quota ===
Modern variants of STV use [[Counting single transferable votes#Surplus vote transfers|fractional transfers]] of ballots to eliminate uncertainty. However, some older implementations of STV with [[Counting single transferable votes#Hare STV the whole-vote method|whole vote reassignment]] cannot handle fractional quotas, and so instead will either [[Ceiling function|round up]], or add one and truncate:<ref name="droop-1881" />

<math>\left\lceil \frac{\text{total votes}}{k+1} \right\rceil \approx \left\lfloor \frac{\text{total votes}}{k+1} + 1 \right\rfloor </math>

This variant of the quota is generally not recommended in the context of modern elections that allow for fractional votes, where it can cause problems in small elections ([[#Common errors|see below]]).<ref name="mw-2007" /><ref name="newland-1980">{{Cite journal |last=Newland |first=Robert A. |date=June 1980 |title=Droop quota and D'Hondt rule |url=http://www.tandfonline.com/doi/abs/10.1080/00344898008459290 |journal=Representation |language=en |volume=20 |issue=80 |pages=21–22 |doi=10.1080/00344898008459290 |issn=0034-4893|url-access=subscription }}</ref> However, it is the most commonly-used definition in legislative codes worldwide.{{cn|date=January 2025}}

=== Derivation ===
The Droop quota can be derived by considering what would happen if {{Math|''k''}} candidates (here 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, if each quota winner's share of the vote equals {{Math|{{frac|1|''k''+1}}}}, while all unelected candidates' share of the vote, taken together, is at most {{Math|{{frac|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.<ref name="droop-1881" />

==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 [[Spoilt vote|spoiled]].

The total number of valid votes is 100, and there are 3 seats. The Droop quota is therefore <math display="inline"> \frac{100}{3+1} = 25 </math>.<ref name="gallagher-1992"/> These votes are as follows:

{| class="wikitable"
!
!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 {{Tick}}
* '''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 vote]]s that can be transferred to their second choice, Hamilton. The tallies therefore become:
* '''Washington''': 25 {{Tick}}
* '''Hamilton''': 30{{Tick}}
* '''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; generally, ties are broken by taking the [[limit (mathematics)|limit]] of the results as the quota approaches the exact Droop quota.

== Common errors ==
There is a great deal of confusion among legislators and political observers about the correct form of the Droop quota.<ref name="dancisin-2013">{{Cite journal |last=Dančišin |first=Vladimír |date=2013 |title=Misinterpretation of the Hagenbach-Bischoff quota |journal=Annales Scientia Politica |volume=2 |issue=1 |pages=76}}</ref> At least six different versions appear in various legal codes or definitions of the quota, all varying by [[Fencepost error|one vote]].<ref name="dancisin-2013" /> The [[Electoral Reform Society|ERS]] handbook on STV has advised against such variants since at least 1976, as they can cause problems with proportionality in small elections.<ref name="mw-2007" /><ref name="newland-1980" />  In addition, it means that vote totals cannot be [[scale invariance|summarized into percentages]], because the winning candidate may depend on the choice of [[unit of measurement|unit]] or total number of ballots (not just their distribution across candidates).<ref name="mw-2007" /><ref name="newland-1980" /> Common variants of the Droop quota include:

<math>\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
        &&\Bigl\lfloor \frac{\text{votes}}{\text{seats}+1}\Bigr\rfloor + 1 \\
    \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{Inadmissible:} && \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}</math>

A quota being "inadmissable" refers to the possibility that more could achieve quota than the number of open seats. However preventing such an occurrence is not necessary. The archaic rounded-off form of the Droop quota (votes/seats+1, plus 1, rounded down) was traditionally seen as needed in the context of modern fractional transfer systems, and it was believed that any smaller portion of the votes, such as exact Droop, would not work because it would be possible for one more candidate than there are winners to reach the quota.<ref name="dancisin-2013" /> s 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.<ref name="mw-2007" /><ref name="newland-1980" /> Due to this misunderstanding, Ireland, Malta and Australia have used Droop's original quota - votes/seats+1, plus 1 - for the last hundred years.

The two variants in the first line come from Droop's discussion in the context of [[Thomas Hare (political reformer)|Hare]]'s STV proposal. Hare assumed that to calculate election results, physical ballots would be reshuffled across piles, and did not consider the possibility of fractional votes. In such a situation, rounding the number of votes up (or, alternatively, adding one and rounding down{{efn|The two are only different when the quotient produced by the number of votes divided by one more than the number of seats is exactly a whole number.}}) introduces as little error as possible, while maintaining the [[Electoral quota#Admissible quotas|admissibility of the quota]], by ensuring that no more can achieve quota than just the number of seats available.<ref name="dancisin-2013"/><ref name="droop-1881" />

=== 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 exactly-proportional system (i.e. one where each voter is represented equally).

The confusion between the two quotas originates from a [[Off-by-one error|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 [[Majority|bare majority]] are [[excess vote]]s.<ref name="droop-1881"></ref>

==Comparison with Hare==

The Hare quota gives more proportional outcomes on average because it is [[seat bias|statistically unbiased]].<ref name="pukelsheim-2017-bias">{{Citation |last=Pukelsheim |first=Friedrich |title=Favoring Some at the Expense of Others: Seat Biases |date=2017 |work=Proportional Representation: Apportionment Methods and Their Applications |pages=127–147 |editor-last=Pukelsheim |editor-first=Friedrich |url=https://doi.org/10.1007/978-3-319-64707-4_7 |access-date=2024-05-10 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-64707-4_7 |isbn=978-3-319-64707-4|url-access=subscription }}</ref> By contrast, the Droop quota is more [[Seat bias|biased towards large parties]] than any other [[Electoral quota|admissible quota]].<ref name="pukelsheim-2017-bias" /> As a result, the Droop quota is the quota most likely to produce [[minority rule]] by a [[Plurality (voting)|plurality party]], where a party representing less than half of the voters may take majority of seats in a constituency.<ref name="pukelsheim-2017-bias" /> However, the Droop quota has the advantage that any party receiving more than half the votes will receive at least half of all seats.

==See also==
* [[List of democracy and elections-related topics]]

== Notes ==
{{notelist}}

==References==
{{reflist}}

==Sources==
* {{Cite book|title = Robert's Rules of Order Newly Revised|last = Robert|first = Henry M.|publisher = Da Capo Press|year = 2011|isbn = 978-0-306-82020-5|location = Philadelphia, Pennsylvania|pages = 4|edition = 11th|display-authors = et al.}}

==Further reading==
* {{cite book |first=Henry Richmond |last=Droop |title=On the Political and Social Effects of Different Methods of Electing Representatives |location=London |year=1869 }}

{{voting systems}}
{{Majorities and quotas}}

[[Category:Single transferable vote]]
[[Category:Electoral system quotas]]
[[Category:Apportionment methods]]