Editing Droop quota (section)

== 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" />