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In the study of electoral systems, the Droop quota (sometimes called the Hagenbach-Bischoff, Britton, or Newland-Britton quota^[1]^[a]) 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, which is a basic component of single transferable voting, a form of proportional representation.^[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.^[6] It is also used in South Africa to allocate seats by the largest remainder method.^[7]^[8]

Standard Formula

The Droop quota for a $k$ -winner election is given by the expression:^[1]^[9]^[10]^[11]^[12]^[13]

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

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 more than one Droop quota's worth of votes is therefore guaranteed to win a seat.^[14]

Rounding

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 or round down. For example:^[4]

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

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.^[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]^[15]

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.^[16] These votes are as follows:

preferences marked	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
Jefferson: 25
Burr: 20
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: 25
Burr: 20
Hamilton: 29Y

Hamilton is elected, so his excess votes are redistributed. Thanks to the four vote transfer from Hamilton, Jefferson accumulates 29 votes to Burr's 20 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), Burr would have been elected to a seat. But under STV he did not collect any transfers and Jefferson was seen as 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.

Common errors

There are at least six different versions of the Droop quota to appear in various legal codes or definitions of the quota.^[17] Some claim that, depending on which version is used, a failure of proportionality in small elections may arise.^[1]^[15] Common variants include:

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

Droop and Hagenbach-Bischoff derived new quota as a replacement for the Hare quota (votes/seats). Their quota was meant to produce more proportional result by having the quota as low as thought to be possible. Their quota was basically votes/seats plus 1, plus 1, the formula on the left on the first row.

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 (the variant on top right). Hagenbach-Bischoff went to votes/seats +1, rounded up, the variant in the middle of the top row.^[4] 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).^[17]

Some hold the misconception that these rounded-off variants of the Droop and Hagenbach-Bischoff quota are still needed, despite the use of fractions in fractional STV systems, now common today.

As well, it is un-necessary to ensure the quota is larger than vote/seats plus 1, as in the historical examples, the variant on the second row, and the formula on the right on the bottom row. 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 votes/seats plus 1, rounded down (the left variant on the third row), it is possible for one more candidate to reach the quota than there are seats to fill.^[17] 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.^[1]^[15] 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,^[18] by promoting representation of smaller parties, although 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.^[18] The Droop quota sometimes allows a party representing less than half of the voters to take a majority of seats in a constituency.^[18]^[4]

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).^[2]

References

^ ^a ^b ^c ^d ^e Lundell, Jonathan; Hill, ID (October 2007). "Notes on the Droop quota" (PDF). Voting Matters (24): 3–6.
^ 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.
^ "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 ^g ^h 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.
^ "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.
^ 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.
^ ^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.

@@ Line 8: / Line 8: @@
 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 [[Largest remainder method|quota-based]] systems such as [[Single transferable vote|STV]] or [[expanding approvals rule|expanding approvals]], these excess votes can be transferred to other candidates, preventing them from [[Wasted vote|being wasted]].<ref name=":2"></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=":2"></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]], which is a basic component of [[single transferable voting]], a form of [[Proportional representation|proportional representation]].<ref name=":2"></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>https://electoral.gov.mt/ElectionResults/General</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>
@@ Line 21: / Line 21: @@
 === Rounding ===
-Modern variants of STV use [[Counting single transferable votes#Surplus vote transfers|fractional transfers]] of ballots to eliminate uncertainty. However, STV elections with [[Counting single transferable votes#Hare STV the whole-vote method|whole vote reassignment]] cannot handle fractional quotas, and so instead will [[Ceiling function|round up]]:<ref name=":2" />
+Modern variants of STV use [[Counting single transferable votes#Surplus vote transfers|fractional transfers]] of ballots to eliminate uncertainty. However, STV elections with [[Counting single transferable votes#Hare STV the whole-vote method|whole vote reassignment]] cannot handle fractional quotas, and so instead will [[Ceiling function|round up]] or [[Floor function|round down]]. For example:<ref name=":2" />
 <math>\left\lceil \frac{\text{total votes}}{k+1} \right\rceil </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=":0" />
 === Derivation ===
-The Droop quota can be derived by considering what would happen if {{Math|''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 {{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=":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.<ref name=":0" /><ref name=":3">{{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}}</ref>
+The Droop quota can be derived by considering what would happen if {{Math|''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 {{Math|{{frac|1|''k''+1}}}} plus 1, while all unelected candidates' share of the vote, taken together, would be less than {{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=":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.<ref name=":0" /><ref name=":3">{{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}}</ref>
 ==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 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 [[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>{{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> These votes are as follows:
+The total number of valid votes is 102, and there are 3 seats. The Droop quota is therefore <math display="inline"> \frac{102}{3+1} = 25.5 </math>. Rounded up, that is 26.<ref>{{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> These votes are as follows:
 {| class="wikitable"
+!preferences marked
-!
 !45 voters
 !20 voters
@@ Line 63: / Line 61: @@
 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
+* '''Hamilton''': 10
+Only Washington has at least 26 votes. As a result, he is declared elected. Washington has 19 [[excess vote]]s that are now transferred to their second choice, Hamilton. The tallies therefore become:
+* '''Washington''': 26 {{Tick}}
 * '''Jefferson''': 25
+* '''Burr''': 20
+* '''Hamilton''': 29{{Tick}}
+Hamilton is elected, so his excess votes are redistributed. Thanks to the four vote transfer from Hamilton, Jefferson accumulates 29 votes to Burr's 20 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), Burr would have been elected to a seat. But under STV he did not collect any transfers and Jefferson was seen as the more generally supported candidate.
-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.
+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.
-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=":1">{{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=":1" /> Some say such versions are  incorrect, and can cause a failure of proportionality in small elections.<ref name=":0" /><ref name=":3" />
+There are at least six different versions of the Droop quota to appear in various legal codes or definitions of the quota.<ref name=":1">{{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> Some claim that, depending on which version is used, a failure of proportionality in small elections may arise.<ref name=":0" /><ref name=":3" />
 Common variants include:
 <math>\begin{array}{rlrl}
-    \text{Historical:} && \left\lceil  \frac{\text{votes}}{\text{seats}+1} \right\rceil
+    \text{Historical:} && \phantom{\Bigl\lfloor} \frac{\text{votes}}{\text{seats}+1} + 1    \phantom{\Bigr\rfloor} && \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}
+    \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}</math>
+Droop and Hagenbach-Bischoff derived new quota as a replacement for the Hare quota (votes/seats). Their quota was meant to produce more proportional result by having the quota as low as thought to be possible. Their quota was basically votes/seats plus 1, plus 1, the formula on the left on the first row.
-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.<ref name=":2" /> 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 <math>m V</math>, the number of votes, by <math>n+1</math>" (where ''n'' is the number of seats).<ref name=":1" /> In such a situation, rounding the number of votes upwards introduces as little error as possible, while maintaining the [[Electoral quota#Admissible quotas|admissibility of the quota]].<ref name=":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 (the variant on top right). Hagenbach-Bischoff went to votes/seats +1, rounded up, the variant in the middle of the top row.<ref name=":2" /> Hagenbach-Bischoff proposed a quota that is "the whole number next greater than the quotient obtained by dividing <math>m V</math>, the number of votes, by <math>n+1</math>" (where ''n'' is the number of seats).<ref name=":1" />
+Some hold the misconception that these rounded-off variants of the Droop and Hagenbach-Bischoff quota are still needed, despite the use of fractions in fractional STV systems, now common today.
-Some hold the misconception that the archaic form of the Droop quota is still needed in the context of modern fractional transfer systems, because when using the exact Droop quota, it is possible for one more candidate than there are winners to reach the quota.<ref name=":1" /> 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.<ref name=":0" /><ref name=":3" />
+As well, it is un-necessary to ensure the quota is larger than vote/seats plus 1, as in the historical examples, the variant on the second row, and the formula on the right on the bottom row. 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 votes/seats plus 1, rounded down (the left variant on the third row), it is possible for one more candidate to reach the quota than there are seats to fill.<ref name=":1" /> 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.<ref name=":0" /><ref name=":3" /> 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. However, some jurisdictions fail to correctly specify this in their election administration laws.{{Cn|date=May 2024}}
+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.{{Cn|date=May 2024}}
 === 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.
-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,<ref name=":43">{{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}}</ref> although sometimes under Hare a majority group will be denied the majority of seats. By contrast, the Droop quota is more [[Seat bias|biased towards large parties]] than any other [[Electoral quota|admissible quota]].<ref name=":43" /> As a result, the Droop rule allows for [[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=":43" />
+As a result, the Hare quota is said to give somewhat more proportional outcomes,<ref name=":43">{{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}}</ref> by promoting representation of smaller parties, although 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 [[Seat bias|biased towards large parties]] than any other [[Electoral quota|admissible quota]].<ref name=":43" /> The Droop quota sometimes allows a party representing less than half of the voters to take a majority of seats in a constituency.<ref name=":43" /><ref name=":2"></ref>
-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=":2"></ref>
 The Droop quota is today the most popular quota for STV elections.{{cn|date=April 2024}}

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