Electoral quota#Admissible quotas

Two commonly-used quotas are the Hare and Droop quotas. According to some the Hare quota is unbiased in the way it hands out seats so it is seen as more proportional than the Droop quota, which tends to be biased towards larger parties or at least kinder to large parties than the Hare quota, as a larger part of an amount consisting of a large number of votes is considered surplus and transferred under Droop than under Hare.{{Cite book |last=Lijphart |first=Arend |title=Electoral Systems and Party Systems: A Study of Twenty-Seven Democracies, 1945-1990 |publisher=Oxford University Press |year=1994 |chapter=Appendix A: Proportional Representation Formulas |chapter-url=https://janda.org/c24/Readings/Lijphart/Lijphart.html}}{{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}}

However, the Hare does not always allocate a majority of seats to a party with a majority of the votes.[https://www.electoral-reform.org.uk/finding-the-finish-line-how-to-set-the-quota-under-stv/] Droop quota guarantees that a party that wins a majority of votes in a district will win a majority of the seats in the district.{{cite book |last1=Balinski |first1=Michel L. |url=https://archive.org/details/fairrepresentati00bali |title=Fair Representation: Meeting the Ideal of One Man, One Vote |last2=Young |first2=H. Peyton |publisher=Yale University Press |year=1982 |isbn=0-300-02724-9 |location=New Haven |url-access=registration}}{{Citation |last=Pukelsheim |first=Friedrich |title=Tracing Peculiarities: Vote Thresholds and Majority Clauses |date=2017 |work=Proportional Representation: Apportionment Methods and Their Applications |pages=207–223 |editor-last=Pukelsheim |editor-first=Friedrich |url=https://doi.org/10.1007/978-3-319-64707-4_11 |access-date=2024-05-10 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-64707-4_11 |isbn=978-3-319-64707-4}}

{{main|Hare quota}}

The Hare quota (also known as the simple quota or Hamilton's quota) is the most commonly-used quota for apportionments using the largest remainder method of party-list representation. It was proposd by Thomas Hare in his first STV proposals.When Hare was devising STV, he envisioned all of the United Kingdom being used as a single district, in which case the difference between Hare and Droop would not be noticeable. see Mr. Hare's Reform Bill Simplified and Explained By Henry Fawcett, p. 12-13 https://books.google.ca/books?id=EKh70Td2dQ4C&redir_esc=y accessed April 16, 2025 The Hare quota is given by the expression:

$\backslash frac\{\backslash text\{votes\}\}\{\backslash text\{seats\}\}$

Specifically, the Hare quota is unbiased in the number of seats it hands out. It does suffer the disadvantage that it sometimes allocates only a minority of seats to a party with a majority of votes.{{Cite book |last=Humphreys |title=Proportional Representation |year=1911 |pages=138}}[https://www.electoral-reform.org.uk/finding-the-finish-line-how-to-set-the-quota-under-stv/]

In at least one proportional representation system where the largest remainder method is used, the Hare quota has been manipulated by running candidates on many small lists, allowing each list to pick up a single remainder seat.See for example the 2012 election in Hong Kong Island where the DAB ran as two lists and gained twice as many seats as the single-list Civic despite receiving fewer votes in total: [https://www.nytimes.com/2012/09/11/world/asia/hong-kong-voting-for-legislature-is-heavy.html?pagewanted=all New York Times report] It is not clear that this is the fault of the Hare quota or in fact the election system that was used.

{{main|Droop quota}}

The Droop quota is used in most single transferable vote (STV) elections today and is occasionally used in elections held under the largest remainder method of party-list proportional representation (list PR). As well, it is identical to the Hagenbach-Bishoff quota, which is used to allocate seats by party in some list PR systems.DANČIŠIN, Misinterpretation of the Hagenbach-Bischoff quota. https://www.unipo.sk/public/media/18214/09%20Dancisin.pdf

The Droop quota is given by the expression:{{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}}

: $\backslash frac\{\backslash text\{total\; votes\}\}\{\backslash text\{total\; seats\}+1\}$ plus 1 and rounded down.{{Cite book |last=Henry R. Droop |title="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) |pages=7–46}}

It was first proposed in 1868 by the English lawyer and mathematician Henry Richmond Droop (1831–1884), who identified it as the minimum amount of support that would not possibly be achieved by too many compared to the number of seats in a district in semiproportional voting systems such as SNTV, leading him to propose it as an alternative to the Hare quota.Henry Richmond Droop, [http://www.votingmatters.org.uk/ISSUE24/I24P3.pdf "On methods of electing representatives"] in the Journal of the Statistical Society of London Vol. 44 No. 2 (June 1881) pp.141-196 [Discussion, 197-202], reprinted in Voting matters Issue 24 (October 2007) pp.7–46.

While Hare quota makes it more difficult for a large party to take its full share of the seats, even denying a majority party a majority of seats, the Droop quota does not disadvantage larger parties.{{cite web |title=Notes on the Political Consequences of Electoral Laws by Lijphart, Arend, American Political Science Review Vol. 84, No 2 1990 |url=http://users.ox.ac.uk/~sann2300/041102-ceg-electoral-consequences-lijphart.shtml |url-status=dead |archive-url=https://web.archive.org/web/20060516204603/http://users.ox.ac.uk/~sann2300/041102-ceg-electoral-consequences-lijphart.shtml |archive-date=2006-05-16 |access-date=2006-05-16}} Some say the Droop quota may go too far in that regard, saying it is the most-biased possible quota that can still be considered to be proportional. This is somewhat softened by the fact that the smaller the quota is (Droop is smaller than Hare), a small party requires fewer votes to take a seat.

Today the Droop quota is used in almost all STV elections, including those in the Republic of Ireland, Malta, Australia, Northern Ireland, and India.{{Cite book |last=Farrell and McAllister |title=Australian Electoral System |pages=24, 60-61}}

In some implementations, a "uniform quota" is simply set by law – any candidate receiving that set number of votes is declared elected, with surplus transferred away.

Something like this system was used in New York City from 1937 to 1947, where seats were allocated to each borough based on voter turnout and then each candidate that surpassed set number of votes was declared elected, and enough others that came close to fill up the borough seats.

Under such a system, the number of representatives elected varied from election to election depending on voter turnout. Under NYC's STV, total seats on council varied: 1937 New York City Council election 26 seats, 1939 New York City Council election 21 seats, 1941 26 seats, 1943 17 seats, and 1945 23 seats.https://repository.library.georgetown.edu/bitstream/handle/10822/1044631/Santucci_georgetown_0076D_13763.pdf?sequence=1&isAllowed=y

Like when Hare and Droop quotas are used, during the use of uniform quota, seats may be allocated to candidates who do not have full quota.

• Quota rule
• Single transferable vote
• Proportional representation
• Largest remainders method
• Instant-runoff voting

{{Voting systems}}

Category:Electoral system quotas