Electoral quota

In proportional representation systems, an electoral quota is the number of votes a candidate needs to be guaranteed election. They are used in some systems where a formula other than plurality is used to allocate seats.

Generally quotas are set at a level that is guaranteed to apportion only as many seats as are available in the legislature. When the electorate is divided into separate districts, the quota is commonly set by reference to valid votes cast in the district.

The quota may be set at a number between:^[1]

${\frac {\text{votes}}{{\text{seats}}+1}}\leq {\text{quota}}\leq {\frac {\text{votes}}{{\text{seats}}-1}}$

The smallest quota given above, votes/seats+1, is sometimes defended. Such a quota may be workable as long as rules are in place for dealing with situations where two or more tied candidates are competing for a lesser number of seats.

The common quotas used in single transferable voting elections are such that no more can achieve quota than the number of seats in the district.

Common quotas

There are two commonly-used quotas: the Hare and Droop quotas. The Hare quota is unbiased in the number of seats it hands out, and so is more proportional than the Droop quota (which tends to be biased towards larger parties);^[2]^[3]

However, the Hare sometimes does not allocate a majority of seats to a party with a majority of the votes.[1] 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.^[4]^[5]

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 used by Thomas Hare in his first proposals for STV. It is given by the expression:

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

Specifically, the Hare quota is unbiased in the number of seats it hands out.^[2] It does suffer the disadvantage that it can allocate only a minority of seats to a party with a majority of votes.^[6][2]

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.^[7] It is not clear that this is the fault of the Hare quota or in fact the election system that was used.

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.^[8]

The Droop quota is given by the expression:^[1]^[9]

{\frac {\text{total votes}}{{\text{total seats}}+1}}

plus 1 and rounded down.^[10]

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.^[11]

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.^[12] 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.^[1]

Today the Droop quota is used in almost all STV elections, including those in the Republic of Ireland, Malta, Australia, Northern Ireland, and India.^[13]

Uncommon quotas

Uniform quota

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.^[14]

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.

References

^ ^a ^b ^c Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Quota Methods of Apportionment: Divide and Rank", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 95–105, doi:10.1007/978-3-319-64707-4_5, ISBN 978-3-319-64707-4, retrieved 2024-05-10
^ ^a ^b Lijphart, Arend (1994). "Appendix A: Proportional Representation Formulas". Electoral Systems and Party Systems: A Study of Twenty-Seven Democracies, 1945-1990. Oxford University Press.
^ 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
^ Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0-300-02724-9.
^ Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Tracing Peculiarities: Vote Thresholds and Majority Clauses", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 207–223, doi:10.1007/978-3-319-64707-4_11, ISBN 978-3-319-64707-4, retrieved 2024-05-10
^ Humphreys (1911). Proportional Representation. p. 138.
^ 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: New York Times report
^ DANČIŠIN, Misinterpretation of the Hagenbach-Bischoff quota. https://www.unipo.sk/public/media/18214/09%20Dancisin.pdf
^ Woodall, Douglass. "Properties of Preferential Election Rules". Voting Matters (3).
^ 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.
^ Henry Richmond Droop, "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.
^ "Notes on the Political Consequences of Electoral Laws by Lijphart, Arend, American Political Science Review Vol. 84, No 2 1990". Archived from the original on 2006-05-16. Retrieved 2006-05-16.
^ Farrell and McAllister. Australian Electoral System. pp. 24, 60–61.
^ https://repository.library.georgetown.edu/bitstream/handle/10822/1044631/Santucci_georgetown_0076D_13763.pdf?sequence=1&isAllowed=y

[:1-1] Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Quota Methods of Apportionment: Divide and Rank", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 95–105, doi:10.1007/978-3-319-64707-4_5, ISBN 978-3-319-64707-4, retrieved 2024-05-10

[:0-2] Lijphart, Arend (1994). "Appendix A: Proportional Representation Formulas". Electoral Systems and Party Systems: A Study of Twenty-Seven Democracies, 1945-1990. Oxford University Press.

[:43-3] 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

[:0222-4] Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0-300-02724-9.

[:3-5] Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Tracing Peculiarities: Vote Thresholds and Majority Clauses", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 207–223, doi:10.1007/978-3-319-64707-4_11, ISBN 978-3-319-64707-4, retrieved 2024-05-10

[6] Humphreys (1911). Proportional Representation. p. 138.

[7] 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: New York Times report

[8] DANČIŠIN, Misinterpretation of the Hagenbach-Bischoff quota. https://www.unipo.sk/public/media/18214/09%20Dancisin.pdf

[9] Woodall, Douglass. "Properties of Preferential Election Rules". Voting Matters (3).

[10] 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.

[RHD2-11] Henry Richmond Droop, "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.

[:2-12] "Notes on the Political Consequences of Electoral Laws by Lijphart, Arend, American Political Science Review Vol. 84, No 2 1990". Archived from the original on 2006-05-16. Retrieved 2006-05-16.

[13] Farrell and McAllister. Australian Electoral System. pp. 24, 60–61.

[14] ttps://repository.library.georgetown.edu/bitstream/handle/10822/1044631/Santucci_georgetown_0076D_13763.pdf?sequence=1&isAllowed=y

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