View source for Quota method

[[Category:Party-list proportional representation]]
[[Category:Apportionment methods]]
{{Short description|Proportional-representation voting system}}
{{more citations needed|date=November 2011}} {{Electoral systems}}

The '''quota''' or '''divide-and-rank methods''' make up a category of [[Apportionment (politics)|apportionment rules]], i.e. algorithms for allocating seats in a legislative body among multiple groups (e.g. [[Political party|parties]] or [[Federal state|federal states]]). The quota methods begin by calculating an [[Entitlement (fair division)|entitlement]] (basic number of seats) for each party, by dividing their vote totals by an [[electoral quota]] (a fixed number of votes needed to win a seat, as a unit). Then, leftover seats, if any are allocated by rounding up the apportionment for some parties. These rules are typically contrasted with the more popular [[Highest averages method|highest averages methods]] (also called divisor methods).<ref name=":12">{{Citation |last=Pukelsheim |first=Friedrich |title=Quota Methods of Apportionment: Divide and Rank |date=2017 |work=Proportional Representation: Apportionment Methods and Their Applications |pages=95–105 |editor-last=Pukelsheim |editor-first=Friedrich |url=https://doi.org/10.1007/978-3-319-64707-4_5 |access-date=2024-05-10 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-64707-4_5 |isbn=978-3-319-64707-4}}</ref>

By far the most common quota method are the '''largest remainders''' or '''quota-shift methods''', which assign any leftover seats to the "plurality" winners (the parties with the largest [[Remainder|remainders]], i.e. most leftover votes).<ref name=":02">{{cite book |last=Tannenbaum |first=Peter |url=http://www.mypearsonstore.com/bookstore/product.asp?isbn=9780321568038 |title=Excursions in Modern Mathematics |publisher=Prentice Hall |year=2010 |isbn=978-0-321-56803-8 |location=New York |pages=128}}</ref> 

When using the [[Hare quota]], this rule is called '''[[Alexander Hamilton|Hamilton]]'s method''', and is the third-most common apportionment rule worldwide (after [[Jefferson's method]] and [[Webster's method]]).<ref name=":12" />

Despite their intuitive definition, quota methods are generally disfavored by [[Social choice theory|social choice theorists]] as a result of [[Apportionment paradox|apportionment paradoxes]].<ref name=":12" /><ref name=":52">{{Citation |last=Pukelsheim |first=Friedrich |title=Securing System Consistency: Coherence and Paradoxes |date=2017 |work=Proportional Representation: Apportionment Methods and Their Applications |pages=159–183 |editor-last=Pukelsheim |editor-first=Friedrich |url=https://doi.org/10.1007/978-3-319-64707-4_9 |access-date=2024-05-10 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-64707-4_9 |isbn=978-3-319-64707-4}}</ref> In particular, the largest remainder methods exhibit the [[Participation criterion|no-show paradox]], i.e. voting ''for'' a party can cause it to ''lose'' seats.<ref name=":52" /><ref name=":02222">{{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}}</ref> The largest remainders methods are also vulnerable to [[Spoiler effect|spoiler effects]] and can fail [[Resource monotonicity|resource]] or [[house monotonicity]], which says that increasing the number of seats in a legislature should not cause a party to lose a seat (a situation known as an [[Alabama paradox]]).<ref name=":52" /><ref name=":02222" />{{Rp|Cor.4.3.1}}

==Method==
The largest remainder method divides each party's vote total by a ''quota''. Usually, quota is derived by dividing the number of valid votes cast, by the number of seats. The result for each party will consist of an [[integer]] part plus a [[Fraction (mathematics)|fractional]] [[remainder]]. Each party is first allocated a number of seats equal to their integer. This will generally leave some remainder seats unallocated. To apportion these seats, the parties are then ranked on the basis of their fractional remainders, and the parties with the largest remainders are each allocated one additional seat until all seats have been allocated. This gives the method its name - largest remainder.

Largest remainder methods produces similar results to [[single transferable vote]] or the [[quota Borda system]], where voters organize themselves into [[Solid coalition|solid coalitions]]. The [[single transferable vote]] or the [[quota Borda system]]<nowiki/>behave like the largest-remainders method when voters all behave like strict partisans (i.e. only mark preferences for candidates of one party).<ref>{{Cite journal |last=Gallagher |first=Michael |date=1992 |title=Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities |url=https://www.jstor.org/stable/194023 |journal=British Journal of Political Science |volume=22 |issue=4 |pages=469–496 |issn=0007-1234}}</ref>

==Quotas==
{{Main|Electoral quota}}
There are several possible choices for the [[electoral quota]]. The choice of quota affects the properties of the corresponding largest remainder method, and particularly the [[seat bias]]. Smaller quotas allow small parties to pick up seats, while larger quotas leave behind more votes. A somewhat counterintuitive result of this is that a ''larger'' quota will always be more favorable to ''smaller'' parties.<ref>{{Cite journal |last=Gallagher |first=Michael |date=1992 |title=Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities |url=https://www.jstor.org/stable/194023 |journal=British Journal of Political Science |volume=22 |issue=4 |pages=469–496 |issn=0007-1234}}</ref> A party hoping to win multiple seats sees fewer votes captured by a single popular candidate when the quota is small.

The two most common quotas are the [[Hare quota]] and the [[Droop quota]]. The use of a particular quota with one of the largest remainder methods is often abbreviated as "LR-[quota name]", such as "LR-Droop".<ref>{{Cite book |last1=Gallagher |first1=Michael |url=https://books.google.com/books?id=Igdj1P4vBwMC |title=The Politics of Electoral Systems |last2=Mitchell |first2=Paul |date=2005-09-15 |publisher=OUP Oxford |isbn=978-0-19-153151-4 |language=en}}</ref>

The Hare (or simple) quota is defined as follows:

: <math>\frac{\text{total votes}}{\text{total seats}}</math>

LR-Hare is sometimes called Hamilton's method, named after [[Alexander Hamilton]], who devised the method in 1792.<ref>{{Cite book |last=Eerik Lagerspetz |url=https://books.google.com/books?id=RNcLCwAAQBAJ&q=alexander+hamilton+invented+the+largest+remainder+method&pg=PA130 |title=Social Choice and Democratic Values |date=26 November 2015 |publisher=Springer |isbn=9783319232614 |series=Studies in Choice and Welfare |access-date=2017-08-17}}</ref>

The [[Droop quota]] is given by:

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

and is applied to elections in [[South Africa]].{{Cn|date=August 2024}}

The Hare quota is more generous to less-popular parties and the Droop quota to more-popular parties. Specifically, the Hare quota is [[Unbiased estimate|''unbiased'']] in the number of seats it hands out, and so is more proportional than the Droop quota (which tends to give more seats to larger parties). The Hare suffers the disproportionality that it sometimes allocates a majority of seats to a party with less than a majority of votes in a district.<ref>{{Cite book |last=Humphreys |title=Proportional Representation |year=1911 |pages=138}}</ref>

==Examples==
The following example allocates 10 seats using the largest-remainder method by Droop quota.
{| class="wikitable"
!Party
!Votes
![[Entitlement (fair division)|Entitlement]]
!Remainder
!Total seats
|-
!Yellows
|47,000
|5.170
|0.170
|5
|-
!Whites
|16,000
|1.760
|0.760
|2
|-
!Reds
|15,800
|1.738
|0.738
|2
|-
!Greens
|12,000
|1.320
|0.320
|1
|-
!Blues
|6,100
|0.671
|0.671
|0
|-
!Pinks
|3,100
|0.341
|0.341
|0
|-
!'''Total'''
|'''100,000'''
|'''10'''
|'''3'''
|'''0.341'''
|}

== Pros and cons ==
It is easy for a voter to understand how the largest remainder method allocates seats. Moreover, the largest remainder method satisfies the [[quota rule]] (each party's seats are equal to its ideal share of seats, either rounded up or rounded down) and was designed to satisfy that criterion. However, this comes at the cost of greater inequalities in the [[seats-to-votes ratio]], which can violate the principle of [[one man, one vote]].

However, a greater concern for social choice theorists, and the primary cause behind its abandonment in many countries, is the tendency of such rules to produce erratic or irrational behaviors called [[Apportionment paradox|apportionment paradoxes]]:

* ''Increasing'' the number of seats in a legislature can ''decrease'' a party's apportionment of seats, called the [[Alabama paradox]].
* Adding more parties to the legislature can cause a bizarre kind of [[spoiler effect]] called the [[New states paradox|new state paradox]].
** When Congress first admitted [[Oklahoma]] to the Union, the House was expanded by 5 seats, equal to Oklahoma's apportionment, to ensure it would not affect the seats for any existing states. However, when the full apportionment was recalculated, the House was stunned to learn Oklahoma's entry had caused New York to lose a seat to Maine, despite there being no change in either state's population.<ref name="Caulfield22">{{cite journal |last=Caulfield |first=Michael J. |date=November 2010 |title=Apportioning Representatives in the United States Congress – Paradoxes of Apportionment |url=http://www.maa.org/publications/periodicals/convergence/apportioning-representatives-in-the-united-states-congress-paradoxes-of-apportionment |journal=Convergence |publisher=Mathematical Association of America |doi=10.4169/loci003163}}</ref><ref name="Stein200822">{{cite book |last=Stein |first=James D. |title=How Math Explains the World: A Guide to the Power of Numbers, from Car Repair to Modern Physics |publisher=Smithsonian Books |year=2008 |isbn=9780061241765 |location=New York}}</ref>{{rp|232–233}}
** By the same token, apportionments may depend on the precise order in which the apportionment is calculated. For example, identifying winning independents first and electing them, then apportioning the remaining seats, will produce a different result from treating each independent as if they were their own party and then computing a single overall apportionment.<ref name=":52" />

Such paradoxes also have the additional drawback of making it difficult or impossible to generalize  procedure to more complex apportionment problems such as [[Biproportional apportionment|biproportional apportionments]] or [[Vote linkage|partial vote linkage]]. This is in part responsible for the extreme complexity of administering elections by quota-based rules like the single transferable vote (see [[counting single transferable votes]]).

=== Alabama paradox ===
The [[Alabama paradox]] is when an ''increase'' in the total number of seats leads to a ''decrease'' in the number of seats allocated to a certain party. In the example below, when the number of seats to be allocated is increased from 25 to 26, parties D and E end up with fewer seats, despite their entitlements increasing.

With 25 seats, the results are:
{| class="wikitable"
!Party
!A
!B
!C
!D
!E
!F
!Total
|-
!Votes
|1500
|1500
|900
|500
|500
|200
|5100
|-
!Quotas received
|7.35
|7.35
|4.41
|2.45
|2.45
|0.98
|25
|-
!Automatic seats
|7
|7
|4
|2
|2
|0
|22
|-
!Remainder
|0.35
|0.35
|0.41
|0.45
|0.45
|0.98
|
|-
!Surplus seats
|0
|0
|0
|1
|1
|1
|3
|-
!Total seats
|7
|7
|4
|'''''3'''''
|'''''3'''''
|1
|25
|}
With 26 seats, the results are:
{| class="wikitable"
!Party
!A
!B
!C
!D
!E
!F
!Total
|-
!Votes
|1500
|1500
|900
|500
|500
|200
|5100
|-
!Quotas received
|7.65
|7.65
|4.59
|2.55
|2.55
|1.02
|26
|-
!Automatic seats
|7
|7
|4
|2
|2
|1
|23
|-
!Remainder
|0.65
|0.65
|0.59
|0.55
|0.55
|0.02
|
|-
!Surplus seats
|1
|1
|1
|0
|0
|0
|3
|-
!Total seats
|8
|8
|5
|'''''2'''''
|'''''2'''''
|1
|26
|}

==References==
<references responsive="0"></references>

==External links==

* [http://www.cut-the-knot.org/Curriculum/SocialScience/AHamilton.shtml Hamilton method experimentation applet] at [[cut-the-knot]]

{{voting systems}}

[[Category:Voting theory]]