View source for Huntington–Hill method

{{Short description|Proportional electoral system}}
{{Electoral systems}}{{United States House of Representatives}}
The '''Huntington–Hill method''', sometimes called '''method of equal proportions''', is a [[highest averages method]] for assigning seats in a [[legislature]] to [[political parties]] or [[State (administrative division)|states]].<ref name=":0">{{cite web |title=Congressional Apportionment |url=http://www.nationalatlas.gov/articles/boundaries/a_conApport.html#six |url-status=dead |archive-url=https://web.archive.org/web/20090228091000/http://www.nationalatlas.gov/articles/boundaries/a_conApport.html#six |archive-date=2009-02-28 |access-date=2009-02-14 |publisher=NationalAtlas.gov}}</ref> Since 1941, this method has been used to apportion the 435 seats in the United States House of Representatives following the completion of each decennial [[United States Census Bureau|census]].<ref>{{Cite web|title=U.S. Code Title 2, Section 2a: Reapportionment of Representatives.|url=https://www.law.cornell.edu/uscode/text/2/2a}}</ref><ref>{{Cite web|title=Computing Apportionment|url=https://www.census.gov/topics/public-sector/congressional-apportionment/about/computing.html|access-date=2021-04-26|publisher=[[United States Census Bureau]]|language=EN-US}}</ref>  

The method minimizes the [[relative difference]] in the number of constituents represented by each legislator. In other words, the method selects the allocation such that no transfer of a seat from one state to another can reduce the percent error in representation for both states.<ref name=":0" />

==Apportionment method==

In this method, as a first step, each of the 50 states is given its one guaranteed seat in the House of Representatives, leaving 385 seats to be assigned. The remaining seats are allocated one at a time, to the state with the highest ''average district population'', to bring its district population down''.'' However, it is not clear if we should calculate the average ''before'' or ''after'' allocating an additional seat, and the two procedures give different results. Huntington-Hill uses a [[continuity correction]] as a compromise, given by taking the [[geometric mean]] of both divisors, i.e.:<ref name=":422">{{Citation |last=Pukelsheim |first=Friedrich |title=Divisor Methods of Apportionment: Divide and Round |date=2017 |work=Proportional Representation: Apportionment Methods and Their Applications |pages=71–93 |editor-last=Pukelsheim |editor-first=Friedrich |url=https://doi.org/10.1007/978-3-319-64707-4_4 |access-date=2021-09-01 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-64707-4_4 |isbn=978-3-319-64707-4|url-access=subscription }}</ref>

:<math> A_{n} = \frac{P}{\sqrt{n(n+1)}} </math>

where ''P'' is the population of the state, and ''n'' is the number of seats it currently holds before the possible allocation of the next seat.

Consider the reapportionment following the 2010 U.S. census: after every state is given one seat: 

# The largest value of ''A''<sub>1</sub> corresponds to the largest state, California, which is allocated seat 51. 
# The 52nd seat goes to Texas, the 2nd largest state, because its ''A''<sub>1</sub> priority value is larger than the ''A<sub>n</sub>'' of any other state. 
# The 53rd seat goes back to California because its ''A''<sub>2</sub> priority value is larger than the ''A<sub>n</sub>'' of any other state. 
# The 54th seat goes to New York because its ''A''<sub>1</sub> priority value is larger than the ''A<sub>n</sub>'' of any other state at this point. 

This process continues until all remaining seats are assigned. Each time a state is assigned a seat, ''n'' is incremented by 1, causing its priority value to be reduced.

== Division by zero ==
Unlike the [[D'Hondt method|D'Hondt]] and [[Sainte-Laguë method|Sainte-Laguë]] systems, which allow the allocation of seats by calculating successive quotients right away, the Huntington–Hill system requires each party or state have at least one seat to avoid a [[division by zero]] error.<ref name=":4222">{{Citation |last=Pukelsheim |first=Friedrich |title=Divisor Methods of Apportionment: Divide and Round |date=2017 |work=Proportional Representation: Apportionment Methods and Their Applications |pages=71–93 |editor-last=Pukelsheim |editor-first=Friedrich |url=https://doi.org/10.1007/978-3-319-64707-4_4 |access-date=2021-09-01 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-64707-4_4 |isbn=978-3-319-64707-4|url-access=subscription }}</ref> In the U.S. House of Representatives, this is ensured by guaranteeing each state at least one seat;<ref name=":4222" /> in [[Party-list proportional representation|party-list representation]], small parties would likely be eliminated using some [[electoral threshold]], or the first divisor can be modified.

== Examples ==
Consider an example to distribute 8 seats between three parties A, B, C having respectively 100,000, 80,000 and 30,000 voices.

Each eligible party is assigned one seat. With all the initial seats assigned, the remaining five seats are distributed by a priority number calculated as follows. Each eligible party's (Parties A, B, and C) total votes is divided by {{math|{{sqrt|2 {{cdot}} 1}} ≈ 1.41}}, then by approximately 2.45, 3.46, 4.47, 5.48, 6.48, 7.48, and 8.49. The 5 highest entries, marked with asterisks, range from '''70,711''' down to '''28,868'''. For each, the corresponding party gets another seat.

{| class="wikitable"
! '''Denominator''' || {{sqrt|1·2}} ≈<br>1.41 || {{sqrt|2·3}} ≈<br>2.45 || {{sqrt|3·4}} ≈<br>3.46 || {{sqrt|4·5}} ≈<br>4.47 || {{sqrt|5·6}} ≈<br>5.48 || {{sqrt|6·7}} ≈<br>6.48 || {{sqrt|7·8}} ≈<br>7.48 || {{sqrt|8·9}} ≈<br>8.49 || Initial<br>seats || Seats<br>won (*) || Total<br>Seats || Ideal<br>seats
|-
|| Party A || '''70,711*''' || '''40,825*''' || '''28,868*''' || 22,361 || 18,257 || 15,430 || 13,363 || 11,785
| align="center" | 1 
| align="center" | '''3''' 
| align="center" | 4
| align="center" | 3.8
|-
|| Party B || '''56,569*'''  || '''32,660*''' || 23,094 || 17,889 || 14,606 || 12,344 || 10,690 || 9,428
| align="center" | 1 
| align="center" | '''2''' 
| align="center" | 3
| align="center" | 3.0
|-
|| Party C || 21,213  || 12,247 || 8,660 || 6,708 || 5,477 || 4,629 || 4,009 || 3,536
| align="center" | 1 
| align="center" | '''0''' 
| align="center" | 1
| align="center" | 1.1
|}

=== Knesset example ===
The [[Knesset]] ([[Israel]]'s unicameral legislature), are elected by party-list representation with apportionment by the D'Hondt method.{{efn|name=Bader-Ofer|The method used for the 20th Knesset was actually a modified D'Hondt, called the Bader-Ofer method.  This modification allows for spare vote agreements between parties.<ref>{{cite web | title = With Bader-Ofer method, not every ballot counts | publisher = The Jerusalem Post | url = https://www.jpost.com/israel-elections/with-bader-ofer-method-not-every-ballot-counts-394027 | access-date = 2021-05-04 }}</ref>}} Had the Huntington–Hill method, rather than the D'Hondt method, been used to apportion seats following the elections to the [[20th Knesset]], held in 2015, the 120 seats in the 20th Knesset would have been apportioned as follows:

{| class=wikitable style="text-align: center;"
!rowspan=3 colspan=2|Party
!rowspan=3|Votes
!colspan=3 style='border-style: solid solid none   none;' | Huntington–Hill
!style='border-style: solid solid none   none;' | D'Hondt{{efn|name=Bader-Ofer}}
!rowspan=3|+/–
|-
!colspan=3 style='border-style: none  none  solid  solid;'| (hypothetical)
!style='border-style: none  none  solid  solid;' | (actual)
|-
!Last priority{{efn|This is each party's last priority number which resulted in a seat being gained by the party.  Likud gained the last seat (the 120th seat allocated).  Each priority number in this column is greater than any priority number in the Next Priority column.}}
!Next priority{{efn|This is each party's next priority number which would result in a seat being gained by the party.  Kulanu would have gained the next seat (if there were 121 seats in the Knesset).  Each priority number in this column is less than any priority number in the Last Priority column.}}
!style="width: 5em;" | Seats
!style="width: 5em;" | Seats
|-
|width=2px bgcolor="{{party color|Likud}}" |
|align=left|[[Likud]]||985,408||33408||32313||30||30||0
|-
|width=2px bgcolor=#642240 |
|align=left|[[Zionist Union]]||786,313||33468||32101||24||24||0
|-
|width=2px bgcolor="{{party color|Joint List}}"|
|align=left|[[Joint List]]||446,583||35755||33103||13||13||0
|-
| bgcolor="{{party color|Yesh Atid}}" |
|align=left|[[Yesh Atid]]||371,602||35431||32344||11||11||0
|-
| bgcolor="{{party color|Kulanu}}" |
|align=left|[[Kulanu]]||315,360||37166||33242||9||10||–1 <!-- This means that switching from D'Hondt to Huntington-Hill would have lost this party a seat. -->
|-
| bgcolor="{{party color|The Jewish Home}}" |
|align=left|[[The Jewish Home]]||283,910||33459||29927||9||8||+1 <!-- This means that switching from D'Hondt to Huntington-Hill would have gained this party a seat. -->
|-
| bgcolor="{{party color|Shas}}" |
|align=left|[[Shas]]||241,613||37282||32287||7||7||0
|-
| bgcolor="{{party color|Yisrael Beiteinu}}" |
|align=left|[[Yisrael Beiteinu]]||214,906||39236||33161||6||6||0
|-
| bgcolor="{{party color|United Torah Judaism}}" |
|align=left|[[United Torah Judaism]]||210,143||38367||32426||6||6||0
|-
| bgcolor="{{party color|Meretz}}" |
|align=left|[[Meretz]]||165,529||37013||30221||5||5||0
|-
| style="text-align:left;" colspan="8"|Source: [https://web.archive.org/web/20150615082949/http://www.votes20.gov.il/ CEC]
|}

Compared with the actual apportionment, [[Kulanu]] would have lost one seat, while [[The Jewish Home]] would have gained one seat.

== See also ==
* [[Edward Vermilye Huntington]]
* [[Highest averages method]]
* [[Joseph Adna Hill]]

==Notes==
{{notelist}}

==References==
{{reflist}}

{{Majorities and quotas}}

{{DEFAULTSORT:Huntington-Hill Method}}
[[Category:Apportionment methods]]