Huntington–Hill method

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 states.^[1] 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 census.^[2]^[3]

The method minimizes the relative difference in the number of constituents represented by each legislator. In other words, the method selects the algorithm such that no transfer of a seat from one state to another can reduce the percent error in representation for both states.^[1]

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.:^[4]

A_{n}={\frac {P}{\sqrt {n(n+1)}}}

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 A₁ corresponds to the largest state, California, which is allocated seat 51.
The 52nd seat goes to Texas, the 2nd largest state, because its A₁ priority value is larger than the A_n of any other state.
The 53rd seat goes back to California because its A₂ priority value is larger than the A_n of any other state.
The 54th seat goes to New York because its A₁ priority value is larger than the A_n 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 and 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.^[5] In the U.S. House of Representatives, this is ensured by guaranteeing each state at least one seat;^[5] in 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 $\sqrt 2 • 1 \approx 1.41$ , 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.

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

Knesset example

The Knesset (Israel's unicameral legislature), are elected by party-list representation with apportionment by the D'Hondt method.^[a] 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:

Party		Votes	Huntington–Hill			D'Hondt^[a]	+/–
			(hypothetical)			(actual)
			Last priority^[b]	Next priority^[c]	Seats	Seats
	Likud	985,408	33408	32313	30	30	0
	Zionist Union	786,313	33468	32101	24	24	0
	Joint List	446,583	35755	33103	13	13	0
	Yesh Atid	371,602	35431	32344	11	11	0
	Kulanu	315,360	37166	33242	9	10	–1
	The Jewish Home	283,910	33459	29927	9	8	+1
	Shas	241,613	37282	32287	7	7	0
	Yisrael Beiteinu	214,906	39236	33161	6	6	0
	United Torah Judaism	210,143	38367	32426	6	6	0
	Meretz	165,529	37013	30221	5	5	0
Source: CEC

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

Notes

^ ^a ^b 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.^[6]
^ 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.
^ 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.

References

^ ^a ^b "Congressional Apportionment". NationalAtlas.gov. Archived from the original on 2009-02-28. Retrieved 2009-02-14.
^ "U.S. Code Title 2, Section 2a: Reapportionment of Representatives".
^ "Computing Apportionment". United States Census Bureau. Retrieved 2021-04-26.
^ Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Divisor Methods of Apportionment: Divide and Round", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 71–93, doi:10.1007/978-3-319-64707-4_4, ISBN 978-3-319-64707-4, retrieved 2021-09-01
^ ^a ^b Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Divisor Methods of Apportionment: Divide and Round", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 71–93, doi:10.1007/978-3-319-64707-4_4, ISBN 978-3-319-64707-4, retrieved 2021-09-01
^ "With Bader-Ofer method, not every ballot counts". The Jerusalem Post. Retrieved 2021-05-04.

[Bader-Ofer-7] 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.^[6]

[8] 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.

[9] 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.

[:0-1] "Congressional Apportionment". NationalAtlas.gov. Archived from the original on 2009-02-28. Retrieved 2009-02-14.

[2] "U.S. Code Title 2, Section 2a: Reapportionment of Representatives".

[3] "Computing Apportionment". United States Census Bureau. Retrieved 2021-04-26.

[:422-4] Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Divisor Methods of Apportionment: Divide and Round", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 71–93, doi:10.1007/978-3-319-64707-4_4, ISBN 978-3-319-64707-4, retrieved 2021-09-01

[:4222-5] Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Divisor Methods of Apportionment: Divide and Round", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 71–93, doi:10.1007/978-3-319-64707-4_4, ISBN 978-3-319-64707-4, retrieved 2021-09-01

[6] "With Bader-Ofer method, not every ballot counts". The Jerusalem Post. Retrieved 2021-05-04.

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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