Highest averages method

The highest averages method is the name for a variety of ways to allocate seats proportionally for representative assemblies with party list voting systems.

An alternative to this method is the largest remainder method, which uses a minimum quota which can be calculated in a number of ways.

Overview

The highest averages method requires the number of votes for each party to be divided successively by a series of divisors. This produces a table of quotients, or averages, with a row for each divisor and a column for each party. The $n$'th seat is allocated to the party whose column contains the $n$'th largest entry in this table, up to the total number of seats available.^[1] See examples below.

Individual methods

D'Hondt method

The most widely used is the D'Hondt formula, using the divisors 1, 2, 3, 4, etc.^[2] This system tends to give larger parties a slightly larger portion of seats than their portion of the electorate, and thus guarantees that a party with a majority of voters will get at least half of the seats.

Sainte-Laguë method

The Sainte-Laguë method divides the number of votes for each party by the odd numbers (1, 3, 5, 7 etc.) and is sometimes considered "more proportional" than D'Hondt in terms of a comparison between a party's share of the total vote and its share of the seat allocation. This system can favour smaller parties over larger parties and so encourage splits. Dividing the votes numbers by 0.5, 1.5, 2.5, 3.5 etc. yields the same result.

The Sainte-Laguë method is sometimes modified by increasing the first divisor to e.g. 1.4, to discourage very small parties gaining their first seat "too cheaply".

Imperiali

Another highest average method is called Imperiali (not to be confused with the Imperiali quota which is a Largest remainder method). The divisors are 2, 3, 4, etc. It is designed to disfavor the smallest parties, akin to a "cutoff", and is used only in Belgian municipal elections.

Huntington-Hill method

In the Huntington-Hill method, the divisors are given by $\sqrt{n(n+1)}$, which makes sense only if every party is guaranteed at least one seat: this is used for allotting seats in the US House of Representatives to the states. (This is not an election, of course.)

Danish method

The Danish method is used in Danish elections to allocate each party's compensatory seats (or levelling seats) at the electoral province level to individual multi-member constituencies. It divides the number of votes received by a party in a multi-member constituency by the growing divisors (1, 4, 7, 10, etc.). Alternatively, dividing the votes numbers by 0.33, 1.33, 2.33, 3.33 etc. yields the same result. This system purposely attempts to allocate seats equally rather than proportionately.^[3]

Quota system

In addition to the procedure above, highest averages methods can be conceived of in a different way. For an election, a quota is calculated, usually the total number of votes cast divided by the number of seats to be allocated (the Hare quota). Parties are then allocated seats by determining how many quotas they have won, by dividing their vote totals by the quota. Where a party wins a fraction of a quota, this can be rounded down or rounded to the nearest whole number. Rounding down is equivalent to using the D'Hondt method, while rounding to the nearest whole number is equivalent to the Sainte-Laguë method. However, because of the rounding, this will not necessarily result in the desired number of seats being filled. In that case, the quota may be adjusted up or down until the number of seats after rounding is equal to the desired number.

The tables used in the D'Hondt or Sainte-Laguë methods can then be viewed as calculating the highest quota possible to round off to a given number of seats. For example, the quotient which wins the first seat in a D'Hondt calculation is the highest quota possible to have one party's vote, when rounded down, be greater than 1 quota and thus allocate 1 seat. The quotient for the second round is the highest divisor possible to have a total of 2 seats allocated, and so on.

Comparison between the D'Hondt and Sainte-Laguë methods

D'Hondt and Sainte-Laguë allow different strategies by parties looking to maximize their seat allocation. D'Hondt can favor the merging of parties, while Sainte-Laguë can favor splitting parties (modified Saint-Laguë reduces the splitting advantage). In the following example, under D'Hondt the Yellows and Greens combined would gain an additional seat if they merged, while under Saint-Laguë the Yellows would gain if they split into six lists with about 7,833 votes each.

Examples

	D'Hondt method							Sainte-Laguë method (unmodified)
party	Yellow	White	Red	Green	Blue	Pink		Yellow	White	Red	Green	Blue	Pink
votes	47,000	16,000	15,900	12,000	6,000	3,100		47,000	16,000	15,900	12,000	6,000	3,100
mandate	quotient
1	47,000	16,000	15,900	12,000	6,000	3,100		47,000	16,000	15,900	12,000	6,000	3,100
2	23,500	8,000	7,950	6,000	3,000	1,550		15,667	5,333	5,300	4,000	2,000	1,033
3	15,667	5,333	5,300	4,000	2,000	1,033		9,400	3,200	3,180	2,400	1,200	620
4	11,750	4,000	3,975	3,000	1,500	775		6,714	2,857	2,271	1,714	875	443
5	9,400	3,200	3,180	2,400	1,200	620		5,222	1,778	1,767	1,333	667	333
6	7,833	2,667	2,650	2,000	1,000	517		4,273	1,454	1,445	1,091	545	282
seat	seat allocation
1	47,000							47,000
2	23,500								16,000
3		16,000								15,900
4			15,900					15,667
5	15,667										12,000
6				12,000				9,400
7	11,750							6,714
8	9,400											6,000
9		8,000							5,333
10			7,950							5,300

With the modification, the methods are initially more similar:

	D'Hondt method							Sainte-Laguë method (modified)
parties	Yellows	Whites	Reds	Greens	Blues	Pinks		Yellows	Whites	Reds	Greens	Blues	Pinks
votes	47,000	16,000	15,900	12,000	6,000	3,100		47,000	16,000	15,900	12,000	6,000	3,100
mandate	quotient
1	47,000	16,000	15,900	12,000	6,000	3,100		33,571	11,429	11,357	8,571	4,286	2,214
2	23,500	8,000	7,950	6,000	3,000	1,550		15,667	5,333	5,300	4,000	2,000	1,033
3	15,667	5,333	5,300	4,000	2,000	1,033		9,400	3,200	3,180	2,400	1,200	620
4	11,750	4,000	3,975	3,000	1,500	775		6,714	2,857	2,271	1,714	875	443
5	9,400	3,200	3,180	2,400	1,200	620		5,222	1,778	1,767	1,333	667	333
6	7,833	2,667	2,650	2,000	1,000	517		4,273	1,454	1,445	1,091	545	282
seat	seat allocation
1	47,000							33,571
2	23,500							15,667
3		16,000							11,429
4			15,900							11,357
5	15,667							9,400
6				12,000							8,571
7	11,750							6,714
8	9,400								5,333
9		8,000								5,300
10			7,950					5,222

References

1. Norris, Pippa (2004). Electoral Engineering: Voting Rules and Political Behavior. Cambridge University Press. p. 51. ISBN 0-521-82977-1.
2. Gallagher, Michael (1991). "Proportionality, disproportionality and electoral systems" (pdf). Electoral Studies 10 (1). doi:10.1016/0261-3794(91)90004-C. Retrieved 30 January 2016.
3. The Parliamentary Electoral System in Denmark PDF