Highest averages method

The highest averages method is one way of allocating seats proportionally for representative assemblies with party list voting systems.

The "highest averages method" requires the number of votes for each party to be divided successively by a series of divisors, and seats are allocated to parties that secure the highest resulting quotient or average, up to the total number of seats available. The most widely used is the d'Hondt formula, using the divisors 1,2,3,4... The Sainte-Laguë method divides the votes with odd numbers (1,3,5,7 etc). The Sainte-Laguë method can also be modified, for instance by the replacement of the first divisor by 1.4, which in small constituencies has the effect of prioritizing proportionality for larger parties over smaller ones at the allocation of the first few seats.

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 used only in Belgian municipal elections. 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 (while this is not strictly speaking an election, it nevertheless uses a highest average method).

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.

An alternative to the "highest averages method" is the largest remainder method, which use a minimum quota which can be calculated in a number of ways.

Comparison between the "d'Hondt" and "Sainte-Laguë" methods

The unmodified "Sainte-Laguë" method shows differences for the first mandates

	d'Hondt method							unmodified Sainte-Laguë method
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		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							modified Sainte-Laguë method
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