Highest averages method

The highest averages method or divisor method is the name for a variety of ways to allocate seats proportionally for representative assemblies with party list voting systems. It 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 nth seat is allocated to the party whose column contains the nth largest entry in this table, up to the total number of seats available.[1]

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

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.

Webster/Sainte-Laguë method

The Webster/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 though it can lead to a party with a majority of votes winning fewer than half the seats. 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 Webster/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 1, 1.5, 2, 2.5, 3, 3.5 and so on. It is designed to disfavor the smallest parties, akin to a "cutoff", and is used only in Belgian municipal elections. This method (unlike other listed methods) is not strictly proportional, if a perfectly proportional allocation exists, it is not guaranteed to find it.

Huntington–Hill method

In the Huntington–Hill method, the divisors are given by , which makes sense only if every party is guaranteed at least one seat: this effect can be achieved by disqualifying parties receiving fewer votes than a specified quota. This method is used for allotting seats in the US House of Representatives among the states.

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 divisors growing by step equal to 3 (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]

Adams's method

Adams's method was conceived by John Quincy Adams for apportioning seats of the House to states.[4] He perceived Jefferson's method to allocate too few seats to smaller states. It can be described as the inverse of Jefferson's method; it awards a seat to the party that has the most votes per seat before the seat is added.

Adams's method uses as the divisor.[5] Like the Huntington-Hill method, this results in a value of 0 for the first seats to be appointed for each party, resulting in an average of ∞. It can only violate the lower quota rule.[6] This occurs in the example below.

Without a threshold, all parties that have received at least one vote, also receive a seat, with the obvious exception of cases where there are more parties than seats. This property can be desirable, for example when apportioning seats to electoral districts. As long as there are at least as many seats as districts, all districts are represented. In a party-list proportional representation election, it may result in very small parties receiving seats. Furthermore, quota rule violations in the pure Adams's method are very common.[7] These problems may be solved by introducing an electoral threshold.[5]

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, Sainte-Laguë, Huntington–Hill and Adams's methods

D'Hondt, Sainte-Laguë and Huntington-Hill allow different strategies by parties looking to maximize their seat allocation. D'Hondt and Huntington–Hill can favor the merging of parties, while Sainte-Laguë can favor splitting parties (modified Saint-Laguë reduces the splitting advantage).

Examples

In these examples, under D'Hondt and Huntington–Hill the Yellows and Greens combined would gain an additional seat if they merged, while under Sainte-Laguë the Yellows would gain if they split into six lists with about 7,833 votes each.

The total vote is 100,000. There are 10 seats. The Huntington–Hill method threshold is 10,000, which is 1/10 of the total vote.

D'Hondt method Sainte-Laguë method (unmodified) Sainte-Laguë method (modified) Huntington–Hill method Pure Adams's method Adams's method with threshold = 1
party YellowWhiteRedGreenBluePink YellowWhiteRedGreenBluePink YellowWhiteRedGreenBluePink YellowWhiteRedGreenBluePink Yellow White Red Green Blue Pink Yellow White Red Green Blue Pink
votes 47,00016,00015,90012,0006,0003,100 47,00016,00015,90012,0006,0003,100 47,00016,00015,90012,0006,0003,100 47,000 16,000 15,900 12,000 6,000 3,100 47,000 16,000 15,900 12,000 6,000 3,100 47,000 16,000 15,900 12,000 6,000 3,100
seats 5 2 2 1 0 0 4 2 2 1 1 0 5 2 2 1 0 0 5 2 2 1 0 0 3 2 2 1 1 1 4 2 2 2 0 0
votes/seat 9,400 8,000 7,950 12,000 11,750 8,000 7,950 12,000 6,000 9,400 8,000 7,950 12,000 9,400 8,000 7,950 12,000 15,667 8,000 7,950 12,000 6,000 3,100 11,750 8,000 7,950 6,000
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 33,571 11,429 11,357 8,571 4,286 2,214 ∞ ∞ ∞ ∞ excluded ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ excluded
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 15,667 5,333 5,300 4,000 2,000 1,033 33,234 11,314 11,243 8,485 47,000 16,000 15,900 12,000 6,000 3,100 47,000 16,000 15,900 12,000
3 15,667 5,333 5,300 4,000 2,000 1,033 9,400 3,200 3,180 2,400 1,200 620 9,400 3,200 3,180 2,400 1,200 620 19,187 6,531 6,491 4,898 23,500 8,000 7,950 6,000 3,000 1,550 23,500 8,000 7,950 6,000
4 11,750 4,000 3,975 3,000 1,500 775 6,714 2,857 2,271 1,714 875 443 6,714 2,857 2,271 1,714 875 443 13,567 4,618 4,589 3,464 15,667 5,333 5,300 4,000 2,000 1,033 15,667 5,333 5,300 4,000
5 9,400 3,200 3,180 2,400 1,200 620 5,222 1,778 1,767 1,333 667 333 5,222 1,778 1,767 1,333 667 333 10,509 3,577 3,555 2,683 11,750 4,000 3,975 3,000 1,500 775 11,750 4,000 3,975 3,000
6 7,833 2,667 2,650 2,000 1,000 517 4,273 1,454 1,445 1,091 545 282 4,273 1,454 1,445 1,091 545 282 8,580 2,921 2,902 2,190 9,400 3,200 3,180 2,400 1,200 620 9,400 3,200 3,180 2,400
seat seat allocation
1 47,000 47,000 33,571 ∞ excluded ∞ ∞ excluded
2 23,500 16,000 15,667 ∞ ∞ ∞
3 16,000 15,900 11,429 ∞ ∞ ∞
4 15,900 15,667 11,357 ∞ ∞ ∞
5 15,667 12,000 9,400 33,234 ∞ 47,000
6 12,000 9,400 8,571 19,187 ∞ 23,500
7 11,750 6,714 6,714 13,567 47,000 16,000
8 9,400 6,000 5,333 11,314 23,500 15,900
9 8,000 5,333 5,300 11,243 16,000 15,667
10 7,950 5,300 5,222 10,509 15,900 12,000

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. Archived from the original (pdf) on 4 March 2016. Retrieved 30 January 2016.
  3. "The Parliamentary Electoral System in Denmark".
  4. "Apportioning Representatives in the United States Congress - Adams' Method of Apportionment | Mathematical Association of America". www.maa.org. Retrieved 2020-11-11.
  5. Gallagher, Michael (1992). "Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities" (PDF). British Journal of Political Science. 22 (4): 469–496. ISSN 0007-1234.
  6. Iian, Smythe (July 10, 2015). "MATH 1340 — Mathematics & Politics" (PDF). Retrieved November 11, 2020.
  7. Ichimori, Tetsuo (2010). "New apportionment methods and their quota property". JSIAM Letters. 2 (0): 33–36. doi:10.14495/jsiaml.2.33. ISSN 1883-0617.
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