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 th seat is allocated to the party whose column contains the th largest entry in this table, up to the total number of seats available.
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. 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 formula_1, 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.
Adams's method.
Adams's method was conceived by John Quincy Adams for apportioning seats of the House to states. 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 formula_2 as the divisor. 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. 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. These problems may be solved by introducing an electoral threshold.
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.