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. 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.
An alternative to this method is the largest remainder method, which uses a minimum quota which can be calculated in a number of ways.
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.
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".
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.
In the Huntington-Hill method, the divisors are given by , 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.)
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