Highest averages method
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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.^{[1]}
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.
SainteLaguë method
The SainteLaguë 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 SainteLaguë 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.
HuntingtonHill method
In the HuntingtonHill 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.)
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 multimember constituencies. It divides the number of votes received by a party in a multimember 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 SainteLaguë 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 SainteLaguë 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 SainteLaguë methods
D'Hondt and SainteLaguë allow different strategies by parties looking to maximize their seat allocation. D'Hondt can favor the merging of parties, while SainteLaguë can favor splitting parties (modified SaintLaguë 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 SaintLaguë the Yellows would gain if they split into six lists with about 7,833 votes each.
Examples
D'Hondt method  SainteLaguë 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  SainteLaguë 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
 ↑ Norris, Pippa (2004). Electoral Engineering: Voting Rules and Political Behavior. Cambridge University Press. p. 51. ISBN 0521829771.
 ↑ Gallagher, Michael (1991). "Proportionality, disproportionality and electoral systems" (pdf). Electoral Studies. 10 (1). doi:10.1016/02613794(91)90004C. Retrieved 30 January 2016.
 ↑ "The Parliamentary Electoral System in Denmark".