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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 SainteLaguë method divides the votes with odd numbers (1,3,5,7 etc). The SainteLaguë 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 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 (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 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.
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.
d'Hondt method  unmodified SainteLaguë 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 
d'Hondt method  modified SainteLaguë 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 
