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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 Sainte-Laguë method divides the votes with odd numbers (1,3,5,7 etc). The Sainte-Laguë 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 Huntington-Hill method, the divisors are given by \sqrt{n(n+1)}, 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 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.

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.

Comparison between the d'Hondt and Sainte-Laguë methods

The unmodified Sainte-Laguë method shows differences for the first mandates

d'Hondt method   unmodified Sainte-Laguë 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      

With the modification, the methods are initially more similar

d'Hondt method   modified Sainte-Laguë 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          







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