Largest remainder method
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The largest remainder method (also known as HareNiemeyer method or as Vinton's method^{[1]}) is one way of allocating seats proportionally for representative assemblies with party list voting systems. It contrasts with the highest averages method.
Method
The largest remainder method requires the numbers of votes for each party to be divided by a quota representing the number of votes required for a seat (i.e. usually the total number of votes cast divided by the number of seats, or some similar formula). The result for each party will usually consist of an integer part plus a fractional remainder. Each party is first allocated a number of seats equal to their integer. This will generally leave some seats unallocated: the parties are then ranked on the basis of the fractional remainders, and the parties with the largest remainders are each allocated one additional seat until all the seats have been allocated. This gives the method its name.
Quotas
There are several possibilities for the quota. The most common are: the Hare quota and the Droop quota.
The Hare (or simple) Quota is defined as follows
The Hamilton method of apportionment is actually a largestremainder method which uses the Hare Quota. It is named after Alexander Hamilton, who invented the largestremainder method in 1792. It was first adopted to apportion the U.S. House of Representatives every ten years between 1852 and 1900. It is used for legislative elections in Russia (with a 7% exclusion threshold since 2007), Ukraine (5% threshold), Tunisia,^{[2]} Namibia and Hong Kong.
The Droop quota is the integer part of
and is applied in elections in South Africa. The HagenbachBischoff quota is virtually identical, being
either used as a fraction or rounded up.
The Hare quota tends to be slightly more generous to less popular parties and the Droop quota to more popular parties. This means that Hare can arguably be considered more proportional than Droop quota^{[3]}^{[4]}^{[5]}^{[6]}^{[7]} although it is more likely to give fewer than half the seats to a list with more than half the vote.
The Imperiali quota
is rarely used since it suffers from the defect that it might result in more seats being allocated than there are available (this can also occur with the HagenbachBischoff quota but it is very unlikely, and it is impossible with the Hare and Droop quotas). This will certainly happen if there are only two parties. In such a case, it is usual to increase the quota until the number of candidates elected is equal to the number of seats available, in effect changing the voting system to the Jefferson apportionment formula (see D'Hondt method).
Examples
These examples take an election to allocate 10 seats where there are 100,000 votes.
Hare quota
Party  Yellows  Whites  Reds  Greens  Blues  Pinks  Total 

Votes  47,000  16,000  15,800  12,000  6,100  3,100  100,000 
Seats  10  
Hare Quota  10,000  
Votes/Quota  4.70  1.60  1.58  1.20  0.61  0.31  
Automatic seats  4  1  1  1  0  0  7 
Remainder  0.70  0.60  0.58  0.20  0.61  0.31  
Highest Remainder Seats  1  1  0  0  1  0  3 
Total Seats  5  2  1  1  1  0  10 
Droop quota
Party  Yellows  Whites  Reds  Greens  Blues  Pinks  Total 

Votes  47,000  16,000  15,800  12,000  6,100  3,100  100,000 
Seats  10+1=11  
Droop Quota  9,091  
Votes/Quota  5.170  1.760  1.738  1.320  0.671  0.341  
Automatic seats  5  1  1  1  0  0  8 
Remainder  0.170  0.760  0.738  0.320  0.671  0.341  
Highest Remainder Seats  0  1  1  0  0  0  2 
Total Seats  5  2  2  1  0  0  10 
Pros and cons
It is relatively easy for a voter to understand how the largest remainder method allocates seats. The Hare quota gives an advantage to smaller parties while the Droop quota favours larger parties.^{[8]} However, whether a list gets an extra seat or not may well depend on how the remaining votes are distributed among other parties: it is quite possible for a party to make a slight percentage gain yet lose a seat if the votes for other parties also change. A related feature is that increasing the number of seats may cause a party to lose a seat (the socalled Alabama paradox). The highest averages methods avoid this latter paradox but since no apportionment method is entirely free from paradox,^{[9]} they introduce others like quota violation.^{[10]}
Technical evaluation and paradoxes
The largest remainder method is the only apportionment that satisfies the quota rule; in fact, it is designed to satisfy this criterion. However, it comes at the cost of paradoxical behaviour. The Alabama paradox is exhibited when an increase in seats apportioned leads to a decrease in the number of seats allocated to a certain party. In the example below, when the number of seats to be allocated is increased from 25 to 26 (with the number of votes held constant), parties D and E counterintuitively end up with fewer seats.
With 25 seats, the results are:
Party  A  B  C  D  E  F  Total 

Votes  1500  1500  900  500  500  200  5100 
Seats  25  
Hare Quota  204  
Quotas Received  7.35  7.35  4.41  2.45  2.45  0.98  
Automatic seats  7  7  4  2  2  0  22 
Remainder  0.35  0.35  0.41  0.45  0.45  0.98  
Surplus seats  0  0  0  1  1  1  3 
Total Seats  7  7  4  3  3  1  25 
With 26 seats, the results are:
Party  A  B  C  D  E  F  Total 

Votes  1500  1500  900  500  500  200  5100 
Seats  26  
Hare Quota  196  
Quotas Received  7.65  7.65  4.59  2.55  2.55  1.02  
Automatic seats  7  7  4  2  2  1  23 
Remainder  0.65  0.65  0.59  0.55  0.55  0.02  
Surplus seats  1  1  1  0  0  0  3 
Total Seats  8  8  5  2  2  1  26 
See also
References
 ↑ Tannenbaum, Peter (2010). Excursions in Modern Mathematics. New York: Prentice Hall. p. 128. ISBN 9780321568038.
 ↑ "2". Proposed Basic Law on Elections and Referendums  Tunisia (Nonofficial translation to English). International IDEA. 26 January 2014. p. 25. Retrieved 9 August 2015.
 ↑ http://www.parl.gc.ca/Content/LOP/researchpublications/bp334e.pdf
 ↑ http://polmeth.wustl.edu/polanalysis/vol/8/PA84381388.pdf
 ↑ http://www.dur.ac.uk/john.ashworth/EPCS/Papers/Suojanen.pdf
 ↑ http://users.ox.ac.uk/~sann2300/041102cegelectoralconsequenceslijphart.shtml
 ↑ http://janda.org/c24/Readings/Lijphart/Lijphart.html
 ↑ See for example the 2012 election in Hong Kong Island where the DAB ran as two lists and gained twice as many seats as the singlelist Civic despite receiving fewer votes in total: New York Times report
 ↑ Balinski, Michel; H. Peyton Young (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. Yale Univ Pr. ISBN 0300027249.
 ↑ Messner; et al. "RangeVoting: Apportionment and rounding schemes". Retrieved 20140202.