Largest remainder method

The largest remainder method (also known as Hare–Niemeyer method, Hamilton 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 various highest averages methods (also known as divisor methods).

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

${\displaystyle {\frac {\text{total votes}}{\text{total seats}}}}$

It is used for legislative elections in Russia (with a 5% exclusion threshold since 2016), Ukraine (5% threshold), Tunisia,[2] Taiwan (5% threshold), Namibia and Hong Kong. The Hamilton method of apportionment is actually a largest-remainder method which uses the Hare Quota. It is named after Alexander Hamilton, who invented the largest-remainder method in 1792.[3] It was first adopted to apportion the U.S. House of Representatives every ten years between 1852 and 1900.

The Droop quota is the integer part of

${\displaystyle 1+{\frac {\text{total votes}}{1+{\text{total seats}}}}}$

and is applied in elections in South Africa. The Hagenbach-Bischoff quota is virtually identical, being

${\displaystyle {\frac {\text{total votes}}{1+{\text{total seats}}}}}$

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. [4][5][6][7][8]

The Imperiali quota

${\displaystyle {\frac {\text{total votes}}{2+{\text{total seats}}}}}$

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 Hagenbach-Bischoff 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.[9] 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 so-called Alabama paradox). The highest averages methods avoid this latter paradox; but since no apportionment method is entirely free from paradox,[10] they introduce others like quota violation.[11]

Technical evaluation and paradoxes

The largest remainder method satisfies the quota rule (each party's seats amount to its ideal share of seats, either rounded up or rounded down) and was designed to satisfy that criterion. However, that 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

References

1. Tannenbaum, Peter (2010). Excursions in Modern Mathematics. New York: Prentice Hall. p. 128. ISBN 978-0-321-56803-8.
2. "2". Proposed Basic Law on Elections and Referendums - Tunisia (Non-official translation to English). International IDEA. 26 January 2014. p. 25. Retrieved 9 August 2015.
3. Eerik Lagerspetz. Social Choice and Democratic Values. Studies in Choice and Welfare. Springer. Retrieved 2017-08-17.
4. http://www.parl.gc.ca/Content/LOP/researchpublications/bp334-e.pdf
5. "Archived copy" (PDF). Archived from the original (PDF) on 2006-09-01. Retrieved 2006-09-01.CS1 maint: archived copy as title (link)
6. "Archived copy" (PDF). Archived from the original (PDF) on 2007-09-26. Retrieved 2007-09-26.CS1 maint: archived copy as title (link)
7. "Archived copy". Archived from the original on 2006-05-16. Retrieved 2006-05-16.CS1 maint: archived copy as title (link)
8. http://janda.org/c24/Readings/Lijphart/Lijphart.html
9. 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 single-list Civic despite receiving fewer votes in total: New York Times report
10. Balinski, Michel; H. Peyton Young (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. Yale Univ Pr. ISBN 0-300-02724-9.
11. Messner; et al. "RangeVoting: Apportionment and rounding schemes". Retrieved 2014-02-02.

External links

• Hamilton method experimentation applet at cut-the-knot