Largest remainders method

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The largest remainders methods [1] are one way of allocating seats proportionally for representative assemblies based on party list voting systems, or for allocating . They contrast with the more popular highest averages methods (also known as divisor methods), which are the preferred system for allocating

Contents

When using the Hare quota (described below), the method is known as the Hare–Niemeyer method, Hamilton's method, or occasionally as Vinton's 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. Usually, this is given by the total number of votes cast, divided by the number of seats. 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 remainder seats unallocated. To apportion these seats, the parties are then ranked on the basis of their fractional remainders, and the parties with the largest remainders are each allocated one additional seat until all seats have been allocated. This gives the method its name.

Quotas

There are several possible choices for the electoral quota; the choice of quota affects the properties of the , with smaller quotas leaving fewer seats left over for small parties to pick up, and larger quotas leaving more seats.

The two most common quotas are the Hare quota and the Droop quota. The use of a particular quota with the largest remainders method is often abbreviated as "LR-[quota name]", such as "LR-Droop". [2]

The Hare (or simple) quota is defined as follows:

It is used for legislative elections in Russia (with a 5% exclusion threshold since 2016), Ukraine (5% threshold), Bulgaria (4% threshold), Lithuania (5% threshold for party and 7% threshold for coalition), Tunisia, [3] Taiwan (5% threshold), Namibia and Hong Kong. LR-Hare is sometimes called Hamilton's method, named after Alexander Hamilton, who devised the method in 1792. [4]

The Droop quota is given by:

and is applied to elections in South Africa.

The Hare quota is more generous to less popular parties and the Droop quota to more popular parties. Specifically, the Hare quota is unbiased in the number of seats it hands out, and so is more proportional than the Droop quota (which tends to be biased towards larger parties). [5] [6] [7] [8]

Examples

These examples take an election to allocate 10 seats where there are 100,000 votes.

Hare quota

PartyYellowsWhitesRedsGreensBluesPinksTotal
Votes47,00016,00015,80012,0006,1003,100100,000
Seats10
Hare Quota10,000
Votes/Quota4.701.601.581.200.610.31
Automatic seats4111007
Remainder0.700.600.580.200.610.31
Highest-remainder seats1100103
Total seats52111010

Droop quota

PartyYellowsWhitesRedsGreensBluesPinksTotal
Votes47,00016,00015,80012,0006,1003,100100,000
Seats10+1=11
Droop quota9,091
Votes/quota5.1701.7601.7381.3200.6710.341
Automatic seats5111008
Remainder0.1700.7600.7380.3200.6710.341
Highest-remainder seats0110002
Total seats52210010

Pros and cons

It is easy for a voter to understand how the largest remainder method allocates seats. The Hare quota gives no advantage to larger or smaller parties, while the Droop quota is biased in favor of larger parties. [9] However, in small legislatures with no threshold, the Hare quota can be manipulated by running candidates on many small lists, allowing each list to pick up a single remainder seat. [10]

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, though at the cost of 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:

PartyABCDEFTotal
Votes150015009005005002005100
Seats25
Hare quota204
Quotas received7.357.354.412.452.450.98
Automatic seats77422022
Remainder0.350.350.410.450.450.98
Surplus seats0001113
Total seats77433125

With 26 seats, the results are:

PartyABCDEFTotal
Votes150015009005005002005100
Seats26
Hare quota196
Quotas received7.657.654.592.552.551.02
Automatic seats77422123
Remainder0.650.650.590.550.550.02
Surplus seats1110003
Total seats88522126

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References

  1. ↑ Tannenbaum, Peter (2010). Excursions in Modern Mathematics. New York: Prentice Hall. p. 128. ISBN   978-0-321-56803-8.
  2. ↑ Gallagher, Michael; Mitchell, Paul (2005-09-15). The Politics of Electoral Systems. OUP Oxford. ISBN   978-0-19-153151-4.
  3. ↑ "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.
  4. ↑ Eerik Lagerspetz (26 November 2015). Social Choice and Democratic Values. Studies in Choice and Welfare. Springer. ISBN   9783319232614 . Retrieved 2017-08-17.
  5. ↑ "Archived copy" (PDF). Archived from the original (PDF) on 2015-09-24. Retrieved 2011-05-08.{{cite web}}: CS1 maint: archived copy as title (link)
  6. ↑ "Archived copy" (PDF). Archived from the original (PDF) on 2006-09-01. Retrieved 2006-09-01.{{cite web}}: CS1 maint: archived copy as title (link)
  7. ↑ "Archived copy" (PDF). Archived from the original (PDF) on 2007-09-26. Retrieved 2007-09-26.{{cite web}}: CS1 maint: archived copy as title (link)
  8. ↑ "Lipjhart on PR formulas".
  9. ↑ "Notes on the Political Consequences of Electoral Laws by Lijphart, Arend, American Political Science Review Vol. 84, No 2 1990". Archived from the original on 2006-05-16. Retrieved 2006-05-16.
  10. ↑ 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
  11. ↑ Balinski, Michel; H. Peyton Young (1982). Fair Representation: Meeting the Ideal of One Man, One Vote . Yale Univ Pr. ISBN   0-300-02724-9.
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