Largest remainders method

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The largest remainder methods or quota methods are methods of allocating seats proportionally that are based on calculating a quota, i.e. a certain number of votes needed to be guaranteed a seat in parliament. Then, any leftover seats are handed over to "plurality" winners (the parties with the largest remainders, i.e. the most "leftover" votes). [1] They are typically contrasted with the more popular highest averages methods (also called divisor methods). [2]

Contents

Divisor methods are generally preferred by social choice theorists to the largest remainder methods because they are less susceptible to apportionment paradoxes. [2] [3] In particular, divisor methods satisfy population monotonicity, i.e. voting for a party can never cause it to lose seats. [3] Such population paradoxes occur by increasing the electoral quota, which can cause different states' remainders to respond erratically. [4] Divisor methods also satisfy resource or house monotonicity, which says that increasing the number of seats in a legislature should not cause a state to lose a seat (a situation known as an Alabama paradox). [3] [4] : Cor.4.3.1 

When using the Hare quota, the method is known as the Hare–Niemeyer or Hamilton method.

Method

The largest remainder methods require the numbers of votes for each party to be divided by a quota representing the number of votes required to win 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 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.

Largest remainder methods can also be used to apportion votes among solid coalitions, as in the case of the single transferable vote, which becomes the largest-remainders method when voters are all partisans (i.e. only rank candidates of their own party). [5]

Quotas

There are several possible choices for the electoral quota; the choice of quota affects the properties of the corresponding largest remainder method, with smaller quotas leaving fewer seats left over for small parties to pick up, and larger quotas leaving more seats. As a result, a larger quota is, somewhat counterintuitively, always more favorable to smaller parties. [6]

The two most common quotas are the Hare quota and the Droop quota. The use of a particular quota with one of the largest remainder methods is often abbreviated as "LR-[quota name]", such as "LR-Droop". [7]

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, [8] 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. [9]

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).

Examples

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

Hare quota

[1]

Droop quota

PartyYellowsWhitesRedsGreensBluesPinksTotal
Votes47,00016,00015,80012,0006,1003,100100,000
Seats (divisor)10 (10+1=11)
Droop quota9,091
Ideal seats5.1701.7601.7381.3200.6710.341
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. [10] 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. [11]

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 Alabama paradox). The highest averages methods avoid this latter paradox, though at the cost of very rare quota violations. [12]

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, this comes at the cost of paradoxical behavior. The Alabama paradox is when an increase in the total number of seats 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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In mathematics, economics, and social choice theory, the highest averages method, also called the divisor method, is an apportionment algorithm best-known for its use in proportional representation. Divisor algorithms seek to fairly divide a legislature between several groups, such as political parties or states.

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In the study of apportionment, the Harequota is the number of voters represented by each legislator under a system of proportional representation. In these voting systems, the quota is the number of votes that guarantees a candidate, or a party in some cases, captures a seat. The Hare quota is the total number of votes divided by the number of seats to be filled.

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  1. Example: proportional by party and by region
  2. Then, as nearly as possible given the totals for each region and each party:

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In mathematics and political science, the quota rule describes a desired property of a proportional apportionment or election method. It says the number of seats allocated to a party should fall inside the ideal region i.e. somewhere between the floor and ceiling of the ideal value. For example, if a party should ideally receive 10.56 seats in parliament, the quota rule says that when the seats are allotted, the party may get either 10 or 11 seats. The most common apportionment methods, the highest averages methods, violate the quota rule.

Mathematics of apportionment describes mathematical principles and algorithms for fair allocation of identical items among parties with different entitlements. Such principles are used to apportion seats in parliaments among federal states or political parties. See apportionment (politics) for the more concrete principles and issues related to apportionment, and apportionment by country for practical methods used around the world.

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State-population monotonicity is a property of apportionment methods, which are methods of allocating seats in a parliament among federal states or political parties. The property says that if the population of State A increases faster than that of State B, then State A should not lose any seats to State B. Apportionment methods violating this rule are called population paradoxes.

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Coherence, also called uniformity or consistency, is a criterion for evaluating rules for fair division. Coherence requires that the outcome of a fairness rule is fair not only for the overall problem, but also for each sub-problem. Every part of a fair division should be fair.

Vote-ratio monotonicity (VRM) is a property of apportionment methods, which are methods of allocating seats in a parliament among political parties. The property says that, if the ratio between the number of votes won by party A to the number of votes won by party B increases, then it should NOT happen that party A loses a seat while party B gains a seat.

Balance or balancedness is a property of apportionment methods, which are methods of allocating identical items between among agens, such as dividing seats in a parliament among political parties or federal states. The property says that, if two agents have exactly the same entitlements, then the number of items they receive should differ by at most one. So if two parties win the same number of votes, or two states have the same populations, then the number of seats they receive should differ by at most one.

References

  1. 1 2 Tannenbaum, Peter (2010). Excursions in Modern Mathematics. New York: Prentice Hall. p. 128. ISBN   978-0-321-56803-8.
  2. 1 2 Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Quota Methods of Apportionment: Divide and Rank", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 95–105, doi:10.1007/978-3-319-64707-4_5, ISBN   978-3-319-64707-4 , retrieved 2024-05-10
  3. 1 2 3 Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Securing System Consistency: Coherence and Paradoxes", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 159–183, doi:10.1007/978-3-319-64707-4_9, ISBN   978-3-319-64707-4 , retrieved 2024-05-10
  4. 1 2 Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote . New Haven: Yale University Press. ISBN   0-300-02724-9.
  5. ↑ Gallagher, Michael (1992). "Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities". British Journal of Political Science. 22 (4): 469–496. ISSN   0007-1234.
  6. ↑ Gallagher, Michael (1992). "Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities". British Journal of Political Science. 22 (4): 469–496. ISSN   0007-1234.
  7. ↑ Gallagher, Michael; Mitchell, Paul (2005-09-15). The Politics of Electoral Systems. OUP Oxford. ISBN   978-0-19-153151-4.
  8. ↑ "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.
  9. ↑ Eerik Lagerspetz (26 November 2015). Social Choice and Democratic Values. Studies in Choice and Welfare. Springer. ISBN   9783319232614 . Retrieved 2017-08-17.
  10. ↑ "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.
  11. ↑ 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
  12. ↑ 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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