Quota rule

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In mathematics and political science, the quotarule describes a desired property of proportional apportionment methods. It says that the number of seats allocated to a party should be equal to their entitlement plus or minus one. [1] [2] [note 1] The ideal number of seats for a party, called their seat entitlement, is calculated by multiplying each party's share of the vote by the total number of seats. Equivalently, it is equal to the number of votes divided by the Hare quota. For example, if a party receives 10.56% of the vote, and there are 100 seats in a parliament, the quota rule says that when all 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 in situations where upholding it would cause a population paradox, although unbiased apportionment rules like Webster's method do so only rarely.

Contents

Mathematics

The entitlement for a party (the number of seats the party would ideally get) is:

The lower frame is then the entitlement rounded down to the nearest integer while the upper frame is the entitlement rounded up. The frame rule states that the only two allocations that a party can receive should be either the lower or upper frame. [1] If at any time an allocation gives a party a greater or lesser number of seats than the upper or lower frame, that allocation (and by extension, the method used to allocate it) is said to be in violation of the quota rule.

Example

If there are 5 available seats in the council of a club with 300 members, and party A has 106 members, then the entitlement for party A is . The lower frame for party A is 1, because 1.8 rounded down equal 1. The upper frame, 1.8 rounded up, is 2. Therefore, the quota rule states that the only two allocations allowed for party A are 1 or 2 seats on the council. If there is a second party, B, that has 137 members, then the quota rule states that party B gets , rounded up and down equals either 2 or 3 seats. Finally, a party C with the remaining 57 members of the club has a entitlement of , which means its allocated seats should be either 0 or 1. In all cases, the method for actually allocating the seats determines whether an allocation violates the quota rule, which in this case would mean giving party A any seats other than 1 or 2, giving party B any other than 2 or 3, or giving party C any other than 0 or 1 seat.

Relation to apportionment paradoxes

The Balinski–Young theorem proved in 1980 that if an apportionment method satisfies the quota rule, it must fail to satisfy some apportionment paradox. [3] For instance, although largest remainder method satisfies the quota rule, it violates the Alabama paradox and the population paradox. The theorem itself is broken up into several different proofs that cover a wide number of circumstances. [4]

Specifically, there are two main statements that apply to the quota rule:

Use in apportionment methods

Different methods for allocating seats may or may not satisfy the quota rule. While many methods do violate the quota rule, it is sometimes preferable to violate the rule very rarely than to violate some other apportionment paradox; some sophisticated methods violate the rule so rarely that it has not ever happened in a real apportionment, while some methods that never violate the quota rule violate other paradoxes in much more serious fashions.

The largest remainder method does satisfy the quota rule. The method works by assigning each party its seat quota, rounded down. Then, the surplus seats are given to the party with the largest fractional part, until there are no more surplus seats. Because it is impossible to give more than one surplus seat to a party, every party will always be equal to its lower or upper frame. [5]

The D'Hondt method, also known as the Jefferson method [6] sometimes violates the quota rule by allocating more seats than the upper frame allowed. [7] Since Jefferson was the first method used for Congressional apportionment in the United States, this violation led to a substantial problem where larger states often received more representatives than smaller states, which was not corrected until Webster's method was implemented in 1842. Although Webster's method can in theory violate the quota rule, such occurrences are extremely rare. [8]

Notes

  1. ↑ The entitlement for a party is sometimes called their seat quota, leading to the term "quota rule"; such seat quotas should not be confused with the unrelated concept of an electoral quota.

Related Research Articles

In the study of electoral systems, the Droop quota is the minimum number of votes needed for a party or candidate to guarantee they will win at least one seat in a legislature.

The D'Hondt method, also called the Jefferson method or the greatest divisors method, is an apportionment method for allocating seats in parliaments among federal states, or in proportional representation among political parties. It belongs to the class of highest-averages methods. Compared to ideal proportional representation, the D'Hondt method reduces somewhat the political fragmentation for smaller electoral district sizes, where it favors larger political parties over small parties.

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An apportionment paradox is a situation where an apportionment—a rule for dividing discrete objects according to some proportional relationship—produces results that violate notions of common sense or fairness.

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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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Apportionment in the Hellenic Parliament refers to those provisions of the Greek electoral law relating to the distribution of Greece's 300 parliamentary seats to the parliamentary constituencies, as well as to the method of seat allocation in Greek legislative elections for the various political parties. The electoral law was codified for the first time through a 2012 Presidential Decree. Articles 1, 2, and 3 deal with how the parliamentary seats are allocated to the various constituencies, while articles 99 and 100 legislate the method of parliamentary apportionment for political parties in an election. In both cases, Greece uses the largest remainder method.

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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. Apportionments violating this rule are called population paradoxes.

Seat bias is a property describing methods of apportionment. These are methods used to allocate seats in a parliament among federal states or among political parties. A method is biased if it systematically favors small parties over large parties, or vice versa. There are several mathematical measures of bias, which can disagree slightly.

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.

Optimal apportionment is an approach to apportionment that is based on mathematical optimization.

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.

In apportionment theory, rank-index methods are a set of apportionment methods that generalize the divisor method. These have also been called Huntington methods, since they generalize an idea by Edward Vermilye Huntington.

References

  1. 1 2 Michael J. Caulfield. "Apportioning Representatives in the United States Congress - The Quota Rule" Archived 2019-05-22 at the Wayback Machine . MAA Publications. Retrieved October 22, 2018
  2. ↑ Alan Stein. Apportionment Methods Retrieved December 9, 2018
  3. ↑ Beth-Allyn Osikiewicz, Ph.D. Impossibilities of Apportionment Archived 2020-09-29 at the Wayback Machine Retrieved October 23, 2018.
  4. 1 2 3 M.L. Balinski and H.P. Young. (1980). "The Theory of Apportionment" Archived 2024-07-31 at the Wayback Machine . Retrieved October 23 2018
  5. ↑ Hilary Freeman. "Apportionment" Archived 2018-09-20 at the Wayback Machine . Retrieved October 22 2018
  6. ↑ "Apportionment 2" Retrieved October 22, 2018.
  7. ↑ Jefferson’s Method Archived 2021-01-20 at the Wayback Machine Retrieved October 22, 2018.
  8. ↑ Ghidewon Abay Asmerom. Apportionment. Lecture 4. Archived 2020-09-27 at the Wayback Machine Retrieved October 23, 2018.
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