Quota method
 | This article needs additional citations for verification. (November 2011) |
| A joint Politics and Economics series |
| Social choice and electoral systems |
|---|
 |
 Mathematics portal |
The quota or divide-and-rank methods make up a category of apportionment rules, i.e. algorithms for allocating seats in a legislative body among multiple groups (e.g. parties or federal states). The quota methods begin by calculating an entitlement (basic number of seats) for each party, by dividing their vote totals by an electoral quota (a fixed number of votes needed to win a seat, as a unit). Then, leftover seats, if any are allocated by rounding up the apportionment for some parties. These rules are typically contrasted with the more popular highest averages methods (also called divisor methods).[1]
By far the most common quota method are the largest remainders or quota-shift methods, which assign any leftover seats to the "plurality" winners (the parties with the largest remainders, i.e. most leftover votes).[2]
When using the Hare quota, this rule is called Hamilton's method, and is the third-most common apportionment rule worldwide (after Jefferson's method and Webster's method).[1]
Despite their intuitive definition, quota methods are generally disfavored by social choice theorists as a result of apportionment paradoxes.[1][3] In particular, the largest remainder methods exhibit the no-show paradox, i.e. voting for a party can cause it to lose seats.[3][4] The largest remainders methods are also vulnerable to spoiler effects and can fail resource or house monotonicity, which says that increasing the number of seats in a legislature should not cause a party to lose a seat (a situation known as an Alabama paradox).[3][4]: Cor.4.3.1
Method
[edit]The quota method divides each party's vote total by a quota. Usually, the quota is calculated by dividing the total number of valid votes by the total number of seats to be allocated. This division results in each party having a number consisting of an integer part plus a fractional remainder. Initially, each party is allocated a number of seats equal to the integer part of this division. After this initial allocation, there are usually some leftover seats remaining unallocated.
To allocate the leftover seats, parties are ranked according to the size of their fractional remainders. The parties with the largest fractional remainders are each given one additional seat until all seats have been assigned. This process is why the method is often called the largest remainder method.
The quota method ensures that the total seats allocated matches the total available seats, but it may suffer from apportionment paradoxes such as the Alabama paradox or the no-show paradox.)
The Hamilton method is a specific application of the quota method that uses the Hare quota, which is calculated by dividing the total number of valid votes by the number of seats to be filled. Under this method, each party's vote total is divided by the Hare quota to determine its initial seat allocation (the integer part), and the remaining seats are distributed to parties with the largest fractional remainders.
The Hamilton method, named after Alexander Hamilton, was proposed for seat apportionment in the United States House of Representatives. Despite its simplicity, it can exhibit paradoxes such as the Alabama paradox. [5].
Quotas
[edit]There are several possible choices for the electoral quota. The choice of quota affects the properties of the corresponding largest remainder method, and particularly the seat bias. Smaller quotas allow small parties to pick up seats, while larger quotas leave behind more votes. A somewhat counterintuitive result of this is that a larger quota will always be more favorable to smaller parties.[6] A party hoping to win multiple seats sees fewer votes captured by a single popular candidate when the quota is small.
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:
LR-Hare is sometimes called Hamilton's method, named after Alexander Hamilton, who devised the method in 1792.[8]
The Droop quota is given by:
and is applied to elections in South Africa.[citation needed]
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 give more seats to larger parties). The Hare suffers the disproportionality that it sometimes allocates a majority of seats to a party with less than a majority of votes in a district.[9]
Examples
[edit]The following example allocates 11 seats using the largest-remainder method by Droop quota.
| Party | Votes | Entitlement | Remainder | Total seats |
|---|---|---|---|---|
| Yellows | 47,000 | 5.170 | 0.170 | 5 |
| Whites | 16,000 | 1.760 | 0.760 | 2 |
| Reds | 15,800 | 1.738 | 0.738 | 2 |
| Greens | 12,000 | 1.320 | 0.320 | 1 |
| Blues | 6,100 | 0.671 | 0.671 | 1 |
| Pinks | 3,100 | 0.341 | 0.341 | 0 |
| Total | 100,000 | 8/11 | 3 | 11 |
The following example allocates 11 seats using the largest-remainder method by Hare quota (Hamilton method).
| Party | Votes | Entitlement | Remainder | Total seats |
|---|---|---|---|---|
| Yellows | 47,000 | 5.170 | 0.170 | 5 |
| Whites | 16,000 | 1.760 | 0.760 | 2 |
| Reds | 15,800 | 1.738 | 0.738 | 2 |
| Greens | 12,000 | 1.320 | 0.320 | 1 |
| Blues | 6,100 | 0.671 | 0.671 | 1 |
| Pinks | 3,100 | 0.341 | 0.341 | 0 |
| Total | 100,000 | 11.000 | 0 | 11 |
Pros and cons
[edit]It is easy for a voter to understand how the largest remainder method allocates seats. Moreover, the largest remainder method satisfies the quota rule (each party's seats are equal 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 greater inequalities in the seats-to-votes ratio, which can violate the principle of one man, one vote.
However, a greater concern for social choice theorists, and the primary cause behind its abandonment in many countries, is the tendency of such rules to produce erratic or irrational behaviors called apportionment paradoxes:
- Increasing the number of seats in a legislature can decrease a party's apportionment of seats, called the Alabama paradox.
- Adding more parties to the legislature can cause a bizarre kind of spoiler effect called the new state paradox.
- When Congress first admitted Oklahoma to the Union, the House was expanded by 5 seats, equal to Oklahoma's apportionment, to ensure it would not affect the seats for any existing states. However, when the full apportionment was recalculated, the House was stunned to learn Oklahoma's entry had caused New York to lose a seat to Maine, despite there being no change in either state's population.[10][11]: 232–233
- By the same token, apportionments may depend on the precise order in which the apportionment is calculated. For example, identifying winning independents first and electing them, then apportioning the remaining seats, will produce a different result from treating each independent as if they were their own party and then computing a single overall apportionment.[3]
Such paradoxes also have the additional drawback of making it difficult or impossible to generalize procedure to more complex apportionment problems such as biproportional apportionments or partial vote linkage. This is in part responsible for the extreme complexity of administering elections by quota-based rules like the single transferable vote (see counting single transferable votes).
Alabama paradox
[edit]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, parties D and E end up with fewer seats, despite their entitlements increasing.
With 25 seats, the results are:
| Party | A | B | C | D | E | F | Total |
|---|---|---|---|---|---|---|---|
| Votes | 1500 | 1500 | 900 | 500 | 500 | 200 | 5100 |
| Quotas received | 7.35 | 7.35 | 4.41 | 2.45 | 2.45 | 0.98 | 25 |
| 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 |
| Quotas received | 7.65 | 7.65 | 4.59 | 2.55 | 2.55 | 1.02 | 26 |
| 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
[edit]- ^ a b c 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
- ^ Tannenbaum, Peter (2010). Excursions in Modern Mathematics. New York: Prentice Hall. p. 128. ISBN 978-0-321-56803-8.
- ^ a b c d 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
- ^ a b 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.
- ^ Gallagher, Michael (1992). "Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities". British Journal of Political Science. 22 (4): 469–496. doi:10.1017/S0007123400006499. ISSN 0007-1234. JSTOR 194023.
- ^ Gallagher, Michael (1992). "Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities". British Journal of Political Science. 22 (4): 469–496. doi:10.1017/S0007123400006499. ISSN 0007-1234. JSTOR 194023.
- ^ Gallagher, Michael; Mitchell, Paul (2005-09-15). The Politics of Electoral Systems. OUP Oxford. ISBN 978-0-19-153151-4.
- ^ Eerik Lagerspetz (26 November 2015). Social Choice and Democratic Values. Studies in Choice and Welfare. Springer. ISBN 9783319232614. Retrieved 2017-08-17.
- ^ Humphreys (1911). Proportional Representation. p. 138.
- ^ Caulfield, Michael J. (November 2010). "Apportioning Representatives in the United States Congress – Paradoxes of Apportionment". Convergence. Mathematical Association of America. doi:10.4169/loci003163 (inactive 29 December 2024).
{{cite journal}}: CS1 maint: DOI inactive as of December 2024 (link) - ^ Stein, James D. (2008). How Math Explains the World: A Guide to the Power of Numbers, from Car Repair to Modern Physics. New York: Smithsonian Books. ISBN 9780061241765.