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Thequota ordivide-and-rank methods make up a category ofapportionment rules, i.e. algorithms for allocating seats in a legislative body among multiple groups (e.g.parties orfederal states). The quota methods begin by calculating anentitlement (basic number of seats) for each party, by dividing their vote totals by anelectoral 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 popularhighest averages methods (also called divisor methods).^[1]

By far the most common quota method are thelargest remainders orquota-shift methods, which assign any leftover seats to the "plurality" winners (the parties with the largestremainders, i.e. most leftover votes).^[2]

When using theHare quota, this rule is calledHamilton's method, and is the third-most common apportionment rule worldwide (afterJefferson's method andWebster's method).^[1]

Despite their intuitive definition, quota methods are generally disfavored bysocial choice theorists as a result ofapportionment paradoxes.^[1][3] In particular, the largest remainder methods exhibit theno-show paradox, i.e. votingfor a party can cause it tolose seats.^[3][4] The largest remainders methods are also vulnerable tospoiler effects and can failresource orhouse 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 anAlabama paradox).^{[3][4]: Cor.4.3.1}

The largest remainder method divides each party's vote total by aquota. Usually, quota is derived by dividing the number of valid votes cast, by the number of seats. The result for each party will consist of aninteger part plus afractionalremainder. 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.

Largest remainder methods produces similar results tosingle transferable vote or thequota Borda system, where voters organize themselves intosolid coalitions. Thesingle transferable vote or thequota Borda system behave like the largest-remainders method when voters all behave like strict partisans (i.e. only mark preferences for candidates of one party).^[5]

There are several possible choices for theelectoral quota. The choice of quota affects the properties of the corresponding largest remainder method, and particularly theseat 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 alarger quota will always be more favorable tosmaller 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 theHare quota and theDroop 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:

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

LR-Hare is sometimes called Hamilton's method, named afterAlexander Hamilton, who devised the method in 1792.^[8]

TheDroop quota is given by:

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

and is applied to elections inSouth 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 isunbiased 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]

The following example allocates 11 seats using the largest-remainder method by Hare quota.

It is easy for a voter to understand how the largest remainder method allocates seats. Moreover, the largest remainder method satisfies thequota 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 theseats-to-votes ratio, which can violate the principle ofone 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 calledapportionment paradoxes:

• Increasing the number of seats in a legislature candecrease a party's apportionment of seats, called theAlabama paradox.
• Adding more parties to the legislature can cause a bizarre kind ofspoiler effect called thenew state paradox.
  • When Congress first admittedOklahoma 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 asbiproportional apportionments orpartial vote linkage. This is in part responsible for the extreme complexity of administering elections by quota-based rules like the single transferable vote (seecounting single transferable votes).

TheAlabama paradox is when anincrease in the total number of seats leads to adecrease 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:

With 26 seats, the results are:

1. ^^abcPukelsheim, 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, retrieved2024-05-10{{citation}}: CS1 maint: work parameter with ISBN (link)
2. ^Tannenbaum, Peter (2010).Excursions in Modern Mathematics. New York: Prentice Hall. p. 128.ISBN 978-0-321-56803-8.
3. ^^abcdPukelsheim, 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, retrieved2024-05-10{{citation}}: CS1 maint: work parameter with ISBN (link)
4. ^^abBalinski, 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.doi:10.1017/S0007123400006499.ISSN 0007-1234.JSTOR 194023.
6. ^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.
7. ^Gallagher, Michael; Mitchell, Paul (2005-09-15).The Politics of Electoral Systems. OUP Oxford.ISBN 978-0-19-153151-4.
8. ^Eerik Lagerspetz (26 November 2015).Social Choice and Democratic Values. Studies in Choice and Welfare. Springer.ISBN 9783319232614. Retrieved2017-08-17.
9. ^Humphreys (1911).Proportional Representation. p. 138.
10. ^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 1 July 2025).{{cite journal}}: CS1 maint: DOI inactive as of July 2025 (link)
11. ^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.

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• Party-list proportional representation
• Apportionment methods
• Voting theory

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