Quota method

The quota methods are a family of apportionment rules, i.e. algorithms for calculating the . They are based on calculating a certain electoral quota, i.e. a given number of votes needed to win a seat. This is used to calculate each party's seat entitlement. Every party is assigned the integer portion of this entitlement, before any seats left over are distributed according to a specified rule.

By far the most common kind of quota method is the largest remainders method, which assigns any leftover seats 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] When using the Hare quota, the method is called Hamilton's method; it is the second-most common apportionment rule worldwide, after Jefferson's method.^[2]

Despite their intuitive appeal, most social choice theorists discourage the use of quota methods, rather than divisor methods, because of their greater susceptibility to apportionment paradoxes.^[2]^[3] In particular, the largest remainder methods exhibit the no-show paradox, i.e. voting for a party can cause it to lose seats, by increasing the size of the electoral quota.^[3] Such population paradoxes occur by increasing the electoral quota, which can cause different states' remainders to respond erratically.^[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 state to lose a seat (a situation known as an Alabama paradox).^[3]^[4]^{: Cor.4.3.1}

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 or the quota Borda system, both of which behave like the largest-remainders method when voters all behave like strict 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, and particularly the seat bias. Smaller quotas leave behind fewer seats for small parties (with less than a full quota) to pick up, while larger quotas leave behind more seats. A somewhat counterintuitive result of this is that a larger quota will always be 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:

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

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:

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

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 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

Party	Yellows	Whites	Reds	Greens	Blues	Pinks	Total
Votes	47,000	16,000	15,800	12,000	6,100	3,100	100,000
Seats (divisor)							10 (10+1=11)
Droop quota							9,091
Ideal seats	5.170	1.760	1.738	1.320	0.671	0.341
Automatic seats	5	1	1	1	0	0	8
Remainder	0.170	0.760	0.738	0.320	0.671	0.341
Highest-remainder seats	0	1	1	0	0	0	2
Total seats	5	2	2	1	0	0	10

Pros and cons

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 their abandonment in most countries, is the tendency of such rules to produce bizarre or irrational behaviors called apportionment paradoxes:

Increasing the number of seats may decrease a particular party's apportionment, a problem called the Alabama paradox.
Adding more parties to the 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 exiting state. apportionment. However, when the full apportionment was recalculated, the House was stunned to learn that New York had lost a seat to Maine.^[9]^[10]^{: 232–233}
- By the same token, the results of an apportionment for all parties can depend on the exact order of calculations. If the number of independents who win a seat is calculated first, and an apportionment is then computed excluding their seats, the results may be different from those reached by treating each independent as if they were in their own political party.
Voting for a party can cause it to lose seats, a situation called a negative vote weight or no-show paradox.
One state can gain population relative to another, e.g. go from having 5× as many people to 6× as many people, yet lose a seat to the slower-growing state.

The highest averages methods avoid all the paradoxes discussed above, with the exception of quota violations. However, quota violations in low-bias methods like Webster's method tend to be both mild and extremely rare.^[11]

Alabama paradox

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:

Party	A	B	C	D	E	F	Total
Votes	1500	1500	900	500	500	200	5100
Seats							25
Hare quota							204
Quotas received	7.35	7.35	4.41	2.45	2.45	0.98
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
Seats							26
Hare quota							196
Quotas received	7.65	7.65	4.59	2.55	2.55	1.02
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

^ ^a ^b Tannenbaum, Peter (2010). Excursions in Modern Mathematics. New York: Prentice Hall. p. 128. ISBN 978-0-321-56803-8.
^ ^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
^ ^a ^b ^c 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. ISSN 0007-1234.
^ 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.
^ 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.
^ Caulfield, Michael J. (November 2010). "Apportioning Representatives in the United States Congress – Paradoxes of Apportionment". Convergence. Mathematical Association of America. doi:10.4169/loci003163.
^ 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.
^ Balinski, Michel; H. Peyton Young (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. Yale Univ Pr. ISBN 0-300-02724-9.

External links

Hamilton method experimentation applet at cut-the-knot

[:0-1] Tannenbaum, Peter (2010). Excursions in Modern Mathematics. New York: Prentice Hall. p. 128. ISBN 978-0-321-56803-8.

[: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

[:5-3] 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

[:0222-4] 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] Eerik Lagerspetz (26 November 2015). Social Choice and Democratic Values. Studies in Choice and Welfare. Springer. ISBN 9783319232614. Retrieved 2017-08-17.

[Caulfield2-9] Caulfield, Michael J. (November 2010). "Apportioning Representatives in the United States Congress – Paradoxes of Apportionment". Convergence. Mathematical Association of America. doi:10.4169/loci003163.

[Stein20082-10] 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.

[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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