D'Hondt method: Difference between revisions
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Change D'Hondt to d'Hondt per the Oxford Reference: https://www.oxfordreference.com/view/10.1093/oi/authority.20110803095715357. Do some more cleanup and add Phragmén's method.
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m (Change D'Hondt to d'Hondt per the Oxford Reference: https://www.oxfordreference.com/view/10.1093/oi/authority.20110803095715357. Do some more cleanup and add Phragmén's method.) |
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{{Wikipedia}}
The '''d'Hondt method''' or the Jefferson method (both are equivalent, but described differently) is a highest averages method for allocating seats. This system favors large parties slightly more than the other popular [[divisor method]], [[Sainte-Laguë method|Sainte-Laguë]], does. The method described is named in the United States after Thomas Jefferson, who introduced the method for proportional allocation of seats in the United States House of Representatives in 1792, and in Europe after Belgian mathematician Victor
It is used in
▲The '''d'Hondt method''' or the Jefferson method (both are equivalent, but described differently) is a highest averages method for allocating seats. This system favors large parties slightly more than the other popular [[divisor method]], [[Sainte-Laguë method|Sainte-Laguë]], does. The method described is named in the United States after Thomas Jefferson, who introduced the method for proportional allocation of seats in the United States House of Representatives in 1792, and in Europe after Belgian mathematician Victor D'Hondt, who described the methodology in 1878.
▲It is used in: Argentina, Austria, Bulgaria, Chile, Denmark (for local elections), Finland, Israel, the Netherlands, Poland, Portugal and Spain, as well as elections to the European Parliament in some countries. The method is named after Belgian mathematician [[Victor d'Hondt]]. Jefferson's method is named after Thomas Jefferson, and was used to apportion the U.S. House of Representatives between 1792 and 1840.
==Allocation==
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The rationale behind this procedure (and the Sainte-Laguë procedure) is to allocate seats in proportion to the number of votes a list received, by maintaining the ratio of votes received to seats allocated as close as possible. This makes it possible for parties having relatively few votes to be represented.
One way to think of the formula is that it first shows who can win the most votes for the first seat, and then if the party that won the first seat can plug more votes per seat for two seats than any other party can for even one seat, it gives that same party two seats; otherwise it gives another party one seat. This logic repeats until all seats have been allocated.
==Example==
As a simple example, if there are 2 seats to be filled, with Party A having 300 votes and Party B having 290 votes, then Party A wins the first seat, with their new vote total becoming 150 votes (calculated as 300/((1)+1) = 300/2). This means Party A now has 150 votes and Party B has 290 votes, so Party B wins the second seat, and the procedure is over.
A larger example (<font color="#FF0000">red</font> indicates that party won a seat in that round because it had the most votes of any party in that round; this table can be thought of as going in "rounds", with the first round showing how many votes each party had, and each successive round showing how many votes each party had after applying the
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<td><div align="right"></div></td>
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==Variations==
The Hagenbach-Bischoff system is equivalent to, and is a faster way of doing
In some cases, a [[election threshold|threshold]] or ''barrage'' is set, and any list which does not receive that threshold will not have any seats allocated to it, even if it received enough votes to otherwise have been rewarded with a seat. Examples of countries using this threshold are Israel (1.5%) and Belgium (5%, on regional basis).
Some systems allow parties to associate their lists together into a single ''cartel'' in order to overcome the threshold, while some systems set a separate threshold for cartels. Smaller parties often form pre-election
== Jefferson's method ==
Jefferson's method is equivalent to
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== Computational complexity ==
Let <math>s</math> be the number of seats and <math>r</math> be the number of parties. The standard sequential allocation procedure determines the outcome in <math>O(s \log r)</math> time. More sophisticated algorithms can determine the outcome in <math>O(r \log r)</math> time.<ref name="Gall 2003 pp. 325–333">{{cite journal | last=Gall | first=Françoise Le | title=Determination of the modes of a Multinomial distribution | journal=Statistics & Probability Letters | publisher=Elsevier BV | volume=62 | issue=4 | year=2003 | issn=0167-7152 | doi=10.1016/s0167-7152(02)00430-3 | pages=332-333}}</ref><ref name="White Hendy 2010 pp. 63–68">{{cite journal | last=White | first=W.T.J. | last2=Hendy | first2=M.D. | title=A fast and simple algorithm for finding the modes of a multinomial distribution | journal=Statistics & Probability Letters | publisher=Elsevier BV | volume=80 | issue=1 | year=2010 | issn=0167-7152 | doi=10.1016/j.spl.2009.09.013 | pages=63–68}}</ref>
== Extensions of theory ==
One of the only ranked PR methods that reduces to d'Hondt in its [[party list case]] is [[Schulze STV]]. Several [[cardinal PR]] methods reduce to d'Hondt if certain divisors are used. Some of these are:
* [[Phragmen's voting rules|Phragmén's method]]
* [[Reweighted Range Voting|Reweighted Range voting]]
* [[Sequential proportional approval voting]]
* [[Single distributed vote]]
== Notes ==
Parties can generally guarantee themselves at least as many seats as they would get in
A 10 B 7 C 4 D 3
A gets the first seat. Now, we can see the vote totals as:
'''A 5 A 5''' B 7 C 4 D 3
In other words, now that A got the first seat, they are "looking to play" for two seats; they are seeing if they can secure two seats no matter what the other parties do with their votes. Notice that the idea of the division is that if the other parties try to take the one seat that A has, then A can re-fuse its votes to ensure they have more votes than the other parties. B gets the second seat and the votes become:
A 5 A 5 '''B 3.5 B 3.5''' C 4 D 3
So here it is seen that A gets the third seat. This is because by splitting their votes evenly between two seats, they can put more votes per seat than any of the other parties can put for two or even one seat (i.e. Party C and Party D can't even put 5 votes in for one seat). The votes would then become
'''A 3.333 A 3.333 A 3.333''' B 3.5 B 3.5 C 4 D 3
etc.
The reason d'Hondt guarantees every party at least as many seats as they have more voters than that number of HB quotas, thus satisfying the [[Droop proportionality criterion]], is because there is always 1 HB quota more than the number of seats, so when a party has more voters than k HB quotas, it can divide its votes to do more than a quota per seat for k seats, whereas all other parties combined can at most do just under (((total number of seats + 1)-k)/(number of seats - (k - 1))) i.e. the most votes the other parties can have divided by the number of seats they're trying to take. For example, if there are 10 seats to be filled, and one party has over 3 HB quotas, then they can take at least 3 seats because they can do over 1 quota per seat for 3,and for all other parties to take at least (10-(3-1))=8 seats (the minimum required for them to deny the other party 3 seats, since you can't get 3 seats if 8 out of 10 seats are already allotted), the other parties can do at most just under ((10+1)-3)/8)=1 quota per seat for 8 seats.
The divisor in d'Hondt will always be equal to or smaller than a [[Hare quota]], because that is the largest divisor possible such that there are only as many winners as seats to be filled.
== References ==
<references />
[[Category:Party list theory]]
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