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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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
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'''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
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etc.
The reason
The divisor in
One easy way to do
▲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 ==
▲One easy way to do D'Hondt in certain simple examples is to compare all parties except the party with the fewest votes (the last-place party) with the last-place party; if all other parties can split more votes per seat than the last-place party has for even one seat such that all other parties would be able to win a combined number of seats equal to or greater than the number of seats to be filled, then the last-place party can be eliminated, and this procedure repeated, to find a minimum number of seats each party must win in D'Hondt. Example: Suppose there are 8 seats to be filled, and 4 parties, A through D, with the votes being (in descending order) A: 10, B: 8, C: 3, and D: 2. Start by dividing every party's votes by just over the last-place party's (D's) vote total (just over 2). Each party can put more than 2 votes per seat for this number of seats: A: 4 (10/2 = 5, which moved down to the next-closest integer is 4), B: 3 (8/2 = 4, shifted down = 3), and C: 1 (3/2 = 1.5 shifted down = 1). In total, these parties have 4 + 3 + 1 seats; this is the number of seats desired, therefore, this is the final result.<br />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.
<references />
[[Category:Party list theory]]
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