D'Hondt method: Difference between revisions

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 Dd'Hondt, who described the methodology in 1878.

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 Dd'Hondt procedure):<table width="100%" border="0" cellspacing="0" cellpadding="0">

The Hagenbach-Bischoff system is equivalent to, and is a faster way of doing Dd'Hondt: It works by first assigning each party as many seats as they have [[Hagenbach-Bischoff quota|Hagenbach-Bischoff quotas]], and then running Dd'Hondt with the recognition of the seats already won by each party.

Jefferson's method is equivalent to Dd'Hondt, but is described differently: <blockquote>Choose a divisor D. A state with population N (or a political party with N seats) is entitled to floor(N/D) seats. If the number of seats allocated equals the size of the legislative body, then use the apportionment just calculated. Otherwise, choose a new value for D and try again.</blockquote>Example: In 1790, the U.S. had 15 states. For the purpose of allocating seats in the House of Representatives, the state populations were as follows:

One of the only ranked PR methods that reduces to Dd'Hondt in its [[party list case]] is [[Schulze STV]]. Several [[cardinal PR]] methods reduce to Dd'Hondt if certain divisors are used. Some of whichthese are:

Parties can generally guarantee themselves at least as many seats as they would get in Dd'Hondt in any PR method by doing [[Vote management|vote management]]. In fact, an alternative way to visualize Dd'Hondt is to see how many seats each party could guaranteeably win if doing vote management. One thing that can help in this visualization is that all [[Highest averages method|Highest averages methods]] pass [[House monotonicity criterion|House monotonicity]], therefore they can all be visualized as simply filling one seat at a time with no change to the result. Example:

The reason Dd'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 Dd'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.

One easy way to do Dd'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 Dd'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.