D'Hondt method

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, 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. Jefferson's method was used to apportion the U.S. House of Representatives between 1792 and 1840.

Allocation
After all the votes have been tallied, successive quotients are calculated for each list. The formula for the quotient (which is related to a Hagenbach-Bischoff quota) is


 * $$\text{quot} = \frac{V}{s+1}$$

where: Whichever list has the highest quotient gets the next seat allocated, and their quotient is recalculated given their new seat total. The process is repeated until all seats have been allocated.
 * V is the total number of votes that party received, and
 * s is the number of seats that party has been allocated so far, initially 0 for all parties.

The order in which seats allocated to a list are then allocated to individuals on the list is irrelevant to the allocation procedure. It may be internal to the party (a closed list system) or the voters may have influence over it through various methods (an open list system).

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 (red 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 d'Hondt procedure):

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

In some cases, a 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 coalitions to make sure they get past the election threshold.

Jefferson's method
Jefferson's method is equivalent to d'Hondt, but is described differently: "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."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: Suppose that there were to be 60 seats in the House.

If a divisor of 55 000 is used, the resulting apportionment is

Computational complexity
Let $$s$$ be the number of seats and $$r$$ be the number of parties. The standard sequential allocation procedure determines the outcome in $$O(s \log r)$$ time. More sophisticated algorithms can determine the outcome in $$O(r \log r)$$ time.

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:


 * Phragmén's method
 * Reweighted Range voting
 * Sequential proportional approval voting
 * Single distributed vote