# D'Hondt method: Map

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The d'Hondt method (mathematically but not operationally equivalent to Jefferson's method, and Bader-Ofer method) is a highest averages method for allocating seats in party-list proportional representation. The method is named after Belgian mathematician Victor d'Hondt. This system is less proportional than the other popular divisor method, Sainte-Laguë, because d'Hondt slightly favors large parties and coalitions over scattered small parties.

Legislatures using this system include those of Albania, Argentina, Austria, Belgium, Brazil, Bulgaria, Chile, Colombia, Croatia, Czech Republic, Denmark, East Timor, Ecuador, Estonia, Finland, Hungary, Iceland, Israel, Japan, Republic of Macedonia, Republic of Moldova, Montenegro, the Netherlands, Northern Ireland, Paraguay, Poland, Portugal, Romania, Scotland, Serbia, Slovenia, Spain, Turkey, Venezuela and Wales.

The system has also been used in Northern Ireland to allocate the ministerial positions in the Northern Ireland Executive, for the 'top-up' seats in the London Assembly, in some countries during elections to the European Parliament, and during the 1997 Constitution-era for allocating party-list parliamentary seats in Thailand. A modified form was used for elections in the Australian Capital Territory Legislative Assembly but abandoned in favour of the Hare-Clark system. The system is also used in practice for the allocation between political groups of a large number of posts (Vice Presidents, committee chairmen and vice-chairmen, delegation chairmen and vice-chairmen) in the European Parliament.

## Allocation

In a closed list system, each voter casts a single vote for the party of their choice. In an open list system, the voter votes for a candidate personally, but the vote is principally counted as a vote for the candidate's party.

After all the votes have been tallied, successive quotients or 'averages' are calculated for each list. The formula for the quotient is \textstyle\frac{V}{s+1}, where:
• V is the total number of votes that list received; and
• s is the number of seats that party has been allocated so far (initially 0 for all parties in a list only ballot, but includes the number of seats already won where combined with a separate ballot, as happens in Wales and Scotland).

Whichever list has the highest quotient or average 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.

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.

## Example

 Party A Party B Party C Party D Party E Votes 340,000 280,000 160,000 60,000 40,000 Percentage of votes 38.6% 31.8% 18.2% 6.8% 4.5% Seat 1 340,000 280,000 160,000 60,000 40,000 Seat 2 170,000 280,000 160,000 60,000 40,000 Seat 3 170,000 140,000 160,000 60,000 40,000 Seat 4 113,333 140,000 160,000 60,000 40,000 Seat 5 113,333 140,000 80,000 60,000 40,000 Seat 6 113,333 93,333 80,000 60,000 40,000 Seat 7 85,000 93,333 80,000 60,000 40,000 Seat 8 85,000 70,000 80,000 60,000 40,000 Seat 9 68,000 70,000 80,000 60,000 40,000 Total Seats 4 3 2 0 0 Votes per Seat 85,000 93,333 80,000 N/A N/A Percentage of seats 44.4% 33.3% 22.2% 0.0% 0.0%

## Statistical unbiasedness

The d'Hondt method has the following notable mathematical property: if the proportion of the votes received by each party is entirely unknown, i.e., is a random variable uniformly distributed on the n-dimensional simplex (where n+1 is the total number of parties competing for the election), then the distribution of seats is also entirely unknown, i.e., each partition of the total number of seats among the parties is equally likely. This can be said to be a condition of unbiasedness.

## D'Hondt and Jefferson

The d'Hondt method is equivalent to the Jefferson method (named after the U.S. statesman Thomas Jefferson) in that they always give the same results, but the method of calculating the apportionment is different. The Jefferson method, invented in 1792 for U.S. congressional apportionment rather than elections, uses a quota as in the largest remainder method but the quota (called a divisor) is adjusted as necessary so that the resulting quotients, disregarding any fractional remainders, sum to the required total (so the two methods share the additional property of not using all numbers, whether of state populations or of party votes, in the apportioning of seats). One of a range of quotas will accomplish this, and applied to the above example of party lists this extends as integers from 85,001 to 93,333, the highest number always being the same as the last average to which the d'Hondt method awards a seat if it is used rather than the Jefferson method, and the lowest number being the next average plus one.

## Variations

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 (2%), Spain (3%), Turkey (10%), Poland (5%, or 8% for coalitions), Iceland, Romania and Serbia (5%) and Belgium (5%, on regional basis). In the Netherlands, a party must win enough votes for one full seat (note that this is not necessary in plain d'Hondt), which with 150 seats in the lower chamber gives an effective threshold of 0.67%. In Estonia, candidates receiving the simple quota in their electoral districts are considered elected, but in the second (district level) and third round of counting (nationwide, modified d'Hondt method) mandates are awarded only to candidate lists receiving more than the threshold of 5% of the votes nationally.

The method can cause a hidden threshold. In Finland's parliamentary elections, there is no official threshold, but the effective threshold is gaining one seat. The country is divided into districts with different numbers of representatives, so there is a hidden threshold, different in each district. The largest district, Uusimaa with 33 representatives, has a hidden threshold of 3%, while the smallest district, South Savo with 6 representatives, has a hidden threshold of 14%. This favors large parties in the small districts.In Croatia, the official threshold is 5% for parties and coalitions. However, since the country is divided in 10 voting districts with 14 elected representatives each, sometimes the threshold can be higher, depending on the number of "fallen lists" (lists that don't get at least 5%). If many votes are lost in this manner, a list that gets barely more than 5% will still get a seat, whereas if there is a small number of parties that all pass the threshold, the actual ("natural") threshold is close to 7.15%.

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. In the Netherlands, cartels (lijstverbindingen) cannot be used to overcome the threshold, but they do influence the distribution of remainder seats; thus, smaller parties can use them to get a chance which is more like that of the big parties.

In French municipal and regional elections, the d'Hondt method is used to attribute a number of council seats; however, a fixed proportion of them (50% for municipal elections, 25% for regional elections) is automatically given to the list with the greatest number of votes, to ensure that it has a working majority: this is called the "majority bonus" (prime à la majorité), and only the remainder of the seats is distributed proportionally (including to the list which has already received the majority bonus).

The d'Hondt method can also be used in conjunction with a quota formula to allocate most seats, applying the d'Hondt method to allocate any remaining seats to get a result identical to that achieved by the standard d'Hondt formula. This variation is known as the Hagenbach-Bischoff System, and is the formula frequently used when a country's electoral system is referred to simply as 'd'Hondt'.

In the election of Legislative Assembly of Macau, a modified D'Hondt method is used. The formula for the quotient in this system is \textstyle\frac{V}{2^{s}}.

## References

1. Aurel Croissant and Daniel J. Pojar, Jr., Quo Vadis Thailand? Thai Politics after the 2005 Parliamentary Election, Strategic Insights, Volume IV, Issue 6 (June 2005)
2. Oikeusministeriö. Suhteellisuuden parantaminen eduskuntavaaleissa. http://www.om.fi/uploads/p0yt86h0difo.pdf

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