D'Hondt method

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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 described is named after Belgian mathematician Victor D'Hondt who described it in 1878. In comparison with the Sainte-Laguë method, d'Hondt slightly favours large parties and coalitions over scattered small parties.^[1]

Legislatures using this system include those of Albania, Argentina, Austria, Belgium, Brazil, Bulgaria, Cambodia, Cape Verde, Chile, Colombia, Croatia, Czech Republic, Denmark, East Timor, Ecuador, Estonia, Finland, Guatemala, Hungary, Iceland, Israel, Japan, Luxembourg, Republic of Macedonia, Republic of Moldova, Montenegro, the Netherlands, Northern Ireland, Paraguay, Poland, Portugal, Romania, Scotland, Serbia, Slovenia, Spain, Turkey, Uruguay 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.^[2] 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

The total votes cast for each party in the electoral district is divided, first by 1, then by 2, then 3, right up to the total number of seats to be allocated for the district/constituency. Say there are P parties and S seats. Then create a grid of numbers, with P rows and S columns, where the entry in the i-th row and j-th column is the number of votes won by the party i divided by j. For the purposes of this explanation, call the highest S entries in the entire grid 'special entries'; each party is given as many seats as there are special entries in its row. The entries are sometimes called 'distribution figures'.

Example: if 8 seats are to be allocated, divide each party's total votes by 1, then by 2, 3, 4, 5, 6, 7, and 8. An example is given in the grid below. The 8 highest distribution figures are highlighted in bold, ranging from 100,000 down to 25,000. For each distribution figure in bold, the corresponding party gets a seat.

	/1	/2	/3	/4	/5	/6	/7	/8	Seats won
Party A	100,000	50,000	33,333	25,000	20,000	16,666	14,286	12,500	4
Party B	80,000	40,000	26,666	20,000	16,000	13,333	11,428	10,000	3
Party C	30,000	15,000	10,000	7,500	6,000	5,000	4,286	3,750	1
Party D	20,000	10,000	6,666	5,000	4,000	3,333	2,857	2,500	0

Government formation: This is repeated for each district/constituency in the country. Often the party with the highest number of seats will get to select the chief negotiators to form a government and is awarded the Prime Minister's seat. Its seats may be majority, minority, or a coalition, according to whether voters have spread their votes across few or many parties.

Proportionality: D'Hondt does not produce absolute/pure proportionality: in relation to their total vote, the two larger parties are slightly advantaged. Party C gets what it deserves, and Party D received too few votes for a seat. To dispute Party A's fourth seat (*25,000) Party D would have needed a minimum of 25,000 votes. So the seat allocation across these 4 parties is a fair, if not absolutely proportional, reflection of their vote.

Party system produced: The method tends to allow for 3 nationwide parties to be present in parliament, or sometimes four if voters spread their vote more evenly across the four leading parties. However, regionally-based parties with concentrated support in certain districts may well get enough votes there to win a handful of seats in parliament. In other words the D'Hondt method also allows for regional-nationalist parties to be represented.

District magnitude effects: With the D'Hondt seat allocation method, district magnitude or constituency size has quite an effect. If our model constituency, above, had only 5 seats, they would all go to Parties A (*100,000, *50,000, *33,333) and B (*80,000, *40,000) and Party C would not get the (*30,000) seat, being short of 3,334 votes to take it from Party A. And in the opposite case, if the district contained just one more seat, then Party D would have a chance of getting it (contending with A and B for the 20,000 distribution number, the next highest in the table). So the larger the number of seats in a district/constituency, the more likely that a fourth nationwide party will get a seat - as long as voters spread their votes.

From party seat to winning candidate: How the parties' allocate the seats they have won to their candidates: in 'Closed List' PR systems, parties tend to present a list of candidates for all available seats (as if they could win them all), ranked by order of priority. Then the top candidates receive the seats won in order of their rank. In an 'Open List' system, in addition to the party's order, voters can add their own ranking of candidates to show where they disagree with the party's ranking. Or the party may have listed their candidates without any preference, leaving it to the voters to rank them.

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. Jefferson devised the method in 1792 for the U.S. congressional apportionment pursuant to the First United States Census, with the objective being the proportional distribution of seats in the House of Representatives among the states rather than distributing seats in a legislature among parties pursuant to an election (the problems are functionally equivalent if one puts states in the place of parties and population in place of votes). Jefferson's method 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 have otherwise been rewarded with a seat. Examples of countries using this threshold are Denmark and Israel (2%), Spain and Montenegro (3%), Slovenia (4%), Hungary (5%), Russia(7%), Turkey (10%), Poland (5%, or 8% for coalitions), Romania, Czech Republic 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%.^[3] 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 into 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. ^ Pukelsheim, Friedrich (2007). "Seat bias formulas in proportional representation systems". 4th ecpr General Conference. http://www.essex.ac.uk/ecpr/events/generalconference/pisa/papers/PP996.pdf. 
2. ^ 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)
3. ^ Oikeusministeriö. Suhteellisuuden parantaminen eduskuntavaaleissa. http://www.om.fi/uploads/p0yt86h0difo.pdf

External links

• Simulator Election calculus simulator based on the modified D'Hondt system
• Calculations using the plain d'Hondt method