D'Hondt method
The D'Hondt method, also called the Jefferson method or the greatest divisors method, is an apportionment method for allocating seats in parliaments among federal states, or in proportional representation among political parties. It belongs to the class of highest-averages methods. Compared to ideal proportional representation, the D'Hondt method reduces somewhat the political fragmentation for smaller electoral district sizes, where it favors larger political parties over small parties.
The method was first described in 1792 by American Secretary of State and later President of the United States Thomas Jefferson. It was re-invented independently in 1878 by Belgian mathematician Victor D'Hondt, which is the reason for its two different names.
Motivation
Proportional representation systems aim to allocate seats to parties approximately in proportion to the number of votes received. For example, if a party wins one-third of the votes then it should gain about one-third of the seats. In general, exact proportionality is not possible because these divisions produce fractional numbers of seats. As a result, several methods, of which the D'Hondt method is one, have been devised which ensure that the parties' seat allocations, which are of whole numbers, are as proportional as possible.Although all of these methods approximate proportionality, they do so by minimizing different kinds of disproportionality.
The D'Hondt method minimizes the largest seats-to-votes ratio. Empirical studies based on other, more popular concepts of disproportionality show that the D'Hondt method is one of the least proportional among the proportional representation methods. The D'Hondt favours large parties and coalitions over small parties due to strategic voting. In comparison, the Sainte-Laguë method reduces the disproportional bias towards large parties and it generally has a more equal seats-to-votes ratio for different sized parties.
The axiomatic properties of the D'Hondt method were studied and they proved that the D'Hondt method is a consistent and monotone method that reduces political fragmentation by encouraging coalitions. A method is consistent if it treats parties that received tied votes equally. Monotonicity means that the number of seats provided to any state or party will not decrease if the house size increases.
Procedure
After all the votes have been tallied, successive quotients are calculated for each party. The party with the largest quotient wins one seat, and its quotient is recalculated. This is repeated until the required number of seats is filled. The formula for the quotient is:where:
- is the total number of votes that party received, and
- is the number of seats that party has been allocated so far, initially zero for all parties.
Alternatively, the procedure can be reversed by starting with a house apportionment that assigns "too many seats" to every party, then removing legislators one at a time from the most-overrepresented party.
Example
In this example, 230,000 voters decide the disposition of eight seats among four parties. Since eight seats are to be allocated, each party's total votes are divided by one, then by two, three, and four. The eight highest entries range from 100,000 down to 25,000. For each, the corresponding party gets a seat. Note that in round one, the quotient shown in the table, as derived from the formula, is precisely the number of votes returned in the ballot.| Round | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Seats won |
| Party A quotient seats after round | 100,000 1 | 50,000 1 | 50,000 2 | 33,333 2 | 33,333 3 | 25,000 3 | 25,000 3 | 25,000 4 | 4 |
| Party B quotient seats after round | 80,000 0 | 80,000 1 | 40,000 1 | 40,000 2 | 26,667 2 | 26,667 2 | 26,667 3 | 20,000 3 | 3 |
| Party C quotient seats after round | 30,000 0 | 30,000 0 | 30,000 0 | 30,000 0 | 30,000 0 | 30,000 1 | 15,000 1 | 15,000 1 | 1 |
| Party D quotient seats after round | 20,000 0 | 20,000 0 | 20,000 0 | 20,000 0 | 20,000 0 | 20,000 0 | 20,000 0 | 20,000 0 | 0 |
While in this example, parties B, C, and D formed a coalition against Party A: Party A received 3 seats instead of 4 due to the coalition having 30,000 more votes than Party A.
| Round | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Seats won |
| Party A quotient seats after round | 100,000 0 | 100,000 1 | 50,000 1 | 50,000 2 | 33,333 2 | 33,333 3 | 25,000 3 | 25,000 3 | 3 |
| Coalition B-C-D quotient seats after round | 130,000 1 | 65,000 1 | 65,000 2 | 43,333 2 | 43,333 3 | 32,500 3 | 32,500 4 | 26,000 5 | 5 |
The chart below shows an easy way to perform the calculation. Each party's vote is divided by 1, 2, 3, or 4 in consecutive columns, then the 8 highest values resulting are selected. The quantity of highest values in each row is the number of seats won.
For comparison, the "True proportion" column shows the exact fractional numbers of seats due, calculated in proportion to the number of votes received. The slight favouring of the largest party over the smallest is apparent.
Further examples
A worked-through example for non-experts relating to the 2019 European Parliament election in the United Kingdom written by Christina Pagel for UK in a Changing Europe is available.A more mathematically detailed example has been written by British mathematician Professor Helen Wilson.
Approximate proportionality under D'Hondt
The D'Hondt method approximates proportionality by minimizing the largest seats-to-votes ratio among all parties.This ratio is also known as the advantage ratio. In contrast, the average seats-to-votes ratio is optimized by the Webster/Sainte-Laguë method.
For party, where is the overall number of parties, the advantage ratio is
where
- – the seat share of party,,
- – the vote share of party,.
captures how over-represented is the most over-represented party.
The D'Hondt method assigns seats so that this ratio attains its smallest possible value,
where is a seat allocation
from the set of all allowed seat allocations.
Thanks to this, as shown by Juraj Medzihorsky, the D'Hondt method splits the votes into exactly proportionally represented ones and residual ones. The overall fraction of residual votes is
The residuals of party are
For illustration, continue with the above example of four parties. The advantage ratios of the four parties are 1.2 for A, 1.1 for B, 1 for C, and 0 for D. The reciprocal of the largest advantage ratio is. The residuals as shares of the total vote are 0% for A, 2.2% for B, 2.2% for C, and 8.7% for party D. Their sum is 13%, i.e.,. The decomposition of the votes into represented and residual ones is shown in the table below.
| Party | Vote share | Seat share | Advantage ratio | Residual votes | Represented votes |
| A | 43.5% | 50.0% | 1.15 | 0.0% | 43.5% |
| B | 34.8% | 37.5% | 1.08 | 2.2% | 32.6% |
| C | 13.0% | 12.5% | 0.96 | 2.2% | 10.9% |
| D | 8.7% | 0.0% | 0.00 | 8.7% | 0.0% |
| Total | 100% | 100% | — | 13% | 87% |
Jefferson and D'Hondt
The Jefferson and the D'Hondt methods are equivalent. They always give the same results, but the methods of presenting the calculation are different.The method was first described in 1792 by US Secretary of State and future President Thomas Jefferson, in a letter to George Washington regarding the apportionment of seats in the United States House of Representatives pursuant to the First United States Census:
Washington had exercised his first veto power on a bill that introduced a new plan for dividing seats in the House of Representatives that would have increased the number of seats for northern states. Ten days after the veto, Congress passed a new method of apportionment, later known as Jefferson's Method. It was used to achieve the proportional distribution of seats in the House of Representatives among the states until 1842.
It was also invented independently in 1878 in Europe, by Belgian mathematician Victor D'Hondt, who wrote in his publication Système pratique et raisonné de représentation proportionnelle, published in Brussels in 1882:
The system can be used both for distributing seats in a legislature among states pursuant to populations or among parties pursuant to an election result. The tasks are mathematically equivalent, putting states in the place of parties and population in place of votes. In some countries, the Jefferson system is known by the names of local politicians or experts who introduced them locally. For example, it is known in Israel as the Bader–Ofer system.
Jefferson's method uses a quota, as in the largest remainder method. The divisor is chosen as necessary so that the resulting quotients, disregarding any fractional remainders, sum to the required total; in other words, pick a number so that there is no need to examine the remainders. Any number in one range of quotas will accomplish this, with the highest number in the range always being the same as the lowest number used by the D'Hondt method to award a seat, and the lowest number in the range being the smallest number larger than the next number which would award a seat in the D'Hondt calculations.
Applied to the above example of party lists, this range extends as integers from 20,001 to 25,000. More precisely, any number n for which 20,000 < n ≤ 25,000 can be used.