D'Hondt method 
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Part of the 
Plurality/majoritarian 

Other systems and related theory 

The D'Hondt method^{[a]} or the Jefferson method is a
Proportional representation systems aim to allocate seats to parties approximately in proportion to the number of votes received. For example, if a party wins onethird of the votes then it should gain about onethird 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.^{[1]} Although all of these methods approximate proportionality, they do so by minimizing different kinds of disproportionality. The D'Hondt method minimizes the number of votes that need to be left aside so that the remaining votes are represented exactly proportionally. Only the D'Hondt method (and methods equivalent to it) minimizes this disproportionality.^{[2]} 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 slightly favours large
The axiomatic properties of the D'Hondt method were studied and they proved that the D'Hondt method is the unique consistent, monotone, stable, and balanced method that encourages coalitions.^{[8]}^{[9]} A method is consistent if it treats parties which received tied vote equally. By monotonicity, the number of seats provided to any state or party will not decrease if the house size increases. A method is stable if two merged parties would neither gain nor lose more than one seat. By coalition encouragement of the D'Hondt method, any alliance cannot lose the seat.
Legislatures using this system include those of
The system has also been used for the "topup" seats in the
After all the votes have been tallied, successive
where:
The total votes cast for each party in the electoral district is divided, first by 1, then by 2, then 3, up to the total number of seats to be allocated for the district/constituency. Say there are p parties and s seats. Then a grid of numbers can be created, with p rows and s columns, where the entry in the ith row and jth column is the number of votes won by the ith party, divided by j. The s winning entries are the s highest numbers in the whole grid; each party is given as many seats as there are winning entries in its row.
In this example, 230,000 voters decide the disposition of 8 seats among 4 parties. Since 8 seats are to be allocated, each party's total votes is divided by 1, then by 2, 3, and 4 (and then, if necessary, by 5, 6, 7, and so on). The 8 highest entries, marked with asterisks, range from 100,000 down to 25,000. For each, the corresponding party gets a seat.
For comparison, the "True proportion" column shows the exact fractional numbers of seats due, calculated in proportion to the number of votes received. (For example, 100,000/230,000 × 8 = 3.48) The slight favouring of the largest party over the smallest is apparent.
round
(1 seat per round) 
1  2  3  4  5  6  7  8  Seats won
(bold) 

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 
The chart below shows an easy way to perform the calculation:
Denominator  /1  /2  /3  /4  Seats won (*) 
True proportion 

Party A  100,000*  50,000*  33,333*  25,000*  4  3.5 
Party B  80,000*  40,000*  26,667*  20,000  3  2.8 
Party C  30,000*  15,000  10,000  7,500  1  1.0 
Party D  20,000  10,000  6,667  5,000  0  0.7 
Total  8  8 
A workedthrough example for nonexperts relating to the 2019 elections in the UK for the European Parliament written by
A more mathematically detailed example has been written by British mathematician Professor
The D'Hondt method approximates proportionality by minimizing the largest seatstovotes ratio among all parties.^{[15]} This ratio is also known as the advantage ratio. For party , where is the overall number of parties, the advantage ratio is
where
The largest advantage ratio,
captures how overrepresented is the most overrepresented 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^{[2]}, the D'Hondt method splits the votes into exactly proportionally represented ones and residual ones, minimizing the overall amount of the residuals in the process. 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% 
Seat allocation of eight seats under the D'Hondt method. 