# D'Hondt method

The D'Hondt method[a] or the Jefferson method is a highest averages method for allocating seats, and is thus a type of party-list proportional representation. 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. There are two forms: closed list (under which a party selects the order of election of their candidates) and open list (under which voters' choices determine the order).

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.[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 parties and coalitions over scattered small parties.[3][4][5][6] In comparison, the Webster/Sainte-Laguë method, a divisor method, reduces the reward to large parties, and it generally has benefited middle-size parties at the expense of both large and small parties.[7]

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 Albania, Angola, Argentina, Armenia, Aruba, Austria, Belgium, Bolivia, Brazil, Burundi, Cambodia, Cape Verde, Chile, Colombia, Croatia, the Czech Republic, Denmark, the Dominican Republic, East Timor, Ecuador, El Salvador, Estonia, Fiji, Finland, Guatemala, Hungary, Iceland, Israel, Japan, Kosovo, Luxembourg, Moldova, Monaco, Montenegro, Mozambique, Netherlands, Nicaragua, North Macedonia, Northern Ireland, Paraguay, Peru, Poland, Portugal, Romania, San Marino, Scotland, Serbia, Slovenia, Spain, Switzerland, Turkey, Uruguay, Venezuela and Wales.

The system has also been used for the "top-up" seats in the London Assembly; in some countries for elections to the European Parliament; and during the 1997 Constitution era to allocate party-list parliamentary seats in Thailand.[10] A modified form was used for elections in the Australian Capital Territory Legislative Assembly, but this was abandoned in favour of the Hare–Clark electoral system. The system is also used in practice for the allocation between political groups of numerous posts (Vice Presidents, committee chairmen and vice-chairmen, delegation chairmen and vice-chairmen) in the European Parliament and for the allocation of ministers in the Northern Ireland Assembly.[11]

## Allocation

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[12][1]

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

where:

• 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 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.

### Example

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

#### Further examples

A worked-through example for non-experts relating to the 2019 elections in the UK for the European Parliament written by Christina Pagel is available as an online article with the institute UK in a Changing Europe[13].

A more mathematically detailed example has been written by British mathematician Professor Helen Wilson.[14]

### Approximate proportionality under D'Hondt

The D'Hondt method approximates proportionality by minimizing the largest seats-to-votes ratio among all parties.[15] This ratio is also known as the advantage ratio. For party ${\displaystyle p\in \{1,\dots ,P\}}$, where ${\displaystyle P}$ is the overall number of parties, the advantage ratio is

${\displaystyle a_{p}={\frac {s_{p}}{v_{p}}},}$

where

${\displaystyle s_{p}}$ – the seat share of party ${\displaystyle p}$, ${\displaystyle s_{p}\in [0,1],\;\sum _{p}s_{p}=1}$,
${\displaystyle v_{p}}$ – the vote share of party ${\displaystyle p}$, ${\displaystyle v_{p}\in [0,1],\;\sum _{p}v_{p}=1}$.

The largest advantage ratio,

${\displaystyle \delta =\max _{p}a_{p},}$

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,

${\displaystyle \delta ^{*}=\min _{\mathbf {s} \in {\mathcal {S}}}\max _{p}a_{p}}$,

where ${\displaystyle \mathbf {s} =\{s_{1},\dots ,s_{P}\}}$ is a seat allocation from the set of all allowed seat allocations ${\displaystyle {\mathcal {S}}}$. 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

${\displaystyle \pi ^{*}=1-{\frac {1}{\delta ^{*}}}}$.

The residuals of party ${\displaystyle p}$ are

${\displaystyle r_{p}=v_{p}-(1-\pi ^{*})s_{p},\;r_{p}\in [0,v_{p}],\sum _{p}\,r_{p}=\pi ^{*}}$.

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 ${\displaystyle 1/1.15=0.87=1-\pi ^{*}}$. 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.,${\displaystyle 1-0.87=0.13}$. The decomposition of the votes into represented and residual ones is shown in the table below.

Party Vote
share
Seat
share