The
d'Hondt method (mathematically but not
operationally equivalent to
Jefferson's method,
and
BaderOfer method) is a
highest averages method for
allocating seats in
partylist proportional
representation.
The method is named after Belgian
mathematician Victor d'Hondt.
This system is less proportional than the other popular divisor
method,
SainteLaguë,
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 'topup' seats in the London Assembly, in some countries during
elections to the
European Parliament, and during the 1997 Constitutionera for
allocating partylist parliamentary seats in Thailand. A modified form was used for elections in the
Australian Capital Territory Legislative
Assembly but abandoned in favour of the HareClark 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
vicechairmen, delegation chairmen and vicechairmen) 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 SainteLaguë
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
ndimensional
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 preelection
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
HagenbachBischoff
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}}.
External links
References
 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)
 Oikeusministeriö. Suhteellisuuden parantaminen
eduskuntavaaleissa. http://www.om.fi/uploads/p0yt86h0difo.pdf