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Plurality and majority systems
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The plurality system is the simplest means of determining the outcome
of an election. To win, a candidate need only poll more votes than any
other single opponent; he need not, as required by the majority formula,
poll more votes than the combined opposition. The more candidates
contesting a constituency seat, the greater the probability that the
winning candidate will receive only a minority of the votes cast.
Countries using the plurality formula for national legislative elections
include Canada, Great Britain, India, and the United States. Countries
with plurality systems usually have had two main parties.

Under
the majority system, the party or candidate winning more than 50 percent
of the vote in a constituency is awarded the contested seat. A
difficulty in systems with the absolute-majority criterion is that it
may not be satisfied in contests in which there are more than two
candidates. Several variants of the majority formula have been developed
to address this problem. In Australia the alternative, or preferential,
vote is used in lower-house elections. Voters rank the candidates on an
alternative-preference ballot. If a majority is not achieved by
first-preference votes, the weakest candidate is eliminated, and that
candidate’s votes are redistributed to the other candidates according to
the second preference on the ballot. This redistributive process is
repeated until one candidate has collected a majority of the votes. In
France a double-ballot system is employed for National Assembly
elections. If no candidate secures a majority in the first round of
elections, another round is required. In the second round, only those
candidates securing the votes of at least one-eighth of the registered
electorate in the first round may compete, and the candidate securing a
plurality of the popular vote in the second round is declared the
winner. Some candidates eligible for the second round withdraw their
candidacy and endorse one of the leading candidates. In contrast to the
two-party norm of the plurality system, France has what some analysts
have called a “two-bloc” system, in which the main parties of the left
and the main parties of the right compete against each other in the
first round of an election to be the representative of their respective
ideological group and then ally with one another to maximize their
bloc’s representation in the second round. An infrequently used variant
is the supplementary-vote system, which was instituted for London
mayoral elections. Under this system, voters rank their top two
preferences; in the event that no candidate wins a majority of
first-preference votes, all ballots not indicating the top two vote
getters as either a first or a second choice are discarded, and the
combination of first and second preferences is used to determine the
winner. Majority formulas usually are applied only within single-member
electoral constituencies.

The majority and the plurality formulas
do not always distribute legislative seats in proportion to the share
of the popular vote won by the competing parties. Both formulas tend to
reward the strongest party disproportionately and to handicap weaker
parties, though these parties may escape the inequities of the system if
their support is regionally concentrated. For example, in national
elections in Britain in 2001, the Labour Party captured more than
three-fifths of the seats in the House of Commons, even though it won
barely two-fifths of the popular vote; in contrast, the Conservative
Party won one-fourth of the seats with nearly one-third of the vote.
Third-party representation varied considerably; whereas the Liberal
Democrats, whose support was spread throughout the country, captured 8
percent of the seats with more than 18 percent of the vote, the Plaid
Cymru, whose support is concentrated wholly in Wales, won 0.7 percent of
the vote and 0.7 percent of the seats. The plurality formula usually,
though not always, distorts the distribution of seats more than the
majority system.
Proportional representation

Proportional
representation requires that the distribution of seats broadly be
proportional to the distribution of the popular vote among competing
political parties. It seeks to overcome the disproportionalities that
result from majority and plurality formulas and to create a
representative body that reflects the distribution of opinion within the
electorate. Because of the use of multimember constituencies in
proportional representation, parties with neither a majority nor a
plurality of the popular vote can still win legislative representation.
Consequentially, the number of political parties represented in the
legislature often is large; for example, in Israel there are usually
more than 10 parties in the Knesset.
Although approximated in many
systems, proportionality can never be perfectly realized. Not
surprisingly, the outcomes of proportional systems usually are more
proportional than those of plurality or majority systems. Nevertheless, a
number of factors can generate disproportional outcomes even under
proportional representation. The single most important factor
determining the actual proportionality of a proportional system is the
“district magnitude”—that is, the number of candidates that an
individual constituency elects. The larger the number of seats per
electoral district, the more proportional the outcome. A second
important factor is the specific formula used to translate votes into
seats. There are two basic types of formula: single transferable vote
and party-list proportional representation.
Single transferable vote
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Developed in the 19th century in Denmark and in Britain, the single
transferable vote formula—or Hare system, after one of its English
developers, Thomas Hare—employs a ballot that allows the voter to rank
candidates in order of preference. When the ballots are counted, any
candidate receiving the necessary quota of first preference
votes—calculated as one plus the number of votes divided by the number
of seats plus one—is awarded a seat. In the electoral calculations,
votes received by a winning candidate in excess of the quota are
transferred to other candidates according to the second preference
marked on the ballot. Any candidate who then achieves the necessary
quota is also awarded a seat. This process is repeated, with subsequent
surpluses also being transferred, until all the remaining seats have
been awarded. Five-member constituencies are considered optimal for the
operation of the single transferable vote system.
Because it
involves the aggregation of ranked preferences, the single transferable
vote formula necessitates complex electoral computations. This
complexity, as well as the fact that it limits the influence of
political parties, probably accounts for its infrequent use; it has been
used in Northern Ireland, Ireland, and Malta and in the selection of
the Australian and South African senates. The characteristic of the Hare
formula that distinguishes it from other proportional representation
formulas is its emphasis on candidates, not parties. The party
affiliation of the candidates has no bearing on the computations. The
success of minor parties varies considerably; small centrist parties
usually benefit from the vote transfers, but small extremist parties
usually are penalized.
Party-list proportional representation

The basic difference between the single transferable vote formula and
list systems—which predominate in elections in western Europe and Latin
America—is that, in the latter, voters generally choose among
party-compiled lists of candidates rather than among individual
candidates. Although voters may have some limited choice among
individual candidates, electoral computations are made on the basis of
party affiliation, and seats are awarded on the basis of party rather
than candidate totals. The seats that a party wins are allocated to its
candidates in the order in which they appear on the party list. Several
types of electoral formulas are used, but there are two main types:
largest-average and greatest-remainder formulas.

In the
largest-average formula, the available seats are awarded one at a time
to the party with the largest average number of votes as determined by
dividing the number of votes won by the party by the number of seats the
party has been awarded plus a certain integer, depending upon the
method used. Each time a party wins a seat, the divisor for that party
increases by the same integer, which thus reduces its chances of winning
the next seat. Under all methods, the first seat is awarded to the
party with the largest absolute number of votes, since, no seats having
been allocated, the average vote total as determined by the formula will
be largest for this party. Under the d’Hondt method, named after its
Belgian inventor, Victor d’Hondt, the average is determined by dividing
the number of votes by the number of seats plus one. Thus, after the
first seat is awarded, the number of votes won by that party is divided
by two (equal to the initial divisor plus one), and similarly for the
party awarded the second seat, and so on. Under the so-called
Sainte-Laguë method, developed by Andre Sainte-Laguë of France, only odd
numbers are used. After a party has won its first seat, its vote total
is divided by three; after it wins subsequent seats, the divisor is
increased by two. The d’Hondt formula is used in Austria, Belgium,
Finland, and the Netherlands, and the Sainte-Laguë method is used in
Denmark, Norway, and Sweden.

The d’Hondt formula has a slight
tendency to overreward large parties and to reduce the ability of small
parties to gain legislative representation. In contrast, the
Sainte-Laguë method reduces the reward to large parties, and it
generally has benefited middle-size parties at the expense of both large
and small parties. Proposals have been made to divide lists by
fractions (e.g., 1.4, 2.5, etc.) rather than integers to provide the
most proportional result possible.

The greatest-remainder method
first establishes a quota that is necessary for a party to receive
representation. Formulas vary, but they are generally some variation of
dividing the total vote in the district by the number of seats. The
total popular vote won by each party is divided by the quota, and a seat
is awarded as many times as the party total contains the full quota. If
all the seats are awarded in this manner, the election is complete.
However, such an outcome is unlikely. Seats that are not won by full
quotas subsequently are awarded to the parties with the largest
remainder of votes after the quota has been subtracted from each party’s
total vote for each seat it was awarded. Seats are distributed
sequentially to the parties with the largest remainder until all the
district’s allocated seats have been awarded.

Minor parties
generally fare better under the greatest-remainder formula than under
the largest-average formula. The greatest-remainder formula is used in
Israel and Luxembourg and for some seats in the Danish Folketing. Prior
to 1994 Italy used a special variant of the greatest-remainder formula,
called the Imperiali formula, whereby the electoral quota was
established by dividing the total popular vote by the number of seats
plus two. This modification increased the legislative representation of
small parties but led to a greater distortion of the proportional ideal.
The proportionality of outcomes also can be diluted by the imposition
of an electoral threshold that requires a political party to exceed some
minimum percentage of the vote to receive representation. Designed to
limit the political success of small extremist parties, such thresholds
can constitute significant obstacles to representation. The threshold
varies by country, having been set at 4 percent in Sweden, 5 percent in
Germany, and 10 percent in Turkey.
Hybrid systems

In some
countries, the majoritarian and proportional systems are combined into
what are called mixed-member proportional or additional-members systems.
Although there are a number of variants, all mixed-member proportional
systems elect some representatives by proportional representation and
the remainder by a nonproportional formula. The classic example of the
hybrid system is the German Bundestag, which combines the personal link
between representatives and voters with proportionality. The German
constitution provides for the election of half the country’s
parliamentarians by proportional representation and half by simple
plurality voting in single-member constituencies. Each voter casts two
ballots. The first vote (Erstimme) is cast for an individual to
represent a constituency (Wahlkreise); the candidate receiving the most
votes wins the election. The second vote (Zweitstimme) is cast for a
regional party list. The results of the second vote determine the
overall political complexion of the Bundestag. All parties that receive
at least 5 percent of the national vote—or win at least three
constituencies—are allocated seats on the basis of the percentage of
votes that they receive. The votes of parties not receiving
representation are reapportioned to the larger parties on the basis of
their share of the vote. During the 1990s, a number of countries adopted
variants of the German system, including Italy, Japan, New Zealand, and
several eastern European countries (e.g., Hungary, Russia, and
Ukraine). A hybrid system also was adopted by the British government for
devolved assemblies in Scotland and Wales. One of the chief differences
between mixed-member systems is the percentage of seats allocated by
proportional and majoritarian methods. For example, in Italy and Japan,
respectively, roughly three-fourths and three-fifths of all seats are
apportioned through constituency elections.

A country’s choice of
electoral system, like its conception of representation, generally
reflects its particular cultural, social, historical, and political
circumstances. Majority or plural methods of voting are most likely to
be acceptable in relatively stable political cultures. In such cultures,
fluctuations in electoral support from one election to the next reduce
polarization and encourage political centrism. Thus, the “winner take
all” implications of the majority or plurality formulas are not
experienced as unduly deprivational or restrictive. In contrast,
proportional representation is more likely to be found in societies with
traditional ethnic, linguistic, and religious cleavages or in societies
that have experienced class and ideological conflicts.