Open Journal of Political Science
2012. Vol.2, No.1, 9-16
Published Online April 2012 in SciRes (
Copyright © 2012 SciRes. 9
Disproportionality and Party System Fragmentation:
Does Assembly Size Matter?
Krister Lundell
Department of Political Science, Åbo Akademi University, Åbo, Finland
Email: krister.lund ell
Received October 28th, 2011; revised December 10th, 2011; accepted January 5th, 2012
The article examines the impact of assembly size on the degree of disproportionality and party system
fragmentation. The hypothesis is as follows: assembly size has a negative effect on the degree of dispro-
portionality and a positive effect on the effective number of parties in systems with single-member dis-
tricts—in proportional electoral systems, by contrast, such a pattern does not exist. In PR systems, notably
the average effective threshold supersedes assembly size in explaining the degree of disproportionality
and the effective number of parties. Electoral thresholds, ordinal ballots and apparentement, which also
have some impact on disproportionality and party system fragmentation in proportional elections, are ab-
sent in systems with single-member districts (with the exception of ordinal ballots in alternative vote sys-
tems). Moreover, the district magnitude does not vary between electoral districts and countries. Therefore,
assembly size is a significant factor in majoritarian systems. The empirical analysis of 550 elections in
democratic countries provides support for the hypothesis.
Keywords: Electoral System; Assembly Size; Effective Threshold; Disproportionality; Party System
Fragmentation; District Magnitude
The purpose of the study is to examine the impact of assem-
bly size on the disproportionality of election results and party
system fragmentation. In the first systematic analysis of conse-
quences of electoral systems, Douglas W. Rae referred to as-
sembly size as the “generally neglected variable” (1967: p. 158).
Yet, assembly size was not included in the analysis. Arend
Lijphart has described the size of legislature as “an integral and
legitimate part of the electoral system” (1994: p. 12). Rein Taa-
gepera (2007) maintains that the degree of inclusiveness (i.e.
the number of seat-winning parties in elections) of a simple
electoral system is dependent on mainly three components: the
electoral formula, the average district magnitude and the as-
sembly size. In this article, it is argued and demonstrated—both
theoretically and empirically—that assembly size negatively
affects the degree of disproportionality and positively influ-
ences party system fragmentation only in systems with sin-
gle-member districts; thereby modifying, supplementing and
correcting the arguments and evidence presented in earlier re-
search. In proportional systems, the effect of assembly size is
superseded in particular by the effect of district magnitude and
formal electoral thresholds; it also competes with the impact of
ballot structure and apparentement, i.e. the opportunity of com-
bining party lists. In systems with single-member districts, by
contrast, assembly size matters, because the district magnitude
is one in each district, there are no formal electoral thresholds,
and apparentement is not applied. Hence, several of the ele-
ments that affect disproportionality and party system fragmen-
tation in proportional systems are absent or do not vary in ma-
joritarian systems, and therefore assembly size becomes impor-
tant in the latter.
There is wide agreement among scholars that district magni-
tude is the single most important determinant of the degree of
disproportionality in elections (e.g. Jones, 1993; Rae, 1967;
Sartori, 1986; Taagepera & Shugart, 1989). The degree of dis-
proportionality, in turn, largely affects the degree of party sys-
tem fragmentation. The following example illustrates why dis-
trict magnitude supersedes assembly size regarding the chances
for small parties to gain representation.
50 seats are allocated by a proportional formula in 25
two-member districts. The legislature decides to improve the
chances of representation for small parties by increasing the
number of two-member districts from 25 to 50. Hence there are
now 100 seats in the legislature. However, the threshold that
guarantees a seat in any given district is still as high as one
third of the votes in each district. Small parties would succeed
much better, if the district magnitude was increased from two to
four in each of the 25 original districts. In such a case, one fifth
of the votes would guarantee representation in a district. What
is more, one nationwide 100-member district would guarantee,
to all intents and purposes, a proportional seat allocation be-
tween a large number of parties. All other things being equal,
the degree of disproportionality directly varies with district
Following this introduction, the theory of assembly size and
electoral consequences is elaborated, and the hypothesis of the
study is presented. Thereafter, the relevant elements of electoral
systems are dealt with, beginning with the effective threshold.
In the following three sections, external variables, the depend-
ent variables and the dataset are described. The empirical
analysis consists of three parts: the total research population,
the sub-set of systems with single-member districts (SMD) and
the sub-set of proportional systems.
The Impact of Assembly Size: Theory and
Earlier Research
The size of legislatures to a great extent varies between the
countries of the world. Obviously, differences in assembly size
are related to differences in country size. The population of a
small country may be represented by a few dozens of deci-
sion-makers, whereas a large assembly is needed in a country
with a numerous population in order to be representative. Taa-
gepera has established a causal link between population and
assembly size (Taagepera, 1972; see also Taagepera, 2007;
Taagepera & Shugart, 1989). The number of seats in the lower
(or the only) chamber of the national assembly corresponds
rather well to the cube root of the population—this is called the
“cube root law of assembly sizes”. It predicts lower houses of
100 seats for countries with one million people, and 1000 seats
for countries with 1000 million people. From this follows that
assembly size varies in accordance with population size in
logarithm form rather than population figures as such. Lijphart
argues that assembly size matters only in small legislatures.
“While the technical possibility of achieving perfect or near-
perfect proportionality keeps improving as assembly size goes
up”, he says, “only the smallest assembly sizes entail serious
restrictions on proportionality” (Lijphart, 1994: p. 102). There-
fore assembly size also has to be analyzed in logarithm form.
Assembly size is defined as the number of elected seats in the
lower or only chamber of the national legislative body.
Taagepera (2007) maintains that the degree of inclusiveness
(openness to small parties) of a simple electoral system, that is,
the level of party system fragmentation, largely depends on
three components of the electoral system: assembly size, dis-
trict magnitude and seat allocation formula. The level of party
system fragmentation, in turn, is to a great extent dependent on
the degree of disproportionality of elections. Plurality/majority
elections usually produce more disproportional seat allocation
than elections that employ list PR allocation formulas. Assem-
bly size, district magnitude and seat allocation formula can be
combined in different ways, and the direction of the effect on
system openness is compactly expressed in Joseph Colomer’s
“micro-mega rule”: the small prefer the large, and the large
prefer the small (Colomer, 2004). It means that large assem-
blies, large electoral district magnitudes, and list PR allocation
formulas with a large quota or large gaps between successive
divisors are advantageous to small parties, and vice versa.
Lijphart maintains that “if electoral systems are defined as
methods of translating votes into seats, the total number of seats
available for this translation appears to be an integral and le-
gitimate part of the systems of translation” (1994: p. 12). He
presents an example of four parties winning 41, 29, 17 and 13
per cent of the national vote in a PR election. The chances of
proportional seat allocation considerably improve for a ten-
member legislature in comparison with a five-member assem-
bly, and perfect proportionality may be achieved, at least in
principle, for a 100-member legislative body. This relationship
is not as evident in plurality elections, he says, because plurality
systems do not aim at proportionality (Lijphart, 1994). Not-
withstanding, by referring to Taagepera (1973), Lijphart argues
that there is a connection between assembly size and dispropor-
tionality in plurality systems as well. He thus finds it theoreti-
cally justified to regard assembly size as one of the important
elements of electoral systems (Lijphart, 1994).
On the face of it, Lijphart’s reasoning of assembly size and
proportionality in PR elections makes sense. Yet, he mentions
nothing about the constituency structure. If the election is con-
ducted “at large” in a nationwide constituency, the degree of
proportionality is indeed dependent on the size of the legisla-
ture—however, in this case, assembly size is equal to the dis-
trict magnitude. To be sure, if the district magnitude is five in
all three legislatures mentioned above, greater proportionality
would be achieved in the 100-member legislature than in the
smaller ones, but this is, in turn, due to the larger number of
constituencies rather than assembly size. In fact, a higher de-
gree of proportionality in the 100-member legislature is inevi-
table unless the relative strength between the four parties is
approximately the same in each of the twenty districts.
That being said, assembly size has an effect on the degree of
disproportionality and the number of parties. A theoretical ex-
planation of the impact of assembly size on the degree of pro-
portionality is provided by means of the cube law, which holds
that votes divided in a ratio of a:b between two parties in a
plurality election results in a seat allocation in the ratio of a³:b³
(Taagepera, 1973). Disproportionality increases as the number
of votes increases and/or assembly size decreases. Lijphart
(1990) presents empirical evidence for this proposition in an
analysis of plurality elections in the small assemblies of the
Eastern Caribbean states. The factor mainly responsible for the
high degree of disproportionality, he says, is the small size of
the representative body. In 43 elections of totally 68 included in
the analysis, the second largest party won a maximum of two
seats. In almost one third of the elections, the second party
gained no representation. These circumstances are, in addition
to disproportional seat allocation, a consequence of the small
number of seats in the legislature. Since parliamentary elections
are contested in a small number of constituencies, a very lim-
ited number of seats can be won by other parties than the larg-
est one.
In a quasi-experiment of the British elections in 1983, Taa-
gepera and Shugart (1989) demonstrate that a reduction of the
number of districts reduces the effective number of parliamen-
tary parties. Neighboring districts were fused two by two and
election results were recalculated. The procedure was repeated
until there was merely one seat left. In the actual assembly, the
effective number of parties was 1.98. The number dropped to
1.86 in a legislature with half the size. In a legislature with 79
seats, the effective number of parties was 1.70, whereas an
11-member assembly returned the value of 1.53. Taagepera
(2007) points out that in reality voters would most likely give
up on smaller parties, and as a consequence the number of both
elective and legislative parties would be even further reduced.
The two analyses above concern only plurality elections. By
contrast, Lijphart’s research population in Electoral Systems
and Party Systems (1994) consists mainly of PR elections. In a
regression analysis of 57 PR systems, he finds that assembly
size has a significant, negative effect on the degree of dispro-
portionality, albeit not as strong as that of the effective thresh-
old and electoral formula. Yet, apparentement and ballot struc-
ture are not included in the model. Assembly size does not in-
fluence the effective number of parties. In a larger sample of 69
electoral systems, the significant effect of assembly size re-
mains when apparentement and presidentialism are controlled
for. However, this sample is not restricted to PR systems.
Moreover, he finds some evidence that assembly size is related
to the effective number of parliamentary but not elective parties
(Lijphart, 1994). Also, there is a difference between small and
Copyright © 2012 SciRes.
large assemblies: the larger the size of the assembly, the smaller
the effect of assembly size on the degree of disproportionality
tends to be. In legislatures with more than one hundred mem-
bers, there is no association (1994; see also Gallagher &
Mitchell, 2005). Lijphart comes to this conclusion by compar-
ing mean values of four categories of assembly size. The la rgest
difference is observed between the smallest legislatures with
less than one hundred members and the next category of legis-
latures with 100 - 200 seats.
The hypothesis of the present work is as follows: assembly
size has a negative effect on the degree of disproportionality
and a positive effect on the effective number of parties in sys-
tems with single-member districts. In proportional systems,
such a pattern does not exist. The influence of assembly size in
PR systems is superseded foremost by the impact of the effec-
tive threshold (which is thoroughly dealt with in the next sec-
tion). In SMD systems, assembly size plays an important role
since the district magnitude (in practice transformed into the
average effective threshold) does not vary. In proportional sys-
tems, by contrast, the effective threshold is more important than
assembly size, because some systems have a larger average
district magnitude as other systems, and several PR countries
apply electoral thresholds that pose a barrier to small parties.
Moreover, ballot structure and the opportunity of combining
party lists are of some significance in proportional systems. In
other words, the influence of assembly size on disproportional-
ity and party system fragmentation in the total research popula-
tion is assumed to be due to its significance among SMD sys-
The Effective Threshold
Several studies have proven that of all electoral system
components the district magnitude has the strongest effect on
the degree of disproportionality (e.g. Jones, 1993; Rae, 1967;
Sartori, 1986; Taagepera & Shugart, 1989). The degree of dis-
proportionality, in turn, is a main determinant of party system
fragmentation. As Taagepera says, “(a)rguably the most impor-
tant aspect of an electoral system is the degree of squeeze it
puts on representation of small parties, which influences the
number of parties and is reflected in deviation from propor-
tional representation” (1998a: p. 393). Electoral thresholds have
a similar effect on disproportionality as small district magni-
tudes. Countries with large districts often apply electoral
thresholds, whereas countries with small districts do not need
legal thresholds in order to counteract a highly fragmented
party system. Legal thresholds may be applied at the district
level, as in e.g. Spain (3 percent) and Belgium (5 percent), and
for the allocation of seats at upper district levels, as in e.g.
Norway (4 percent)—yet most often they are used at the na-
tional level, a s in e.g. It aly (4 percent) and Slovakia (5 percent).
However, the effects of district magnitude and electoral thresh-
olds cannot be separately analyzed, because they covariate to a
great extent (Anckar, 1997). On the other hand, as Lijphart
points out, “…these two variables can be seen as functionally
equivalent: they both set limits to the representation of small
parties” (1997: p. 73). Taagepera and Shugart (1989) were the
first to create a combined measure, called the effective magni-
tude. Lijphart has presented modified versions of this measure,
using the term effective threshold instead of effective magni-
tude (1994, 1997). The interpretation is similar: a legal thresh-
old has approximately the equivalent effect to a certain district
magnitude, that is, the effective magnitude, and vice versa.
The effective threshold is a rough estimate and a midpoint in
a range between no representation and full representation, the
so called thresholds of exclusion and inclusion (Lijphart, 1994).
If a party passes the threshold of inclusion (a minimum per-
centage of the vote under the most favorable conditions), it
becomes possible for it to win a seat in a district. When a party
passes the threshold of exclusion (a maximum perce ntage of t he
vote under the most unfavorable circumstances), it is guaran-
teed to win a seat in that particular district. However, the effec-
tive thresholds usually vary between districts. In addition, these
thresholds are also to some extent affected by the electoral
formula and the number of parties that compete. Lijphart (1994)
and Taagepera and Shugart (1989) deal with these problems in
the followi ng way : they assume that , first, the numbe r of part ies
is roughly the same as the district magnitude, second, the elec-
toral formulas are roughly averaged and, third, the effective
threshold is half-way between the upper and the lower thresh-
olds (Taagepera, 1998b).1
Since the present work aims at modifying and complement-
ing the argument put forward by Lijphart, I shall use a similar
measure of effective threshold, based on the average district
magnitude. The effective threshold is calculated as follows: T =
75%/(M + 1).2 Hence, if seas are allocated in a four-member
district, the effective threshold is 75%/(4 + 1) = 15%. It means
that a party that gets 13 percent of the vote in a four-member
district will probably not win a seat in that district. However, if
a party got 13 percent of the vote nationwide in one hundred
four-member districts, it surely would win several seats. The
effective threshold at the national level is not the same as the
effective threshold at the district level. If a party fails to cross
the national effective threshold, it is likely to be considerably
underrepresented, but it will not necessarily fail to win any
representation at all (Lijphart, 1994). Moreover, it is a question
of what we are trying to predict from the effective threshold.
Do we want to know the threshold which a party has to cross in
order to obtain any representation at the national level or do we
want to relate the effective threshold to nationwide dispropor-
tionality and fragmentation? As in Lijphart’s study Electoral
Systems and Party Systems (1994), the national effective th-
reshold calculated on the basis of the average district magnitude
is more appropriate here, because we are estimating the impact
on nationwide disproportionality and party system fragmenta-
Taagepera (2002) has constructed a formula that measures
the effective national threshold, i.e. the vote share which con-
stitutes a fifty-fifty chance of securing a seat in the legislature.
The key variables of the formula are average district magnitude,
total assembly size and the number of electoral districts. How-
ever, with regard to disproportionality and party system frag-
mentation, the formula understates the importance of average
district magnitude and overstates the significance of the number
of districts (Gallagher & Mitchell, 2005). Moreover, a formula
that includes the total assembly size is not appropriate in the
present study, because the influence of the effective threshold is
1The nationwide inclusion threshold depends on the magnitude of the small-
est district, whereas the nationwide exclusion threshold depends on all dis-
tricts (Taagepera, 1998b).
2This is a simplified version of the formula that Lijphart applied in Electora
Systems and Party Systems(1994). In a sh or t no t e published th ree years later
Lijphart (1997) says that he would now opt for this streamlined formula,
originally suggested by Taagepera.
Copyright © 2012 SciRes. 11
compared to that of assembly size. If we are interested in dis-
proportionality and fragmentation, Lijphart (1994) and Taa-
gepera and Shugart (1989) suggest that the average magnitude
of the country as a whole should be applied.3 Taagepera refers
to the formula presented above as “an overall estimate for the
mean threshold of minimal representation” (2007: p. 246). The
formula is an empirical expression with roots in theory (Taa-
gepera, 2007). For a large part of the countries, it is relatively
unproblematic to establish the average effective threshold. In
some countries, however, legal thresholds, adjustment (com-
pensatory) seats and combined electoral systems must be taken
into account when calculating the effective threshold. Details
are provided in the Appendix.
Other Relevant Elements of the Electoral System
The electoral formula is one of the main elements of an elec-
toral system, yet it is not included among the independent vari-
ables in the regression analysis. It strongly correlates with the
effective threshold which means that, if included in the same
model, it would most likely distort the results. Rather, electoral
formula is dealt with by analyzing two sub-sets of elections,
namely systems with single-member districts and proportional
systems. The former category consists of the single-member
plurality formula, the two-round system and the alternative vote.
The block vote version of plurality systems is not included in
that sample, because it employs multi-member districts. The
sub-set of proportional systems consists of proportional list
systems, the single transferable vote and mixed-member pro-
portional systems. Despite combining two electoral formulas,
the last mentioned is included among PR systems, because the
final seat allocation is proportional. In addition to all electoral
formulas included in these two sub-sets, mixed-member ma-
joritarian systems, the block vote and the single-non-transfer-
able vote are part of the total research population.
Ballot structure and apparentement are also of some impor-
tance with regard to electoral consequences. Both variables
may be classified in different ways (see e.g. Cox, 1997). As for
disproportionality and fragmentation, the most central feature
of ballot structure is whether the voter may cast one or several
votes. We can thereby distinguish between categorical and
ordinal ballots. The theoretical link between ballot structure and
disproportionality is indirect, relating to the two-way relation-
ship between disproportionality and the party system. Dispro-
portionality decreases multipartism but, to some degree, multi-
partism also increases disproportionality (Lijphart, 1994). Ap-
parentement implies that parties are allowed to connect their
lists, and the combined vote total is used in the initial seat allo-
cation. This opportunity improves the chances for small parties
which otherwise would be considerably disadvantaged. Lijphart
(1994) has found that apparentement has a positive impact on
proportionality when the effective threshold is controlled for,
whereas Anckar (1998) has reported a positive, significant ef-
fect of apparentement on the effective number of parties.
External Variables
Party system structure is also related to other factors than
those that are part of the electoral system; particularly country
size, presidentialism and cultural heterogeneity. Yet, country
size cannot be included in the regression models since both
population and area strongly correlate with assembly size.4 In
presidential elections, which may be seen as single-member
district elections in the country as a whole, votes are mainly
cast for candidates of the two largest parties, and this pattern is
assumed to be reflected in parliamentary elections. However,
the association between presidentialism and party system frag-
mentation is dependent on two conditions: the presidential elec-
tions must be conducted according to the plurality rule, i.e. in a
single round, and they must be held at the same time as parlia-
mentary elections (Shugart & Carey, 1992). Presidentialism is
also assumed to have an indirect effect on disproportionality.
Because of a relatively small effective number of parties in
governmental systems as described above, the degree of dis-
proportionality is expected to be smaller as well (Jones, 1993;
Lijphart, 1994). Hence, in the empirical analysis, presidential-
ism is controlled for. I shall apply Sartori’s definition of presi-
dentialism: the president 1) must be popularly elected, 2) can-
not be discharged by a parliamentary vote, and 3) appoints as
well as directs the government (Sartori, 1994).
The occurrence of several ethnic groups is assumed to influ-
ence the party system structure in plurality systems. Through-
out the years, some qualifications have been added to Du-
verger’s (1964) theory of plurality elections and two-party sys-
tems. Perhaps the most important exception is concerned with
countries in which ethnic and other minorities are regionally
concentrated. If ethnic minorities constitute a majority of the
population in some regions, “third” parties may in some elec-
toral districts obtain a fair-sized share of seats (Sartori, 1994).
Indeed, Duverger (1964) acknowledged that the contesting
parties may be different in different parts of the country,
thereby making a fragmented party system at the national level
possible. Ethnic heterogeneity is controlled for in the sub-set of
SMD systems.5
3Another related problem concerns the effective threshold in countries with
districts of unequal magnitude. Taagepera (1998a) uses Finland as an exam-
le. The average magnitude is 13.3 (200 seats allocated in 15 districts),
which returns an effective threshold of 5.2 per cent. However, the three
smallest districts in 1983 had a magnitude of 1, 7, and 8 whereas the two
largest ones had 20 and 27 seats. In the largest di strict, the effecti ve thresh-
old was as low as 2.7 per cent, and thi s is the threshold that a small p arty had
to cross in order to get a seat in the parliament. Taagepera concludes that
ecause of large deviation in magnitude, party system fragmentation in
Finland is greater than it would be if all districts were of equal size. So why
do we not calculate the effective threshold on the basis of the largest district
then? First of all, in a worldwide comparison, it is difficult to obtain reliable
information on the size of districts in all countries. Secondly, and more
importantly, the results would be distorted if we only consider one district.
Small parties might get one or two seats in the two largest districts in
Finland but that is not always the case. The largest district is usually the one
that includes the capital. Quite often, however, small ethnic parties are
regionally concentrated in smaller districts elsewhere (Anckar, 2002).
Therefore, the average magnitude and threshold is a better solution than
paying attention only to the lar g est district.
Dependent Variables
Disproportionality and party system fragmentation are the
dependent variables of the study. Disproportionality is meas-
ured by means of Gallagher’s least-squares index (1991), cal-
culated in the following way: the difference between vote and
seat shares for all parties are initially squared and then added.
4In the research population of the study, the correlation coefficient between
assembly size and logged population is 0.82. Almost perfect correlation
prevails between logged assembly size and l ogged pop ulation: 0.92.
5Values on ethnic heterogeneity provided by Anckar, Eriksson and Leskinen
(2002) are applied. The degree of fragmentation is calculated according to
the index of ethnic fractionalization proposed by Rae and Taylor (1 970).
Copyright © 2012 SciRes.
Thereafter, the sum is divided by two and the square root of this
value is taken. Higher values indicate higher degrees of dis-
proportionality. Party system fragmentation is measured by
Laakso and Taagepera’s (1979) index, “the effective number of
parties”, calculated as one divided by the sum of squared seat
shares for each party. The index may also be calculated on the
basis of vote shares. Lijphart (1994) included two measures: the
effective number of elective parties based on vote shares and
the effective number of parliamentary parties based on seat
shares. I include only the latter because the analysis focuses on
party system structure in the legislature. Furthermore, in Li-
jphart’s study, electoral system elements are more strongly
related to the effective number of parliamentary parties than the
effective number of elective parties.
When studying effects of electoral systems, some level of
democracy must prevail. The most widely used source for de-
termining the level of democracy is Freedom House’s annual
survey of political rights and civil liberties. For each of the two
dimensions, a scale ranging from one to seven is applied, with
one representing the most democratic and seven the least de-
mocratic. In addition, countries are categorised as “free”, “part ly
free” and “not free”. The latter classification is used for select-
ing the research population of the study. First, only elections
conducted during years of “free” status are included. Second, in
order to qualify for inclusion, a country must have been classi-
fied as “free” or “partly free” each year since t he previous elec-
tions. The analyzed time period is from 1972 to 2008, i.e. ever
since Freedom House has provided ratings of the level of de-
mocracy. All parliamentary elections (to the lower or only
chamber) that fulfil the criteria above are included in the analy-
sis,6 which results in a total research population of 550 elec-
tions.7 The sub-sets of SMD systems and PR systems consist of
166 and 315 elections, respectively.
Empirical Analysis
The Total Research Population
The empirical analysis begins with the total research popula-
tion. Assembly size negatively correlates with the degree of
disproportionality and positively with the effective number of
parliamentary parties. The coefficients are –0.315 and 0.263,
respectively, and significant at the 0.001-level. Nevertheless,
the effective threshold is much more strongly associated with
the dependent variables; the correlation coefficients are 0.687
and –0.543, respectively. Ballot structure and apparentement
are also negatively associated with disproportionality and posi-
tively with party system fragmentation.
In Table 1, multivariate patterns are given, applying OLS
regression analysis. Assembly si ze has a negative impact on the
degree of disproportionality and a positive impact on the effect-
Table 1.
The effect of assembly size, effective threshold, ballot structure, ap-
parentement and presidentialism on disproportionality and the effective
number of parliamentary parties, OLS regression.
Degree of
disproportionality Effective nu mber of
parliame ntary parties
(Constant) 7.058
6.029*** 2.892
Assembly size (log) –2.401
Effective t h reshold .311
Ballot structu re
(dummy) –.248
(dummy) .407
(dummy) .012
R-square .495 .375
F-sig. *** ***
N 530 543
Note: In each c ell, from to p d ownwar ds , figures indicate the regression coefficient,
the standardized regression coefficient and the T-value, ***=sig. < 0.001; Sources:
Anckar (2002); Chronicle of Parliamentary Elections; Countries of the World;
Inter-Parliamentary Union; Lijphart (1994); Parties and Elections in Europe.
tive number of parliamentary parties when other relevant vari-
ables are controlled for. However, the most important variable
with regard to both disproportionality and fragmentation is the
effective threshold. Apparentement also has a stronger effect
than assembly size on the effective number parties. Ballot
structure has no independent effect on the dependent variables.
As mentioned earlier, Lijphart maintains that assembly size
affects the degree of disproportionality in small assemblies but
not in large ones. In Lijphart’s analysis (1994), no influence of
assembly size was found in a multiple regression with regard to
the effective number of parties. Although the difference be-
tween small and large legislatures has already been taken into
account by using assembly size in logarithm form, I have also
run a separate analysis (not presented in table format) in a sam-
ple of assemblies with less than one hundred seats. There is a
strong bivariate association between assembly size and the
degree of disproportionality. However, in the multivariate
analysis, assembly size has no impact—the effective threshold
is the sole significant variable. Yet, it does not falsify Lijphart’s
statement, because his sample consisted of PR systems only.
Lijphart excluded majoritarian systems because in his research
population of established democracies, majoritarian elections
were conducted mostly in very large assemblies. Therefore we
have to return to the matter when the sub-set of PR systems is
6Countries that lack political parties (Federated States of Micronesia, Mar-
shall Is lands, Nauru, P alau an d Tuval u) are ex clud ed fro m the anal ysis since
they have no party systems, and consequently no values on the dependent
variables of the study. Another four countries (Kiribati, Papua New Guinea,
Samoa and S olo mon Islan d s) are ex clud ed becau se i n mos t electi on s, a l arg e
part of the successful candidates have been elected as independents.
7All data used in this study can be received from the author upon request.
Some data, mainly on the degree of disproportionality, are missing. There-
fore, N in the regression analysis of the total research population is smaller
than 550.
Systems with Single-Member Districts
Next, SMD systems are separately analyzed. Presidentialism
and ethnic heterogeneity are included as external control vari-
ables. Ballot structure is the only electoral system characteristic
that we need to control for when analyzing the effect of assem-
bly size in this sample, because no formal electoral thresholds
Copyright © 2012 SciRes. 13
are applied, apparentement does not exist in SMD systems, and
the average district magnitude in each country is one. Assembly
size correlates more strongly with the dependent variables in
SMD systems than in the total research population; the correla-
tion coefficients are –0.372 and 0.473, respectively, and sig-
nificant at the 0.001-level. As shown in Table 2, the effect of
assembly size on disproportionality remains when other factors
are controlled for, yet the impact of ballot structure and espe-
cially presidentialism is stronger. Concerning the effective
number of parliamentary parties, assembly size is the most
important variable.
Proportional Systems
There is a significant bivariate association between assembly
size and party system fragmentation in the sub-set of propor-
tional systems, yet not as strong as that between the effective
threshold and fragmentation. The effective threshold strongly
correlates with disproportionality—in contrast, no bivariate
relationship exists between assembly size and disproportional-
ity. Worthy of attention is also that the effective number of
parties is somewhat more strongly related to apparentement
than the effective threshold.
Results of the regression analysis are given in Table 3. As
hypothesized, the negative effect on disproportionality and the
positive impact on the effective number of parties of assembly
size that was found in both the total research population and
among systems with single-member districts does not exist in
proportional systems when the other variables are controlled for.
The impact of the effective threshold is significant with regard
to both dependent variables. Surprisingly, there is a significant
positive effect of assembly size on the degree of disproportion-
ality. Most likely, this is a spurious relationship caused by the
Table 2.
The effect of assembly size, ballot structure, presidentialism and ethnic
heterogeneity on disproportionality and the effective number of parlia-
mentary parties in systems with single-member districts, OLS regres-
Degree of
disproportionality Effective nu mber of
parliame ntary parties
(Constant) 21.962
9.884*** .957
Assembly s ize
(log) –2.675
Ballot structu re
(dummy) –6.258
(dummy) –9.051
heterogeneity –3.409
R-square .261 .337
F-sig. *** ***
N 164 166
Note: In each c ell, from to p d ownwar ds , figures indicate the regression coefficient,
the standardized regression coefficient and the T-value, ***=sig. < 0.001; Sources:
Anckar (2002); Anckar, Erikss on, & Les kinen (2002); Chro ni cle of Parli ament ary
Elections; Countries of the World; Inter-Parliamentary Union; Lijphart (1994);
Parties and Elections in Europe.
Table 3.
The effect of assembly size, effective threshold, ballot structure, ap-
parentement and presidentialism on disproportionality and the effective
number of parliamentary parties in proportional systems, OLS regres-
Degree of
disproportionality Effective number of
parliame ntary parties
(Constant) –1.011
–1.234 3.306
Assembly size (log) 1.259
Effective t h reshold .307
Ballot structu re
(dummy) –.979
(dummy) .229
(dummy) .632
R-square .327 .149
F-sig. *** ***
N 314 315
Note: In each c ell, from to p d ownwar ds , figures indicate the regression coefficient,
the standardized regression coefficient and the T-value, ***=sig. < 0.001, **=sig. <
0.01; Sources: Anckar (2002); Chronicle of Parliamentary Elections; Countries o f
the World; Inter-Parliamentary Union; Lijphart (1994); Parties and Elections in
interplay of several variables, since, first, assembly size and
disproportionality do not correlate, and, second, there is no
theoretical explanation of this kind of association. The effective
threshold explains most of the variation, whereas apparente-
ment is of no significance with regard to the degree of dispro-
portionality. Instead, ballot structure influences disproportion-
ality among proportional electoral systems. Elections in which
the voters have more than one vote produce more proportional
results than elections with a categorical ballot. In a sample (not
presented in table format) with legislatures that have less than
one hundred seats, assembly size has no effect on the degree of
disproportionality. Effective threshold and ballot structure are
significant at the 0.001-level and the 0.05-level, respectively.
The fact that apparentement possesses most explanatory
power regarding the effective number of parliamentary parties
in the PR sample calls for some further investigation. Seven
countries provide the opportunity of combining p arty lists:
Finland, Israel, Lithuania, the Netherlands, Poland, Romania
and Switzerland. Together they constitute 54 cases. The aver-
age effective number of parliamentary parties in this group of
countries is 4.86, compared to 3.60 among the remaining 262
cases of proportional elections. The mean effective threshold in
the countries with apparentement is 7.25, whereas the other
group returns the mean value of 4.39. However, there is a very
large difference in district magnitude between the groups,
which is not transformed into a similar difference in the effec-
tive threshold. The average district magnitude among appar-
entement countries is as large as 60.9, compared to 18.3 among
elections without the opportunity of combining lists. On the
basis of previous research, we know that district magnitude is
Copyright © 2012 SciRes.
Copyright © 2012 SciRes. 15
the single most influential element with regard to electoral sys-
tem consequences. These circumstances most likely explain
why the explanatory power of apparentement surpasses the
effective threshold in this sample.
Concluding Remarks
In Electoral Systems and Party Systems, Lijphart (1994) ar-
gued that the assumed influence of assembly size foremost
concerns PR systems, and that a possible effect on dispropor-
tionality is less plausible in non-PR systems. The empirical
findings here suggest that it is just the opposite. In majoritarian
systems, increasing assembly size results in a lower degree of
disproportionality and a higher degree of party system frag-
mentation. In proportional systems, these tendencies do not
exist. The difference between PR and majoritarian systems
arises because PR systems consist of elements that are absent or
do not vary in systems with single-member districts. PR coun-
tries use multi-member districts of varying size, and many of
them apply electoral thresholds that restrict the chances of rep-
resentation for small parties. In addition, some PR systems
provide the opportunity of combining party lists, and some use
ordinal ballots. These elements also have some impact on the
degree of disproportionality and the effective number of parties.
With the exception of ordinal ballots in alternative vote systems,
all these elements are absent in countries with single-member
districts, and therefore assembly size becomes much more im-
portant in majoritarian systems.
To be sure, in SMD systems, the size of the legislature is di-
rectly related to the number of districts. The chances for small
parties to win a seat increase as the number of constituencies
increases, particularly if their support is regionally concentrated.
Accordingly, on the basis of these results, we might as well
conclude that the number of districts is decisive in SMD sys-
tems since the number of districts is equal to assembly size.
Here, however, focus has been on the influence of assembly
size and differences between majoritarian and PR systems in
this respect. In proportional electoral systems, ballot structure,
apparentement, and, in particular, the effective threshold su-
persede assembly size as determinants of the degree of dispro-
portionality and the effective number of parties. In electoral
systems with single-member districts, by contrast, assembly
size remains a significant factor.
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Determining the Effective Threshold in Countries
with Formal Electoral Thresholds, Adjustment
(Compensatory) Seats and Mixed Electoral Systems
If the formal electoral threshold is higher than the effective
threshold based on average magnitude, the former becomes the
actual effective threshold. This is the case in e.g. Israel where
the effective threshold would be 0.62 percent without a legal
threshold of 2 percent.
The use of compensatory seats implies that the final seat al-
location is based on the results nationwide; any disproportion-
ality at the lower district level is corrected at the upper level.
However, it presupposes that the number of compensatory seats,
the total number of seats in the legislature, and the districts at
the lower level are sufficiently large. In most countries that
apply compensatory seats, the upper level determines the effec-
tive threshold. A formal threshold is often applied at the upper
level; in Norway, for example, a party needs 4 percent of the
total vote in order to be entitled to compensatory seats. Since
this formal threshold is lower than the effective threshold based
on the average magnitude at the lower level, it becomes the
actual effective threshold. The same applies to countries such
as Sweden, where a legal threshold is present at both the upper
and the lower level: the 4 per cent formal threshold at the upper
level, which is used for allocating compensatory seats, consti-
tutes the effec t ive threshol d .
However, in some countries with compensatory seats, parties
get more than one shot at qualifying for representation. They
may circumvent the legal threshold by winning a sufficient
number of seats in the constituencies. Gallagher and Mitchell
(2005) discuss the cases of Austria and Denmark and conclude
that it is very rare for a party to qualify for the alternative route
if they have not also crossed the formal threshold of 4 and 2
percent, respectively. Accordingly, the legal thresholds may
well be treated as effective thresholds. Another kind of adjust-
ment seats is called additional seats. The allocation of addi-
tional seats is not dependent on the results at the lower level
and hence does not affect the overall proportionality to any
noticeable extent. Therefore the upper level district is added to
the lower level districts when the average magnitude and the
effective threshold are calculated. This strategy is also used for
countries where compensatory seats only partially compensate
for the disproportionality at the lower level.
There are two kinds of mixed electoral systems: mixed-
member proportional (MMP) where any disproportionality in
the lower tier is eliminated through seat allocation in the upper
tier, and mixed-member ma joritarian (MMM) where there is no
link between the tiers. In MMP systems, calculation of the ef-
fective threshold is based on the upper tier, usually determined
by a legal threshold. However, some MMP countries provide
alternative routes to parliamentary representation. In Germany,
parties qualify for upper tier seats if they win either 5 percent
of the list votes or three single-member districts. In New Zea-
land, a party has to be successful in only one single-member
district in order to circumvent the 5 percent threshold at the
upper level. Again, as in the corresponding cases with adjust-
ment seats, Gallagher and Mitchell (2005) come to the conclu-
sion that the legal threshold at the upper level represents the
national effective threshold. The authors also point out that
formal thresholds at the district level in Spain and Belgium do
not affect their national effective thresholds (Gallagher &
Mitchell, 2005).
At a first glance, MMM sy stems could be tre ated similarly as
MMP systems; i.e. to let the upper tier determine the effective
threshold. However, since there is no link between the tiers,
small parties are substantially under-represented in the final
seat allocation. Therefore, it is reasonable to take into account
the relative weight of both tiers when calculating the effective
threshold. In both tiers, the effective threshold is multiplied by
the number of seats in the tiers, respectively. These values are
added and the sum is divided by the total number of seats.
Hungary (as of 1994) and Italy (from 1994 to 2001) have ap-
plied MMM systems with partial compensation (Shugart &
Wattenberg 2001). In Hungary votes cast for unsuccessful can-
didates in the lower tier are added to their parties’ list votes. In
the Italian mixed system, the transfer went from party lists to
candidates in the single-member districts. The transfer of votes
makes it difficult to determine how large a vote share is needed
in order to win a seat in each tier. In my analysis, I rely on Gal-
lagher and Mitchell’s (2005) estimate of “fair” representation,
which in both cases is the legal threshold in the list tier, i.e. 5
percent in Hungary and 4 percent in Italy.
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