2 Introduction: Goals and Methods
elections almost automatically call the PR system being used a "complex form of PR'! In fact, with only a few exceptions, PR systems can be classified and described in relatively simple and straightforward terms. One of the reasons for the unnecessary confusion surrounding electoral systems is that both electoral engineers and students of electoral systems have used confused terminologies-with the same term sometimes being used for different practices and the same practice referred to by different terms. I shall try to clarify and simplify the basic terms, and I shall present the principal properties of the various electoral systems in clearly defined categories so as to facilitate comparisons among them as well as the systematic testing of their political
consequences-'
WHICH ELECTORAL SYSTEMS?
This analysis covers the electoral systems used in twenty-seven democracies from 1945 to 1990-that is, those used in the vast majority of the free and democratic parliamentary elections (at the national level in countries larger than mini-states) that have ever been conducted. Of the twenty-seven democracies, twenty-four are the world's most durable democracies with a history of free elections without major interruptions since 1945 or shortly thereafter. They are the four most populous countries of Western Europe (the United Kingdom, France, Germany, and Italy), the five Nordic states (Sweden, Norway, Denmark, Finland, and Iceland), the three Benelux states (The Netherlands, Belgium, and Luxembourg), four other smaller democracies (Ireland, Austria, Switzerland, and Malta), and eight countries outside Europe (the United States, Canada, Costa Rica, Israel, India, Japan, Australia,
and New Zealand).
To these I added Spain, Portugal, and Greece, although they do not qualify under the criterion of long-term and uninterrupted democracy.2 On the other hand, they have been democratic since the mid-1970s and are generally judged to be stable and consolidated democracies; moreover, it seems appropriate to consider them alongside the other West European democracies. Malta is another somewhat doubtful case for inclusion since it did not become independent until 1964, but it did conduct free

Introduction: Goals and Methods 3
universal-suffrage elections as an internally self-governing territory from 1947. It also offers the advantage of providing a second example, in addition to Ireland, of ^e unusual single transferable vote (STV) form of PR. Finally, a practical advantage of including Malta as well as Spain, Portugal, ^nd Greece is that their election data are available in the Internati^^ Almanac of Electoral History, the handbook that serves as the major source of election data for this study.3
As the year in which the Seeoi^ y^ld War ended, 1945 is a conventional starling-point for studies in the social sciences- It is especially appropriate for this stu^y because, prior to 1945, many of the countries listed above were not democratic or democratic for only a short period (such as Germany, Italy, and Japan), not yet independent (India and Israel ^ ^ ^ have fully democratic elections with universal suffrage g^ce women did not have the right to vote (France and B^g^^ 0^ course, even after 1945, two of our countries contin^ ^ have serious restrictions on the right to vote: the United States until the passage of the Voting Rights Act in 1965 and Switzerland until the adoption of women's voting rights in 1971. Most of our countries conducted elections in 1945 or 1946; for the others, the starling-point is the first election after 1945 followed ^y an uninterrupted period of regular free elections lasting until ^e end of 1990.
The year 1990 was chosen as the ^nd of the period under analysis for practical reasons-the availably ^ accurate and comparable election data. However, three el^^g ^eld before the end of 1990-the November 1989 electi^ ^ ^-^ ^ ^e December 1990 elections in Germany and Denmark-could not be included because the necessary election information was still missing when the data analysis had passed a ci-ii^ai point. There may also be some symbolic significance in ending ^e analysis just prior to the 1990 all-German election because ^is was an election in a partly
new and different country and, ev^n more significantly, it marked the end of the post-Second Worl<j y^. ^
The electoral systems to be analy^ ^e those for national lower-^ouse elections (or, in the case of Unicameral parliaments, the elec-"ons of the one chamber) in the co^tnes and the period indicated.
and T^8 that a11 other nationa' (upper-house and presidential) "a ail subnational elections are excluded, even when they are
^'rect popular elections. The only exception is that, for the twelve moers of the European Corr^^y the elections to the

4 Introduction: Goals and Methods ;
European Parliament are included. For the purpose of this analy-
sis, I am treating the European Parliament as a set of national
mini-parliaments. This does not represent a correct view of the actual operation of the European Parliament, but it is an accurateinterpretation of the way it is elected-by twelve different elec- toral systems that are generally much more closely related to the twelve national parliamentary electoral systems than they are to
each other. Vernon Bogdanor writes that even the third cycle of
European elections, held in 1989, again 'proved to be, as in 1979
and 1984, primarily an arena for a set of national contests'.4 And
Hermann Schmitt cites survey data showing that most voters con-
tinue to think of the parties competing for election to the Euro-
pean Parliament as national parties, and that they would also 'prefer
to have parties in the European Parliament structured along na- tional rather than on political-ideological lines'.5
In addition to enriching the data base for this study, the inclusion of the Euro-elections has two special advantages. First, it provides examples of the election of relatively small representative bodies: all of the twelve countries have European Parliament delegationsthat are considerably smaller than the lower or only chambers of their national parliaments. Second, it offers good opportunities -5 for controlled comparison, because for most countries the elec- toral systems for the national parliaments and for the national
mini-parliaments in the European Parliament resemble each other closely but are not completely identical.
Table 1.1 lists the 350 parliamentary and 34 Euro-elections that form the empirical basis of this comparative study. In almost all
cases, all the votes cast and all seats at stake in an election are
included in the analysis. However, I made a few exceptions to this
general rule in order to make the comparisons of votes and seats
as accurate as possible. For instance, I excluded all uncontested
seats for which no votes were cast or recorded (mainly in coun- tries with majoritarian election systems but also in Ireland and Switzerland) and seats filled by indirect election (the West Berlin representatives in the Bundestag and in the European Parliament).
In order not to confuse the effects of different electoral systems,
I excluded the few STV districts (both votes and seats) from Bri-
tish elections, which have been mainly plurality elections; the four
two-member and three-member STV districts in the 1945 parlia-
oTV mentary election and the Northern Ireland three-member a^

^ ^ -a ^o0 ls � j district used for the 1979, 1984, and 1989 Euro-e lections. I also

r0 ^ f
w j In contrast with Douglas W. Rae's classic study, in which elections serve as the units of analysis,6 my cases are seventy electoral systems, defined as sets of essentially unchanged election rules under which one or more successive elections are conducted. Elections held under the same electoral system are regarded as repeated observations of the operation of a single electoral system. For instance, Finland provides only one electoral system under which its thirteen parliamentary elections were held, whereas Germany i provides six different electoral systems, four Bundestag and two ^ European Parliament systems, that guided its total of fourteen ?i j elections.
I, My variables are the basic characteristics of electoral systems, j measures of disproportionality, and measures of multipartism and ^ E of the production of majority parties. I follow two basic multivariate >o | approaches. One is a comparable-cases strategy that concentrates �; on within-country variations when more than one electoral system ' is used in the same country; this permits the examination of the ^ effect of changing one aspect of an electoral system while the : system remains the same in other respects. Additionally, the effect [ of small changes within electoral systems-changes that are not M ^ | sufficiently important to signify changes of the electoral system- I will be examined.
^ The second basic strategy relies on a cross-sectional research ^ !. design in which cross-tabulations as well as multiple correlation ^ e and regression are applied to the seventy electoral systems. How-S J ^er, what is an advantage for the first strategy-having attractive " ,^ ^rnparable cases in the form of more than one electoral system
oa .J r ln tne ^me country-presents somewhat of a problem for the
-5 t J ^ond strategy, because it means that some of the cases are not ^ H I "ipletely independent- For this reason, 1 shall also present an

analysis of fifty-three, instead of seventy, cases by combining those electoral systems in the same country where such a combination is at all possible and justifiable.
A different but at least equally crucial aspect of my research strategy was the combination of a collaborative project with, as its product, a single-authored book. Experts on, and usually in, each of the twenty-seven countries, supplied me with vital information, interpretation, and feedback on the rules and operation of their countries' electoral systems. My goal was to find the optimal mix of the pooled wisdom provided by joint scholarship with the consistency of having one author and researcher-in-chief-
Finally, a major methodological goal of this book is to promote replication. As the following chapters will repeatedly show, a host of important decisions must be made with regard to classification, measurement, and other methodological matters. I shall always explicitly defend my choices, and, in many instances, I shall also show the results that would have been yielded if different measures and methods had been used. But I want to make it as easy as possible for my readers to reanalyse the data according to the alternatives that they prefer. For this purpose, all of the basic data are easily available. The detailed characteristics of the seventy electoral systems are given in the tables of Chapter 2. The measures of disproportionality and multipartism for the same seventy electoral systems are listed in Appendix B. If readers want to do a more thorough replication, they can obtain the disproportionality and multipartism measures for each of the 384 elections from the author.7 Finally, all of the raw election data may be found in a few easily accessible sources: the International Almanac of Electoral History together with the 1989 and 1990 updates in the European Journal of Political Research (for all of the countries except India and Costa Rica) and the two volumes Europe Voles 2 and Europe Voles 3 (for the European Parliament elections).8 Appendix C contains a list of corrections and clarifications concerning these data as well as the election figures for India and Costa Rica.
OVERVIEW AND PREVIEW
Chapter 2 will give a detailed description and classification of th� seventy electoral systems. It will also highlight general patterns

(such as the high frequency of the use of list PR and of the d'Hondt formula) and general trends (such as the increasing use of more proportional methods and the increasing use of two-tier districting systems). Chapter 3 will examine the concepts of electoral disproportionality, multipartism, and majority-party generation, and will discuss the advantages and disadvantages of the different operational measures that have been proposed. How to measure disproportionality presents the most serious problem, but I shall show not only that my preferred measure-Michael Gallagher's least-squares index-offers a good solution, but also that values of the different measures advocated by other scholars correlate highly with those of the least-squares measure.
The next three chapters examine the relationships between the electoral system variables, disproportionality, multipartism, and majority-party generation. Chapter 4 does so by examining within-country variation, and Chapter 5 by means of a cross-sectional design. Chapter 6 extends the analysis by examining the potential explanatory power of four additional elements of electoral systems: ballot structure, malapportionment, presidentialism, and apparentement. My main conclusions will be that, of the five dependent variables, disproportionality is the one that can be explained best in terms of the electoral system characteristics, and that the strongest explanation of the various dependent variables is provided by what I shall call the 'effective threshold', a combination of district magnitude and electoral thresholds. The effect of the electoral system on multipartism is more modest but still very important, and the explanatory power of the other electoral system variables-the electoral formula, assembly size, apparentement, ballot structure, and presidentialism-is also more modest but, again, not at all negligible. Chapter 7 will conclude by examining some of the practical lessons that electoral engineers and reformers can learn from my findings.

2
Electoral Systems: Types, Patterns, Trends
THE foremost purpose of this book is to analyse the political effects of electoral systems. The first step that needs to be taken towards that goal is the description and classification of the electoral systems. This is usually done in terms of the different 'dimensions' of electoral systems-a practice that I shall also adopt for the description of the seventy electoral systems of our twenty-seven countries between 1945 and 1990 in the bulk of this chapter. The last three sections will deal with the empirical relations between the dimensions and with general patterns and trends in the development of electoral systems.
DIMENSIONS OF ELECTORAL SYSTEMS
There is broad agreement among electoral system experts that the two most important dimensions of electoral systems, with major consequences for the proportionality of election outcomes and for party systems, are the electoral formula and the district magnitude.' Three main types of electoral formulas and a large number of subtypes within each of these are usually distinguished: majoritar-ian formulas (with plurality, two-ballot systems, and the alternative vote as the main subtypes), PR (classified further into largest remainders, highest averages, and single transferable vote formulas), and semi-proportional systems (such as the cumulative vote and the limited vote). The purpose of the introduction of PR "1 many countries was to achieve greater proportionality and better minority representation than the earlier majoritarian electoral
methods had produced-
District magnitude is defined as the number of representatives
elected in a district (constituency). One of the best-known findings

of Douglas W. Rae's 1967 study The Political Consequences of Electoral Laws-the first systematic comparative analysis of the effects of electoral systems on disproportionality and multipartism, which has been a major source of inspiration for this book-was the extremely strong influence of district magnitude.2 Rae modestly attributes this proposition to James Hogan who wrote in 1945: 'the decisive point in P.R. is the size [magnitude] of the constituencies: the larger the constituency, that is, the greater the number of members which it elects, the more closely will the result approximate to proportionality.'3 And twenty years earlier, George Horwill had already referred to district magnitude as 'the all-important factor'.4
In PR systems, proportionality-and the chances for small parties to gain representation-are necessarily very limited when there are only two or three representatives per district, but increase dramatically when magnitude increases. In countries with multi-member districts, district magnitude tends to vary; in this study, therefore, magnitude will usually refer to the average district magnitude. It can be calculated very simply by dividing the total number of seats in the legislature (to which I shall henceforth refer as the assembly size) by the number of districts. Because of the importance of this dimension, all three variables-average magnitude, number of districts, and assembly size-will be listed for each electoral system in the tables in this chapter that provide the basic information on our seventy electoral systems. As will be discussed shortly, assembly size is also an important factor in its own right.
One complication with regard to magnitude is that there may be two, or even more, levels of districts; for instance, a country with PR elections may be divided into, say, ten or twenty electoral districts, but may also have a national district that is superimposed on the lower-level districts. This type of system, for which Rae has coined the term 'complex districting',5 will be explained in due course.
Another important dimension of electoral systems is the electoral mreshold, that is, a minimum level of support which a party needs in order to gain representation. If the electoral law provides for ^ch a threshold, it is usually applied at the national level (indicated by N in the tables), but it may also be imposed at the district t"). or at an in-between, regional (R) level, and the minimum
a^ oe denned in terms of a certain number of votes, a percentage

of the votes, or some other criterion such as the winning of at least one seat in a lower-level district in order to be eligible for seats in
the higher-level district.
Not all electoral systems have such legal thresholds-in fact, most do not-but, as Rein Taagepera and Matthew S. Shugart have pointed out, even in the absence of an explicit legal threshold, an actual threshold is implied by the other two dimensions of the electoral system, especially by district magnitude.6 Low magnitudes have the same effect as high thresholds: both limit proportionality and the opportunities for small parties to win seats; as magnitudes increase and thresholds decrease, proportionality and the chances for small parties improve. In other words, legal thresholds and district magnitudes can be seen as two sides of the same coin. Accordingly, I shall often treat these two dimensions as one variable. All magnitudes and legal thresholds can be converted into a single operational indicator: the effective threshold, stated in terms of a percentage of the total national vote. How the effective threshold is calculated will be explained later-
The fourth major dimension on which this study will focus is assembly size-that is, the total number of seats in the legislature. Rae calls attention to this 'generally neglected variable', but he does not enter it into his empirical analysis.7 Its effect has not been studied systematically by other electoral system analysts either, perhaps because they have tended to see it as a factor external to the electoral system, that is, as merely a characteristic of legislatures elected according to particular electoral systems rather than as a characteristic of electoral systems as such. However, if electoral systems are defined as methods of translating votes into seats. the total numbers of seats available for this translation appears to be an integral and legitimate part of the systems of translation.
In any case, there can be no doubt that assembly size can have a strong influence on proportionality and on the degree of multi-partism. For instance, if four parties win 41, 29,17, and 13 per cent of the national vote in a PR election-to use the example that I shall also use in Appendix A to illustrate the operation of different PR formulas-there is no way in which the allocation of seats can be handled with a high degree of proportionality if the election is to a mini-legislature with only five seats; the chances of a proportional allocation improve considerably for a ten-member legislature; and perfect proportionality could be achieved, at least u*

principle, for a 100-member legislative body. Of course, the same pattern theoretically applies to non-PR systems as well, but since these systems do not even aim at proportionality, the hypothesis that assembly size may have a significant additional effect on their degree of disproportionality may seem less plausible. Nevertheless, Taagepera has found that, in plurality elections, the degree of disproportionality does tend to increase, all other factors being equal, as the size of the legislature decreases.8 In short, there is ample theoretical Justification to include assembly size as one of the important dimensions of electoral systems.
As already stated in the previous chapter, I define an electoral system as a set of essentially unchanged election rules under which one or more successive elections are conducted in a particular democracy. This definition can now be refined by stating it in terms of the four major dimensions of electoral systems: if there is a significant change on one or more of the four dimensions, this means that a new electoral system must be distinguished. A further refinement is needed in order to define precisely what counts as significant change. The electoral formula is a discrete variable;
hence any change in the formula can be recognized easily and will be regarded as a significant change. In two-tier districting systems, the criterion will be a change in formula at what I shall define later as the decisive tier. However, since the other three dimensions are continuous variables, exact cut-off points have to be specified.
For all three, I propose a 20 per cent criterion: 20 per cent or greater change in district magnitude (in two-tiered districting systems, the magnitude at the more important upper level will be counted), 20 per cent or more change in the national legal threshold (or the adoption of such a threshold where none existed before), and 20 per cent or greater change in assembly size. For instance, a change in legal threshold from 5 to 6 or more, in district magnitude from 10 to 12 or more, or in assembly size from 200 to ^40 or more (or the other way around) will be regarded as changes lhat create a different electoral system.9 This criterion is neces-^nly arbitrary; cut-off points anywhere between 10 per cent and 5 P" ^nt would also be reasonable and legitimate. By selecting
e Natively high value of 20 per cent, I am deliberately opting 0 e"" on the side of caution; in particular, [ want to avoid artifi-y inflating the number of cases (electoral systems) for the

analysis by creating two or more cases that are too strongly alike and that really should be treated as a single case. Chapters 4 and 5 will examine, respectively, the effects of changes within electoral systems and of changes in a smaller set of cases generated by combining relatively similar cases; in other words, these analyses will first relax and then tighten the 20 per cent criterion, and will therefore provide a check of whether 20 per cent is too strict or too lenient as the cut-off point.
FOUR OTHER ELECTORAL SYSTEM VARIABLES
The above four dimensions provide the framework for the description and classification of the seventy electoral systems in this chapter and will also be the main independent variables in the analysis of the effects of these electoral systems in Chapters 4 and 5. In addition, I shall pay some attention to four minor, but not necessarily negligible, aspects of electoral systems and test their political consequences: ballot structure, malapportionment, the difference between legislative elections in parliamentary and in presidential systems, and the possibility of linked lists.
First, ballot structure is one of Rae's three basic dimensions of electoral systems along with formula and magnitude. (Rae does not consider thresholds and assembly size.) Ballot structure can be either categorical, if the voter can give his or her vote to one party only, which is the case in most electoral systems, or ordinal if the voter can divide his or her vote among two or more parties. (The term 'ordinal ballot structure' is somewhat misleading because it includes, but is not limited to, systems in which voters rank order two or more parties.) Rae hypothesizes that ordinal ballots, by allowing vote dispersion, may encourage multipartism, but finds that his evidence (for twenty countries in the period from 1945 to 1964) contradicts his hypothesis-10 However, since the hypothesis is not implausible, it is worth retesting it against the much broader empirical evidence of our seventy electoral systems.
Second, in his recent analysis of the proportional or dispropof" tional effects of different electoral formulas, Michael Gallaghs1' rightly warns his readers that other dimensions of electoral systems

may also affect the degree of proportionality of election outcomes:
in addition to district magnitude and thresholds (he does not mention the factor of assembly size), he points to 'the possibility of malapportionment'." In single-member district systems, malapportionment means that the districts have substantially unequal voting populations; malapportioned multi-member districts have magnitudes that are not commensurate with their voting populations. Obviously, malapportionment may systematically favour one or more parties and therefore contribute to electoral disproportionality. Malapportionment often takes the form of rural or regional overrepresentation. It has not been a serious problem in most of our long-term democracies during the post-Second World War era, but its possible influence is also worth testing.
Third, Shugart has shown that presidential systems can have an important effect on legislative elections if presidential elections are by plurality and if legislative elections are held at the same time: large parties have an advantage in presidential races since smaller parties do not have much of a chance to have one of their candidates elected, and this advantage tends to carry over into the legislative elections.12 Hence, presidentiatism tends to discourage multipartism. Because our set of countries includes only two presidential systems (the United States and Costa Rica), it does not offer an optimal opportunity to test this hypothesis, but semi-presidential systems (France, Finlaflnd, and Portugal) and parliamentary systems with directly elected presidents (Austria, Iceland, and Ireland) may be hypothesized to have similar effects.
The fourth variable that I shall examine pertains especially to PR systems in which voters choose among competing party lists. In several of these, parties are allowed formally to link or connect their lists, which means that their combined vote total will be used in the initial allocation of seats. A set of such inter-party connected lists is usually referred to by the French term apparenlemenl. As Andrew McLaren Carstairs has pointed out, since almost all electoral systems, including PR, in practice favour the larger parties o some extent, 'the question of whether or not apparentement is Permitted can be of great importance to the smaller parties'.13 ^al other electoral systems have features that are functionally o ^^nt to appareniemenl. Along with ballot structure, mal-Pportionment, and presidentialism. it will be tested in Chapter 6.

MAJORITAR1AN ELECTION SYSTEMS
Table 2.1 lists the twelve majoriiarian election systems that have operated in seven of our countries during the 1945-90 period. Six of these countries used only majoritarian electoral systems, and the basic facts concerning their entire electoral system history is contained in the table: Canada, New Zealand, and the United States used the same system throughout the period, and Australia, India, and the United Kingdom, while undergoing a significant change on one dimension, stayed within the confines of majoritarianism. France is the only country in the table with only two (of its six) electoral systems in Table 2.1.
When countries have used two or more systems, they are numbered in chronological order; for instance, IND1 is the first system used in India in the 1952 and 1957 elections, and IND2 is the Indian system for the elections from 1962 to 1984; and the two French systems are labelled FRA3 and FRA6 because two non-majoritarian systems occurred before FRA3 and again between FRA3 and FRA6. For countries with European Parliament elections (all of which took place at the end of our period, between 1979 and 1989), these Euro-election systems are identified by their chronological numbers and also, for the sake of maximum clarity, by the letter 'E\ For example, UK1 is the system for House of Commons elections and UK2E the system for electing British representatives to the European Parliament. I shall use the same conventions in the tables for PR and other electoral systems later on in this chapter. Furthermore, all of these tables will also list the number of elections in each electoral system and the time-span during which these elections took place (in the second column).
Two further general conventions will be used in order to make these tables as clear and informative as possible. One is that all integers indicate exact and unchanging numbers; all other numbers indicate averages. For instance, the district magnitudes of 1 In Table 2.1 mean that in these electoral systems all districts in aU elections were, without exception, single-member districts, whereas the three district magnitudes of 1.00 indicate the use of some, but very few, two-member or multi-member districts. Second, it is noted which entries indicate approximations. An example is the plurality
formula for US House of Representatives elections; this has indeed been the usual formula, but the majority-runoff method has also been used (in Louisiana, where the first stage of the election is referred to as the 'non-partisan primary', and in Georgia). All values of the effective threshold in Table 2.1 are also indicated as approximations; the reasons for using these approximations and the definition of the term 'effective threshold' will be given later on during the discussion of PR systems. (It is also worth recalling the exclusions specified in Table 1.1; in particular, the numbers of districts and the assembly sizes are based on contested seats only.)
Of the many majoritarian formulas that exist in theory, Table 2.1 shows that only three have been in actual use in our set of countries between 1945 and 1990: plurality, majority-plurality, and the alternative vote.14 The plurality formula-often also called the first-past-the-post (FPTP) or relative majority method-stipulates that, in single-member districts, voters can cast one vote each and that the candidate with the most votes wins. (In two-member districts, voters have two votes and the two candidates with the most votes win; and so on.) Five countries have used plurality and have used it almost without exceptions: Canada, India, New Zealand, the United Kingdom, and the United States.15
The French Fifth Republic provides the only instance of the two-ballot majority-plurality formula. Here the rule is that a majority (that is, an absolute majority-more than half of the valid votes) is required for election on the first ballot; if the first ballot does not produce a winner, a second ballot is conducted and the candidate with the most votes wins, even if he or she wins only a plurality of the votes. The second ballot can have more than two candidates, but the usual second-ballot pattern in France is a con-te'st between two principal candidates, because the weakest candidates are forced to withdraw and other candidates may withdraw voluntarily in favour of stronger candidates of allied parties. However, the majority-plurality formula should be distinguished from the majority-runoff in which the second round of the election is restricted to the top two candidates from the first round; it may therefore be characterized as the majority-majority formula, "1 contrast with the French majority-plurality method. The majority-runoff has not been used in our set of countries for legislative elections (with the small exception of some US Congressional

elections, noted above), but it is used for direct presidential elections in France, Portugal, and Austria.16
Australia is the only country that has used the alternative vote. Voters are asked to list the candidates in order of their preference. If a candidate receives an absolute majority of first preferences, he or she is elected; if not, the weakest candidate is eliminated, and his or her ballots are redistributed among the remaining candidates according to these ballots' second preferences; this process continues until a majority winner emerges. As a simple example, let us assume that there are four candidates {A, 5, C, and D) receiving, respectively, 41, 29, 17, and 13 per cent of the voters' first preferences; since no candidate has received a majority of the first preferences, candidate D is eliminated. Let us further assume that the second preferences on D's ballots are for C; this means that, after the second round of counting, C now has 30 per cent of the vote, A 41 per cent, and B 29 per cent. B is therefore eliminated next, and in the third round of counting, the contest is between A and C-one of whom will be the winner. The alternative vote, which in Australia is usually referred to as 'preferential voting', may be thought of as a refinement of the majority-runoff formula in (he sense that weak candidates are eliminated one at a time (instead of all but the top two candidates at the same time) and that voters do not have to go to the polls twice.17
The plurality systems are listed first in Table 2.1 (followed by majority-plurality and alternative vote) and, within the plurality group, the systems are listed in decreasing order of district magnitude. The most striking characteristic of these magnitudes is that, with the exception of the first Indian system, they are either exactly 1 or very close to 1; that is, single-member districts have been the rule and two-member or larger multi-member districts ^ry infrequent exceptions. The only instance of substantial numbers of larger than single-member districts occurred in the 1952 and 1957 Indian elections: slightly more than a third of the ^eats were in two-member districts (and in one three-member "'strict in 1952). The next on the list is Canada, which had two
wo-member districts in the nine elections from 1945 to 1968, aiding an average district magnitude for all fifteen elections of
'gntly less than 1.005, rounded to 1.00 in Table 2.1. The United
lh^ ^s had between one and ^ree two-member districts in Congressional elections from 1946 to 1968-as welt as one

eight-member district in 1962 (the state of Alabama)-for an overall average magnitude of 1.003.'8 And the United Kingdom had fifteen two-member districts in 1945, yielding an average magnitude of 1.002 for all of its post-war parliamentary elections.
As the foregoing already implies, it is also striking that all larger than single-member districts were abolished everywhere: in the United Kingdom after the 1945 election, in India after 1957, and both in Canada and in the United States after 1968. From 1970 on, only single-member districts survived.
All majoritarian systems make it difficult for small parties to gain representation (unless they are geographically concentrated), because they need to win majorities or pluralities of the vote in electoral districts. For this reason, all majoritarian systems tend to systematically favour the larger parties, to produce disproportional election outcomes, and to discourage multipartism-19 District magnitudes larger than 1 tend to reinforce these tendencies; at the extreme, a single at-large (nation-wide) district would, assuming strict party-tine voting, give all legislative seats to the plurality or majority party. For instance, if the 435 members of the US House of Representatives were elected in one 435-member district, with each voter having 435 votes and casting these votes for either 435 Democratic or 435 Republican candidates, the House would end up consisting of either 435 Democrats or 435 Republicans. It is therefore a very important characteristic of the majoritarian systems in Table 2.1 that they are largely single-member district systems. Single-member districts do not make majoritarian systems into proportional ones, but they do limit the degree of dispro-portionality. The exact degrees of disproportionality and of the discouragement of multipartism that remains will be analysed in Chapters 4 and 5.
Given the prevalence of single-member districts, the number of districts in all majoritarian election systems is large-in fact, equal or almost equal to the assembly size in most cases. In most countries, the size of the assembly has remained very stable, especially in the United States where a membership of 435 Re' presentatives was maintained throughout the period with the exception of the two elections after the admission of Hawaii and Alaska when it was temporarily raised to 437. At the other extreme, Australia's House of Representatives doubled in size fro01 1946 to the late 1980s. France's National Assembly was expanded

by about 17 per cent from the 1981 two-ballot election to the 1986 PR contest, and the larger size was retained when the double-ballot was readopted for the 1988 election.
Finally, since majoritarian election systems are inherently unfavourable for small parties, they do not need-and generally do not use-legal thresholds. The one exception, as Table 2.1 shows, is the threshold that French election law has set for access to the second ballot. In 1958 and 1962, candidates with less than 5 per cent of the district vote in the first round were barred from the second ballot; this was raised to 10 per cent of the eligible electorate (approximately 13 per cent of the valid votes) for the next three elections and to 12.5 per cent, again of the eligible voters (about 17 per cent of the valid votes), before the 1978 election. However, both in France and in the other majoritarian systems, parties need many more votes in order to get elected to the legislature in significant numbers and not to be severely underrep-resented. For this reason, I estimate the 'effective threshold'-a term to be denned more precisely in the next section-for all majoritarian systems to be about 35 per cent.
PR: SINGLE-TIER DISTRICTING AND D'HONDT
PR systems are the most common type of electoral systems; fifty-two of our total of seventy-almost three-fourths-unambiguously fit this category. Moreover, as I shall show later, the remaining six "on-PR and non-majoritarian systems (in Japan, Greece, and France) are closer to PR than to majoritarian systems and five (in Japan and Greece) can be interpreted as PR systems. I shall present the fifty-two straightforward PR systems in four tables, two for the ^ngle-tier and two for the two-tier systems.
Table 2.2 lists the systems that, within the PR family, are the ^ost common: those using one-tier districting and the d'Hondt "rmula. What was said about the majoritarian formulas also applies
l0 Pi? f
*"o rormulas: many more have been invented-and even more ca" be imagined-than are in actual use- In addition to the most _equemly used d'Hondt formula, only six PR formulas (and a few
lern closely "^"^le these) have been used in all of the PR sys-s during the 1945-90 period: modified Sainte-Lague (which,

like d'Hondt, is a highest averages or divisor system), four largest remainders or quota systems (using the Hare. Droop, and two Imperial; quotas), and the single transferable vote (STV, which always uses the Droop quota). +he highest averages and largest remainders (LR) systems are list PR systems in which voters vote for lists of candidates (although they may also be able to express a preference for one or more candidates within their preferred list), in contrast with STV in which they cast a preferential vote for individual candidates.
Among the highest averages formulas, the d'Hondt method (which uses the divisor series 1, 2, 3, 4, etc.) is the least proportional and systematically favours the larger parties. It contrasts with the pure Sainte-Lague formula (using the odd-integer divisor series 1, 3, 5, 7, etc.) which approximates proportionality very closely and treats large and small parties in a perfectly even-handed way. In practice, the Sainte-Lague formula is used only in a modified form in which the first divisor is raised from 1 to 1.4, thereby making it harder for small parties to gain their first seats-and hence reducing the proportionality of the election result to some extent.20
The oldest and best known of the LR systems uses the Hare quota, which is the total number of valid votes cast (V) divided by the district magnitude (M, the number of seats available in the district): V/M.21 Parties are given as many seats as they have won quotas, and any remaining seats are given to the parties with the largest remainders of votes. The Hare quota is impartial as between small and large parties and tends to yield closely proportional results. Less proportional outcomes are produced by the Droop quota which divides the votes by M+l, the normal Imperiali quota which uses M + 2, and the reinforced Imperiali quota which uses M + 3 as the denominator. The use of these lower quotas means that there will be fewer remaining seats to be allocated- and hence also more wastage of remaining votes, which is especially "armful to the smaller parties and results in a decrease in proportionality. The Imperiali quotas are so low that there will often "ot be any remaining seats. Whenever the quota is lowered to such an extent that all seats can be assigned without the use of remaining voles, the outcome becomes exactly the same as that of '"c d-Hondt formula.22
STV '
" ^ a preferential rather than a list system but, if voters

cast mainly party-line votes or if most of the inter-party crossover votes offset each other-a simplifying but not unrealistic assumption-its results can be compared to those of LR. All STV systems need to select a quota that elects a candidate and, in principle, any of the quotas discussed above could be used. In practice, however, STV systems invariably use the Droop quota.
To sum up, as far as their effects on the proportionality of the electoral outcome and on multipartism are concerned, the differences cul across the broad categories of divisor, quota, and STV systems. The d'Hondt and LR-Imperiali systems are the least proportional and systematically favour the larger parties; modified Sainte-Lague, LR-Droop, and STV form an intermediate category;
and LR-Hare is the most proportional formula. These tendencies are explained in greater detail in Appendix A, which also provides more detailed descriptions and examples of the operation of the
different formulas.
By definition, PR requires multi-member districts, that is, a
district magnitude of at least 2 seats.23 In order to achieve a minimum of proportionality, however, the magnitude should be considerably larger than 2 and, as argued in the beginning of this chapter, magnitude impacts the degree of proportionality and the chances for small parties very strongly. Table 2.2 presents the twenty-one d'Hondt single-tier districting systems in increasing order of magnitude. The smallest average magnitude among these systems is above 5 seats, in the immediate post-war French elections, but magnitudes vary greatly-up to 150 seats in the Netherlands since 1956. About half have the maximum magnitude allowed by their assembly size: a single at-large (nation-wide) district. This means that they combine the least proportional formula with the most proportional magnitude, In the case of Luxembourg's Euro-elections, the magnitude is still only 6 seats, since only a total of 6 seats are available in this 'assembly'-the smallest assembly size among all of our electoral systems. However, the other ten systems with at-large elections also have the largest magnitudes (and are all listed in the bottom half of the table); six of these are systems for Euro-elections, and the other four are the extremely large-magnitude Dutch and Israeli election systems. The number of districts in the other systems range widely, from 2 to 102.
The large magnitudes are partly offset again by the use of legal thresholds. Eight of the electoral systems shown in Table 2.2 have

such thresholds, but the majority do not. However, as already indicated in the beginning of this chapter, even in the absence of an explicit legal threshold, the district magnitude and the electoral formula, especially magnitude, effectively imply a barrier to smaller parties- For instance, in a small district with a magnitude of 5 seats (like the average district in France in 1945-6), it is easy to see that one-fifth of the votes is sufficient for winning a seat, but that this is very unlikely or even impossible with only one-tenth of the votes- It is more difficult, however, to find the exact equivalent:
for a given average district magnitude, what is the effective threshold at the national level?
EFFECTIVE THRESHOLDS
There are three problems in determining the effective threshold. First, the threshold implied by district magnitude is not one specific percentage but a range of possibilities between the so-called thresholds of representation and exclusion. The threshold of representation (or inclusion) is the minimum percentage of the vote that can earn a party a seat under the most favourable circumstances; the threshold of exclusion is the maximum percentage of the vote that, under the most unfavourable conditions, may be insufficient for a party to win a seat. Another way of portraying these two thresholds is as a lower and an upper threshold: if a party passes the lower threshold, it becomes possible for it to win a seat; when it passes the upper threshold, it is guaranteed to win a seat.
Plurality single-member district systems can provide the simplest illustration of these thresholds. Assume such a district in which five candidates compete. The lower threshold is 20 per cent because a candidate can win with slightly more than this vote percentage in the most favourable situation of the other four candidates evenly splitting the other votes (each receiving just "nder 20 per cent of the vote). The higher threshold is 50 per cent ln tne "^t unfavourable situation of our candidate being faced by
ne very strong candidate; now only 50 per cent plus one vote guarantees election. A simple PR illustration is the following: a
fee-member district, three parties, and the d'Hondt formula.

The lower threshold is 20 per cent since it is possible for a party to win a seat with just over this percentage of the vote if the other two parties are kind enough to split the rest of the votes evenly, each receiving just beiow 40 per cent of the vote (or to receive just below 60 per cent and just below 20 per cent respectively). The higher threshold is 25 per cent: by exceeding this percentage slightly, a party will win a seat even in the most unfavourable case of one of the other parties garnering all of the other votes, that is, almost
75 per cent.
In addition to the problem of determining the exact threshold in the range between the upper and lower thresholds, there are two additional problems. One is that, while these thresholds are largely determined by the district magnitude, they are also influenced to some extent by the electoral formula and the number of political parties that compete. Second, both the magnitude and the number of parties may vary considerably from district to district.
In order to deal with these problems, I shall follow Taagepera and Shugart's lead, although my final solution will be slightly different from theirs.24 They suggest a series of useful and reasonable approximations: that the number of parties be assumed to be about the same as the district magnitude, that the average magnitude for the system as a whole be used, that the formulas also be roughly averaged, and, most importantly, that the effective threshold be assumed to be half-way between the upper and the lower thresholds. Under the first of these assumptions, the upper threshold is almost the same for all formulas: it is equal to or slightly below the Droop quota, that is (expressed as a percentage), 100%/(M + 1).
Unfortunately, the lower threshold varies much more for the different formulas. Taagepera and Shugart pick the lower threshold for the LR-Hare formula: 100%/Mp (where p is the number of parties). This yields too low an estimate for three reasons. One is that the LR-Hare threshold of representation is not only the lowest of all of the formulas but much lower than the others, especially d'Hondt. Second, the tow LR-Hare threshold occurs only in the highly exceptional situation of all parties having very small remainders, which allows a smalt party to win a seat with a fraction of a Hare quota; for instance, in a district with 10 seats and 10 parties and a vote distribution of 91 per cent for one big party and about 1 per cent for the other 9 small parties, one of these small parties can win a seat with just above 1 per cent of the vote. The

more normal situation is for the average remainder to be half of a Hare quota-and therefore also for the lower threshold to be one-half the Hare quota: 100%/2A/. Third, since LR-Hare is itself an unusual formula, it makes more sense to use the lower threshold of the most common formula, namely d'Hondt. As it happens, the d'Hondt lower threshold is only slightly higher than the more normal LR-Hare threshold just estimated."
This 100%/2M threshold therefore appears to be the natural candidate to be used for the average lower threshold. The effective threshold now becomes the mean of the upper threshold-100%/ (M + l)-and the lower threshold- 100%!2M- or:
50% 50%
(M+\) 2M
It should be noted that the Taagepera-Shugart effective threshold, based on the same Droop quota that I use but on the much lower LR-Hare threshold of representation, turns out to be appreciably lower than my effective threshold: after some more streamlining, Taagepera and Shugart arrive at the attractively simple effective threshold of 50%/Af. It is worth noting further that their effective threshold is the same as my lower threshold (the threshold of representation).
In order to determine which of the two alternatives offers the closest equivalent to the formal thresholds imposed by electoral laws, I compared two groups of PR systems. The first group consists of the twenty systems that have clear legal thresholds, independent of the. values of their district magnitudes and independent of any assumptions about whether the lower or middle thresholds should be chosen as the effective thresholds.26 The second group consists of thirty-seven systems whose effective thresholds are inferred entirely or partly from district magnitudes or where assuming a low threshold, like the threshold of representation, versus a middle threshold makes a difference in the calculations (as in tne Belgian and first Austrian systems to be discussed later). In ^e first group, I regressed the percentage of disproportionality fusing the least-squares index, my principal measure of dispro-Portionality to be explained in the next chapter) on the effective "reshold, and I found a regression coefficient of 0.42; this means at ^or every percentage increase in the effective threshold,

disproportionality increased by 0.42 per cent. I repeated this operation for the second group using alternatively the lower Taagepera-Shugart threshold and my effective threshold. The regression coefficients were 0-50 and 0.40 respectively-showing that the latter measure is the closer equivalent. When the electoral formula (d'Hondt and LR-Imperiati versus all other formulas) and assembly size (logged) were also entered into the equations, the regression coefficient was 0.42 in the first group and 0.54 and 0.42 respectively in the second group-confirming the better equivalence of my measure of effective magnitude.27
Another way of judging the two alternative measures of effective threshold is to examine which one yields the higher correlations with the various measures of disproportionality and multipartism for all fifty-seven PR systems and for our universe of electoral systems. Here my findings are that it does not make a great deal of difference whether the Taagepera-Shugart measure or my measure is used (see Chapter 5). One plausible explanation of the relatively strong relationships between the Taagepera-Shugart threshold and mullipartism is that small parties may be encouraged not just by the prospect of being proportionally represented but by the hope of gaining any representation at all, even if it is well below full proportionality.
Three further comments on effective thresholds are in order. One is that neither the Taagepera-Shugart nor my effective threshold works well for plurality and majority systems. For M = 1, both equations yield the value of 50 per cent-which is obviously the upper threshold, above which victory is guaranteed, instead of an average between upper and lower thresholds. As in the case of PR systems, it is much easier to determine the upper than the lower threshold because the latter is strongly influenced by the number of candidates in the race. If we assume a relatively small number of candidates, say four of five, the lower threshold is about 20 to 25 per cent-yielding an effective threshold, half-way between the upper and lower limits, of about 35 per cent- This rough but reasonable estimate is used for all of the majoritarian systems in Table 2.1-including the early Indian system with the slightly higher M of 1.21, and also including the Australian majority system where my assumption is that candidate with 35 per cent of the first preferences has a reasonable chance of being elected with the help of second preferences transferred from weaker candidates." However,

in order to emphasize the roughness of this estimate, it is given as a round number without decimals.29
The second comment concerns the effective thresholds in PR systems: in some cases, these can be given with a high degree of precision (particularly when there is a national legal threshold expressed in percentage terms), but when they have to be calculated from average district magnitudes or on the basis of other criteria (of which some examples will be discussed shortly), they are also rather rough estimates. Hence-in contrast with average district magnitudes, numbers of districts, and assembly sizes, which can all be determined very accurately-the values of the effective thresholds are given to only a single decimal place- The one exception is the Dutch electoral system since 1956 (at the bottom of Table 2.2) in which the national legal threshold, and therefore also the effective threshold, is exactly two-thirds oC 1 per cent.
Finally, it is worth re-emphasizing that all effective thresholds except national legal thresholds are not only rough estimates but also midpoints in a range between no representation and full representation. Hence, falling short of such an effective threshold does not necessarily entail getting no representation at all-as it does when the threshold is a national legal barrier-but being substantially underrepresented.30
In Table 2.2, the effective threshold for each system is the larger of the value computed from the average magnitude and the legal threshold, if any. The two district-level thresholds are applied in districts with such a low average magnitude that the national effective threshold is actually higher. In the 1986 French case, the 5 per cent district threshold was meaningless in the 93, out of the total of 96, districts with magnitudes of about 14 or fewer seats- In the Spanish parliamentary election system, the 3 per cent district threshold becomes effective only for district magnitudes above about 24 seats: of the 52 districts, only Barcelona and Madrid have greater magnitudes. Similarly, the national effective threshold has "een much higher than the district-level legal threshold in the two French majority-plurality systems (see Table 2.1 above). The legal threshold here is the minimum vote in the first round that entitles a candidate to compete in the second round.3'
^e six national legal thresholds in Table 2.2 are all higher than tne Active thresholds implied by the district magnitudes, although '" the Dutch and Israeli cases, the legal threshold does not raise

the barrier a great deal. The two Israeli systems near the bottom of Table 2.2 provide a good example: the 1 per cent legal threshold adopted after the 1949 election did raise the effective threshold, but only from 0.6 to 1 per cent.12 The French and German Euro-election systems are examples of a much stronger boost from an implied threshold of only around 1 per cent to a legal threshold of 5 per cent.
PR: SINGLE-TIER DISTRICTING AND NON-D'HONDT FORMULAS
The other single-tier districting systems-those that do not use the d'Hondt formula-are presented in Table 2.3- The fact that there are only eleven systems in this table, compared with twenty-one in Table 2.2, is a good indication of the popularity of the least proportional d'Hondt method. And fewer than half of the non-d'Hondt systems use the most proportional LR-Hare formula. (For its Euro-elections, Greece has used a procedure not quite identical with, but closely akin to, LR-Hare.)33 On th<; other hand, the district magnitudes of these non-d'Hondt systems, while displaying almost the same range as those using d'Hondt, are by and large appreciably lower. The lowest are in the four STV systems;
one important reason is that, in these preferential systems, high magnitudes are impractical because these entail large numbers of candidates-which impose heavy burdens on the voters who have to rank order these candidates- We find the highest magnitudes, as before, in the systems with at-large elections.
The systems are listed in increasing order of district magnitude. Because most of them do not have legal thresholds, all but one of the effective thresholds shown in the table are in decreasing order, The one exception is the system for the 1989 German Euro-election;
Germany switched from the d'Hondt to the LR-Hare formula but maintained the relatively high 5 per cent national threshold.
PR; TWO-TIER DISTRICTING SYSTEMS
The remaining twenty-one PR systems are somewhat more complicated, mainly because they use two tiers of districts but also

because many of them have legal thresholds that are not easy to translate into effective thresholds. The basic rationale for two-tier districting is to combine the advantage of reasonably close voter-representative contact offered by smaller districts with the advantage of greater proportionality and minority representation offered by larger districts.34
Two types of two-tier methods can be distinguished: remainder-transfer and adjustment-seats systems. The first is used by the seven electoral systems in Table 2.4. In the lower-tier districts, one of the LR formulas is applied, but instead of allocating the remaining seats to the parties with the highest remainders of votes in these districts, all remaining vbtes and seats are transferred to, and allocated in. higher-tier districts. The fourteen systems shown in Table 2.5 belong to the second type: here the districts at the lower level are used for the initial allocation of seats, but the final allocation takes place at the higher level on the basis of all of the votes cast in all of the lower-tier districts that together make up the higher-tier district. Most commonly, a certain number of adjustment seats are provided at the higher level in order to even out the disproportionalities that may have occurred at the tower level. (The numbers of these adjustment seats can be calculated easily by subtracting the total of the lower-tier seats-the number of districts times the average magnitude at the lower level-from the total number of seats, i.e., the assembly size.)35
The tables report the basic characteristics for both tiers, with the more important higher level listed first. In fact, with regard to the proportionality of the election outcome and the opportunities for small parlies, the upper level is the decisive level. The one exception concerns the electoral formula in remainder-transfer systems. Here the formula at the lower level predominates: no higher-tier formula is able to favour systematically the larger over the smaller parties, since the parties with the highest totals of remaining votes are not necessarily the largest parties. What is of crucial importance for the proportionality of the outcome is how many seats will be available at the higher level-which is determined by the lower-tier formula. Only LR-Hare at the lower level produces a sufficient number of remaining seats for full proportionality. The seven remainder-transfer systems exhibit the entire range of LR. formulas: in decreasing order of proportionality, LI^ Hare, LR-Droop, partly LR-Droop and partly normal LR-Impen^

(in the smaller and larger districts respectively in the first postwar Italian election), normal LR-Imperiali. and reinforced LR-Imperiali-with only two countries, Italy and Austria, providing instances of all of these formulas-36
In the adjustment-seats systems, the higher-tier formulas are decisive. Like the decisive formulas in the remainder-transfer systems, they range from the least proportional to the most proportional methods. Most are divisor methods (d'Hondt in Germany and Iceland, and modified Sainle-Lague in Sweden and Norway) but LR-Hare has also been used fairly frequently (in Germany since 1987 and in all Danish parliamentary elections). Malta introduced a contingent higher tier before the 1987 election: if the party winning a majority of the first preference votes does not win a majority of the lower-tier seats, it receives a sufficient number of upper-level adjustment seats to ensure it a parliamentary majority. This provision became operative in the 1987 election when the Nationalist Party had to be awarded four adjustment seats to turn its narrow national vote majority into a majority of parliamentary seats. This method does not fit any of the standard PR formulas, but it comes closer to LR-Hare than to any of the other methods.37
In keeping with the basic rationale of two-tier districting, the district magnitudes at the lower l^vel are fairly small, usually less than 10 seats; Italy and, since 1971, Austria are the major exceptions. Germany has taken the idea of small lower-tier districts, providing close voter-representative contact, to its logical extreme by adopting single-member districts at the lower level. The other side of the coin, however, is that this requires a relatively large number of upper-tier seats (or the purpose of proportional adjustment. In all of the two-tier systems (assuming that, in the adjustment-seats systems, there are enough adjustment seats), the effects of small magnitude at the lower level are overridden at the higher level. At the upper level, the district magnitudes are all sizeable, ranging from a minimum of well over 20 seats to the huge national district of more than 600 seats in recent Italian elections. In about two-thirds of the two-tier systems, the upper-tier district is a national at-large district.
Without legal thresholds, such large upper-tier districts offer from very good to near-perfect proportionality and excellent op-porlunilics for even very small parlies. Four of these systems have indeed operated without legal thresholds. In the case of the l94o

Italian upper-tier district of 556 seats, this yields the lowest effective threshold-only 0.1 per cent-of any of our electoral systems. It is therefore not surprising that most of the two-tier systems do have legal thresholds. These tend to be more complex than the thresholds in single-tier systems. In order to translate them into effective thresholds, two further assumptions need to be made. One is that party support is distributed evenly across a country instead of being regionally concentrated. The other concerns the frequent use of multiple criteria for barring small parties from participating in the allocation of seats at the higher tier. When these are alternative criteria (for instance, in recent German elections, winning either 5 per cent of the national vote or three seats in the lower-tier single-member districts), the criterion that is the easiest to satisfy becomes the basis for determining the effective threshold. When they are joint criteria (for instance, in recent Italian elections, winning both 300,000 votes nationally and at least pne seat at the lower tier), the effective threshold must be based on the stricter requirement.
On the basis of these assumptions, about half of the two-tier systems can be assigned effective thresholds fairly easily. For the four systems without any legal thresholds, the effective thresholds can be calculated simply from the upper-tier district magnitude (Italy in 1946, the Italian Euro-election system, Greece in 1989-90, and Malta in 1987). For the six systems with legal thresholds expressed in terms of a minimum percentage of the national vote, this percentage automatically becomes the effective threshold (the three German systems from 1953 on, Denmark since 1964, Sweden since 1970, and Norway in 1989). For Denmark from 1953 to 1960, the national threshold of 60,000 votes represented an average of approximately 2.6 per cent of the total valid vote in these three elections and hence a 2.6 per cent effective threshold. And for the first German electoral system in 1949, the 5 per cent threshold applied at the regional (Land) level translates, on the assumption of an even spread of party support, into a national effective threshold of 5 per cent. (This example shows that the assumption 01 even distribution of party support is based on an average ^'tuation. Uneven support can obviously help a small party: with per cent support in one half and 3 per cent support in the other 11 of a country, a party would not meet a national 5 per cent reshold but would meet the regional 5 per cent barrier in half of
the country. But it could also hurt: if the percentages were 7 per cent and 4 per cent respectively, the full national minimum would be met, but the regional minimum would be met in only half of
the country.)
The remaining eight systems have legal thresholds formulated in more complex special rules (marked 'SR' in Tables 2.4 and 2.5). Four patterns can be distinguished:
1. The legal threshold for receiving seats in the national higher-level district is that a party has already won at least one seat at the lower level. In the two Italian systems since 1948, a small party has been able to do so by receiving at least the respective Imperial! quota of the votes in the largest lower-level district, namely Rome;
this required an average of 2.6 per cent of the vote in 1948 and 1953, hut only an average of 2.0 percent later when the quota was changed from the reinforced Imperial! to the slightly higher normal Imperial! quota, but the magnitude of the Rome district was increased considerably. In the two Icelandic systems, the requirement of winning at least one lower-level seat could be achieved most easily by winning a seat in the Reykjavik district with respectively 8 and an average of 12.22 seats. The effective thresholds for
these magnitudes are 8.7 and 5.8 per cent.
2. The legal threshold for receiving seats in the national district is to have won a certain minimum number of voles in one or more specified areas. The one example here is Denmark from 1945 to 1953: parties needed to win a Hare quota of the total national vote in one of the country's three regions. This could be achieved most easily in Jutland, where about 42 per cent of all votes were cast, and where about 1.6 per cent of the regional vote equalled the
national Hare quota.
3. The legal threshold for receiving seats in higher-level regional districts-not a national district, in contrast with the first pattern -is winning at least one seat in one of the region's lower-level districts. The two Austrian electoral systems, with initially four. and later two upper-tier districts, belong to this type- In order to convert this rule into a national effective threshold, another average assumption has to be made: between the situation where a party barely fails lo win any lower-tier seats and is hence completely excluded from representation, and the situation where the party just manages to win such scats in all of the higher-tier districts and therefore fully participates in the proportional allocation of seats.

iippp ^^ point is meeting this requirement in half of the ^ats a &r lrlcts (l-eoo from which about half of the total assembly Iti the ^ d) that have the largest-magnitude lower districts. 3 TV rst Austrian system, this required enough strength to win ^istrir-t ^l11013 ln a ten-member and later in an eleven-member hole) Hp^16 ''"S an effective threshold of 8.5 per cent; this thres-Systp Fe^kl sharply to about 2.6 per cent in the second Austrian 36 IQ 4'"�^^ Hare quota in a much larger district varying from
4 p. seals in six elections was sufficient. the in,^*^'1^ Belgian system resembles the Austrian except for t^uota, 6r "^imum required at the lower tier-0-66 of a Hare bgr "(- i^^ot a full Hare or Droop quota-and the larger num-ItigH, c oc ^wer-tier and upper-tier districts. For the rest, the ts ^P c '^"^erting the Belgian rules into an effective threshold ^ thirt ss in the Austrian case. Meeting this requirement in Perni.ic """^tnber or, more usually, a fourteen-member district able s P^Y to share in the allocation of about half of the avail-^e?-^ seats.39
Cretin .'^Tables 2.2 and 2.3 were organized in terms of in-eff^.. s ^"^ict magnitude. This corresponds with decreasing 4ggi., ^^sholds except where these are overridden by higher ^lon o es 'ds, Because the various legal thresholds are so com-Qver..--) ^^ier systems and because they clearly and strongly ^lec. ^^ffects of the high-magnitude upper-tier districts, the ^rdp,. ^y^ems in Tables 2.4 and 2.5 are listed in decreasing ^ai ri, e '^ive threshold-which is the most important feature 'm-g^shes these systems from each other.
INT
^^EDIATE SYSTEMS: SEMI-PR, REINFORCED PR, AND MIXED PR-MAJORITY
^ ^ . . ^ar^ 'a "^g six electoral systems do not fit either the major-
^'Ufo .r ^K categories: semi-PR in the two Japanese systems, ^i>^ ect ^ in three Greek systems from 1974 to 1985, and a ^6 V} ^ an(! majority in the French system in 1951 and Suffi0^^!", I shall argue that five of the six can be regarded
^^Par^"1^ similar t0 PR that t^y can be included in the litLvt analyses of all PR systems.

40 Electoral Systems
The Japanese limited vote (LV) and single non-transferable vote (SNTV) systems are usually referred to as semi-PR systems, and SNTV is usually regarded as a special case of LV. Voters cast their votes for individual candidates and, as in plurality systems, the candidates with the most votes win. However, unlike in plurality systems, the voters do not have as many votes as there are seats in the district (and districts have to have at least two seats):
this is the reason why the formula is called the 'limited' vote. The more limited the number of votes each voter has, and the larger the number of seats at stake, the more LV tends to deviate from plurality and the more it resembles PR. In the 1946 LV election in Japan, each voter had only two voles in districts with 4 to 10 seats, and only three in districts with 11 to 14 seats- SNTV is the special case of LV where the number of votes cast by each voter is reduced to one. In Japan from 1947 on, SNTV has been applied in districts with an average of almost four seats. Table 2.6 presents the vital statistics of the two Japanese electoral systems.
LV and SNTV offer good opportunities for minority representation. The SNTV threshold of exclusion (the upper threshold, above which a candidate is guaranteed a seat) is the Droop quota:
20.2 per cent in the average Japanese electoral district in all elections from 1947 on. The LV upper threshold in the 1946 election was a similar 20.5 per cent.40 LV and SNTV have the unusual property of having an extremely low threshold of representation (the threshold above which it becomes possible to win a seat): the most extreme example in, say, a three-member district would be one candidate receiving all but two of the votes, and hence obviously being elected, and two other candidates receiving one vote each-and also winning seats' For this reason, Japan has imposed a legal threshold equalling one-fourth of a Hare quota at the district level. These are still relatively low thresholds-and much lower than the effective thresholds calculated on the basis of the average district magnitudes.
In many respects, including the average district magnitude, Japanese SNTV resembles Irish STV. The principal difference, of course, is that SNTV appears to be less proportional because no votes can be transferred. However, this disproportionality does not stem from the usual cause of discrimination against the smaller parties. In fact, the non-transfer of votes among candidates tends to present a considerable problem for the larger parties: a large

party has to make sure not to nominate too many candidates (which may cause these candidates to lose in spite of a high total vote for the party's candidates) and to have its voters cast their votes as evenly as possible for its candidates. In contrast, a small party only needs to nominate one candidate in order to maximize its chances of winning a seat. And, in LV systems, a small party only needs to nominate as many candidates as the number of votes that each voter has. Therefore, as far as their political effects are concerned, SNTV and LV can be regarded more legitimately as unusual forms of PR and not highly proportional forms of PR-but more as a result of their relatively small magnitudes and high effective thresholds than because of their electoral formulas-than as non-PR systems. Unless specifically stated otherwise. I shall include them in all future comparisons of PR systems, and I shall group them together with the Irish and Maltese STV systems.41
As noted in Chapter 1, PR systems are all too readily characterized as highly complex. But this description does fit Greek reinforced PR from 1974 to 1985.42 These three systems are also quite idiosyncratic, but they can still be made comparable to the mainstream PR systems. Let me use the first Greek system, used in the 1974 election, as the basic example. For clarity's sake, I shall focus on the principal rules and omit the many minor details
and special provisions.
Superficially, the system looks like a four-tier remainder-transfer system: seats not allocated at lower tiers by the Hare quota are transferred to higher tiers (with the exception of the fourth tier, consisting of the 12 so-called 'State seats', which are awarded separately on the basis of the parties' national vote totals). The big difference with remainder-transfer systems is that the remaining seals are transferred, but not the remaining votes. At the middle and high tiers, the remaining seats are allocated on the basis of the parties' vote totals instead of their remaining votes. This means that, in a typical lower-tier district with five seats and four sizeable parties (a reasonable assumption for the Greek situation), the average remainder would be half a Hare quota, and the total remaining votes would add up to two Hare quotas: only three seats would be allocated, and all of the remaining votes would be lost. The crucial point to understand here is that this system effectively operates like d'Hondt (which, as explained earlier in this chapter and in Appendix A, also disregards all remaining

votes) in a district that is considerably smaller than its formal district magnitude.
At the middle tier, this process is repeated: the seats transferred to this level are allocated on the basis of the parties' votes and Hare quotas in nine districts. And, at the third tier, the transferred seats are again allocated on the same basis, but now all still remaining seats are given to the largest party-a formula much closer to d'Hondt than LR-Hare. At these two levels, an additional disadvantage for smalt parties is the 17 per cent national threshold.43 At each tier (including, as indicated above, the highest tier of State seats), the results are calculated on the basis of the parties' vote totals. This means-the second crucial point that must be emphasized-that the parliamentary election takes the form of four separate and parallel elections of four mini-assemblies.
The 1974 Greek system is presented in these terms in Table 2.7. The lower districts have a formal average magnitude of 5.14 seats (the total assembly size of 300 seats less the 12 State seats, divided by the 56 districts), but 2 seats are assumed to go to the second level-which means that the estimated true district magnitude is only 3.14 and that, while the quota that is applied is the Hare quota, the true formula is not LR-Hare but d'Hondt. At the next level, there are now assumed to be 112 seats in 9 districts-an average formal magnitude of 12.44 but an estimated true magnitude of only 10.44. And, at the third level, the still remaining estimated number of 18 seats are allocated.'" The effective threshold at each level is based on the district magnitude or the legal threshold of 17 per cent, whichever is higher. The overall characteristics of the system are the dominant formula (d'Hondt) and the weighted averages (weighted according to the number of seats allocated at each level) of the effective thresholds and the 'assembly sizes' of the four parallel mini-assemblies.
The description of the second and fourth Greek systems in Table 2.7 follows the same logic. The only important change in the second system was the substitution of the Droop for the Hare quota at the lower tier. Assuming the same typical lower-tier district with "ve seats and four sizeable parties, the average remainder is now half a Droop quota and the total remaining votes, adding up to two Droop quotas, are still lost, but, because of the lower quota, four instead of three seats can be allocated. This was a major change ecause it made the system considerably less disproporlional-by

increasing the lower-tier magnitude by an estimated one seat (and hence decreasing the effective threshold at this tier as well as the weighted mean for the whole system) and by increasing the weighted assembly size by more than a third. The main change in the fourth Greek system was the abolition of the 17 per cent legal threshold-again a substantial shift away from disproportionality because it lowered the effective thresholds at the three higher levels and, as a result, also the weighted average.45 Recast in terms of these measurements, the three Greek systems can be compared with the other PR systems. In spite of the deceptive label of 'reinforced' PR, these systems are not highly proportional-as a result of the use of d'Hondt, low district magnitudes, and high effective thresholds-but still, like the Japanese systems, permitting an appreciable degree of proportionality and minority representation.
Finally, the French electoral system used in the 1951 and 1956 elections may also be called a reinforced PR system-reinforced not to help the largest parties, as in Greece, but the medium-sized parties in the political centre. Unlike the Greek systems, unfortunately, it cannot be made amenable to comparative analysis together with the other PR systems.
It was engineered by the centre parties in order to maximize their own representation and to discriminate against the big parties on the left and right, the Communists and Gaullists. One of the devices they used for this purpose was apparentement: the possibility of linking two or more party lists, and of thereby gaining the advantage that majoritarian and most PR systems give to large parties, but without having to present joint lists. And while appareniemenss could in principle be negotiated between any two or more parties, they constituted a much more feasible option for the centre parties than for the extreme right and left, The second device was the majority principle: if one party would win an absolute vote majority in a multi-member district, that party would win all seats; failing a one-party victory (an unlikely outcome in a "mill-party system), all seats would be given to the apparenfemeni ^th a majority of the votes. If neither type of majority materialized, the system would revert to PR with the d'Hondt formula, but their "Pparentemenss would still give the centre parties the same advan-^e that d'Hondt gives to the larger parties. This was the system ^erywhere except in the eight electoral districts in the Paris

region, where the centre parties were too weak to be able to profit from the majority rule. Hence the very opposite system was engineered: no apparentements, no majority rule, and LR-Hare in relatively large districts.
Table 2.8 provides the basic facts for the 1951-6 French system. For Paris, the system was a straightforward LR-Hare system. For the rest of the country, the results are broken down according to whether PR-d'Hondt or the majority rule operated in the districts. The figures are averages for the two elections. The majority rule came into force in 40 districts, with 173 seats, in 1951, but in only 11 districts, with 59 seats, in 1956.46 The average district magnitude of the majority-rule districts (4.84 seats) as well as the range of magnitudes of these districts (from 2 to 10 seats) appear to contradict my earlier statement concerning the rarity of the use of majoritarian formulas in larger than single-member districts. However, in only one case was there a majority party that won all the seats-in a two-member district in 1951; all other majority winners were majority apparentements of two or more parties which then divided up the seats won among themselves according to the respective strengths of their separate party lists. (For this reason, I have computed the effective threshold for the majority districts as if they were PR districts, instead of assigning them the arbitrary 35 per cent attributed to the other majoritarian systems.)
Although more than three-fourths of the seats were allocated by PR, the majority-rule component in these elections was still so strong and its application so interwoven with PR in the areas outside of Paris, that it is impossible to disentangle them. Moreover, the two PR formulas belong to opposite extremes. For these reasons, the mixed French system used in the 1951 and 1956 elections will have to be left out of most of the analyses in Chapters 4 and 5, for instance, when the effects of PR and majoritarian systems are compared and when PR systems are compared with each other.
GENERAL PATTERNS
By presenting the seventy electoral systems in terms of groups of systems with similar key characteristics (majoritarian versus

PR systems, d'Hondt versus other PR formulas, one-tier versus two-tier systems) and, within each group, according to other important features (district magnitude and effective threshold). I have already implicitly pointed at some of the general patterns in the electoral systems used by the twenty-seven stable democracies in the 1945-90 period (for their national first-chamber or only-chamber elections). In this section, 1 shall treat these general
patterns in an explicit and systematic manner.
The most striking general aspect of the electoral formulas is
their widely different frequency of application. Of the three major categories, PR has been used in about three-fourths of the systems:
of the seventy systems, fifty-two are unambiguously PR, and this number rises to fifty-five if the three reinforced pR systems of Greece are added. Majoritarian formulas have been used in twelve systems, and semi-PR only twice, in Japan. Within the broad major-itarian and PR categories, some formulas have never been used -even such well-known possibilities as the majority run-off47 the pure Sainte-Lague, the STV with a quota other than the Droop quota, and the cumulative vote48-while, among those that have been in use, two account for more than half of the cases; plurality has been far more prevalent than the other two majoritarian formulas together (in seven out of twelve systems and, even more strikingly, in five out of seven countries), and d'Hondt has been used more often than all of the other divisor, quota, and STV systems combined (in twenty-seven out of fifty-two PR systems and, if reinforced PR is added, in thirty out of mly-five PR systems).
The same general pattern of uneven usage also occurs with regard to district magnitudes. Majoritarian formulas can in principle be applied in districts ranging from single-member to at-large. In practice, single-member districts have been the rule, two-member districts have been rare, larger multi-member districts even more exceptional, and at-large elections have never been used. The theoretical range for PR systems is from two-member districts to at-large, and most of this range has actually been used, but the lowest magnitudes of between 2 and 5 seals have been rare. Of the fifty-two unambiguous PR systems, only two have used average magnitudes (the higher-tier magnitudes in the case of two-tier districts) of less than 5 seals, and only fifteen have used average magnitudes of less than 10 scats. This means that both majoritarian and PR systems have avoided district magnitudes thai seriously

limit proportionality and raise disproportionality. Two general conclusions emphasized by Rae are that all electoral systems tend to be disproportional, but that some (especially majoritarian ones) tend to be more disproportional than others (especially PR).49 A third, partly contradictory, conclusion could be added; as a result of their choice of district magnitudes, all electoral systems are reasonably proportional-or at least far less disproportional than they
could potentially be made to be. We do find many relatively small lower-tier districts in two-tier
PR systems, but their effects are overridden at the higher level. As stated earlier, the most important reason for instituting two-tier districting is to combine the advantage of closer voter-representative contact in smaller districts with the greater proportionality of larger districts. Comparing single-tier and two-tier systems, we would therefore generally expect lower-tier magnitudes to be lower and upper-tier magnitudes to be higher than the magnitudes of one-tier systems. This is indeed the case. The means for the twenty two-tier systems are 8.28 and 207.83 seats, compared with a mean of 35.70 seats in the thirty-two one-tier systems. The medians can express these differences more sensitively, they are 6.37 and 91-50 in the two-tier systems, and 12.20 in the single-tier
systems. Legal thresholds can take away the proportionalizing effect of
large district magnitude again and, not surprisingly, thresholds are most common in two-tier systems and in high-magnitude single-tier systems. These legal thresholds tend not to be excessively high, however; the 17 per cent thresholds in the Greek reinforced PR. systems are exceptional. The highest legal threshold among the fifty-two unambiguous PR systems is only 8.7 per cent. It is instructive to compare the effective thresholds of these systems:
the average effective threshold of the twenty-eight PR systems that do not have legal thresholds (that is, where the effective threshold is entirely based on the district magnitude) is 7.5 per tent; in the twenty-four systems with legal thresholds it is 3.8 per ^nt. The medians are 8.4 and 4.0 per cent respectively. This means lhat while legal thresholds do raise the effective thresholds, they d0 not raise them to the level, or even close to the level, of the
^stems without legal thresholds.
The fourth and final major dimension of the electoral system n ^hich this study focuses-assembly size-varies a great deal.

The parliaments (lower or only houses) range in total membership from 40 in Malta until the mid-1950s to an average of 632.85 in the United Kingdom during the entire 1945-90 period (650 in the 1987 election). The sizes of the national delegations to the European Parliament range from 6 to 81. These numbers are closely related to population sizes: large countries tend to have larger parliaments than smaller countries, and the larger members of the European Community have larger Euro-detegations than the smaller members-although the smaller countries are still considerably overrepresented. Taagepera has suggested and proved an even more specific and quite remarkable relationship: the cube root law of assembly sizes- This law holds that assembly size tends to be roughly the cube root of the population size.50 The delegations to the European Parliament are all considerably smaller than the national Parliaments, of course-closer to a fourth root than a cube root relationship.51
EMPIRICAL LINKS AMONG THE DIMENSIONS
In Chapters 4 and 5, I shall analyse the influence of the electoral system dimensions on proportionality and multipartism. I shall do so by means of multivariate comparisons in order to control for any empirical relationships among the independent variables themselves. At this point, however, let us take a direct look at the mutual relations of these independent variables: the electoral formula, the effective threshold (as a composite variable based on legal thresholds and district magnitudes), and assembly size. As in later chapters, I shal! include Japanese semi-PR and Greek reinforced PR among the PR systems (reinforced PR as a d'Hondt and semi-PR as a non-d'Hondt formula), but I shall also report the results for the fifty-two unambiguous PR systems without semi-PR and reinforced PR (by means of endnotes); the latter option never materially affects the results. The mixed system used by France in the 1951-6 period will be omitted.
The strongest relationship is between the two major categories of electoral formula (majoritarian versus PR systems) on the one hand and the effective threshold on the other. The twelve rnaJor-itarian systems all have an effective threshold of 35 per ce

compared with an average effective threshold of only 6.6 per cent for the fifty-seven PR systems; the respective medians are 35 and 5 per cent.52 As explained earlier, the 35 per cent threshold for the majoritarian systems is an arbitrarily assigned estimate, and a reasonable argument could be made that the estimate should be lower, perhaps as low as 30 per cent. However, even this lower percentage clearly does not change the stark contrast between the majoritarian and PR systems in this respect.
One might plausibly surmise that, within the broad category of PR systems, there would be a similar difference between the less proportional (d'Hondt and LR-Imperiali) and the more proportional formulas (all other formulas, including the combination of LR-Droop and LR-Imperiali used in Italy in 1946). This turns out not to be the case. The average effective threshold of 6.5 per cent in the d'Hondt and LR-Imperiali systems is actually lower than the 6.7 per cent threshold in the other PR systems, but the difference is slight and not statistically significant. The medians are an identical 5.0 per cent."
The majoritarian-PR dichotomy is also related, but much less strongly, to assembly size. Table 2.1, which lists the majoritarian systems, suggests such a relationship because it includes some of the largest countries with, consequently, the largest assemblies:
India, the United States, the United Kingdom, and France. The average assembly size of the majoritarian systems is indeed well above that of the PR systems: about 323 compared with about 202 members. The respective medians are even farther apart: about 352 versus 152 members-'"1 However, the correlation coefficient between the majoritarian-PR contrast as a dummy variable and assembly size is only 0.25 (statistically significant at the 5 per cent level in a two-tailed test, but only barely). Because of the wide range of assembly sizes with a concentration of cases at the lower end of the range, it is more appropriate to use the logged than the raw assembly sizes. This reduces the correlation coefficient slightly to 0.23 (which is no longer statistically significant). Nevertheless, 'he important substantive conclusion is that the tendency to dis-Proportionality of systems with majoritarian formulas is to a small extent compensated by their larger assembly sizes.
^gain, there is no corresponding difference between the more and [ess proportional PR formulas. The mean assembly size of the
"^dt and LR-Imperiali systems is about 211, and of the other

PR systems about 191 members. The medians are an almost identical 150 and 152 respectively."
To turn to the third leg of the triad, we would expect a positive relationship between the effective threshold and assembly size on the basis of our earlier findings of positive relationships between the majoriiarian-PR difference and both effective threshold and assembly size. For all 69 cases, the correlation coefficient is indeed a positive 0-22. and the correlation between logged assembly size and effective threshold is a similarly positive 0.19, but neither correlation is statistically significant. Among the fifty-seven PR systems, the two dimensions are almost completely unrelated.
The only strong relationship that we have discovered among our three electoral system dimensions, therefore, is the link between electoral formula and effective threshold, and this relationship is strong only if the formula is denned in terms of the majoritarian-PR dichotomy. One the other hand, the relationship is so strong (the correlation coefficient is a highly significant 0.92) that this finding has major consequences for the multivariate analysis in Chapter 5: in order to avoid the problem of multi-collinearity, the two variables cannot be entered together as independent variables in any multivariate regression equations.
TRENDS
One of the best-known generalizations about electoral systems is that they tend to be very stable and to resist change. In particular, as Dieter Nohlen has emphasized, 'fundamental changes are rare and arise only in extraordinary historical situations'.56 The most fundamental change that Nohlen has in mind is the shift from plurality to PR or vice versa. Indeed, in our universe of twenty-seven countries from 1945 to 1990, this kind of change has not Just been rare but completely absent. And only one country-France-has experienced changes back and forth between a majontarian system and PR.
As far as less fundamental changes are concerned, our twenty-seven countries do show considerable variability by producinB seventy different electoral systems-an average of more than two and a half electoral systems per country. And, while these may "�

be what Nohlen calls 'fundamental' changes, they are not minor changes either: they entail clear changes in electoral formula and/ or changes of at least 20 per cent on the other dimensions. But the countries differ considerably with regard to their predilection for change: the number of electoral systems per country ranges from one to six.
Three broad categories, based on the presentation of the electoral system characteristics in Tables 2.1 to 2.8, can be distinguished. The first consists of countries that had only one electoral system during the entire 1945-90 period: three plurality countries (Canada, New Zealand, and the United States) and two PR countries (Finland and Switzerland). To these should be added the six countries in which the only change was the adoption of a new system for the European Parliament elections: Belgium, Ireland, Luxembourg, Portugal, Spain, and the United Kingdom. Because the sizes of the Euro-delegations was set at a level far below the sizes of the national parliaments, this change necessarily produced a new electoral system according to my criteria. It should be noted, however, that only Ireland and the United Kingdom adopted Euro-election systems that are true miniatures of their parliamentary election systems; the other four also adjusted their effective thresholds.
The second broad group consists of countries that changed but did not completely overhaul their electoral systems: two with major-itarian systems (Australia and India), one with semi-PR (Japan), and five PR countries (Austria, Costa Rica, Iceland, Italy, and the Netherlands). Each of these countries could be easily accommodated within the same table earlier in this chapter. Two countries fit this description except for their Euro-elections: Denmark and Germany. They can therefore also be placed in this middle category (although Germany had no less than six different electoral systems).
Six countries that experienced the most radical changes, and whose parliamentary election systems had to be included in more than one table, make up the final group. France and Greece are ihe clear leaders with major shifts from PR to majority-plurality systems and vice versa (France) and from reinforced PR to a highly Pfoportional form of PR (Greece); moreover, Greece used five Afferent systems from 1974 to 1990-which is, in relative terms, a much larger number than the six used by France (and by Ger-y) 1" a time-span of more than forty years. The others arc

Israel, Malta, Norway, and Sweden. The Norwegian and Swedish cases are especially important because their reforms represent broader trends; the establishment of two-tier districting systems (also adopted by Malta on a contingency basis), the abandonment of the d'Hondt in favour of a more proportional formula (as in Germany and Greece), and the adoption of a 4 per cent national threshold. As far as the last reform is concerned, Germany was the only country in the 1950s that had a 5 per cent national threshold for its parliamentary elections; since then, national thresholds of 4 or 5 per cent have been adopted not only by Norway and Sweden but also, for their Euro-elections, by France, Germany, and the Netherlands.
The above changes point to a trend of greater proportionality in electoral systems. Let us examine these trends systematically in terms of the three basic dimensions. The clearest patterns appear with regard to assembly size. The only significant (that is, 20 per cent or greater) changes in the total memberships of the national parliaments are increases: in Australia (twice), Germany, Malta, the Netherlands, and Sweden.57 On the other hand, all of the new systems adopted for the Euro-elections entailed substantial decreases.
The patterns of changes in electoral formula and significant (20 per cent or greater) changes in effective threshold are more complex. They are shown in Table 2.9, subdivided according to (1) changes in parliamentary election systems versus changes to or in Euro-election systems and (2) changes to more versus less proportional features. Several instances of change appear twice; for instance, the shift from the first to the second Austrian electoral system entailed the adoption of both a more proportional formula and a lower threshold. For the sake of simplicity, however, the major system shifts in France are listed only as changes in formula. Overall, the trend has been to greater proportionality; adoptions of more proportional formulas and thresholds for parliamentary elections have been twice as numerous as the adoption of less proportional rules, and almost the same statement can be made for the adoption of Euro-election rules. No plurality countries are included in Table 2-9. but it is worth recalling that plurality systems have also tended to greater proportionality-or, more accurately, less disproportionality-as a result of the universal abolition of multi-member districts-

Another way of summarizing the major trends in systems for
parliamentary elections is to compare the system used for the first with that for the last election in each country in the 1945-90 period. Of the sixteen countries that underwent changes, only nve ended up with a less proportional system than they started out with: France, Denmark, Israel, Italy, and Japan. Moreover, with the exception of France and Japan, the shifts away from proportionality were relatively small, mainly involving minor increases in effective thresholds. Eleven countries ended up with more proportional systems: Australia, Austria, Costa Rica, Germany, Greece, Iceland, India, Malta, the Netherlands, Norway, and Sweden.
It is by no means certain, of course, that these trends will persist. For the 1992 elections, Israel increased its national threshold from 1 to 1.5 per cent, and Malta did not have to use its contingent upper tier. Following its highly proportional 1990 election, Greece has already changed its electoral law back to reinforced PR. And in 1992 Austria adopted a new electoral law with a national threshold of 4 per cent-higher than its previous effective threshold of 2.6 per cent (see Table 2.4). However, regardless of the strength of the trend towards greater proportionality, it is clear that many countries are making, if not fundamental reforms, at least major adjustments in their electoral laws. It is important to describe these not just in terms of their basic tendencies towards proportionality or disproportionality-as I have done throughout this chapter -but to measure the influence of the three electoral system dimensions on the proportionality of election outcomes and on multipartism as precisely as possible. This will be the task of Chapters 4 and 5, after the various indices of proportionality and mullipartism are presented in Chapter 3.

3
Disproportionality, Multipartism, and Majority Victories
THE two main political consequences of electoral systems on which this study focuses are (1) their effects on the proportionality or disproportionality of the electoral outcomes and (2) their effects on the party system, particularly the degree of multipartism and the tendency to generate majority victories. Four measures will be proposed and applied for each of these effects.
Disproportionality means the deviation of parties' seat shares from their vote shares, and it appears prima facie to be a simple and straightforward concept, while multipartism and other party system characteristics appear to be considerably more complex and multifaceted. Rather surprisingly, however, the question of how best to measure disproportionatity has been much more difficult and controversial than the question of how to measure the key party system characteristics. As a consequence, the four indices of disproportionality that I shall present in this chapter are alternative ways of trying to measure the same phenomenon. I shall argue that one of them-the least-squares measure-is preferable to the others, and I shall rely on it as my principal measure of disproportionalily in Chapters 4 and 5. But 1 shall also occasionally report the results for the other indices, and the values �f all four are listed in Appendix B-allowing readers who prefer one of the alternatives to do their own reanalysis of the data with their favourite index. In contrast, the four measures of party system characteristics are measures of different, albeit not unrelated, aspects of the party systems: the effective number of elective parlies, the effective number of parliamentary parties, the tendency of the electoral system to manufacture a parliamentary majority for par-^s that have not received majority support from the voters, and e tendency to generate a parliamentary majority party regard-
ess of Nether the party's majority of the seats was manufactured or earned.

152 Electoral Engineering
chosen which will hopefully guide the new democracy's elections for a long time (or, in the case of a redemocratizing country, a new system that will hopefully work better than the old one), it is important to examine alt of the options as well as their advantages and disadvantages. This examination should include a careful look at the great variety of electoral rules and experiences of the world's stable and consolidated democracies. Therefore, to the extent that this study of seventy electoral systems in twenty-seven democracies will have some practical utility, it may have more to offer to electoral engineers in the new democracies than in these twenty-seven old democracies.

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