The Logic of Simple Electoral Systems

Predicting Party Sizes

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The Logic of Simple Electoral Systems

Rein Taagepera

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Preface

This book uses electoral systems to predict a significant aspect of partysystems—the number and size distribution of parties, and some of theconsequences. It builds on Seats and Votes (Taagepera and Shugart 1989),but it is a very different book. This is so because of the marked advances inour understanding of the connection between institutional inputs, suchas electoral systems, and political outputs, such as the number of partiesand the duration of governmental cabinets.

During the last 15 years, many important works have been publishedthat reflect advances in the study of electoral systems. Farrell (2001)has a most thorough description of the multiplicity of electoral systemsused, including their history. The handbook edited by Colomer (2004a)focuses on the origins of electoral systems and regularities in the choiceamong them, presenting numerous case studies. Gallagher and Mitchell(2006) describe electoral systems in 22 countries. Lijphart (1994) studiessystematically the effect of institutional inputs on the number of partiesand proportionality between seats and votes. Reynolds, Reilly, and Ellis(2005) offer practical advice on the strong and weak aspects of the varioussystems, and Diamond and Plattner (2006) evaluate the outcomes forconsolidation of democracy.

Katz (1997), Lijphart (1999), Powell (2000), and Reynolds (2002) ana-lyze the implications of elections as a core part of democracy, eachfrom a very different angle. Cox (1997), Blais (2000), and Norris (2004)connect institutions to political behavior, stressing the various forms ofstrategic coordination. Shugart and Carey (1992) and Jones (1995) studythe impact of electoral systems in presidential regimes. Monroe (2007a,2007b) reconsiders application of social choice theory to electoral systems,introduces new measures of bias and responsiveness, and applies theresults to institutional engineering in democracies.

Specific electoral rules and/or geographic areas have been addressed inmore detail by Grofman et al. (1999) on single non-transferable vote inEast Asia, Bowler and Grofman (2000) on single transferable vote, Shugart

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Preface

and Wattenberg (2001) on mixed-member proportional systems, Grofmanand Lijphart (2002) on proportional representation in Nordic countries,Elklit (1997) on emerging democracies, and Reynolds (1999) on SouthernAfrica. Darcy, Welch, and Clark (1994) and Henig and Henig (2001) haveinvestigated the effect of electoral systems on women’s representation,and Rule and Zimmerman (1994) cover women and minorities.

The ability to carry out analyses of worldwide scope has dependedon the availability of electoral data. Among data collections, Mackieand Rose (1974, 1982, 1991, 1997) has been the major workhorse forlong-established democracies. Nohlen, Krennerich, and Thibaut (1999),Nohlen, Gotz, and Hartmann (2001), and Nohlen (2005) have completedthe gap for Africa, Asia-Pacific and the Americas, respectively. A compara-ble collection for East Central Europe (Nohlen and Kasapovic 1996) seemsto be available, as yet, only in German.

How does the present book complement this extensive work? It focuseson the very simplest electoral systems. In that narrow slice, it specifies theaverage relationships between institutional inputs and the party politicaloutputs to an unprecedented degree. The results may be of interest to thepractitioners of politics, for the following reason.

Political scientists knew a long time ago that certain electoral systemstend to restrict the number of parties and lengthen cabinet duration, butthey did not quite know by how much, and why by precisely that much.Suppose a practicing politician felt that cabinets in her country did notlast long enough so as to implement long-term policy. We could tell herthat reducing the number of parties could help prolong cabinet duration,and this reduction in the number of parties, in turn, could be obtained byreducing the ‘district magnitude’. The latter means the number of seatsallocated within the same district. The smaller it is, the tougher it usuallymakes for small parties to obtain representation.

Yet, we would have been at a loss, 20 years ago, if the practitionercontinued to ask: ‘To what level should the district magnitude be reduced,if we want to double the average lifetime of cabinets?’ Back then, wecould just recommend that the politicians try some change in the givendirection, and see if it is too little or too much. If they overdid the changein district magnitude, we would not know it until several decades later,because individual cabinets vary in duration. Only the mean durationover a long time span is strongly determined by institutional constraints.

In contrast, we can now make explicit predictions, in the case of stabledemocracies, when the electoral rules are rather simple. New Zealandused to have a mean cabinet duration of 6.3 years, which implies that

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the same party often remained in office after new elections. In 1996,New Zealand changed electoral rules and ushered in an era of short-livedcoalition cabinets. It will be seen that a logical model predicts a new meancabinet duration of 2.0 years, within ±30 percent. This means somewherebetween 1.4 and 2.6 years. In 1996–2002, the actual mean duration was1.4 years. If the model holds, this mean is likely to increase slightly in thefuture. We shall see.

It can be seen that I am willing to go on record with a specific pre-diction. This book presents models which such predictions are based on.These models tie cabinet duration and the number and size of parties toinstitutional inputs. The book also presents actual evidence. Predictionbecomes hazier for new democracies where the political culture is still ina flux. Even for mature democracies, prediction becomes unmanageablewhen electoral rules are rather complex—as they often are. Yet, comparedto 1989, we have shifted from no ability to predict in any quantitativedetail to some such ability. This advance gives us hope that, within thenext 20 years, further strides will be made. The present book will helpdo so, by pointing out some promising directions, and by presenting aunified picture of the methods that have brought us that far.

My method owes more to my Ph.D. (see Taagepera and Williams 1966)and other work in physics (Taagepera and Nurmia 1961; Taagepera, Storey,and McNeill 1961) than to what currently prevails in comparative polit-ical science. In physics, one tries to break up a problem into smallerpieces. It often results in a sequence of equations of varied formats, asdictated by the nature of the issue on hand, each equation involving onlya few variables and even fewer constants. Interaction of variables followsthe basic format D ← C ← B ← A. There are no alternatives. The sameconstant values often recur in many equations and have names—they arehere to stay.

In contrast, today’s political scientists often throw a large number ofinput variables into a simple regression equation that is either linear orfollows a limited number of other habitual patterns. The same researchermay present several alternate regressions, with some variables included orexcluded. These are parallel expressions for the same output, rather thansequential: D ← A or D ← (A, B) or D ← (B, C) or D ← (A, B, C). Theseare alternatives. The number of constants and coefficients exceeds thenumber of variables, and their numerical values are rarely used again,once published—they are dead on arrival into print. Many distinguishedexceptions occur, but regression equations are presently the main patternin comparative political science.

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Preface

The physicist goes first after the most general, and only gradually fleshesit out with more realistic details. In contrast, today’s political scientistsoften want to include all possible factors at once. Does a colleague suggestanother variable? The answer may well be: ‘OK, I’ll put it in the regres-sion.’ I observed precisely this response at a conference, January 14, 2005,where the speaker entered all his variables linearly. If it were a physicsconference, the reaction might rather be: ‘OK, I’ll try to fit it logically intothe model when I get to work on a second approximation.’

I definitely prefer the sequential approach, using only a few variablesat each stage, connected by an equation the mathematical format ofwhich is based on logical expectations guided by empirical evidence. Thismethod is described in more detail in Beyond Regression (Taagepera 2008).Most often the resulting format is nonlinear. Many of my colleagues inpolitical science understand this method and make use of it (see Coleman2007; Colomer 2007; Grofman 2007). Hopefully, the results presentedhere will enable others to appreciate the power this method adds to themore usual approaches in today’s political science.

Still, some colleagues may not be convinced. After raising a slew ofdetailed issues, an anonymous reviewer for Taagepera and Allik (2006)candidly noted: ‘Perhaps I have continued problems with this paperbecause I am skeptical that there is much of value operating at such a highlevel of generality . . . huge amounts of real-world variation are consignedto nowhere.’ Actually, we consigned these variations to a much betterplace than nowhere, namely to the next-level analysis. While ferretingout the universal, science does not ignore detail, but it does introducesome hierarchy in approach to detail.

The reviewer continued: ‘The pattern the paper identifies, even thoughit can be modeled in a convincing way, may simply be a contingentsummary of the particular real-world data used.’ Here we reach the coreof a general unease about my approach, among some colleagues. If mymodels fit, they supposedly must fit for the wrong reason, even if thehidden artifact cannot be pinned down! We shall see. This book makes anumber of specific predictions that can turn out false.

In contrast, all too many studies in political science are safe againstbeing proved false because they only predict the broad direction of achange, while leaving its precise amount unspecified. Or they present aprobability distribution, without specifying the expectation value, as quan-tum physicists call it. It roughly means the value at which there is a 50-50probability of the next actual case being below or above the predictedvalue. This expectation value is what I have in mind when predicting a

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Preface

mean cabinet duration of 2.0 years for New Zealand, with a specified rangeof error. It is not a rigidly ‘deterministic’ prediction—it only expresses theaverage expectation.

This is how we should present our results, if we want political science tobe considered relevant by the public and by the practitioners of politics.King, Tomz, and Wittenberg (2000) have rightly criticized substantivelyambiguous conclusions like ‘The coefficient on education was statisticallysignificant at the 0.05 level.’ A decision-maker would be hard put to makeuse of such research. If one expects political practitioners to make useof what political scientists publish, this gap between jargon and usableresults should be a matter of concern.

This brings me to another aspect where this book may differ fromTaagepera and Shugart (1989). Meanwhile, I was drawn into active politicsthree times. I participated in Estonia’s constitutional assembly in 1991–2.In late 1992, I ran for President of Estonia, ending a respectable third(Taagepera 1993). And in 2001, I was the founding chair of a new partywhich, soon after the end of my tenure, went on to win the parliamentaryelections, before crashing (see Taagepera 2006). I never was a hands-onmanager of a political organization or campaign, but I was a high-levelparticipant and acquired some feeling for time pressures faced by sleeplesspoliticians. They may need to decide on an institutional feature withouttime for detailed study of alternatives and their possible consequences.Scholars choose their problems; problems choose their politicians. Canwe present scholarly results in a way politicians could use? This book triesto open all chapters with a section ‘For the practitioner of politics’.

What can we expect from electoral systems? If these systems are simple, wecan already predict quite a lot, at a level of precision useful to the politicalpractitioner. In the case of more complex electoral systems, we are stillfar from specific advice. This book tries to establish the basis for tacklingevermore complex cases.

There is some tension between these two goals—the clear-cut recipes thepractitioner needs, and the more tentative reasoning that gropes towardthe unknown. At the start of the chapters, I present the practical recipefirst, in a simple form which risks overstating the level of certainty ofthe claims. After all, the practitioner is often under pressure to act prettyquickly on the basis of the best current evidence, rather than wait forbetter evidence in years to come. Thereafter, in the body of the chapter,I shift to the scholarly mood, supplying theoretical considerations andempirical evidence. I try to distinguish between what is widely usable andwhat is more technical, shifting the latter to appendices at the end of the

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chapters. This is where I also express doubts about my own findings andpoint out deviating cases and trends.

My main emphasis is on elections for and party strengths in the lower oronly chamber of legislative assemblies. As a spin-off from assembly mod-els, a model also results for seat allocation among regions and countries infederal and supranational assemblies—such as the European Parliament.Presidential elections enter only as a conceptual limit case when an assem-bly is gradually reduced in size.

Party system specialists may be disappointed that this book says solittle about the internal structure of parties and their interaction in asystem. Our knowledge about parties continues to expand (among themost recent books, see Ware 1996; Mair 1997; Bowler, Farrell, and Katz1999; Dalton and Wattenberg 2000; Gallagher, Laver, and Mair 2000;Gunther, Montero, and Linz 2002; Webb and Farrell 2002; Cross 2004;Adams, Merrill, and Grofman 2005; Katz and Crotty 2006). The structureand interaction of parties should affect their size distribution. By howmuch they do, however, does not seem to have reached the stage ofoperational prediction. Establishing and isolating the mean impact ofelectoral systems, as this book does, serves to narrow the range of whatremains to be explained by other factors.

Many people deserve thanks for helping this book to come into exis-tence. My wife Mare kept bugging me about writing the book whenother academic and intellectual concerns tended to take precedence. Heractivity in chemistry and knowledge space theory approaches to teachingof science maintained my contact with practices in natural sciences. ClaireM. Croft at Oxford University Press urged me to bring my post-1989 worktogether in book form, and Elizabeth Suffling, Tanya Dean, and MickBelson edited it into a technically superb form. Bernard Grofman (2003)has been an early exponent of my approach, and his inseparable associate,A Wuffle, has supplied quotes to introduce two subparts of the book.Arend Lijphart, Mathew S. Shugart, Lorenzo de Sio, Daniel Bochsler, EvaldMikkel, Russ Dalton, and Anthony McGann have commented on variousdrafts and/or helped in other ways. Undergraduate and graduate studentsat University of California, Irvine and at University of Tartu in Estoniahave wittingly or unwittingly raised new questions about various aspectsof content and style. Indeed, several of them (Mirjam Allik, John Ensch,and Allan Sikk) have become coauthors of detailed studies condensedin this volume. Mirjam Allik also finalized most of the graphs. I thankthem all.

R.T.

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Contents

List of Figures xiiiList of Tables xvList of Symbols xix

1. How Electoral Systems Matter 1

Part I. Rules and Tools

2. The Origins and Components of Electoral Systems 13

3. Electoral Systems—Simple and Complex 23

4. The Number and Balance of Parties 47

5. Deviation from Proportional Representation andProportionality Profiles 65

6. Openness to Small Parties: The Micro-Mega Rule andthe Seat Product 83

Part II. The Duvergerian Macro-Agenda: How Simple ElectoralSystems Affect Party Sizes and Politics

7. The Duvergerian Agenda 101

8. The Number of Seat-Winning Parties and theLargest Seat Share 115

9. Seat Shares of All Parties and the Effective Number of Parties 143

10. The Mean Duration of Cabinets 165

11. How to Simplify Complex Electoral Systems 177

12. Size and Politics 187

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Contents

13. The Law of Minority Attrition 201

14. The Institutional Impact on Votes and Deviation from PR 225

Part III. Implications and Broader Agenda

15. Thresholds of Representation and the Number of PertinentElectoral Parties 241

16. Seat Allocation in Federal Second Chambers andthe Assemblies of the European Union 255

17. What Can We Expect from Electoral Laws? 269

Appendix: Detecting Factors Other than the Seat Product 287

References 293Index 307

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List of Figures

1.1. The opposite impacts of electoral systems and current party politics 3

2.1. When district magnitude changes, PR and plurality rules affectlargest party bonus in opposite ways 20

4.1. Balance vs. effective number of legislative parties in 25 countries,1985–1996 52

4.2. Effective number of legislative parties vs. effective number ofelectoral parties 54

5.1. Deviation from proportional representation vs. effective numberof electoral parties 68

5.2. Proportionality profiles for FPTP elections in New Zealand and the USA 71

5.3. Proportionality profile for Two-Rounds and PR elections in France 72

5.4. Proportionality profile for PR elections in Finland 73

5.5. Proportionality profile for Mixed-Member Proportional electionsin Germany 74

6.1. Vote shares at which parties tend to win their first seat vs. districtmagnitude, for various PR formulas 88

7.1. The opposite impacts of current politics and electoral system 107

7.2. The macro-Duvergerian agenda, as of 2007 109

8.1. The number of seat-winning parties vs. the seat product MS 118

8.2. The median seat share of the largest party vs. the number ofseat-winning parties 123

8.3. The median seat share of the largest party vs. assembly size, for 30single-seat systems—predictive model and regression line 126

8.4. The median seat share of the largest party vs. seat product MS for46 single- and multi-seat systems—predictive model and regression line 129

9.1. Actual average seat shares of parties ranked by size vs. largest seat share 146

9.2. Average seat shares of parties ranked by size vs. largest seatshare—politically adjusted predictive model and actual data 151

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List of Figures

9.3. Effective number of legislative parties vs. seat productMS—predictive model and regression line 153

10.1. Mean cabinet duration vs. effective number of legislativeparties—predictive model and regression line 169

10.2. Mean cabinet duration vs. seat product MS—predictive modeland regression line 171

13.1. Seat shares vs. vote shares for FPTP with high disproportionalityexponents—attrition law and Caribbean data 208

13.2. Actual seat shares vs. those calculated from the attrition law, forFPTP systems with high disproportionality exponents 210

13.3. Actual effective numbers of legislative parties vs. those calculatedfrom the attrition law, for FPTP systems with highdisproportionality exponents 212

13.4. Actual deviations from PR vs. those calculated from the attritionlaw, for two-party FPTP systems with high disproportionalityexponents 213

15.1. Nationwide number of seat-winning parties vs. average thresholdof representation 249

16.1. Number of subunit-based second chamber seats vs. the geometricmean of first chamber size and the number of subunits 259

16.2. Seat and voting weight distribution in the European Parliamentand the Council of the EU in 1995—predictive model and actualvalues 264

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List of Tables

3.1. Possible seat allocation rules in a single-seat district 25

3.2. Example of basic seat allocation options in a single-seat district 26

3.3. Allocation of 6 seats by d’Hondt divisors (1, 2, 3, . . . ) 31

3.4. Allocation of seats in a 6-seat district, by various quota and divisorformulas 33

3.5. Example of seat allocation by single transferable vote (STV) in a5-seat district 35

3.6. Established democracies 1945–90—number of electoral systemsand the total number of elections in which they were used 44

3.7. Electoral systems and British–French heritage 45

5.1. Satisfaction of the Taagepera and Grofman (2003) criteria by threeindices of deviation from PR 77

5.2. Mean values of deviation indices D1 and D∞, for given meanvalues of D2 79

5.3. An example where Loosemore–Hanby’s deviation index D1 lookstoo low 80

5.4. A counterexample where Loosemore–Hanby’s deviation index D1

no longer looks too low 80

5.5. Examples where Loosemore–Hanby’s deviation index D1 may lookpreferable to Gallagher’s D2 81

5.6. Which pattern would correspond to one-half of maximumdeviation from PR? 82

6.1. Effect of district magnitude and seat allocation formula on thedistribution of seats in a district where the percentage vote sharesare 48+, 25−, 13−, 9−, 4, and 1+ 86

6.2. Magnitudes at which parties with percentage vote shares 48+, 25−,13−, 9−, 4, and 1+ would win their first seat under variousallocation formulas 87

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List of Tables

6.3. Effect of district magnitude and seat allocation formula ondeviation from PR and on the effective number of parties, in adistrict where the percentage vote shares are 48+, 25−, 13−, 9−,4, and 1+ 89

6.4. The seat product and the resulting expected number ofseat-winning parties in the assembly 92

6.5. Tentative values of allocation formula exponent F in the seatproduct, for various PR formulas 93

8.1. The actual number of seat-winning parties, the expected number(based on district magnitude and assembly size), and their ratio 117

8.2. Assembly size (S) and the largest party’s seat share (s1), forsingle-seat district systems 125

8.3. District magnitude (M), assembly size (S), and the largest party’sseat share (s1), for multi-seat PR systems 128

8.4. Complexity of electoral systems and deviation of the largest seatshare from the model s1 = 1/(MS)1/8 130

9.1. Actual average seat shares of parties ranked by size vs. largest share 145

9.2. Seat shares of parties ranked by size, for given largestshare—probabilistic model, politically adjusted model, and theactual world averages 150

9.3. Effective number of parties for given largest share 161

9.4. Entropy-based effective number of parties 164

11.1. Actual district magnitudes (M) and effective magnitudes derivedfrom Meff = (N6/S)F, for stable democracies with relatively simpleelectoral systems in 1945–96. 180

11.2. Effective magnitudes for complex electoral systems, with outputMeff calculated from Meff = (N6/S)F 182

12.1. Predicted largest seat shares and effective numbers of parties, atselected populations and district magnitudes 191

12.2. Total and per capita party memberships at selected populationsand district magnitudes—empirical approximations 194

13.1. Women’s share in US public office 203

13.2. Caribbean countries with unusually high disproportionality exponents 207

13.3. Volleyball scores and the law of minority attrition 222

13.4. Selection constant for women’s attrition in US politics seems to be 3.5 222

14.1. From votes to seats, and back to votes—hypothetical vote shares,with S = 100 and n = 3.00 228

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List of Tables

14.2. Predicting the number of electoral parties from assembly size, forhigh responsiveness FPTP systems 230

14.3. Predicting the deviation from PR (Gallagher’s D2) from assemblysize, for high responsiveness FPTP systems 232

15.1. District-level thresholds of minimal representation for variousseat allocation formulas—general and for a 6-seat district 243

15.2. Sample constellations (in %) where the party shown in boldnarrowly wins or narrowly fails to win a seat in a 6-seat district,using the Sainte-Laguë seat allocation formula 243

15.3. Number of ‘pertinent’ electoral parties (p′) and resultingthresholds of representation (in %), if p′ = M1/2 + 2M1/4 andTR = (TI TE )1/2 246

16.1. The number of seats in the European Parliament—prediction bythe cube root law and the actual number 260

16.2. Total voting weights in the Council of the EuropeanUnion—prediction by S = P 1/6T1/2 and actual 260

16.3. Characteristics of seat allocations in the Council of the EuropeanUnion and the European Parliament 263

16.4. Incongruent seat allocations for the European Parliamentelections of 2004, compared to population in 2000 265

A.1. Residuals of the number of seat-winning parties (N0) 288

A.2. Residuals of the largest seat shares (s1) 290

A.3. Residuals of the largest seat shares (s1), effective numbers ofparties (N), and mean cabinet durations (C) 291

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List of Symbols

This list of symbols is not intended to be complete, but it includes the mostimportant, especially those used in several chapters. Numbers refer to the chapterin which the symbol first occurs.

A aggregate of an electoral system, 6

AV alternative vote, 3

a advantage ratio, seat–vote ratio, 5

B index of balance, 4

BC Borda count, 3

BV block vote, 3

b break-even point, 5

C mean duration of cabinets, in years, 10; number of Electoral Collegemembers, 13

D1 deviation from PR, Loosemore–Hanby measure, 5

D2 deviation from PR, Gallagher measure, 5

D∞ deviation from PR, Lijphart (1994) measure, 5

d divisor gap in seat allocation formula, 6

E number of electoral districts, 13

F formula exponent in seat product, 6; first chamber size, 16

FPTP first-past-the-post, single-seat plurality, 2

H entropy, 4

k conversion exponent between largest seat and vote share, 14; generalconstant elsewhere

L literate fraction of population, 12

LR largest remainders, 3

LV limited vote, 3

M district magnitude, 2

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List of Symbols

Meff effective magnitude at given assembly size, 2

MMP mixed-member proportional, 2

MS seat product, 6

m total membership of parties, 12

N, N2 effective number of components, parties, 4

NS effective number of parties, based on seat shares, 4

NV effective number of parties, based on vote shares, 4

N0 number of seat-winning parties, 4

N1 entropy-based number of parties, 4

N∞ number of parties that profit from small party abandonment, 4

n disproportionality exponent, 7, 13; general constant elsewhere

OLS ordinary least squares method of linear regression, 8

P population, 12

PBV party block vote, 3

PR proportional representation, 2

p number of seat-winning parties in a district, 8

p′ number of parties running in a district, 15

q0 simple quota for seat allocation, 3

q1 Hagenbach–Bischoff quota for seat allocation, 3

qDroop Droop quota for seat allocation, 3

qi general quota for seat allocation, 3

R number of registered parties, 12

R2 linear correlation coefficient squared, 8

r number of registered parties, 12

S assembly size, total number of seats, 2; second chamber size, 16

SNTV single non-transferable vote, 3

STV single transferable vote, 3

s, s1 fractional seat share of the largest party, 8

si fractional seat share of ith party, 3

T threshold of votes to win a seat, 8; number of territorial subunits, 16

TE threshold of exclusion, 15

TI threshold of inclusion, 15

TR Two-Rounds elections, 3

t total seat share of third parties, 12; turnout, 17

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List of Symbols

UV Unlimited Vote, 3

V total number of votes, 13

VA aggregate volatility, 5

VI individual volatility, 5

V1 volatility, Pedersen measure, 5

V2 volatility, Gallagher measure, 5

V∞ volatility, Lijphart (1994) measure, 5

v, v1 fractional vote share of the largest party, 4

vi fractional vote share of ith party, 4

W working-age fraction of population, 12

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1

How Electoral Systems Matter

For the practitioner of politics:

� Electoral systems help determine how many parties a country has,how cohesive they are, who forms the government, and how long thegovernment cabinets tend to last.

� Electoral systems are expressed in electoral laws. Their impact dependson the way politicians and voters make use of these laws.

� At times, flawed electoral laws can undo democracy or lead to staleness.

Who governs? Electoral systems matter in democracies because they affectthe answer to this question Robert Dahl (1961) posed in a different con-text. In the January 2006 Palestinian elections, the electoral system usedgave Hamas 70 percent of the seats and hence threw the Palestinian–Israelirelations into turmoil. Yet Hamas received only about 45 percent of thelist votes, as against about 41 percent for the more moderate Fatah.1 Withproportional representation rules, no party would have won an absolutemajority of the seats, leading to a more balanced coalition government.In contrast, the actual, heavily majoritarian electoral system was boundto boost the seats for whichever party received even slightly more votes.The answer to the question ‘Who governs?’ was determined as much bythe electoral system as by popular votes.

Elections are one way to determine who the leaders will be. This methodis more peaceful than fighting it out, more credible in modern times than

1 Matthew S. Shugart, ‘The magnitude of the Hamas sweep: The electoral system didit’ (http://fruitsandvotes.com, visited on February 28, 2006), calculates the Hamas list voteas 44.5 percent. The complex multi-seat nontransferable vote system, with many otherembellishments, makes an exact count for candidates difficult. See also Steven Hill, ‘Votesystem gave Hamas huge victory’, Hartford Courant, February 8, 2006, The Prague Post, February15, 2006.

1

http://fruitsandvotes.com


claims of divine favor, and more systematic than estimating the loudnessof noise made by various factions at an open-air meeting. Only transfer ofpower from parent to offspring can compete with elections in orderlinessof procedure; and in the modern world, elections have become a morewidespread practice. The supposed goal is to have the ‘people’ expresstheir will.

By electoral system, we mean the set of rules that specify how voterscan express their preferences (ballot structure) and how the votes aretranslated into seats. The system must specify at least the number of areaswhere this translation takes place (electoral districts), the number of seatsallocated in each of these areas (district magnitude), and the seat allocationformula. All this will be discussed in more detail later.

This book deals only with elections that offer some choice. It bypassesfake elections where a single candidate for a given post is given total oroverwhelming governmental support, while other candidates are openlyblocked or covertly undermined. It also largely overlooks pathologies ofelectoral practices such as malapportionment and gerrymander, exceptfor pointing out which electoral systems are more conducive to suchmanipulation.

The physical conditions of elections matter, such as ease of registrationof voters and candidates, location and opening times of polling stations,the timing of elections, and ballot design—see Mozaffar and Schedler(2002) and Reynolds and Steenbergen 2006). It is presumed in this bookthat such conditions of electoral governance are satisfactory. My onlyconcern is to explain, in what are considered fair elections, how electoralsystems affect the translation of votes into seats, how the results also affectthe distribution of the votes in the next elections, and what it meansfor party systems. Moreover, the book largely limits itself to first or onlychambers of legislatures, except for one chapter on second chambers andsupranational assemblies, plus incidental comments on presidential andlocal elections.

This scope may look narrow, but translation of votes into seats bydifferent electoral systems can lead to drastically different outcomes. Wealready saw what it meant for Palestine. Also, with a different electoralsystem (and traditions in applying it), a mere 36.3 percent of the totalvote would not have made Salvador Allende president of Chile in 1970,and Chile’s history could have taken a very different course.

Around 1930, the vote shares of the British Liberals and the IcelandicProgressives were practically the same: 23.4 percent for the Liberals in1929 and 23.9 percent for the Progressives in 1933. But the rules for

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ELECTORAL SEATS VOTES POLITICS SYSTEMS DISTRIBUTION DISTRIBUTION & PARTIES

POLITICAL CULTURE

Figure 1.1. The opposite impacts of electoral systems and current party politics

allocating assembly seats on the basis of popular votes differed. TheIcelandic Progressives won 33 percent of the seats and played a leadingrole in the country’s politics, while the British Liberals won less than10 percent of the seats. The resulting disappointment affected the votesin the next election and sent the Liberals down to near-oblivion.

Thus, electoral systems can sometimes make or break a party—or evena country. In less spectacular ways, they affect party strengths in therepresentative assembly and the resulting composition of the governingcabinet. They can encourage the rise of new parties, bringing in new bloodbut possibly leading to excessive fractionalization, or they can squeezeout all but two parties, bringing clarity of choice but possibly leading toeventual staleness. It is well worth discovering in quantitative detail howelectoral systems and related institutions affect the translation of votesinto seats.

Electoral Systems, Seats, Votes, and Party Politics

Figure 1.1 shows the opposite impacts of electoral systems and partypolitics on the distribution of seats and votes among parties. Electoralsystems restrict directly the way seats can be distributed. In particular,when single-seat districts are used, only one party can win a seat in thegiven district. The impact on votes is more remote. When a party fails toobtain seats in several elections, it may lose votes because voters give upon it, or it may decide not to run in the given district. The impact onparty system and hence on politics in general is even more remote. Still,if a party fails to win seats all across the country, over many elections,it may fold, reducing the number of parties among which the voters canchoose.

The impacts of the existing party system and current politics are attenu-ated in the reverse direction. The total number of meaningful parties maybe limited by the workings of the electoral system, but current politicsdetermines which parties obtain how many votes. The impact of current

3


politics on the seats distribution is weaker, as the electoral system mayrestrict the number of parties that can win seats. Still, current politicsdetermines which parties win seats. Finally, current politics has no impactat all on the electoral system, most of the times. Yet, infrequently, it hasa major impact, when a new electoral system is worked out from scratch,or when protest against the existing electoral system builds up for anyreason.

At all stages, political culture plays a role. The same electoral lawsplay out differently in different political cultures, shaping different partysystems. Along with the initial party system, political culture shapesthe adoption of electoral laws. If stable electoral and party systems suc-ceed in lasting over a long time, this experience itself can alter the ini-tial political culture—a connection not shown in Figure 1.1. This bookmentions political culture rarely, but not because I underestimate it. Ijust do the relatively easy things first, and political culture is harder totackle.

In the study of current politics, votes come first, and seats follow—thearrows at the top of Figure 1.1. This direction may look natural, but it isreversed when we study the impact of electoral systems. Now seats arerestricted directly, and restrictions on votes follow in a slow and diffuseway—the arrows at the bottom of Figure 1.1. Recognition of such reversalis essential for elucidating the impact of electoral systems.

The Limiting Frames of Political Games

Politics takes place in time and space—both the immutable physical spaceand the institutional space that politics can alter, but with much inertia.The physical size of polities matters for their functioning, as stressedearly on by Robert Dahl and Edward Tufte (1973). Institutional size alsoplaces constraints on politics. For instance, in a five-seat electoral district,at least one party and at most five parties can win seats. Within thesebounds, politics is not predetermined, but the limiting frame still restrictsthe political game. It is rare for one party to win all seats in a five-seatdistrict, while such an outcome is inevitable in a single-seat district. Thisobservation may look obvious and hence pointless, but it will be seen thatit leads to far-reaching consequences.

Institutions are containers within which the political processes takeplace. Containers matter. True, the content matters more, and containersdo not decide what is poured into them. But if they leak, crack, overflow,

4


or corrode, they do affect the outcome. Indirectly, they even affect thecontent, because one learns from experience not to pour, for instance, theproverbial new wine into old wineskins. It would be false dichotomy toask whether containers matter or not. It is a question of how much theymatter, and how.

So it is with political institutions. An excellent institutional frameworkcannot compensate for flawed political culture, but inadequate institu-tions can make it worse. Such a risk is high when political culture iscorrosively intolerant and does not value cooperation and compromise.To maximize stability, institutions should be congruent with politicalculture, to use Harry Eckstein’s terminology (Eckstein 1966, 1998), butnot so congruent as to help perpetuate an undemocratic culture. Electoralsystems are part of such institutions.

Electoral Systems and Party Systems

Here I use ‘electoral systems’ with some hesitation. In systems theory,a system divides the world into external and internal, and it has somecapacity to restore internal equilibrium when disturbed by external fac-tors. If so, then one could speak of an electoral system only when theelectoral rules have been embedded in a political culture where voters andpoliticians have acquired reasonable skills in handling the rules to theirenlightened self-interest, which includes most actors’ long-term interestin preserving a modicum of stability. Such skills are based on experience.A set of electoral rules can be promulgated as laws overnight, but it takesseveral electoral cycles for politicians and voters to learn how to handlethese laws to their best advantage. Hence electoral rules become a stablelimiting frame for the electoral game only when they have been used afair number of times.

In this light, should we define ‘electoral systems’ as not only a set ofrules but also include the skills people exert in using them? There is somemerit in such a definition (Taagepera 1998a), but it also leads to newdifficulties in telling electoral and party systems apart. Therefore, I adhereto the generally accepted definition of electoral system as the set of rulesthat govern ballot structure and seat allocation.

Electoral system thus defined is inextricably intertwined with partysystem. Even the earliest election in a new democracy is bound to takeplace in the context of some constellation of proto-parties, but to talk ofa party system truly serves a purpose only when some degree of stability

5


has set in regarding the identity, size, and interaction of parties. Earlyparty constellations are often kaleidoscopic configurations of individualpoliticians, devoid of anything akin to a system. During early democra-tization, major parties may vanish completely and new ones may arise.Thus, the early party constellations can be even more fleeting than akaleidoscope, where at least the pieces remain the same (Grofman, Mikkel,and Taagepera 2000). Such party constellations become a party systemonly slowly.

What is involved in a party system? It is more than just the num-ber and sizes of parties. It also includes their interactions. Peter Mair(1997: 214–20) offers two convincing examples of decoupling betweenparliamentary strengths of parties and their interaction patterns regardinggovernment formation and maintenance. A long-standing feature of theIrish party system was Fianna Fail’s refusal to engage in coalition cabinets,which constrained the voters to vote either for Fianna Fail and single-party cabinets or for coalition cabinets by ‘The Rest’. When a varietyof reasons induced Fianna Fail to participate in coalitions, starting in1989, the entire pattern of possible combinations expanded. Party systemchanged without any change in electoral system or any appreciable shiftin seat shares of parties.

Denmark is Mair’s contrary example (1997) of party system remainingthe same despite shifts in party strengths. Instead of previous 5 parties,10 parties won seats in 1973, and the combined vote share of the estab-lished 5 dropped from 93 percent to 65. Yet the interaction pattern ofparties changed little, as another minority cabinet replaced the previousone. Note, however, that the electoral system did not change either—onlythe electoral outcomes did.

The fact that a party system also involves interaction patterns amongparties does not do away with the importance of the number and sizes ofparties. A two-party system offers inherently different options, comparedto a multiparty system. The shift in Ireland 1989 and the non-shift inDenmark 1973 both played themselves out within the usual range ofoptions available in multiparty systems. When describing party systems,it would needlessly be limiting to claim either that only party sizes matteror, conversely, that party sizes do not matter at all.

The initial electoral system can play a major role in determining theparty system, but it is not the only factor. This book does focus on theimpact of electoral systems on representation and party systems, becausethe workings of electoral systems are relatively well understood qualita-tively and also with some quantitative rigor. But we should remember

6


that historical and cultural factors may produce a different party systemon the basis of the same electoral system.

Electoral systems affect politics, but they are also products of politics.Political pressures can alter them. This is well known, but after an ini-tial bow to this two-way causality, most researchers treat electoral sys-tems as causes of party systems rather than results. Consider the famousDuverger’s law (to which I will return), saying that plurality rule for seatallocation tends to produce a two-party system. How often does thisallocation rule produce a two-party constellation, and how often does itresult from a preexisting two-party constellation? Indeed, if the dawnof democracy in a given country finds the decision-makers divided intotwo parties, these parties may wish to choose the plurality rule so as toblock entry of new competitors. If, on the contrary, the initial decision-makers are split into many parties, they may wish to play it safe andadopt proportional representation (PR) so as to reduce their risk of totalelimination.

Only recently this issue has been addressed systematically (Boix 1999;Benoit 2002; 2004; Colomer 2005). Party constellations do tend to precedeand determine the electoral systems. Once in place, though, the electoralsystem helps to preserve the initial party constellation and to freeze it intoa party system. To avoid causal implications in either direction, we mayreword Duverger’s law: ‘Seat allocation by plurality rule tends to go withtwo major parties.’

Chess Rules and Electoral Rules

Electoral laws establish the rules for how the electoral game is carried outand how the winners are determined. In this, they are somewhat akin tochess rules (Taagepera 1998a). But there is one marked difference. Chessrules are extraneous to the game, while electoral rules are interwoven withthe game. In his classic Fights, Games, and Debates (1960), Anatol Rapoportimagines going to a statistics-oriented person to analyze chess. The latterreports items like the distribution of duration of games and the attritionrates of chess figures at successive moves. Rapoport, however, mumbles:‘But is this what we want to know about chess?’ In particular, does thisenable us to play better chess? Guess not.

Still, such statistical information on chess would be of interest. Itcertainly would, if proposals arose for changing the chess rules. Wouldthe change make the game boringly long or, to the contrary, awkwardly

7


short? But even then, the rules would not be part of the game. Beforesitting down at the chessboard, a player will not negotiate for fairer rulesto prevail on the chessboard, threatening otherwise with boycott. Noplayer will declare that, if s/he wins, s/he will change the rules. Theserules are quite constant in space and time. They define the game ratherthan being part of the game. The loser cannot claim that the rules werebiased.

Electoral rules also define the game, but they are part of it. They varyin space and time. Losers can blame them, and at times do. Change inelectoral rules can be part of an election platform. Because these rules canbe changed through political processes, the statistical and logical analysisof the properties of electoral systems is part of the study of politics, whilethe study of the consequences of various conceivable chess rules is notpart of learning chess.

This is not to deny the strategic aspects of politics, which are subjectto game-theoretical approaches and conditioned by political culture andvarious path-dependent factors. One need not even claim equal impor-tance for institutional aspects and for electoral systems in particular. Theyare merely the limiting frames for political games. A good electoral systemcannot save a polity where many other institutions, attitudes, and policieshave broken down. And on the other hand, a healthy polity can findways to compensate for a poor electoral system. However, an inadequateelectoral system can contribute to crisis in the case of shaky polities—andmost polities have their fragile aspects and periods.

My approach to electoral systems is very much in line with what madeRapoport ask ‘But is this chess?’ For chess, the response would be ‘No’, butfor the study of politics, it is ‘Yes’, because here the rules are themselvespart of the game.

The Study of Electoral Systems

Within political science, electoral studies are a relatively mature field ofstudy. They are located at the core of political science:

Although there are many concerns of political science that do not center aroundelections, the study of democratic practices—to which elections indisputably arecentral—is certainly one of the most crucial topics for the discipline as a whole.The study of elections is more than the study of electoral systems, and the studyof electoral systems is more than ‘seats and votes’, but the numerical values of

8


seats and votes for individual political parties and candidates are among the mostimportant quantitative indicators that we, as political scientists, employ in ourwork. (Shugart 2006)

For political scientists, electoral laws offer a further attraction: the pos-sibility of institutional engineering. For the given votes, one can cal-culate the extent to which different electoral laws would have alteredthe composition of the representative assembly, and one can proposechanges in laws. Of course, under different laws voters may have voteddifferently. For instance, a shift from plurality to PR may encouragevoters to shift to third parties. Such tendencies also must be taken intoaccount.

Actually, fundamental changes in electoral laws are infrequent, becausethey usually require agreement by representatives chosen under the oldlaws—and why should they change laws that served them well in gettingelected? Still, electoral laws may well be more conducive to institutionalengineering than institutions firmly stipulated in constitutions, not tomention political culture.

The quantitative nature of many features of electoral systems—thenumbers of seats and votes, precise allocation algorithms, and the like—may attract those political scientists who yearn to discover quantitativeregularities akin to those that have paid off in natural sciences. For thesame reason, electoral studies may repel those who consider the studyof politics an art rather than a science, or at most a science that thriveson richness of details rather than broad generalizations—zoology ratherthan molecular biology. Students of politics are largely reduced to non-repeatable observations in vivo instead of repeatable in vitro laboratorytests. Hence any general scientific laws in politics, if they exist at all, arebound to be hidden, submerged underneath considerable random scatterin data. This scatter may easily be construed as absence of general laws.This book, however, presents evidence that logical models can be con-structed in the context of electoral systems, and that they lead to specificquantitative predictions, which are confirmed empirically by the averagesof many elections, and even more by averages of many electoral systems.

This book first reviews the typology of electoral systems and intro-duces some analytical tools that make a comparative study of electoralsystems possible. This overview of rules and tools is a prelude to the‘Duvergerian agenda’ which has dominated the electoral studies for thelast half-century—the attempt to express the impact of the main featuresof electoral systems on representation and party system.

9


The central part of the book presents recent advances in the macro-scopic aspect of the Duvergerian agenda. These advances help us under-stand the logic of simple electoral systems to the degree that specific quan-titative predictions can be made for the average of many elections carriedout under the same rules. Individual elections, of course, can vary wildly,just as daily weather can vary within a well-defined climatic pattern. Allthis applies to simple systems. We are still far from being able to predict indetail the impact of complex electoral systems, but we have made markedprogress during the last few decades.

In this light, the final part of the book broadens the agenda and asks:What can we expect from electoral laws? It briefly describes advances instudying more complex electoral systems and lays out the agenda forextending our predictive ability from simple to complex systems. Oncewe have a firm grip on the impact of institutions, we can separate it fromthe impact of political culture and study the latter in relative isolationfrom confusing side effects. We are then in a position to attempt todesign electoral laws so as to obtain specific average outcomes—and alsoto have a sober awareness of the limits to our ability to design. Theaforementioned quantitative nature of many features of electoral systemsenables us to build and test logical quantitative models more extensivelythan has been the case in other studies of politics. Can we transfer someof the methodology developed for electoral systems? The book concludeswith this provocative issue: To what extent can electoral studies supply a‘Rosetta Stone’ to some other parts of political science?

10

Part I

Rules and Tools


2

The Origins and Components ofElectoral Systems


� When choosing an electoral system, a main trade-off is between deci-siveness of government and representation of various minority views.

� As long as you keep the electoral system simple, its average effect canbe predicted to a fair extent, and this book is of some use.

� When electoral systems are made complex, no one can predict theiractual workings—and you kid yourself if you think that you can.

� Try remembering future. Do not push for laws that favor large partiesjust because your party is large now—it may shrink.

� Do not change electoral laws frequently. Allow an understanding todevelop of how the electoral system works.

Elections are one way to determine who the leaders will be. But whodetermines what the rules for elections should be, and what are theoptions? How well does the resulting electoral system satisfy the originalintent? The choice of electoral system is affected by so many contradictoryconcerns that the choices made in specific historical instances could havehardly been predicted, although some outcomes were more likely thansome others (cf. Colomer 2004b). The path from devising electoral lawsto a mature understanding of how the resulting electoral system works isalso wrought with uncertainties.

The first third of the book deals with ‘rules and tools’. It describes thevariety of electoral rules devised that combine into electoral systems usedin various countries. It also presents the analytic tools needed to measure

13

Rules and Tools

inputs, such as the total number of seats, and outputs, such as the numberof parties.

Basic Choices

The main concern is balance between decisiveness of government andrepresentation of various minority views. Electoral systems that pushtoward a two-party system and hence one-party cabinets may promotedecisiveness of government. This outcome has been claimed for seatallocation by plurality in single-seat districts, often designated as first-past-the-post (FPTP). The desire for maximally PR, on the one hand,is best satisfied by a PR seat allocation rule applied nationwide. Onecan have both one-party cabinets and PR only if the political culturespontaneously develops just two parties of any appreciable size. This wasthe case in post-World War II Austria, despite its PR rules, but it occursrarely. On the other hand, some political cultures may miss out both ondecisiveness and on proportionality, having many small parties and yetlarge deviations from PR.

One may also be concerned about party cohesion, which is weakenedby some electoral systems, about voters having a personal representative,about regional, ethnic, and women’s representation, and so on. Colomer(2004b) stresses the desire to avoid the worst possible outcome for thelargest party. If this party feels safe against electoral reverses, it may pushfor FPTP. But if it feels insecure, it may opt for PR as insurance againstcatastrophic loss.

In new democracies, two considerations emerge stronger than in theestablished ones. One is legitimacy of electoral laws. If these laws are per-ceived as unfair, for whatever reason, founded or unfounded, then democ-racy is in trouble. The other aspect is the cost of elections, both in termsof money and expert labor. Some electoral systems are appreciably costlierthan some others, and new democracies, in particular, are often strappedfor funds and skilled administrators (Reynolds, Reilly, and Ellis 2005).

Electoral Laws Are Often Chosen or Changed in a Messy Way

Electoral laws are made by humans. They depend on the constellation ofpolitical forces and opinions at the time they are adopted. A foundingassembly consisting of one or two major blocs may prefer FPTP so as to

14

Components of Electoral Systems

freeze out newcomers, while a fractured assembly may pick some formof PR so as to enable all groupings to survive (Colomer 2004b, 2005). Yetmany other factors and concerns also enter.

It may look hard-boiled realism to declare that self-interest of theoriginal decision-makers determines the choice of electoral laws whendemocracy is introduced (or reintroduced). However, such a claim retroac-tively explains away whatever the outcome happens to be, and hence itexplains nothing (Taagepera 1998a). People decide on what is in theirinterest on varied and sometimes fleeting grounds. Winning the nextelection is a major concern, but it can conflict with long-term interests(including preservation of stability), ideological preferences (includingadvice by foreign advisers that belong to the same ideological strain), andthe force of habit. Which of these will overrule the others in defining‘self-interest’?

The means used to achieve one’s presumed self-interest can be mis-informed and hence counterproductive (Kaminski 2002; Andrews andJackman 2005). Thus, during the liberalization processes in the Soviet-dominated area of the late 1980s, the old communist regimes preferred tokeep the Soviet electoral rules, which favor the largest party even whenapplied honestly. The Communists did so not only by force of habit butalso because they expected to be the largest party. It turned out to bea catastrophic misjudgment in many countries. In Palestine 2006, Fatahmay have made the same miscalculation.

The predominant forces may stick to the pseudo-democratic electionrules inherited from the preceding political regime, either because theyare unaware of the alternatives or because they rationally try to balancethe merits of the existing rules against the costs and risks of innovation.Thus, most ex-British colonies adopted the British FPTP. Little did theyrealize that what produces a fairly balanced two-party representation inthe British Chamber of the Commons of some 600 members can producelopsided one-party predominance in the 20-seat assembly of a small islandnation. Such nations often ended up with a decimated parliamentaryopposition. Similarly, several post-Soviet states maintained the Sovietelectoral rules for a while, which required high participation, allowedvoting against all candidates, yet required absolute majority to win. Whatformally worked in Soviet one-candidate pseudo-elections led in inde-pendent Ukraine to interminable repeat elections, with participation everdecreasing. Some seats remained vacant permanently.

Countries may return to a tradition interrupted by dictatorship or for-eign occupation. Thus Zambia’s Third Republic returned to the rules of the

15

Rules and Tools

multiparty First Republic, after the de jure one-party state of the SecondRepublic. Earlier tradition itself may offer contradictory options. Thus,Estonia’s choice in 1992 was influenced both by its ultra-proportionalrules of the 1920s and by the disproportional rules adopted in 1938, inreaction to excessive multipartism.

Such reaction to undesirable aspects of the existing electoral system isa major motive for changing it. Time pressures may be less than duringthe original introduction of democracy, but overreaction to the existingsystem may cause an excessive shift in a different direction.

The role of random happenings should not be ignored. As Nigel Roberts(1997) asks regarding New Zealand: ‘What would have happened if DavidLange had not made an inadvertent pledge during the 1987 election tohold a binding referendum on the question of electoral reform?’ But forthis irretrievable slip of the tongue, the ball might not have started rolling.If this could happen in stable New Zealand, then how often may thechoice of the initial electoral rules in new democracies have been decidedby who happened to be at which meeting, and in what mood?

With 20-20 hindsight, one can always invoke ‘self-interest’ so as toperfectly explain away this conglomeration of desire to win, yet followtradition, avoid new thinking and information gathering, satisfy foreignideological sponsors, and maintain some idealistic concern about futurestability—all this combined with miscalculation and chance happenings.

The process of determining the electoral laws often starts with compet-ing simple formats being proposed, for example FPTP or nationwide PR. Acompromise between the two may be negotiated, for example the ‘MixedMember Proportional’ (MMP) system in West Germany in the late 1940s.If one of the basic formats carries, the losing side may try to introduceamendments. For instance, if nationwide PR promises representation totiny parties, a legal threshold of some percentage of votes may be pro-posed so as to block them. Regional parties, however, who are majorplayers in their respective regions, may oppose such a limitation, aimedchiefly at tiny nationwide parties. It may then be decided that the legalthreshold does not apply to parties that satisfy certain local requirements.

In the course of such wrangling, a superficially strong stipulation maybe gutted by subtle further additions. Blocking and enabling measuresmay reach such complexity that no one can predict the actual pattern ofoutcomes. At this point, the electoral systems expert might well give up,but opposing politicians may still believe that they have outfoxed eachother. The actual pattern, as it takes shape during several elections, maynot satisfy anyone.

16


If so, what opportunity does all that leave for supposedly rational adviceby neutral experts on electoral systems? It is not up to the experts toquestion the motivational basis of the desired outcome. They can onlyhelp avoid misconceived means to reach the desired ends. They can ask‘Which kind of results do you want?’ and then point out to what extentthe rules under consideration may ensure or defeat the stated goals. Tosome extent, one can design for a two-party system and the long lastingcabinets that tend to result. One can also design for maximal PR, at thecost of relatively short-lived multiparty coalition cabinets. But it is almostimpossible to design simultaneously for near-perfect PR, yet long-lastingcabinets. And only simple electoral systems lead to somewhat predictableparty systems.


Elections can apply to one position (president), a few (local council)or several hundred (parliament). Voters may have to voice unqualifiedsupport for one or several candidates (‘categorical ballot’), or they maybe able to rank candidates (‘ordinal ballot’). Details of electoral laws thathave been or could be used are given in Chapter 3. Here the basic choicesare outlined.

Some most fundamental choices that pertain to elections are outsidethe electoral laws as such. Every democratic country needs electoral lawsfor the first or only chamber of its legislative assembly. If a presidentialregime is chosen, it also needs laws for presidential elections. If a two-chamber assembly is chosen, both chambers need election or selectionrules. Some aspects of electing a president or selecting a second chamberare addressed in later chapters. Here I consider only the inevitable first oronly chamber.

The first question is: How many seats should such a chamber have?Large countries are almost bound to have more seats (in line with a logicalrelationship presented in Chapter 12), but there is some leeway. Giventhat smaller assemblies offer less room for variety, the choice of assemblysize (S) affects the chances of smaller parties. For the given election,assembly size is usually fixed in advance, but in some systems it fluctuatesslightly, depending on the outcome of the election.

The next question is: Into how many electoral districts should the coun-try be divided? Electoral districts mean the areas within which popularvotes are converted into assembly seats. The number of seats allocated

17

Rules and Tools

within a district is called district magnitude (M). It is arguably the singlemost important number for election outcomes. One can have numeroussingle-seat districts where M = 1, or fewer multi-seat districts where M > 1.The limit is one nationwide district where district magnitude equalsassembly size: M = S. All districts need not be of equal magnitude, andoverlapping districts also occur.

The next or rather concurrent issue is the seat allocation formula withinthe district. It is tied in with ballot structure. The voter may be given one ormore votes. If only the first preferences are taken into account, the voter isasked to cast one or several categorical votes (categorical ballot). If secondand later preferences are also taken into account, the voter is asked to rankthe candidates (ordinal ballot). The allocation formula stipulates how theresulting votes are to be converted into seats. At the one extreme, all seatsin the district may be given to the party with the most votes (pluralityrule). At the other extreme, one could use a PR formula that favors thesmallest parties and takes into account second preferences and supportfor specific candidates.

Assembly size, district magnitude, and seat allocation formula (plus the cor-responding ballot structure) are the three indispensable features regardingwhich a choice cannot be avoided, if one wants to allocate seats on thebasis of votes. Further features can be added, such as legal thresholds forminimum representation. A mix of district magnitudes and allocationformulas can be introduced. Several rounds of voting can be used. Severaltiers (levels) of seat allocation can be used, going beyond the basic dis-tricts. Such additions are frosting on the cake rather than indispensableingredients.

Seats can be allocated on the basis of votes for party lists or votesfor individual candidates. The two options can also be mixed. Insteadof having only the choice of party lists (closed lists), voters mayhave the option to voice preferences for one or more candidates ona list (open lists), or they may be even required to vote for a spe-cific candidate. The so-called panachage (literally, cocktail) may evenenable them to vote for a list but also mix in candidates from otherlists.

In sum, choosing an electoral system involves three inevitable choices(S, M, and allocation formula) and numerous optional ones. The ways tocombine and mix them are infinite in principle and extremely numerousin practice. One can promulgate electoral laws, but the resulting partysystem may differ from the expected. Thus, with FPTP, most voters in mostcountries tend to vote for the two largest nationwide parties, but in some

18


countries regional parties may subsist, or there may be large numbers ofsuccessful independents.

Single-seat districts may look simple, but they still offer several choicesfor seat allocation, to be discussed in Chapter 3. In multi-seat districtsthe options multiply. Seat allocation can be made on the basis of votesfor individual candidates or votes for party lists. Voters can have onlyone vote or as many as M, the number of seats at stake. If votes arefor individual candidates, transfer of votes among candidates may bepossible, when second preferences are marked on the ballot. When partylists are used (usually with one vote per voter) the basic choice is betweenplurality rule and one of the many PR seat allocation formulas.

Multi-seat plurality favors the largest party nationwide. This advantageis already marked in the case of single-seat districts, but it grows with dis-trict magnitude. With a single nationwide district (M = S), this advantagebecomes absolute: The largest party wins all the seats in the assembly.Because of this lopsided advantage, multi-seat plurality is rarely used indistricts of more than 2 or 3 seats.

The effect of magnitude is reversed when a PR formula is used. Onecomes closest to ideal PR when the entire country forms a single hugedistrict. Here, a decreasing district magnitude increases the large partyadvantage and hurts the small parties. Proportionality is the least when aPR rule is applied in a single-seat district. Here, the plurality and PR rulesmeet and lead to the same outcomes. Indeed, ‘single-seat plurality’ couldas well be called ‘single-seat PR’! This is why I prefer to designate it asFPTP, a relatively neutral term between plurality and PR.

Figure 2.1 shows the overall picture for party lists in single- and multi-seat districts. The contrast between plurality and PR allocation rules isextreme for a nationwide single district (M = S). Here plurality rule wouldassign all S seats in the assembly to the winning list, while PR rules wouldproduce highly proportional outcomes. As the electorate is divided intoincreasingly smaller districts (M < S), the contrast between the outcomesof plurality and PR rules softens, until they yield the same outcome in thecase of single-seat districts (M = 1).

Simple Electoral Systems

The more complex the laws are, the more we are in uncharted territory,for several reasons. Voters may react to complex laws in different ways.Also, the more complex the laws, the fewer past cases with similar laws we

19

Rules and Tools

FPTPM = 1Markedlargestpartybonus

M = SAll seats go to thelargest party

M = SHardly any largestparty bonus

PLURALITYRULE

PR RULE

Direction of bonus growth

Figure 2.1. When district magnitude changes, PR and plurality rules affect largestparty bonus in opposite ways

have in the world, so as to draw empirical lessons. Lastly, logical predictivemodels also become so complex that they can offer little guidance.

The simplest family of electoral systems is the one where a total of Sseats are allocated to closed lists in a single round, in districts of equalmagnitude M and according to a standard PR formula. For M = 1, it boilsdown to FPTP. Semi-proportional or plurality formulas do not count assimple. Two parameters, M and S, largely suffice to specify a simplesystem. The electoral formula also affects the outcome, especially whenM ranges from 2 to 5. Still, changes in magnitude matter markedly more(as is shown in Chapter 6).

Apart from FPTP elections that involve no primaries (e.g. UK), perfectlysimple electoral systems are rare. However, the specific impact of electoralsystems on the translation of votes into seats is easiest to investigate forsimple systems. This is what this book focuses on.

How Easy Should It Be to Alter the Electoral Laws?

Should the electoral laws be specified in the constitution, making themhard to change? Or should they be regular laws that the national assem-bly can alter fairly easily? It depends on how easy it is to change theconstitution. Practices vary, and consistency may not prevail even withinthe same country. Thus the Estonian Constitution of 1992 delves in detailon the procedure for electing the figurehead president but specifies onlya vague ‘principle of proportionality’ for the election of the real powercenter, the national assembly. Similar vagueness has led to heated debatesin the Czech Republic on how semi-proportional the electoral formula

20


can become without violating the constitutional norm of proportionality(Novák and Lebeda 2005).

The more the constitution spells out the details of the electoral system,the more difficult it becomes to change electoral laws that have provedinadequate. On the other hand, excessive ease can lead to opportunisticchanges by an unpopular government who hopes to soften the blow at thenext elections. Such was the case in France 1986 where the long-standingTwo-Rounds rule in single-seat districts was replaced by multi-seat PR butwas immediately reintroduced by the incoming majority.

If electoral laws are changed too frequently, no stable pattern has timeto develop. Maybe it should take at least a 55 or 60 percent majority tochange the electoral law—or simple majority in two successive assemblies(Arendt Lijphart, private communication).

On Terminology

Varying terminology continues to plague electoral studies. Districts aresometime called constituencies. Some traditional texts talk of single-member and multi-member districts. However, electoral rules allocateseats or memberships to candidates or parties—they do not allocate ‘mem-bers’ as such. It is hence more logical to talk of single-seat and multi-seatdistricts.

The terms electoral ‘rules’, ‘formulas’, ‘formats’, ‘laws’, ‘arrangements’,‘systems’, and ‘design’ have at times been used on the very same page,almost as synonyms, but not quite. One might talk of the ‘FPTP rule bywhich the major party dominance is enhanced’, or the ‘FPTP laws accord-ing to which the major party dominance is enhanced’, or the ‘FPTP systemin which the major party dominance is enhanced’. Different authors haveused these and other terms in slightly different senses. Therefore, theirdefinitions for the purposes of this book should be given. My definitionsneed not be more functional, but they clarify what the terms mean here.

The focus of this book is on the effect of those rules by which votesare translated into seats and also the rules on how voters can expresstheir votes—categorical or ranked ballot, number of votes per voter, openor closed list, etc. Apart from these ballot structure and seat allocationrules, there are other rules governing the process of elections in a broadersense: voting rights, registration of voters, calling the election, candidatenomination, campaigning, advertising, opinion polls, and distribution ofpolling places (Farrell 2001: 3; Reynolds, Reilly, and Ellis 2005: 5). The

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study of such rules is outside the scope of this book, even while I recognizetheir impact on the votes.

Hence, ‘electoral rules’ or ‘electoral system’ stands in this book forballot structure and seat allocation rules, unless otherwise specified. I use‘seat allocation formula’ when this is what I mean, rather than ‘electoralformula’, which sometimes has been used as a synonym for electoral rulesin a broader sense.

In individual countries, the rules that form the ‘electoral system’ findexpression in ‘electoral laws’ specific to the country. Features that formpart of the same law in one country may belong to separate laws inanother. For analytic purposes, we overlook such differences—we comparethe workings of similar ballot structures and seat allocation rules acrosscountries, regardless of the legal formats. The electoral system is embedded inthe electoral laws of the particular country. I take ‘electoral arrangements’or ‘format’ to mean the way the electoral system is embedded in laws, butI rarely use these terms. Finally, I use ‘electoral design’ to mean future-oriented institutional engineering. It is not a mere synonym for existingsystem or laws.

In previous literature, the consensus has been that ‘electoral system’stands for a set of rules that are mutually consistent and completelyspecify the ballot structure and seat allocation. Explicitly or implicitly,this applies to Lijphart (1994: 1, 7), Farrell (2001: 3), Colomer (2004b: 3),Reynolds, Reilly, and Ellis (2005: 5), and Taagepera and Shugart (1989: xi).Farrell (2001) uses ‘electoral laws’ for what I designate as rules for electionsin the broad sense, regulating everything from voting rights to opinionpolls. Among these laws, there is one ‘set of rules which deal with theprocess of election itself: how citizens vote, the style of ballot paper, themethod of counting, the final determination of who is elected . . . This iselectoral system . . . ’ Farrell (2001: 3). In contrast to Farrell, authors likeLijphart (1994), Colomer (2004b), Reynolds, Reilly, and Ellis (2005), andTaagepera and Shugart (1989) avoid the term ‘electoral laws’.

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3

Electoral Systems—Simple and Complex


� To allocate seats to candidates or parties, laws must specify at leastthe following: the total number of seats in the assembly (or its firstchamber—assembly size), the number of seats allocated in each electoraldistrict (district magnitude), how these seats are allocated (allocation for-mula), and how a voter can express her/his preferences (ballot structure).

� Assembly size depends strongly on population size.� District magnitude can be as low as 1 (single-seat districts) or as high as

assembly size.� The simplest seat allocation formulas are d’Hondt and Sainte-Laguë

divisors, and Hare quota plus largest remainders. For single-seat dis-tricts, these PR formulas are reduced to first-past-the-post, where thecandidate with the most votes wins.

� With these formulas, the larger the district magnitude, the more pro-portional the seat shares are to the vote shares, and the more partiesmay be represented.

� The smaller the district magnitude, the larger the seat share of thelargest party tends to be, and one-party cabinets become more likely.

� Optional features include legal thresholds, Two-Rounds elections, eachvoter having several votes, and voters ranking candidates. The advan-tages of complex and composite electoral systems may be real or imag-ined. Either way, they make it harder to predict the number of partiesand the average proportionality of seats to votes.

At the minimum, a full set of electoral rules must stipulate the follow-ing: the total size of the representative body, the magnitude of electoral

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districts, the seat allocation formula, and the corresponding ballot struc-ture. To these, Lijphart (1994: 1) adds electoral threshold. However, nolegal threshold needs to be stipulated (cf. Lijphart 1994: 11), and the‘effective threshold’ inherent in district magnitude (to be discussed later)cannot be prescribed separately from district magnitude and seat alloca-tion formula.

As further factors, Lijphart (1994: 15) adds malapportionment, presi-dentialism, and apparentement. These three are discussed toward the endof this chapter. Farrell (2001), Colomer (2004b), and Reynolds, Reilly, andEllis (2005) do not explicitly list the indispensable ingredients of electoralsystems, nor did Taagepera and Shugart (1989). Assembly size dependsstrongly on population size. It is the most undervalued factor—it occursin the subject index only in Taagepera and Shugart (1989) and Lijphart(1994).

This chapter first describes the systems that are basic in that they donot add any further ingredients to the basic three, do not mix the basicformulas or systems, and apply the basic rules in a fairly uniform way.Thereafter, more complex or composite systems will be surveyed. I focuson the most widely used electoral systems. Farrell (2001) describes theseand others in more detail, and Reynolds, Reilly, and Ellis (2005) indicatehow they have worked out in specific countries. Colomer (2004b) presentsa most thoughtful account on how and why the various approachesdeveloped over history, with preferences shifting from unanimity anddrawing by lots toward majority and then toward PR.

The basic systems include what I call the simplest family of electoralsystems (Chapter 2) and also some others that offer more elaborate alloca-tion formulas or ballot structures, but without mixing them. In line withthe scheme in Figure 2.1, the basic systems are divided into groups bydistrict magnitude and by allocation formula: single-seat districts, multi-seat districts with plurality-oriented seat allocation formulas, and multi-seat districts with PR-oriented seat allocation formulas. The latter, in turn,may be party centered or candidate centered.

Single-Seat Districts (M = 1)

When the country is divided into single-seat districts, then the basicchoices are two. The candidate with the most votes (plurality or ‘relativemajority’) could be declared the winner. This is traditionally called FPTPformula. Alternatively, absolute majority (more than 50 percent of the

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Table 3.1. Possible seat allocation rules in a single-seat district

Categorical ballot(marking one candidate)

Ordinal ballot(ranking several)

Plurality First-past-the-post Borda CountMajority Two-Rounds Alternative Vote

votes) could be required. In either case, the voter can be asked to casteither a categorical ballot for one candidate or to rank the candidates, asshown in Table 3.1.

In FPTP systems, the candidate with the most votes wins. This systemis widely used in British-heritage countries and tends to produce or pre-serve a two-party system with a fair balance between government andopposition. However, assembly size can make a difference. Large countrieswith large assemblies such as the United Kingdom (S around 650) andIndia (S around 550) can have more than two significant parties in theassembly, while in small countries like Barbados (S = 26) one party tendsto have about 70 percent of the seats, leaving a weak opposition withonly 30 percent (Taagepera and Ensch 2006). The reasons are discussed inChapter 8. Among the systems with single-seat districts, FPTP qualifies asa simple electoral system. All others are more complex.

In Two-Rounds (TR, ‘Second-Round Runoff’) systems, if no candidatereaches 50 percent of the votes, the two candidates with the most votesgo into a second round of elections, where one of them is bound to reach50 percent of valid ballots. Few stable democracies use this system (Birch2003), but it is fairly widespread in Africa and Asia (see tables in Reynolds,Reilly, and Ellis 2005: 30–1). France requires 50 percent in the first roundbut only a plurality in the second (‘Two-Rounds Majority-Plurality’). Thenumber of parties running in the first round can be large, but the secondround tends to focus on only a few parties.

In Alternative Vote (AV, ‘Majority Preferential’, ‘Instant Runoff’), votersmay or must rank all candidates. When the votes are counted, the weakestcandidate is eliminated and his voters’ votes are transferred according totheir second preferences. The process is repeated, if necessary, possiblyinvolving some voters’ third and lower preferences. When only two candi-dates remain, one of them is bound to have at least 50 percent. AlternativeVote has been used for nearly 100 years in Australia, where a two-and-a-half party system has taken root (Farrell and McAllister 2006). Instead ofeliminating the least popular candidate (the one with the fewest first placevotes), the most disliked candidate (the one with the most last place votes)

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Rules and Tools

could be dropped. This variant has not been used but has theoreticaladvantages (Grofman and Feld 2004).

In Borda count (BC), the ranked votes are weighted. With 4 candidatesrunning, every first preference receives 3 points, every second preference2 points, and every third preference 1 point. These weighted votes areadded up, and the candidate with plurality wins. As Jean-Charles de Bordahimself put it 200 years ago, it is a good system ‘only for honest men’(Colomer 2004b: 30), because it is highly susceptible to strategic voting.Only two Pacific countries use variants of BC: Nauru and Kiribati (Reilly2002).

Intermediary approaches are possible, especially with Two-Rounds.Recall that France has a majority-plurality system. In some intermediaryapproaches, winning in the first round may require only 40 percent ofthe votes or being sufficiently ahead of the next-ranking candidate (e.g.by 10 percent votes). All these options also apply to direct presidentialelections, where M = 1 by definition.

The workings of these systems are illustrated in Table 3.2. There are 100voters and 4 candidates, assumed to line up on the simplistic left-rightideological scale. When voters are asked to rank candidates, I will assumethat their second preference is the candidate closest to their first choice.In the case of equal closeness, they are assumed to split their secondpreferences evenly between the candidates to the left and to the right oftheir first preference. Assume the first preferences are as shown in the firstline in Table 3.2. The rest follows from these simplifying assumptions.

Left has the most first preference votes and wins by the FPTP rule. Bythe Two-Rounds majority rule, the second round pits Left against Right,the centrist voters shift to their ideologically closest candidates, and Rightwins. Borda count multiplies the first preferences by 3, second preferences

Table 3.2. Example of basic seat allocation options in a single-seat district

Left Center-Left Center-Right Right Total

First or only preference 33 14 24 29 100Second preference 14/2 = 7 33 + 24/2 = 45 14/2 + 29 = 36 24/2 = 12 100Third preference 14/2 = 7 24/2 + 29 = 41 33 + 14/2 = 40 24/2 = 12 100

First-past-the-post 33 winsTwo Rounds, 2nd round 33 + 14 = 49 Eliminated Eliminated 29 + 24 = 51 winsBorda Count, total points 120 173 184 wins 123 600Alternative Vote,

2nd stage 33 + 14/2 = 40 Eliminated 14/2 + 24 = 31 29 1003rd stage 40 — 31 + 29 = 60 wins Eliminated 100

26


by 2, and third preferences by 1, and then adds these points. Center-Rightnarrowly surpasses Center-Left and wins.

Finally, by the Alternative Vote rule, the process is more complex.Since no one reaches 50 percent, the weakest candidate is eliminated—theCenter-Left. His votes are reallocated according to the second preferences.Still, no one reaches 50 percent. The weakest candidate now is Right,narrowly surpassed by Center-Right, thanks to the boost of second prefer-ences from Center-Left. With Right eliminated, her votes are reallocatedaccording to the second preferences, and Center-Right wins by a largemargin of 60.

Thus, depending on the seat allocation formula chosen, almost anycandidate could win in this particular sample, chosen to illustrate theimportance of the allocation formula. In most actual cases, many formu-las yield the same result.

It is most important to realize that the actual election outcomes are notrule-blind. The seat allocation formula is known before the elections takeplace, and parties and voters will adjust. The outcome depends on howwell they can coordinate (Cox 1997). When the rule is FPTP, then Center-Right would effectively play a ‘spoiler’ role, enabling Left to win. Hence, ifthe opinion polls offer a realistic idea of the relative strengths of the can-didates, Center-Right might drop out so as not to split the right. If Rightand Center-Right fail to coordinate in such a way in the first election,the Left victory may teach them to present a joint candidate in the nextelection. The latter step, in turn, will force Left and Center-Left to choosebetween presenting a joint candidate and facing sure defeat. This is howFPTP pushes the party system toward two dominant parties, as claimed byDuverger’s law, but it may take time, and exceptions outnumber the caseswhere a balanced nationwide two-party system develops.

The other allocation formulas exert less pressure toward concentration.In Two-Rounds, many candidates may continue to run in the first roundand the losers may bargain with their support prior to the second round.In Alternative Vote, voters do not have to worry about playing a ‘spoiler’role. The voter may express support for her/his favorite, even if the latterhas no chance to win, and then mark as second preference the preferredone among the top candidates. However, suppose two rightist parties andonly one leftist party run. If ranking is mandatory, leftist voters are forcedto mark a rightist as their second preference. To give them another option,the leftist party may induce a weak centrist candidate to run.

In BC, if the opinion polls enable the Left voters to anticipate theoutcome, they may tilt victory to Center-Left by strategically marking

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Rules and Tools

their third preference as Right, so that the total points for Center-Rightdrop to 151. Anticipating this Left ploy, Right may respond in kind,reducing the Center-Left points to 144. Or Right may even induce a fakeleftist candidate to run so as to act as a spoiler. The resulting uncertaintiesare the main reason for why BC is little used.

Multi-Seat Districts: Overview

Seat allocation in multi-seat districts can be carried out on the basis ofvotes for individual candidates or votes for lists presented by parties (orother groups). A large number of combinations are possible, such as thefollowing.

When voters vote for individual candidates, each of them may haveonly one vote or as many as M, the number of seats at stake. If theyhave several votes, they may or may not be allowed to place them allon the same candidate (cumulation). If voters have only one vote, theymay be allowed (or even required) to rank candidates, making transferof votes among candidates possible. This way the votes for the losingcandidates and also the superfluous votes for the top candidates wouldnot be ‘wasted’ but would help these voters’ second preference candidates.Finally, various forms of Approval Ballot permit voting for up to M candi-dates, or even more.

When party lists are used (usually with one vote per voter), the basicchoice is between plurality, where the party with the most votes winsall the M seats in the district, and one of the many PR seat allocationformulas to be described later. Intermediary ‘semi-proportional’ formulasalso exist.

Multi-Seat Districts with Plurality-Oriented SeatAllocation Formulas

One can run party lists in multi-seat districts and allocate all seats to thelist with the most votes. Stable democracies have largely abandoned suchParty Block Vote (PBV, ‘Block Ballot’), because it boosts the advantage of thelargest party nationwide even more than FPTP, weakening the oppositionto the point of making it ineffective.

In contrast to such a party-centered approach, one can formally ignorethe existence of parties, focus on candidates, and give each voter as many

28


votes as there are seats in the district. Such Unlimited Vote (UV, ‘UnlimitedBallot’) enables a voter to spread her votes among, say, the candidates shedeems the most honest, regardless of ideology or party. However, if thereis strong party loyalty, UV becomes akin to PBV, decimating all oppositionto the dominating party. Unlimited Vote is used in many local electionsin the USA, which are formally run on non-party basis.

One can alleviate major party dominance by allowing Cumulative Vote(‘Cumulative Ballot’), where the minority can load their votes heavilyon a few candidates, or by shifting from UV to Limited Vote (LV, ‘Lim-ited Ballot’), meaning that for M seats in the district, each voter hasless than M votes (to be used with or without cumulating). ‘MultipleNon-Transferable Vote’ is a term that could mean either LV or UV. Thiswas the ingredient that made Hamas the big winner in Palestine 2006(cf. Chapter 1). As the number of votes per voter is reduced, the dominantparty voters find it ever harder to hoard all the seats in the district. Whenthe number of votes per voter is reduced to the square root of districtmagnitude (M 0.5), seat shares seem to become fairly proportional to thevote shares, although no theoretical proof to that effect seems to exist.

Approval Voting (‘Approval Ballot’) amounts to unlimited vote carriedto the extreme: Vote for as many candidates as you please, and the Mcandidates with the most votes win. Jean-Charles de Borda might againsay that it is a good system ‘only for honest men’, those who tolerantlyapprove of their less-preferred candidates too. When the electorate ispolarized, however, and voters vote only for their own party’s candidates,the plurality party wins all the seats. Even worse, an intolerant minoritycan prevail over a tolerant majority of 60 percent who also approves ofsome of their less-preferred candidates. Approval voting easily turns intodisapproval voting.

Multi-Seat Districts with PR-OrientedFormulas—Party Centered

Here, the goal is to make the seat shares of parties reasonably proportionalto their vote shares. The voter is given one categorical vote, to be cast for aparty list (‘closed list’) or for a candidate within the list (‘open list’). Eitherway, parties receive seats on the basis of their total votes (List PR). Theseseats go to the candidates at the top of the closed list, or to the candidateswith the most personal votes in the case of open list. Intermediary waysto allocate seats within the party are also used. The issue of intraparty

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Rules and Tools

seat allocation is important, for example for representation of women(Matland and Taylor 1997) and local interests (Crisp et al. 2004; Shugart,Valdini, and Suominen 2005), but this book largely bypasses it.

How should the seats be allocated among the parties? The easiestapproach to understand might be Simple Quota and Largest Remainders,called more briefly Hare-LR. It is used for instance in Costa Rica, and itworks as follows. If M seats are available, a share 100%/M of the votesshould entitle a party to a seat. This is the simple or Hare quota (also calledexact or Hamilton quota), designated here as q0 for reasons of systematicsthat will become apparent later. Parties receive as many seats as theyhave full quotas of votes. These quotas are subtracted from the total voteshares, leaving almost always remainders of votes, and some seats alsoremain unallocated. Such seats are allocated to the parties with the largestremainders. Any party with a remainder of more than half-quota (q0/2) islikely to receive such a remainder seat, but it depends on how the voteshappen to be distributed among the other parties.

The quickest way to determine the Hare-LR allocation is to multiply thefractional vote shares of parties by M. The integer parts of these productsrepresent quota seats, and the remaining seats go to parties with thelargest decimal parts.

So as to avoid having too many remainder seats, we could reduce thequota. One might consider 100%/(M + 1), designated here as q1. Now itis possible to allocate more seats by full quotas—or even all of them.But one runs a tiny risk of allocating more seats than the district has.Suppose M = 4, so that q1 = 20 percent. If party votes happen to be exactly60.00, 20.00, and 20.00 percent, 5 seats would be allocated! To guardagainst this admittedly unlikely outcome, the Hagenbach-Bischoff quotaadds 1 vote to the total votes, before calculating the quota, while Droopquota adds 1 vote to the quota itself (NOT 1 percent of all votes!), leadingto qDroop = 100%/(M + 1) + 1 vote. This single vote makes over-allocationof seats impossible. For practical purposes, the Droop and Hagenbach-Bischoff quotas are identical to q1.

Some electoral systems (e.g. formerly in Italy) have not worried aboutover-allocation and have used q1, and even q2 = 100%/(M + 2) and q3 =100%/(M + 3), both called Imperiali quotas. In case of over-allocation, thelucky district simply receives extra seats in the representative assembly,until the next election.

Quotas larger than simple quota can also be devised, for example, q−1 =100%/(M − 1), and so on, but they are not used in practice. Somewhatcounterintuitively, small quotas favor large parties, while large quotas favor

30


small parties. This is easiest to see by considering extreme cases. Supposeagain that M = 4, and we decide to make use of q−3 = 100%/(M − 3) sothat the quota is 100 percent. No one receives a quota seat, and the fourlargest parties receive one remainder seat each, even when the vote sharesare as unbalanced as 60, 30, 7, 2, and 1. Of course, considering q−3 isunrealistic, but unrealistic extreme cases offer a powerful conceptual toolin disciplines such as physics. I will use this tool extensively, and it isexplained in Beyond Regresssion (Taagepera 2008).

Fixed quotas of a specified number of votes have also been used. Thenthe total number of seats a district receives depends on turnout. Thismight be an incentive to go and vote.

The basic philosophy for all quotas plus largest remainders is subtraction:Each time a seat is allocated to a party, a specified amount of votes is sub-tracted from its total votes. Instead, one can also use a divisor philosophy:Each time a party is allocated a seat, divide its total votes by a specifiedamount, before the allocation of the next seat is considered. Such divisorsare described next.

The most widely used divisors are the d’Hondt (Jefferson) divisors, 1, 2,3, 4, . . . They work as shown in Table 3.3. Suppose the party percentage ofvote shares in the district are exactly 48, 25, 13, 9, 4, and 1, and M = 6seats are to be allocated. First, we divide all vote shares symbolically by 1,which does not alter them. Next, we allocate the first seat to the largestshare, 48 percent, as indicated by (1) next to ‘48’ in Table 3.3. But we alsodivide this share by 2, reducing it to 24. As we compare the new shares,the second-largest party’s 25 percent exceeds the largest party’s 24, andhence it receives the second seat, while its share is divided by 2. The nexttwo seats go again to the largest party, with its share divided by 3 andthen by 4. (A common mistake students make at this point is to divide 24by 3, instead of dividing the original 48 by 3.) The fifth and sixth seatsgo to the third- and second-largest parties, whose quotients (13 and 12.5,respectively) narrowly surpass the largest party’s 12.

Table 3.3. Allocation of 6 seats by d’Hondt divisors (1, 2, 3, . . . )

votes, % 48 (1) 25 (2) 13 (5) 9 4 148/2 = 24 (3) 25/2 = 12.5 (6) 13/2 = 7.548/3 = 16 (4)48/4 = 12

seats 3 2 1 0 0 0seats, % 50 33 20 0 0 0

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In this particular case, all major parties are overpaid: their seat sharesexceed their vote shares. In general, however, d’Hondt divisors tend tofavor the largest party. As M increases, this advantage lessens. Finland hasused d’Hondt divisors for a full 100 years (with mean M = 14), and othercurrent examples include Switzerland, Luxembourg, Spain, and Portugal.

Various other divisors can also be used. Faster increase in divisorsreduces large party advantage. Sainte-Laguë (Webster) divisors (1, 3, 5,7, . . . ) abolish this advantage, and the so-called Danish divisors (1, 4,7, 10, . . . ) actually favor the smaller parties (as is shown in Chapter 6).To increase their seat shares, large parties might then split their listsstrategically, but it would be risky to do so, because fake splits may becomereal. At the extreme, one could use huge divisor gaps such as 1, 51, 101,151, . . . Then all M largest parties may win one seat each.

In the other direction, slowly increasing divisors favor heavily thelargest party. Imperiali divisors (1, 1.5, 2, 2.5, . . . ) have been used. (Donot confuse them with the aforementioned Imperiali quotas!) The divisorseries with the slowest increase would be 1, 1, 1, 1, . . . where the largestparty wins all the seats. Thus, surprisingly, multi-seat plurality rule sur-faces as the extreme member of the divisor family of the PR formulas.

One can also devise divisors that tend to favor middle-sized parties. TheModified Sainte-Laguë divisors (1.4, 3, 5, 7, . . . ) are used in Norway andSweden. Here the initial divisor 1.4 (instead of 1) makes it hard for smallparties to receive their first seat. The quaintest divisors ever used mightbe the ‘modified d’Hondt’ divisors used in Estonia: 10.9, 20.9, 30.9, 40.9,50.9, . . . They are equivalent to 1, 1.87, 2.69, 3.48, 4.26, . . .

To illustrate the effect of the various quota and divisor approaches,Table 3.4 shows the allocations of seats when vote shares are again exactly48, 25, 13, 9, 4, and 1 percent, as in Table 3.3, and the district has 6 seats.

The perfectly proportional seat share, as shown at the top of Table 3.4,is fractional and can only be approximated. Visibly, allocations 3, 2, 1,and 3, 1, 1, 1, which occur in the center of the table, come closest to pro-portionality, and these are the only rules used fairly widely. (Operationalmeasures for deviation from PR are presented in Chapter 5.) Allocationformulas at the top of the table tend to overpay the largest party and arerarely used. Those at the bottom tend to overpay the small parties and arehardly ever used.

Given that large quotas allocate all too many seats by largest remain-ders, while small quotas risk over-allocation of quota seats, one may lookfor a sufficient quota. Start with Droop quota and reduce the quota grad-ually, until all seats are allocated by quota, with no need to consider the

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Table 3.4. Allocation of seats in a 6-seat district, by various quota and divisor formulas

Votes, % 48 25 13 9 4 1

Perfectly proportional seat share 2.88 1.50 0.78 0.54 0.24 0.06

Steady divisors (1, 1, 1, 1, . . . ) = plurality 6 0 0 0 0 0Imperiali divisors (1, 1.5, 2, 2.5, . . . ) 4 2 0 0 0 0Modified d’Hondt (1, 1.87, 2.69, 3.48, . . . ) 4 2 0 0 0 0Imperiali quota q3 = 100%/(M + 3) = 11.1% 4 2 1 [overallocation!]Imperiali quota q2 = 100%/(M + 2) = 12.5% 3 2 1 0 0 0D’Hondt divisors (1, 2, 3, 4, . . . ) 3 2 1 0 0 0Modified Sainte-Laguë div. (1.4, 3, 5, 7, . . . ) 3 2 1 0 0 0Droop/Hagenbach-B. quota q1 = 14.3% 3 2 1 0 0 0Sainte-Laguë divisors (1, 3, 5, 7, . . . ) 3 1 1 1 0 0Hare quota q0 = 100%/M = 16.7% 3 1 1 1 0 0Danish divisors (1, 4, 7, 10, . . . ) 3 1 1 1 0 0Quota q−1 = 100%/(M − 1) = 20% 3 1 1 1 0 0Quota q−2 = 100%/(M − 2) = 25% 2 1 1 1 1 0Quota q−3 = 100%/(M − 3) = 33.3% 2 1 1 1 1 0Divisors 1, 51, 101, 151, . . . 1 1 1 1 1 1Quota q−4 = 100%/(M − 4) = 50% 1 1 1 1 1 1

remainders. The result may surprise: The remainderless quota is equivalentto d’Hondt divisors. In other words, d’Hondt represents the sufficient quota(Colomer 2004b: 44).

Thus, the d’Hondt formula occupies a central position on the landscapeof List PR formulas: It is at the crossroads of quota and divisor methods.This is how Thomas Jefferson actually came to define what later cameto be called d’Hondt divisors in electoral studies (Colomer 2004a: 44).Like Alexander Hamilton, who first defined the simple quota, and DanielWebster, who first defined the Sainte-Laguë divisors, Jefferson was con-cerned with seat allocation to the US states according to their populations.All these approaches were reinvented in Europe when the need arose toallocate seats to parties according to their votes.

As district magnitude increases, all allocation formulas in the centralrange of Table 3.4 tend to produce seat allocations closer to perfect PR,and the choice of the particular formula makes less of a difference (seeChapter 6). In principle, these allocation formulas can be applied to anydistrict magnitude, up to and including nationwide allocation.

In the opposite direction, what happens if these formulas are applied tosingle-seat districts? All of them allocate the only seat at stake to the partywith the most votes, and hence they become equivalent to FPTP. In thissense, drawing a line between List PR in multi-seat districts and FPTP isartificial. FPTP is a limiting case of List PR.

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The description of seat allocation formulas in various countries oftenrisks confusion. The d’Hondt procedure can be speeded up by first allocat-ing seats by full Droop or Hagenbach-Bischoff quotas and then, insteadof using largest remainders, switching to d’Hondt. This is how the Swisselectoral law describes the procedure. The outcome is pure d’Hondt. Yetdescribing it as ‘the Hagenbach–Bischoff formula’ may leave the mis-taken impression that the Hagenbach–Bischoff quota is used with largestremainders.

Multi-Seat Districts with PR-OrientedFormulas—Candidate Centered

Rather than force voters to vote for parties as blocs, one may wish tolet them express preferences for specific candidates, regardless of partyaffiliations. The aforementioned LV allows a voter to vote for severalcandidates. It seems to achieve a reasonable degree of PR among partieswhen the number of votes per voter does not exceed the square root ofdistrict magnitude (M 0.5).

The most limited number of votes per voter would be one vote. Calledsingle non-transferable vote (SNTV), this may be the simplest method thatcould be applied in multi-seat districts. In a district with M seats, the Mcandidates with the most votes win. In analogy to FPTP, SNTV may beconsidered an ‘mth past the post’ system (Reed and Bolland 1999). Sim-plicity is desirable, but SNTV has a unique drawback. It is the only majormulti-seat system that penalizes parties for running too many candidates.This is illustrated by the following example.

Consider a five-seat district. Based on opinion polls and previous elec-tion results, party votes (in percent) are expected to be A: 45, B: 13, C: 28,and D: 14. Assume party A fields 2 candidates, who split the vote 23 to 22,and party C also fields 2 candidates, who split the vote 16 and 12. A wouldwin 2 seats and each other party 1 seat. However, A is underpaid, with 45percent of the votes and only 40 percent of the seats. It might considerrunning 3 candidates and would win 3 seats, if the votes are dividedevenly among the candidates (15, 15, 15). But if one of the candidatesis overly popular, so that the split is 23, 11, 11, then A could end up withonly 1 seat. The same could happen, if the shares of the 3 candidates areroughly equal but the total vote for A falls below expectations.

Because of such coordination problems, SNTV is rarely applied in dis-tricts of more than 3–5 seats. Such a low district magnitude reduces the

34


degree of proportionality to the point where SNTV is often called semi-proportional. The limited proportionality, however, is more a matter ofmagnitude than of the allocation formula as such (Cox 1996). Japan,South Korea, and Taiwan have used SNTV, but Japan abandoned it in 1994(see Grofman et al.1999).

Coordination dilemmas can be avoided, if voters are asked to rank can-didates. Stipulate that it takes a Droop quota worth of votes to win a seat.Second and later preferences are used to transfer excess votes of popularcandidates to other candidates. Called single transferable vote (STV), thismethod is the multi-seat equivalent of the aforementioned AlternativeVote. While the latter uses only elimination of weaker candidates, multi-seat districts also impose the need to transfer excess votes of the successfulcandidates. The STV is used in Ireland, Malta, and the Australian Senate(Bowler and Grofman 2000).

Table 3.5 follows up on the previous example of an unlucky vote con-stellation, where SNTV would allocate the largest party only 1 seat out of5. How would STV allocate these seats? We will assume that the secondpreferences go to the candidates of the same party, or to the ideologicallyclosest party. Droop quota for M = 5 is 100%/6 = 16.7 percent. (The extra1 vote in the Droop quota can be neglected.) Any candidate who reachesthis quota wins a seat. Her excess votes are allocated according to hervoters’ second preferences. If this helps further candidates to reach a

Table 3.5. Example of seat allocation by single transferable vote (STV) in a 5-seatdistrict

Candidates A1 A2 A3 B C1 C2 D

First preference votes (%) 23.0 11.0 11.0 13.0 16.0 12.0 14.0Quota allocation −16.7Remainder transfer (assumed) 6.3 → +4.3 +2.0New totals 15.3 13.0

Elimination of the weakest +9.0 ← −12.0 → 3.0New totals 26.0 17.0

Quota allocations −16.7 −16.7Remainder transfers +8.6 ← 8.3 ← 0.3New totals 21.6

Quota allocation −16.7Remainder transfers +4.9 ← 4.9New totals 17.9

Quota allocation −16.7Residual remainders 15.3 1.2 [they add up to one Droop quota.]Seats for parties 2 seats 1 seat 1 seat 1 seat

Note: Assume that candidates are listed in the order of placement on left-right scale.

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full quota, the process is repeated. If not, then the weakest candidate iseliminated, and his votes are allocated according to his voters’ secondpreferences. In later stages, third and fourth preferences may come intoplay. In this particular example, the largest party wins 2 seats, as it didwith SNTV when not taking risks and fielding only 2 candidates.

Visibly, the STV procedure is more complex than those previouslydescribed—and I have omitted some details that can make it evenmessier—see Farrell, Mackerras, and McAllister (1996) and Bowler andGrofman (2000). With M = 5, it is quite possible that 6 parties mightrun. Since there is no penalty for fielding many candidates, in contrast toSNTV, there might be an average of 4 candidates per party, for a total of 24.It is hard to rank that many candidates in a meaningful way. Hence, STV israrely used in districts with more than 5 seats. This low district magnitudeimpedes approach to perfect PR. Yet, STV offers maximal freedom ofchoice to the voters, without fear that one’s vote might be wasted. Forinstance, if a voter’s main concern is to enhance women’s representation,he could express high preference for all female candidates. Thus, STVmay have considerable philosophical appeal, and computer programs canhandle the technical aspects easily.

What happens when the SNTV or STV procedures are applied in single-seat districts? The SNTV is then reduced to FPTP, similarly to List PRformulas. The STV, however, is reduced to Alternative Vote. In sum,the traditional distinction between single- and multi-seat districts is notneeded in the analysis of the impact of electoral systems on party systems.Indeed, such distinction makes analysis harder. Single-seat districts aremerely the limiting cases of multi-seat districts.

Complex and Composite Electoral Systems

All electoral systems previously described offer only one district magni-tude and one seat allocation formula, even while this formula might bequite involved. The possibilities for electoral design multiply when dis-trict magnitude varies from district to district, when legal thresholds areintroduced to constrain the workings of the basic allocation formula, orwhen different allocation formulas are applied in sequence or in parallel.These and some other practices are reviewed next.

The prime task of this book, however, is to explain in depth the effectsof the simplest of the basic electoral systems. From this viewpoint, whatfollows can be bypassed, but it is needed, if one wants to obtain a picture

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of systems actually used. Indeed, tabulation at the end of the chaptershows that a fair proportion of actual electoral systems involve complexor composite features.

Unequal District Magnitudes

When multi-seat districts are used in a country, they almost never havethe same magnitude, because they tend to correspond to historical–geographical subunits. Typically, M would be higher in major cities andlower in rural areas, while in small ethnically distinct areas some single-seat districts might be thrown in. Thus, most magnitudes in Finland areclose to the arithmetic mean of 14, but extremes have ranged at timesfrom 3 to 28, plus one single-seat district in the autonomous ÅlandIslands. Habitually, the mean magnitude is used to characterize the elec-toral system, but variation may have political consequences. The direc-tion and nature of these consequences emerges best when we consider ahypothetical extreme case.

Suppose a country with an assembly of 200 seats has one district ofM = 100 around the capital city and 100 districts of M = 1 in the sparselysettled countryside. All districts use the same List PR formula, which in thesingle-seat districts amounts to FPTP. The arithmetic mean magnitude isvery close to 2, but the resulting party system is likely to be quite differentfrom that in a country of 100 districts of M = 2. Two-seat districts mayenable 2 or 3 parties to receive seats. In contrast, the district of M = 100may offer opportunities for a dozen of parties. With such a power base,some small parties may also try their luck in the single-seat districts. Insum, the country with all seats at M = 2 is likely to have a two- or three-party system, while the country with variation in M is likely to have amultiparty system.

One may devise measures of average M different from the arithmeticmean, so as to magnify the impact of the largest districts (see Appendixof Taagepera 1998b). But no such averaging may circumvent a featurenoticed by Monroe and Rose (2002): Uneven distribution of district mag-nitudes can bias party representation. A simple illustration follows.

Assume that our aforementioned hypothetical country has only twomajor parties, one having 70 percent support in the urban district butonly 30 percent in the rural districts, and the other having the propor-tions reversed. Thus, the overall voting strengths of the two parties areequal and amount to 50 percent (assuming no malapportionment). Their

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representation in the assembly, however, is lopsided. The rural party winsall the rural FPTP seats, plus its proportional share of 30 seats in the urbandistrict, for a total of 130 seats. The urban party only wins its proportionalshare of seventy seats in the urban district. This extreme example isunlikely to materialize. But to a lesser degree, the same tendencies dooccur in actual countries that use a wide range of district magnitudes, asshown by Monroe and Rose (2002) for Spain.

Compared to more complex and composite electoral systems, unevenM has been considered a minor deviation from simple electoral systems.Yet it can significantly alter the survival chances of small parties. It canalso alter the seat balance of major parties.

Legal Thresholds

While PR for parties may be considered desirable in general, a profusionof tiny parties is not. Therefore, limits on minimal representation areimposed in many countries that use List PR in large magnitude districtsor even nationwide. Typically, parties below a given threshold of votesare not entitled to participate in seat allocation. The legal threshold usedmay be a low as 0.67 percent (The Netherlands) or as high as 5 percent(Germany). Some countries apply even higher thresholds to alliance listsof several parties.

It matters whether the threshold applies nationwide or in individualdistricts. Suppose a party has 4.9 percent of the nationwide votes. Anationwide threshold of 5 percent would bar it from obtaining seats.The same threshold applied in individual districts, however, would notprevent it from winning seats in those districts where the party has morethan 5 percent of the votes.

District magnitude as such imposes an effective threshold. For example,when M = 5, it is nearly impossible for a party to win a seat with lessthan 10 percent of the votes. These effective thresholds are calculated later(Chapter 15). For the moment, it is important to realize that a district-level legal threshold may block small parties in large districts while havingno impact whatsoever in small districts. Take Spain, with many five-seatdistricts but also huge districts in Barcelona and Madrid. Only the latterare affected by Spain’s district-level legal threshold of 5 percent, becausethe effective thresholds inherent in small district magnitudes are largerthan that.

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Legal thresholds can be expressed in terms other than a percentage ofvotes. It may be a fixed number of votes, or a threshold of 2 seats in theassembly, so as to block one-seat parties and independents. On the otherhand, legal thresholds may contain loopholes. Thus, the 5 percent bar inGermany does not apply to parties that win at least 3 seats in single-seatdistricts.

Legal Majorities

Sometimes minimal representation for the largest party is prescribed, soas to enhance governability. It so happened in Malta that the largest partyby votes obtained fewer seats than the next-largest party, whose voteswere placed more advantageously across the districts. As a reaction to thisanomaly, Malta now stipulates that a party with more than 50 percentvotes must obtain more than 50 percent of the seats. If needed, extra seatsare added to the usual number.

In Italian municipal elections, parties with sufficient pluralities aregiven 60 percent of the seats in large cities and 66.7 percent in smallmunicipalities, before seats are allocated to other parties. In contrast toMalta, the total size of the assembly does not change.

Multiple Tiers

When simple quota is applied in multi-seat districts, one may decide toallocate seats only by full quota. The remainder votes and seats are trans-ferred to larger second-tier super-districts or even a single nationwide dis-trict. There, allocation formula may change. Thus, Belgium used to switchfrom quota to d’Hondt. The overall outcome would be equivalent to pured’Hondt in Belgian super-districts, except for limitations on small partiesand alliance lists, in the upper tier. Some countries have three tiers, andGreece has even four, with complex limitations—see Lijphart (1994: 44).

When too many seats are deemed to go to the second tier, one mayalleviate the full quota rule in the districts and allocate seats by largestremainders, as long as these remainders surpass 0.9 or 0.75 of the fullquota. Estonia introduced such a relaxation around 2000. Note that thenumber of seats allocated in the upper tier(s) varies from election toelection and is not known ahead of the time. One can also assign a fixednumber of seats to each part of the system. Then we have a compositesystem, as discussed next, rather than a multitier one.

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Rules and Tools

Composite Systems: Single versus Double Ballot, Parallelversus Compensatory

A country that uses single-seat districts may offer smaller parties a chanceby superimposing a parallel system that operates by List PR. The numberof such mixed systems has been increasing, and the variety of options islarge—see Massicotte and Blais (1999).

Sometimes the voter has a single ballot for both systems—for example,in Mexico, Taiwan, South Korea, and the Italian Senate (Ferrara 2006).This puts a voter who prefers a small party in a bind. If she votes forthe small party, her vote has no effect on the outcome in the single-seatdistrict, where only the largest parties have a chance. If she votes for alarge party, her preferred small party may lose a PR seat.

Such dilemmas are avoided by giving each voter separate ballots forsingle-seat and large-magnitude districts. A voter who prefers a smallparty would vote for this party in the large district but may well votestrategically for one of the two largest parties in the single-seat district.Nationwide allocation, in turn, can proceed either in parallel or in acompensatory way, as explained below. The two approaches, which mayyield very different outcomes, are often confused.

Seats can be allocated separately in two parallel systems. When there are100 single-seat districts and one nationwide district of M = 20, then smallparty representation remains symbolic. But overall semi-PR results whenthe large district has 100 seats. Then small parties win about one-half oftheir proportional shares, while the two largest parties still maintain thehefty bonus gained in the single-seat districts. Italy and Japan shifted inthe 1990s to such parallel systems.

One may go a step further and use the vote shares in the nationwidedistrict to establish nationwide PR. Here we have a compensatory system.Parties that win many small-district seats receive accordingly fewer seatsin the nationwide allocation. If the nationwide seats are few, say, 20 outof a total of 120, they may not suffice to restore nationwide PR. But ifthere are as many nationwide seats as small-district seats, then one comesclose to perfect PR, unless blocked by a legal threshold—which usually isthe case.

Germany uses such a compensatory MMP system (see Shugart andWattenberg 2001). It has also been called Personalized PR, and Addi-tional Member system, although the latter term conjures the image ofparallel systems that do not interact. Note that compensatory systems

40


could also function with a single ballot per voter. The small party voterwould not have any impact on who wins in the local small district,but it would not matter in the nationwide allocation by compensatoryPR.

One can also complement small List PR districts with a fixed numberof nationwide compensatory seats. Usually, small party access is subjectto limitations. In Iceland, only parties with at least one district seat canreceive compensatory seats. Denmark sets a legal threshold of 2 percentvotes, but with complex loopholes.

Intra-List Party Competition and Apparentement

Most List PR systems favor larger parties even at medium district magni-tudes, and the effect becomes strong at low magnitudes. This creates anincentive for parties to present joint lists in specific districts or nation-wide, if electoral law allows it. The effect is similar when separate listsare used, but they are declared to count together for the purposes of seatallocation—a format traditionally called apparentement. Switzerland is along-standing example. Detailed arrangements vary.

Competition among allied parties remains possible, if the voter votesfor a specific candidate. Such alliances become crucial in Chile, whereM = 2. The logic of two-seat districts, similar to Duverger’s law, forces theparties to form two large blocks. Within the block, each party presents onecandidate (it would be self-defeating to run more than one). The votes forall candidates in the block are added and supply the basis for allocationof seats by d’Hondt. This means that in order to win both seats, a blockmust have more than twice the votes of the other block, which is rarelythe case. Within the block, the seat(s) won go to the candidate(s) with themost votes.

The same method could be used in single-seat districts, if candidateswere allowed to make alliances. The seat would go to the candidate withplurality vote within the alliance with plurality in the district. In analogywith the FPTP, it could be called FHFT—First Horse in the First Team. Sucha system would push the parties toward formation of two major blocks ofparties, while preserving many parties. This approach has been used forpresidential elections in Uruguay. ‘Desistance pacts’ among Italian parties(Gambetta and Warner 2004: 246) also amount to an informal applicationof ‘FHFT’.

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Party-Candidate Cocktails (Panachage)

We have seen that in the presence of party lists, a voter may be required tovote for a party only (closed list), for a party and optionally for a candidatewithin a party (open list), or for a specific candidate within the party (themost open list). Panachage (literally: cocktail) carries the option one stepfurther: The voter may vote for a party but also for a specific candidatein another party. Why allow such a mix? By this means a supporter of ahopelessly small party can cast a symbolic vote for this party, while stillgiving support to the most acceptable candidate in a major party. Usedin Switzerland and Luxembourg, it may be seen as a step from List PRtoward STV.

Primary Elections

Parties can (or may be required by law) carry out intraparty electionsso as to determine the rank order of candidates on a closed list—or theonly candidate in the case of single-seat districts. Such primaries may belimited to formal party members, or they may be open to the public, theonly limit being that one can participate in only one party’s primary. Inthe USA, where ‘bipartisan gerrymander’ (see next section) often makesone party’s primary the only election with a real choice, proposals haverisen to have a joint primary for all parties. The two top candidateswould advance to the actual election. This system would be similar toTwo-Rounds, except that the two finalists might belong to the sameparty.

Thus the line between intraparty elections (that might be considered aprivate affair) and public elections is not clear-cut. Comparative studies ofprimaries are as yet few. This is one area where more work is needed.

Pathologies of Electoral Systems: Malapportionmentand Gerrymander

Electoral fraud can take place in many ways. Parties and candidates canbe prevented from running or campaigning and advertising. Voters canbe intimidated or bought, and votes can be miscounted. But short ofsuch blatant fraud, various games can also be played with magnitudesand boundaries of electoral districts.

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Malapportionment means that some districts have too few or too manyseats, compared to their population. During rapid urbanization, malap-portionment can arise spontaneously, when the countryside becomesdepopulated but still preserves its assembly seats. Reapportionment onthe basis of new census results may be delayed, if parties who stand tolose from it are strong in the existing assembly. The larger the districtmagnitude, the easier it is to reapportion in an incremental way. Indeed,malapportionment becomes impossible when a nationwide single districtis used.

Gerrymander means drawing single-seat district borders in such a way asto assure safe districts (say, 60 percent majorities) to the party in chargeof districting, while leaving the other major party with wastefully largelosing minorities (say, 40 percent) in those districts and with wastefullyhuge winning majorities (say, 80 percent) in other districts. It started withGovernor Elbridge Gerry’s somewhat salamander-shaped district in Massa-chusetts, 1812, which came to be called Gerry’s Mander. Many districts incontemporary USA look like salamanders that have gone through a meatgrinder, with almost disconnected pieces scattered around—examples areshown in Rush and Engstrom (2001: 5–13). Gerrymander is a specificproblem of single-seat districts. It is hard to carry out with low-magnitudemulti-seat districts and impossible when the magnitude is large.

Bipartisan gerrymander is a development of recent decades in the USA.The two major parties agree to divide the state into districts safe for eitherof them. Effectively, voters no longer choose the assembly members butassembly members choose their voters. The only election with a meaning-ful choice is the locally dominant party’s primary. As a result, nearly allincumbents who run are reelected. Furthermore, appealing to the centerof the general public (the median voter) no longer is a winning strategy.One has to appeal to the median voter of the locally dominant party.This way, bipartisan gerrymander may have contributed to an increasein ideological polarization of the US House, although contrary evidence(Brunell 2006) also exists.

Which Electoral Systems are used the Most?

Lijphart (1994) analyzed 70 different electoral systems, used in 27 moststable democracies, from 1945 to 1990. The countries range from devel-oped countries to Costa Rica and India. Also included were elections to theEuropean Parliament (EP). A change in electoral system sometimes meant

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Table 3.6. Established democracies 1945–90

Systems Elections

Basic (plus legal thresholds and unequal district magnitudes)First-past-the-post 7 78M = 1, alternative vote 3 19M = 1, majority–plurality 2 8List PR, d’Hondt formula 21 94List PR, Hare-LR formula 5 38List PR, modified Sainte-Laguë 2 15Single transferable vote 4 27Single nontransferable vote 2 18

CompositePR, with two-tier districting 7 31PR, two tiers and adjustment seats 13 68PR, four-tier districting 3 4Mixed PR-majority 1 2

TOTAL 70 402

Note: Number of electoral systems and the total number of elections in which they were used.

Source: Tables in Lijphart (1994: 16–47).

a thorough reversal (e.g. from FPTP to large-magnitude PR), but oftenit was a relatively minor shift within the same basic framework. Somesystems lasted only one or two elections, before they were changed, whilethe USA with its frequent House elections went through 23 elections. Thefrequency distribution by type of electoral rule is shown in Table 3.6. Insum, 26 percent of the elections used single-seat districts, mostly FPTP,while 37 percent used List PR, mostly d’Hondt. Only 11 percent used STVor SNTV, and 26 percent used composite systems, mostly combining two-tier districting and adjustment seats.

From the 1950s to the 1990s, Golder (2005) records a vast expansion ofdemocratic elections, and the relative shares of various systems fluctuate.By and far, the proportion of majoritarian systems has been steady around36 percent, and there has been a shift away from pure PR (down from41 percent in the 1950s to 28 percent in the 1990s) in favor of mixedsystems (up from 14 to 22 percent).

In the early 2000s, Reynolds, Reilly, and Ellis (2005: 30) counted 68established democracies in sovereign or autonomous areas of the world.Their electoral systems were distributed as follows: Single-seat districts39.7 percent, mostly FPTP; List PR 30.9 percent; STV or SNTV 5.8 percent;Block Vote 11.8 percent; various composites 11.8 percent. Block Vote owesits rather large share to small British islands such as Guernsey, Jersey, andMan. If one goes by population shares, India and the USA bring the FPTP

44


share to 70 percent. On the other hand, 61 percent of the new democra-cies of the recent decades have chosen List PR. When existing systems arechanged, the trend also is away from plurality/majority systems towardPR and mixed systems (Soudriette and Ellis 2006).

What determines the choice of electoral systems? It makes eminentsense to consider the constellation of political forces at the moment ofdecision. It is in the interest of a two-party founding body to chooseFPTP and of a multiparty body to choose PR. However, colonial historylooks like an even more powerful determinant. Of all countries that haveelection rules, 36 percent have some British heritage, but this share shootsup to 87 percent for the FPTP countries and drops to 6 percent for theList PR countries. Also, only 13 percent of all countries have some Frenchheritage, but 45 percent of the Two-Rounds countries do.

Many other countries used FPTP or Two-Rounds in the past, but mostof those with no British/French heritage have later switched to List PRor mixed systems. Strikingly, while colonialism has receded into moredistant past, its impact on electoral systems has become more marked.

The details of this Franco-British factor are shown in Table 3.7, basedon Reynolds, Reilly, and Ellis (2005: 32 and 166–73). Many of the 199countries and territories listed have poor or nonexistent democratic cre-dentials. Authoritarian regimes can subvert any electoral laws for theirnoncompetitive fake elections, but they disproportionately tend to preferTwo-Rounds.

Table 3.7. Electoral systems and British–French heritage

Electoralsystem

Totalcases

Britishheritage

Frenchheritage

Othercases

Britishshare(%)

Frenchshare(%)

Other’sshare(%)

Total 199 72 26 101 36 13 51

British heritage favoritesFPTP 47 41 2 4 87 4 9AV 3 3 0 0 100 0 0STV 2 2 0 0 100 0 0LV & BC 2 2 0 0 100 0 0BV & PBV 19 11 4 4 58 21 21SNTV 4 2 0 2 50 0 50

French heritage favoriteTR 22 2 10 10 9 45 45

Favorites of the rest of the worldList PR 70 4 6 60 6 9 86Parallel 21 3 4 14 14 19 67MMP 9 2 0 7 22 0 78

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The prevalence of FPTP in the British-heritage countries, from St Kittsto India, is striking, as is its current absence in the rest of the world.Alternate Vote, STV, LV, and modified BC occur in very small numbers,but all those cases happen to be in the British-heritage one-third of allthe countries. To a lesser degree, this also applies to Block Vote, PBV,and SNTV. As noted, Two-Rounds occurs with relatively high frequencyin French-heritage countries. The rest of the world uses preponderantlyList PR, with an increasing sprinkle of parallel systems and MMP.

Conclusions

The central purpose of this book is to elucidate regularities in the impactof electoral systems on party systems. This effort has to start with the sim-plest electoral systems. Even some of the electoral systems designated hereas basic surpass as yet our present capabilities for building quantitativelypredictive models. This applies even more to those systems designatedas complex and composite. Pathologies of electoral systems are a sourceof increased random noise. While building logical models of manageablesimplicity, one has to overlook many such complications. At the sametime, we must be aware of the simplifications made, so as not to mistakethe models for the real world. Only then can we use these models forprediction—and know the limits on these predictions.

Many claims made in this chapter and in the preceding ones needlogical proof and empirical testing. For this we need operational tools.What exactly do we mean when talking of a smaller or a larger number ofparties, or of deviation from PR? Some such analytic tools are presentedin the next chapters.

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4

The Number and Balance of Parties


� When some parties have many seats and some have few, we need ameaningful ‘effective’ number of parties, so as to compare the effects ofelectoral systems on party systems.

� The standard way to express the effective number of parties is to convertto fractional seat shares, square them, add, and take the inverse. Thus,for seat shares 50-40-10, the effective number is N = 1/[(0.5 × 0.5) +(0.4 × 0.4) + (0.1 × 0.1)] = 2.38.

� The same can be done with vote shares.� This method is not ideal, but all others are worse.� Effective number can be complemented by a measure of balance in

party sizes.

The number of parties is among the most frequent numbers in politicalanalysis, and it is central to the study of party systems. A party systeminvolves, of course, much more than the mere number of parties, but it isimpossible to describe it without giving some idea of how many playersare involved.

But what is a meaningful number of parties in an assembly, when someparties have many seats and some have few? Also, what is a meaningfulnumber of parties in an election, when some of them obtain many morevotes than some others? Description of electoral systems in Chapter 3 hasincluded hints at how they affect the number of parties. It is time to mea-sure it. The main part of this chapter presents what is needed to measurethe number of parties in an informed way, for various purposes. Chapterappendix addresses various methodological issues that some readers maywish to bypass.

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We begin with the number of legislative parties—those in the assembly.Three ways to measure this number will be introduced, plus a resultingindex of balance, so as to characterize a mix of large and small parties.These measures are then used to classify party systems. Extension toelectoral parties leads to a comparison of the numbers based on votesand seats. Thereafter, the effective number of components is expandedto uses beyond electoral systems. In chapter appendix, the discussionbecomes somewhat more technical, as I justify the choice of indices,indicate relationships among them, and discuss the quest for a numberof relevant parties.

Basic Indices of Number and Balance of Legislative Parties

It often suffices to talk of two-party or two-and-a-half party systems, andso on. But in the face of large, small, and tiny parties, we need a moreprecise measure for cross-country comparisons of institutional effects, andalso for detecting gradual changes within a country. Three approaches tothe number of parties are useful either for practical purposes or for con-struction of predictive models. As a specific example, suppose 100 seatsare distributed among six parties as 48-25-13-9-4-1. How many parties arethere?

The simplest way is to count the seat-winning parties,meaning thosewho are represented in the assembly with at least one seat. I will call thisnumber N0, for reasons of systematics explained in chapter appendix:

N0 = the number of seat-winning parties.

In our example, N0 = 6. This is the largest number of parties that couldpossibly be claimed for this constellation. Its obvious shortcoming is thatwe may not feel the system really has 6 meaningful parties. We shouldtake into account at least the relative size of the largest party. Indeed,the inverse of the largest fractional share represents the smallest numberof parties that could be claimed for a party constellation (see chapterappendix). I will call this number N∞, for reasons of systematics:

N∞ =1s1

= inverse of the largest fractional share.

In our specific example, s1 = 0.48, hence N∞ = 1/0.48 = 2.08. This maylook more realistic than N0 = 6, but we may feel that it underestimatesthe number of parties, given that more than two parties can have someimpact.

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The effective number of parties yields a value intermediary between N0

and N∞. Called N2 for reasons of systematics, it is calculated as

N2 =1

�(si )2= inverse sum of squared fractional shares.

Here si stands for the fractional seat share of ith party. In ourspecific example, N2 = 1/[0.482 + 0.252 + 0.132 + 0.092 + 0.042 + 0.012] =1/0.3196 = 3.13. For some purposes, this party system acts as if it werecomposed of 3 equal-sized parties, plus a minor fourth party. This doesnot imply that the largest three parties matter fully, the fourth only alittle, and the rest not at all—the meaning is more subtle. Introduced intopolitical science by Laakso and Taagepera (1979), the effective numberhas ‘become the most widely used measure of the number of parties’(Lijphart 1994: 70). ‘It is now the standard measure of how concentratedvote shares are in electoral contests’ (Cox 1997: 29).

The formula above requires that all seats be first converted into frac-tional seat shares, but one can bypass this stage. Suppose seats in an80-seat assembly are distributed 45-34-1 (as in New Zealand in 1943).Rather than convert into fractional shares, one can calculate the effectivenumber as N2 = 802/ (452 + 342 + 12) = 2.01. The following formula can beused with the numbers of seats as well as with percentages:

N2 =(�Si )2

�(Si )2.

Here Si is the number of seats for ith party. Note the beautiful symmetryof this expression. The numerator and denominator include exactly thesame symbols, with only the parentheses shifted.

N∞ cannot be larger than N2, and N0 cannot be smaller:

N0 ≥ N2 ≥ N∞.

They are equal when all seat-winning parties have equal shares. For exam-ple, for 25-25-25-25, all three measures yield 4.00.

We use the effective number more often than any other measure.Therefore, I will from now on designate it simply as N, unless otherwisespecified:

N = N2.

The formula for the effective number is ‘operational’ in that it can beapplied mechanically to any constellation of fractional shares. But theformula does not tell us which shares we should feed in. Should the

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German CDU and its Bavarian ally CSU be counted as a single party or twosister parties, given their noncompetition in elections and cooperation ingovernment? In the reverse direction, factions inside the Japanese LiberalDemocratic Party could be seen as ‘parties within the party’ (Reed andBolland 1999). Such judgment calls are up to the researcher, regardless ofhow the number of parties is defined.

Lijphart (1999: 69–74) settles such dilemmas by calculating N in bothways and taking the arithmetic mean. This is a reasonable solution whenthe two horns of the dilemma are fairly close. But Chile presents diffi-culties. There, the effective number of major voting blocks is close to 2.However, the effective number of mutually competing parties within theblocks is many times higher. Both numbers make sense, in different ways,but their mean might not make any sense.

For a given effective number of parties, the actual shares of parties maybe quite equal or highly unequal. The following index of balance may be asuitable measure of such variation (Taagepera 2005):

B =logN∞logN0

=−logs1

logN0.

This index takes the ratio of the logarithms of the minimal (N∞) andmaximal (N0) estimates of the number of parties. It is also the logarithmof the largest fractional share divided by the logarithm of the number ofseat-winning parties, with a negative sign.

The index of balance can range from nearly 0 to 1. It is 1 for full balance,meaning that all seat-winning parties have equal shares. It approaches0 for utter imbalance. Most constellations have a balance around 0.5.Our sample constellation 48-25-13-9-4-1 yields B = 0.41, meaning thatit is slightly less balanced than most constellations. Among the variousconstellations that all lead to N = 3.00, 34-33-33 has a high balance ofB = 0.98, while a low balance of B = 0.18 is reached with 57-1 plus 21parties at 2 percent. Odd as the definition of B may look, the nextsection shows that it leads to some agreement with intuitive notionsabout balance.

Mapping Party Systems, Using Number and Balance of Parties

Characterizing the types of party systems has concerned students of partypolitics for a long time—see the excellent overview by Steven Wolinetz(2006). A long strand of researchers, extending from Jean Blondel (1968)

50


to Allan Siaroff (2000), has considered the number of parties and therelative balance among them the basic criteria for classifying party sys-tems. Of course, several other criteria also enter. The nature of alternationamong ruling parties must be considered (Mair 1997). Alternation may benil (permanent hegemonic party), partial (when some coalition partnersare replaced), or wholesale (ruling party or coalition fully replaced). Thedegree of ideological polarization can be mild or strong (Sartori 1976).Patterns of opposition can be competitive, cooperative, or coalescent(Dahl 1966). Still, with few exceptions (Gunther and Diamond 2003),these specifications complement the number and balance of parties ratherthan replacing them. In the words of Wolinetz (2006: 60): ‘Relationshipsdepend on numbers.’

Blondel (1968) had to depend on impressionistic estimates of the num-ber of parties rather than an operational measure. Siaroff (2000) madeuse of the effective number of parties but had to depend on quali-tative estimates of balance. With the benefit of the index of balance,we can now operationalize both measures, as exemplified in Figure 4.1.Here balance is graphed against the effective number in 25 stable partysystems, from 1985 to 1996. The average indicators for two to fourelections are shown, as derived from data in Mackie and Rose (1997).Cases with and without absolute majorities are shown with differentsymbols.

Approximately in line with Siaroff’s typology (2000), Figure 4.1 dis-tinguishes regions that correspond to two-party, two-and-a-half to three-party, multiparty, and highly multiparty systems. The line B = 0.50 offersa convenient separation line between relatively balanced and unbalancedsystems, with about half the cases falling on either side of the line.Balance is a matter of degree, though. In many ways, Denmark is closerto the Netherlands, across the B = 0.5 line, than to Italy, in the sameconventional region.

Extreme lack of balance is rare, although it could occur for any effectivenumber. Also rare is near-perfect balance, of which Malta is the only casein Figure 4.1. If a third party had won even a single seat, Malta’s indexof balance would have tumbled to around 0.65, close to the USA. Givensuch a possibility, balance may look very unstable, but this is not so. Overtwo to four elections in 1985–96, it is observed to remain stable within± 0.03 units, while the effective number at times changes by more than1 unit. As the number of parties increases, balance tends to be restrictedto an ever narrowing zone around B = 0.5. Deviations from this mean arepossible but rare.

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Rules and Tools

AULNZ

USA

MAL

GREPORCAN

JPN

UK

AUT

BEL

SWI

ITA

SPA

IRE

FRA

GER

SWENORISR DEN

NET

FINLUX

ICE

One-party

Two-party

Three-party

Multi-party

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

Effective number of legislative parties

Bal

ance

s1 > 0.5

s1 < 0.5

B = −log2/log(1−N/4)

Con

cept

ually

forb

idde

n ar

ea

BALANCED

UNBALANCED

Highlymulti-party

2 3 4 5 6 7 8

Figure 4.1. Balance vs. effective number of legislative parties in 25 countries,1985–1996

At the left and top of the graph, conceptually forbidden areas areshown. Indeed, for less than two effective parties, high balance becomesimpossible. Also, very high balance is impossible for most values of theeffective number of parties, given that the definition of B involves N0,which comes in integer numbers. An empty one-party zone at N < 1.7and high imbalance is shown in the graph. Among the stable democraciesby Lijphart’s criteria (1999), Botswana would be located there. African

52


party systems in general tend to have low balance, with a few dominantlarge parties surrounded by numerous small ones (Mozaffar, Scarritt, andGalaich 2003).

Sister parties present problems in Germany and Australia. Should theybe counted together or separately? Both options are shown in Figure 4.1and are seen to matter relatively little for the German CDU and CSU. Forthe National and Liberal Parties in Australia, however, the way to countthem makes a difference. Such judgment calls cannot be avoided.

Spain and Japan illustrate the limits of a classification of party sys-tems on the basis of number and balance alone. Both countries havelarge major parties plus many minor ones, meaning low balance. Theyare located close to each other in Figure 4.1. Their alternation patterns,however, differ. In Japan, Liberal Democratic Party continued being thepredominant party, while in Spain People’s Party relieved the Socialistsas the largest party. A quantitative measure of alternation remains tobe worked out. It would represent a third dimension, orthogonal to Nand B.

For given N and B, can we tell whether the largest party has absolutemajority? It can be shown that majority never prevails above the dashedcurve B = −log2/log(1 − N/4), shown in Figure 4.1. Below this curve,majority becomes increasingly likely as balance decreases, but it neverbecomes a certainty at N larger than 2. Even when N is as low as 2.2 andB is as low as 0.20, this combination still could arise from 2 parties barelyshort of 50 percent plus a smattering of parties with 1 seat each—unlikelyas such a combination would be. Note, however, that a party can remainpredominant even while not enjoying absolute majority at the moment.Hence the characterization of party system as such need not depend onabsolute majority. In Japan, majority materialized in 1986 and 1990, butnot in 1993. It certainly affected the moment’s politics but not the broadframework.

Legislative Versus Electoral Parties

The effective number could also be calculated on the basis of vote sharesof parties:

NV =1

�(vi )2.

Here vi stands for the fractional vote share of the ith party. To distinguishbetween the number of legislative and electoral parties, we can use the

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Rules and Tools

N s=

N v

N s=

N v− 0.4

N s=

N v− 0.

8

SPA 1977−89

IND 1962−84

FRA 1958−81

0

1

2

3

4

5

6

0Nv

Ns

PR

M = 1

1 2 3 4 5 6

Figure 4.2. Effective number of legislative parties vs. effective number of electoralparties

notation NS for seats and NV for votes when the need arises. These num-bers are sometimes referred to as effective number of parliamentary parties(ENPP) and effective number of electoral parties (ENEP), respectively, butsingle symbols with subscripts are more in line with scientific notation.Indeed, ENP risks to be mistaken for multiplication of the quantities E ,N, and P .

The effective number based on votes (NV) almost always exceeds theone based on seats (NS), although exceptions occur. Figure 4.2 showsthe average NS over many elections graphed against the average NV. Itincludes those 37 electoral systems distinguished by Lijphart (1994: 17–47 and 160–2) that involved at least 3 national elections.1 The effectivenumber of parties is rarely much less than 2.0—overwhelming dominance

1 The effective numbers data are given in Lijphart (1994:160-2), labeled ENEP and ENPP.The number of elections is listed in other tables (Lijphart 1994:17-47). I have omitted theelectoral systems that were discarded after one or two elections, because the first electionafter a change of system is atypical. Also omitted were elections to the European Parliament,because the perceived weakness of this body affects voting behavior. When the electoraloutcome is considered unimportant, voters are more willing to waste their vote on minor

54


by one party is unusual in stable democracies. The effective numberrarely exceeds 5.5 for electoral parties or 5.0 for legislative parties. Mostdifferences between the two range from 0 to 0.8, and the median is 0.4(Taagepera and Shugart 1989: 84). Hence, for most electoral systems,

NS = NV − 0.4 ± 0.4.

This relationship is purely empirical. It obviously could not apply whenNV is less than 1.4, as NS would fall below 1. The gap exceeds 0.8 mainlyin early, unsettled electoral systems, where more parties run than canrealistically expect to win seats under the given electoral laws. Only India1962–84, France 1958–81, and Spain 1977–89 have a persistent gap ofmore than 0.8 between NV and NS. The latter two countries are amongthose where malapportionment in favor of rural areas has been heavy(Lijphart 1994: 128).

Effective Number of Components: Seats, Votes, Polities, andPower Shares

All that has been said about seat shares also applies to shares of votes,except that it would be difficult to specify N0, the number of parties thatreceive at least one vote. Actually, we would be more interested in thenumber of ‘serious’ parties that run, but this number is hard to define (asis discussed in Chapter 15). The task becomes even more difficult whennumerous independents run, some of them seriously. In some ways, eachindependent acts as a separate party, yet they lack many characteristics ofparties.

The effective number of components other than parties can be usefuloutside the realm of electoral and party studies. This approach applieswhenever well-defined components add up to a well-defined total. Onecan measure the effective number of polities in the world, based on theirareas or populations (Taagepera 1997a). Over the last 5,000 years, theirnumber has fallen from close to a million to around 20. One could alsomeasure the effective number of car manufacturers in a country or theeffective number of ethnic groups or religions—provided one can agreeon which groups represent different ethnies or religions. The numberobviously differs depending on whether one enters the fractional shares

parties that fail to win seats. As a result, the gap between vote- and seat-based effectivenumbers of parties is often much wider in Euroelections than in national elections.

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of Muslims and Christians as such, or whether one enters the Sunni, Shia,Catholic, Orthodox, and Protestant shares separately.

In addition to seat or vote shares, one could also consider the effec-tive number of parties based on shares of power, using for instance thestandardized Banzhaf indices (Dumont and Caulier 2006). This may bringus closer to operationalizing the measurement of Sartori’s ‘relevant par-ties’ (1976), to be discussed in the chapter appendix. The power indexapproach presumes that minimal winning coalitions are the norm for gov-ernment formation. The actual distribution of cabinets in stable democ-racies, however, shows an appreciable incidence of minority cabinets andoversized (i.e. larger than minimal winning) coalitions (Lijphart 1999: 98):

Minimal winning, one-party 37%Minimal winning coalitions 25%Minority cabinets 17%Oversized coalitions 21%.

Nonetheless, the power index approach is well worth further investiga-tion. The choice depends on the goal. For the purposes of momentarygovernmental power, a constellation 53-47 means that the larger partyhas all the governmental power, as reflected by the power-based effectivenumber, NP = 1.00, rather than by the seats-based effective number, NS =1.99. But the smaller party may well become the dominant one, comenext elections. Here the seats-based NS would reflect better the nature ofthe party system, compared to power-based NP, which remains at 1.00 afterpower transition to the other party.

Conclusion

This completes the overview of how to apply the effective number andbalance of parties (or other components) in various ways. What fol-lows in the chapter appendix may not be needed to make use of thesemeasures but matters from the methodological viewpoint. Several weakaspects of the effective number of parties, defined as N = 1/�(si )2, havebeen pointed out by Taagepera and Shugart (1989: 259), Molinar (1991),Taagepera (1997b, 1999a), and Dunleavy and Boucek (2003), as is dis-cussed in appendix. Still, for most purposes the effective number remainspreferable to any of the alternatives offered. The only way to improve on itsignificantly would be to supplement it with a second indicator. It couldbe the index of balance B = logN∞/logN0), or the inverse of the largest

56


share, or Sartori’s number of relevant parties (1976) (discussed in chapterappendix).

The fractional shares that enter the equation for N can be shares ofvotes, seats, power, or something else, depending on the purpose. In thisbook, N without a subscript stands for the effective number of partiesbased on seat shares. Mapping the party systems on the basis of theeffective number and balance of legislative parties leads to a picture closeto some previous intuitive classifications.

Appendix to Chapter 4

What options do we have for measuring the number of parties in the first place,and what are their limitations? How do the various measures interrelate? Are theyinherently different, or are they different variants of the same master equation?Could the number and balance of parties be combined into a single super-index?Last but not the least, could we feed in something else, apart from the sheer size ofparties, in a quest for a number of relevant parties? It may look at times that I amgoing very slow and making a fuss about the obvious, but the superficially obvioushas sometimes unexpected implications.

How to measure the number of parties

First, let us consider some properties of the three standard ways to measure thenumber of parties. The number of seat-winning parties (N0) looks simple, but thesame value N0 = 6 could represent 20-20-20-20-10-10 or 90-2-2-2-2-2. Moreover,in the presence of numerous tiny parties and independents, many data collectionsmay lump as many as 20 percent of the seats under the label ‘Others’. When 9 seatsare reported as ‘Others’, it could mean 9 seats for a single ephemeral party or oneseat each for 9 parties or independents. In the absence of any other knowledge,the square root of the seats for ‘Others’ might be added to the explicitly namedseat-winning parties.

Apart from such uncertainty, N0 is the largest number of parties that conceivablycould be claimed for the given constellation. At the other extreme, the inverseof the largest fractional share (N∞ = 1/s1) is the smallest number that couldbe claimed, for the following reason. Consider 25-25-25-25. Clearly, there are1/0.25 = 4 parties. Suppose now that we are given 25- . . . - . . . and are told that noneof the unknown shares surpasses 25 percent. The smallest number of parties wecould possibly guess at is 4. If we guessed 3, at least one of the other shares wouldhave to surpass 25 percent.

Extending this observation to any largest fractional seat shares usually leadsto inverses that are not integers. Consider a largest share of 30 percent. Then

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N∞ = 1/0.30 = 3.33. This means 1+1+1+1/3. The corresponding constellation withthe largest shares, and hence arguably with the fewest possible number of parties, is30-30-30-10, where 10 is indeed one-third of the large shares. It can be shown thatsuch an interpretation of the fractional part of N∞ always applies. The nice aspectof such a definition of the number of parties is that, whenever N∞ is less than 2, weknow that one party has absolute majority. The complementary weakness of N∞ isthat it is not sensitive at all to the distribution of shares beyond the first, so thatN∞ = 2.08 could mean a balanced 48-48-4 or an unbalanced 48-10-10-10-10-10-2.Note that N∞ is not affected by the ‘Others’ category. In this sense it is a moreoperational measure than N0—less is left to the judgment of the person doing thecalculation.

In the presence of large and small parties, the effective number of parties,N = N2 = 1/�(si )2, is an intermediary measure between N0 and N∞. It saves us anarbitrary decision on which parties matter, by applying self-weighting. This meansthat, for seat shares 50-40-10, each fractional share is given a weight proportionalto its size:

0.50 × 0.50 = 0.250.40 × 0.40 = 0.160.10 × 0.10 = 0.01,

The third party contributes very little to the resulting sum of 0.42. We do nothave to exclude it by setting some arbitrary threshold—the small party effectivelyeliminates itself, through self-weighting. The effective number N = 1/0.42 = 2.38indicates that for many purposes (but not all!) the party system acts as if it werecomposed of 2 equal-sized parties, plus a smaller party. The constellation thatfits that description exactly and leads to N = 2.38 is 45.3-45.3-9.4. The algebraicconnection between the fractional part (0.38) of N and the seat share of theresidual party (0.094) exists but is quite complex. For our previous example 48-25-13-9-4-1, the simple equivalent with the same value of N = 3.13 would be 32.6-32.6-32.6-2.2, meaning 4 seat-winning parties rather than the actual 6.

The minimal number of parties needed to form a majority coalition can usuallybe estimated from N as follows. Take one-half of the effective number of partiesand round it off to the closest integer (Taagepera 2002a). In 50-40-10, we have2.38/2=1.19, which rounds off to 1. For 48-25-13-9-4-1, one-half of the effectivenumber (3.13/2 = 1.57) rounds off to 2, but it does so only narrowly, reflecting thefact that just a small increase in the largest party share would enable it to forma single party majority cabinet. This visualization of N fails sometimes, but suchcases are rare in practice.

The effective number of parties conveys some intuitive meaning as long as allparties are roughly of the same size. But consider the constellation 53-15-10-10-10-2, where N = 3.00. In what sense are there 3 effective parties, rather than 1 or5? The visualization of at least N/2 parties being needed for majority coalition also

58


fails here. Indeed, 3.00/2 = 1.50 technically rounds off to 2, while actually oneparty alone has absolute majority.

This leads us to a shortcoming of N: It does not always tell us whether one partyhas an absolute majority. When N < 2, the largest share is bound to be more than50 percent. When N exceeds 4, the largest share is bound to be less than 50 percent.But when 2 < N < 4, we do not know—and this is the range where the effectivenumber most often falls (cf. Figure 4.2). From this viewpoint, we are better servedby N∞ = 1/s1, but at the cost of not distinguishing between cases such as 53-47,where the next election might reverse the power relationship, and 53-15-10-10-10-2, where the largest party is likely to remain dominant even if loses majority.Both have N∞ = 1.89.

In contrast to N∞, the effective number N is sensitive to the distribution ofshares beyond the first. The constellation 48-48-2 has N = 2.17, while 48-10-10-10-10-10-2 has N = 3.56, although both have N∞ = 2.08. But this also means thata residual category of ‘Others’ can affect N. Suppose we have 48-25-13-4-1, plus9 seats for ‘Others’. If these 9 seats go to a single ephemeral party, then N = 3.13.If they go to 9 separate parties or independent candidates, then N = 3.20. In theabsence of any further information, one might take the mean of these extremes,leading to N = 3.16. It can be seen that the possible error is small, unless ‘Others’include a large share of the seats. In that case, more refined approaches exist (seeTaagepera 1997b).

Approximate relationships between the three measuresof the number of parties

This section presents approximate relationships that prevail between N0, N, andN∞, on the average. Logical proof and empirical evidence for these relationships aregiven in later chapters. If we only knew the largest seat share and hence N∞, ourbest guess for the effective number would be N∞ with exponent 4/3 = 1.33:

N ≈(

1s1

)4/3

= N4/3∞ .

Our best guess for the number of seat-winning parties would be the square of N∞:

N0 ≈(

1s1

)2

= N2∞.

Do not expect exact agreement in individual cases! Thus, for 48-25-13-9-4-1, weactually have N = 3.13, but N4/3

∞ would lead to N ≈ (1/0.48)4/3 = 2.084/3 = 2.66.Also, since 2.082 = 4.3, we would guess that 4 or 5 parties would win seats, whenthe actual number is 6. The point is that, even in the absence of any otherinformation but the largest share, we can still estimate the effective number ofparties and the number of seat-winning parties, although with an appreciable

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Rules and Tools

error. Even with such limited information, we can do better than saying ‘We donot know’. This book makes systematic use of such an ‘ignorance-based’ approach(Taagepera 1999b) to develop useful predictive models.

Conversely, if we only knew that N0 parties won seats, our best guess for theeffective number would be

N ≈ N2/30 ,

and for N∞ it would be the square root of N0:

N∞ ≈ N1/20 .

Thus, for 6 parties winning seats, our best guesses would be 62/3 ≈ 3.30 for theeffective number of parties, instead of 3.13 for our specific example. We wouldalso guess at 61/2 ≈ 2.45 for N∞, which leads to 1/2.45 = 0.41 = 41% for the largestshare, instead of the actual 48 percent.

If we knew both the number of seat-winning parties and the largest seat share,our best guess for the effective number would be the geometric mean of thetwo separate guesses. In our specific example, (2.66 × 3.30)1/2 = 2.96. Comparedto either of the separate guesses, this is closer to the actual 3.13, as one wouldexpect: The more information we have, the better our guess is likely to become.

Do not use these coarse approximations, if you have more detailed information on thedistribution of seats! But also have the courage to make the most out of incompleteinformation, rather than say, ‘I don’t know’. This is one of the most valuablelessons I learned from my Ph.D. work in physics. Indeed, nuclear physicist EnricoFermi (after whom element Fermium is named) reputedly challenged his studentswith a social-sciency question: ‘How many piano tuners are there in the City ofNew York?’ The idea was to show how much knowledge could be deduced fromapparently total ignorance, so as to guide one’s research into the right ballpark.Indeed, with minimal coaching, most of my undergraduates get the number ofpiano tuners within a factor of 2 of the census figure.

Systematics of the number of parties

The expressions for N0, N2, and N∞ all derive from the same master equation(Laakso and Taagepera 1979):

Na = [�(si )a]1/(1−a),

where the parameter a can range from 0 to ∞. The larger the value of a, the moreheavily the largest share weighs in, compared to the smaller shares. When a = 0, allseat-winning parties weigh in at equal weight of 1, so that the formula yields thetotal number of parties, N0. When a tends to ∞, only the largest share matters, andthe formula yields the inverse of this share, N∞. As a is increased, the value of Na

60


decreases. When a = 2, the effective number N2 = 1/�(si )2 results. Obviously, nomeasure of the number of parties that exceeds N0 would make sense. The masterequation also suggests that no measure that falls short of N∞ = 1/s1 would makesense.

The master equation is undefined at a = 1, but its limit as a tends toward 1 iswell defined. N1 is the inverse of the product of all fractional shares, raised to theexponents equal to themselves:

N1 =1∏ssii

.

It can also be expressed as

N1 = eH,

where H is the system entropy, as defined in physics:

H = −�si lnsi .

While physicists use natural logarithms in the definition of entropy, informa-tion scientists prefer logarithms to the base 2. It does not matter; either way,N1 = 1/

∏ssii .

The connection to entropy is remarkable and possibly surprising. This mea-sure of the number of parties has been occasionally used ever since 1960 (seeTaagepera and Shugart 1989: 260). It has great philosophical appeal, in view of theimportance of entropy as a unifying concept that extends from thermodynamicsto information science. Unfortunately, it has one practical drawback. Its values,located in-between those of N0 and N2, are overly sensitive to the presence oftiny parties and independents, which data sources tend to group under the label‘Others’. Thus, it may not be even possible to calculate N1 with sufficient precision,and it exaggerates the number of parties, compared to our intuitive notions. Forthese reasons, N2=1/�(si )2 has proved preferable, as a measure of the number ofparties. It is also slightly easier to calculate N2, and some desirable consequencesof using N2 will emerge in later chapters.

The quest for a single super-index to characterize party systems

It would be nice to have a single number that all alone would express all thereis to characterize about a party system. Unfortunately, no single measure of thenumber of parties can be satisfactory. The hard fact is that no single number canindicate both the central tendency and the variation around it. What does thisabstract statement mean?

Consider a normal distribution with a mean of 1,000 units. If the individualvalues range around this mean with a standard deviation of only 1 unit, then

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the mean pretty much tells the entire story. But if the standard deviation is 100units, then this standard deviation needs to be specified so as to warn us aboutthe appreciable scatter. Similarly, the effective number is a measure of centraltendency. It alone will do, when all parties are of roughly equal size. But if theyare widely out of balance, then the effective number cannot tell us so and needs acomplementary index to show how balanced the party sizes are.

In particular, an effective number between 2 and 4 does not tell us with certaintywhether one party has absolute majority. To be on the safe side, one mightsupplement N in such cases with N∞ (Taagepera 1999a). This approach was usedby Siaroff (2003) to investigate two-and-a-half-party systems. But in most casesthe second number conveys little extra information, given the heavy colinearityof N and N∞ (remember the approximation N2 ≈ N4/3

∞ !). This is why the index ofbalance was introduced. True, B still does not clearly tell us whether one party hasmajority, and it is slightly colinear with N, for the following reasons.

Although B ranges in principle from 0 to 1, very low values of B can be reachedonly with a huge number of tiny parties, or with a very large largest party. Forexample, 99-1 is characterized by N = 1.02 and B = 0.01. At the opposite extreme,values of B exceeding 0.5 become impossible when the largest share exceeds21/2/2 = 0.7071. Due to these limitations, some colinearity between B and Nremains—in contrast to 0 colinearity, in principle, between the mean and standarddeviation of a normal distribution. But this colinearity is minimal, compared to theone between N∞ and N.

All the preceding applies to the balance of legislative parties. Estimation of thebalance of electoral parties is made difficult because the number of parties thatobtain at least a minimum of votes is hard to stipulate. I will return to this issue inChapter 15.

No single number can indicate both the central tendency and the variationaround it. Nonetheless, the idea of having a single number to characterize a partysystem is so attractive that the optimistic quest for such a perfect measure haslasted for many decades, and probably will continue. Apart from the entropy-basedmeasure, it has produced two indices that boil down to a mix of N and N∞.

Molinar’s index NP (1991) can be shown (Taagepera 1999a) to amount toNP = 1 + N − (N/N∞)2. For 50-50, we have of course N = N∞ = NP = 2.00. If oneof these parties breaks up, leading to 50-25-25, then N∞ remains the same, N goesup to 2.67, but NP actually goes down to 1.89. The latter shift may well reflect theincreased degree of predominance of the largest party and the resulting imbalance,but hardly the number of parties as such. While avoiding some problems of N,index NP introduces several others (see Taagepera 1999a).

Dunleavy and Boucek (2003) propose an index Nb that amounts to averaging Nand N∞. This average, Nb = (N + N∞)/2, involves, of course, all the same problemsthese authors rightly criticize in N, attenuated by a half. Still, given that N hasproblems that N∞ does not have, and vice versa, their mean might conceivablyalleviate both sets of problems to a sufficient degree. Unfortunately, this is not

62


always the case. A major shortcoming of N is that, in the range 2 < N < 4, itgives no hint on whether the largest party has absolute majority. Within thereduced range 2 < Nb < 3, Nb preserves this indeterminacy. For instance, for theconstellation 52-8-8-8-8-8-8, where N2 = 3.24 and N∞ = 1.92, we get Nb = 2.58,which, like N, gives no hint about the absolute majority of the largest party.

To repeat a statement that the proponents of Nb found hard to accept: ‘Onemay harbor the illusion that by judicious combination of N and N∞ (plus possiblysomething else) one might achieve a super-index that satisfies all desiderata. Thisis about as wishful as hoping to combine the mean and the standard deviation intoa single measure. Two numbers are inherently able to transmit more informationthan one.’ (Taagepera 1999a)

The number of relevant parties

All the approaches discussed up to now are mechanical in the sense that they donot take into account anything but the number of seats parties have. This is thebest one can do in the absence of any other knowledge. Here, we face both the strengthand the weakness of such a mechanical approach. Its strength lies in enablingus to compare a large number of cases, across countries and within countries atdifferent times, without the need to dig up huge amounts of detailed informationand then risk getting lost in the multiplicity of considerations. The weakness of themechanical approach lies in not making use of extra knowledge that would colorthe picture. After all, some small centrist parties matter more than some extremistlarger parties, for coalition formation and other purposes.

Giovanni Sartori (1976) proposed a very different approach to the number ofparties: the number of relevant parties, designated here as R. His basic criterion forrelevance is whether a party has entered into governing coalitions. Some largeparties, such as the Italian Communist Party, have at times been perenniallyexcluded from coalitions on grounds of ideological incompatibility. Yet they arerelevant, because their exclusion makes formation of majority cabinets difficult,and minority cabinet survival depends on the tacit support by such excludedparties. Therefore, Sartori also includes in his count of relevant parties thosecapable of ‘blackmail’.

Sartori counts the relevant parties, without assigning different weights to highlyor only marginally relevant ones. Expert opinions may differ on whether toinclude a party with marginal coalition or blackmail ability. This means thatthe number of relevant parties (R) is less ‘operational’ than the effective numberof parties (N). ‘Operational’ means that any one who carries out the prescribedoperation (such as N = 1/�s2

i ) obtains the same result. Note that I do not claimthat even N is fully operational, because opinions may disagree on whether toinclude affiliated parties such as the German CDU and CSU jointly or separately.

Excluding the blackmail parties, Sartori (1976: 300–3) also offers a measure ofthe number of government-relevant parties alone, designated here as G. In a private

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communication to Sartori (January 27, 2000), I used the data in his Parties andParty Systems (1976) to try to establish some average relationships between R, G,and the effective numberN. These relationships depend on several of assumptionson the use of which Sartori has not corrected me.

The central pattern for the relationship between Sartori’s numbers ofgovernment-relevant parties (G) and all relevant parties (R) is around R = G + 1.5,meaning that most often there are 1 to 2 relevant parties that are rarely included ingovernment support. The opposite extremes are Austria’s classical grand coalitions(R = G) and Norway and Italy (R = G + 3). The latter two also differ from eachother. Norway truly had three relevant center-right parties in addition to the usualone-party social-democrat cabinet. In Italy, one large blackmail party indirectlymade more parties relevant for government support in the following way. When 30percent of seats are excluded from the game, majority cabinet formation requires50 of the remaining 70, rather than 50 of 100, making even minor parties desirablepartners. This accounts for the large total number of relevant parties Sartori findsin Italy.

For the relationship between Sartori’s number of government-relevant partiesand the effective number of parties (N), the central pattern is G = N − 1, meaningvery roughly that one of the N largest parties is most often outside the cabinet.However, in a number of cases G = N, meaning roughly that either all large partiesare included (Austria) or large parties left out of the cabinet are balanced by rathersmall parties being included (West Germany, Italy, and most of all, France IV).At the other extreme, in Norway, Sweden, and Canada G falls short of (N − 1),reflecting unusually marked size disparity between the largest party and the 2 or3 next-largest ones. Such disparity would imply a low value of B, the index ofbalance.

Finally, the relationship between Sartori’s total number of relevant parties andthe effective number of parties varies. While R and N agree in two-party constella-tions, R is larger than N for many multiparty constellations. The gap increases asthe number of parties increases. Among the multiparty systems, Switzerland andthe Netherlands are closest to R = N. At the other extreme, R exceeds N most inItaly, with Norway and France IV and V coming next. Italy and Norway are ofcourse the outliers regarding R = G + 1.5 too. This means that in these countries Ris out of whack not only with N but also with Sartori’s own number of government-relevant parties.

In sum, these various ways to express the number of parties (N, R, and G) partlytell the same story, while partly illuminating different aspects of party systems.This book prefers to use N, because it is highly operational and is found to beconnected to institutional factors. For cases that deviate from the general patternsthus predicted, it might be worthwhile to consider by how much the number ofrelevant parties differs from the effective number.

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5

Deviation from ProportionalRepresentation and ProportionalityProfiles


� Excessive deviation from proportional representation may hurt thedemocratic legitimacy of the regime.

� Two alternatives predominate for measuring the deviation from PR.� The simplest is to add all the differences between seat and vote shares

of each party, then divide by 2.� This measure (D1) ranges from 2 to 7 percent for most of the stable PR

systems, and from 6 to 24 percent for most of the first-past-the-postsystems.

� By a fancier measure (D2), these ranges shift to 1 to 5 percent and 5 to17 percent, respectively.

� Malapportionment of districts can increase the deviation from PRappreciably.

� Proportionality profiles are a way to show which parties are advantagedand which are disadvantaged.

� Volatility of votes from one election to the next can be measured thesame way as the deviation from PR.

This chapter deals with measurement of deviation from PR, meaning thedeviation of seat shares of parties from their vote shares. One can considerthe overall deviation for all parties and characterize it by a number. Onecan also consider each party separately and graph the resulting propor-tionality profiles of parties.

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Rules and Tools

Two other features have the same mathematical format as deviationfrom PR: volatility of votes from one election to the next, and the extentof ticket splitting, when voters have more than one ballot. All thesephenomena deal with measuring deviation from a norm. Deviation fromPR is 0 when seat shares equal vote shares, which represent the norm.Volatility is 0 when vote shares in the next election equal those in theprevious one, which is taken as the norm. The extent of ticket splittingis 0 when all voters vote exactly for the same parties with all their ballots.Here, any ballot could be taken as the norm for the others.

Basic Indices of Deviation from PR and Volatility

Two ways to measure deviation from PR, volatility, and ticket splittinghave dominated. Both start with the difference between the actual shareand the norm, for each party, but then they process these differences indifferent ways. Loosemore and Hanby (1971) introduced into politicalanalysis the index of deviation that I will designate as D1, for reasonsof systematics explained in chapter appendix. For deviation from PR, it is

D1 =12

�|si − vi |.

Here si is the ith party’s seat share, and vi is its vote share. The index canrange in principle from 0 to 1 (or 100 percent). Note that |si − vi | = |vi − si |is never negative. D1 dominated until Gallagher (1991) introduced what Iwill designate as D2:

D2 =[

12

�(si − vi )2]1/2

.

It has often been designated as the ‘least square’ index, but this is amisnomer. The index does involve squaring a difference but no min-imization procedure so as to find some ‘least’ squares. D2 can rangefrom 0 to 1 (100 percent), but whenever more than two parties havenonzero deviations the upper limit actually remains below 1—an awk-ward feature to be discussed in chapter appendix. When only two par-ties have nonzero deviations, the one gaining what the other is losing,then D1 and D2 have the same value. But when more than two par-ties have nonzero deviations, then D1 is bound to be larger than D2.In sum,

D1 ≥ D2.

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Deviation from PR

It is possible, though rare, that one of these indices increases while theother decreases, from one election to the next.

A third possible index is simply the largest single difference for anyparty. For reasons of systematics (see chapter appendix), I will designate itas D∞:

D∞ = max|si − vi |.The largest |si − vi | may or may not pertain to the largest party. At timesa third party loses more than the largest party wins, because some ofthe gain goes to the second-largest party. Along with D1 andD2, Lijphart(1994: 62) proposed D∞ for consideration. He labeled these three indices,D, LSq, and LD, respectively and published their average values forthe aforementioned electoral systems in various countries and periods(Lijphart 1994: 160–2). In most cases, the values of D2 and D∞ are fairlyclose:

D2 ≈ D∞.

D2 tends to fall slightly short of the largest single seat–vote difference butcan exceed it occasionally. It is always the case that

D1 ≥ D∞.

For volatility, the corresponding equations are

V1 =12

�|v1i − v0i |,

V2 =[

12

�(v1i − v0i )2]1/2

,

V∞ = max |v1i − v0i |.Here the subscripts 1 and 0 refer to two elections. V1 is often called thePedersen index (Pedersen 1979), though Przeworski (1975) foreshadowedit, under the name disinstitutionalization.

In the case of deviation from PR, Gallagher’s D2 rapidly displaced D1

during the 1990s as the most popular index. Discussion of concepts inchapter appendix suggests that this shift may not have been as justifiedas it looked at the time. Maybe D1 is actually preferable. But goingwith the flow, I will use D2 for applications that follow, despite myreservations.

For volatility, V1 has continued to reign. The same seems to be the casefor ticket splitting. In the following, I will focus mainly on deviation

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Rules and Tools

D 2= 5.5(N v

− 1)

D2 = 15(Nv − 1)/Nv2

FRA 1958−81

JPN 1947−90

SPA 1977−89

ICE 1946−59

0

2

4

6

8

10

12

14

16

18

1

Nv

D2

(%)

PR

M = 1

Malapportioned

Conceptual

2 3 4 5 6

Figure 5.1. Deviation from proportional representation vs. effective number ofelectoral parties

from PR. Chapter appendix will address the general issue of how tomeasure deviation from a norm, how the various indices are related, andthe relative advantages of D1 and D2.

Empirical Patterns of Deviation from PR

One may wonder whether the number of parties that run affects deviationfrom PR. More parties could lead to more deviation from PR, because morevotes might be wasted on small parties that fail to get seats. However,party systems are likely to adjust themselves to electoral systems. Highdeviation from PR would push less successful parties toward giving up,which would reduce deviation. Thus, an informal ‘law of conservationof D’ (Taagepera and Shugart 1989: 123) might prevail, where politicalculture would decide which equilibrium level of deviation is consideredtolerable.

Figure 5.1 shows the empirical picture when Gallagher’s D2 is graphedagainst the effective number of electoral parties (NV). The aforemen-tioned 37 electoral systems studied by Lijphart (1994: 160–2) are used,which involve at least 3 national elections. While the number of partiestends to remain stable or change gradually over long periods, deviationfrom PR can fluctuate wildly from one election to the next (see graphin Taagepera and Shugart 1989: 109). Thus only averages over at least

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Deviation from PR

3 elections are meaningful. Three categories of systems are shown withdistinct symbols: single-seat districts with FPTP or Alternative Vote; multi-seat PR; and some of the systems where Lijphart (1994: 128) observes highmalapportionment in favor of rural areas: France 1958–81, Spain 1977–89,Iceland 1946–59, and Japan 1947–90.

For FPTP and AV, deviations are mainly around 10 percent but sur-pass 16 percent in India. D2 tends to increase steeply with increas-ing number of parties, roughly following a pattern D2 = 5.5%(NV − 1).Note that D2 must be 0 when a single party runs (NV = 1), becauseit is bound to receive all the votes given to any party, as well as allthe seats. This conceptual ‘anchor point’ is shown with a triangle inFigure 5.1, and the approximate trend line is made to pass through it.The importance of such anchor points is discussed in Beyond Regression(Taagepera 2008).

Deviation is less than 5 percent for multi-seat PR systems not subjectto high malapportionment. In contrast to FPTP/AV, deviation tends todecrease slightly with increasing number of parties. The average patternis roughly D2 = 15%(NV − 1)/N2

V.Why use such a curved approximation rather than a simple straight

line? A straight line would predict a deviation of about 5 percent whena single party runs, which does not make any sense. Conceptually,0 percent is required. The linear fit would also predict a negativedeviation at very large numbers of parties. Although such largenumbers might never be reached in actual party systems, it is goodscientific practice to avoid conceptual inconsistencies even in extremesituations. The curve D2 = 15%(NV − 1)/N2

V is the simplest one that passesthrough the anchor point at NV = 1 and also remains above 0 even athigh NV.

Apart from respecting the anchor point at NV = 1, the curves shownin Figure 5.1 are empirical. D2 tends to range from 1 to 5 percent forstable PR systems, and from 5 to 17 percent for stable FPTP/AV systems.Systems with high malapportionment (France, Spain, Iceland, and Japan)fall outside the FPTP/SV and multi-seat PR patterns observed.

When Loosemore–Hanby’s D1 is used, the pattern of Figure 5.1 becomesmore diffuse but is preserved. The curves (not shown) shift upward. D1

tends to range from 2 to 7 percent for stable PR systems, and from 6 to 24percent for stable FPTP systems.

Lijphart (1999: 169) graphs the effective number of legislative parties(NS) against D2, for 36 most stable democracies. Here D2 is found todecrease with increasing NS both for PR and for FPTP/AV. It is unclear

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whether the reversal of the pattern for FPTP/AV is due to the use of NS orwhether the inclusion of many small FPTP countries alters the picture.

Proportionality Profiles

Indices of deviation from PR characterize the entire electoral system,which is useful for comparative purposes, but they do not tell us howthe system affects parties of different sizes. In particular, they do not tellus whether the system advantages large or small parties. In Chapter 3,electoral systems were characterized by their inputs—institutions andlaws—but similar laws sometimes produce different outcomes in differ-ent countries—or even during different periods in the same country. Itwould be useful to have a way to characterize electoral systems by theiroutputs, including their degree of proportionality for individual parties inindividual elections. This is what the proportionality profiles are about.

The advantage ratio or seat–vote ratio of a given party in a given electionis defined as the ratio of its seat shares to vote shares:

a =(% seats)(% votes)

.

Proportionality profile is the graph of advantage ratio versus vote share,for each party, in one or several elections (Taagepera and Laakso 1980).Figures 5.2–5.5 present some typical profiles, using data from Mackie andRose (1991, 1997). Note that all curves that try to express the mean pro-portionality profile must respect two conceptual anchor points. A partywith almost 0 votes will win 0 seats, and hence a = 0 when v = 0. At theother extreme, a party with all votes will win all the seats, so that a = 1when v = 1. In between these two anchor points, smaller parties tend tobe underpaid (a < 1) and larger parties overpaid (a > 1).

The share of votes at which the average pattern crosses the horizontalline a = 1 is the break-even point (b). Ideally, no party with vote sharebelow b should be overpaid, and no party with vote share above b shouldbe overpaid. An operational definition for b might balance the numberof errors in both directions (Taagepera and Shugart 1989: 258). Then thebreak-even point is the vote share such that there are as many cases witha > 1 and v < b (top left quadrant) than there are cases with a < 1 andv > b (bottom right quadrant).

Figure 5.2 shows the profiles for 10 elections in New Zealand (1966–93)that preceded the shift to PR, and also for 10 elections in the USA (1976–94) during a similar period. Both countries used FPTP, but in the USA

70

Deviation from PR

Two

oppo

nent

sO

ne o

ppon

ents

a = 100%/v

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0Votes (%)

Adv

atag

e ra

tio

NZ 1966−93, b ~ 33%USA 1976−94, b ~ 49% FORBIDDEN

AREA

20 40 60 80 100

Figure 5.2. Proportionality profiles for FPTP elections in New Zealand and the USA

practically only two parties competed, while in New Zealand third partieskept challenging the big two, especially toward the end of the period. Itwill be seen later (Chapter 13) that the expected profiles can be calculatedfor FPTP, depending on the number of opponents the given party faces.Figure 5.2 shows these expected curves in the case of one and twoopponents, respectively. These curves assume that the so-called cube lawapplies (see Chapter 13). With only one opponent, this means a = v2/[v3 +(1 − v)3]. With two equal-sized opponents, a = v2/[v3 + 0.25(1 − v)3].

With one opponent, the break-even point comes at 50 percent. Themaximum overpayment is 33 percent (a = 1.33), and it would occur at67 percent of the votes. At extremely high vote shares the advantage ratiois bound to decrease, because a is limited to a = 100%/v—otherwise theparty would have more than 100 percent of the seats. This conceptualmaximum curve for a is shown in Figure 5.2, and the region above it isindicated as conceptually ‘Forbidden Area’. The US data visibly hug theone-opponent curve. The empirical break-even point is around b = 49%.

With two equal-sized opponents, the break-even point is lowered to33.3 percent. In the face of a split opposition, a party can achieve muchhigher overpayments, and at lower vote shares. The maximum is a = 1.61,and it would occur at a vote share of 52.5 percent. The New Zealanddata are mostly scattered between the one-opponent and two-opponent

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Rules and Tools

0.0

0.5

1.0

1.5

2.0

2.5

0

Votes (%)

Adv

anta

ge r

atio

M = 1

PR

10 20 30 40

Figure 5.3. Proportionality profile for Two-Rounds and PR elections in France

curves. The empirical break-even point is no higher than 33 percent. Mostparties with less than 14 percent votes receive no seats.

Figure 5.3 shows the profile for 10 elections held in France (1958–93)after the introduction of Two-Rounds majority-plurality rule in single-seat districts. Vote shares in the first (or only) round are used, as thisindicates the voters’ first preferences. A different symbol is used for 1986,when PR was used. No theoretical curves are available. An approximatebest-fit curve is sketched in. The outcomes are extremely scattered, asseat allocation depends on nationwide and local deals prior to the sec-ond round. Depending on their ability to cut deals, some parties with20 percent votes won more than 40 percent of the seats, while some otherswon as little as 2 percent. According to the operational definition of thebreak-even point, b ≈ 16%, but the wide scatter renders the very notionof a break-even point almost pointless.

As for the PR election in 1986, the data points are almost as widelyscattered as for the Two-Rounds elections. This scatter reflects the lowaverage district magnitude (M = 5.8), the high legal threshold of 5 percent

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Deviation from PR

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

Votes (%)

Adv

anta

ge r

atio

10 20 30

Figure 5.4. Proportionality profile for PR elections in Finland

applied at district level, plus the inability of many parties to learn how tocope with the new rules.

Figure 5.4 shows the profile of a typical List PR system with numerousparties: Finland 1979–95. Only five elections are shown, so as not toclutter the graph, but the pattern is almost identical for the preceding fiveelections (1962–75). District magnitudes varied widely around a mean ofM = 14, and local alliances (apparentement) were allowed. No theoreticalcurves are shown. The operationally defined break-even point is around6 percent—much lower than for the FPTP systems—and the maximumoverpayment of large parties remains modest.

Without alliances and without some extra-large districts, the profilemight follow the curve shown in Figure 5.4, with b = 11%. However,large parties are willing to strike alliances with tiny parties (v < 6%),so as to shift seats away from their major competitors. When a tinyparty thus wins even a few seats, its advantage ratio can exceed 1.0—the more so, the smaller its nationwide vote share. In Figure 5.4, thisregion of well-allied tiny parties appears as a distinct cluster at the topleft.

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Rules and Tools

0

0.2

0.4

0.6

0.8

1

1.2

0

Votes (%)

Adv

atag

e ra

tio

5 10 15 20 25 30 35 40 45 50

Figure 5.5. Proportionality profile for Mixed-Member Proportional electionsin Germany

Finally, Figure 5.5 shows the profile for eight elections in West Germany(1961–87) after its electoral laws stabilized in the late 1950s but prior toGerman unification. This is the typical profile for essentially nationwideallocation by PR, subject to a legal threshold of 5 percent nationwidevotes. It is a step function: a is 0 for V < 5%, and a is slightly above 1.0 forV > 5%—1.03 on the average. The degree of advantage of the successfulparties depends on the vote share (v0) that went to the parties that did notsurpass the threshold in that particular election: a = 100%/(100% − v0).

How do advantage ratios of individual parties tie in with the systemwidedeviation from PR? Since a = 1 means perfectly PR, we should considerthe deviations 1 − a. All these deviations can be turned positive by takingeither the absolute value |1 − a| or the square (1 − a)2. We cannot just taketheir mean for all parties, large and small, because tiny parties would carrytoo much weight. Once we weight the deviations 1 − a by the respectivevote shares, we are back to D1 in the case of |1 − a|. By weighting thesquares (1 − a2) with the squares of the respective vote shares, we areessentially back to D2.

A proportionality profile is a ‘snapshot’ of the given electoral system. Itindicates at a glance the average impact of the electoral system on largeand small parties and the degree of scatter around this average. Oddities

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Deviation from PR

are quickly spotted, such as the high advantage of some tiny parties inFinland. If we truly understand the functioning of an electoral system,then we should be able to predict its proportionality profile withoutlooking at any data. The theoretical curves in Figure 5.2 hint at someability to do so. This challenge is addressed toward the end of the book(Chapter 14).

Volatility and Ticket Splitting

When measuring volatility, V1 has continued to rule, and no shift to V2

seems to have been considered. Volatility is usually measured on the basisof votes in consecutive elections. But it can also be measured on the basisof seats instead of votes, or on the basis of nonconsecutive elections (e.g.20 years apart, so as to evaluate long-term change), etc. Splits and mergersof parties make it sometimes hard to decide whether the voters are volatilerather than parties (Sikk 2005).

It is important to distinguish between individual and aggregate volatil-ity. Individual volatility (VI) can be established only by exit polls orinterviews, and the result depends on the quality of voters’ recollec-tions of how they voted several years earlier. Aggregate volatility (VA)refers to changes in party total vote shares. It can easily be measuredwhenever party votes data are available, but it hides individual shiftsthat cancel out. If as many voters switch from party A to party B asvice versa, then the aggregate volatility is 0. In the opposite direction,aggregate volatility could be as high as individual volatility, but nohigher.

To repeat, for given individual volatility, the corresponding aggregatevolatility can be as high as the individual, or as low as 0. In the absence ofany other knowledge, our best guess is halfway between these limits:

VA ≈ 0.5VI.

It is somewhat risky to reverse the direction. But when all we know isaggregate volatility—which is usually the case—our best guess for indi-vidual volatility still might be double the aggregate volatility: VI ≈ 2VA.

The study of volatility as well as ticket splitting remains outside thescope of this book. I only point out that the methods of measurement arethe same as for deviation from PR, with similar dilemmas.

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Conclusion

No perfect way to measure deviation from PR has been found. As alreadyobserved for the number of parties, a single index cannot express allthe details. Chapter appendix shows that two indices offer comparableadvantages. Fortunately, in most actual cases they tell pretty much thesame story. In the case of FPTP, deviation from PR may tend to increasewith increasing number of parties running, while the reverse may be truefor PR in large-magnitude districts. Proportionality profiles offer a wayto express visually the differential impact of an electoral system on thefortunes of individual parties, large and small.


What follows matters from the methodological viewpoint, although it may not beneeded to make use of proportionality profiles and indices of deviation. What arethe options for measuring deviation from a norm? How do the various measuresinterrelate? Are they inherently different, or are they different variants of the samemaster equation? What are the advantages and limitations of the two main indices(D1 and D2), and what are the alternatives that may work better for some purposes?These questions of methodology are similar to those that arose for the number ofparties. I will consider these issues from the viewpoint of shares of votes (vi ) andseats (si ) of parties. Volatility and ticket splitting may present different problems.

How to measure deviation from a norm

How do we measure deviation from a norm? While 0 deviation is clearly defined assi = vi , maximum deviation is hazier. It could be argued that maximum deviationis reached when parties with votes have no seats, and parties with no votes haveall the seats. But should such maximum deviation be construed as ∞, leading toa scale from 0 to ∞, or as 100 percent deviation, leading to a scale from 0 to 1?Here we compare two numbers, si and vi , which leads to possibilities that did notarise when establishing the number of parties, based on si alone. Should we firstmodify si and vi separately (by squaring them, for instance), before subtracting ordividing them?

In view of such multiplicity of options, it should come as no surprise thata recent review (Taagepera and Grofman 2003) found that 19 different indiceshave been proposed. For choosing among them, Taagepera and Grofman (2003)proposed 12 criteria, which an ideal measure of deviation should satisfy—seeTable 5.1. They partly overlap with criteria proposed earlier by Monroe (1994).

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Deviation from PR

Table 5.1. Satisfaction of the Taagepera and Grofman (2003) criteria bythree indices of deviation from PR

D1 D2 D∞

Theory-inspired criteria

1. Is informationally complete (makes use of all si and vi ) 1 1 02. Uses data for all parties uniformly 1 1 13. Uses si and vi symmetrically 1 1 14. Varies between 0 and 1 (or 0 and 100 percent) 1 15. Has value 0 when si = vi for all parties. 1 1 16. Has value 1 (or 100 percent) when parties with no votes 1 0 0

have all the seats.7. Satisfies Dalton’s principle of transfers (for differences) 0.5 1 0.5

Practical criteria8. Does not include the number of parties 1 1 19. Is insensitive to lumping of residuals 0 1 0

10. Is simple to compute 1 0.5 111. Is insensitive to shift from fractional to per cent shares 1 1 112. Apart from si and vi , does not include any other inputs 1 1 1

TOTAL SCORE 10.5 10.5 8.5

7a. Satisfies Dalton’s principle of transfers (for ratios) 1 0 0.5ALTERNATE TOTAL SCORE 11 9.5 8.5

Some criteria may look self-evident, but each of them is violated by at least 1 ofthe 19 measures proposed.

Among these criteria, Dalton’s principle needs explanation (see Monroe 1994).When a seat is transferred from a richer component to a poorer one, the indexof deviation should increase (the strong form of Dalton’s principle) or, at least, itshould not decrease (the weak form). ‘Richer’ here refers to the party that is moreoverpaid, or at least less underpaid, in terms of seat shares as compared to voteshares. It seems straightforward, but there is a hitch. To evaluate relative over-or underpayment, do we compare the differences (si − vi ) or the ratios (si/vi )? Itcan happen that one party has a higher difference while another has a higherratio. Which one is ‘richer’ in such a case? The difference criterion is usuallyaccepted. If so, then D1 satisfies the weak form but fails the strong form ofDalton’s principle, whileD2 does satisfy the strong form too. This is the outcomeshown in the main body of Table 5.1. However, the ratio criterion sounds asreasonable. By the ratio criterion, it turns out (Taagepera and Grofman 2003)that it is D1 that satisfies the strong form, while D2 fails to satisfy even the weakform!

Among the 19 indices considered by Taagepera and Grofman (2003), Table5.1 shows only 3. Loosemore–Hanby’s D1 (1971) and Gallagher’s D2 (1991) areincluded because they reach the highest total scores (10.5 of a possible 12). Thusthey look equally adequate but not ideal. However, if we use the ratio criterion fortransfers, instead of differences, then D1 increases to 11.0 and widely surpasses D2,

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which drops to 9.5. Then D2 would share the second and third places with Giniindex (not discussed here), which advances from 8.5 to 9.5.

The single largest deviation, D∞, is included because it is part of the samesystematics, to be presented next. By the criteria shown, D∞ (total score 8.5) issurpassed by 4 and equaled by 5 other indices, so it is not among better themeasures.

Systematics of deviation from a norm

The simplest measure of deviation from a norm might indeed be the largestdeviation alone, meaning D∞. Its disadvantage is that whatever happens to therest of the parties does not matter. This problem is akin to the one we encounteredpreviously when considering the inverse of the largest share as a measure of thenumber of parties (N∞).

The next simplest approach might be D1. It runs into trouble with Dalton’sprinciple, at least when the difference criterion is used. It is also felt to overem-phasize the importance of numerous small deviations as compared to one largeone. This problem is akin to what disqualifies N0 and weakens the entropy-basedN1 as measures of the number of parties.

Finally, like N2 previously, D2 represents an intermediary approach, taking intoaccount all the deviations but giving more weight to the larger ones. Actually, allthree indices, D1, D2, and D∞, are special cases of a master equation, presentedhere for the first time:

Dk =[

12

�|si − vi |k]1/k

.

Any k < 1 would emphasize smaller deviations at the expense of larger ones, whichis hardly desirable, and k = 1 is on the verge of doing so. At the other extreme, as ktends toward ∞, Dk tends ∞ the largest deviation alone, neglecting all others. Asa compromise, k = 2 leads to D2, which boosts the impact of larger deviations. Thesmaller deviations eliminate themselves through squaring, much the same waysmaller parties do in N2.

If there are only two parties, then one’s loss is the other’s gain, so that D1 = D2 =D∞. When more than two parties have nonzero deviations, then the broad patternis the following. As k increases beyond 1, Dk at first decreases and then starts toincrease again. This means that

D1 > D2 [more than two nonzero deviations]

is inevitable, while D∞ may or may not catch up with D2. It depends on whetherthe square of the largest deviation exceeds the sum of squares of the other devia-tions. For instance, when D1 = 10%, individual deviations 8-2-5-5 lead to D∞ > D2,while 7-3-5-5 lead to D∞ < D2. Whenever at least two parties are underpaid and

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Deviation from PR

Table 5.2. Mean values of deviation indices D1 and D∞, for given mean valuesof D2

Range of D2 Mean D2(%) D1(%) 100(D2/100).863 D∞(%) 126(D2/100)1.058

0–1.99% 1.45 2.65 2.58 1.47 1.422.00–4.99 3.23 5.11 5.17 3.33 3.335.00–9.99 7.23 10.36 10.36 8.10 7.81

10.00–17 12.28 16.32 16.37 14.09 14.33

two other parties are overpaid in terms of seats, then D1 is bound to be larger thanD∞.

To evaluate the empirical relationships among the three indices, we can useagain those 37 electoral systems studied by Lijphart (1994: 160–2), which involvedat least 3 national elections. Table 5.2 divides them into groups by size of D2.

As expected, D1 is always larger than D2. The ratio of D1 to D2 decreases withincreasing deviation from PR, while their difference increases. Consider somesimple cases. If 8 parties had equal deviations (4 gaining and 4 losing), it can beshown that the ratio D1/D2 would have to be exactly 2. The observed ratio at lowdeviations (1.83) comes close to this simple equivalent constellation. On the otherhand, for 4 parties with equal deviations, we would have D1/D2 = 20.5 = 1.41. Theobserved ratio at the highest deviations (1.30) drops somewhat below this simpleequivalent. Table 5.2 shows that the overall pattern is well approximated by

D1 = 100(

D2

100

).863

[D1 and D2 in percent].

This means that in about one-half the cases the actual value of D1 is expected to fallbelow D1=100(D2/100).863, and in about one-half the cases it is expected to surpassit. This equation is empirical. It can be expected to express the average relationshipbetween D1 and D2, as long as most small deviations result from many partiescompeting, while most large deviations result from 3 to 4 parties competing.

It can also be seen in Table 5.2 that D∞ tends to be larger than D2, althoughvalues smaller than D∞ are also observed at low deviations (D2 < 5%). In contrastto D1/D2, the ratio of D∞ to D2 increases with increasing deviation, as does theirdifference. Table 5.2 shows that the overall pattern is well approximated by D∞ =126(D2/100)1.058, with D∞ and D2 in percent. I have not worked out the theoreticaljustification behind this empirical approximation.

The advantages and disadvantages of Loosemore–Hanby’sD1 and Gallagher’s D2

The fact that D2 practically always is less than D1 is often considered an advantage.This is based on the feeling that D1 exaggerates the total degree of deviation whennumerous parties have tiny deviations. One might offer the examples in Table 5.3.

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Table 5.3. An example where Loosemore–Hanby’s devia-tion index D1 looks too low

Case A Case B

votes (%) 55 45 10 . . . 10 10 . . . 10seats (%) 60 40 11 . . . 11 9 . . . 9

D1 = 5.0% D1 = 5.0%D2 = 5.0% D2 = 2.24%

D∞ = 5.0% D∞ = 1.0%

In case A, a single party falls short by 5 percent and a single other party hasa corresponding bonus. We have the impression of a 5 percent deviation, andthe indices D1, D2, and even D∞ all confirm this expectation. Now suppose that10 parties receive 10 percent votes each, but 5 of them win 11 percent of theseats each, while the other 5 win only 9 percent each (case B). These many tinyindividual deviations may look like negligible random fluctuations, but they stilladd up to D1 = 5%—the same value we have in case A. In contrast, D2 distinguishesbetween cases A and B. It is 5 percent for A but only 2.24 percent for B. D∞ dropseven further, to 1 percent. It would seem that D2 can distinguish between a fewsignificant and many random deviations, while D1 cannot.

But I have cheated. Cases A and B differ in more than concentration of deviation.Let us introduce cases C and D, shown in Table 5.4. First compare case C toprevious case A. All indices are exactly the same, but in case A we saw a nonrandomdeviation (in contrast to randomness in case B), while case C can be explained onlyas a large random deviation—cf. the large scatter in advantage ratios for France, inFigure 5.3. How come? In case A, we were influenced by our expectation of thelarger party obtaining a bonus and the smaller party being penalized, because thisis the usual pattern. This expectation is extraneous to measurement of deviationas such.

Next, how would we feel about case D, where 5 parties with 15 percent votesreceive a 1 percent bonus, while 5 parties with 5 percent votes have a 1 percentpenalty? The indices are exactly the same as in case B above, and all individualdeviations are as tiny as in case B, but now the bonuses and penalties all go in the

Table 5.4. A counterexample where Loosemore–Hanby’sdeviation indexD1no longer looks too low

Case C Case D

votes (%) 50 50 15 . . . 15 5 . . . 5seats (%) 45 55 16 . . . 16 4 . . . 4

D1 = 5.0% D1 = 5.0%D2 = 5.0% D2 = 2.24%D∞ = 5.0% D∞ = 1.0%

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Deviation from PR

Table 5.5. Examples where Loosemore–Hanby’s deviationindexD1 may look preferable to Gallagher’s D2

Case E Case F Case G

votes (%) 100 0 50 50 0 50 50 0 0 . . . 0seats (%) 0 100 0 0 100 0 0 1 1 1 . . .

D1 = 100% D1 = 100% D1 = 100%D2 = 100% D2 = 86.6% D2 = 50.5%

D∞ = 100% D∞ = 100% D∞ = 50%

expected directions. Are we still certain that D1 = 5.0% is excessive and D2 = 2.24%is adequate, in case D? There is a systematic shift of 5 percent.

Even if we decide that the lower value of D2 for cases B and D, as compared tocases A and C, is justified, it partly results from an awkward artifact: When manyparties are present, the possible scale for D2 stops much short of 100 percent. Thehypothetical examples in Table 5.5 clarify this claim.

Suppose an absolute monarch decides to have elections for a 100-seat assembly.Suppose one party receives all the votes, but the monarch still decides to assignall the seats to a royal party that received 0 votes (case E). Clearly, this outcomerepresents 100 percent deviation from PR, and all 3 indices lead to such a result.Now suppose two parties split the vote, but neither receives any seats (case F). Isthe deviation from PR now alleviated, just because the vote is less concentrated?D1 and D∞ respond ‘no’: Deviation remains at 100 percent. But D2 tells us ‘yes’:Deviation is reduced to 86.6 percent.

Now suppose the monarch does not assign the seats to a royal party but to100 individuals, who formally are independents and hence in many ways areequivalent to 100 separate parties (case G). D1 tells us that deviation remains at100 percent. But D2 says deviation is further reduced to a mere 50.5 percent. D∞also says that deviation is only 50 percent. Thus, D2 and D∞ hide the blunt factthat none of those parties who received votes got any seats. They would be theautocrat’s preferred indices!

In sum, the very attentiveness to concentration of deviation that makes Gal-lagher’s D2 look more attractive than Loosemore–Hanby’s D1 in case B leadsto gross underestimation of total deviation in case G. No satisfactory methodhas been devised to correct for this discrepancy—and this is not for lack oftrying.

Up to now, we have defined zero deviation from PR, and also what wouldconstitute full deviation. To specify the central part of a scale anchored by theseextremes, we should further ask which pattern of deviation would correspond toone-half of maximum deviation. It might be argued that this is a situation whereone-half of the votes is converted by perfect PR, while the other half is convertedby utmost lack of PR (cases H and I shown in Table 5.6). Once more, the outcome

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Table 5.6. Which pattern corresponds to a half of maximumdeviation from PR?

Case H Case I

votes (%) 50 50 0 25 25 25 25 0 0seats (%) 50 0 50 25 25 0 0 25 25

D1 = 50% D1 = 50%D2 = 50% D2 = 35.4%

D∞ = 50% D∞ = 25%

depends on the number of parties. In case H, all 3 indices yield 50 percent. In caseI, only D1 does. D2 yields 35.4 percent, while D∞ drops to 25 percent.

Recall that both D1 and D2 were found to be superior to any other measuresproposed, when it comes to the desiderata listed in Table 5.1, yet neither satisfiesall of them. When Dalton’s principle is applied with the ratio criterion, D1 stronglysurpasses D2. It also yields reasonable values in most cases above, while the valuesof D2 are counterintuitive for cases F, G, and I. The only situation where D1 seemsinferior is lack of discrimination between cases A and B—but this contrast becomessuspect when comparing cases C and D. Moreover, when it comes to logical modelbuilding, D1 has been easier to visualize (cf. Taagepera and Shugart 1989: 109–11).In sum, the shift from D1 to the more complex D2 in the electoral studies of the1990s may not be as justified as it looked at the time. Maybe D1 is preferable.

Actually, we might need an index that takes into account the directionality ofshifts—an index that labels large party bonuses and small party penalties positive,while declaring large party penalties and small party bonuses negative. Considerthe index

d = �(si − vi )(

vi− 1NV

),

where NV is the effective number of electoral parties. It would express expecteddirectionality in most cases, but not for cases E–G in Table 5.5. Furthermore, I donot know how to normalize it with respect to conceptual limits.

Going even deeper, the very quest for a single ideal measure to characterizeall aspects of deviation from a norm might be as futile as the quest for a singlesuper-index to characterize the number of parties in all its aspects (see Chapter 4).Here, the analogy with central tendency and variation around it does not apply.Something else is needed.

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6

Openness to Small Parties: TheMicro-Mega Rule and the Seat Product


� For inclusive representation of even small parties, it helps to have largeassemblies, large district magnitudes, and large quotas or large gapsbetween divisors in formulas for allocating seats.

� Conversely, large parties would prefer small assemblies, magnitudes,and quotas—but only if they are absolutely certain to stay large.

� Worldwide tendency has been to play it safe and move toward moreinclusive representation.

� The number of parties increases with increasing ‘Seat Product’—thenumber of the seats in the assembly × the number of seats in theaverage district.

� When seats are allocated by plurality in multi-seat districts, however,the number of parties decreases with increasing number of seats perdistrict.

Now that the ways to measure the number of parties and deviation fromPR have been specified, we can proceed to sense more quantitativelyhow the three indispensable components of electoral systems affect theopenness of the system to small parties. These components are assemblysize, district magnitude, and seat allocation formula. The degree of inclu-siveness (openness to smaller parties) of a simple electoral system largelydepends on the combination of these three institutional inputs.

The direction of the impact of these inputs on system openness iscompactly worded in Josep Colomer’s ‘micro-mega rule’ (2004b: 3). Goingbeyond directionality, their quantitative impact is expressed by the ‘seat

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product’, first explicitly proposed in this book. This chapter also offerssome illustrative cases and other evidence that may help in getting a feelfor how district magnitude, seat allocation formula, and assembly sizeaffect the number and sizes of parties.

This focus on electoral system openness to small parties involves novalue judgment. I do not claim that more inclusive systems are better. AllI say is that electoral systems differ in their degrees of openness, and thatthese differences impact the style and nature of politics. The golden meanlies somewhere between extreme closure and extreme openness, and itslocation may vary, depending on circumstances. Offering a measure ofinstitutional openness, in the form of seat product, merely helps us toplace a given system on the openness scale.

Colomer’s Micro-Mega Rule

Large assemblies, large electoral district magnitudes, and List PR allocationformulas with a large quota or large gaps between successive divisors—allthese enhance openings for small parties. Conversely, it would seem tobe in the interest of large parties to keep the competition out by havingsmall assemblies, small district magnitudes, and small quotas or smallgaps between divisors. While such knowledge has diffusely been aroundfor some time, Colomer (2004b: 3) compresses it in a felicitous ‘micro-mega rule’:

The small prefer the large, and the large prefer the small.

He extends it to large parties preferring a single allocation formula, whileit is in the interest of small parties to have composite systems wheredifferent parts have different formulas. Such variety tends to offer furtherentry points to small parties. If it is true that ‘the small prefer the large’,then the reverse should be in the interest of the large parties, except forone major reservation.

The longer democracies last, the more the generalization above needsqualification. Over time, even large parties experience moments of weak-ness and learn to appreciate less risky formulas—larger assemblies, districtmagnitudes, and quotas, plus composite systems. Therefore, the seculartrend has been to shift from nationwide winner-take-all toward ever-more inclusive electoral systems, as extensively documented by Colomer(2004b: 53–62). So, over time, the micro-mega rule might become:

The small prefer the large, and the large hesitate preferring the small.

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Micro-Mega Rule and Seat Product

Having defined in the preceding chapters how to measure the number ofparties and the deviation from PR, we can now illustrate quantitativelythe way small parties stand to profit from large district magnitudes andfrom seat allocation formulas that have large quotas or large gaps betweendivisors. It is more difficult to illustrate the impact of assembly sizes.Detailed formulas and empirical evidence will be given in later chapters.But a reasonably chosen illustration may help in getting a feel for what isinvolved.

The Effect of District Magnitude and Seat Allocation Formula:An Illustrative Example

Consider again the distribution of percentage votes 48, 25, 13, 9, 4, 1,as used in Table 3.3. That table compared the effects of various electoralformulas when M = 6. Now we gradually increase the district magni-tude, from 1 to above 100, and ask how the seats are allocated on thebasis of three different formulas. Hare quota (with largest remainders)is the largest quota used in practice, and it should favor the smallerparties, according to Colomer’s rule. Among the widely used divisor rules,d’Hondt has the smallest gaps between divisors and thus should favor thelarger parties. Table 3.3 confirmed it, at one specific district magnitude.All other formulas widely used in practice fit in-between Hare-LR andd’Hondt. Among these, Sainte-Laguë divisors will be seen to emerge asarguably the most proportional seat allocation formula.

How do these three formulas affect the seat allocation as magnitude isgradually increased? This is shown in Table 6.1, where percentage of voteshares are assumed to be 48+, 25−, 13−, 9−, 4, and 1+. To avoid ties inallocation of seats, a tiny amount is added (+) or subtracted from (−) frominteger percentages.

It can be seen that Hare-LR often gives fewer seats to the largest partyand more to the small ones, compared to d’Hondt. Sainte-Laguë mostoften follows the Hare pattern but sometimes agrees with d’Hondt and inrare cases differs from both (M = 20, 51, shown for this purpose, and alsoM = 25, not shown in the table). Entries in bold script for Sainte-Laguëhighlight these deviations from Hare.

The Hare-LR result for the party with 4 percent votes at M = 9 and10 (shown in bold script) highlights the so-called Alabama paradox: Thisparty wins a seat when 9 seats are to be allocated, but loses it again whenthe total is raised to 10 seats! It wins it back at M = 11 (not shown in

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Table 6.1. Effect of district magnitude and seat allocation formula on the distributionof seats in a district where the percentage vote shares are 48+, 25−, 13−, 9−, 4,and 1+

M Hare-LR Sainte-Laguë d’Hondt

1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 02 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 03 2 1 0 0 0 0 2 1 0 0 0 0 2 1 0 0 0 04 2 1 1 0 0 0 2 1 1 0 0 0 3 1 0 0 0 05 2 1 1 1 0 0 3 1 1 0 0 0 3 1 1 0 0 06 3 1 1 1 0 0 3 1 1 1 0 0 3 2 1 0 0 07 3 2 1 1 0 0 3 2 1 1 0 0 4 2 1 0 0 08 4 2 1 1 0 0 4 2 1 1 0 0 5 2 1 0 0 09 4 2 1 1 1 0 5 2 1 1 0 0 5 2 1 1 0 0

10 5 3 1 1 0 0 5 3 1 1 0 0 5 3 1 1 0 020 10 5 2 2 1 0 9 5 3 2 1 0 11 5 2 2 0 030 14 8 4 3 1 0 14 8 4 3 1 0 15 8 4 2 1 051 24 13 7 5 2 0 24 13 7 4 2 1 25 13 7 4 2 070 34 17 9 6 3 1 34 17 9 6 3 1 35 18 9 6 2 0

100 48 25 13 9 4 1 48 25 13 9 4 1 48 25 13 9 4 1106 51 26 14 10 4 1 51 26 14 10 4 1 51 27 14 9 4 1

Table 6.1), but loses it once more at M = 12. Only from M = 13 on doesthis party steadily win a seat. Similarly, the party with 1 percent votes winsa seat when 25, 33, or 48–50 seats are allocated but loses it again whenthe total is raised. Only from M = 53 on does the 1 percent party steadilywin a seat.

The name of the Alabama paradox originates in the late 1800s, whenAlabama did lose a seat in the US House when the House size wasincreased, even while Alabama’s share in the US population did notdecrease. Thus the Alabama paradox is of concern when quota formulasare applied to seat allocation among territorial units. It is a minor inconve-nience in practice when applied to allocation of seats to parties. However,it becomes a nuisance when one tries to answer the simple theoreticalquestion: At what magnitude does a party with a given vote share stand achance to win a seat? For divisor formulas, the answer is clear. For Hare-LR,the answer ‘25 seats’ would be here superficially correct for the tiny party,yet misleading, because the first seat becomes safe only above M = 53.

Table 6.2 shows the magnitudes at which the parties in Table 6.1 wouldobtain at least one seat under various allocation formulas. Added to theprevious three formulas are Imperiali (1, 1.5, 2, 2.5, . . . ) and Danish (1,4, 7, . . . ) divisors. For divisor formulas, d represents the gap betweensuccessive divisors (starting from 1). Sufficiently large parties obtain min-imal representation almost regardless of the allocation formula used. The

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Table 6.2. Magnitudes at which parties with percentage vote shares 48+, 25−, 13−,9−, 4, and 1+ would win their first seat under various allocation formulas

Votes Danish Hare-LR Sainte-Laguë d’Hondt Imperialishares (d = 3) (d = 2) (d = 1) (d = 0.5)

48+ 1 1 1 1 125− 2 2 2 2 313− 3 4 4 5 9

9− 5 5 6 9 154 9 9, 11, 13 13 24 431+ 33 25, 33, 48–50, 53 49 96 189

100/d 33.3 — 50 100 200

success of the small parties, however, depends very much on the alloca-tion formula. The tiny 1 percent party can win its first seat only whendistrict magnitude approaches M = 100/d. This means a magnitude closeto 33 for Danish divisors but close to 200 for Imperiali.

Indeed, it can be shown that tiny parties win their first seat when theirvote share approaches v = 1/(Md), meaning simple quota 1/M divided bythe divisor gap. With d’Hondt (d = 1), small parties need almost a fullquota to win a seat. With Sainte-Laguë (d = 2), they need only one-halfof the simple quota, while with Imperiali (d = 0.5), they need double thesimple quota. When the Alabama paradox cases are overlooked, Hare-LRis close to Sainte-Laguë. The effects of various other quotas remain to bestudied.

Figure 6.1 makes the results in Table 6.2 more explicit, graphing thepercentage of votes at which the first seat tends to be won against districtmagnitude. Both variables are on logarithmic scales. The lines shown aretheoretical (from Chapter 15). The quasi-data points are from Table 6.2,omitting the Alabama paradox cases.

The striking feature is that the Sainte-Laguë pattern (shown as roundsymbols) is a straight line at all district magnitudes. This means thatparties always tend to win their first seat when their vote share reachesone-half of the simple quota. The same is true for Hare-LR (small crosses),except for the Alabama paradox cases. In contrast, allocations with otherformulas start out at the Sainte-Laguë line but then gradually shift,at higher magnitudes—upwards for d < 2 (slanted crosses for d’Hondt,squares for Imperiali) and downwards for d > 2 (triangles for Danish).From this viewpoint, Sainte-Laguë appears central among all divisorrules. So might Hare-LR be among all quota rules, when overlookingthe Alabama paradox. On different mathematical grounds, Balinski and

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Danish lim

it

Sainte-Laguëlimit

d’Hondt lim

it

Imperiali lim

it

1

10

100

1,000100101

District magnitude

Vot

es (

%)

at w

hich

the

first

sea

t is

won

Figure 6.1. Vote shares at which parties tend to win their first seat vs. districtmagnitude, for various PR formulas

Young (2001) also consider Sainte-Laguë the most proportional formula.So do Schuster et al. (2003), both on theoretical and on empirical grounds,along with Hare-LR.

Table 6.3 shows the deviation from PR (Gallagher’s measure, D2) andthe effective number of parties for the seat allocations in Table 6.1. Asdistrict magnitude increases, deviation from PR at first decreases rapidly,with occasional minor reversals. It later decreases slowly, as D2 approachesthe conceptual limit of D2 = 0. Hare-LR always produces deviations at leastas low as d’Hondt, and often appreciably lower. Sainte-Laguë most oftenagrees with Hare but occasionally with d’Hondt. The vote shares used arebound to yield perfect lack of deviation (D2 = 0.00) at M = 100, but D2

increases again at higher magnitudes, though only slightly. Perfect lack ofdeviation is seen to be extremely rare.

The effective number of legislative parties is low at low magnitudes.When Hare-LR or Sainte-Laguë is used, it quickly catches up with the

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Table 6.3. Effect of district magnitude and seat allocation formula ondeviation from PR and on the effective number of parties, in a districtwhere the percentage vote shares are 48+, 25−, 13−, 9−, 4, and 1+

M Deviation from PR (D2) Effective number (NS)

Hare-LR Sainte-Laguë d’Hondt Hare-LR Sainte-Laguë d’Hondt

1 42.4 42.4 42.4 1.00 1.00 1.002 21.2 21.2 21.2 2.00 2.00 2.003 18.5 18.5 18.5 1.80 1.80 1.804 11.0 11.0 22.3 2.67 2.67 1.605 11.7 12.6 12.6 3.57 2.27 2.276 9.0 9.0 9.6 3.00 3.00 2.577 6.5 6.5 11.1 3.27 3.27 2.338 4.1 4.1 12.4 2.91 2.91 2.139 6.3 6.6 6.6 3.52 2.61 2.61

10 5.3 5.3 5.3 2.78 2.78 2.7820 1.3 2.8 6.2 3.16 3.33 2.6030 2.6 2.6 2.6 3.15 3.15 2.9051 1.3 1.4 1.4 3.16 3.19 3.0170 1.3 1.3 1.9 3.12 3.12 2.93

100 0.0 0.0 0.0 3.13 3.13 3.13106 0.5 0.5 0.5 3.13 3.13 3.10

Note: For deviation, disagreements between Hare-LR and Sainte-Laguë are shown in bold script.For effective number, values higher than the one based on votes (NV = 3.13) are shown in bold.

effective number of electoral parties (3.13) and at times even surpassesit. Such overbeats are indicated in bold script. Large overbeats are fewerfor Sainte-Laguë, although they exceed Hare at M = 20 and 51. At highermagnitudes, Hare and Sainte-Laguë yield NS = NV, on the average. Thisequality suggests that these rules favor large and small parties equally.With d’Hondt, in contrast, NS approaches NV but never surpasses it. Theremaining gap indicates that d’Hondt maintains a large party advantage,however small, even at large M.

The Effect of District Magnitude and Seat Allocation Formula:Generalization

The conclusions drawn from the illustrative example could be derivedthrough more formal mathematical procedures. Doing so would gain inrespectability but lose in readability and intuitive understanding. So I willleave it at that, at least until Chapter 15.

The seat allocation formula matters for the fortunes of parties. At thesame district magnitude, the smallest parties need almost twice the votesto land their first seat with d’Hondt, as compared to Sainte-Laguë. Yet

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district magnitude matters much more, as long as one sticks to the usualPR formulas. (Shifting to plurality allocation rule in multi-seat districtsreverses the effect!) Altering the magnitude by a factor of 2 (meaningmultiplying or dividing by 2) affects small parties more than any shiftamong the usual PR formulas could—and even larger changes in M areavailable, given a possible range from M = 1 to M = 100 and even higher.At M = 5, a party with 5 percent votes has little chance of winning a seateven under Sainte-Laguë. At M = 20, it is bound to win a seat even underd’Hondt.

In this qualified sense, district magnitude is the decisive factor, as longas all seats are allocated within districts, by some PR allocation rule. Aspointed out in Figure 2.1, allocation rule can reverse the effect of magni-tude when allocation by plurality is included. It is obviously possible tooverrule the effect of district magnitude by district level legal thresholdsor apparentement (which is relatively rare), or when the seat allocationprocess is extended beyond the basic districts (which is frequent).

The illustrative example used in the previous section has a major weak-ness. It tacitly presumes that vote distribution is unaffected by districtmagnitude. This is not so. The distribution 48-25-13-9-4-1 is realistic fora district of magnitude around 10, embedded among similar neighboringdistricts where the two smallest parties might be doing better than inthe given district. If the actual district magnitude is 1 or 2, however,only 2 or 3 parties or blocs are likely to survive. On the other hand, ifthe actual M is around 100 (with no legal threshold), then even moreparties than 6 are likely to try their luck, and the 48 percent party wouldbe subject to heavy centrifugal forces. Such interaction between districtmagnitude and the size distribution of parties needs to be put on aless impressionistic foundation, by empirical measurement and predic-tive model construction. This will be done in the central part of thisbook.

The Effect of Assembly Size

The effect of district magnitude and seat allocation formula could beillustrated with a hypothetical vote distribution. This is not possiblewith assembly size. We could look for actual cases where everything butassembly size is the same, or carry out some quasi-experiments on whatwould happen if we reduced an actual assembly size. Neither approach isconclusive, but they help.

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Grofman and Handley (1989) considered the percentage of blacks in theHouses and Senates of the US states. The electorates and electoral rulesfor the two bodies are basically the same, the only difference being thatthe House usually has more members. The percentage of blacks is foundto be higher in the Houses than in the respective Senates. While thisresult does not refer to parties, it suggests that if an ethnic or ideologicalminority decides to form a party, it would have more success in largerassemblies.

The following quasi-experiment could be carried out. Take a countrywith single-seat districts and plurality rule, where the vote distributionin each district is known. Fuse neighboring districts two by two, applyplurality rule, and see how many seats parties would win in the combineddistricts. Fuse the resulting districts once more, and so on. Such a quasi-experiment was carried out for the British elections of 1983 (Kaskla andTaagepera 1988; cf. Taagepera and Shugart 1989: 173–4). Reducing thenumber of districts reduces the effective number of parties in the Houseof Commons as follows:

Votes 2.93 effective parties633 seats 1.98 (actual assembly)316 seats 1.86

79 seats 1.7011 seats 1.531 seat 1.00

This quasi-experiment presumes that reducing assembly size would notappreciably alter the voting pattern. In reality, it would. Voters wouldgive up on smaller parties, reducing the effective number of electoralparties, and thus possibly depressing the number of legislative partieseven further. The figure for 11 seats compares well with the pattern insome tiny island countries such as St Vincent & Grenadines, where theparliament in 1974–84 had 13 seats and NS ranged from 1.35 to 1.74 evenwhile the identity of the dominant party changed (calculations based ondata in Nohlen 1993: 701) When the assembly is reduced to one seat, NS

is bound to fall to 1.00. Taking this anchor point into account supportsthe qualitative idea that reducing assembly size must tend to reduce thenumber of parties.

The two studies presented do not amount to systematic empirical evi-dence, but they may help to give a feeling for the trend. A predictivemodel will be presented and tested later on.

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The Seat Product of Simple Electoral Systems

District magnitude, seat allocation formula, and assembly size can becombined into a ‘seat product’ that characterizes the degree of opennessor inclusiveness of a simple electoral system. Inspired by Colomer’s micro-mega rule (2004b: 3), it is first made explicit in this book. Here, I run aheadof the evidence to be presented later on. However, it may help to knowin which direction we are proceeding. In some ways, the seat productrepresents the comprehensive outcome of my trying to make sense ofelectoral systems, over 40 years.

The single number that best characterizes any simple electoral system isthe product of the number of the seats in the assembly (S) and the numberof seats in the average district (mean district magnitude, M), the lattermodulated by the seat allocation formula through a ‘formula exponent’(F ):

Seat Product = MF S.

For the usual PR formulas, F is around +1. It is slightly smaller for d’Hondtthan for Sainte-Laguë or Hare-LR, by an amount to be specified later. Forplurality formula, F = −1. For semi-proportional formulas, intermediaryvalues of F would apply. In the case of FPTP, M = 1, and thus the valueassigned to F does not matter. Using F = −1 would stress the fact thatFPTP is the most proportional outcome to which plurality formula couldlead, while using F = +1 would remind us that FPTP is the least propor-tional outcome to which a PR formula could lead. When overlookingmulti-seat plurality, the seat product basically amounts simply to MS.

The seat product incorporates all three indispensable components of asimple electoral system. What range do its numerical values cover, andwhat do they mean? Some typical values of the seat product are shown inTable 6.4, along with its fourth root. The latter represents the number of

Table 6.4. The seat product and the resulting expected number ofseat-winning parties in the assembly

MFS N0 = (MFS)1/4

Any country with nationwide plurality 1 1If UK had plurality in M = 10 districts 65 2.8Actual UK, FPTP in 650 districts 650 5.0Malta, 65 seats by PR in M = 5 districts 325 4.2Finland, 200 seats by PR in M = 14 districts 2,800 7.3Netherlands, 150 seats, nationwide PR 22,500 12.2

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parties expected to win at least one seat in the assembly (N0), as will beproposed and tested later, in Chapter 8.

It has been noted earlier that the most and the least proportionaloutcomes both occur when there is a single nationwide district: M = S(cf. Figure 2.1). In this case, the plurality formula yields a seat product ofS−1S = 1, expressing the fact that only one party can obtain seats. At theother extreme, M = S with a PR formula leads to a seat product S2, whichbecomes quite large in the case of large assemblies. This large numbersuggests that many parties can obtain representation, and Chapter 8 willspecify, how many. The seat product implies that a 625-seat assemblyelected by FPTP is as inclusive (open to small parties) as a 25-seat assemblyelected by nationwide PR, given that the seat product is 625 in both cases.Such a claim may look debatable at first glance, but it will be substantiatedin later chapters.

The exponent F in the seat product MF S is not merely a symbolicformulation that can take only the values +1 for PR and −1 for plurality.There are ways to specify its values for different proportional and semi-proportional seat allocation formulas. These specifications have not yetbeen worked out in detail, but the general method for doing so is alreadyapparent. The tentative results are shown in Table 6.5. See chapter appen-dix for the reasoning behind it. At the level of a single district, S = M, andhence the seat product is MF +1. This value is also shown in the table, andsome implications are presented in the chapter appendix.

By this count, the use of d’Hondt would reduce the seat productappreciably, compared to Sainte-Laguë or Hare-LR. For Finland, it woulddrop from 2,800 to 1,030, reducing the expected number of seat-winningparties from 7.3 to 5.7. For the Netherlands, the cut is even more drastic,from 22,500 to 3,350, reducing the seat-winning parties from 12.2 to7.6—and the 0.67 percent legal threshold would reduce it even further.However, it is still uncertain whether the specific PR formula really has

Table 6.5. Tentative values of allocation formula exponent Fin the seat product, for various PR formulas

d F F + 1

Plurality 0 −1.00 0.00Imperiali divisors 0.5 0.32 1.32D’Hondt 1 0.62 1.62Sainte-Laguë, Hare-LR 2 1.00 2.00Danish divisors 3 1.26 2.26

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Rules and Tools

that much impact. Hence, unless otherwise indicated, I will use F = 1 forthe d’Hondt systems too.

Conclusion

The examples presented help visualize how large assemblies, large districtmagnitudes, and large gaps between divisors (or large quotas) help thesmall parties, and how small assemblies, magnitudes, and divisor gaps orquotas help the large parties. This tendency is expressed qualitatively inColomer’s macro-mega rule and quantitatively in the dependence of thenumber of parties on the seat product.

Inclusiveness (openness to small parties) of an electoral system isreflected in the number of parties that make it to the assembly. Whicheverway one measures this number, it tends to increase with increasing seatproduct MF S. This formula suggests that district magnitude and assemblysize enter basically on an equal basis, but the seat allocation formulaenters as a modulator of district magnitude that can reverse the directionof its impact. The small (parties) prefer the large (assemblies, district mag-nitudes, divisor gaps, and quotas) because all these contribute to increaseMF S.


This appendix explains how I estimated the values of the formula exponent (F ),as offered in Table 6.5. It also wonders whether the square root of the seat productis the basic building block for evaluating the properties of electoral systems.

How to estimate the formula exponent in the seat product

Which seat allocation formula should be taken to represent perfect proportion-ality (F = 1.00)? As ‘remainderless quota’, d’Hondt has considerable appeal, butit keeps the effective number of legislative parties below the effective number ofelectoral parties even at large magnitudes. Thus, it is not quite neutral regardinglarge and small parties. Deviation from PR offers limited guidance, because it hasbeen pointed out (e.g. Cox and Shugart 1991; Gallagher 1991) that a measure ofdeviation can be constructed on the basis of any seat allocation formula so as tomake that particular formula look the most proportional. The effective number ofparties seems more neutral in this respect. When one takes the effective numberbased on votes as the standard, Table 6.3 shows that d’Hondt restricts the number

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of parties at all district magnitudes, while Hare-RL and Sainte-Laguë fluctuate fairlyevenly between under- and over-representation. Moreover, these two formulas alsolook central in Figure 6.1.

Thus the choice seems to be between Hare-RL and Sainte-Laguë, as the bench-mark (F = 1.00) for comparing the ‘PR-ness’ of all seat allocation formulas. Hare-LRis simple to visualize, but the Alabama paradox makes it inconvenient. Therefore,it would seem that Sainte-Laguë is the most suitable benchmark. Recall that thisagrees with the theoretical arguments by Balinski and Young (2001) and Schusteret al. (2003) who consider Sainte-Laguë the most proportional formula.

There is another reason to focus on divisor formulas. As pointed out in Table 3.4,divisors offer a link to plurality (divisor gap d = 0) at the one extreme and to ‘everyparty gets a seat’ (huge divisor gaps) at the other. Quotas do not offer such a widerange without risking overallocation. The usefulness of this continuous variable,the divisor gap, emerges as one graphs the effective number of parties versusdistrict magnitude and considers what happens, depending on the allocationformula used.

This is what Kaskla and Taagepera (1988) did in a quasi-experiment that groupedthe British single-seat electoral districts two by two, four by four, and so on.Treating them as multi-seat districts, they applied various seat allocation formulasto the resulting vote shares based on the actual votes in 1983. The resulting curves,N versus M, are shown in Taagepera and Shugart (1989: 265). The broad pattern isas follows.

We have two benchmarks for the effective number of parties. One is NV, whichdepends on the vote constellation. The other is NS at M = 1, designated here asNFPTP, which is lower than NV and tends to increase with increasing assembly size.This dependence on S is addressed in Chapter 9. Here we ask what happens to NS

at constant assembly size, as the district magnitude is increased.With multi-seat plurality (d = 0), NS decreases below NFPTP, until it drops to 1.00

at M = S. With d’Hondt (d = 1), it increases above NFPTP, slowly approaching NV

but never consistently reaching it. With Sainte-Laguë (d = 2), it quickly reachesNV and by M = 10 exceeds it, before slowly approaching NV again, down fromhigher values. The extent of this over beat increases, as the divisor gap is increasedto Danish divisors (d = 3) and beyond. But what would happen at tiny non-zerodivisors such as d = 0.2? At low M, they behave like plurality, lowering NS belowNFPTP, but later the pattern reverses itself—NS surpasses NFPTP and approachesNV, thus behaving more like a PR formula. Around M = 10, Ns is back to thelevel of NFPTP, as if the effect of the d = 0.2 formula were not affected by districtmagnitude.

In this light, if we take Sainte-Laguë to correspond to pure PR (F = 1.00) inthe seat product MF S, then the Danish divisors should be assigned a value of Fsomewhat higher than 1.00 so as to express their tendency to favor small parties.D’Hondt should have a value of F lower than 1.00, and Imperiali an even lowervalue. Multi-seat plurality calls for F = −1 so that the seat product is reduced to

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Rules and Tools

1 at M = S. In between, a divisor close to 0.2 would correspond to F = 0. Thisassignment depends on taking M = 10 as the magnitude where the divisor effectis required to be nil, compared to FPTP. The precise value of d may depend onassembly size. Here we have a first approximation.

In sum, we must have F = −1 for d = 0 (plurality) and F = +1 for d = 2 (Sainte-Laguë). For d = 1 (d’Hondt) we would expect a value of F moderately below 1, andfor d = 3 (Danish) moderately above 1. These demands are roughly satisfied, if weset the relationship between F and d to

F = 2(

d2

)k

− 1.

The parameter k depends on the value of d at which we want F to assume thevalue 0. If d = 0.2 for F = 0, then the equation above leads to k = log0.5/log0.1 =0.30. Hence the tentative connection between divisor gap and allocation formulaexponent F might be around

F = 2(

d2

)0.3

− 1.

This is only a preliminary estimate. No constant divisor gap exists for which pro-portionality would be independent of magnitude. One can only balance decreas-ing proportionality at low magnitudes and increasing proportionality at highmagnitudes. Empirical usefulness of the formula above remains to be tested, and Ihave as yet no theoretical justification. Empirically, it would lead to the values ofallocation formula exponent F listed in Table 6.5. This table focuses on divisorformulas, but the general approach used should apply to quota formulas withvarious quota values, taking Hare-LR to correspond to F = 1.00.

Implication of seat product for a single district

When considering a single district of M seats, one would have to set the totalnumber of seats also at M, so that the seat product becomesMF M = MF +1. Withplurality formula (F = −1), this equation leads to M0 = 1, as it should, as one partywins all the seats. With a perfect PR formula (F = 1), we get M2.

The values of F in Table 6.5 would imply the following. The same degree of pro-portionality that prevails with Sainte-Laguë at magnitude M should prevail alreadyat a lower magnitude M(2/2.26) = M0.88 with Danish divisors, at a higher magnitudeM(2/1.62) = M1.23 with d’Hondt, and at a still higher magnitude M2/1.32 = M1.52

with Imperiali. (With plurality, this magnitude would be pushed to infinity.) Aquick comparison with Tables 6.1 to 6.3 shows both agreement and disagreement,depending on the value of M and also on the criteria of proportionality used(deviation from PR, either D1 or D2, or closeness of the seat-based effective numberto the vote-based).

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The aggregate of a simple electoral system

In some contexts, the square root of the seat product appears as the underlyingbasic building block for expressing the impact of an electoral system. I will desig-nate it as the ‘aggregate’ (A) of the electoral system:

A = (MF S)1/2.

Subsequent chapters will present the logical arguments and observational tests forrelationships between the electoral system aggregate and the various measures ofthe number of parties, as highlighted in Chapter 4. With PR or FPTP (F = 1), thefollowing applies, on the average:

N0 = the number of seat-winning parties = A1/2 (see Chapter 8).

N2 = the effective number of legislative parties = A1/3 (see Chapter 9).

N∞ = inverse of the largest seat share = 1/s1 = A1/4 (see Chapter 8).

These are, respectively, the square, cube, and fourth roots of the electoral systemaggregate. Consequently, the average relationships among these measures of thenumber of parties form a remarkable series with exponents 1, 2, 3, and 4:

N4∞ = N3

2 = N20 = A1.

What it implies is that, never mind how one tries to measure the number of parties,it still boils down to the same aggregate of assembly size and district magnitude,the latter critically modulated by the seat allocation formula.

What does A represent? Allocation by plurality in a single nationwide districtleads to A = 1, expressing the fact that only one party can win seats in the assembly.At the other extreme, M = S with PR allocation yields A = S. It expresses the factthat up to S parties can conceivably win seats in the assembly—even while such anoutcome is highly unlikely. Thus, at both extremes, the electoral system aggregateequals the maximum number of parties that conceivably could win a seat.

Unfortunately, such a simple interpretation of the aggregate no longer applieswhen the country is divided into several districts (M < S). Take for instance thecase M = 1, where PR and plurality rules both yield A = S1/2. The number of partieswinning seats could still be as high as S, if a different party (or independent)wins in each district. As long as a PR allocation formula is applied, the conceiv-able maximum number of seat-winning parties remains the same, S, regardlessof whether district magnitude is S or 1 or anything in-between. We may sensethat the likely number of seat-winning parties will go down as district magnitudedecreases. This is what N0 = A1/2 expresses. Thus, for FPTP (M = 1), the likeliestnumber of seat-winning parties would be A1/2 = S1/4. With 2-seat districts, thenumber of seat-winning parties would be expected to go slightly up with PR rules,to (2S)1/4 = 1.19S1/4, and down with plurality, to (S/2)1/4 = 0.84S1/4.

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Rules and Tools

It still is not clear whether the seat product or the aggregate should be seen asthe centerpiece of an electoral system, and so I present them both. As seen above,the aggregate has appealing theoretical features, but in the chapters that follow,the product MS fits in naturally into my prose, while the aggregate does not. Also,it would be hard to get a political practitioner to be involved with something thatincludes a square root! So I use MS more often than A.

The order in which discoveries are made is rarely the logical order in which it isconvenient to present them in retrospect. It is evident that multi-seat plurality ruleis even less favorable to small parties than FPTP. But that it acts on the number ofparties as if it changed a magnitude M into 1/M—this intuition has bugged me forseveral decades, without my being able to prove it. The relationships between theseat product MS and the various measures of the number of parties were graduallydeveloped and tested in articles published in 1993–2006. A final correction for theeffective number of legislative parties was made while writing this book. Only thencould the pieces come together into the dual notion of seat product and aggregateof an electoral system.

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Part II

The Duvergerian Macro-Agenda:How Simple Electoral SystemsAffect Party Sizes and Politics

Ask not what the electoral rules can do for your country, ask whatthose rules can do on the average.

A Wuffle


7

The Duvergerian Agenda


� Starting with the characteristics of the electoral system, the ‘Duverg-erian agenda’ aims at predicting the average seat and vote sharedistributions of parties, their effective number, and deviation fromPR.

� In the case of simple electoral systems, such prediction has becomepossible for seat shares and cabinet duration.

� Simple electoral systems are those using a usual PR formula or first-past-the-post, so that assembly size and district magnitude tell the wholestory.

� The following chapters will offer specific formulas and tell when theycan or cannot be used.

Duverger’s law has been mentioned from the very first chapter on.Maurice Duverger (1951, 1954) highlighted the possibility of predictablerelationships between electoral systems and political outcomes. Thesearch for such regularities has been called the ‘Duvergerian agenda’(Shugart 2006: 28), and it arguably has formed the core of the field ofelectoral studies during the late 1900s. This search looks for answers toquestions like: How does the electoral system shape the party system? Towhat extent are voters’ choices affected by electoral rules? And what arethe processes that cause the relationships found?

The very idea of the existence of predictable relationships between elec-toral systems and party political consequences remains controversial, butit keeps revolving. Our aim should be to go beyond qualitative statementsand establish quantitative average patterns. Semi-quantitative answers

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The Duvergerian Macro-Agenda

such as Colomer’s micro-mega rule (Chapter 6) are useful guides, but oneshould get as specific as possible.

This chapter specifies the core idea of the Duvergerian approach andtraces its development over the last fifty years. It indicates the broadresearch agenda this approach suggests and the opportunities the simplestelectoral systems present, in particular. A road map results for the severalfollowing chapters, which form the centerpiece of this book.

The Core Idea of the Duvergerian Approach

One broad idea underlies the line of inquiry that received a major boostfrom Duverger’s work, although Duverger himself expressed it in a nar-rower form. When the electoral system is simple:

average distribution of party sizes depends on the number of seatsavailable.

Directly, this means the number of seats in the electoral district. Single-seat districts restrict the number of parties more than do multi-seat dis-tricts. However, the total number of seats in the representative assemblymatters, because more seats offer more room for variety. It is possible tohave more than 10 parties in a 500-seat assembly, but not in the 10-seatnational assembly of St Kitts and Nevis. At the same district magnitude,a larger assembly is likely to have more parties, all other factor being thesame. Once we add the impact of the seat allocation formula, Colomer’smicro-mega rule and the notion of seat product emerge (cf. Chapter 6).

The chapters that follow will deduce and test the logical consequencesof this dependence of party system on the number of seats availablein the assembly and the district. Within the Duvergerian agenda, thisis the ‘macro’ part in that it deals with system-level variables (Shugart2006). The complementary ‘micro’ part tries to elucidate how such macro-changes emerge from decisions made on the individual level, by votersand politicians. Before proceeding to the resulting research agenda, thischapter summarizes the history of the Duvergerian thought.

Duverger’s Law and Hypothesis: Mechanicaland Psychological Effects

The study of electoral systems began with advocacy pieces for specific setsof rules, such as those written by Borda (cf. Colomer 2004b: 30), Hare

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(1859), and Mill (1861). This tradition continued up to the mid-1900s.(For details, see Taagepera and Shugart 1989: 47–50 or Colomer 2004b).A major analytical landmark was reached with Maurice Duverger’s work.Duverger (1951, 1954) was the first to announce clearly what came to becalled Duverger’s law and hypothesis (Riker 1982), making a connectionbetween electoral and party systems. Avoiding implications of unidirec-tional causality, they can be worded as follows:

(1) Seat allocation by FPTP tends to go with two major parties (‘law’).

(2) PR formulas in multi-seat districts tend to go with more than twomajor parties (‘hypothesis’, because more exceptions were encoun-tered).

Note that the Duverger statements (law and hypothesis) involve only oneparameter, district magnitude. This means they address only the systems Ihave called simple. They say nothing about elections with run-offs, tiers,legal thresholds, ordinal ballots, or any other complications, unless suchcomplex systems are somehow reduced to analogous simple systems, ashas been tried in different ways by Taagepera and Shugart (1989) andLijphart (1994).

These statements can be made more specific thanks to improved opera-tionalization of the notions involved. Rae (1967) coined the term district‘magnitude’ and applied it to systematic worldwide analysis. Laakso andTaagepera (1979) introduced the effective number of parties. In thoseterms, we could interpret Duverger’s law to mean ‘M = 1 goes with 1.5 <

N < 2.5’, while Duverger’s hypothesis means ‘M ≥ 2 goes with N > 2.5’.Actually, as district magnitude increases from M = 1 to M = S (nation-

wide single district), the number of parties tends to increase graduallyand at a decreasing rate, as first shown graphically in Taagepera andShugart (1989: 144). In this light, the discontinuity between the law andhypothesis should be removed, leading to a single function N = f (M)for the average pattern. Taagepera and Shugart (1989: 144, 153) offeredempirical equations, but no logical explanation for the patterns observedcould be found. These equations no longer should be used, because mod-els with a stronger conceptual foundation are presented in Chapters 9and 14.

What produces the outcomes noted by Duverger? Low district magni-tudes (and M = 1 in particular) arguably put a squeeze on the numberof parties in two ways. In any single-seat district with plurality, one ofthe two largest parties nationwide will win, unless a third party has a

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local concentration of votes quite different from its nationwide degreeof support. This is the so-called Duverger mechanical effect. Hence thirdparty votes most often are ‘wasted’ (for the purpose of winning seats), sothat these parties are underpaid, nationwide. Correspondingly, the twolargest parties will be overpaid in terms of seats. This effect is observedinstantaneously, for any given election, once the seat and vote shares arecompared. In this sense, it is ‘mechanical’.

In contrast, the so-called Duverger psychological effect develops slowly,over several elections. The mechanical effect means that votes for thirdparties are effectively wasted in most districts. In the next election, somevoters are tempted to abandon such parties, except in the few districtswhere the third party won or came close. With reduced votes, such partiesstand to win still fewer seats in the next election, causing even furthervoters to give up on them. Thus, third parties are gradually eliminated,unless they have local strongholds. But even there, voters may hesitatebetween a preferred third party and a tolerable major party, which hasmore chances to form the cabinet and bring resources to the district.

The psychological effect is often presented in terms of voter strategies,but it also works on politicians and contributors. Anticipating anotherdefeat and lacking resources, a third party may desist from running ina district even before its former voters have a chance to abandon it.Financial contributors may be hard to find, and few people may volun-teer to campaign for a lost cause. ‘Scholars disagree over which of thesecausal mechanisms—strategic voting in the mass electorate or strategiccontributing in the elite strata—is the more important. . . . In my view,both kinds of resource concentration are important. Elites typically actfirst’ (Cox 1997: 30).

The Duverger effects apply foremost at the district level. This is wherethe seat is lost or won and where the votes are wasted or not, regardlessof nationwide results. Voters have no direct reason to abandon a thirdparty nationwide who won in their own district—or only narrowly lostand could win in the next election. The extension of the psychologicaleffect to the nationwide scene need not follow, but it often does, if votersperceive the third party representatives as ineffective in the assembly.Third parties have vanished in the USA, but have survived and even madea comeback in the United Kingdom. In Canada, Duverger’s law operatesat the provincial level, but the two dominant parties are not the same inall provinces, leading to a more scattered nationwide pattern. The pictureis even more diverse in India.

104


Many non-Duvergerian factors may counter the psychological effect(and other strategic considerations) in UK and elsewhere. When partiesare successful in subnational or supranational elections where PR is used(such as elections to the Scottish Assembly or the Parliament of the Euro-pean Union), these parties are motivated to show their flag in the FPTPelections too. When parties are publicly financed, it may pay to run evenwhen few or no seats are won. The advent of TV may enhance third partyvisibility, compared to printed press, and the Internet makes intrapartycontacts less expensive for parties with dispersed memberships. When themajor party programs converge toward the middle voter (Downs 1957),they may come to look so similar in the eyes of third party voters thatneither party may be seen as a ‘lesser evil’ worth a strategic shift awayfrom the third party.

Strategic considerations, such as those in the classical Duverger effects,are not limited to single-seat districts. As district magnitude increases,these effects are attenuated but still restrict the smaller parties. If a countryhas larger and smaller districts, smaller parties may decide to run only inthe larger districts. Thus in Finland 1962–83, about 7.5 parties ran in thesmallest districts (M = 7 to 11), while about 8.5 did in the largest (M = 17to 27). The effective number of electoral parties was around 4.6 in thesmallest and around 5.1 in the largest (Taagepera and Shugart 1989: 119).Once again, parties may give up on voters before voters have a chanceto abandon the parties. It is hard to sort out the strategic considerationsof voters, a party’s anticipation of such considerations, and the party’sown dilemma between concentration of resources and showing the flagin many districts.

The Broad Duvergerian Agenda

The Duvergerian agenda consists of explaining and predicting the resultsand causes of Duverger’s effects. It includes micro and macro aspects. Amicro dimension underlies the psychological effect and related strategicconsiderations. It involves the individual decisions of voters, party leadersand contributors in what Cox (1997) calls strategic coordination. Reed(1991) observed that, in Japanese SNTV elections, M + 1 ‘serious’ candi-dates tend to run in a district with M seats. Cox (1997: 99) proposed anextended M + 1 rule as a direct generalization of Duverger’s law and testedit in various ways. This issue will be revisited in Chapter 15.

105


The longstanding macroscopic approach tries to make use of the restric-tions imposed by electoral rules (low district magnitude, in particular) toexplain the number and size distribution of parties, as well as the degreeof disproportionality of seats to votes. The number of political cleavagesor ‘issue dimensions’ is also taken into account, to the extent it can beestimated independently of party differences. In a recent overview ofelectoral systems, Shugart (2006) considers the macro dimension of theDuvergerian agenda the ‘core of the core’ of electoral studies.

Since 1980, advances in the study of simple electoral systems for par-liamentary elections have been such that Shugart (2006) feels that ‘theagenda of proportionality and number of parties is largely closed’ andneeds only fine-tuning. But it is always risky to call an agenda closed.Around 1900, just prior to the birth of relativity and quantum mechanics,many considered physics a closed field.

True, the ‘core of cores’ of the Duvergerian agenda has been investigatedto the point where meaningful spin-offs have become possible towardsystematic investigation of more complex electoral systems, intrapartyimpact of electoral rules, and the effects of ‘second-order’ rules such asclosed versus open lists. But this need not mean that the core issues areresolved to a satisfactory degree. This book presents recent findings in themacro-Duvergerian realm and indicates that quite a lot still remains to bedone.

The Duverger statements, important as they are historically, still giveonly semiquantitative answers. In FPTP systems, should we visualizeDuverger’s law more as 52-48 or as 50-40-10? Saying that it could meaneither and that more precision is not needed would restrict prediction tothe level politicians can handle on their own, without a need for politicalscientists. It surely would be of interest to know which electoral lawsare more conducive to 52-48 rather than to 50-40-10. Saying that suchprecision cannot be achieved would mean giving up before even trying.For beginners, we can determine the empirical world average pattern forall FPTP systems at various assembly sizes. Once we have such a baseline,we can measure by how much individual countries deviate from it, andask why. Alternatively or concurrently, we may ask what patterns wewould expect on logical grounds.

In multi-seat PR systems, similar questions need answers. Duverger’shypothesis merely predicts more than two significant parties. But howmany, and what are their most likely relative sizes? Obviously, districtmagnitude matters, as approximated by outdated empirical equationsoffered by Taagepera and Shugart (1989: 144, 153). But if, at a given

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magnitude, N is 3.00, does it more often imply three equal parties (34-33-3) or one large party and several smaller ones (45-29-21-16)? And whyis the empirical relationship what it is, rather than some other expressionin M? There must be a reason for the specific form, and we would not bescientists if we did not try to find it.

Colomer’s micro-mega rule expands the scope of Duverger’s statements,by going beyond district magnitude and including assembly size and seatallocation formula. But it still is only semi-quantitative. For the tiniestparties, Colomer’s recipe is clear: Promote as large assemblies, districtmagnitudes, and quotas as possible. But what would be the optimalcombinations for a medium-sized party?

In retrospect, no progress could be made as long as the dominantpicture was votes determining the seats, with the electoral system a blackbox in-between the votes and seats (cf. Taagepera and Shugart 1989: 64,202). The erroneous central idea was that this black box determines howvotes are translated into seats:

VOTESELECTORAL SYSTEM−−−−−−−−−−−−−−−−→ SEATS. [WRONG]

The breakthrough came with the realization (Taagepera and Shugart 1993)that the electoral system is not an intervening control box between votesand seats. Rather, votes and electoral system both affect seats, from oppo-site directions:

VOTES −−−−→ SEATS ←−−−− ELECTORAL SYSTEM. [RIGHT]

This may seem a minor difference in visualizing the Duvergerian idea.However, the first format leads to a dead end, while the second oneopened up a way to crack the Duvergerian nut. The basic idea wasexpressed in Figure 1.1. Figure 7.1 introduces the mechanical and psy-chological effects. For simplicity, it omits political culture.

ELECTORALSYSTEM

SEATSDISTRIB-UTION

VOTESDISTRIB-UTION

MECHANICALEFFECT

PSYCHOLOGICALEFFECT

CURRENTPOLITICS,CULTURE,HISTORY

Figure 7.1. The opposite impacts of current politics and electoral system

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For individual elections, votes come first, based on current politicsand, more remotely, on the country’s historical peculiarities (‘path depen-dence’). These votes will determine the seats, in conjunction with themechanical effect of the electoral system. For the average of manyelections, however, the causal arrow reverses its direction. Through themechanical effect, electoral system pressures the distribution of seats toconform to what best fits in with the total number of seats available.Through the psychological effect, the electoral system eventually alsoimpacts the distribution of votes, possibly counteracting culture andhistory.

Indeed, voters are no longer free to vote for a seventh-ranking party,if the electoral system has deprived it of seats and it has stopped toexist. As a result, the average of many elections in many countriesusing similar electoral systems may produce a predictable pattern. Doother factors matter, such as a country’s historical tradition and cul-ture, and the moment’s political events? Of course they do. But theycan be addressed only when the more universal patterns have beenelucidated.

Actually, the impact of the electoral system reaches even further thanvotes. Expanding on findings by Anderson and Guillory (1997) andKlingemann (1999), Andrew Drummond (2006) has documented thatit affects political attitudes. Those voters whose preferred parties winelections tend to view elections, government and even democracy morefavorably than the losers. On the average, majoritarian electoral systemscreate more losers, and the loss itself is starker. As a result, overall evalua-tion of politics and democracy tends to be lower in majoritarian systemsand higher in PR systems.

Going even further, Lijphart (1999: 270–300) brings evidence thatpeople in consensual countries, largely defined by PR electoral system,not only are more satisfied with democracy but also have less socialviolence, more political equality, and higher women’s representation.Lijphart (1999: 258–70) also challenges the widespread belief that con-sensual countries are supposedly less efficient economically.

Consensus politics involves aspects beyond electoral system, so someof these differences may not derive from electoral systems as such. Itcould well be that countries with inherently consensual political culturestend to choose multi-seat PR rather than FPTP in the first place. It wouldbe harder to claim that people initially satisfied with democracy choosemulti-seat PR. This book will not investigate such cultural effects of elec-toral systems.

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The Macro-Duvergerian Agenda for Simple Electoral Systems

The overall road map for the central part of this book is shown inFigure 7.2, to be discussed in some detail. It expands and clarifies ascheme first offered in Taagepera (2001), tracing the connections to be

FOUNDINGPARTIES POPULATION

DISTRICTMAGNITUDE

ASSEMBLYSIZE

EFFECTIVENUMBER OF

PARLIAMENTARYPARTIES

OTHERSEAT

SHARES

VOTESHARES

CABINETDURATION

DEVIATIONFROM PR

EFFECTIVENUMBER OFELECTORALPARTIES

Self-Preservation

STRATEGICCONSIDERATIONS

DISPROPORTIONALITYEXPONENT

LARGESTSEAT

SHARE

ELECTORATE

Figure 7.2. The macro-Duvergerian agenda, as of 2007

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specified in the following chapters. All this applies only to simple electoralsystems—those that include no features beyond assembly size, a fairlyuniform magnitude for districts, and seat allocation according to a usualPR formula (which boils down to FPTP when M = 1).

Thick arrows indicate definitions, such as defining the effective num-ber on the basis of seat shares of parties. Thin arrows indicate connec-tions for which we have quantitatively predictive models. Dashed arrowsshow conceptual connections for which more than fine-tuning is needed,because even the broadest form of the quantitative model is fuzzy ormissing. Only downward arrows are shown, because this is the predomi-nant direction of causality under usual conditions, but mutual interactionis not to be excluded. For instance, politics usually has little effect onpopulation size, but when politics leads to secession or annexation, it hasa major effect.

Starting from the top of Figure 7.2, population strongly constrainsassembly size. It will be seen (Chapter 12) that a cube relationship pre-vails: Assemblies of 100 seats tend to go with 1,000,000 people, whileisland countries with little more than 1,000 people have little more than10 seats. In democracies, electorate is almost proportional to population.

District magnitude is wide open in principle. The founding parties canchoose any magnitude, from M = 1 (FPTP) to M = S, the assembly size. Par-liamentary parties can later initiate changes. Actually, choice is restrictedby the self-preservation instinct of parties, as condensed in Colomer’smicro-mega rule. True, in new democracies the dominant parties may beshort-sighted and act contrary to their long-term interests. Boix (1999)and Colomer (2005) have advanced our knowledge of how parties chooseelectoral systems, but a quantitatively predictive model still eludes us. Oneof the best rules of thumb still is ‘British heritage → FPTP, no Franco-British heritage → List PR’ (cf. Chapter 3).

District magnitude places constraints on the number of parties thatcan win seats in the district. The expected average number of seat-winningparties in one district can be determined, using the ignorance-based modelapproach (cf. Taagepera 1999b, 2003). Adding the constraints exerted byassembly size, the same approach enables us to predict the average num-ber of seat-winning parties nationwide one could expect in the absence ofany other information. This is where the seat product MS (cf. Chapter 6)emerges, as shown in Chapter 8. For clarity, Figure 7.2 omits the numberof seat-winning parties.

The number of seat-winning parties, in turn, constrains the seat shareof the largest party. The largest share cannot be more than 100 percent of

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the entire assembly, nor can it be less than the average share, which is theinverse of the total number of seat-winning parties. The ignorance-basedmodel approach allows us to determine the expected average seat share ofthe largest party, based only on the number of seat-winning parties, itselfbased on the seat product MS. It is a purely institutional model, up to thispoint, and it fits the data, on the average. This model is also developedand tested in Chapter 8.

Subtracting the expected seat share of the largest party from the totalyields the range in which the second largest share can lie. On this basisthe entire most likely distribution of seat shares of parties can be inferred, inthe absence of any other information. However, at this stage, the observedaverage distribution differs from the one predicted by the purely institu-tional model. For the first time, we have to introduce a non-institutionalparameter to account for strategic and other factors that hurt the smallerparties. A part of their inherent support is shifted to major parties. Agood fit to the observed average seat share distribution is obtained inChapter 9 when the transfer parameter is set around a half.

The effective number of legislative parties can be estimated in two ways. Itcan be calculated on the basis of the estimated seat shares of all parties.Alternatively, it can be deduced from the largest share alone. Indeed,the largest share places constraints on the value the effective numberof parties can take. While this approach is less precise than the onebased on the shares of all parties, it has the advantage of being purelyinstitutional, bypassing the strategic considerations that work against thesmall parties. Thus it enables us to make a prediction (in Chapter 9) forthe effective number of legislative parties, based on the seat product MSalone.

A major payoff comes in Chapter 10, when the effective number oflegislative parties, in turn, is connected to duration of governmentalcabinets. This time, the predictive model does not use the ignorance-basedapproach but optimizes the number of communication channels. Theoverall outcome is a specific prediction regarding average cabinet duration,made solely on the basis of the seat product MS plus one empiricallydetermined constant.

By this time, the ignorance-based approach has been repeated so manytimes that the impact of district magnitude and assembly size is quitedistant and can be expected to be completely blurred out by other politicaland cultural factors. The wonder is that this is not the case. The institu-tional effect on cabinet duration is still evident—and it has the predictedfunctional form.

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Thus, this sequence of predictive models actually offers a baseline forinformed institutional engineering. Population largely fixes the assemblysize, but district magnitude can be modified so as to alter average cabinetduration not only in the desired direction but also to the desired degree.However, the given country’s historical tendency to deviate from world-wide averages must be corrected for. With the same seat product, Italy willhave shorter lasting cabinets than Spain.

What about the third ingredient of the micro-mega rule, the seatallocation formula? As long as one keeps away from multi-seat plural-ity, the allocation formula affects the impact of district magnitude toa relatively minor extent (cf. Figure 6.1 and Table 6.5). The followingchapters study in detail the links in the concatenation that extends fromdistrict magnitude to cabinet duration. At the same district magnitude, weshould expect the d’Hondt systems to deviate from the average towardthe direction less favorable to small parties–fewer seat-winning parties,lower effective number of parties, and longer cabinet durations. The Hare-LR and Sainte-Laguë systems should fit better. The difference, however, issmall and hard to detect.

This concludes the study of the largely mechanical effect of electoralsystems on the distribution of seats—the left half of Figure 7.1. This partof the macro-Duvergerian agenda is now largely closed, except for con-nections to population and founding parties. Details about the effectivenumber of parties need to be fine-tuned. Future emphasis would be onextending the theory from simple to more complex electoral systems.

The largely strategic (‘psychological’) impact of electoral systems on thedistribution of votes, on the other hand—the central part of Figure 7.1—still remains wide open. This is the region at the bottom right of Fig-ure 7.2. Institutions are bound to impact votes in a fuzzier way than seats.An electoral system can block the seventh-largest party from getting anyseats, but it cannot prevent people from voting for this party, if they reallyinsist and the party refuses to fold.

How are seat shares connected to vote shares of parties? This is the thindashed horizontal line at lower right of Figure 7.2. Here the relationshipwas first worked out in the opposite direction, going from votes to seats.This work started a century ago with the so-called cube law of Anglo-Saxon elections that applied to FPTP systems and empirically connectedthe seat ratios of two parties to their vote ratios. The empirical cube lawwas extended into a theory-based seat-vote equation (Taagepera 1973,1986) that applied to PR elections, too. It can now be seen as partof a broader law of minority attrition (Chapter 13). The disproportionality

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exponent that enters here depends on the number of voters (electorate)and seats (assembly size). It is strongly conditioned by district magni-tude. Throughout the path from seat-winning parties to cabinet duration,assembly size and district magnitude play an essentially symmetrical rolein the form of the seat product. In contrast, their role is cardinally differ-ent in the law of minority attrition.

This law can be reversed to go from seats to votes. (Only this direction isshown in Figure 7.2.) This way, the vote shares, too, can be inferred fromassembly size and district magnitude alone, but with increasing blur, plusa new difficulty. Parties with few votes are easily predicted to win no seats.But how does one go in the reverse direction and estimate the vote sharesof parties that run and do not get any seats? Ways to work out the mostprobable distributions need refining.

Once this major link has been completed (Chapter 14), the effectivenumber of electoral parties follows from vote shares. It is found to be moremanageable, however, to estimate the largest vote share from the largestseat share and the seat-vote equation, and use it to estimate the effectivenumber of electoral parties.

The difference between the largest seat and vote shares also enables usto estimate at least some of the indices of deviation from PR. By this time,one can expect that the distant connection to the institutional factors(assembly size and district magnitude) would largely be overridden byother political and cultural factors. Surprisingly, some faint connectioncan still be detected, but full testing remains to be done. Only at thatpoint could we say that the macro-Duvergerian agenda is closed, as far asthe simple electoral systems are concerned, apart from fine-tuning.

Conclusion

The quantitative study of the relations between votes, seats, and electoralsystems took off with the observation of the ‘cube law’ at the beginningof the twentieth century. It received a major boost with Maurice Duvergerin the mid-1900s, to the point that the core of electoral studies duringthe most recent 50 years has largely consisted in trying to implement theDuvergerian agenda.

The core idea of the Duvergerian approach is that, when the electoralsystem is simple, the average distribution of party seat shares depends onthe number of seats available. As the next chapters document in detail, itcan now be specified that this distribution depends on the product of

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the number of seats available in the assembly and in the district. Whenone shifts from legislative to electoral parties, assembly size and districtmagnitude begin to enter asymmetrically. Plurality rule and complex andcompound electoral systems need separate treatment.

The macro-Duvergerian agenda has recently made marked advances,and for simple electoral systems it might become closed in the near future.This would mean that the average seat and vote share distributions and theresulting measures of number of parties and deviation from PR could beinferred from the characteristics of the electoral system. Proportionalityprofiles could be predicted. Thereafter, the macro-Duvergerian agendawould focus on elucidation of more complex electoral systems and elec-tions other than nationwide legislative elections. The micro-Duvergerianagenda remains to be developed to the point where quantitative predic-tions can supplement postdictions and the macro-level phenomena canbe explained through micro-level processes.

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8

The Number of Seat-Winning Partiesand the Largest Seat Share


� The quantity to watch is the product of the number of seats in theassembly and the number of seats allocated in the average district.

� The larger this ‘seat product’, the larger the number of parties in theassembly.

� The larger the seat product, the smaller the seat share of the largestparty.

� If you wish to increase the largest seat share by one-tenth (0.1), yourbest bet is to multiply (1+0.1) by itself 8 times, which yields 2.14. Thisis by how much you must divide the present seat product. This meansthat either you split the present districts into two smaller districts oryou cut the assembly size by a half.

� If you wish to reduce the largest seat share by one-tenth, your best betis to multiply (1 − 0.1) by itself 8 times, which yields 0.43. This is byhow much you must divide the present seat product. To do so, you canincrease district magnitude or assembly size or both.

� This way to calculate is based on a logical model that agrees with theworld average. It is approximate, because other factors enter.

� At the same average district magnitude, widely unequal districts usuallyreduce the seat share of the largest party.

Here we start from nothing but district magnitude and assembly size, andpresume FPTP or a standard PR seat allocation rule. On that institutionalbasis alone, we predict how many parties are likely to win at least oneseat, and how large the seat share of the largest party is likely to be,

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on the average. If you think it cannot be that simple, you have lotsof company, but I will try to explain why it works. The world averagesupplies a comparison point for individual countries.

The main body of the chapter offers what is needed to apply thepredictive models developed, along with evidence about their degree ofvalidity for simple electoral systems. It concludes with implications forinstitutional engineering. Most derivations of models are given in chapterappendix, along with more technical and philosophical concerns.

The Number of Seat-Winning Parties

The logical model to be presented claims that the most likely number ofparties (p) that win at least one seat in an isolated district of M seats is(Taagepera and Shugart 1993)

p = M1/2,

when List PR is used. It obviously also applies to FPTP. Nationwide, whenan assembly of S seats is elected in districts of M seats, the most likelynumber of seat-winning parties (N0) is, according to this model,

N0 = (MS)1/4.

This means that, with a large numbers of cases, we expect one-half ofthem to fall above and one-half below the value N0 = (MS)1/4. No pre-diction is made about individual elections, where the number of seat-winning parties could deviate widely from the model. The mean for manyelections using a simple electoral system (List PR or FPTP) is expected tobe within a factor of 2 of the model, that is, no more than the double andno less than a half of the predicted number of seat-winning parties. Wecannot expect to be closer, because country-specific factors beyond theseat product also enter. Finally, the mean of many electoral systems withthe same seat product is expected to be close to the model, because herethe country-specific factors should cancel out.

Before presenting the quantitatively predictive model, let us see howwell it works. All those elections in Mackie and Rose (1991) were consid-ered where at least four elections were carried out under the same rulesand all seats were allocated in districts. Values for the resulting thirty elec-toral systems are tabulated in Taagepera (2002b). Table 8.1 compares theactual and predicted numbers of seat-winning parties, using the geometricmeans for electoral systems with the same seat allocation formula.

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Table 8.1. The actual number of seat-winning parties, the expectednumber (based on district magnitude and assembly size), and theirratio

Electoral formula No. of systems Actual N0 N0 = (MS)1/4 Ratio

FPTP 9 3.57 3.87 0.92Modified Sainte-Laguë 2 5.99 6.29 0.95STV 3 4.29 3.95 1.09D’Hondt 8 6.96 6.12 1.14Alternate Vote 2 3.54 3.05 1.16SNTV (Japan) 1 10.0 6.6 1.5Two Rounds 5 7.03 4.04 1.74

All M = 1 systems 16 4.41 3.82 1.15All M > 1 systems 14 6.30 5.63 1.12All systems 30 5.21 4.58 1.14

Data source: Taagepera (2002b).

The electoral systems are listed in Table 8.1 in the order of increasingratios of the actual and expected numbers of seat-winning parties. Thedeviations of these ratios from 1.00 indicate the degree of fit of the model.The means of simple electoral systems fit within 15 percent. The modelneed not fit for more complex systems, but for STV and Alternate Vote,it still does. Only for Two-Rounds systems and the single case of SNTVwould we need to look for factors beyond the impact of the seat product.Among all 30 electoral systems, the actual-to-expected ratio is the lowestfor the USA 1938–88 (0.54) and the highest for two Two-Rounds systems,Germany 1871–1912 (3.0) and the Netherlands 1888–1913 (2.0). SeeTaagepera (2002b) for individual electoral systems and various method-ological issues.

Figure 8.1 uses the same data to graph the actual N0 for 30 electoralsystems against the seat product MS, both on logarithmic scales. The meanexpectation line N0 = (MS)1/4 is shown, as well as the lines at one-half andat twice these values. The data points are expected to be crowded in thecenter of the zone delineated by the latter two lines. If the model doesnot hold, the data points could be scattered all over the place or theycould all be above or below the predicted line. Except for two Two-Roundssystems (Germany 1871–1912 and the Netherlands 1888–1913), systemaverages are located in the expected zone. Thus the model has somemerit.

Any acceptable model must include the anchor point MS = 1 → N0 = 1(also shown in Figure 8.1), because one seat obviously goes to one andonly one party. The best-fitting line (not shown) that passes through this

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1

10

100

10,0001,000100101

Seat Product (MS )

Num

ber

of S

eat-

Win

ning

Par

ties

(N0)

N0 = 2(MS).25

N0 = (MS).25

N0 = (MS).25/2

USA 1938−

NET 1888−

GER 1871−SPA

IRE

SWIJPN

M > 1

M = 1

Figure 8.1. The number of seat-winning parties vs. the seat product MS

Data source: Taagepera (2002b).

anchor point corresponds to N0 = (MS)0.26 rather than the predicted N0 =(MS)0.25. The difference is a minor one.

Note that the criteria of agreement with the model differ for predic-tive models and for postdictive statistical data fits (see Taagepera 2008).For the latter, measures of scatter, such as R2, are the main criteria. Forpredictive models, in contrast, closeness to the predicted value is whatmatters. The visibly low R2 in Figure 8.1 is due to the relatively shortrange that MS can take on the logarithmic scale, but all simple systemsare within the expected zone. Indeed, if the FPTP systems were testedseparately, R2 would be close to 0, due to the short range of MS=S, but thepredictive model would still be confirmed within an 8 percent averagedeviation from the model.

The Number of Seat-Winning Parties: The Model forSingle District

Having shown that the model reflects reality, I now present the first partof the model, the one for a single district, in some length, because it

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introduces the broad idea of ‘ignorance-based models’(Taagepera 1999b),explained in more detail in Beyond Regression (Taagepera 2008). It is impor-tant to clarify this broad idea, as it will be used repeatedly later on. Thenationwide model for seat-winning parties is placed in chapter appendix,as will subsequent models based on this approach.

Consider an electoral district with 100 seats, such as the Netherlandsactually had in 1918–52 in its first chamber—a single nationwide district.How many parties are likely to win seats, and what is the likely averagenumber of seats per party? Assume a usual PR formula, so as to excludemulti-seat plurality. Also assume that each seat is allocated to one specificparty.

At one extreme, a single party could win all the seats. At the other,100 parties could win one seat each. Both extreme outcomes are unlikelyunder PR rules, but they are conceptually possible. In contrast, a numberof parties below 1 or above 100 is logically impossible. In the absence ofany other information, if we had to hazard a guess, we would minimizethe maximum error by choosing the mean between the logical limits.For reasons explained in Taagepera (2008), the geometric mean shouldbe used, because both limits are positive and, moreover, vary by severalorders of magnitude. Hence we would guess at 10 parties to win seats.

We could approach the problem from a different angle, asking: What isthe likely average number of seats per party? This number, too, can rangefrom 1 (when 100 parties win 1 seat each) to 100 (when one party wins allthe seats). The geometric mean is 10 seats per party. The two approachesyield congruent answers: 10 parties, each winning an average of 10 seats,amount to a total of 100 seats, which is the actual number. This may lookobvious, so why belabor it? The point is that it would not work with thearithmetic mean.

Indeed, some colleagues have wondered why I try to avoid the good oldarithmetic mean. So let us use it. The arithmetic mean of 1 and 100 partieswinning at least one seat each yields 50.5 seat-winning parties. If someonetold you that a country is likely to have as many as 50 parties in its 100-seat assembly, would your common sense accept it? Never mind, let uscontinue. The average number of seats per party can also range from 1 to100, so the arithmetic mean is 50.5 seats per party. However, an averageof 50.5 parties winning an average of 50.5 seats per party would amountto a total of 2,550 seats rather than 100! Thus, using the arithmetic meanwould lead to logical inconsistency. Only the geometric mean avoids it,for reasons given in Taagepera (2008). Maybe the present specific examplemakes it more believable on an intuitive level.

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Actually, 8 to 17 parties won seats in the 9 nationwide PR electionswhere the Netherlands had 100 seats in its first chamber (1918–52). Themean number of seats per party ranged from 5.9 to 12.5. The geometricmean was 10.29 parties winning seats, and the geometric mean seat sharewas 9.72 seats per party—pretty close to the expected 10. This exampleshows that a guess based on conceptual limits can be appreciably off foran individual election, yet can be close for the average of several elections.

Let us now generalize for districts of any magnitude M, excluding amulti-seat plurality allocation rule. Designate by p the number of seat-winning parties within one district. The conceivably allowed range for p is1 ≤ p ≤ M. In the absence of any other information, our best guess for thenumber of parties that win at least one seat is the geometric mean of theextremes:

p = M1/2.

This is a mathematically elegant expression. The reaction of many politi-cal scientists may well be the one described by Steven Reed (1996):

Political scientists are traditionally less comfortable with the assumption of math-ematical elegance than are physicists. We tend to be more comfortable with thepresumption that ‘things are more complicated than that’. More importantly, weexpect some behavioral model to underlie our theories. There is no particularbehavioral basis to Taagepera’s theory. It does not depend on rational voters orstrategic political parties. It is less a ’political’ theory than a mechanical one.Taagepera and Shugart see this characteristic as a strength (Taagepera and Shugart1993: 456). However, theories without actors seem more appropriate for physicsthan for political science... When we have an accurate equation produced in thisfashion, what is it exactly that we know? (Reed 1996: 73)

In the sense of German Verstehen, intuitive insight into what is goingon, we may have little. Yet, we have ability to predict, on a definitelynonempirical basis, and this is not to be discounted. Accordingly, Reedconcludes:

One must be able to isolate mechanical effects before one can properly evaluatebehavioral and political effects. Mechanical effects may not be as fascinating asreal politics, but they are equally important. (Reed 1996: 80)

To paraphrase Winston Churchill’s dictum about democracy, p = M1/2 isthe worst possible prediction one could make—except for all others. Thisprediction is pulled almost completely from thin air, but all the otherswould be completely so, in the absence of further information. If furtherinformation should yield a different outcome, we would be happy to

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accept it. Such information could come in two forms. One is the empiricalmedian of a large number of cases at the same M. The other is a logicalargument of a political nature. In the absence of either, p = M1/2 is thebest we can do. Note that for single-seat districts (M = 1), p = M1/2 yieldsp = 1, as it certainly should. This is an essential anchor point. What-ever format should be proposed for p = f (M), it must yield p = 1 whenM = 1.

For a single case with M > 1, no firm prediction is made. The actualfigure may be far off from p = M1/2. But for a large number of cases, ourbest guess is that one half of them would be above and the other halfbelow the curve p = M1/2. I call this best guess the expectation value. In thewords of a classical text on quantum mechanics:

The expectation value is the mathematical expectation (in the sense of probabilitytheory) for the result of a single measurement, or it is the average of the results ofa large number of measurements . . . (Schiff 1955: 24)

It is a notion useful in quantum mechanics, among others, but it involvesnothing specific to physics. If I had to predict, I would predict the expec-tation value, because everything else would be even less justified, prior toreceiving any further information. This is the meaning of ‘expectation’and ‘prediction’ in this chapter, and the following.

All this applies to one isolated district, such as a single nationwide one.In a country with many multi-seat districts, one can expect p = M1/2

to underestimate the number of seat-winning parties, for the followingreason. There is a difference between a district of M = 25 within a largercountry and a small country where all 25 assembly seats are elected withinthe same district. In the small country, all parties are generated withinthe district. In the large country, large nationwide parties with no readyconstituency within the given district may still run, bringing in fundsand personnel from elsewhere, and occasionally winning at least one ofthe 25 seats. In order to show the flag throughout the country, they mayshift resources to such districts even when this diversion might reducetheir seats in their strongholds. If so, then the number of seat-winningparties in the given district may exceed 251/2 = 5.

This phenomenon does not increase the number of parties that winseats nationwide, since it refers only to large parties. Parties that barelystand to win one or a few seats in the assembly will try to concentrate theirresources in their strongholds. Taagepera and Shugart (1993) offer a morecomplex model that accounts for the likely impact of nationwide politicsin the districts. It has not been fully tested, because we are more interested

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in the number of parties that win seats nationwide, and there, the effect ofdistrict-center interaction cancels out. The extension of the model to thenationwide scene, N0 = (MS)1/4, is given in chapter appendix. It followsfrom the conceptual limits of 1 to M parties winning seats in a districtand 1 to S parties winning seats in an assembly of S seats.

The Largest Seat Share

Connecting the seat share of the largest party (s1) to the electoral systeminvolves two steps. First, s1 is connected to the number of seat-winningparties (N0), yielding

s1 = 1/N1/20 = N−1/2

0 ,

or, in an equivalent but more symmetric form,

s1N1/20 = 1.

Second, the connection to the seat product is almost automatic, throughN0 = (MS)1/4:

s1 =1

(MS)1/8,

or, in an equivalent but more symmetric form,

s1(MS)1/8 = 1.

The derivation of the predictive model is given in chapter appendix. Itstarts with the observation that the largest party wins at least the averagenumber of seats and at most close to all seats. The rest follows. Here, themodel will be tested in three stages:

(1) largest share versus the number of seat-winning parties,(2) largest share versus the seat product for single-seat systems,(3) largest share versus the seat product for multi-seat systems.

The first stage is interconnected with the index of balance presented inChapter 4.

The Largest Share, the Number of Seat-Winning Parties, andthe Index of Balance

The model s1 = 1/N1/20 has been tested (Taagepera 2005) with those 604

elections in Mackie and Rose (1991, 1997) where the fuzzy ‘Others’

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1.0

0.5

0.3

0.2

0.1 1

Seatshareof thelargestparty

(s1)

The number of seat-winning parties (N0)

s1 = 1/N00.5

s1 = 1/N0

s1 = 1

FORBIDDENAREA

2 3 5 10 20

Figure 8.2. The median seat share of the largest party vs. the number of seat-winning partiesSource: Reprinted, with modified labels, by permission of Sage Publications Ltd from ReinTaagepera, ’Conservation of Balance in the Size of Parties’, Party Politics, 11: 283–98 (© SagePublications, 2005).

category did not include more than one seat. This meant 24 countries.The overall median of the product N1/2

0 s1 is 0.985—only 1.5 percent offthe predicted 1.000.

Figure 8.2 shows the degree of agreement at different numbers of seat-winning parties. Both s1 and N0 are graphed on logarithmic scales, sos1 = 1/N1/2

0 becomes a straight line. This median relationship is confirmedwhen 3–12 parties win seats. Deviations occur when only two parties winseats (93 elections), or more than 12 (29 elections). These deviations arediscussed in chapter appendix.

Shown as thick lines in Figure 8.2 are the conceptual limits of s1 at givenN0. The lower limit (s1 = 1/N0) occurs when all seat-winning parties haveequal shares. The upper limit (close to s1 = 1) occurs when the largest partyhas almost everything. The expected average line is the geometric meanof these extremes.

One unintended result of testing s1 = 1/N1/20 was development of the

index of balance (Taagepera 2005), introduced in Chapter 4 as a means tocomplement the effective number of parties in somewhat the same way asstandard deviation complements the mean of a normal distribution. This

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index was defined as

B =−logs1

logN0=

logN∞logN0

.

Reconsider Figure 8.2 in this light. The lower limit (s1 = 1/N0) correspondsto perfect balance (B = 1). The upper limit (close to s1 = 1) correspondsto near-complete imbalance (B ≈ 0). What Figure 8.2 says is that partysystems have a median balance of 0.5, except at N0 = 2, where thereis more balance, and at very high N0, where there is more imbalancebetween the largest party and the tiny ones.

In sum, if the number-share conservation s1N1/20 = 1 holds, then a

balance of 0.5 results. These are two ways to express the same averagerelationship. Extremely unbalanced and balanced distributions of seatscan occur and do, as shown in chapter appendix. It is just that half-balanced distributions are more frequent.

The Largest Share versus the Seat Productfor Single-Seat Systems

The clearest cases to test are systems with single-seat districts, becausemany actual single-seat systems are truly simple FPTP. With M = 1, themodel s1 = 1/(MS)1/8 is reduced to

s1 =1

(S)1/8[FPTP]

or, more symmetrically

s1S1/8 = 1. [FPTP]

Based on detailed data in Taagepera and Ensch (2006), Table 8.2 showsthe mean assembly sizes, largest seat shares and products s1S1/8 for FPTP,Two-Rounds (TR) and Alternate Vote (AV) systems. For FPTP, systems withsmall, medium, and large assemblies are shown separately. Taagepera andEnsch (2006) tabulate the country, time period and the number of elec-tions for each system. As a rule, only systems with at least 5 elections wereaccepted. However, in order to extend the range to very small assemblies,some systems with only 3 or 4 elections were included for assemblies with10 to 42 seats.

For the 24 FPTP systems, the mean product s1S1/8, expected to be 1.00, isoff by only 1.1 percent. For groups by assembly sizes, the geometric meansare off by ±4 percent at most, and these errors look random, meaning that

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Table 8.2. Assembly size (S) and the largest party’s seat share (s1), for single-seatdistrict systems

Number of elections Geometric means for

S s1 s1 S1/8

8 FPTP systems with S = 10 to 24 37 17.04 .676 0.9638 FPTP systems with S = 26 to 68 71 43.25 .653 1.0458 FPTP systems with S = 75 to 643 209 178.11 .537 1.027All 24 FPTP systems 317 1.0115 TR systems with S = 100 to 508 42 261.95 .437 0.8761 AV system 31 106.3 .497 0.890All 30 single-seat systems 390 0.985Presidential elections 1 1.000 1.000

Data source: Taagepera and Ensch (2006).

no systematic dependence on S is left unaccounted. Hence the model isconfirmed within ±1 percent. The lowest mean s1S1/8 for an individualsystem is 0.68, for the 3 elections in St Kitts 1980–9 (S = 10.3). Thehighest is 1.29, for the 7 elections in Botswana 1965–94 (S = 33.2). The84 elections in the USA 1828–1994 are also markedly off (s1S1/8 = 1.22).

The mean for 5 Two-Rounds systems, where the model is not expectedto apply, deviates much more, falling below 1.00 by 13 percent. Theindividual systems vary widely, with s1S1/8 ranging from 0.58 for the 13elections in Germany 1871–1912 (S = 396) to 1.70 for the 6 elections inItaly 1895–1913 (S = 508). On the average, Two-Rounds seems to reducethe seat share of the largest party, when controlling for assembly size.

In the single case of Alternate Vote (Australia, 31 elections in 1919–93), the low value of s1S1/8 (0.89) depends on counting the Liberal andCountry/National seats separately, like Mackie and Rose (1991, 1997) do.If the Coalition of Liberal and Country/National parties were counted as asingle party, like Nohlen, Gotz, and Hartmann (2001) do, the mean largestshare would increases to 59.2 percent and s1S1/8 = 1.06.

For all 30 single-seat systems, the mean product s1S1/8, expected toequal 1.00, is off by only 1.5 percent. Thus the model is confirmedwithin ±1.5 percent even when including the Two-Rounds systems,which can expected to deviate from the model for simple electoral sys-tems. At the bottom, Table 8.2 also includes the presidential systems, asa reminder that the model also applies to them, as an extreme limitingcase.

Figure 8.3 (from Taagepera and Ensch 2006) shows the largest seatshares of the 30 single-seat systems graphed against assembly sizes, bothon logarithmic scales, so that s1 = 1/(S)1/8 becomes a straight line. The

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1,00010010

0.2

Mean S

Means1

0.4

0.6

0.8

1.0s1 = S –0.125, R2 = 0.245 ‘Model’

s1 = 0.915S –0.108

R 2 = 0.256 ‘Best-Fit’

Italy 1895−1913, 6

Netherlands 1888−1913, 8

Germany 1871−1912, 13

St Kitts 1980−9, 3

Figure 8.3. The median seat share of the largest party vs. assembly size, for 30single-seat systems—predictive model and regression lineSource: Reprinted from Electoral Studies, 25, R. Taagepera and J. Ensch, ‘Institutional Determi-nates of the Largest Seat Share’, 760–75, © 2006 Elsevier Ltd., with permission from Elsevier.

ordinary least squares (OLS) best fit of the logarithms corresponds to

s1 = 0.915S−0.108. [best fit, R2 = 0.256]

It is almost superimposed to the predicted line, for which the R2 ispractically as high:

s1 = 1.00S−0.125. [predictive model, R2 = 0.245]

The low R2 comes largely from the scatter of the Two-Rounds systems.Recall that R2 matters for postdictive analysis, because it is the onlymeasure of quality of fit one has there. Here, however, the main questionis whether the predicted average curve agrees with the actual best fitwithin the random scatter of data. It visibly does when assemblies are onthe small side. The largest seat shares are larger than predicted for mostassemblies of more than 200 seats. One must ask whether this deviationfrom the model is random or systematic.

The case of India, not included in the Taagepera and Ensch (2006) data-set, may be instructive in respect. Here I use data from Nohlen, Gotz, andHartmann (2001: I: 577–9). From 1951 to 1984, India had a very largedominant party (geometric mean s1 = 0.671). With mean S = 517.2 for the8 elections, it yields an extremely high s1S1/8 = 1.465. Yet for the next

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5 elections (1989–99) the mean s1 drops sharply to 0.348. With S = 543, ityields a very low s1S1/8 = 0.764. The overall mean is still on the high side(1.140), but the shift tells us that it may be too early to conclude that themodel needs adjustment at high assembly sizes.

It was observed in Chapter 6 that it is hard to devise a hypotheticalexample to illustrate how assembly size affects the number and size distri-bution of parties. Figure 8.3 supplies evidence: Reduced assembly size doesenhance largest party predominance, at least in single-seat systems. Butsuch directional model is the easy part, and it cannot offer quantitativeprediction. The real triumph is here that one specific decreasing curve wasoffered as the best possible guess, on purely theoretical grounds, and theactual best fit curve is almost superimposed to it.

The Largest Seat Share versus the Seat Product forMulti-Seat Systems

Now we proceed to test the full model, s1 = 1/(MS)1/8 or s1(MS)1/8 = 1,using multi-seat PR systems. Taagepera and Ensch (2006) could locateonly 10 systems where all seats were allocated within districts, for at least5 elections. Even most of these systems deviate from the ideal of simplesystems. In 6 systems List PR with divisors was used, mostly d’Hondt,but also modified Sainte-Laguë. Complexities included varying districtmagnitudes, legal thresholds, and district level apparentement. In 4 othersystems, candidate-centered PR was used: STV, SNTV, or List PR approach-ing nonlist through extensive panachage and cumulation of multiple votesper voter (Switzerland).

In order to extend the range of the seat product MS, elections in anationwide single district are of high interest, but all such systems involvelegal thresholds. Relaxing the criteria even further, 6 nationwide singledistrict systems were included, with legal thresholds ranging from 0.67to 5 percent. It is shown in Chapter 15 that the restraining effect of anationwide threshold T corresponds approximately to that of a districtmagnitude of M = (75%/T) − 1. Instead of M = S, this adjusted magnitudewas used for nationwide PR with thresholds.

The geometric means for these three groupings are shown in Table 8.3.See Taagepera and Ensch (2006) for data on individual systems. In the rela-tively simple list PR systems, mean s1(MS)1/8 exceeds the predicted 1.00 by9 percent, individual systems ranging from 0.94 (Luxembourg 1919–94) to1.30 (Spain 1977–96). The mean for the more complex systems is within

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Table 8.3. District magnitude (M), assembly size (S), and the largest party’s seat share(s1), for multi-seat PR systems

Number of elections Geometric means for

M S MS s1 s1(MS)1/8

List PR in districts (6cases)

106 9.39 171.8 1613 .432 1.086

Candidate-centeredPR in distr. (4)

88 4.90 152.3 746 .428 0.971

Nationwide PR +threshold (6)a

68 (42.0)a 196.7 8270 .362 1.119

All 16 multi-seatsystems

262 1.068

All 46 systems(M > 1andM = 1)

652 1.013

Data source: Taagepera and Ensch (2007).a In nationwide single district with legal threshold T , M = S is adjusted to M = (75%/T ) − 1.

3 percent of the expectation, but the range is wide, from 0.67(Switzerland 1919–95) to 1.28 (Japan 1928–93). In nationwide PR sys-tems, the mean s1(MS)1/8 formally exceeds 1.00 by 12 percent. Thisfigure depends very much on the adequacy of the threshold correctionM = (75%/T) − 1.

For all 16 PR systems in Table 8.3, the mean excess over s1(MS)1/8

is 7 percent. Finally, joining the single-seat and multi-seat systems andtaking the geometric mean for all 46 systems in Tables 8.2 and 8.3 yieldss1(MS)1/8 = 1.013—only 1.5 percent above the predicted 1.00.

If the seat product for the d’Hondt systems were taken as M0.62S ratherthan MS (as suggested in Chapter 6), then s1(M0.62S)1/8 = 0.978, on theaverage. Overall agreement with the expected 1.000 would become evenbetter. However, the method for estimating the exponent F in MF Sshould be refined before it can be used with any confidence.

Figure 8.4 shows the largest seat shares of all 46 systems (single-seat andmulti-seat) graphed against the seat product, both on logarithmic scales,so that s1 = 1/(MS)1/8 becomes a straight line. The corresponding data aretabulated in Appendix to the book. The OLS best fit of the logarithms ofM and S separately corresponds to

s1 = 0.847M−0.113S−0.090. [best fit, R2 = 0.54]

This is to be compared to the predicted relationship, for which the R2 ispractically as high:

s1 = 1.00M−0.125S−0.125. [theoretical model, R2 = 0.51]

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100.2

0.4

0.6

0.8

1.0

Mean MS

Means1

100 1,000 10,000

Italy 1895−1913, 6s1 = (MS)–0.125

R 2 = 0.509 ‘Model’

St Kitts 1980−89, 3

s1 = 0.847M –0.113S –0.090

R2 = 0.538 ‘Best-Fit’ Netherlands 1888−1913, 8

Germany 1871−1912, 13 Switzerland 1919−25, 21

Figure 8.4. The median seat share of the largest party vs. seat product MS for 46single- and multi-seat systems—predictive model and regression lineNote: Squares: M = 1; Triangles: M > 1.Source: Reprinted from Electoral Studies, 25, R. Taagepera and J. Ensch, ‘Institutional Determi-nants of the Largest Seat Share’, 760–75, © 2006 Elsevier Ltd., with permission from Elsevier.

These two lines are almost superimposed. (How does one graph on a scaleof MS when M and S have different exponents? See Taagepera and Ensch2006) Maximum error is under 3 percent.) Here R2 is higher than in Figure8.3, simply because the range of MS is wider. But again, the question is nothow high is R2, but whether the predicted curve agrees with the actualbest fit within the random scatter of data. It visibly does.

Table 8.4 lists the various electoral systems in the order of increas-ing mean deviation from the predicted largest share. The main featuresadding complexity to the systems are also shown. Two-Rounds falls shortof the model and has by far the widest range of variation. Connect-ing first-round votes to seats, largely won in the second round, mayadd unpredictability. As mentioned earlier, the figure for Alternate Votedepends heavily on how to count parties in Australia; both alternatives areshown in Table 8.4. The effect of different candidate-centered PR systemsmay differ. The mean of this mixed category happens to agree with themodel.

The FPTP approaches the ideal simple system more than any othercategory, and the deviation from the model is minimal on the average.Individual systems, however, can deviate appreciably. Multi-district ListPR is another category simple in principle, but in practice many additional

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Table 8.4. Complexity of electoral systems and deviation of the largest seat share fromthe model s1 = 1/(MS)1/8

Allocation formula Factors of complexity Deviation

Mean Range

Two-Rounds Loose connection betweenfirst-round

−12% −42 to +70 %

votes and second-round seats.Alternate Vote Individual cross-party

preferences.−11/ + 6% —

Candidate-centeredPR in districts

Individual cross-partypreferences.

−3% −33 to +28%

FPTP Simple. Some primaries andgerrymander.

−1% −32 to +29 %

Multi-district list PR Uneven M, apparentement, legalthresholds.

+9 % −6 to +30 %

Nationwide PR Legal thresholds, their effect onM conventionally adjusted.

+12% −7 to +33 %

features enter, and they apparently tend to reinforce the largest party. Thisis surprising at first look, because most factors of complexity would seemto favor the smaller parties. However, if the exponent F in MF S weretaken as less than 1.0 for the relatively many cases using d’Hondt, thenthe excess for the largest share would decrease. Systems with nationwidePR also look unexpectedly favorable to the largest party, but here theoutcome depends on the accuracy of the conventional way to adjustmagnitude in view of legal thresholds.

The model treats district magnitude and assembly size in a symmetricalway, but the extent of their ranges differs. The limited range of assemblysizes shows up when the logarithm of the largest seat share is regressedagainst the logarithm of S alone. Here R2 = 0.28. When regressed againstthe logarithm of M alone, R2 increases to 0.43. Regression against both Mand S leads to R2 = 0.54 (Taagepera and Ensch 2006). Thus assembly sizestill matters in practice, but district magnitude matters more.

Implications for Institutional Engineering

Now we are approaching payoff for institutional engineering. Connectingthe largest seat share to institutional characteristics would enable us togo beyond trial-and-error. The number of all seat-winning parties is oflittle interest to the practical politician, because it heavily depends on the

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tiniest parties that barely win a seat or two and have almost no impact onpolitics. But the largest seat share matters for government formation andsurvival. When it is less than 50 percent, it influences the number andweight of potential coalition partners the largest party can choose. If thelargest party remains in opposition, its seat share influences its blockingability. When the largest share surpasses 50 percent, the extent of theexcess still makes a difference by enhancing the ruling party’s clout, yetalso encouraging factions within it. Frequent changes in electoral systemsare not desirable. I still believe the following:

A major purpose of elections is to supply a stable institutional framework for theexpression of various viewpoints. Even if imperfect, a long-established existingelectoral system may satisfy this purpose better than could a new and unfamiliarsystem, even if it were inherently more advantageous. . . . Familiarity breeds stabil-ity. . . . Major electoral reforms should not be undertaken lightly. (Taagepera andShugart 1989: 218)

However, if the urge to change becomes strong, the model presented heremay help to fine-tune change so as not to overdo it.

For institutional engineering, the quantity to watch is the seatproduct—the product of the number of seats in the assembly and thenumber of seats allocated in the average district. The lower the seatproduct, the larger the seat share of the largest party. To increase theaverage share of the largest party, one can lower either district magnitudeor assembly size, or lower both to a more moderate degree.

The simple model presented here goes beyond mere directionality ofadjustment. It tells us by how much we should change the seat prod-uct, for a desired effect. The model indicates the world average of thelargest share, for a given combination of district magnitude and assemblysize.

It would be simple-minded, however, to base institutional engineeringin a given country solely on a universal model, without taking intoaccount the country’s peculiarities. If a country has exceeded the worldaverage in the past, it is likely to continue to do so when the electoralsystem is modified. A corresponding adjustment term should be intro-duced into the worldwide model. We do not have to know which factorscause the need for adjustment. They could be due to political culture,institutional features other than M and S, or something else. Chances arethat they will continue to exert a similar influence on the altered electoralsystem. In other words, the effective seat product could differ somewhatfrom the product of M and S, depending on other factors.

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Consider Finland, for instance. Its largest seat share during the last50 years has been around 27 percent, which is rather low. Indeed, itis only 3/4 of the worldwide expectation. Here, the relationship is nots1 = 1/(MS)1/8. It can be expected to be around s1 = 0.75/(MS)1/8, subjectto some assumptions. Suppose Finland wishes to raise the largest share toan average around 32 percent, changing nothing but district magnitude.One should not resort to the full worldwide model. There is no needeven to calculate the adjusted s1 = 0.75/(MS)1/8 for the existing and thedesired conditions. It suffices to observe that the desired increase wouldmultiply the present largest share by 32/27=1.185. To obtain this outcome,the model suggests that MS should be divided by (1.185)8 = 3.89. If theassembly size is not changed, this would mean going from the present 14districts at mean M = 14.3 to 54 districts at mean M = 3.7.

The present assembly of 200 seats is relatively large for Finland’s popula-tion of 5 million (see Chapter 12). If it can be reduced to 150 (0.75 of thepresent), then M can be reduced only by a factor of 3.89 × 0.75 = 2.92,meaning 31 districts at mean M = 4.9. Either change would obviouslywhittle down or eliminate some smaller parties. Whether this would beacceptable is up to the decision-makers. The political scientist can onlyoffer alternative projections.

Apart from assembly size and the mean district magnitude, otheraspects of the electoral system must also be considered. The largest districtin Finland at times has had 27 seats—double the mean. This district iswhere many small parties win their only seat. Simply making the districtmagnitudes more uniform could reduce the number of seat-winning par-ties by a factor of (2)1/4 = 1.19, that is, by 19 percent. It would boost thelargest share by (2)1/8 = 1.09, meaning a 9 percent increase, from 27 to29.5.

Some small parties in Finland win seats thanks to district-level alliances.If such alliances are prohibited, further small parties would be squeezedout, and the largest seat share would increase by an amount hard toestimate. In sum, when the electoral system includes stipulations that gobeyond the simplest possible, changing those extra features can go a longway to alter the seat share of the largest party.

Political inertia that opposes any change in the electoral system isconsiderable. After all, those in position to decide profited from theexisting system—they got elected. It is usually easier to make the rulesmore favorable to small parties. The large parties might not object, as thisbuys them insurance in case they themselves lose popularity (Colomer2004b). Going in the reverse direction (as presented above) could meet

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vociferous opposition by smaller parties. However, if the proliferationof parties should become manifestly excessive, then public opinion maydemand a sharp cutback, for instance by going from PR to FPTP. A grossover-correction may result.

This is where the present model could become useful. For a reasonablysimple electoral system in a country with some past democratic record,we have advanced much beyond the qualitative advice ‘To have fewerand larger parties, reduce district magnitude’. We can now tell roughly byhow much it should be reduced. Yes, it is ‘roughly’, but this is better thanno estimate at all of the degree of reduction needed.

Estimation is more difficult when the country has not had any demo-cratic elections, because then we do not know the direction of localcorrectives to the universal average pattern. The best we could do is tocompare with neighboring democratic countries with somewhat similarcultures.

All bets are off when politicians choose to go beyond the simplest for-mat and insert all sorts of complex and mutually contradictory features.Then we cannot calculate with any precision. In particular, if the choiceis Two-Rounds, anything can happen. If you want predictability, keep itsimple.


This appendix presents the derivations for the nationwide number of seat-winningparties and the largest seat share. It offers further insights into the minimalmeasure of the number of parties (N∞) and index of balance. It wonders aboutthe apparent symmetry of the roles of M and S in the seat product. Finally, itpoints out that nothing obliges the real world to follow the probabilistic averagesas expressed in the models presented. Then why do they fit, nonetheless?

The number of seat-winning parties nationwide: The model

Consider the number of parties that are likely to win at least one seat in anassembly of S members. Assume a simple electoral system where all representativesare elected in districts of uniform magnitude M, using some usual PR formula(or FPTP, when M = 1).

The nationwide number of seat-winning parties (N0) is at least equal to thenumber of such parties (p) in a single district, which itself can conceivably rangefrom 1 to M, with an expected mean of M1/2. (At least as a first approximation,the impact of nationwide politics in districts is ignored, since it would cancel out,

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nationwide.) Even if approximately M1/2 parties win seats in each district, thesemay not be the same parties in all the districts. Hence the nationwide number canbe expected to be larger than in any district: N0 > p = M1/2. The most favorablecondition for small parties to win at least one seat would be, if the entire countrywere made a single nationwide district of magnitude S. Then the number of seat-winning parties could range from 1 to S, with an expected mean of S1/2. Anysubdividing of the country into several districts is bound to reduce the chances ofthe smallest parties. Hence N0 < S1/2.

In sum, M1/2 < N0 < S1/2. If nothing else is known besides M and S, thenthe best guess for N0 is the one that balances the district level and nationwideconstraints. The geometric mean of the extremes is

N0 = (MS)1/4.

This is the point where the seat product MS, announced in Chapter 6, emergesfrom purely probabilistic considerations. In this sense, it is a pivotal point.Remarkably, M and S play a symmetrical role in predicting the number of seat-wining parties, as long as we avoid multi-seat plurality and complex electoralsystems. This model agrees with the following two anchor points. When M = S(nationwide single district), the model yields N0 = S1/2, as it should. When S = 1,M is also bound to be 1, given that M ≤ S, so that N0 = 1 results, as it wellshould.

When only one seat is at stake, it means presidential rather than assemblyelections. The two elections differ in a number of ways, but presidential electionsalso offer similarities with elections in an M = 1 district to fill an assembly seat.The same seat allocation formulas offer themselves, even while the importanceof presidential elections may affect the choice of a formula. Hence we shouldbe worried if a predictive model for the number of seat-winning parties in theassembly did not predict correctly the outcome of presidential elections. Thepresent model does correctly predict N0 = 1 for S = 1.

In the case of FPTP elections for an assembly (M = 1, S > 1), the model predictsN0 = S1/4—the number of seat-winning parties is simply the fourth root of assem-bly size. This is relatively easy to test, given the large number of FPTP electionresults available.

For countries with several multi-seat districts, testing becomes more difficultbecause hardly any truly simple PR system exists. If nothing else, district magni-tude varies from district to district. Malta seems to be the only country wherea uniform district magnitude of M = 5 has been maintained over a long time.Malta held the assembly size constant at S = 40 for the 5 elections held in 1947–55. The model predicts that (5 × 40)1/4 = 3.76 parties would win seats. The actualfigures ranged widely, from 2 to 6, but their geometric mean was 3.73, close to theprediction. Actually, so close an agreement is plain luck. With only five elections

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to average, one would be happy if the prediction were within plus or minus oneparty—especially given that Malta used STV rather than categorical List PR.

How the largest share relates to the number of seat-winning parties:The model

There must be an average relationship between the number of seat-winning partiesand the seat share of the largest party, because many seat-winning parties wouldrestrict the number of seats that could go to the largest. Vice versa, a small largestparty leaves many more opportunities to the small parties. Let us specify theconceptual limits.

When N0 parties win at least one seat each, the average fractional share for aseat-winning party is 1/N0. The largest share (s1) obviously must at least equal thisaverage. It also must fall slightly short of the total (1), so as to leave N0 − 1 seats tothe other parties. So the largest share cannot be larger than (S − N0 + 1)/S. We canneglect N0 and simply say that s1 < 1, if assembly size S is much larger than N0, asit usually is. The conceptual limits on the largest share then are 1/N0 ≤ s1 < 1. Inthe absence of any other information, our best guess for s1 is the geometric meanof the conceptual limits (Taagepera and Shugart 1993):

s1 = 1/N1/20 = N−1/2

0 .

Conversely,

N0 = 1/s21 = s−2

1 .

A symmetrical form that does not take a stand on which variable influences theother is

s21 N0 = 1 or s1 N1/2

0 = 1.

This is a kind of a ‘law of conservation’, in that the value of this product remainsunchanged. Quantities that are conserved during a transformation are of consid-erable interest in physics, and might be so in social sciences. The ‘number-shareconservation’ developed here applies much more broadly than just to seats in anassembly. For instance, it enables us to predict the shares of the largest federalsubunits, once the number of subunits is given. For the populations as well asareas in the USA, Canada, and Australia, it works within ±20 percent (Taagepera1999b).

In the absence of any other knowledge, whenever a well-defined total is dividedamong N0 components, the fractional share of the largest component multi-plied by the square root of the number of components is expected to be 1.

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Some of the apparent consequences of number-share conservation may seem hardto accept. Suppose an external factor increases N0—e.g. a party splits up. Does thecurrent largest share now feel pressure, so to say, to change so as to conserve theproduct around 1? Conversely, if some external factor increases the largest share,does the number of components feel pressure to change downwards?

Such a dilemma actually arises regarding the Central Australian and Cana-dian Northern territories, which are distinct but not full-fledged federal subunits.Should they be counted among the N0 components, when estimating the popula-tions of New South Wales and Ontario, respectively? And how could the latterpopulations depend on such number games? Similar questions may arise forparties that run as semi-coalitions. It so happens that counting the aforementionedterritories as separate components leads to an underestimate of the populations ofthe largest subunit, while discounting them leads to an overestimate—but bothare in the ±20 percent range. The apparent problem is akin to the one that ariseswith normal distributions: If some property is normally distributed in severalsubspecies, how can it also be normally distributed for the entire species? But mostoften it is.

Party-based elections offer a possibility to test the number-share conservationwith literally hundreds of cases: The largest seat share in every election wherethe number of seat-winning parties can be specified. It has nothing to do with theparticular electoral system, because the number-share conservation is as universal arelationship as normal distribution. Such testing of a proposed law of conservationis of interest by itself. What makes it even more interesting for the study ofelectoral systems is that the number of seat-winning parties can be estimated fromthe product MS. If so, then the largest seat share, too, can be estimated from purelyinstitutional data (M and S), to the extent that s1 N1/2

0 = 1 applies.The variation around the median thus estimated is of course considerable, espe-

cially in the middle ranges of the variables. Consider the data used in Figure 8.2.With 4 parties winning seats, the median prediction is s1 = 0.500. The actualmedian of the 79 cases is 0.495, even though the distribution is wide:

Range of s1 0.30–0.39 0.40–0.49 0.50–0.59 0.60–0.69 0.70–0.79 0.80–0.89

Number of cases 12 32 26 4 3 2

Conversely, for s1 ranging from 0.45 to 0.54, the median prediction is N0 = 4. Theactual median of the 162 cases is 4, with the following distribution:

Range of N0 2 3 4 5 6 7 8–9 10–12 13–16

Number of cases 23 42 30 28 15 9 6 5 4

Visibly, s1 N1/20 = 1 cannot be expected to work in single cases any better than the

weight of one particular British woman can be inferred from the average weight ofBritish females. But the average prediction is quite precise, without any input ofdata.

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Methodological issues remain. The distributions above are visibly not normal—as one would expect for variables that cannot take negative values (cf. Taagepera2008). Thus arithmetic means would be misleading. But these distributions are notclearly lognormal either, so that the use of geometric mean is also questionable.This is why the median was used, as it makes no assumptions about the form ofthe distribution. Another remaining problem is the direction of testing. Figure 8.2shows the median s1 at given N0, as if the number of seat-winning parties weredriving the largest share, rather than vice versa. What would result from a reversegraphing of median N0 at given s1? We run into difficulties because empirical N0

comes in integer values only.The discrepancy in Figure 8.2 at N0 = 2 arises from a hidden assumption that fails

at very low N0. Consider our starting point: 1/N0 ≤ s1 < 1. Taking the geometricmean of these conceptual limits implicitly presumes that the distribution of logs1 is symmetrical, so that extreme cases on both sides occur with equal frequency.If so, then the outcome would be 71-24. However, when only two parties achieverepresentation, political competition may push toward a balance between them.Hence constellations close to the lower limit (50-50) are politically quite plausible,while constellations approaching the upper limit (about 99-1) are unlikely indemocracies. Assuming a distribution that reflects these considerations producesthe observed median value of s1 when only two parties win seats (see Taagepera2005).

The deviation from s1 N1/20 = 1 at high N0 in Figure 8.2 is harder to explain.

Beyond 12 seat-winning parties, further tiny parties or independents winning aseat or two no longer seem to affect the largest share. This brings us to the thornyissue of how to count the independents.

In one sense, an independent representative is equivalent to a minor partywith a single seat. Hence the independent should be counted as one moreseat-winning party. But there is a difference between a party with nationwideorganization and ambitions that happens to win only one seat nationwide, andan independent candidate who concentrates solely on one particular district.Parties tend to have claims of representing some ideology or interest. Indepen-dents have fewer such claims and, once elected, often tend to coalesce witha larger party to an extent small parties cannot afford without losing theirraison d’être.

In electoral studies the issue becomes salient when a large proportion of seatsare occupied by independents, as has happened in Ireland and Japan, in particular.(For this reason, all elections in Ireland had to be eliminated from the testing ofs1 N1/2

0 = 1.) Assuming that these countries had 20–50 distinct parties makes littlesense, but pretending that the seats occupied by independents do not exist leadsto equally odd results. In sum, the question of how to count the independents asparts of a party system matters when there are many of them. Here more work isneeded.

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The minimal measure of the number of parties and index of balance

It was shown in Chapter 4 that the inverse of the largest share (N∞ = 1/s1) is theminimal measure of the number of parties for a given seats constellation. Theaverage connection relationship s2

1 N0 = 1 can now be recast as N0/N2∞ = 1. This

means that, on the average, N∞ is the square root of N0, and, conversely, N0 is thesquare of N∞:

N∞ = N1/20 and N0 = N2

∞.

These relationships (hinted at in appendix to Chapter 4) represent a probabilisticaverage. It can be seen from Figure 8.2 that this average holds when N0 ranges from3 to 12, meaning N∞ ranging from 1.7 to 3.5 or the largest share ranging from 29to 58 percent. This covers most of the usual range. One has to be careful, however,in the case of highly multiparty systems where even the largest share falls muchbelow 30 percent.

Given that N0 = (MS)1/4, the minimal measures of the number of parties canalso be tied to institutions:

N∞ = N1/20 = (MS)1/4.

The formula for balance itself can be recast in terms of the two extreme measuresof the number of parties. Balance is the ratio of logarithms of the minimal andmaximal measures of the number of parties:

B =logN∞logN0

.

We have been successful in connecting the minimal and maximal measures ofthe number of parties to the seat product and might expect the same to be thecase for balance—but here we fail. The balance for particular countries cannotbe predicted on the basis of institutions. Indeed, the average relationship N0 = N2

∞leads to B = 0.5, which certainly holds as a world average. This is the only thing wecan predict about balance, for any M and S. In other words, the index of balancefor a given country precisely tells us by how much they deviate from the balancethat could be expected, given their assembly size and district magnitude. It is asecond-order measure, like deviation from PR.

How the largest seat share relates to the seat product

When all seats are allocated within districts of fairly equal magnitudes, itwas shown that, on the average, N0 = (MS)1/4 parties can be expected to winseats. Upon testing, this prediction held within 15 percent for FPTP and theusual PR systems. It was also shown that s1 = 1/N1/2

0 is to be expected onquite universal grounds, and this expectation is confirmed, with some deviation

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when only two parties win seats. Connecting the two equations immediatelyyields

s1 =1

(MS)1/8

or, in a more symmetric form,

s1(MS)1/8 = 1.

This is another sort of a law of conservation: The product of the largest seatshare and the 8th root of the seat product is constant at 1. A symmetrical formu-lation avoids the issue of which variables are dependent, and which are ‘inde-pendent’. In the following, the largest seat share is treated as dependent oninstitutions, but the model only posits interdependence. A dominant party thathappens to become unusually large may conceivably exert pressure to changethe electoral system so as to ensure its continued domination. If so, thenthe largest share becomes the driving force, and MS becomes the ‘dependent’part.

Oddly but pleasantly, at this stage we leave behind one methodologicalheadache of the previous stages of model construction. Counting the number ofseat-winning parties, be it at district or nationwide level, is made hard by lackof detailed data on the smallest parties, often lumped as ‘Others’, and the fuzzynature of independents. But now we are dealing only with the largest seat share,for which clear data are much easier to locate. The only dilemmas involve semi-joint largest parties like the CDU and CSU in Germany and fractured parties likeLDP in Japan.

Seat product—the main characteristic of a simple electoral system

According to the models N0= (MS)1/4 and s1 = 1/(MS)1/8, district magnitude andassembly size play a strikingly symmetrical role. An increase in one would com-pensate for a decrease in the other. The same number of seat-winning partieswould be expected for a large 625-seat assembly elected from single-seat districts(S = 625, M = 1) and for a tiny 25-seat assembly elected in a single nationwidedistrict (S = M = 25). In both, N0 = (MS)1/4 = 5.5 parties are expected to win seats.Is there really such equivalence? And regardless of how many parties win one or afew seats, does such equivalence extend to the effective number of parties? It willbe seen that it does.

Such equivalence makes the seat product MS the single most important indicatorto characterize a simple electoral system. Just as district magnitude (along with theseat allocation formula) characterizes the effect of the electoral system in a singledistrict, the product MS characterizes the nationwide effect of the electoral system(along with the allocation formula).

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The seat product could be used at the district level, where it becomes simply themagnitude, modified by F . It was noted in Taagepera and Shugart (1989: 118, 139–41) that the empirical relationships of deviation from PR and break-even pointwith magnitude involved the square root of M, as if this were some sort of a basicbuilding block. This square root of M would correspond, at district level, to whatwas called electoral system aggregate in appendix to Chapter 6.

But if district magnitude and assembly size play symmetrical roles, how comethe impact of magnitude on electoral outcomes was recognized a century ago,while the impact of assembly size is still questioned? The answer is that the widthsof the ranges that M and S can take are not at all symmetrical. M can vary overmore than two orders of magnitude—from 1 to 100 and beyond. Since 1001/4 =3.16, having nationwide PR instead of FPTP can triple the number of seat-winningparties. In contrast, S in the 30 systems condensed in Table 8.1 varies barely overone order of magnitude—25–628. Going from an assembly of 25 seats to 628 wouldonly double the number of seat-winning parties, given that (628/25)1/4 = 2.24.

Moreover, the actual options when choosing an assembly size are limited by thepopulation to be represented (as is explained in Chapter 12). Median countrieshave presently around 10 million people, and their assemblies rarely have lessthan 100 or more than 400 seats. Having a 400-seat assembly instead of 100 canboost the number of seat-winning parties only by (400/100)1/4 = 1.41, meaning41 percent. Such a change is dwarfed by what one can achieve by changing districtmagnitude. It is hence no wonder that the impact of S does not catch the eye aseasily as the impact of M.

It need not work, but it does. Why?

When I expected 10 parties to win seats in a 100-seat PR district, it was notbecause of some positive arguments. There was only a negative reason: Any otherexpectation would be even harder to justify, in the absence of any further information.It need not materialize, even as average, when further information arrives. Itis merely the only guess we are justified to make under complete informationblackout, apart from conceptual limits. The same goes for all subsequent stagesof the model, up to the largest seat share in the assembly. If the term ‘prediction’has slipped into the model building, its meaning is the following: I cannot becertain it actually is so, but if I had to make a quantitative prediction, yes, I canmake one (and only this one!), rather than say ‘I don’t know’.

As information is added, we could fully expect to find that the empirical averagepattern differs from the simple model. The direction and extent of deviation fromthe model may help us locate factors of a political nature that could explain thedeviation. Yes, reality can be expected to deviate from a model built on nothingelse than conceptual limits. But we are in for a surprise: For the simplest electoralsystems we can find, the simple model does work quite well. Deviations from themodel arguably increase at the rate the systems turn more complex.

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A second surprise is that the model fits even better for the largest seat share thanfor the number of seat-winning parties. Indeed, for the simplest system (FPTP),the number of seat-winning parties deviates from the model by 8 percent on theaverage, while the largest share deviates only by 1 percent. For all systems tested,the deviations are 14 percent and less than 1.5 percent, respectively. Yet, in order tobuild up the model for the largest share, we played the ignorance-based card threetimes over. We did it twice for the number of seat-winning parties—at district andassembly levels—and for a third time when going from the number of seats to thelargest share. We started on a thin limb to begin with, and then we took more andmore risks. It should work less well as stages are added.

It should work less well at each successive stage, because random error accumu-lates. Moreover, sooner or later, some universal sociopolitical factor may enter soas to tilt the world average away from the purely ignorance-based best guess. Suchfactors do enter, indeed, for individual electoral systems. Imperial Germany falls42 percent short of the expectation for the largest share, while pre-World War IItaly exceeds it by 70 percent. With such a wide observed range, the world averageitself could easily be anywhere between 80 and 120 percent off the simple model.Why is it within 1 percent for the simplest category (FPTP) and within 1.5 percentfor the mean of all systems?

True, when testing for the largest seat share, we circumvented the problemof independents and other ‘Others’ that bedeviled measurement of the numberof seat-winning parties. Also, somewhat different sets of electoral systems wereused in the two studies. Still, improvement of fit with the simple model remainspuzzling. Maybe the micro-Duvergerian specialists can explain it. The fact is thatthe simple model seems to apply, as a worldwide average, without need for majoradjustments. In this particular case, sociopolitical nature turns out to be as simple,on the average, as it possibly could be. Individual countries need correction terms,but the world does not.

One may feel like protesting: ‘This is outrageous. It cannot be that simple anddevoid of political content. There must be some artifact.’ Indeed, several reviewersfor the studies condensed here suggested that many other factors could affect thelargest seat share. They then went on to claim that overlooking these factors mayhave led to an artificially good agreement with the simple model. But how couldthat be? Overlooking significant factors reduces agreement almost by definition ofwhat is significant. Occam’s razor played a role in development of natural sciences:Omit what is not absolutely needed. This is no time to blunt Occam’s razor inpolitical science.

Instead of remaining in denial, one is better off by accepting the outrageouslysimple model as a baseline and focusing on the political features that makeindividual polities deviate from it. This is where many other factors enter—political, cultural, and other institutional. What is it that makes the Swiss largestparty at any given election so small and the US largest party so large, comparedto expectations based on district magnitude and assembly size? How could we

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predict the quantitative degree of such deviations? What is it that makes the Two-Rounds systems so unpredictable—and what should we include to make it morepredictable?

To the extent that the ignorance-based model fits the world average for thesimplest systems, we are tempted to go beyond the initial claim that ‘This is thebest guess we can make in the absence of any further information.’ We may betempted to become more assertive: ‘This is no longer a guess but a firm prediction,because it has worked previously.’ It would be risky. We should preserve somehumility, in view of our inability to predict for individual systems. As further dataaccumulate, a small but significant deviation from the simple model may appearfor the average of even the simplest electoral systems.

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9

Seat Shares of All Parties and theEffective Number of Parties


� The quantity to watch is again the seat product—the number of seatsin the assembly × the number of seats allocated in the average district.

� The lower the seat product, the lower the effective number of parties inthe assembly.

� If you wish to reduce the effective number of parties by one-tenth, yourbest bet is to multiply 1 − 0.1 = 0.9 by itself 6 times, which yields 0.53.This is by how much you must multiply the present seat product. Todo so, you can cut the present districts into two smaller districts or cutthe assembly size by a half. You can also reduce both district magnitudeand assembly size by about 30 percent.

� This way to calculate is based on a logical model that agrees with theworld average. It is approximate, because other factors enter, but this isyour best bet.

� The average seat shares of second-largest and third-largest parties alsocan be calculated from the seat product.

� At the same average district magnitude, unequal districts usuallyincrease the effective number of parties.

Chapter 8 established a model that predicts the largest seat share in simpleelectoral systems on the basis of the seat product. Here the model isextended to seat shares of parties at all other ranks by size. This is notabout sizes of specific parties but parties that happen to occupy a givenrank by size at a given election. The following questions are of interest.Does political competition tend to place the two largest parties in a special

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category by size, far larger than the third parties, or do seat shares taper offgradually? Could either pattern predominate, depending on the size of thelargest share? How is the resulting effective number of parties connectedto the seat product?

Here the other seat shares are calculated in terms of an interveningvariable: the largest seat share. Since the latter can be expressed in terms ofthe seat product MS, so can the other shares, in principle. However, thisis messy mathematically, because the equations that connect the otherseat shares to the largest are quite involved. Mathematics as such doesnot become more complex, compared to the previous chapter, but simpleexpressions pile up.

The effective number of parties is the most widely used single numberto characterize a party system. Fortunately, at this stage of cumulation,the mathematics becomes simpler again, because the probabilisticallyexpected pattern can be approximated by another simple function of theseat product.

The Empirical Pattern of Seat Share Distribution

The first task is to find out what the empirical pattern looks like. At givenseat share of the largest party, what is the typical share that goes to thesecond-largest party, and so on? Note that we do not follow the patternof one specific party but parties that happen to have a given rank by size,at a given election.

For individual elections the possibilities are wide open. The second-largest party may tie with the largest (e.g. The Netherlands 1901, 1905,1909, both largest parties at 25.0 percent), or it may have as few seats asthe third-largest party (e.g. Germany 1898: 25.7 −14.1 −14.1 −. . . ; Italy1900: 81.1 −6.7 −6.6 −. . . ). Some countries may follow a steady patternthat deviates from the worldwide average—but we can establish such adeviation only after we determine the average, to serve as a benchmark. Insome other countries, the distribution may vary widely from one electionto the next. A major shift in the fortunes of one specific party may or maynot alter the size distribution, compared to the previous election. Whenthe Conservatives in Canada plummeted from 57.3 percent of the seats in1988 to 0.7 percent in 1993, the ranked distribution shifted merely from57.3 −28.1-14.6 to 60.0 −18.3 −17.6 −3.1 −0.7 −0.3.

All this variation nonetheless occurs around some worldwide average,which may have considerable inertia over space and time. It may also have

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The Effective Number of Parties

Table 9.1. Actual average seat shares of parties ranked by size vs. largest share

Rank Seat shares, for given largest share (±2.5%)

1 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.02 17.8 22.4 24.8 27.8 29.9 32.0 37.1 40.6 35.3 29.9 25.43 14.9 19.0 17.5 18.2 14.8 13.7 8.0 4.1 3.3 3.4 2.94 13.3 12.3 11.9 9.5 9.0 6.3 3.5 0.2 0.8 1.4 0.75 11.2 7.2 6.6 4.9 3.8 2.1 1.2 0.1 0.4 0.3 0.56 8.8 5.3 3.8 2.1 1.6 0.6 0.2 0.0 0.1 0.0 0.37 5.5 3.8 2.2 0.9 0.5 0.2 0.1 0.0 0.1 0.0 0.28 2.8 1.8 1.4 0.7 0.2 0.19 2.2 1.2 0.8 0.4 0.1

10 1.7 0.8 0.5 0.2 0.111 0.8 0.5 0.2 0.112 0.4 0.3 0.1 0.113 0.3 0.2 0.1 0.114 0.1 0.1 0.115 0.1 0.116 0.1SUM 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0

Source: Smoothed from Taagepera and Allik (2006).

a logical explanation in terms of statistics and/or what democratic politicsis about, at its most universal. This is why establishing the worldwideaverage pattern is of interest.

Taagepera and Allik (2006) used essentially all of the more than 700elections recorded in Mackie and Rose (1991, 1997), regardless of thecomplexity of the electoral system used. The elections were grouped byintervals of the largest share (s1), and in each interval the arithmetic meanwas calculated for the second-largest party, the third-largest, and so on.

Table 9.1 shows the empirical results for largest shares up to 70 percent.Beyond 70 percent, few cases are available, and hence the averages comewith a large random error. Compared to data published in Taageperaand Allik (2006), Table 9.1 converts these data to 5 percent intervals ofthe largest share. A few more parties than predicted by N0 = 1/s2

1 gainoccasional representation, though mostly with an average of less thanone seat per election.

Figure 9.1 presents the graph of the original data, reproduced fromTaagepera and Allik (2006). The format used is an extension of whathas been called the Nagayama triangle. Nagayama (1997) graphed thevote shares of the second-running contestant against the vote shares ofthe top contestant. The total of the two shares is 100 percent at most(s1 + s2 = 1 = 100%), and the second-largest share can at most equal thelargest (s2 = s1). These lines determine a triangle that delimits the allowed

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s2 = 0.8s1

s2 = 0.8(100 − s1)

s2 = s1s1 + s

2= 100%

0%

10%

20%

30%

40%

50%

0%

Largest party seat share

Sea

t sha

res

First- Second- Third- Fourth- Fifth- Sixth- Seventh-largest

10% 20% 30% 50%40% 60% 70% 90%80% 100%

Figure 9.1. Actual average seat shares of parties ranked by size vs. largest seat share

Source: Reprinted from Electoral Studies, 25, R. Taagepera and M. Allik, ‘Seat Share Distributionof Parties: Models and Empirical Patterns’, 696–713, © 2006 Elsevier Ltd., with permissionfrom Elsevier.

zone for any election results. Reed (2001) popularized this format, andGrofman et al. (2004) investigated thoroughly its theoretical properties.Taagepera (2004) extended its use from candidates to parties, from voteshares to seat shares, and—most relevant here—to third-ranking parties,and so on.

In addition to the upper limits for the second-largest party (s2 = s1 ands2 + s1 = 1 = 100%, respectively), Figure 9.1 shows the empirical patternsfor parties at all ranks by size, up to s7. Also shown, as a dashed line, isan approximation for the pattern of the second-largest party. When thelargest share is less than 50 percent, the second-largest share tends to be0.8 times the largest share: s2 = 0.8s1. When the largest share surpasses50 percent, the second-largest share tends to be 0.8 times what is left bythe largest fractional share: s2 = 0.8(1 − s1).

At this stage, we do not distinguish between cases with different elec-toral systems that happen to yield the same largest share in a givenelection. The cases with very large largest shares correspond mainly tosingle-seat districts and those with very small largest shares to PR in largedistricts. There is a wide overlap in the center. Work in progress (Taageperaand Laatsit 2007) indicates that the patterns for FPTP and List PR divergeto some extent. What we have in Table 9.1 and Figure 9.1 is the average

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for all electoral systems. Such an approach offers the advantage of a largenumber of cases and the disadvantage of wide dispersal among them. It isa basis for more detailed work to follow.

The broad pattern is clear, except for the blank at largest shares lessthan 20 percent. Here, a thought experiment will help. If the largest sharewere extremely small, then it would take a huge number of parties, someof them with shares almost equal to the largest, to bring the total up to100 percent. For instance, if the largest share were 4 percent, it wouldtake more than 25 parties, many of them close to but none of them largerthan 4 percent. However, within the actual range of the largest share,even the seventh-ranking party (the smallest one shown in Figure 9.1 andTable 9.1) has a larger share than 4 percent. Hence, as the largest shareincreases, all the curves must rise at first, starting out from an anchorpoint at s1 = 0 → si = 0. Thereafter, they must peak and decline when thelargest share becomes sufficiently large. This is the broad common patternshared by parties at all ranks, from the second-largest share to the seventh-largest (and beyond).

But when does the peak occur, and how high does it reach? Further-more, do some curves bunch together more tightly than some others? Atfirst glance at Figure 9.1, it might be tempting to guess that the ith rankingparty peaks when the largest percentage share is 100%/i. In other words,maximum si corresponds to s1 ≈ 1/ i. It is close, but not quite accurate.For the second-largest party, the peak occurs at 55 percent, higher than100%/2 = 50. In contrast, the peaks for the third- and fourth-rankingparties occur at less than 33.3 and 25 percent, respectively.

The heights of the peaks offer no readily visible regularities. In par-ticular, the peak for the second-largest party towers way above all theothers. It hugs the line for the largest party (graphed against itself) whens1 < 0.5 = 50%, but so does the third-largest party when s1 < 0.25 = 25%.Later, the two curves part company, and very drastically so. The shares ofthird- and fourth-largest parties show a minor increase at very large largestshares. This may be an artifact due to a low number of cases.

In sum, the details of the curves are so intricate that it might lookhopeless to determine the reasons and hence the underlying pattern. Butlet us try, anyway. In particular, why is it that we start to have a two-party game, with the third-largest party bunched with minor parties, onlywhen the largest share surpasses 30 percent? Why is it that, at largest share25 percent, it rather looks like 3 parties standing apart from the rest, ratherthan 2?

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The Probabilistic and Politically Adjusted Modelsfor Seat Shares

In the previous chapter, we guessed at the largest fractional share, basedon its conceptual limits: the average share 1/N0 and the near total,1. The result, s1 = 1/N1/2

0 , agreed pretty well with actual values. Wecan repeat the procedure with the second-largest party, using only thepart (1 − s1) left over by the largest party. Details are given in chapterappendix.

Taagepera and Allik (2006) tabulate the complete results, for the largestshare ranging from 14 to 91 percent, and show the resulting graph, againfollowing the Nagayama format. The pattern vaguely agrees with theobserved average pattern when the largest share is small, but it fails toagree when the largest share increases. In particular, the model predictsthat the second-largest party would peak later and at a much lower valuethan it actually does, and the third-largest party is also predicted to peakmuch later.

At this point, we should recall what was said about s1 = 1/N1/20 in

the previous chapter: It is the only guess we can reasonably make, ifconceptual boundaries are all we know. It is an expectation value onlyin this limited sense. We guessed at the number of seat-winning parties,purely on the basis of two institutional constraints (district magnitudeand assembly size), and were surprised to find that it actually worked,with no need to introduce any further political considerations. We wenton to the largest share, and the surprise was repeated. With the shares ofother parties, politics finally catches up with institutions. We need somepolitical input. But we should keep such input at the bare minimum—theleast that we can get away with so as to explain the pattern observed. Thismeans introducing only some broad principle of politics that applies toall democratic systems.

The actual mean seat shares almost always penalize smaller parties infavor of larger parties, compared to the probabilistic expectations. Thetransition point between ‘smaller’ and ‘larger’ parties shifts as the largestshare increases. At s1 = 0.2 = 20%, as many as 6 largest parties exceed theprobabilistic expectation. The number drops to 4 at s1 = 0.3, to 3 at s1 =0.35, and to only 2 at s1 = 0.40. The simplest mathematical function thatcomes close to expressing these observations is the inverse of the largestfractional share, 1/s1.

We encountered 1/s1 earlier (Chapter 4), as one of the ways to expressthe number of parties: N∞. As a measure of the number of parties, it tends

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to be inferior to the usual effective number of parties. But here it acquiresan intriguing special meaning:

N∞ seems to be the number of parties that profit from a politicallyinduced shift of seats to larger parties, compared to probabilisticexpectations.

Recall that the peak value for ith-ranked party was observed to correspondto s1 ≈ 1/ i. In other words, this peak occurs when i = N∞. Peaking meansthat the party shifts from the bonus group when N∞ is large to thepenalized group when N∞ becomes smaller. So the two observations aremutually consistent. There must be a way to explain logically why theshift occurs at rank equal to N∞, but I have not found it yet.

What causes this rather systematic shift? Duverger’s mechanical andpsychological effects immediately come to mind, but here we are deal-ing with something even more general. The Duverger effects are mostmarked for single-seat districts, where the largest share tends to be large.Yet here we observe penalization of the smallest parties even when thelargest share is quite small, which most often corresponds to PR. Suchpenalization may be a major puzzle raised by the discrepancy betweendata and probabilistic expectations in Taagepera and Allik (2006).

Several factors may disadvantage the smallest parties even under PR.Legal thresholds of representation block them in some systems—but onlyin some. More broadly, small parties always suffer from lack of ‘economicsof scale in advertising, raising funds, securing portfolios supplying policybenefits, and so on’ (Cox 1997: 141). Media coverage of minor parties is solimited that some voters could be unaware of the very existence of partieswhose programs might appeal to them. The major factor may consistof strategic devices much broader than Duverger’s psychological effect,devices of the type Cox (1997: 194–6) has called strategic sequencing.Moreover, winning seats is not the only goal the voters have in mind.Even if a preferred party wins seats proportional to its votes, some votersmay still abandon it, if larger parties offer a better chance to be representednot only in the assembly but also in government.

Taagepera and Allik (2006) construct a model to account for the result-ing shift of support from small to large parties. It is condensed in chap-ter appendix. Table 9.2 shows the seat share distributions at three val-ues of the largest share: a low 25 percent, where about 16 parties areexpected to win seats according to N0 = 1/s2

1 ; a median 38 percent, whereabout 7 parties are expected to win seats; and a rather high 50 percent,where about 4 parties are expected to win seats. At each of these values

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Table 9.2. Seat shares of parties ranked by size, for given largest share—probabilisticmodel, politically adjusted model, and the actual world averages

Party Largest share 25.0% Largest share 38.0% Largest share 50.0%

Rank Prob. Polit. Actual Prob. Polit. Actual Prob. Polit. Actual

2 19.4 20.7 22.4 25.4 30.0 29.9 28.9 34.6 37.13 14.9 17.4 19.0 16.5 16.0 17.0 14.9 8.2 8.04 11.3 14.2 12.3 10.1 5.7 7.8 6.2 4.5 3.55 8.5 8.3 7.2 5.9 3.9 3.1 0.0 2.2 1.26 6.3 3.4 5.3 2.9 2.7 1.5 0.5 0.27 4.6 2.8 3.8 1.2 1.7 0.78 3.3 2.2 1.8 0.0 1.0 0.69 2.4 1.7 1.2 0.6 0.5

10 1.6 1.3 0.8 0.3 0.411 1.1 1.0 0.5 0.1 0.312 0.7 0.8 0.3 0.0 0.213 0.4 0.6 0.2 0.114 0.3 0.5 0.115 0.1 0.3 0.116 0.0 0.1 0.0SUM 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0

Source: Data interpolated from Taagepera and Allik (2006).

of the largest share, three distributions are shown: those predicted bypurely probabilistic and politically adjusted models, plus the actual one.Horizontal lines indicate the ranks beyond which parties no longer areexpected to win seats, on the basis of N0 = 1/s2

1 . Bold script indicates thecases where the model differs from the actual average by more than 2.0percentage points.

For the second- to fourth-ranking parties, most predictions by the prob-abilistic model deviate from the actual distribution by more than 2 per-centage points. At s1 = 0.50 = 50%, this model underestimates the second-largest share by 8 percentage points (and the deviation becomes worse forlarger s1, not shown here). The politically adjusted model, though far fromperfect, agrees with the actual distribution within 2 percentage points intwo-thirds of the cases.

Note that the model N0 = 1/s21 predicts the number of seat-winning

parties quite well at s1 = 0.25 = 25%. At 38 percent, many more partiesoccasionally win a seat, resulting in a string of average shares of less than1 percent. The politically adjusted model reproduces this long tail, whilethe probabilistic model does not. The same applies at s1 = 0.50 = 50%,where the fifth-ranking party achieves some minimal representation inmost cases. Here the general deviation from N0 = 1/s2

1 at low number ofseat-winning parties starts to set in.

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s1 + s2 = 100%

s2 = s1

0%

10%

20%

30%

40%

50%

0%Largest party seat share

Sea

t sha

res

Second- Third- Fourth- Empirical second- Empirical third- Empirical fourth-largest

10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Figure 9.2. Average seat shares of parties ranked by size vs. largest seat share—politically adjusted predictive model and actual data

Source: Reprinted from Electoral Studies, 25, R. Taagepera and M. Allik, ‘Seat Share Distributionof Parties: Models and Empirical Patterns’, 696–713, © 2006 Elsevier Ltd., with permissionfrom Elsevier.

The overall degree of fit is shown in Figure 9.2. The model-based curvesare shown along with the actual data (the same as in Figure 9.1), for thesecond- to fourth-largest parties. Consider each curve separately.

The seat shares of the second-largest party agree with the modelwithin ±2 percentage points, as long as the largest share remains below50 percent. This error is within the range of random scatter of datapoints themselves, so the fit is as good as it possibly could be. When thelargest share is between 50 and 60 percent, the model falls short of theactual values by up to 6 percentage points. This deviation looks seriousenough to call for further refinement of the model. At largest sharesbeyond 70 percent, the model predicts a pure two-party constellation, sothat s2 = 1 − s1. The actual curve falls below that expectation by up to 8percentage points, but the number of data points is low and restricted toa few countries, so that the data are questionable.

The seat shares of the third-largest party agree with the model within ±3percentage points, as long as the largest share remains below 55 percent.The model predicts the rise, peak, and decline quite accurately. Whenthe largest share ranges from 55 to 65 percent, the model exceeds theactual values by up to 5 percentage points. At largest shares beyond70 percent, the model predicts complete extinction of the third party,

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while actually its share rises again—but once again, the data themselvesare questionable.

The seat shares of the fourth-largest party agree with the model within±2 percentage points everywhere, except when the largest share is 40 per-cent. The corresponding graphs for fifth- to seventh-largest parties areavailable in Taagepera and Allik (2006). Agreement with the model iswithin ±2 percentage points, which says little, given that the sharesthemselves are below 12 percent and mostly below 5 percent. Here theaccuracy of the empirical data suffers from the presence of the ‘Others’category.

In sum, this graph shows that the politically adjusted model does fit,within the random fluctuation of the empirical data, when the the largestseat share is below 50 percent. For the largest shares above 50 percent, arefinement of the model may in order, along with further data collectionso as to determine the empirical means with more confidence. Table 9.2agrees with the picture described. Taagepera and Allik (2006) offer fulltabulation of numerical values.

The Effective Number of Legislative Parties

We now come to the effective number of legislative parties (N), arguablythe most important single indicator to characterize a party system. Fora given largest seat share, the values of N are restricted to a range ofwhich the upper and lower boundaries are somewhat complex. As shownin chapter appendix, the geometric mean of these extreme values can beapproximated with

N =1

s4/31

.

Combining it with s1 = (MS)−1/8 leads to a mean estimate of the effectivenumber of parties from purely institutional inputs. The effective numberof legislative parties is around the sixth root of the seat product:

N = (MS)1/6.

I tested this approximate model with those twenty-five countries inLijphart’s Patterns of Democracy (1999) in which all seats are allocatedin districts, so that M can be determined. The corresponding data aretabulated in Taagepera and Sikk (2007). The ratio N/(MS)1/6, expected to

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NED

SPAUKMLT

PNG

BOT

MRTN = 1.09MS0.153

R2 = 0.5091

1

10

100

10MS

N

N = (MS)1/6

100,0001,00001,000100

Figure 9.3. Effective number of legislative parties vs. seat product MS—predictivemodel and regression line

Data source: Taagepera and Sikk (2007).

be 1.00, ranges from 0.72 (UK 1945–97) to 2.74 (Papua New Guinea 1977–97). The geometric mean of this ratio is 1.036 for 14 single-seat systemsand 0.953 for the 11 multi-seat systems. For all 25 systems, the geometricmean is 0.999. This is closer to 1.000 than one could hope for.

Using these data, Figure 9.3 shows the mean effective number of leg-islative parties (see data in Appendix to the book) graphed against theseat product MS, using logarithmic scales on both axes. The best linear fitof logarithms corresponds to

N = 1.09(MS)0.153. [observed best fit, R2 = 0.51]

It almost overlaps with the expected

N = 1.00(MS)0.167. [theoretical model]

Visibly, the predictive model fits this particular data-set practically as wellas the postdictive best fit. Lighter lines indicate a half and double theexpected value. Most data points crowd together in the center of the zone,and only Papua New Guinea is outside. Due to approximation involvedin N = 1/s4/3

1 , we would expect some deviation at MS > 3,500, but it seemsinsignificant compared to random variation.

Various other factors besides the seat product also interact with theeffective number of parties. The degree of centralization offers a puzzlingexample. An increasing number of parties has been found to increase

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central government expenditures, roughly as the cube of the largest partyseat share (Mukherjee 2003). In turn, higher political and economic cen-tralization has been found to reduce the effective number of parties inIndia and the USA (Chhibber and Kollman 1998). While the measures ofcentralization and number of parties differ somewhat, it should not makethat much of a difference. Do we have a situation where centralizationand party multiplication keep each other in check? I have no answer.

Conclusions and Implications for Institutional Engineering

The most important conceptual result is that we have logically connectedthe various ways to measure the number of parties to each other and toinstitutional inputs of the electoral system, condensed in the seat product.Three ways to measure the number of parties were pointed out earlier: N0,N2, and N∞. We can now express their average relationship to the seatproduct MS. The basic building block seems to be not the seat productitself but its square root. It is the geometric mean of M and S and mightbe called the aggregate of the electoral system (cf. appendix to Chapter 6):A = (MS)1/2. The averages of N0, N2, and N∞ correspond to the square,cube and 4th roots, respectively, of the electoral system aggregate:

N0 = the number of seat-winning parties = (MS)1/4 = A1/2.

N2 = the effective number of legislative parties = N = (MS)1/6 = A1/3.

N∞ = inverse of the largest seat share = 1/s1 = (MS)1/8 = A1/4.

These relationships among them form a remarkable series:

N4∞ = N3

2 = N20 = A1.

It follows (and this is of course the same as N0 = 1/s21) that

N0 = N2∞.

The systematics of measuring the number of parties, which included N0,N2, and N∞, was first pointed almost 40 years ago (Laakso and Taagepera1979). The approximate relationships shown above, however, do notemerge directly from the definitions of N0, N2, and N∞. The path ofdiscovery, always using the geometric means of upper and lower limits,was the following. First the highest possible way to count the parties(N0) was obtained from MS. Then the lowest possible way (N∞ = 1/s1) was

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calculated from N0, and it was found to be the square of the highest. Theintermediary N2 was obtained from balancing these two extremes.

At the start of the chapter, the following question was posed: Doespolitical competition tend to place the two largest parties in a specialcategory of size, far larger than the third parties, or do seat shares taperoff gradually? We find it depends on the size of the largest share. Electoralsystem divides the parties ranked by size into two groups: relative losersand winners. As the largest share increases, ever fewer parties belong tothe advantaged group. As the largest share surpasses 25 percent, only twowinning parties tower above the rest.

The seat product allows us to make much more specific predictionsfor the distribution of seat shares, for any simple electoral system. Forinstance, in an assembly of 200 seats elected in 8-seat districts, we wouldexpect somewhat more than (1,600)1/4 = 6.3 parties to win seats, on theaverage. The largest party is expected to have about (1,600)−1/8 = 0.40 =40% of the seats. Figure 9.2 and Table 9.2 for politically adjusted sharesthen suggest a distribution around 40-30-16-6-4-2-1-1, with 8 partieswinning seats (for a more detailed table, see Taagepera and Allik 2006).Instead, one may prefer to use the empirically observed distributions inTable 9.1. Then the pattern changes slightly, to 40-30-15-9-4-1.5-0.5, with7 parties winning seats usually, and sometimes a few more. This would beour best guess in the absence of any other information.

The main use of the pattern thus established is to supply the baselinefor characterizing the average seat distributions in a given country. Oncewe find out how the country differs from the worldwide average, we canstart looking for the underlying reasons. This should be a fertile field ofstudy in years to come.

What are the implications for institutional engineering? The observa-tions made in the previous chapter still apply. The quantity to watch isthe seat product. The larger the seat product is, the larger the effectivenumber of parties. To lower the average effective number of parties, onecan lower either district magnitude or assembly size, or lower both to amore moderate degree. Once again, the country’s peculiarities must betaken into account. The effective seat product could differ somewhat fromthe product of M and S, depending on other factors.

To alter an existing effective number of parties, however, there is noneed to go through the worldwide average model. Suppose you wish toreduce the effective number of parties by one-tenth. Your best bet is tomultiply 1 − 0.1 = 0.9 by itself 6 times, which yields 0.53. This is by howmuch you must multiply the present seat product, which roughly means

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dividing the seat product by 2. To do so, you can cut the present districtsof magnitude M into two smaller districts of magnitude M/2. You canalso cut the assembly size by a half, but now it gets a bit tricky. If youkeep the same district boundaries, district magnitudes would be cut intoby a half, and you would reduce the number of parties excessively. Tokeep the same magnitude, you would have to join two existing districts.You can also reduce both district magnitude and assembly size by about30 percent, but district boundaries would have to be redesigned.

More generally, for any desired fraction x of decrease in the effectivenumber of parties, multiply 1 − x by itself 6 times. This is our best bet forthe fraction by which the existing seat product has to be multiplied. Thischange can be obtained by altering either district magnitude, assemblysize, or both—but check that the new M and S do give the desired product.


This appendix derives the predictive models for seat shares and the effective num-ber of parties. It extends the model to the entropy-based variant of the effectivenumber. It also points out a marked conceptual inconsistency in the seat sharemodel, which makes us wonder why the model still fits actual data.

The probabilistic model for seat shares

In the previous chapter, we used the conceptual limits of the largest fractionalshare, average share 1/N0 and almost 1, to estimate the likely mean. The result, s1 =1/N1/2

0 , agrees pretty well with actual values, as long as the largest share remainsless than 63 percent (cf. Figure 8.3), which would correspond to N0 = 2.

Given this success, let us repeat the procedure with the second-largest party.For given largest share, the remaining (N0 − 1) parties account for a total fraction(1 − s1) of all the seats. Hence their average share is (1 − s1)/(N0 − 1). This is thelower limit for the second-largest party. Its upper limit is slightly below (1 − s1), soas to leave a minimal number of seats to the third parties. The geometric mean ofthese limits is

s2 =(1 − s1)

(N0 − 1)1/2.

For the third-largest party, similar reasoning yields

s3 =(1 − s1 − s2)(N0 − 2)1/2

.

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The general formula for the ith ranking party is

si =(1 − �s j )

(N0 − i + 1)1/2,

where the summation ranges from j = 1 to i − 1. Once the largest seat share isgiven, all other shares can be calculated. Note that this approach does not presumea connection between the largest share and the number of parties, such as s1 =1/N1/2

0 . Introducing this relationship would lead to

s2 = s1

[1 − s1

1 + s1

]1/2

.

For integer values of N0, the sum of shares thus calculated is 1, as it should. If wedo calculate N0 from N0 = 1/s2

1 and it has a noninteger value, then we apply theformula to the integer part of N0, and the resulting sum of seat shares is slightlyless than 1. We take the remainder to represent a tiny party whose rank exceedsthe integer part of N0. Taagepera and Allik (2006) include a table of resulting seatshares and graph it.

Political adjustment to the probabilistic model for seat shares

We now assume that the small parties give up a fraction m of their inherent supportbase to larger parties. The tiniest parties may well give up a larger fraction. If so,then m gradually decreases with increasing party size. It would reach 0 when theparty almost accedes the select club of the parties that profit from the shift. Thus mcan be expected to be a function of the relative size of the parties. Determining theshape of this function, however, is difficult. As a first approximation, we assumethat m is the same for all parties that lose support. The conceptual limits on such aconstant are m = 0, when no support is lost, and m = 1, when all support is lost. Inthe absence of any further knowledge, we try first the median value m = 0.5, whichsimplifies calculations. On this basis we establish the total share of seats given upby the small parties.

This total is transferred to the large parties. The number of such parties wasobserved to be close to N∞ = 1/s1. The largest among them may well profit morethan proportionately, but it is hard to estimate how much more. So again, as a firstapproximation, we assume that all such parties profit equally.

The party at the watershed between the losers and the winners occupies a specialposition. This is the party at rank i0 closest to 1/s1. We assume it gains as much asit loses, so that its share remains unchanged. This assumption breaks down whenthe largest share is so large (more than 40 percent) that even the third-largest partystarts losing seats. From this point on, we must assume that there is no middleground between losers and winners.

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The previous probabilistic equation for seat shares was si = (1 − �s j )/[(N0 − i +1)1/2]. It is now adjusted according to the description above. The notation s′

i is usedso as to tell the adjusted shares apart from the unadjusted. Full derivation of thefollowing equations is given in Taagepera and Allik (2006). For small parties, thoseat rank index i > i0, the adjusted seat shares are

s′i = (1 − m)si . [i > i0]

For large parties, those at rank index i < i0, the adjusted seat shares become

s′i = si +

m(1 − �sk)(i0 − 1)

, [i < i0]

where k runs from 1 to i0. The intermediary party (i = i0) undergoes no adjustmentwhen the largest share is small:

s′i = si . [i = i0, s1 < 0.4]

At unadjusted s1 > 0.4 = 40%, this intermediary stage vanishes, and for the twolargest parties the equation becomes

s′i = si +

m(1 − s1 − s2)2

. [2 largest parties, s1 > 0.4]

When the largest share exceeds 50 percent, the number of winners from theadjustment drops below 2, so that even the second-largest party begins to sufferfrom the adjustment. Then

s′1 = s1 + m(1 − s1). [largest party, s1 > 0.50]

Assuming m = 0.5 yields a slightly simplified set, still fairly messy:

s′i = 0.5si [i > i0]

s′i = si + 0.5(1 − �sk)/(i0 − 1) [i < i0]

s′i = si . [i = i0, s1 < 0.40]

s′i = si + (1 − s1 − s2)/4. [2 largest parties, s1 > 0.40]

s′1 = 0.5(1 + s1). [largest party, s1 > 0.50]

This is what the curves in Figure 9.2 and the politically adjusted numbers inTable 9.2 are based on. Work in progress (Taagepera and Laatsit 2007) graphs thecurves for values of m ranging from 0 to 1 and compares them to actual data.

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Critique of the politically adjusted model for seat shares

Once again, this is the worst possible way to account for political adjustment tothe probabilistic model, except for all others, if we want to end up with a specificprediction. We approximate a varying loss function m(s) with a constant. We alsouse a constant level of gain, plus one neutral point that suddenly vanishes whenthe largest share becomes large.

Even before making this gross simplification, we assumed a single function m(s)to account for the multitude of distinct factors that range from strategic sequenc-ing to consequences of economics of scale. Could we feed them in separately? Wecould, as far as algebra is concerned. But we would end up with so many parametersthat our database would not suffice to determine their numerical values. It wouldbe another of those impressive models that are unable to make specific numericalpredictions.

In the face of such complexities, one may encounter advice to give up and limitoneself to qualitative or directionally predictive models, which never are falsifiedbecause their predictions are so fuzzy that everything and its opposite fit in. Interms of quantitative prediction, we would be left with the previous probabilisticmodel, with all its discrepancies. Yet these discrepancies all point in one broaddirection: Compared to PR, the largest parties win, while the smaller ones lose,with a neutral point around the party whose rank is close to N∞ = 1/s1. Evena single parameter, such as m, should go a long way to correct for this broaddiscrepancy. The result would be expected to be closer to the actual pattern,compared to the probabilistic model. If our guess at m = 0.5 is excessive, we wouldobserve an overcorrection, and vice versa. Hence, such a first approximation wouldhelp us to refine the model, either by adjusting the value of m or by suggestinghow to introduce a second parameter. Figure 9.2 indicates that the model canstand refinement at large values of the largest share, where s1 = 1/N1/2

0 breaksdown.

One hazy aspect of the model is that it equivocates between shares of seats andshares of votes, when it talks of the ‘inherent support base’ of a party. For PR, itdoes not matter much, but when seat shares differ appreciably from vote shares, asis the case for FPTP, it might matter. The seat and vote levels are hard to disentangleconceptually, because voters can react only at the vote level (by withholdingvotes), while being motivated to defect by what happens at the seat level: lowrepresentation of the party in assembly and government, plus the concomitantlow press coverage.

Two questions immediately arise: What would the pattern be, if Figure 9.1 wereredone, using vote shares instead of seat shares? And would the patterns be thesame for List PR and for FPTP? Work in progress (Taagepera and Laatsit 2007)indicates that the overall pattern for votes and seats are identical, within therange of random error. Thus the equivocation between votes and seats in modelingpolitical adjustment is less severe than it could have been. The value of m might

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be around 0.25 for List PR and around 0.75 for FPTP—but still with essentially thesame patterns for seats and for votes.

A relatively minor problem with the adjustment is that the procedure includesdiscontinuities. As the largest share increases, the other larger parties make suddentransitions from winner status to neutral and then to loser status. This explainsthe jagged shapes of the curves in Figure 9.2. Such ‘quantum jumps’ are mostlikely due to approximating a smoothly changing loss function m(s) and thecorresponding gain function with constants. Broad agreement with data suggeststhat we are basically on the right path. Hence it may be worth while to invest inworking out a model with smoothly changing m(s).

Further questions arise when the largest share increases beyond 63 percent, sothat applying s1 = 1/N1/2

0 rounds off to N0 = 2. Figure 8.3 shows that s1 = 1/N1/20

no longer applies when N0 = 2. So the model should be reworked for cases with ahegemonic largest party.

But the most severe problem with the adjusted model is one that no commenta-tor has picked up: The model undermines its own foundations. The very startingpoint of the adjustment was that the number of seat-winning parties is close toN0 = 1/s2

1 . But in the process of adjustment, the largest share itself shifts from its‘inherent support level’ (s1) to an adjusted level. This adjusted level is the onlyone we actually observe, according to the model. But if so, then N0 = 1/s2

1 nolonger applies to the observed largest share—it only applies to a hypothetical andunobservable ‘inherent support level’ for the largest party!

The nice part of this paradox is that it explains a discrepancy in Table 9.2.At largest shares 38 and 50 percent, more parties are observed to win minorshares of seats than was expected on the basis of N0 = 1/s2

1 . This is indicated bythe horizontal lines in Table 9.2. Moreover, the adjusted model quite accuratelyreproduces these shares. But how come that the relationship N0 = 1/s2

1 (in itsreversed form s1 = N1/2

0 ) fitted the actually observed values so well in Figure 8.2?At some values of the ‘inherent support level’, the adjusted largest share exceeds itappreciably, and hence both should not fit.

Is the reader confused? It should be so, because I am confused myself. In viewof the agreement with data, this confusion is no reason to give up on the modelpresented. Clumsily, it expresses something real. It will take time to clarify theterms used.

How the effective number of legislative parties connectsto the largest share

We have two ways to estimate the effective number of legislative parties, for agiven largest seat share. We could calculate it from the equations of the politicallyadjusted model. But the result would be algebraically messy. Alternatively, wecould try a shortcut. Observing that the effective number depends most heavily onthe largest share, we could try to estimate the effective number from that largest

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Table 9.3. Effective number of parties for given largest share

s1 (%) 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0Act.N 7.32 5.75 4.93 4.08 3.56 3.05 2.53 2.13 2.06 1.95 1.80

1/s3/21 11.18 8.00 6.09 4.83 3.95 3.31 2.83 2.45 2.15 1.91 1.71

2/s0.91 − 1 7.51 5.96 4.91 4.14 3.56 3.10 2.73 2.43 2.17 1.95 1.76

1/s4/31 8.55 6.35 4.98 4.05 3.39 2.90 2.52 2.22 1.98 1.78 1.61

Source: Calculated from actual mean shares in Table 9.1 and as estimated from N = 1/s3/21 (old model), N =

2/s0.91 − 1 (new more precise model), and N = 1/s4/3

1 (new approximate model).

Note: Deviations of more than 0.3 parties from the actual mean values are shown in bold.

share alone. The values derived from the adjusted model for all seat shares wouldserve as a check on how accurate the shortcut is.

An average connection between the effective number (N) and the largest share(s1) was proposed by Taagepera and Shugart (1993): N = 1/s3/2

1 . Combined withs1 = (MS)−1/8, it leads to N = (MS)3/16. It turns out that N = 1/s3/2

1 overestimates N,except when the largest share exceeds 62 percent. The discrepancy is due to amistake in the model. I first present the old model, because it is simple to follow,and then two versions of the corrected one.

At a given value of s1, the effective number could be almost as low as 1/s1 = N∞.This is the case when all shares are equal. It could also be almost as high as 1/s2

1 =N0. This is the case when all other shares are infinitesimally small. With only thelimits 1/s1 < N < 1/s2

1 known, the best guess is the geometric mean of N∞ and N0:N = 1/s3/2

1 .Taagepera and Allik (2006) compared these estimates with the actual mean val-

ues and found deviations. Similar contrasts can be seen in Table 9.3, as comparedto empirical mean seat shares in Table 9.1. When the largest share is small, 1/s3/2

1

severely exceeds the observed mean N. The observed mean catches up arounds1 = 0.62 and surpasses 1/s3/2

1 when the largest share becomes predominant. Thereason for such a discrepancy is that the simple model overstates both conceptuallimits.

The old model assumed that the minimal effective number, at given largestshare, is reached when all shares are equal. Then N = 1/s1. However, all sharescan be equal only when s1 = 1/2, 1/3, 1/4, etc. At any other values, N cannot go aslow as 1/s1. The difference is marked when s1 is large, and this is why the observedmean N remains larger than 1/s3/2

1 when the largest share exceeds 62 percent.At the other extreme, the old model assumed that maximal N at given largest

share is reached when the shares of all seat-winning parties but the largest areinfinitesimally small. Then N = 1/s2

1 . But the other shares cannot be that small.The smallest conceivable nonzero share is one out of the S seats in the assembly.Yet even 1/S is too low. All shares but the largest being 1/S would imply a numberof seat-winning parties that most often exceeds by far the mean expectation ofN0 = 1/s2

1 parties winning seats. If we limit the number of seat-winning parties

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to N0 = 1/s21 , the effective number of parties is the lowest when all parties apart

from the largest are equal. These equal shares amount to a total of (1 − s1) dividedamong (N0 − 1) parties. The resulting maximum N falls short of the previouslimit, 1/s2

1 . The discrepancy is the largest at small s1. This is why the observedmean N remains larger than 1/s3/2

1 when the largest share is small. The picturechanges when the largest share becomes so large that the N0 = 1/s2

1 projects toless than 2 seat-winning parties. Here we partly have to revert to the earliermodel.

The important outcome is that the resulting intricate relationship between theeffective number of parties and the largest share can be approximated by modelsthat agree with data better than does the previous N = 1/s3/2

1 . The details of tworefined models are given in the next section, and the results are shown in Table 9.3.The two-parameter model

N =2

s0.91

− 1 [refined new model]

agrees with actual mean data within ±0.3 parties at all ranges of the largest share.Note that it respects the conceptual anchor point s1 = 1 → N = 1. The simplifiedone-parameter model

N =1

s4/31

[approximate new model]

agrees with actual mean data within ±0.3 parties only when the largest shareexceeds 27 percent. Still, this is a degree of agreement the previous N = 1/s3/2

1

offers only when the largest share exceeds 43 percent (cf. Table 9.3). When thelargest share is as small as it ever is observed to (around 20 percent), even thenew approximate model exceeds the observed mean by 1.2 parties, but this isappreciably better than the old model’s excess of 3.9 parties.

One may lose in generality by trying to fit too closely to theoretical boundaryconditions which offer a wide permissible zone. So I will use N = 1/s4/3

1 . Combin-ing it with s1 = (MS)−1/8 leads to a simple format for the mean estimate of theeffective number of parties from purely institutional inputs:

N = (MS)1/6.

The effective number of legislative parties is around the sixth root of the seat product.Compared to the old model’s exponent 3/16 = 0.1875, we have shifted to 1/6 =0.1667. For very large seat products (MS > 3,500), it can be expected to overes-timate N by more than 0.3. If this difference matters, N = 2/s0.9

1 − 1 should beconsidered instead of N = 1/s4/3

1 . In most cases, random variation exceeds thislevel.

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The effective number of parties and the largest share:Details of the refined model

This section involves tedious calculations that yield a somewhat wiggly meanrelationship between the effective number of legislative parties and the largestseat share. I have to present these calculations so as to justify the claim that,with just a slight smoothing of the actual average relationship, N = 1/s4/3

1 is theprobabilistically expected value of N. The result-oriented reader, however, maywish to bypass this section.

For a given largest share (s1) and number of seat-winning parties (N0), the largestpossible value of the effective number of parties (N) corresponds to the other(N0 − 1) parties having equal shares (1 − s1)/(N0 − 1):

max N =1

s21 + (N0 − 1)(1 − s1)2/(p − 1)2

=1

s21 + (1 − s1)2/(N0 − 1)

.

As long as N0 = 1/s21 applies, this expression simplifies into

max N =1 + s1

2s21

.

When the largest share is very small, max N approaches 0.5/s21 . This is appreciably

lower than the previous limit 1/s21 .

When N0 = 2, the relationship N0 = 1/s21 no longer applies (cf. Figure 9.2). This

N0 = 2 corresponds to s1 larger than 1/(2.5)0.5 = 0.63. Here maximum possible Ncorresponds to the situation where all parties but the largest have one seat. Usingthe actual number of seats, there are then (S − S1) such small parties in an assemblywith S seats, and max N = S2/(S2

1 + S − S1). Since S21 � S − S1 even for an assembly

as small as 10 seats, this limit reduces itself to the previous max N = S2/S21 = 1/s2

1 .The lower limit is even more complex. Assume that assembly size is sufficiently

large (S � N0), so that the minimum of one seat going to each minor party canbe neglected. Actually, this is the case for all assemblies with at least three seats.Indeed, whenever N0 = 1/s2

1 and s1 = (MS)−1/4 hold, then S � N0 amounts to M S3, which is the case whenever S > 3. When s1 = 1/2, 1/3, 1/4, etc., we can have allshares equal and hence N = 1/s1. For intervening values of the largest share, makingas many shares as possible equal to the largest minimizes N, leaving the remainder asa smaller party.

For 1 < s1 < 1/2, no other party can match the largest. The effective number islowest when the second-largest party is as large as possible, meaning s2 = 1 − s1.The resulting minimal effective number is

min N =1

1 − 2s1 + 2s21

, [1 < s1 < 1/2]

a value larger than 1/s1. At s1 = 0.65, 1/s1 = 1.54, while 1/(1 − 2s1 + 2s21 ) = 1.83.

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For 1/2 < s1 < 1/3, the lowest N corresponds to s2 = s1 and s3 = 1 − 2s1. Then

min N =1

1 − 4s1 + 6s21

. [1/2 < s1 < 1/3]

For the next ranges which are still of practical interest, the equations are

min N =1

1 − 6s1 + 12s21

, [1/3 < s1 < 1/4]

min N =1

1 − 8s1 + 20s21

, [1/4 < s1 < 1/5]

where the deviation from 1/s1 becomes negligible.We should try to fit the geometric means of these minimum and maximum

values with some simple function of s1. They can be well fitted with a two-parameter format N = a/sb

1 − a + 1, which correctly predicts N = 1 when s1 = 1. Thebest fit is close to N = 2/s0.9

1 − 1.However, we should also try to fit with the simpler one-parameter format N =

1/sn1, because that form can be easily combined with s1 = (MS)−1/8 so as to connect

N with the seat product MS in a simple way. This is possible, indeed. Dependingon whether one wishes to emphasize the fit at lower or higher values of the largestshare, the exponent n could be taken as anywhere between 1.30 and 1.45. Thechoice of the simple fraction n = 1/3 = 1.333 agrees with data (cf. Figure 11.4),although it could run into trouble at very high values of the seat product.

Entropy-based effective number of legislative partiesand the largest share

The same approach can be used to estimate any measure of the number of parties,Na = [�(si )a]1/(1−a), with 0 < a < ∞ (cf. Chapter 4). In particular, the entropy-basedN1 = eH might be of interest. The exact equations for minimum and maximumvalues are even more involved than those for N2. The best two-parameter fit isaround N1 = 4/s0.67

1 − 3. The best one-parameter fit, around N1 = 1/s1.61 , is very

coarse—see Table 9.4.

Table 9.4. Entropy-based effective number of parties

s1(%) 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0Act. N1 8.60 7.08 6.10 5.03 4.34 3.68 3.02 2.33 2.36 2.30 2.214/s0.67

1 − 3 8.76 7.13 5.96 5.08 4.39 3.83 3.36 2.97 2.63 2.34 2.081/s1.6

1 13.13 9.19 6.86 5.36 4.33 3.59 3.03 2.60 2.26 1.99 1.77

Source: Calculated from actual mean shares in Table 9.1 and as estimated from two equations.

Note: Deviations of more than 0.3 parties, compared to the actual mean values, are shown in bold.

164

10

The Mean Duration of Cabinets


� The quantity to watch is again the seat product—the product of thenumber of seats in the assembly and the number of seats allocated inthe average district.

� The lower the seat product, the longer the mean duration of govern-ment cabinets.

� If you wish to double the mean duration of cabinets, your best bet is todivide the present seat product by 8. To do so, you can cut each presentdistrict into 8 smaller districts. You can also cut each present districtinto 2 smaller districts and cut assembly size by a half.

� This way to calculate is based on a logical model that agrees with theworld average. It is approximate, because other factors enter, but this isyour best bet.

� A desired increase in cabinet duration does not come free—it reducesrepresentation of small parties, and they will put up a fight. If, asa price for reduction in district magnitude, you agree to introducenew complexities in the electoral rules, then all bets are off regardingconsequences.

Many political and even economical factors may be affected by the effec-tive number of parties, to judge by the numerous empirical studies thatinclude N. Even when significant, most such relationships remain empir-ical. For the mean duration of government cabinets, however, we can alsoestablish a predictive model of why the number of parties should affectit, and by how much. This means that a logical connection is establishedbetween the mean cabinet duration and the effective number of parties.As the latter, in turn, is connected to the seat product, so will cabinet

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duration. Thus, mean cabinet duration can be predicted from institutionalinputs.

The duration of cabinets can make a difference for the nature andquality of governance. If the regular interelection period is 4 years, thena mean cabinet duration of 4 years implies that cabinets tend to lastuntil the next election, while a mean duration of 2 years implies roughlyone cabinet change between two elections. Extremely short-lived cabinetsmake it hard to formulate and implement policy. At the other extreme,cabinets that last past many elections may favor cronyism and stagnation.So the ability to use institutional means to modify the mean cabinet dura-tion by a specified amount could be of interest to political practitioners. Inparticular, when new democracies decide on their institutions, durationof cabinets is often among the concerns of decision-makers, foremost inthe form of what they do not want: ‘Let us avoid short-lived cabinets likethey have in. . . .’

As for the students of politics, Warwick (1994: 139) represents thewidespread view that even mildly short-lived cabinet can put regimesurvival in danger, over the longer run. Dogan disagrees (1989), andLijphart (1999: 130) puts it bluntly: ‘This view is as wrong as it iswidespread.’ Lijphart (1999: 131–9) sees mean cabinet duration as auseful indicator of executive dominance and proceeds to measure itin various ways. To the extent we can explain why some countrieshave shorter durations than some others, we might also understandvarious implications of cabinet duration, such as how it affects regimeperformance.

Nothing will be said here on why some cabinets last longer than someothers, within the same country with stable institutions. This is thesubject of a rich separate literature, reviewed by Laver (2003), whichfocuses on bargaining models based on rational choice. As far as themean duration is concerned, however, bargaining models offer no spe-cific predictions. Thus the chapter on ‘Party systems and cabinet stabil-ity’ in a book by Laver and Shepsle (1996: 195–222) offers ‘two basicconclusions’: Certain bargaining constellations are ‘substantially morestable’, and ‘the model can be used to understand why governmentsmight change tack between elections’ (Laver and Shepsle 1996: 215).They present theory, simulations, and discussion of specific past cases.But if one asks how much duration is expected in a given country, onthe average, they offer no answer. The present book does give such ananswer, with a 50-50 probability of this prediction being high or lowand with an estimate of likely margin of error. This answer is based on

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the number of players (parties) involved, a number itself grounded ininstitutions.

The Inverse Square Law of Cabinet Duration, Relative to theNumber of Parties

That the number of parties affects cabinet duration was documented byLijphart (1984: 124–6) several decades ago. Coalition cabinets, prevalentin multiparty systems, tend to be more short-lived than one-party major-ity cabinets. When Grofman (1989) controlled for the effective number ofparties, the effect of cabinet type (minimal winning or larger or smaller)on cabinet duration largely disappeared. The number of parties and cab-inet types are strongly correlated, and it is easier to visualize the numberof parties imposing cabinet type rather than cabinet type impacting thenumber of parties. A logical connection between the number of partiesand cabinet duration was presented in Taagepera and Shugart (1989: 99–101). The crucial link is the number of communication channels amongthe parties, which can also become conflict channels. Hence more partiesmean more potential conflicts, which can undo a cabinet. The followingmodel emerges.

We may surmise that cabinets break up, on the average, when a certainamount of conflict has accumulated in the political system. It may looksimplistic, but this is actually the only logical guess we could make, in theabsence of any further information, short of sterile ‘We cannot know’. Ifthis assumption is wrong or overly simple, lack of agreement with datawill tell us so. To the extent it holds, the mean duration of cabinets(C) can be expected to be inversely proportional to the frequency ofconflicts ( f ): When conflict frequency doubles, duration is halved. Morebroadly, C = k′/ f , where k′ is a constant. True, it could well be that C isnot proportional to 1/ f . But if so, would it increase at a more or lessthan proportional rate? If we cannot answer this question, our best bet isproportionality.

Conflict frequency itself may depend on the number of conflict chan-nels (c) among parties. Would it be more than proportional to c, or less?Not knowing which way it is, the only defensible guess is proportionality:f = k′′c, where k′′ is a constant. Of course, some channels are less conflict-ual than some others, and the degree of conflict varies over time. This iswhy some individual coalitions last longer than others. But here we areconcerned about the mean degree of conflict per channel.

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If the assembly has n equal-sized parties, then the number of com-munication channels among them is c = n(n − 1)/2, as one can eas-ily check by drawing connecting lines between points. However, thiswould underestimate the number of potential conflict channels. Par-ties are not fully unitary actors. Coalitions sometimes break up becauseof a coalition partner’s internal conflict. If we add an intraparty con-flict channel per party, we would have c = n(n + 1)/2, which, in turn,might be an overestimate. The mean of the two estimates is simplyc = n2/2.

Combining all these links yields C = k/n2, where k is a constant, k =2k′/k′′. Thus the model predicts an inverse square relationship, leaving kto be determined empirically. Since n is a pure number, k must have thesame time units as cabinet duration itself. We will measure C and k inyears.

Actually, all parties are rarely equal-sized. In such cases we will assumethat the effective number of parties (N) is to be used:

C =k

N2.

The choice of the effective number of parties can be disputed, and shouldbe. First, let us see where it leads us. If using N is erroneous, then thepresumed relationship will not be observed.

Using Lijphart’s data (1984) for stable democracies, Taagepera andShugart (1989) found that C = 400 months/N2 = 33 years/N2 predictsmean cabinet duration within a factor of 2, meaning that the observedvalues are within a zone that extends from a half to double the predictedvalue. A journal reviewer once mistook ‘within a factor of 2’ to meanwithin ±2 years. So it is worth stressing: if C = 1 year, then ‘within a factorof 2’ means from 0.5 to 2 years; but if C = 10 years, then it means from5 to 20 years. Then the logarithm of C is within ±log 2 of the predictedvalue.

More extensive data (from Lijphart 1999) leads to C = 39 years/N2

(Taagepera 2003). A reanalysis with slightly corrected data (Taagepera andSikk 2007) puts the best fit with the format C = k/N2 at

C =42 years

N2.

The measure of cabinet duration used was the one devised by Dodd(1976) and designated as ‘average cabinet life I’ by Lijphart (1999:132–3).

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100

MRT

NC = 42/N 2

(R2 = 0.770)

10

1

C(y

ears

)

1

C = 31.3/N1.757

(R 2 = 0.787)

10

Notes:

Thin solid line: best-fit between logarithms. Bold solid line: theoretically based prediction [C = 42 years/N2].Dashed lines: a half and double the expected value.

Figure 10.1. Mean cabinet duration vs. effective number of legislative parties—predictive model and regression line.

Source: Taagepera and Sikk (2007).

The model implies that the product of N2 and cabinet duration isconserved: N2C = k. The mean C and N for every country yield a differentvalue of duration constant k. Estimates of the world mean have movedfrom 33 to 42 years. When calculated on the basis of individual countrydata for the 36 stable democracies in Lijphart (1999), the distributionaround the mean of 42 years is roughly normal, with a standard deviationof 14 years. The lowest individual value is 16 years, and the highest is72 years—except for Switzerland at 445 years! Being more than 3 standarddeviations off justifies the exclusion of Switzerland from the test set.Switzerland is the only non-presidential country where the executive,once empowered by parliament, does not depend on legislative confi-dence (Lijphart 1999: 119–29). Thus some key assumptions of the inversesquare model may not apply to such regimes.

Figure 10.1 (from Taagepera and Sikk 2007) shows cabinet durationfor the remaining 35 democracies in Lijphart (1999) graphed againstthe effective number of parties, both on logarithmic scales, so that theinverse square relationship becomes a straight line. The best linear fit oflogarithms corresponds to

C =31.3 years

N1.76. [observed best fit, R2 = 0.79]

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When the exponent 2 is imposed, the best fit is

C =42 years

N2. [theoretical model, R2 = 0.77]

The two lines almost superimpose, and R2 is barely reduced, so themodel visibly fits. In view of existence of a logical model and its empir-ical confirmation, we have here a law, in the scientific sense of theterm—the inverse square law of cabinet duration, relative to the number ofparties.

It might seem more appropriate to consider only the communicationchannels within the coalition. Why would stresses among the partiesoutside the coalition shorten coalition duration? This question is wellfounded, but unpublished work by Lijphart and me (reported in Taageperaand Shugart 1989: 101–2) yields a surprising result: There is poor correla-tion between the mean coalition duration and the number of parties inthe coalition itself. Parties excluded from the coalition still seem to haveways to affect its duration.

It remains to account for the deviation from C= 42 years/N2 for indi-vidual countries. Here the cabinet types may enter separately from thenumber of parties. Also, balance in the size of parties may play a rolebecause, at the same effective number, party systems with a dominantparty might be more durable. Inclusion of centrist parties also might con-tribute to duration (van Roozendaal 1992), at the same effective numberof parties.

Is there an Inverse Cube Law of Cabinet Duration, Relative tothe Seat Product?

Chapter 9 connected the effective number of parties to the seat prod-uct MS: N = (MS)1/6, on the average. It follows (as first pointed out inTaagepera and Sikk 2007) that cabinet duration has an inverse cube rela-tionship to the seat product, on the average:

C =42 years(MS)1/3

.

Out of the 35 democracies tested in Figure 10.1, district magnitude cannotbe specified for 10 countries, because all seats are not allocated within thedistricts, or further features are introduced, such as large legal thresholds.Figure 10.2 (from Taagepera and Sikk 2007) shows cabinet duration for

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NED

SPA

INDPNG

MRT

BOT

C = 21.848/(MS)0.233

(R2 = 0.300)

1

10

100

100,00010,0001,00010010MS

C (

year

s)

C = 42/(MS)1/3

(R 2 = 0.240)

Notes:

Thin straight line: best-fit between logarithms.Bold straight line: theoretically based prediction [C = 42 years/(MS)1/3].Dashed lines: a half and double the expected value.

Figure 10.2. Mean cabinet duration vs. seat product MS—predictive model andregression line

Source: Taagepera and Sikk (2007).

the remaining 25 democracies graphed against the seat product, both onlogarithmic scales, so that the inverse cube relationship becomes a straightline. See data in Appendix to the book. The best linear fit of logarithmscorresponds to

C =21.8 years(MS)0.233

. [observed best fit, R2 = 0.30]

When the exponent 1/3 is imposed, the best fit is

C =42 years

N0.333. [theoretical model, R2 = 0.24]

Thus, in this case, the expected slope exponent is appreciably above theactual, and R2 is reduced.

In model building, we are now 4 steps removed from the seat prod-uct: MS → N0 → s1 → N → C. Agreement with the model is bound todecreases with each extra step, because further political, cultural, his-torical, and institutional factors can enter, and random variation alsoaccumulates. Consequently, R2 for the best-fit line (for logarithms) canbe expected to decrease, and does so indeed, as we go from C versus N(R2 = 0.79) to C versus s1 (R2 = 0.53) and to C versus MS (R2 = 0.30). Forthe predicted line, the decrease in R2 is even steeper (Taagepera and Sikk2007), from 0.77 to 0.35 and 0.24, respectively.

Such a decrease in R2 would be of little concern, as long as the predictedexponent (slope of the log-log graph) agrees with the model. Actually, as

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one successively regresses C against N, s1, and MS, the empirical slopesincreasingly fall short of the expected: by 12 percent (slope 1.76 comparedto 2.00), 20 percent (2.14 to 2.67) and 30 percent (0.233 to 0.333), respec-tively. Part of this increasing discrepancy is an artifact of one-directionalregressing, as discussed in Beyond Regression (Taagepera 2008). Still, quitea few cases in Figure 10.2 differ from expectation by more than a factorof 2 (Botswana and Spain on the high side, and Mauritius and Papua NewGuinea on the low side).

More data are needed to clarify the issue—and possibly a more refineddefinition of cabinet duration. Taagepera and Sikk (2007) point out theproblem of Mauritius, which deviates markedly from the expected patternin Figures 10.1 and 10.2. The same prime minister stayed in office for14 years but juggled the party composition 7 times, in quite minor ways.By the Dodd (1976) counting rules, this led to a mean cabinet durationof only 2.1 years, which feels low in the face of such a long tenure by thesame prime minister. Maybe the way to measure cabinet duration shouldbe altered in the light of this anomaly.

A general inverse cube law of cabinet duration, relative to the seat product,may well exist, as a universal average, but for the moment it is a possibilityrather than certainty. We may either tentatively accept it, as basis ofuncertain prediction, or we would have to abstain from prediction alto-gether. Note that, in 19 cases of 25, C = 42 years/(MS)1/3 still predicts meancabinet duration within a factor of 2. Moreover, the 6 widely deviant casesare spread evenly in Figure 10.2: Botswana, Spain, and the Netherlands areabove the predicted zone, while Mauritius, Papua New Guinea, and Indiaare below.


Once more, we have reached a remarkably simple connection to institu-tions, this time for a highly visible feature in politics: how often do gov-ernmental cabinets change, on the average. Mean cabinet duration tendsto relate to the seat product as C = 42 years/(MS)1/3 not just empiricallybut for well-defined theoretical reasons. This equation is our best bet ininstitutional engineering, but it should be used with some good sense.

The regular interelection period is around 4 years in most countries. Ifso, then a mean duration of 2 years implies roughly one cabinet crisisbetween two elections, while a mean cabinet duration of 4 years impliesthat cabinets tend to last until the next election. We can now say that

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such a doubling of mean duration corresponds to cutting the electoralsystem magnitude MS by a factor of 2 cubed, meaning by 8.

To double the mean duration, one might split each existing district into8 smaller ones so as to reduce district magnitude. Alternatively, one mightsplit each district into 2 smaller districts, while also reducing the totalassembly size by one-half. The latter step alone would cut the magnitudeof existing districts by a half. With existing districts split, district magni-tude is only one-quarter of the previous. For example, assume an assemblyof S = 128 is elected in 16 districts of M = 8, so that MS = 1,024. Reducingthe assembly to S ′ = 64 seats and splitting the districts into two wouldresult in 32 districts with M′ = 64/32 = 2. Now M′S ′ = 128, which is 1/8 of1,024.

Note that the duration constant k did not enter in the example above.In comparing the relative effect (doubling) of two combinations of M andS in the same country, the constant k cancels out. This is how one shouldalways proceed in institutional engineering, if the country has a previousdemocratic record. If it does not have such a record, the value of k inC = k/N2 could be estimated from the records of democratic neighborswith approximately the same sociopolitical characteristics. Applying theworldwide mean of k = 42 years should be the last resort. It carries a largemargin of uncertainty.

A desired increase in cabinet duration does not come free—it also altersparty constellation. If cabinet duration is to be doubled, MS is to bedivided by 8. However, assembly size and district magnitude are juggled tothat end, the effective number of parties, N = (MS)1/6, must be expectedto go down by a factor of about 81/6 = 1.41. The largest share s1 = (MS)−1/8

is likely to increase by a factor of 81/8 = 1.30, that is, by 30 percent. Thenumber of seat-winning parties, N0 = (MS)1/4, is likely to be reduced by afactor of 81/4 = 1.68, that is by 40 percent. If the existing assembly has10 parties represented, only 10/1.68 = 6 are likely to remain—and thesmallest of them are likely to have their shares of seats reduced. They andthe four parties completely excluded will put up a fight. If they cannotblock the reduction in district magnitude, they will look for ways to coun-terbalance it by introducing various complexities in the electoral rules.Think Italy in the 1990s. Once the electoral rules are made complex, theresulting impact on the number of parties and cabinet duration becomesunpredictable.

Changes in electoral rules are never easy and painless, nor should theybe. I am not here in the business of pointing out tricks of how to pullit off. All I can do is to say: If you manage to change the seat product

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MS in a simple electoral system by a certain amount, up or down, thenspecific degrees of change in party constellation and cabinet duration arethe most likely to follow, within a large but specified margin of error.


What determines the value of the duration constant in C = k/N2?

Over time, the mean estimates of duration constant k in C = k/N2 have shiftedfrom 33 to 42 years. Both values are in the ball park of the length of a full politicalcareer. Extended to N = 1, it would suggest that one-party democracies would stillundergo change in partisan composition of cabinet every 40 years or so.

Given that one-party democracy is unusual, Taagepera and Shugart (1989: 101)approach the issue through an ideal two-party system (N = 2). If the tendency to‘throw the rascals out’ is balanced by the increase in resources that incumbencybrings, then the party in power has a 50 percent chance to win the next elections,a 25 percent chance to win the two next elections, and so on. Assume no earlydissolution. Then the probabilities of a cabinet lasting 1, 2, 3, etc. interelectionperiods (p) are 1, 1/2, 1/4, 1/8, etc., which add up to C = 2p. If p is 4 years, thenthe average duration of the cabinet in an ideal two-party system (N = 2) wouldamount to 8 years. Then k = CN2 = 32 years. Larger values of k would suggest thatincumbent resources more than outweigh incumbent unpopularity.

If we take C = k/N2 dead seriously and use individual country values of C andN to calculate the constant k, the results range from 16 years for Mauritius 1976–97 (a methodologically problematic case) and 24 years for Greece 1974–2004, to71 years for the Netherlands 1946–2002 and 72 years for Botswana 1965–2004(with its nearly one-party democracy). These figures represent cabinet durations‘normalized’ for the effect of N, as physicists would put it, or ‘controlled’ for N, assome social scientists put it. This means that the impact of the number of partieshas been removed, so that the variation in k is either random or due to some otherfactors.

Which other factors might affect mean cabinet duration? With the same numberof parties, it may be presumed that polities with a more transparent politicalculture have longer lasting cabinets. Indeed, rough calculations by Allan Sikk andme suggest that k is higher in countries with less corruption and higher self-expression scores on Inglehart’s (1997) scale. Predictive models for this aspectremain to be worked out.

Could the value of k have something to do with Lijphart’s ‘executives-parties’(1999) dimension? The latter distinguishes between consensual and majoritariansystems. With the same number of parties, consensual polities might be expected tohave longer lasting cabinets. However, when graphing k versus executives-partiesscores for 1945–96 (from Lijphart 1999: 312), hardly any trend can be seen. The

174


geometric mean of the 7 most majoritarian cases (scores below −1.0) is k = 39.4years. For the 7 most consensual cases (scores above +1.0), it is k = 46.5 years. Thedifference, though in the expected direction, is not significant.

The catch here is that the number of parties is not the same. Having a PRelectoral system and a large number of parties are major criteria for Lijphart ratinga country consensual in the first place. Most of the consensual steam goes intoraising the number of parties, which shortens cabinet life, rather than into raisingthe duration constant, which lengthens cabinet life. Hence consensus systems,despite a possibly higher duration constant, tend to have shorter lived cabinets,compared to majoritarian.

Which came first, the model or the facts?

Science consists of interaction between mental constructs and data. In which orderdo they tend to come? Do we first construct the logical model of how things shouldbe and then gather data to test it? Or do we first gather data on how things are,graph them and then look for patterns that beg for a logical explanation?

Cabinet duration offers examples of both. The possible connection between thenumber of parties and cabinet duration was so direct that it made sense to graph Cversus N. For broad reasons explained in Beyond Regression (Taagepera 2008), bothwould be graphed on logarithmic scales. Once this was done, the slope was soblatantly close to −2 that one had to suspect an inverse square relationship evenbefore looking for a logical model.

The reverse direction applies to the cube root law. The path that leads fromthe seat product to cabinet duration is so indirect that one hardly would havethe idea to graph C against MS just for the heck of it. And if someone did, theempirical slope would not have led toward an explanation in terms of a cube root,because the data are too scattered. Here the model C = k/(MS)1/3 definitely camefirst, resulting from the combination of C = k/N2 and N = (MS)1/6, and testing withdata followed.

So it can start with either the egg or the hen, depending on circumstances.Actually, it is a repeated interaction. The idea that the number of parties mightaffect cabinet duration was already a directional model, albeit not yet quantitative.And testing C versus MS might lead to discrepancies that make one modify themodel.

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11

How to Simplify Complex ElectoralSystems


� Resistance to simplifying a complex electoral system is least when theexisting number and size of parties are not altered.

� The existing effective number of parties is most likely to be maintained,when the district magnitude used in the new simple electoral system istaken as the sixth power of the effective number of parties, divided byassembly size.

� The task remains risky, especially when one party is very large and theothers very small. The devil is in the details. But the level of details canbe addressed only once effective magnitude lays out the broad picture.

The stated goal of this book is predicting party sizes on the basis ofelectoral systems. For simple electoral systems, this task is now completedfor parties-in-the-assembly. With only two basic characteristics—assemblysize and district magnitude—we can specify the likeliest number andsize distribution of parties in the assembly, and we get some agreementwith actual data. We can even extend this approach to a major outputof assembly politics—stability of government. Most surprising, the twobasic characteristics play a symmetrical role (as long as multi-seat plu-rality is avoided and the differences among the various PR formulas areoverlooked). They fuse into a single characteristic, the seat product, whichalone predicts the average party constellation for a given simple electoralsystem.

Predicting parties-in-the-assembly completes a half of the macro-Duvergerian agenda—the central and lower left parts of the scheme

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in Figure 7.2. The remaining major task is to predict parties-in-the-electorate—the lower right corner in Figure 7.2. This objective alsoinvolves deviation from PR, at the intersection of seats and votes.

Two side issues also remain: Where do electoral systems come from?What about the nonsimple systems? All too many of the actual electoralsystems are complex. What can we predict about their impact on partysystems? This chapter addresses the complex systems. The next one dealswith population, which strongly determines assembly size and may affectparty politics in other ways, too. The determinants of the other compo-nent of the seat product, district magnitude, remain an open question.

The Notion of Output-Based Effective Magnitude

Effective magnitude can be approached from two opposite directions.One can start from actual electoral laws in a complex electoral systemand skillfully try to evaluate their likely effect on electoral outcomes, ascompared to the effect of a given district magnitude in a simple electoralsystem. This path was followed by Taagepera and Shugart (1989) and, inthe format of effective threshold, by Lijphart (1994). Chapter appendixdescribes the evolution of this approach.

Alternatively, one can take at face value some output of the electoralsystem, such as the effective number of parties, and calculate whichsimple electoral system would be expected to produce it, on the average.This is what is done here. The mean relationship between MS and theeffective number of parties was established as N = (MS)1/6. Reversing ityields

Meff =N6

S.

[N6

S≥ 1

]

This approach applies as long as it turns out that Meff ≥ 1. If not, we willhave to assume seat allocation by plurality, as will be explained soon.

Due to the large exponent 6 of N in the effective magnitude formula,even small random variations in the number of parties would be magni-fied. Suppose a small assembly of 60 seats elected by FPTP is observed tohave N = 2.0. It would lead to an estimate of Meff = 1.07, which largelyagrees with the actual M = 1. But a minor and possibly random shift of 10percent, from N = 2.0 to N = 2.2, would yield Meff = 1.89, suggesting PR in2-seat districts. In the opposite direction, N = 1.8 would yield Meff = 0.567,much less than 1. How should this outcome be interpreted?

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How to Simplify Electoral Systems

We no longer can assume a simple electoral system, meaning List PR orFPTP, but have to keep in mind the full form of the seat product, MF S,

which allows for multi-seat plurality. If N6/S turns out less than 1, wehave to interpret it as use of multi-seat plurality, meaning F = −1. NowN6/S is reversed into S/N6:

Meff = S/N6, plurality. [N6/S < 1]

In the example above, this means Meff = 1/0.567 = 1.76. It rounds off to2 and thus suggests plurality in 2-seat districts, if anything. The generalexpression is Meff = (N6/S)1/F . If F is restricted to values +1 and −1, onecould hesitatingly simplify it to

Meff =(

N6

S

)F

,

where using F = −1 implies allocating seats by plurality rule.The example above indicates that a mere 10 percent random deviation

in the effective number of parties will alter the effective magnitude bya factor of almost 2, because 1.16 = 1.77. An electoral system with 1-seatdistricts can easily seem to have 2-seat districts, with either PR or plurality.Also, a PR system with M = 10 can seem to have 5- or 20-seat districts,judging by the value of the effective number of parties. Only largerdeviations might need further explanation.

Output-Based Effective Magnitude for Systems of KnownDistrict Magnitude

Effective magnitude will first be calculated for relatively simple electoralsystems where district magnitude should play the major role in determin-ing the effective number of parties. This way, we can see how closely theformula reproduces district magnitude. The actual systems used for testingN = (MS)1/6 still have complexities such as unequal district magnitudesand second rounds. Therefore, we should expect discrepancies betweenthe calculated and actual district magnitudes. If certain degrees of dis-agreement with the actual district magnitude correspond to certain typesof complexities, then we might have a way to measure the impact of suchcomplexities.

In this light, the general formula Meff = (N6/S)F is now applied tothose 26 out of Lijphart’s 36 stable democracies (1999) in which all seatsare distributed within districts. None use multi-seat plurality. Table 11.1

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Table 11.1. Actual district magnitudes (M) and effective magnitudes derived fromMeff = (N6/S)F , for stable democracies with relatively simple electoral systems in1945–1996.

Country M S MS N Meff = (N6/S)F Comments

Barbados 1 26 26 1.76 1Trinidad 1 36 36 1.82 1Botswana 1 37 37 1.35 6 Plur. One ethnic party hegemonyBahamas 1 42 42 1.68 2 Plur.Jamaica 1 55 55 1.62 3 Plur. No explanationMauritius 1 68 68 2.71 6 Ethnic parties?New Zealand 1 85 85 1.96 1Papua-NG 1 108 108 5.98 423 Extremely local ethnic partiesAustralia 1 128 128 2.22 1Canada 1 270 270 2.37 1USA 1 435 435 2.40 2France 1 508 508 3.43 3 Two-RoundsIndia 1 542 542 4.11 9 Ethnic-regional partiesUK 1 635 635 2.11 7 Plur. No explanationMalta 5.0 59 294 1.99 1 No explanationCosta Rica 7.8 55 426 2.41 4Ireland 3.5 154 538 2.84 3Luxembourg 14.2 57 809 3.36 25Norway 7.7 154 1190 3.35 9Japan 4.0 486 1940 3.71 5Spain 6.7 350 2330 2.76 1 Despite ethnic

parties and uneven MPortugal 11.3 249 2810 3.33 6Finland 14 200 2940 5.03 81 Uneven M and local alliancesSwitzerland 25 197 4920 5.24 104 Panachage and cumulationIsrael 120 120 14,400 4.55 74Netherlands 140 140 20,000 4.65 72

Sources: Lijphart (1999: 76–7) for effective numbers of parties (N), various sources for assembly sizes (S), andactual magnitudes (M).

presents these countries in the order of increasing MS, except for keepingthe M = 1 systems separate. The calculated values of Meff are rounded offto the nearest integer, with apparent use of multi-seat plurality indicated.In 5 cases of 25, the effective district magnitude is less than the actual bymore than a factor of 2. These are shown in bold. The 6 contrary cases,where the effective district magnitude exceeds the actual by more than afactor of 2, are shown in bold italics.

Some single-seat systems have so few parties that one might think theyhave multi-seat plurality, if the effective number of parties were the onlyinformation on hand. This is the case for Jamaica, Botswana, and, moststrongly, for UK. One would think UK had plurality rule in 7-seat districtsrather than FPTP, and no institutional explanation is in sight. Some othersingle-seat systems have so many parties that one might think they have

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PR in multi-seat districts, if the effective number of parties were the onlyinformation on hand. In France, Two-Rounds may give smaller parties anentry point. In India and Mauritius, local ethnic parties might be seen asa factor, but in this case Canada’s marked local variety should also showup as an unusually high effective magnitude—yet it does not.

The truly extraordinary case is Papua New Guinea, where fraction-alization exceeds what would be expected if PR were used in a singlenationwide district! Indeed, its effective magnitude surpasses by far itsassembly size. No strategic coordination whatsoever seems to take placeamong the various tribes.

On the PR side, Malta and Spain look as if they had single-seat districts.It cannot be pinned on Malta’s use of STV, because Ireland fits almostperfectly (Meff = 3.4 vs. actual M = 3.5). In Spain, the low Meff occursdespite a wide variation in district magnitudes that would push preciselyin the opposite direction. Nor can it be due to ethnic parties as such,because these, too, tend to increase the effective number of parties (cf.India). In the opposite direction, Finland and Switzerland look as if thesecountries consisted of 2 to 3 huge districts. Uneven M and local allianceshelp small parties in Finland, and panachage and cumulation may do thesame in Switzerland, but their effect could hardly be that large. Historicalpath dependence might have to be invoked.

Could the effect of possible random variation in the effective number ofparties be mitigated by basing the estimate of effective magnitude on sev-eral outputs, including the largest party’s seat share and cabinet duration?This can readily be done, converting the previous relationships into formsMeff = 1/(s8

1 S) and Meff = (42 years/C)3/S, respectively. Neither offers moreagreement with the actual magnitude than Meff = N6/S does. The geomet-ric mean of the three approaches does no better than Meff = N6/S alone.Many party systems appear strongly affected by other factors besides theseat product, but the nature of these factors is not readily visible.

Output-Based Effective Magnitudes for Complex Systems

We may now proceed to cases where other features override district mag-nitude, so that the latter can be expected to have little impact on partypolitical outputs. We can still try to use Meff = (N6/S)F to estimate theeffective magnitude, always at given assembly size (cf. chapter appendix),but with huge doubts. Table 11.2 shows the remaining 10 cases amongLijphart’s 36 stable democracies (1999). Countries are shown in the order

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Table 11.2. Effective magnitudes for complex electoral systems, with output Meffcalculated from Meff = (N6/S)F

Country District M S N Output Meff Input Meff Comments

Greece 1974 on 4 300 2.20 3 Plur. 3 ‘Reinforced PR’Austria 1945 on 7/20 175 2.48 1 ∼3.5/20 Two-party traditionGermany 1949 on 1 526 2.93 2 10Sweden 1948 on 8/11 298 3.33 5 9/12Colombia 1958 on — 191 3.32 7 —Venezuela 1959 on — 199 3.38 7 4Italy 1946 on 20 613 4.91 23 ∼20Belgium 1946 on 7 205 4.32 29 12Denmark 1945 on 6/7 170 4.52 29 25Iceland 1946 on 1.5/6 58 3.72 64 60

Sources: Lijphart (1994: 31–44) for the lowest level district magnitudes, Lijphart (1999: 76–7) for effective numbersof parties (N), various sources for assembly sizes (S), and Taagepera and Shugart (1989: 136–9) for estimates ofinput-based Meff.

of increasing effective magnitude. For comparison, the table also showsthe input-based effective magnitudes given for roughly the same periodsin Taagepera and Shugart (1989: 136–9). They broadly agree with theeffective thresholds in Lijphart (1994). When the output- and input-basedeffective magnitudes agree, institutional explanation seems sufficient. Thecases where they disagree are shown in bold in Table 11.2. Here historical–cultural factors may have to be considered.

Greece has had various forms of ‘reinforced PR’, meaning low-magnitude districts plus legal thresholds that at times have reached 30percent for 3-party alliances. The total effect seems stronger than wasdeemed possible by Taagepera and Shugart (1989): it looks close to plu-rality in 3-seat districts rather than PR. The Greek electoral system actslike reinforced plurality rather than ‘reinforced PR’.

Austria has a long tradition of one major sociopolitical cleavage thatsupports a two-party system even while the electoral systems might haveallowed more parties to rise in 1945–70 and definitely made it easy from1971 on. Despite district level M = 7 and M = 20 for the two periods (plusmildly restrictive second tiers), Austria’s number of parties would makeone think that it has single-seat districts.

Germany has effectively nationwide PR, restricted by a 5 percent legalthreshold. Taagepera and Shugart (1989) estimated this restriction to becomparable to having seat allocation in districts of M = 10. The conver-sion formula M = 75%/T − 1 (cf. chapter appendix) suggests M = 75/5 −1 = 14. Actually, Germany looks as if it had PR in 2-seat districts. Incontrast to Austria, a prior two-party tradition cannot be invoked. The

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Erststimme (‘primary vote’) on the ballot being cast for single-seat districtsmight exert a strong psychological effect (as it was intended), despite thecompensatory Zweitstimme (‘secondary vote’) that restores PR.

The input- and output-based effective magnitudes agree for Italy,Denmark, and Iceland. Sweden has fewer parties than the electoral sys-tem would enable it to have, while Venezuela has mildly more. Belgiumlooks like a country with very high magnitude. Here, ethnic split hasproduced more parties than one could expect on the basis of institutionalinputs.

Conclusion and Implications for Institutional Engineering

The seat product MS allows us to calculate the expected party politicaloutput of a simple electoral system. For complex electoral systems, wheredistrict magnitude clearly does not tell everything, one can reverse direc-tion. We can calculate an effective magnitude from known assembly sizeand the actual output in the form of effective number of legislative parties.This would be the district magnitude which, for given assembly size, wouldbe expected to lead to the same effective number of parties, if simpleelectoral rules were used. In some cases, effective magnitude correspondsto multi-seat plurality.

This output-based effective magnitude can be compared with actualdistrict magnitudes—or with input-based effective magnitudes derivedfrom judicious evaluation of the impact of various features in a com-plex electoral system. Such comparisons help us gain confidence in theestimates based on inputs when there is agreement. Disagreements makeus ask questions about causes that go beyond electoral systems—ethnicsaliency and historical path dependence, for instance.

What does output-based effective magnitude mean for institutionalengineering? Suppose it is felt in a country that its existing electoralsystem is overly complex and could stand simplification. Even when mostparties in the assembly share this feeling, they would be leery of intro-ducing changes for fear that their own party may suffer. So it becomesa question of how to simplify the rules without altering the numberand size distribution of parties. This is precisely what the output-basedeffective magnitude is about. Starting from the actual effective number ofparties, the effective magnitude indicates to which simple electoral systemthe given complex system is most like. Thus, effective magnitude is the

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measure of the present system that helps to find the district magnitude tobe used in the simpler system.

It is not that simple, of course. It works best when the present partysystem is well balanced. In the presence of one very large and numerousvery small parties, the effective number of parties does not tell the wholestory. In such a case, one would also want to calculate the effectivemagnitude based on the largest seat share, Meff = 1/(s8

1 S), and try to bal-ance this number with the one obtained from Meff = N6/S. A satisfactorycompromise may or may not be available. The devil is in the details. Butthe level of details can be addressed only once the broad picture has beenadequately laid out.


The quest for input-based effective magnitude

Effective magnitude is a notion that has caused considerable confusion in electoralstudies. Given that I was the one who coined the term, it is up to me to clarify it—and now it can be done. Since the publication of Taagepera and Shugart (1989),there have been times when I exclaimed to myself that the entire concept was self-contradictory, and I began to avoid it. At the same time, I kept receiving inquiriesabout how to determine it for a given complex electoral system. Notions and pro-cedures have a life of their own. They sometimes continue to be used even whiletheir originators have given up on them. The brief history of effective magnitudeis presented here in the hope of reducing the incidence of faulty applications inthe future. The main message is the following:

� Effective magnitude makes sense only in the context of a given assembly size.� For complex electoral systems, effective magnitude is the value to plug into

the seat product MS, in lieu of district magnitude, so as to obtain the observedparty constellation.

The notion of an effective magnitude was introduced by Taagepera and Shugart:‘The concept of magnitude will have to be broadened to include not only thedistrict magnitude but also a nationwide [emphasis added] “effective magnitude” ’(Taagepera and Shugart 1989: 126). We did not call it ‘effective district magnitude’,because we felt it addressed precisely the various complex features of electoralsystems, most of which operate outside the existing districts.

Nonetheless, our chapter title ‘Magnitude: The Decisive Factor’ (Taagepera andShugart 1989: 112) engendered a small cottage industry, claiming that districtmagnitude is not always the decisive factor, because there is the seat allocationformula (plurality!) and all those factors that go beyond the district. But read the

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title again: the word ‘district’ is not there. The review table (Taagepera and Shugart1989: 138–9) explicitly contrasts the two magnitudes, ‘District’ and ‘Effective’.Effective magnitude was meant to transcend the existing districts.

We cannot complain, because we were confused ourselves. We looked for somesort of a magnitude that went beyond the existing districts and would produceroughly the same effect on outputs, such as the number of parties and deviationfrom PR, as the existing complex electoral system. We could not quite pin it down.The crucial shortcoming was ignoring assembly size, even while we pointed outelsewhere (Taagepera and Shugart 1989: 174–5) that it affected the number ofparties. We imagined districts where the number of seats would equal the effectivemagnitude, but left open how many such districts there would be—even while theanswer is self-evident in retrospect: assembly size divided by effective magnitude.For such a fuzzy equivalent system, we tried to develop some guidelines on howto evaluate the impact of legal thresholds, adjustment seats, and multistage seatallocation (Taagepera and Shugart 1989: 135–40, 206–69).

Lijphart (1994) added valuable insights by shifting the focus from effectivemagnitude to effective threshold. A degree of equivalence between district magni-tudes and legal thresholds had been pointed out by Taagepera and Shugart (1989:117, 276–7). Lijphart (1994: 12) stated explicitly that reducing a legal thresholdor increasing the district magnitude ‘can be seen as the two sides of the samecoin’. (The statement obviously implies that all seats are allocated within districts.)Rather than expressing legal thresholds and various complexities in terms of asomewhat equivalent effective magnitude, he expressed district magnitude and allthe rest in terms of a somewhat equivalent effective threshold.

The coarse conversion formula between the ‘two sides of the coin’ shifted fromT = 50%/M (Taagepera and Shugart 1989: 117) to T = 75%/(M + 1). The latterformula was first reported in Lijphart (1994: 183) as private communication byTaagepera, and it was explicated in Taagepera (1998b)—see Chapter 15. The mainflaw was that nationwide and district level legal thresholds were confused andmistakenly treated as equivalent.

The incongruence becomes blatant when the notion of effective threshold isapplied to single-seat districts. Lijphart (1994) estimated it as T = 35%, and the for-mula T = 75%/(M + 1) yields T = 37.5% for M = 1. While such a threshold reflectsreasonably a party’s chances to win in a district (graph in Taagepera 1998b: 399),imagining its equivalence to a nationwide legal threshold of 37.5 percent wouldbe preposterous. In New Zealand 1928, such a nationwide legal threshold wouldhave disqualified even the nationwide winner from obtaining seats!

While less visible, the same incongruence affects the ‘other side of the coin’,effective magnitude. According to T = 75%/(M + 1), a nationwide legal thresholdas high as 37.5 percent would appear to be no more restrictive on parties than hav-ing single-seat districts. This is the point where I began to doubt of the existenceof this phantom, the effective magnitude. In retrospect, the central position of theproduct MS, highlighted already in Taagepera and Shugart (1993), should have led

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to the answer that finally dawned on me while writing the present book. For theeffect of a simple electoral system on the number and seat share distribution ofparties, it is the combination MF S that matters, in the first approximation. For acomplex electoral system with given assembly size and seat allocation formula, onecan look for an effective magnitude that would have roughly the same effect onthe party system as would an actual district magnitude in a simple system.

In retrospect, the input-based estimates of effective magnitude in Taageperaand Shugart (1989: 136–9) coincide reasonably well with output-based ones (cf.Table 11.2), despite ignoring the assembly size factor. How come? Most nationalassemblies have 50–500 members, which is a fairly limited range. The estimateswere instinctively fitted to about 200 members. When one goes to very small orvery large assemblies, more marked differences could arise.

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� Take the cube of the number of seats in the first or only chamber of yournational assembly, and you roughly get your country’s population.

� Unconsciously, assembly sizes are chosen to fit the cube root of popu-lation, because this size minimizes the workload of a representative.

� The population of a country puts broad limits on the size of its nationalassembly and thus limits the options for institutional engineering.

� Smaller countries have fewer registered parties but more party membersper 1,000 population. They may have slightly more durable cabinets.

Is politics different in small and large countries, all other factors being thesame? In their seminal Size and Democracy, Robert Dahl and Edward Tufte(1973) convincingly showed that systematic differences are to be expectedand that they do show up empirically. Some of these differences involvepolitical institutions, such as the size of representative assemblies, whileothers, such as trade/GNP ratio or military capabilities, affect politicsindirectly. This chapter reviews the work done since 1970 on the impactof population on assembly size, cabinet duration, the number of partiesand their memberships.

Overview

A causal link between country population (P ) and assembly size (S) wasestablished by Taagepera (1972). It starts with the number of communi-cation channels—the same basic notion used for the inverse square lawof cabinet duration (Chapter 10). It leads to a cube root law of assembly

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sizes: S = P 1/3, which applies within a factor of 2 to most countrieswith sufficient literacy. This equation predicts assemblies (or first cham-bers) of 100 seats for countries of 1 million people, and 1,000 seats for1,000 million. The model was streamlined and re-tested by Taageperaand Shugart (1989: 173–83). Using analogy with absorption law inphysics, a logical model for trade/GNP ratio was also developed (Taagepera1976; Taagepera and Hayes 1977), but it does not concern us heredirectly.

After the 1970s, interest waned in population (or area) as a factor indomestic politics. Dag and Carsten Anckar (1995) raised again the issue ofsize and democracy, adding insularity as a special factor. Taagepera (1972,1976) had noted that small island countries tend to have assembly sizesbelow the cube root of population, as well as trade/GNP ratios lower thanthose of continental countries of similar population.

Dag Anckar (1997) and Carsten Anckar (1997a, 1998, 2000) focused onthree variables relevant to party politics: the number of parties registered,the vote share of the largest party, and the effective number of electoralparties. The input consisted of population and area (which are highlycorrelated) of 77 states. It was found that an increase in either measureof size tends to go with increases in the number of parties registeredand the effective number of electoral parties, while the vote share of thelargest party tended to decrease correspondingly. These empirical findingscontinue to hold when elections with FPTP and PR rules are consideredseparately.

Steven Weldon (2006) added a complementary empirical observation:The total membership of parties also tends to increase with the electorate(which is very highly correlated with population). However, the per capitaparty membership decreases with increasing electorate, and so may thedegree of activity of these members.

Meanwhile, a different line of research began to stress the significanceof assembly size, without explicitly connecting it to population. Grofmanand Handley (1989) noted differences in the composition of Houses andSenates of US states that could be attributed only to differences in thenumber of members (cf. Chapter 6). Lijphart (1994) found some impactof assembly size on disproportionality and multipartism. Taagepera andShugart (1993) built a multilayered logical model centered on what Inow call the seat product MS, and assembly size emerged as one of thetwo key variables. The indirect connection to population, through theaforementioned cube root law, was noted in Taagepera (2001), but nodirect test was carried out.

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Subsequent sections describe the cube law of assembly sizes and itsimplications for party sizes, as well as the work by Anckar (1998, 2000)and Weldon (2006) on the number and membership of parties. I focuson population (rather than area or electorate) as the central measure ofcountry size. It is not always clear how population influences variouspolitical quantities. It may affect the number of registered parties andtotal party membership directly, or through assembly size, which mayput restraints on party politics.

The Cube Root Law of Assembly Sizes

When the memberships of the first or only chambers of assemblies ofstable democracies are graphed against population, both on logarithmicscales, the empirical slope is visibly around 1/3—see graph in Taageperaand Shugart (1989: 175). The best fit line passes near the point P = 1,S = 1. This is an important conceptual anchor point. A population of1 person would be expected to represent itself. Thus it seems that assemblysize is close to the cube root of population represented:

S = P 1/3.

Taagepera and Shugart (1989: 175) also show that, from 1790 to 1913, theUS House played a continual catch-up game with the cube root law, aspopulation expanded during the intercensal periods. In 1913 the Housesize was frozen, and it is by now only two-thirds of the cube root ofpopulation.

Graphing all national assemblies of the world (Taagepera and Shugart1989: 176) confirms that the cube law pattern applies to one-party orno-party regimes as well as for those with at least some multiparty ele-ments. Yet there is a tendency for assemblies to be smaller than S = P 1/3

when literacy is low and populations are less than 1 million. It may beargued that only literate people can meaningfully participate. Taagepera(1972) defined ‘active population’ as the literate working age population:P ′ = P LW, where P is the total population, L is literacy rate, and W is theworking age fraction of the population. Omission of literate populationpast retirement age is unjustified conceptually but was imposed by avail-ability of data. It leads to moderate underestimation of active population.A logical quantitative model, presented in chapter appendix, predicts thatassembly size would be the cube root of double the active population:

S = (2P ′)1/3.

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When all national assemblies of the world are graphed against P ′ = P LW(Taagepera and Shugart 1989: 178), the fit is good for active populationsabove 0.2 million. For very small populations, assembly sizes still tendto fall short of the model. Notably, island nations tend to have smallerassemblies than expected.

In all countries with extensive literacy, total population tends to beclose to double the working age population, as countries with manychildren tend to have few old people, and vice versa. Therefore, whenliteracy is above 90 percent, little is lost by using S = P 1/3 rather than themore complex expression. When literacy is 75 percent, it is expected toreduce assembly size by 1 − 0.751/3 = 10%.

The logical model applies only to democratic countries, so why wouldnondemocratic regimes pick similar sizes for their puppet assemblies? Itmay be imitation. Communist regimes tended to have twice the usualassembly size for given population, presumably so as to look especiallydemocratic. Most postcommunist democracies have cut down on assem-bly sizes. China continues with a bloated assembly of 3,000, while thecube root of its population of 1,300,000,000 is still only 1,100. Even if agenuine world parliament were formed, the present world population of6 billion would project to only 1,800 seats.

As of now, assembly sizes still fit S = P 1/3 when population exceeds1 million and its growth rate is low. Countries with rapidly growingpopulations seem to play a perennial catch-up game, gradually enlarg-ing their assemblies. Overall, countries with single-seat districts tendto have relatively small assemblies, British counterexample notwith-standing. The two effects, population growth and single-seat districts,combine in many small island countries. By now, we have apprecia-ble time series for growth in population and assembly size. In view ofthe importance of assembly size in affecting the number of parties, athorough reanalysis would be desirable, updating Taagepera and Shugart(1989).

Does the cube root law apply to subnational assemblies? The way themodel is constructed, it should. But empirically, such assemblies tendto fall short. I could find data for all city councils of the capital citiesin the European Union (EU), except Ljubljana and La Valetta. The geo-metric mean for the 23 cities is S = 0.82P 1/3, which is 18 percent belowexpectation. The Houses of the US states also fall short of S = P 1/3. Asfor the US cities, they often elect only 5 to 10 commissioners. Here thephilosophy seems to be elect a city government rather than a deliberativeassembly.

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Table 12.1. Predicted largest seat shares and effective numbers ofparties, at selected populations and district magnitudes

Population s1(%) for N for

M = 1 M = 10 M = 100 M = 1 M = 10 M = 100

10,000 68 51 (38) 1.67 2.45 (3.59)1 million 56 42 32 2.15 3.16 4.64100 million 46 35 26 2.78 4.08 5.99

Do Smaller Countries have Fewer Legislative Parties?

When S is replaced by P 1/3, in line with the cube root law, in previousmodels s1 = (MS)−1/8 for the largest seat share and N = (MS)1/6 for theeffective number of parties, we obtain

s1 = M−1/8 P−1/24

and

N = M1/6 P 1/18.

Table 12.1 shows the corresponding average predictions. No systematictesting has been carried out. For a population of 10,000, the cube root lawindicates a 22-seat assembly, and all actual assemblies fall much short of40 seats; hence the entry for M = 100 in Table 12.1 is unreal.

Have Smaller Countries More Durable Cabinets?

If mean cabinet duration tends to be C = 42 years/(MS)1/3 (cf. Chapter 12)and S = P 1/3, then we would expect

C =42 yearsM1/3 P 1/9

.

This implies that smaller countries would have more durable cabinets,on the average. When single-seat districts are used, we would expect amean duration of 11.7 years in a country of 100,000 and only 5.4 yearsin a country of 100 million—a twofold decrease. However, populationenters as its ninth root, which could make its impact hardly noticeable,compared to that of district magnitude. A country of 100 million using100-seat districts would have an expected cabinet duration of 1.2 years,meaning a much larger 4.5-fold decrease due to M.

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I tested the 14 single-seat systems for which Lijphart (1999) givescabinet durations. The geometric mean cabinet duration is 7.6 years forthe 8 smaller countries (population 0.25 to 4 million in 1993, geometricmean 1.18 million). It is less indeed—5.8 years—for the 6 larger countries(population 17 to 870 million, geometric mean 82 million). The ratio ofdurations for the two groups would be expected to be the ninth root oftheir population ratio, which is 1.6. The actual ratio of durations is only1.3.

While the mean trend goes in the expected direction, it may well beaccidental, given that the scatter of data exceeds by far both means. Itwould take many more data to test whether smaller countries really tendto have more durable cabinets. Such data will become available by 2010,as the Third Wave democracies reach the minimum age of 20 years, thecriterion of stable democracy proposed by Lijphart (1999). We shall see.

How Population could Affect the Number of Parties Registeredand their Memberships

The general format of equations in previous sections is y = kMa P b, wherek, a, and b are constants. Given that Anckar (1998) and Weldon (2006)deal with quantities that can take only positive values, considerationspresented in Beyond Regression (Taagepera 2008) suggest that their datamight be reanalyzed in the same framework, where y stands, succes-sively, for the number of parties registered, their combined member-ship, etc. When logarithms are taken, these equations become linear:log y = log k + a logP + b log M. As in Chapters 8–11, linear regressionsshould be tried with all variables logged, rather than using the variablethemselves or logging only some of them (such as population).

What is the rationale for expecting that, along with population, districtmagnitude may play a role in quantities such as the number of partiesregistered and their memberships? Ever since Duverger’s law was spelledout, it has been clear that seat allocation by FPTP cuts down minor partyrepresentation. When unsuccessful parties go out of business, the numberof parties registered may be expected to decrease. With fewer parties, theircombined membership may also decrease. With PR, in contrast, the largerthe district magnitude, the more parties can win at least a few seats,possibly inducing more parties to register and recruit members.

Even if it should be confirmed that population and district magnitudedo impact the number and membership of parties, it remains to be

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verified whether the changes are proportional to population and magni-tude with some exponent so that the format y = kMa P b applies. And evenif this should be the case, the question would remain: Why do the con-stants have the values they empirically have? Our understanding remainsincomplete until we can answer the ‘why’. This is ongoing research, andonly tentative results can be reported in the next few sections.

Do Small PR Countries have Higher Party Memberships,Per Capita?

Party membership means here the combined membership of all partiesin the country. One might expect that relatively more people join partiesin smaller countries, where party politics is less anonymous and closer tohome. Thus, smaller countries may have higher per capita party member-ships. However, small countries also have smaller assemblies, which limitthe number of successful parties and hence may reduce per capita partymembership. So which way is it? There are two ways to find out. One isto construct more detailed quantitative models so as to see by how muchcoziness could increase political participation and by how much smallassemblies could impede it. These models have not been completed. Theother way is empirical.

A study by Steven Weldon (2006) confirms that relatively more peoplejoin parties in smaller countries. Weldon compares the party membershipto the size of the electorate in 27 countries in the late 1990s. Electorate(P ′′) amounts to approximately 3/4 of the total population. Member-ship density (d, the ratio of membership to electorate) in long-standingdemocracies ranges from 1.6 percent in France and 1.9 percent in UK to23.8 percent in Malta and 27.3 percent in Iceland. Linear regression ofthe logarithms of d and P ′′ corresponds to d = 1,550(P ′′)−0.37 (R2 = 0.51).Austria is a major outlier, with d = 17.7%, despite its medium population.

I prefer to think in terms of total party membership (m) rather thanper capita. Keep in mind that d is in percentage, and assume that elec-torate is roughly three-quarters of the population (P ′′ = 0.75P ). ThenWeldon’s regression line (2006) would correspond to m = 0.75P (d/100) =12(P ′′)+0.63. This would imply that total party membership grows some-what faster than the square root of electorate. The joker in the equationis the constant 12. At P ′′ = 1, one could visualize at most one party witha membership of 1. Yet, when the equation above is carried to this limit,it would imply that an electorate of 1 person would still include 12 party

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members! If the anchor point at P ′′ = 1 is to be respected, the empiricalequation would need some modification, unless some other factor isintroduced, such as district magnitude.

At the same population, more parties are available in PR systems. Henceone might expect that PR systems incite more people to join parties.Weldon (2006) does not consider the possible effect of electoral system,but his data and graph suggest that higher district magnitudes go withhigher per capita party memberships. Shifting from electorate to totalpopulation, preliminary reanalysis of his data suggests that total partymembership (m) might be related to district magnitude (M) and population(P ), approximately as

m = M3/8 P 3/4.

This equation says that total party membership grows slower than pop-ulation, and that larger district magnitudes boost party membership(presumably by boosting the number of parties). This equation satisfiesthe anchor point: P = 1, M = 1 → m = 1. However, I have no theoreticaljustification as yet for these particular exponents. It must have somethingto do with the number of parties. The corresponding per capita partymembership would be

mP

=M3/8

P 1/4.

As population increases, per capita party membership would decreaseas the fourth root of population, meaning fairly slowly. Table 12.2shows the estimates that result from these extremely approximate empir-ical fits. Again, the entry for M = 100 is unreal for a population of10,000.

Weldon (2006) also finds that party member activism seems to decreaseas party membership increases. Larger memberships may reinforce a feel-ing of anonymity and passivity among the rank and file. This finding

Table 12.2. Total and per capita party memberships at selected popula-tions and district magnitudes—empirical approximations

Population Total membership (thou.) Per capita membership (%)

M = 1 M = 10 M = 100 M = 1 M = 10 M = 100

10,000 1.0 2.4 (5.6) 10.0 23.7 (56.2)1 million 31 75 178 3.2 7.5 17.8100 million 1,000 2,371 5,623 1.0 2.4 5.6

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depends crucially on assuming that the subunits are the relevant popula-tion units in federal countries, and the evidence rests on Germany alone.More research remains to be done on the effect of population on partymembership. Recent decrease in party memberships in most democraciesadds another factor that makes the study of population effects moredifficult (Tan 1989, 1997; Dalton and Wattenberg 2000).

Do Smaller Countries have a Lower Number ofParties Registered?

The number of parties registered (r ) might be expected to depend fore-most on ease of registration, which could hardly be expected to be size-related. In some countries, party registration is a matter of sending ina form and paying a symbolic fee, while some other countries set vari-ous further conditions, such as a large founding membership. Still, withsimilar registration requirements, smaller countries would have fewerpoliticians interested in going through the process. Indeed, Anckar (1998,2000) does find a correlation between r and the logarithm of population.

Since both r and P can take only positive values, the format r = aP b

should be preferred to the format r = a + b logP used by Anckar (cf.Taagepera 2008). At P = 1, the number of parties registered could not beabove 1, but it could be much lower. Hence a 1 could be expected.Reanalyzing Anckar’s data along these lines, the best fit of logarithmsshows that the number of parties registered is approximately

r = 0.04 P 3/8.

This equation implies 1.3 registered parties for a country of 10,000 people,7 parties for 1 million, and 40 parties for a country of 100 million. Mostcountries agree within a factor of 2. Countries with a lower number ofparties registered are Mexico, Turkey, South Korea, Japan, and Jamaica. Incontrast to the number of seat-winning and effective parties, the numberof parties registered shows no correlation with district magnitude.

The exponent 3/8 is sufficiently close to 1/3 to recall that assemblysize is S = P 1/3. Hence, one might wonder whether the number of reg-istered parties might be proportional to assembly size. However, whenthe number of registered parties is graphed against assembly size, bothon logarithmic scales, scatter widens. The best symmetric fit is aroundr = 1.6S0.9. The proportionality line r = 0.1S expresses the trend almost aswell, except at very small assemblies of less than 30 seats. The widening

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scatter suggests that the number of parties registered does not have acausal link to assembly size. Both seem to be affected by population(among other factors), separately: r ← P → S rather than P → S → r . Forassembly, minimization of communication channels suggests the expo-nent 1/3. For the number of parties registered, I have no logical model, asyet.

Another puzzling regularity is observed empirically: Total party mem-bership, as measured by Weldon (2006), tends to be proportional to thesquare root of the product of population and the number of partiesregistered, as measured by Anckar (1998):

m = 27(r P )1/2.

It fits within a factor of 2 for most M = 1 and M > 1 systems. I haveno logical explanation, and it does not connect easily with the regular-ities previously observed. More research remains to be done in all thoseaspects.

The Number of Electoral Parties

The number of seat-winning parties, the largest seat share, and the effec-tive number of legislative parties were all found to be connected tothe seat product MS, with S, in turn, connected to population. For theelectoral parties, the number of registered parties has been empiricallyconnected to population. Assuming that parties are registered in orderto run, r represents the number of vote-getting parties. Anckar (1998,2000) has empirically connected population to the largest vote share andthe effective number of electoral parties. A logical connection of thesequantities to institutions is established in Chapter 14, but the extensionto population remains to be worked out.

Third Parties in FPTP Systems

For FPTP systems, the largest seat share is s1 = (1/S)1/8 according to themodel developed in Chapter 8. Thus, its preponderance decreases withincreasing assembly size, and the same is bound to be the case for thesecond-largest share, as confirmed by inspection of Table 9.1. Hence therelative share of third parties should increase with increasing assemblysize.

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Size and Politics

This is found to be so indeed, when the combined seat shares of thirdparties (t = 1 − s1 − s2), as tabulated in Gerring (2005), are graphed againstassembly sizes. For S < 50, the median t is around 4 percent, while itis around 7 percent for 50 < S < 200 and 18 percent for S > 200. Thetheoretical pattern becomes too complex to be calculated. The empiri-cal median fit is close to t = 0.006(S − 2)0.59, where the subtraction of 2reminds us that an assembly of 2 seats could not possibly fit in a thirdparty. Marked deviations are UK (4.3 percent) and USA (0.1 percent)on the low side and Papua New Guinea (54 percent), Solomon Islands(36 percent), and St Kitts (22 percent) on the high side.

To the extent the cube root law holds, the relationship to populationwould be around t = 0.006P 0.2. Gerring (2005) observes a strong correla-tion of third party share with federalism. It is partly a size effect, as federalcountries tend to be on the large side, but partly seems real, as Papua NewGuinea, Solomon Islands, and St Kitts stand out among smaller countriesby being at least mildly federal.

Population and Institutional Engineering

Many other factors besides its size should affect the functioning of arepresentative assembly, and unusual size does not always look openlydysfunctional. Tiny New Hampshire does not seem to suffer from itsunusually large lower chamber of some 400 members. To fit in with thecube root law, the British House of Commons would have to shrink by40 percent, while the US House would have to expand by 40 percent. Yetneither body shows ill effects that could be traced back to their size. Evenif optimizing the channels of communications mattered, the bottom ofthe curve of channels versus assembly size is so flat that a large range ofsizes should be almost equally acceptable.

And yet assemblies seem to be sensitive to the cube root of population.The USA started out in 1790 with a House much smaller than a halfof the cube root norm, but within 40 years it brought the House up tothe cube root of its ever-expanding population—and then stayed close tothe cube root until 1913. Worldwide, as the population increased in mostcountries from 1970 to 1985, assembly sizes went up by more than 10 per-cent in 57 countries, stayed about the same in 44, and went down by morethan 10 per cent in only 4 countries (Taagepera and Shugart 1989: 179).This trend seems to continue (except for reductions in postcommunist

197


countries). It may be felt that assembly sizes could be all over the place,but somehow they hew close to the cube root of population.

What this means for institutional engineering is that countries do havesome leeway in choosing the size of their national assembly, but thereare limits. Most actual assembly sizes are within a factor of 2 of the cuberoot of the populations represented. For a country of 1 million, this zoneextends from 50 to 200 seats. For 64 million, it is 200 to 800, and for343 million, 350 to 1,400 million. Still, most assemblies are closer to thecenters of these zones—as if it mattered.

It was observed earlier that the previous models for the largest seatshare, the effective number of parties and mean cabinet duration can bereformulated in terms of population and district magnitude. I have super-ficially tested only the expected relationship to cabinet duration, C = 42years/(M1/3 P 1/9). Populations are a given that institutional engineers findhard to alter. Thus its connection to legislative parties seems to be of onlyacademic interest, for the time being, but it may change.

The same applies to the population dependence of electoral parties.Here, all findings are recent and empirical. Logical models need to beconstructed and tested. Once this is done, something useful for practicalpurposes may emerge.


The cube root law of assembly sizes: The model

The main purpose of legislative assemblies may be to pass laws, but what theyphysically do most of the time is talking and listening. The original French termparlement comes from parler—to speak. Interacting with constituents and col-leagues, assembly members always face an overload of communication. Minimiz-ing this load is a significant factor for their efficiency. By trial and error, assemblysizes tend to adjust toward maximum efficiency. Consider two extreme cases.

If the assembly is very small, the interaction load within the assembly is low, butthe number of constituents per representative is large. On the other hand, if theassembly is very large, the constituent load decreases, but assembly interactionsgrow even faster. Some intermediary size may be optimal. Let us put it in termsof the number of communication channels—the approach used previously whenmodeling mean cabinet duration (Chapter 10).

Each active member of the population must have a two-way channel (talkingand listening) to a representative, if representative democracy is to have meaning.Channels among constituents also exist, but they do not put a load on the

198

Size and Politics

representative. The active population outside the assembly is (P ′ − S). Thenumber of constituent channels toward each of the S representatives is(P ′ − S)/S = P ′/S − 1. But since they both send and receive information, the totalnumber of constituency channels per representative is

cC =2P ′

S− 2.

Within the assembly, each member communicates with each of the other S − 1members both as speaker and as listener, meaning 2(S − 1) channels per representa-tive. But this is not all. Whenever two members are talking, the other S − 2 mem-bers are interested in what they are talking about and, figuratively, try to listen in.How many channels does a given representative try to monitor? Each of the other(S − 1) representatives is at one end of a channel to each of the (S − 2) remainingrepresentatives, meaning (S − 1)(S − 2) ends of channels or (S − 1)(S − 2)/2 chan-nels. In sum, the number of assembly channels per representative is

cA = 2(S − 1) +(S − 1)(S − 2)

2=

S2

2+

S2

− 1.

Thus the total number of channels making demands on an average assemblymember is

c = cC + cA =2P ′

S+

S2

2+

S2

− 3.

As a check, note that when the population is reduced to 1 (P ′ = S = 1), the numberof channels becomes 0, as it should.

All channels are of course not equally demanding, but as a first approximationwe will assume that they are. Then the load on the assembly member is minimizedwhen c is minimal. Apply differential calculus. The number of communicationchannels is minimized when the differential of c with respect to S is made 0:

dcdS

=d

dS

[2P ′

S+

S2

2+

S2

− 3]

= S − 2P ′

S2+

12

= 0.

This equation can be transformed into

S3 +S2

2− 2P ′ = 0.

For populations of more than 1,000, the central term adds less than 0.5 percent tothe first one and can be neglected. Hence the optimal assembly size is

S = (2P′)1/3. [P

′> 1,000]

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The approximation P = 2P′

holds within 10 percent when literacy is above75 percent. Then, simply,

S = (P )1/3.

Taagepera and Shugart (1989: 181) show graphically how the number of con-stituent channels decreases with increasing assembly size, while the number ofassembly channels increases. The bottom of the curve at medium assembly sizes israther flat, so that deviations from the optimal S by a factor of 2 do not increase theload per representative appreciably. This is the zone where most actual assembliesare observed to lie. Some of the implicit assumptions that enter the model arelisted in Taagepera and Shugart (1989: 181). With these reservations, the cube rootexpression has a theoretical foundation as well as empirical confirmation. Henceit qualifies as a law in the scientific sense.

The various approximations made do not apply when the population repre-sented is extremely small. Disturbingly for the validity of the model, however,actual assemblies tend to fall noticeably below the expectation even for popula-tions as large as 200,000. I have not found a logical reason. As noted, the actualsizes of the assemblies of US states and of the capital cities of the EU also fall belowexpectations. The mean shortfall for the latter is only 18 percent, but it makesone wonder whether the model needs adjustment in the case of non-sovereignassemblies.

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13

The Law of Minority Attrition


� The more important levels have fewer positions—and the share ofminorities goes down. A party with a small share of votes gets an evensmaller share of seats. If women are few in city councils, they are evenfewer in the legislative assembly. The law of minority attrition expressesit quantitatively.

� In first-past-the-post systems, we can calculate the seat shares of allparties from their vote shares. A 1 percentage point increase in votesfor a major party produces an increase of about 3 percentage points inseats, but it depends on the number of voters and seats.

� This ‘responsiveness’ can range from 2 percentage points when theassembly has many seats to 4 when it has few seats. If you feel respon-siveness is too steep or too mild, it could be altered by changingassembly size.

� For a guess at the responsiveness, divide the zeroes in the number ofvoters by the zeroes in the number of seats. For example, with 1,000,000voters and 100 seats, responsiveness ratio is 6/2 = 3.0. For more accu-racy, use logarithms. Country-specific cultural and geographic factorscan alter responsiveness.

� The law of minority attrition can be adjusted for multi-seat plurality,such as the US Electoral College, where responsiveness is even higherthan in FPTP. It also applies to PR systems, but the responsiveness is soclose to 1 (perfect PR) that it may not be of practical interest.

� The law of minority attrition might help determine which part of the‘rubber ceiling’ on women’s advancement is natural and which part issocially imposed.

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We now move on to investigate the impact of electoral systems on thedistribution of votes. As indicated in Figure 7.1, the electoral system has amassive, direct, and largely mechanical effect on the distribution of seats.When it comes to votes, it becomes a diffuse and indirect psychologicaleffect (plus other strategic and logistic effects).

Voters are in principle free to vote for any party, regardless of insti-tutional constraints. However, they might not be free to vote for theseventh-largest party when the FPTP system has driven this party outof existence. And they may be free to vote for the third-largest partyonly while being aware that their vote may not count in a positiveway—and may count negatively by reducing the vote for their secondpreference, who otherwise could have a chance to win. Conversely, withlarge-magnitude PR, one may feel one’s vote counts, but one’s preferredparty may join a coalition cabinet that may soft-pedal the program itemsthat made one vote for that party in the first place (Strøm 1990).

So, electoral systems affect seats immediately and votes in the longterm. However, as one looks at a single election, votes come first, andseats are allocated later. Therefore, it made sense in early electoral studiesto take the votes as given and try to explain the seats in terms of votes.In contrast, this book focused first on the seat share distribution, due torealization that the institutional impact on the average of many electionsoperates in the opposite direction, starting with institutions. This impacton seat distribution has been addressed with some success. It is now timeto ask: What can average seat share distributions tell us about the averagevote shares?

But we have to take a detour. This chapter reviews what can be foundby going from votes to seats. A broad predictive model is presented, thelaw of minority attrition. Only thereafter can Chapter 14 proceed to thereverse approach, trying to predict votes from seats, which themselves areestimated from institutional givens.

The Law of Minority Attrition: Women’s ‘Rubber Ceiling’,Elections, and Volleyball Scores

In the course of sequential selection processes, the categories underrep-resented in the early phases tend to be underrepresented even more inthe later phases, where the positions available are scarcer. Robert Putnam(1976: 33) called it the ‘law of increasing disproportionality’ as one movesup the ladder of authority. Suppose the share of an ethnic minority is

202


Table 13.1. Women’s share in US public office

Public office Number of positions Women’s share (%)

City council members ∼100, 000 20Mayors ∼10, 000 10State lower house members ∼10, 000 10US House of Representatives 435 5US Senate 100 2

Source: Adjusted from Taagepera (1994).

small at the lower echelons, where positions are relatively numerous: totalseats in provincial legislatures, or assistant professorships at universities.If so, then its share tends to be even smaller at higher echelons with fewerseats or positions: national assemblies, or full professorships.

The same goes for women, although they are not a minority in theoverall population, as their careers raise them toward what used to becalled the ‘glass ceiling’ but more recently has been characterized as a‘rubber ceiling’ because it is not firm but offers ever stronger resistanceas one moves up. The pattern is clear in the US politics: ‘The higher onegoes, the fewer women one finds in public office’ (Darcy, Welch, and Clark1994). Table 13.1 shows rough percentages of women in the US publicoffices of the early 1980s. Women’s share decreases as positions are fewer.

Something analogous happens in elections. The number of positions islarge in the electorate. It is much smaller in the assembly elected—andthe seat share of minor parties tends to be smaller than their vote share.The effect is minor in PR systems and marked in plurality systems, but itis there. Now suppose the assembly is an electoral college that chooses thepresident, by plurality vote. The number of positions is reduced to one,and minority attrition is bound to become extreme, as all seat shares butone are reduced to zero.

But the phenomenon of attrition applies even beyond social categories.In volleyball, the total number of points won by either team is relativelylarge compared to the total number of games won, which in the USchampionships can range only from 3 to 5. In terms of Table 13.1, gamesare scarcer ‘positions’ than points—and the losers’ share of total gameswon is smaller than their share of the points earned. The ability of votesto win seats in FPTP elections depends on their location among electoraldistricts. Similarly, points help to win games, depending on their locationamong the game periods.

This brings us to two fundamental questions. Can we formulate log-ically founded predictive models for minority attrition processes such as

203


party votes to seats, volleyball points to games, and women’s shares in citycouncils and national assemblies? And if we do, are the models different,or does the same model fit all phenomena?

To put it differently, are the mechanisms that underlie minority attri-tion in elections specifically political, more broadly social, or imposedby even more general mechanisms? If the detailed patterns of attritionof minorities are similar in elections and volleyball scores, then it wouldsuggest that the underlying mechanisms do not depend on human natureany more than the normal distribution of weights of humans or peasdepends on the specific nature of humans or peas. This is what makes thestudy of volleyball scores worthwhile. The outcome is more important forelections than playing ball.

The brief answer is that the minority attrition processes can indeed beexpressed by a quantitatively predictive model. Moreover, the attritionprocesses involved in volleyball matches and elections do look similar.The same equation comes close to expressing the attrition in both cases,and the only free parameter in this equation is the number of positionsavailable at various stages of selection.

In sum, the basic attrition process may not depend on specifically polit-ical or even broadly human considerations. A certain degree of attrition isinherent to minority–majority relationships. However, political or socialextras can be thrown in so as to reinforce or reverse it. Males are a minorityamong US school teachers, but there is no attrition of this minority whenit comes to advancement to school principals—quite to the contrary! Topphysicians also are males even in societies where most physicians arefemales. The law of minority attrition may supply a baseline so as todetermine which part of attrition is natural and which part is sociallyimposed.

In the following, the law of minority attrition will be presented inthe context of FPTP elections, where it first was developed, under thename of seat–vote equation (Taagepera 1969, 1973). It is later extendedto multi-seat districts with seat allocation by PR and plurality. Analysisof volleyball scores and women’s ‘rubber ceiling’ is given in chapterappendix.

The Law of Minority Attrition for FPTP Systems

How are the seat shares (si , s j ) of two parties, i and j , related to theirvote shares (vi , v j )? When all seats are allocated by plurality in single-seat

204


districts, the expected relationship can be shown to be

si

s j=

(vi

v j

)n

,

where the ‘disproportionality exponent’ (n) is

n =log Vlog S

.

Here V is the total number of voters and S the total number of single-seatdistricts. This is the format in which a special case of the law was firstobserved, around 1910. It was found that

si

s j=

(vi

v j

)3

fits British elections. The relationship came to be called the ‘cube law’ ofAnglo-Saxon elections (Kendall and Stuart 1950). When vote shares oftwo major parties are 60 to 40 percent, then their seat shares tend to bearound 77 to 23. This empirical regularity was observed to apply in severalcountries where FPTP was used. The connection of n to the number ofdistricts and votes was established around 1970 (Taagepera 1969, 1973).How this model is theoretically derived is explained in chapter appendix.

For calculation of seat shares, the following format is more practical:

si =vn

i

�vnk,

where the summation is over all parties that receive votes. Forinstance, suppose the percentage, of votes are 40-35-25, and n = 3.Then s1 = 0.403/(0.403 + 0.353 + 0.253) = 0.064/(0.064 + 0.043 + 0.016) =0.064/0.123 = 0.52 = 52%. I have written it out in detail, because thenthe other seat shares can be computed quickly: s2 = 0.043/0.123 = 35%and s3 = 0.016/0.123 = 13%. Here the largest party gets a bonus, thesecond-largest breaks even, while the third party is heavily penalized.This is Duverger’s mechanical effect spelled out quantitatively.

The equation above is fully equivalent to si/s j = (vi/v j )n, as one caneasily check by dividing si = vn

i /�vnk by s j = vn

j /�vnk . This equation and

n = log V/log S are quite different in nature. The first has little specificallysociopolitical content, as the format yi = f (xi )/� f (xk) applies to manyphysical as well as social phenomena. The second is more specific. Theexponent n reflects the disproportionality of the electoral system. Whenonly two parties contest the seats, n is what has been called the responsive-ness of an electoral system to a shift in votes (Tufte 1973). Suppose both

205


of these parties have close to 50 percent of the votes. Then n = 3.0 meansthat when the vote share of a party increases by 1 percentage point, itsseat share increases by 3 percentage points.

For n = 1, we have perfect proportionality of seat shares to vote shares.The more n increases beyond 1, the more the curve seats-versus-votesbecomes steeper at vi = 50%, meaning more disproportional seat distribu-tion. When n tends to ∞, the curve tends to become vertical at vi = 50%,and we reach utmost disproportionality: The party with the most voteswins all the seats.

The equation n = logV/logS indicates that disproportionality increaseswhen more voters are added but decreases when more single-seat districtsare added. When the cube root law of assembly sizes applies (see Chapter12), then logV/logS ≈ logP/logS = 3. As an example, suppose P = 10 mil-lion. If the cube root law applies, S = (107)1/3 = 215, and log P/log S =7/2.33 = 3.00. Suppose only 5 million people vote. In this case, logV/log S = 6.70/2.33 = 2.87, which is within 5 percent of 3.00.

It follows from si = vni /�vn

k and n = log V/log S that

(log S) log(

si

s j

)= (log V) log

(vi

v j

).

This is the law of minority attrition expressed as a single equation, in theseats–votes context. This equation is symmetric in seats and votes, whichhas important implications, as discussed in chapter appendix.

The law of minority attrition applies beyond national assemblies. Whenn = 1, it would express perfect PR. This situation could formally emergewith FPTP in the extreme case where there are as many seats as voters,with each voter voting for oneself. Then n = logV/logS = 1. Some tradeunion elections actually come close to the limit n = 1, because even a2-worker shop elects a shop steward. Thus the average n for trade unionelections is predicted to be around 1.5. For agreement with data, see graphin Taagepera and Shugart (1989: 163). In national assemblies, n rangesfrom 2.5 to 4.0. Finally, in direct presidential elections with pluralityrule, S = 1 means logS = 0, so that n = logV/logS tends toward ∞—whichcorresponds, as it should, to a seat ratio of 1:0, regardless of the vote ratio.Let us see in more detail what the attrition law can and cannot do.

Testing the Law of Minority Attrition for FPTP Systems

The cube law emerges from the general minority attrition law becausedemocratic assemblies tend to follow the cube root law of assembly sizes.

206


Table 13.2. Caribbean countries with unusually high disproportionalityexponents

Country Period Assembly Size Exponent (n) Two-party elections

Antigua 1980–9 17 3.5 1980, 1989Barbados 1966–91 26 3.5 1971, 1976, 1981, 1986Trinidad 1961–91 36 3.6 1961, 1971, 1986St Lucia 1974–92 17 3.8 1974, 1979, 1992Grenada 1972–84 15 3.9 1972, 1976, 1984St. Vincent 1974–89 13 4.0 1984, 1989

Data source: Nohlen (1993).

When the actual number of seats and valid votes is used, the expecteddisproportionality exponent around 1970 was observed to range from n =2.61 for the relatively large British House of Commons to n = 3.17 in therelatively small House of Representatives in New Zealand (Taagepera andShugart 1989: 166).

Assemblies in small island nations tend to fall below the cube root ofpopulation. This means that the disproportionality index n reaches 3.5and even 4, when they use FPTP (Lijphart 1990). The winner’s advantageis huge, and the opposition is often utterly decimated. These nationsmay think they use the British electoral system, but the outcome is moreextreme because of excessively small assembly sizes.

Table 13.2 shows assembly sizes and the resulting disproportionalityexponents n in 6 small island countries where this exponent is unusuallyhigh, ranging from 3.5 to 4.0. These are the only countries in LatinAmerica and the Caribbean where data listed in Nohlen (1993) lead ton ≥ 3.5. Their assemblies are very small even for their small populations.With such high exponent values, any disagreements with the modelshould stand out the strongest. This is why I will use them as test casesthroughout this chapter. Elections where third party votes are less than5 percent are indicated. These practically pure two-party elections will beseen to fit the attrition law better than multiparty elections.

Figure 13.1 graphs si versus vi for individual elections in these countries,for data in Nohlen (1993), using n = 3.75. Two curves and two sets of datapoints are shown. The ‘one-opponent’ curve represents the predictionof the attrition law when only two parties run. Then the attrition lawbecomes

s =vn

vn + (1 − v)n,

207


s = v

Tw

o op

pone

nts

One

opp

onen

t

0

20

40

60

80

100

0

Votes (%)

Sea

ts (

%)

Two-party contests

Multiparty contests

20 40 60 80 100

Figure 13.1. Seat shares vs. votes shares for FPTP with high disproportionalityexponents—attrition law and Caribbean data

s and v being the seat and vote shares, respectively, of either of the twoparties. The corresponding data for elections where the third party voteshare is below 5 percent are shown with round symbols.

Recall that the disproportionality exponent n (responsiveness) is theslope of the curve at si = vi = 50%, meaning that when the vote share ofa party increases by 1 percentage point around 50 percent, its seat shareincreases by n = 3.75 percentage points. At very low or high vote shares,the seat–vote curve bends so as to respect the conceptual anchor points(0; 0) and (100; 100), along with a third anchor point at (50; 50). This is anexample of a general curve with 3 anchor points, as discussed in BeyondRegression (Taagepera 2008). The data largely agree with the prediction,with some random scatter spread almost evenly on the two sides of thecurve.

208


When more than two significant parties compete, no unique curve, si

versus vi , emerges from si = vni /�vn

k , because the seats of a given partydepend on how the votes are distributed among its competitors. A partywith 40 percent votes can be a big winner when the distribution is40-30-30, while 40-50-10 would make it a relative loser. The outcomesfrom all such constellations can be calculated from si = vn

i /�vnk , but graph-

ical representation becomes more complicated. The ‘two-opponent’ curvein Figure 13.1 shows what the minority attrition law predicts when a partyfaces two opponents with exactly the same vote shares. Here the equationis

s =vn

vn + 21−n(1 − v)n.

A split opposition would greatly benefit the given party. Elections withmore than two significant parties running are shown in Figure 13.1with cross symbols. As expected, they are mostly on the left of the one-opponent curve. Small parties are even on the left of the two-opponentcurve. Surprisingly, they actually tend to receive PR, indicated by theline s = v, which would correspond to n = 1. We will return to this dis-agreement. For the moment, let us observe that the general case cannotgraphed and tested in the format of Figure 13.1, as each data pointcorresponds to a different curve. We have to look for other formats, andit is not easy.

One approach is to graph seat ratios versus vote ratios for all par-ties, always relative to the party with the largest vote share: si/s1 ver-sus vi/v1. This approach was used by Taagepera and Shugart (1989:160–8). Shugart (2007) applies a related format to 208 elections,ranging from Canadian provinces to the Caribbean. He graphs theexpected ratio of seats of the two major parties, calculated as (v1/v2)n,against the actual seat ratio, s1/s2. The scatter is wide, but on theaverage the expected ratio is close to the actual—except for nation-wide elections in India. Here the actual ratio falls steadily muchbelow the expected. In India, both largest parties have been unusuallysmall.

A general shortcoming of any ratio approach is that it boosts relativeerror, and it is especially severe here for the following reason. Whenthe largest share surpasses its expected size by a random amount, thesecond-largest share is likely to be correspondingly smaller by a similaramount. Hence the ratio (v1 + ε)/(v2 − ε) is boosted doubly—by the +ε inthe numerator and by the −ε in the denominator.

209


Actual

= Expec

ted

0

20

40

60

80

100

0

Expected seat share (%)

Act

ua

l se

at s

ha

re (

%)

Two-party contests

Multiparty contests

20 40 60 80 100

Figure 13.2. Actual seat shares vs. those calculated from the attrition law, for FPTPsystems with high disproportionality exponents

The most precise way to test the attrition law might be the following,when more than two parties run. Calculate the expected seat share foreach party directly from si = vn

i /�vnk , using all the actual vote shares

and the exponent value n = logV/logS calculated from the number ofvalid votes in that particular election. Graph the actual seat sharesagainst these expected ones. Because this approach is relatively time-consuming, it has not been used previously, but there may not be any wayaround it.

Figure 13.2 graphs the previous Caribbean data, based on Nohlen(1993), in this format—actual seat shares versus the expected. Again, two-and multiparty contests are shown with distinct symbols. For the two-party contests, limited and balanced scatter around the line ‘Actual =Expected’ shows that data fit well with the predictive model. In contrast,multiparty contests tend to give the largest party fewer seats than would

210


be expected on the basis of the attrition law, with the corresponding boostto the smaller parties.

This one-sided deviation from expectation does not recur in the afore-mentioned larger sample investigated by Shugart (2007), so it is too earlyto declare that the model applies only to almost pure two-party contests.But a thorough testing in the format of Figure 13.2, using all availableFPTP elections, would be called for. We should also examine the concep-tual underpinnings of the model, which may lead to a correction term forthe law of minority attrition.

Using Votes to Predict the Effective Number of LegislativeParties and Deviation from PR

Effective number of legislative parties and deviation from PR are centralto the study of the impact of electoral systems on party systems. How wellcan the attrition law predict these quantities on the basis of votes? I useagain the previous Caribbean data, where the disproportionality constantis extreme, so that large deviations from PR can be expected.

Figure 13.3 shows the actual effective number of legislative parties,based on Nohlen (1993), graphed against the one calculated from thevote shares with the help of the attrition law. The scatter around theline ‘Actual = Expected’ is limited and balanced for two-party contests,showing agreement, on the average. One does not expect an institutionalmodel to fit the outcomes of individual elections but only the overalltendency. For multiparty contests, in contrast, the attrition law steadilyunderestimates the number of legislative parties, in line with Figure 13.2,where this model overestimated the largest seat share. Grenada 1990stands out as an extreme case. Once more, Shugart’s aforementionedstudy (2007) suggests that more data analysis is needed before drawingconclusions.

When we proceed to deviation from PR (Gallagher’s D2), we arereaching the predictive bounds of the attrition law for the followingreason. Deviation from PR involves subtraction of two variables, si − vi ,which boosts random error. When the largest seat share surpasses the voteshare by more than the expected amount, through random fluctuation,then the other seat shares are bound to be correspondingly smaller. Notethat, in contrast, calculation of effective number of parties involves onlyadditions.

211


Gr90

Actual

= Expec

ted

1

1.5

2

2.5

3

1Expected Ns

Act

ualN

sTwo-party contests

Multiparty contests

1.5 2 2.5

Figure 13.3. Actual effective numbers of legislative parties vs. those calculatedfrom the attrition law, for FPTP systems with high disproportionality exponents

For the multiparty contests in Caribbean countries the correlationbetween actual and predicted deviations from PR becomes almost nil,and there is no point in graphing it. For two-party contests, shown inFigure 13.4, the scatter around the line ‘Actual = Expected’ is wide butfairly balanced.

In sum, the law of minority attrition can be used to infer the effectivenumbers of legislative parties from the vote shares of parties, at least forsome types of elections. For deviation from PR, random scatter tends totake over. We are here approaching the limits of predictability based onthis model.

212


Actual

= Exp

ecte

d

0

10

20

30

0

Expected deviation from PR (D2)

Act

ualD

2

10 20 30

Figure 13.4. Actual deviations from PR vs. those calculated from the attrition law,for two-party FPTP systems with high disproportionality exponents

The Law of Minority Attrition for Multi-Seat Districts

The law of minority attrition can be adjusted for multi-seat plurality, suchas the US Electoral College, where responsiveness is even higher thanFPTP. It can also be adjusted for PR systems, where the responsivenessis close to 1 (perfect PR). The respective disproportionality exponents are

n =(

logVlogS

)1/M

[PR]

and

n =logVlogE

. [plurality]

213


Here E stands for the number of electoral districts. At M = 1, both expres-sions yield n = logV/logS. See chapter appendix for elaboration and pos-sible extension to Two-Rounds elections.

Why Some FPTP Contests Deviate from theLaw of Minority Attrition

The century-long awareness of the tendencies expressed in the ‘cubelaw’ may have contributed to its own demise in recent UK elections(Blau 2004), as parties have learned to counteract this natural ten-dency by concentration of resources into the most profitable districts.The result is that the disproportionality exponent has recently beenreduced to a value lower than logV/logS. In the USA, the pattern hasbeen affected in the past by one-party elections in the South and tradi-tional gerrymander elsewhere. More recently, bipartisan gerrymander alsoenters.

Once more, one is reminded that few actual electoral systems are sim-ple. The attrition law still holds for FPTP systems as a unifying first approx-imation that joins trade union elections (n ≈ 1.5), assembly elections insingle-seat districts (n = 2.5 to 4), and direct presidential election, as alimiting case where S = 1 and n tends toward ∞.

The fading of the cube law in Britain may be among the first instanceswhere political science expressly encounters a broad question: Whendoes our understanding of the world alter the world itself? In quantummechanics, the observation of an elementary particle inevitably alterseither its position or momentum (the famous principle of indetermi-nacy), but the problem fades in macroscopic physics. Microorganismsrespond to the invention of antibiotics by mutations that increase theirresistance. Awareness of the law of gravitation helped humans to deviseways to circumvent its impact and build airplanes. When political sciencedevelops laws that describe simple political phenomena, politicians canbe expected to look for loopholes. Their inventiveness can match that ofaeronautical engineers.

Coming now to the Caribbean countries graphed in Figures 13.2 and13.3, we observed that multiparty contests tend to give the largest partyfewer seats than the attrition law would predict, with the correspondingboost to the smaller parties. This deviation suggests that we should testwith more data but also examine the conceptual underpinnings of themodel. This may lead to a correction term for the law of minority attrition.

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One underlying assumption of the attrition law is that the supportof parties in the individual districts follows some ‘regular’ distributionpattern. At the very least, a unimodal distribution is implied. However,the distribution of effective number of electoral parties across districts isfar from unimodal in all four countries graphed by Chhibber and Kollman(2004: 42). The USA comes closest, with a sharp symmetric peak at N = 2.0marred only by uncontested elections that produce a minor spike atN = 1.0. Great Britain has a high one-sided peak at N = 2.0, followed byextremely few cases at N = 2.1 and a low and wide secondary peak atN = 2.5. Canada and India follow vaguely similar patterns.

The striking common feature of the latter three countries is lack of dis-tricts with 2.1 effective parties. Such a value, barely higher than N = 2.0,could result from the presence of a third party of only 3 percent when thetwo major parties are balanced (48.5-48.5-3), or a larger third party, upto 7 percent, when the distribution is fairly lopsided, such as 60 − 33 − 7.The second peak around N = 2.5 would correspond to constellations suchas 44-44-12, with a larger third party. What these bimodal patterns mightexpress is reluctance of third parties to run in districts where they havevery little support. It remains to be determined whether and how theattrition law could be modified so as to account for bimodal distributionsof party strengths.

Conclusion

We have developed a predictive model for FPTP systems to convert fromvote shares of parties to their seat shares. It covers the basic patterns ofa wide variety of elections, from direct and indirect presidential electionsto parliamentary elections with FPTP or List PR, and with a slight mod-ification, multi-seat plurality. In the context of elections, the model isexpressed as a ‘seat–vote equation’ in two senses. It converts votes to seats,and the disproportionality exponent (‘responsiveness’) of the conversiondepends on the total votes and seats, at least in FPTP and List PR.

Beyond elections, this law of minority attrition applies to attrition ofsocial minorities as their careers raise them toward the ‘rubber ceiling’.It can be tested in the unlikely context of volleyball scores.

In the preceding chapters, district magnitude and assembly size playedsuch symmetric roles that their effect could be condensed into a sin-gle indicator, the seat product MS. This convenient package no longerworks in the conversion of votes into seats. Here S enters the expression

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n = logV/logS separate from M. There is no inherent reason why M andS should come in a single package in all aspects of electoral studies. Nowthat we see them part company, there is even more cause to marvel abouttheir staying together in the same package MS for so long—all the wayfrom the number of seat-winning parties to mean cabinet duration.

Once more, it is time to remind ourselves that logical regularities canapply only to averages in simple electoral systems, with equal numbers ofseats and votes per district. The regularities observed depend on a quasi-normal (or at least unimodal) distribution of voting strengths of partiesacross the districts. If a country consists of ethnically distinct regions,and the parties are ethnically based, then FPTP could lead to prettyproportional outcomes (exponent n close to 1) rather than anything likethe cube relationship.

Thus country-specific cultural, geographic, and political factors can alterresponsiveness. The law of minority attrition is but a first approximation.It enables us to calculate the seat shares of all parties from their voteshares, as long as some known and possibly also some unknown assump-tions hold. Thus, it supplies a base line. If (and only if) deviations from itspredictions are observed do we have to ask what further factors enter.


This appendix offers a formal derivation of the law of minority attrition for FPTPsystems, extends it to multi-seat districts, and tests it with volleyball scores andwomen’s shares in politics.

Derivation of the law of minority attrition

My Master’s thesis on ‘The Seat–Vote Equation’ (Taagepera 1969) at the Universityof Delaware first formulated the law of minority attrition and indicated its twoforms—a single symmetric equation or two separate ones. The same year, HenriTheil (1969) offered formal proof that the mechanical effect of FPTP on thetransformation of vote shares into seat shares must follow the format

si

s j=

(vi

v j

)n

.

The important observation was that, among all functions of the form si/s j =F (vi/v j ), this is the only one that does not lead to inconsistencies in the presenceof more than two parties. Here we have an example of the ‘Sherlock Holmes

216


approach’: winnowing out the inconsistent options leaves only one acceptableform. The alternative format si = vn

i /�vnk automatically follows.

An alternative approach is to observe that the format yi = f (xi )� f (xk) resultswhenever the values of yi depend on xi and their sum is 1: yi = f (xi ) and �yi = 1.These are very loose conditions and hence fit many physical and social phenom-ena. Impose the further condition that the ratio of two values of y must be afunction of the ratio of the corresponding values of x. This means yi/yj = F (xi/xj ),as presumed by Theil. Then f (x) = axn is the only satisfactory function f (x). Henceyi = xn

i /�xnk , or in terms of s and v, si = vn

i /�vnk .

But why presume that ratios depend on ratios, as expressed by si/s j = F (vi/v j ),rather than presuming that differences depend on differences, as expressed bysi − s j = F (vi − v j )? Still other combinations of two seat shares might be presumedto depend on the same combination of the corresponding vote shares. It can beshown (Taagepera and Shugart 1989: 185) that the relationship of differences canbe valid only for perfect PR, and that other combinations run into inconsistencies.

Theil (1969) left the value of exponent n open. Its value 3 in the empirical ‘cubelaw’ remained unexplained. It results when a certain degree of district-to-districtvariability is assumed, but this degree of variability itself had to be arbitrarily cho-sen so as to yield n = 3. In contrast, my Master’s thesis, the gist of which reachedprint several years later (Taagepera 1973), started with the following thoughtexperiment that shows that the value of n cannot be 3 under all circumstances.

First, reduce the number of seats available gradually to 1, which correspondsto direct presidential election. Whenever party j has the most votes, it wins thesingle seat, so that the seat ratio must become si/s j = (0/1). Such a ratio can beobtained from the vote ratio (vi/v j ) only when the exponent n is made to tendto ∞. Next, go in the opposite direction and increase the number of seats untilit equals the number of voters (V). Assuming that everyone runs and votes foroneself, perfect PR would prevail. This would require n = 1 in the equation si/s j =(vi/v j )n. It becomes clear that the exponent n must decrease from ∞ to 1, as thenumber of seats (S) increases from 1 to V. The number of seats available matters.

It follows that the number of voters also matters. Suppose we have 100 single-seat districts for a million voters and an exponent larger than 1 is observed toapply—like the cube law. Now, for the same 100 districts, reduce the number ofvoters to 100. Perfect PR (n = 1) must set in, meaning that the exponent decreasesfrom some larger value to 1. Thus, at constant S, the exponent n must decreasewith decreasing number of voters (V). In sum, n is a function of both V and S:n = n(V, S). More specifically, n is an increasing function of V and a decreasingfunction of S.

Next, consider multistage elections, where V voters elect a larger electoral collegeof C electors, who elect the final body of S members. The exponents in the twostages are n(V, C) and n(C, S), respectively. For the total process it is n(V, S). Con-sistency requires that the final outcome must be the same, whether we calculateby stages or directly: V → C → S should be equivalent to V → S. Such consistency

217


requires that n(V, C)n(C, S) = n(V, S). This is possible only when n(V, S) has theform

n(V, S) = f (V)/ f (S)

Here a function of V is divided by the same function of S. It then results fromsi = vn

i /�vnk that

f (S)log(

si

s j

)= f (V)log

(vi

v j

).

This is the minority attrition law, except that we still have to specify the functionf . When S = 1, n = f (V)/ f (S) must tend toward ∞. Hence f (1) = 0 is required.Among the functions that satisfy this condition, the simplest are f (x) = x − 1 andf (x) = logx. Empirical data (cube law, in particular) do not agree with f (x) = x − 1and strongly point in the direction of f (x) = logx. The theoretical argument to thateffect, however, made in Taagepera (1973) and repeated in Taagepera and Shugart(1989: 187), remained debatable. This was a lasting weak link in the theory of theminority attrition law. I now offer the following reasoning.

When S increases from 1 to V, the exponent n decreases from ∞ to 1. Whatwould be the mid-ranges for S and n? The mid-range for S is easy to define. Giventhat S and V can take only positive values and V can be much larger than 1, themeaningful mean of 1 and V is their geometric mean (cf. Taagepera 2008). Hencethe mid-range of S is defined as S = V1/2.

For n, the mid-range is hard to define, because one of the limits tends to ∞.Observe, however, that the minority attrition law is symmetric in seats and votes.It corresponds not only to si = vn

i /�vnk but also to vi = sm

i /�smk , where m = 1/n =

f (S)/ f (V). In contrast to n, this m has a finite range, from m = 0 (when S = 1) tom = 1 (when S = V ). Given that m can become 0, the use of geometric mean isexcluded, and arithmetic mean defines the mid-range in this case as m = 1

2 .Would the two mid-ranges, S = V1/2 and m = 1

2 , correspond to each other? Inprinciple, S = V1/2 could correspond to a value lower or higher than m = 1

2 , butwe have no reasons to prefer one direction to the other. Thus, in the absence ofany further information, the best guess is that f (V1/2)/ f (V) = 1/2. The functionf (V) = logV satisfies this condition. It also satisfies f (1)/ f (V) = 0 and f (V)/ f (V) =1. Furthermore, by taking the mean of S = 1 and S = V1/2, thereafter the meanof S = V1/4 and S = 1, and so on, it becomes clear that f (V) = logV is the onlyacceptable function. Thus

n =logVlogS

.

In the proof of f (V) = log V, did I stack the cards in favor of log V the moment Itook the geometric mean of 1 and V for S but the arithmetic mean of 0 and 1 for m?This was not an ad hoc choice but application of general principles enounced in

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Taagepera (2008) and repeatedly applied in various chapters of the present book.Yes, the cards were stacked in favor of logx—but by nature, not by me.

Combining n = logV/logS with si/s j = (vi/v j )n leads to the format first enouncedaround 1970 (Taagepera 1969, 1973):

logS log(

si

s j

)= logV log

(vi

v j

)= ‘selection process constant’.

At that stage, it was called the seat–vote equation. Its nature as a broader law ofminority attrition was first pointed out in Taagepera and Shugart (1989: 184–5,188), Taagepera (1994).

Such a format symmetric in seats and votes has esthetic appeal and brings intoevidence a combination of party characteristics that remains unchanged as votesare converted into seats (and possibly into seats in an intervening electoral collegealso elected by FPTP). The expression ‘log(sum of all components) times log(ratioof two components)’ remains the same at different stages of the selection process.In physics, this constant would be called a ‘constant of motion’. Here, ‘selectionprocess constant’ seems appropriate. This constant is what makes the symmetricform of the minority attrition law theoretically interesting. For most purposes,however, it is more convenient to plug n = logV/logS into the format si = vn

i /�vnk .

The law of minority attrition for multi-seat districts(and Two-Rounds elections)

With multi-seat districts, a crucial distinction must be made between seat alloca-tion by plurality and PR, symbolized by the notation MF , where F = 1 for PR andF = −1 for plurality. The law of minority attrition can be extended to multi-seat PRand plurality, but in quite different ways. No theory has been proposed for Two-Rounds election, but Dolez and Laurent (2005) have carried out empirical workthat will be discussed.

The minority attrition relationship is extended to multi-seat PR (Taagepera 1986)by proposing that in this case

n =(

logVlogS

)1/M

. [PR]

Combining it with si/s j = (vi/v j )n leads to a single equation which is again sym-metrical in seats and votes:

(logS)1/Mlog(

si

s j

)= (logV)1/Mlog

(vi

v j

)= constant. [PR]

I still cannot prove this formula, but it seems to fit. For M = 1, it reduces itselfto the original minority attrition law for FPTP, as it should. When M increases

219


beyond 1, the Mth root quickly reduces the exponent n to values hardly above then = 1.00 of perfect PR. Consider a 100-seat assembly that represents a populationof 1 million, so that the cube root law of assembly size applies: logV/logS = 3.Even with a district magnitude as low as M = 5, the disproportionality exponent nbecomes as low as 31/5 = 1.25. Taagepera and Shugart (1989: 168) show the degreeof agreement for Finland’s nationwide pattern (where n = 1.07 is predicted) andindividual districts (n = 1.14 predicted). For a 100-seat assembly elected nationwide(M = S = 100), the expression becomes n = (logV/logS)1/S = 30.01 = 1.011. This isclose to perfect PR, but still not quite.

Now consider multi-seat plurality. Plurality rule applied to a single nationwidedistrict allocates all seats to the same party, regardless of the number of seats.Hence, it is the number of electoral districts (E ) rather than the number of seatsthat matters in this case, so that n = logV/logS must be replaced by n = logV/logE .Combining it with si/s j = (vi/v j )n leads to a single equation which no longer issymmetrical in seats and votes:

logS log(

si

s j

)= logE log

(vi

v j

). [Plurality]

For single-seat plurality, S = E , so the distinction does not matter.So it turns out that we have two quite different expressions for disproportional-

ity exponent n, one for plurality and one for PR:

n =log VlogE

, [Plurality]

n =(

logVlogS

)1/M

. [PR]

At M = 1, the two become identical. This M = 1 corresponds to the least propor-tional outcome one could have with List PR rules, and also to the most propor-tional outcome one could have with plurality rule. It would be nice to find a singleunifying expression that includes both of them. Taagepera and Shugart (1989: 169)offered a form that still depends on arbitrary assignment of parameter values, andno progress has been made. The unified attrition law becomes asymmetrical inseats and votes the moment one introduces multi-seat plurality. In view of itsfading popularity, one might as well forget about it, if it were not for the USElectoral College.

In the US Electoral College, all seats in the same state go to the party with themost votes (with few marginal exceptions). Hence the number of electoral districtswhere the plurality rule is applied is the number of states, which has gone from 23in 1820 to the present 51 (including the District of Columbia). As the populationalso increased, the resulting values of n = logV/logE has remained around 5, andthis is close to the median of actual data (Taagepera and Shugart 1989: 162), which

220


are extremely scattered. A complicating feature is the widely unequal number ofseats per state.

The theory of the law of minority attrition has not been extended at all toelectoral systems with ordinal ballot or Two-Rounds. However, Dolez and Laurent(2005) have carried out empirical work on the French legislative elections 1978–2002. They find that, when n = 4, the format si/s j = (vi/v j )n fits for the Left andthe Right taken as blocs. Given that France has S = 555 metropolitan single-seatdistricts, FPTP would be expected to lead to such a high exponent value onlywhen logV = 4logS = 4log555 = 11.0, meaning an electorate of some 100 billionpeople! It remains to be tested whether all Two-Rounds systems lead to extra highdisproportionality exponents. If so, then the mechanisms leading to it would haveto be found. On the other hand, the high exponent could merely be a result ofunusually high nationalization of electoral behavior since the 1960s, as Dolez andLaurent (2005) suggest. If so, it would be in striking contrast to the exponentdecreasing during the same period in UK (Blau 2004).

Volleyball, women, and the law of minority attrition

In volleyball scores the two stages that correspond to votes and seats are thenumber of points (P) and the number of games (G) won by the winner (W) and theloser (L). In contrast to elections, where more than two parties may be involved,each match involves only two teams, so that the symmetric form of the attritionequation becomes an elegant assertion that ‘The log of sums times the log of ratiosis constant’:

log(GW + GL)log(

GW

GL

)= log(PW + PL)log

(PW

PL

)= c.

Here c is a constant, the selection process constant. This constant differs in almostevery individual election, because seats can be divided in a large variety of ways.Volleyball tournament rules, in contrast, allow for only three outcomes, becausethe contest ends when one side has won 3 games. It can only be 3 to 2 or 3 to1 or 3 to 0. Hence there are only three values for the selection process constant,when the latter is based on the games, cG = log(GW + GL)log(GW/GL). If the matchis won 3 to2, cG = log(3 + 2)log(3/2) = 0.123. If it is won 3 to 1, cG = 0.287. If it iswon 3 to 0, log(3/0) tends toward ∞, and so does cG. This case will need specialdiscussion.

Now we can compare the games-based process constants cG to the medianvalues of the selection process constant emerging from the points earned, cP =log(PW + PL)log(PW/PL). We expect that median cP = cG. The actual values of cP

range widely. They even extend to negative values, because a team with fewertotal points can win more games. But my unpublished calculations (Table 13.3)show that the medians of a large number of US championship matches are closeto expectations.

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Table 13.3. Volleyball scores and the law of minorityattrition

Games ratio (GW/GL) 3 : 2 3 : 1 3 : 0Number of matches 119 162 96Expected constant (cG) 0.123 0.287 [→infinity]Observed median cP 0.098 0.224 .390Deviation +26% +28% ??

Given the wide range of actual outcomes, the degree of agreement of the meansis impressive. However, cP falls short of cG by the same percentage in both finitecases, which suggests a possible systematic deviation. The constellation 3:0 offersus a clue. We are here applying an essentially continuous-variable model to a lownumber of integers, and this spells trouble.

As long as the loser wins at least one point, the attrition equation techni-cally predicts that the loser also wins at least a tiny fraction of a game. Butgames are counted in integer numbers. This means that a formal 0.03 games(or even 0.49 games) won by the loser would be rounded off to 0 gameswon. The observed median cP = 0.39 for the 3:0 outcomes corresponds to whatwould be expected if the ratio of games won were about 2.8 to 0.2. Similarly,the observed medians for 3:1 and 3:2 matches correspond to what would beexpected if the ratio of wins were about 2.9 to 1.1 and 2.9 to 2.1, respectively.The structure of the data is such that the larger figure is always rounded offupward and the lower one downward. Hence the gap we observe between cP

and cG. I have not yet found a way to refine the model so as to counteract thistendency.

One difference between elections and volleyball scores is that the total numberof seats to be filled is predetermined (or almost so), while the number of volley-ball games (3, 4, or 5) is known only retroactively, and the winner’s number isalways 3. One has to review Theil’s (1969) and Taagepera’s (1973) possible hiddenassumptions in light of these and other possible differences.

Women’s ‘rubber ceiling’ is more difficult to analyze, because it has been under-going a change over the last 50 years. Should we compare women’s share in theUS Senate to their share among city council members now or 30 years previously,

Table 13.4. Selection constant for women’s attrition in US politics seems to be 3.5

Public office Number of positions Women’s share% Selection constant

City council members ∼100,000 20 ∼3.1Mayors ∼10,000 10 ∼3.8State lower house members ∼10,000 10 ∼3.8US House of Representatives 435 5 3.4US Senate 100 2 3.3

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when the present senators got their starts? Many numbers in Table 13.1 are onlyvague estimates, but let us see what we get when we calculate the selection processconstant at various stages. The equation is c = log(PW + PM)log(PM/PW), whereW stands for women and M for men. For example, for city council members,the constant is around c = log100,000 log(80/20) = 3.1. To the extent that themodel fits, the selection constant should be the same at all other stages too.Table 13.4 shows the results, along with the previous data in Table 13.1. Theselection constant does remain in the range 3.5 ± 0.4, which is encouraging.It would be worthwhile to gather and test more precise data, and for manycountries.

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14

The Institutional Impact on Votes andDeviation from PR


� Using nothing but the product of assembly size and district magnitude,theory-based equations allow us to estimate the average effective num-ber of parties based on votes.

� Deviation from proportional representation is typically 10–20 percentfor first-past-the-post electoral systems. It can be estimated from theproduct of assembly size and district magnitude, within ±4 percentagepoints.

� These results refer to the averages for many elections carried out underthe same electoral laws. In individual elections, the number of partiesand deviation from PR can vary widely.

� When estimating the likely effect of changes in electoral laws on thenumber of parties and deviation from PR, also take into account thepast tendencies in the given country.

Reversing the usual direction of the law of minority attrition enables usto deduce vote shares of parties from their seat shares, which themselvescan be traced back to the impact of district magnitude and assembly size.This reversal is imperfect, and random noise increases. Still, in principle,we should be able to use this step to calculate the average distributionof all vote shares, including the largest, and hence the effective numberof electoral parties. Combining the respective seat and vote shares, weshould also be able to estimate deviation from PR on the basis of seatproduct.

225


The causal chain, however, becomes so extended that only limitedpredictability can be expected. In the Chapter 13, calculation of deviationfrom PR on the basis of actual vote shares and the attrition equation led towide scatter. The scatter can be expected to widen when we use theoreticalseat shares, themselves imperfectly deduced from institutions.

Up to now, this book has complemented predictive modeling withextensive testing, often reported in even more detail in previously pub-lished articles. Now we are in uncharted waters. This chapter indicateshow to approach the problem of converting from seats to votes in theory.It offers illustrative examples, but testing with extensive data remains tobe done. It is a research agenda where many researchers can get involved.

Relationships Between the Ways to Measure the Number ofElectoral Parties

Recall that the various ways to express the number of legislative partiesare related, on the average, as

N4∞ = N3

2 = N20 .

The number of seat-winning parties (N0) was derived from M and S.Later derivations, however, followed from N0 on logical grounds, withno further institutional input. Indeed, the inverse of the largest share (N∞)can be expected to be approximately the square root of the numberof components, whether these components are seat-winning parties orCanadian provinces. The effective number is tied to N∞ in a similarlyabstract way.

What this means is that the logic that leads from N0 to N∞ and N2

applies to vote shares too. Without any further proof needed, we canexpect that the relationship above connects the effective number of elec-toral parties and the inverse of the largest vote share, on the average:

N4V∞ ≈ N3

V2.

The number of vote-getting parties (NV0) is harder to define than theprevious number of seat-winning parties. We are interested in the numberof ‘serious’ electoral parties/candidates, rather than those who obtain amere couple of votes. Various norms have been proposed: add one moreto those parties/candidates that win a seat (Reed 1991; Cox 1997), oradd those parties/candidates who obtain at least 70 percent of the votesneeded to win a seat (Hsieh and Niemi 1999; Niemi and Hsieh 2002). One

226

The Institutional Impact on Votes

operational way would be to define the number of ‘serious’ vote-gettingparties as

NV0 = N2V∞ =

1v2

i

and see whether the resulting values of such a phantom NV0 make anysense. Chapter 15 returns to this issue.

From Seats to Votes: The Attrition Equation Read Backward

The law of minority attrition was deduced going from votes to seats, butits symmetrical form indicates that one can go in the reverse direction,too:

vi =smi

�smk

,

where, in the case of PR elections or FPTP,

m =1n

=(

log Slog V

)1/M

.

Once the average seat shares are estimated from institutional inputs(Chapter 9), we can use the equations above to estimate the average voteshares over many elections with simple electoral rules (Taagepera 2001).This procedure looks simple, but we run into a problem of rounding tointegers (like with volleyball scores in appendix of Chapter 13). Seatscome in integer numbers. When the seat–vote equation predicts 0.54 seatsfor one minor party and 0.44 seats for another, it is easy enough to round0.54 seats to 1 seat and 0.44 seats to 0 seats. But when the rounded figuresfor seats are given, how can one guess at what decimal fractions they wererounded from? How can we get the ‘0.44’ back from ‘0’?

To illustrate the degree of error that can result, let us consider a votedistribution akin to that in many recent UK elections: 40-35-20-5. Forsimplicity, assume that 1 million people vote and 100 FPTP seats are atstake, so that n = 3.00. First, use the seat–vote equation to calculate theseat shares and round them to integer numbers of seats. Then feed theseseats into the reversed seat–vote equation, so as to see how close we get tothe initial vote shares. The results are shown in Table 14.1.

Based on zero seats for party D, the reversed seat–vote equation pre-dicts zero votes for this party. All other vote shares are overestimated

227


Table 14.1. From votes to seats, and back to votes—hypothetical voteshares, with S = 100 and n = 3.00

Party A B C D Eff. N

Votes shares (%) 40 35 20 5 3.08Calculated seats (%) 55.65 37.28 6.96 0.11Number of seats 56 37 7 0 2.20Seat shares (%) 56 37 7 0Calculated votes (%) 42.18 36.73 21.09 0 2.80Difference +2.18 +1.73 +1.09 −5.00

Seats (out of 100.25) 56 37 7 0.25Calculated votes (%) 39.44 34.35 19.72 6.49 3.16Difference −0.56 −0.65 −0.28 +1.49

correspondingly, and the effective number of electoral parties is underesti-mated by 0.28 parties. How can we correct for this rounding down to zero?

Knowing that zero seats for one or several minor parties could havebeen rounded off from as high as 0.49 seats or as low as 0.00 seats, aconservative estimate might assume a single unrepresented party, with avirtual seat share halfway between 0 and 0.50, meaning 0.25. This newstarting point is shown at the bottom part of Table 14.1. Having a seattotal of 100.25 presents no problem, because one could plug either seatshares or numbers into vi = sm

i /�smk , without altering the outcome. Now

the vote share of party D is overestimated, while the other vote shares areunderestimated accordingly. The differences are reduced, though, com-pared to the previous approach, and the effective number of electoralparties is overestimated only by 0.08 parties.

This example suggests that the compound error in estimating the voteshares on the basis of seat shares might usually be only a few percent,but we cannot be sure. There may be more than one unrepresentedparty. Maybe we should assume one party with 0.25 seats and anotherat 0.25/2 = 0.125, and still another at 0.125/2 = 0.0625 seats.

More systematic statistical approaches may be available to estimate thenext values in a decreasing series such as 56-37-7- . . . One would haveto carry out cyclical calculations (actual votes → seats → votes) in var-ious countries, so as to obtain a better sense of the range of error. Thisapplies to FPTP systems. For List PR, the differences between vote andseat shares are bound to be small, but the low threshold of representation(see Chapter 15) sometimes tempts numerous tiny parties to run and fail.The number of votes wasted this way may add up and could be hard toestimate. We are here on untested grounds.

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Predicting the Number of Electoral Parties fromInstitutional Inputs

Previously (Chapter 9), all seat shares could be calculated, starting fromthe largest seat share, itself calculated from the seat product MS. Then theeffective number of legislative parties could be calculated from these seatshares. These calculations were messy, however, and so an approximateshortcut was devised, leading directly from the largest share to the effec-tive number: N = 1/s4/3

1 . It involved a loss of precision of known extent,which was tolerable in comparison with variation caused by other factors.We face a similar choice here. We could use all seat shares to convert intovote shares, but the connection to the seat product would depend oncomplex calculations.

Instead, we can look for a way to connect the largest vote share (v1)alone to the largest seat share (s1). Assume that it can be approximatedas v1 = sk

1, where k > 1, so that the largest vote share is smaller than thelargest seat share. Then

v1 = sk1 = (MS)−k/8

and

NV∞ = (MS)k/8 = (NS∞)k.

The effective number of electoral parties would result automatically:

NV =1

v4/31

= (MS)k/6 = NkS .

This number is larger than the number of legislative parties, NS = (MS)1/6.A phantom number of ‘serious’ vote-getting parties also results:

NV0 = (MS)k/4 = NkS0.

For the usual range of the largest seat share in stable FPTP systems (s1 >

0.3), the theoretical derivation for conversion s1 → v1 is given in chapterappendix. A satisfactory approximation is

k = 1.28n0.21. [FPTP, s1 > 0.3]

Here n is the previous disproportionality exponent n = 1/m = log V/ log S.Given that n > 1, it follows that k > 1, and v1 < s1, as one would expect.

Let us carry out a reality check to see whether any traces of an institu-tional impact on the electoral parties remain, at the level of a quantita-tive prediction. Consider the six Caribbean countries with an unusually

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high disproportionality exponent (responsiveness) of 3.5 to 4.0, as usedin Chapter 13. The mean exponent n = 3.75 for these countries yieldsk = 1.28 (3.75)0.21 = 1.69. Hence, for these FPTP systems,

v1 = s1.691 . [high-exponent FPTP]

For the largest seat share, the geometric mean of all 30 elections in thesecountries (data from Nohlen 1993) is s1 = 72.0%. How well does it predictthe largest vote share? It yields v1 = sk

1 = 0.7201.69 = 0.574 = 57.4%. Theactual geometric mean of the largest vote shares is 55.9 percent. We areoff by only 1.5 percentage points. The following connections to assemblysize follow:

v1 = S−1.69/8 = S−0.21 [high-exponent FPTP]

and hence

NV∞ = S0.21, [M = 1, n = 3.75]

NV = S1.69/6 = S0.28

NV0 = S1.69/4 = S0.42.

Table 14.2 tests for the impact of assembly size for individual countries,arranged in the order of increasing S. The overall geometric means differslightly from the ones above because all countries do not have the samenumber of elections. Going from assembly size to the largest seat sharealready involves a mean relative error of 6 percent. Surprisingly, the meanerror does not increase as we go from legislative to electoral parties,

Table 14.2. Predicting the number of electoral parties from assembly size, for highresponsiveness FPTP systems

Country, no.of elections

Largest seat share Largest vote share Eff. no. el. parties

S S−1/8 Act. s1 S−1.69/8 Act.v1 S1.69/6 Act. NV

St. Vincent, 4 13.5 .722 .819 .577 .597 2.08 2.23Grenada, 4 15 .713 .690 .565 .499 2.14 2.40St. Lucia, 6 17 .702 .629 .550 .547 2.22 2.15Antigua, 3 17 .702 .859 .550 .631 2.22 1.99Barbados, 6 26 .665 .694 .502 .534 2.50 2.15Trinidad, 7 36 .639 .735 .469 .580 2.74 2.22GEOM. MEAN .734 .690 .534 .563 2.31 2.19Relative error +6.4% +5.4% −5.2%

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although the causal chain is longer. For the effective number of electoralparties, the relative error reaches 20 percent for individual countries, butit remains below 6 percent for the mean of the 6 countries. Thus themodel passes this first reality check. Full testing of Duvergerian approachextended to electoral parties remains to be done.

Predicting the Vote Shares of All Partiesfrom Institutional Inputs

The average seat share distribution of all parties, for given largest seatshare, was determined empirically in Chapter 9, and a logical model wasoffered and tested. Work in progress (Taagepera and Laatsit 2007) indicatesthat the empirical pattern, relative to the largest share, is the same for voteshares, within the range of random error. Small parties seem to lose aboutone-half of their inherent support to large parties, whether one goes byseats or by votes. The theoretical model presented seems to apply to seatand vote shares equally well.

However, Taagepera and Laatsit (2007) also graph PR and FPTP electionsseparately. The patterns for seats and votes are again similar, but theydiffer for the two seat allocation rules. With FPTP, small parties seem tolose about three quarters of their inherent support, while with PR theylose only one quarter. The heavier losses in systems subject to Duverger’spsychological effect come as no surprise. However, it is confirmed thatappreciable loss of inherent support for small parties occurs in PR systemstoo.

Predicting Deviation From PR from Institutional Inputs

Deviation from PR (D2, Gallagher’s measure) has previously been foundto correlate with the logarithm of district magnitude (Anckar 1997b). Noequation for the best fit line was given, so we cannot compare with theresults of the following calculations. With this estimation I really go onthe limb, by assuming that D2 ≈ (s1 − v1), which is not always the case.It is true that D2 is often close to the largest seat–vote difference (cf.Chapter 5). But the largest difference also may come from third partylosses, which can exceed the largest party gains when some gain goesto the second-largest party. Moreover, subtractions always boost error,

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Table 14.3. Predicting the deviation from PR (Gallagher’s D2)from assembly size, for high responsiveness FPTP systems

Country S−1/8 − S−1.69/8 Actual D2

(%) (%)

St. Vincent 14.5 17.9Grenada 14.8 15.9St. Lucia 15.2 9.5Antigua 15.2 23.3Barbados 16.3 14.0Trinidad 17.0 14.5GEOM. MEAN 15.5 15.3Relative error −1.2%

as pointed out earlier. With these cautionary notes, we may try to seewhether the following theoretical approximation comes anywhere closeto reality:

D2 ≈ s1 − v1 = s1 − sk1 = (MS)−1/8 − (MS)−k/8.

Table 14.3 shows the outcomes for the 6 Caribbean countries, using thecalculated and actual largest shares in Table 14.2. The geometric meanof the actual deviations from PR is within 1 percent of the theoreticalprediction. This is too good to hold upon more extensive testing, but it isencouraging nonetheless. For individual countries, the arithmetic meandifference between the expected and the actual is 3.9 percentage points,reaching 8.7 for Antigua, where only 3 elections are averaged.

This result is better than one could have expected. It seems that, witha sufficient number of FPTP elections, we may actually start off withnothing but assembly size and still be able to predict the mean deviationfrom PR mostly within 4 percentage points. It remains to be seen to whatextent this result is confirmed with FPTP in larger assemblies and with PRsystems.

Figure 5.1 shows the empirical pattern of D2 versus the effective numberof electoral parties. Given that both D2 and NV can be estimated theoreti-cally from the seat product, we should be able to calculate the theoreticalcurves D2 versus NV and compare them with the empirical relationshipsobserved in Figure 5.1. This task remains to be done.

It can be seen that theoretical estimation of deviation from PR becomescomplicated even for FPTP and may become even more intractablefor multi-seat List PR. All this leaves unexplained a simple empiricalregularity documented by Taagepera and Shugart (1989: 118, 141) for

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Loosemore–Hanby’s D1. On the district level,

D1 =50%M1/2

,

and nationwide,

D1 =25%M1/2

.

One may suspect that the latter might be affected by assembly size.Accounting theoretically for these empirical regularities remains a chal-lenge.

Predicting Proportionality Profiles from Institutional Inputs

Proportionality profiles offer ‘snapshots’ of the actual electoral systems,as illustrated in Figures 5.2–5.5. Such profiles indicate at a glance theaverage impact of the given electoral system on large and small par-ties, and the degree of scatter around this average. If we truly under-stand the functioning of an electoral system, then we should be ableto predict its proportionality profile without looking at any data, juston the basis of electoral rules, institutional givens, and possibly someinformation on political culture. The deductive chain that extendsfrom institutions to advantage ratio is schematically the following (cf.Figure 7.2):

Seat product MS → largest seat share → other seat shares plus

largest vote share → other vote shares → advantage ratios

How close are we to such a stage, even for the simplest electoral systems?An unpublished report (Taagepera 2002c) offers a few early examples.Agreement is satisfactory for New Zealand, Finland, and the Netherlands,while limited for UK—as one might expect on the basis of Table 11.1,where UK looks as if it applied plurality in 7-seat districts (rather thanM = 1). This part of testing remains to be done. It would involve estab-lishing criteria for goodness of prediction, comparing the predicted andactual profiles for many countries, and establishing the main foci wheredisagreement arises in the deductive chain that extends from institutionsto advantage ratios.

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Does this chapter complete the Duvergerian macro-agenda, at least for thesimplest electoral systems? Formally, the last blank corner in Figure 7.2has been filled in—the vote shares, the effective number of electoral par-ties, and deviation from PR. The impact of institutions on votes remainsto be fully tested for FPTP systems with large assemblies, and the verytheory must still be fleshed out for PR systems, so as to determine thevalues of parameter k. Nonetheless, we have made marked headway, asno corner of the overall scheme remains completely blank and the pathfor completing the rest is outlined. At the very least, this chapter offers aresearch agenda for interested scholars.

One significant aspect not included in Figure 7.2 remains to be consid-ered: the various kinds of thresholds of representation. They occasionallyslipped into preceding discussions but need more systematic presentation.This is done in Chapter 15.

As for institutional engineering, the vote shares and the effective num-ber of electoral parties may be of less interest than seat shares and theeffective number of legislative parties, which directly impact politics inthe assembly. But deviation from PR becomes at times subject of politicalconcern, as a large deviation may impinge on perceived fairness of thesystem.

The deviation from PR (D2, Gallagher’s measure) is typically 10–20 per-cent for FPTP systems. It has been seen that, using nothing but the prod-uct of assembly size and district magnitude, theoretically based equationsmay allow us to estimate it within ±4 percentage points. This error refersto the averages for many elections carried out under the same electorallaws. In individual elections, the number of parties and deviation from PRcan vary widely. Thus the precision of prediction is limited, but we canstill be more quantitative than merely predicting the expected direction ofchange, when a country contemplates a change in assembly size or meandistrict magnitude.

Once more, when estimating the likely effect of changes in electorallaws on the number of parties and deviation from PR, one must alsotake into account the past tendencies in the given country. A coun-try with deviation from PR higher than theoretically predicted andhigher than the empirical world average at given seat product can beexpected to maintain a correspondingly higher deviation from PR afteran electoral reform. Other institutions, political culture, and past historymatter.

234



Here the largest vote share is derived from the largest seat share. Also investigatedis the gap between the effective numbers of legislative and electoral parties, andbalance in party sizes is revisited.

Theoretical derivation of the largest vote share from the largest seatshare for FPTP

The reversed attrition equation yields the largest vote share in terms of the seatshares of all parties: v1 = sm

1 /�smk . Calculations are simpler when one considers 1/v1

rather than v1. Then

1v1

=�sm

k

sm1

.

The first task is to approximate the sum �smk as a function of the largest seat share

s1 alone. Assume that the largest party faces N′ equal-sized parties. Then

�smk = sm

1 + N′[

(1 − s1)N′

]m

and

1v1

= �smk

sm1

= 1 + N′[

1 − s1

N′s1

]m

.

What would be a realistic number of parties the largest party faces? Followinga similar line of thought, Taagepera and Shugart (1989: 190) used the effectivenumber of parties but subtracted one, so as to account for the party under con-sideration. This is a clear undercount, and it led to an overestimate of break-evenpoints and distortions in the calculation of proportionality profiles (Taagepera andShugart 1989: 88–91, 191–6). Rather, all seat-winning parties (N0) should be takeninto account. If so, then the largest party faces N′ = N0 − 1 other parties. Moreover,we have seen that N0 itself can be expressed in terms of the largest seat share:N0 = 1/s2

1 . When this assumption is included, the previous equation becomes

1v1

= 1 +

(1s21

− 1

)1−m [1 − s1

s1

]m

.

Upon further extensive simplification (but without approximations!), it becomes

1v1

= 1 +(

1s1

− 1) (

1s1

+ 1)1−m

.

235


Graphing this function, log v1 versus log s1, for the actual range the largest seatshares take in most FPTP elections (0.30 to 0.80) leads to what are for practicalpurposes straight lines for all values of parameter m that occur in FPTP (n = 2.5 to4.0, hence m = 0.4 to 0.25). This means an approximation

v1 = sk1,

where the exponent k depends on n. Note that this expression fits two conceptualanchor points: When s1 = 0 then v1 = 0, and when v1 = 1 then s1 = 1. Graphingk versus n on logarithmic scales yields a straight line which corresponds to theequation

k = 1.28n0.21. [2.5 ≤ n ≤ 4.0]

Is this a truly theoretical result, in view of the empirical-looking fit in this equa-tion? It is theoretical, indeed, because no empirical data enter. Approximations forcomplex theoretical expressions still are theoretical, and there is nothing unusualin such simplification. In quantum mechanics, Schrödinger’s equation can besolved exactly only for the hydrogen atom; all quantum chemistry follows fromjudicious approximations.

Of course, one must specify the range of input values for which a givenapproximation is valid. Here this range is the one that occurs for FPTP elections:2.5 ≤ n ≤ 4.0. Within this range, the values of the largest vote share calculatedfrom v1 = sk

1 and k = 1.28n0.21 differ by no more than ±0.4 percentage points fromthose calculated directly from the exact equation 1/v1 = 1 + (1/s1 − 1)(1/s1 + 1)1−m.The approximation would most likely be different for the PR systems, where1.0 ≤ n ≤ 1.25. I have not calculated it as yet.

The model v1 = sk1 runs into conceptual trouble for presidential elections. S =

M = 1 leads to s1 = 1 and hence v1 = 1. Of course, this value v1 = 1 could be theresult of rounding off from as low as 0.51, but in plurality election a candidate canwin with even less than v1 = 0.49, which would round off to 0. This inconsistencyalso means that v1 = sk

1 cannot be extended to single FPTP electoral districts. Doesit affect countrywide results? At least for hypothetical data in Table 14.1, s1 = 0.56predicts v1 = 39.3%, close to the initial 40 percent.

The gap between the effective numbers of electoral andlegislative parties

It was noted in Chapter 4 that, on the average, NV − NS ≈ 0.4. Can it be theoret-ically explained why this gap tends to be around 0.4 rather than much more ormuch less? Recall that NV = Nk

S and hence NS = N1/kV . Thus theoretical expectation

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would be

NV − NS = NV − N1/kV = (MS)k/6 − (MS)1/6

.

This expression is hard to reduce to a simpler form. For the 6 Caribbean countries,the theoretically predicted gap ranges from 0.54 to 0.92. It vastly exceeds the actualgap. Here the subtraction boosts the relatively small difference between expectedand actual NV , as reported in Table 14.2.

It remains open whether the predictive ability of the model reaches here a limit,or whether a small adjustment could do the trick. For detailed worldwide checkingof the relationship between the effective numbers of electoral and legislativeparties, it might be easier to compare the ratios rather than the differences of thetwo numbers:

NV/NS = (MS)(k−1)/6 = N(1−1/k)V .

Balance in party sizes

Balance in party sizes is one characteristic that has not been connected to institu-tions. As a world average, B = 0.5 fits, and this was the basis for our estimates forthe largest seat share and hence the effective number of parties and cabinet dura-tion. But what causes deviations from this average balance? Looking at Figure 4.1,we may ask which institutions might be common to countries with low balance(UK, Spain, Greece, Portugal, Canada, Ireland, and Italy) in contrast to countrieswith high balance (Malta, Belgium, Austria, Luxembourg, Iceland, and Finland)?

One may note that the low balance group consists mainly of large countries(median population around 40 million), while the high balance group consistsof small countries (median population around 1.5 million). But this is a happen-stance of countries included in the data source (Mackie and Rose 1997). Inclusionof small Caribbean FPTP countries discussed in the present chapter would even outthe population score.

Regardless of population, FPTP countries tend to have few parties and lowbalance between the largest party and the rest. But what about multi-seat electoralsystems that also exhibit a dominant party (Japan, pre-1990 Italy, and Ireland)?And what about Greece, Portugal, and Spain, where major parties take turns inhaving lopsided majorities? Is it just path-dependent political development, orcan we locate institutional features that favor the rise of fleetingly or durablypredominant parties? It remains to be seen.

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Part III

Implications and Broader Agenda

Ask not what the electoral rules can do for your country, ask what yourcountry can do to the electoral rules.

A Wuffle


15

Thresholds of Representation and theNumber of Pertinent Electoral Parties


� Nationwide threshold of minimal representation is the average voteshare needed to win one seat in the assembly. It is close to 38 percentdivided by the square root of the seat product (assembly size timesdistrict magnitude).

� If greater inclusion of political minorities is desired, this threshold canbe lowered by increasing district magnitude and/or assembly size.

Various thresholds of representation have entered previous discussion,and it is time to present them systematically. The break-even point (b)in Chapter 5 represents the threshold of nationwide seat share of a partybreaking even with its vote share. District magnitudes at which partieswith a given vote share obtain their first seat (Chapter 6) hint at athreshold of minimal representation in a district. Such a threshold visiblydepends on the seat allocation formula used, and the differences can beappreciable, as illustrated in Table 6.1. For various usual PR formulas, anoverall average estimate for the threshold of minimal representation in adistrict was invoked in Chapter 8: T = 75%/(M + 1). Legal thresholds ofrepresentation also have been mentioned, from Chapter 3 on.

There is considerable confusion between thresholds at district andnational levels, as well as about the significance of minimal and otherlevels of representation. Some of these thresholds depend on the numberof parties competing. Hence the issue of the number of ‘serious’ electoralparties arises again and will be addressed.

241


District-Level Thresholds of Minimal Representation

Thresholds of representation are vote shares a party must obtain so as towin a specified number or share of seats. Of most interest are the votesneeded to win one seat (minimal representation), the votes needed forbreaking even (seat share equals vote share), and the votes needed towin one-half of all seats (majority). These thresholds depend on the seatallocation formula, and also on the number of parties running. At thedistrict level, they were worked out in the 1970s. At the nationwide level,which is of more interest, progress has been made only since the 1990s.This section deals with the thresholds for minimal representation at thedistrict level.

The inclusion threshold (TI) is defined as the minimum vote share a partyneeds to win its first seat, under the most favorable conditions. For a voteshare v smaller than TI, the probability of winning a seat is 0: P (1) = 0.Conversely, the exclusion threshold (TE) is the maximum vote share withwhich a party still can fail to win its first seat, under the most adverseconditions. For v larger than TE, the probability of winning a seat is 100percent: P (1) = 1. In-between these vote shares, the party may or may notwin a seat, depending on how the other vote shares are distributed amongits competitors. We may expect the probability to increase gradually withthe party’s vote share. We might wish to designate as the mean thresholdof minimal representation (TR) the vote share at which a party has a 50-50chance to win its first seat.

The inclusion and exclusion thresholds can be theoretically calculated,for given district magnitude (M) and number of parties running (p′).For d’Hondt allocation formula, Rokkan (1968) established the inclusionthreshold, and Rae, Hanby, and Loosemore (1971) the exclusion thresh-old. Others followed, such as inclusion threshold for Hare-LR (Laakso1979).

Table 15.1 shows the general formulas for d’Hondt, Sainte-Laguë, andHare-LR, and also the specific values when 6 or 8 parties run in a 6-seatdistrict (the example used in Tables 3.3 and 3.4). For the Sainte-Laguëand Hare exclusion thresholds, many sources list the same value as ford’Hondt, TE = 1/(M + 1), but this is true only if more parties run than thereare seats. If the number of parties is equal or less than M, which is usuallythe case at high M, then the exclusion threshold is lower. At M = 1, allthese thresholds boil down to the FPTP threshold, also shown.

At which vote share would a party have a 50-50 chance to win its firstseat? We have not found a way to calculate it, but it could be around

242

Thresholds of Representation

Table 15.1. District-level thresholds of minimal representation (in fractional shares) forvarious seat allocation formulas—general and for a 6-seat district

Formula d’Hondt Sainte-Laguë Hare-LR FPTP

Inclusion, TI 1/(M + p′ − 1) 1/(2M + p′ − 2) 1/Mp′ 1/p′

Exclusion, TE 1/(M + 1) 1/(2M − p′ + 2) for p′ ≤ M (p′ − 1)/Mp′ for p′ ≤ M1/(M + 1) for p′ > M 1/(M + 1) for p′ > M 1/2

Sample thresholds for M = 6, where 75%/(M + 1) = 10.7%p′ = 2 TI = TE 14.286% 8.333% 8.333%p′ = 6 TI 9.091% 6.250% 2.778%

TE 14.286% 12.500% 13.889%TR arithm. 11.64% 9.38% 8.33%TR geom. 11.40% 8.84% 6.21%

p′ = 8 TI 7.692% 5.556% 2.083%TE 14.286% 14.286% 14.286%TR arithm. 10.99% 9.92% 8.18%TR geom. 10.48% 8.91% 5.46%

the midpoint of the interval where 0 < P (1) < 1. This could mean thearithmetic mean, TR ≈ (TI + TE)/2, or the geometric mean, TR ≈ (TITE)1/2.Both are shown in Table 15.1, for the specific case of M = 6. The outcomesdepend heavily on how many parties run. At given number of parties,d’Hondt has the highest thresholds, and Hare-LR most often has the low-est (except for TE at p′ = 6). However, the inclusion threshold TI = 1/Mp′

for Hare-LR is rather artificial. Even much higher vote shares have a near-zero probability to result in a seat, because this is the range where theAlabama paradox occurs (cf. Chapter 6). Hence the actual 50-50 point forlanding the first seat with Hare-LR is likely to be higher than the meansof inclusion and exclusion thresholds.

What types of constellations lead to threshold outcomes? For Sainte-Laguë, Table 15.2 shows examples for a 6-seat district where a smallparty wins a seat with a vote share barely above the theoretical inclusion

Table 15.2. Sample constellations (in %) where the party shown in bold narrowly winsor narrowly fails to win a seat in a 6-seat district, using the Sainte-Laguë seat allocationformula

p′ = 2 narrowly included 8.34–91.66narrowly excluded 8.33–91.67

p′ = 6 narrowly included 6.255–5 at 18.79 and 6.251–4 at 6.250–68.749narrowly excluded 12.45–4 at 12.50–37.55

p′ = 8 narrowly included 5.56–2 at 5.55–5 at 16.668narrowly excluded 14.27–6 at 14.28–0.05

243


threshold or fails to win a seat with a vote share barely below the theoret-ical exclusion threshold.

The Number of Pertinent Electoral Parties in a District

All these theoretical thresholds of inclusion, as well as some thresholdsof exclusion, depend on the number of parties competing, which is notan institutional input. The sticky part is that, at this point, we do notknow how many parties would typically run in a meaningful way, noreven what we should mean by ‘meaningful’. As pointed out in Chapter14, this number is of interest for various other purposes too, but its likelyvalue is hard to deduce from institutional inputs. Moreover, it is alsohard to measure it empirically, even retroactively. While the number ofseat-winning parties is fairly clear, determination of the ‘number of vote-getting parties’ is complicated by parties that do obtain a few votes butstill cannot be considered serious or pertinent to the process.

Reed (1991, 2003) considered the number of candidates who win ornarrowly fail to win in Japanese SNTV elections. He observed that thenumber of such ‘serious’ or ‘viable’ candidates that run in a district withM seats tends to be M + 1. Gallagher (2001) confirmed the findings for theJapanese second chamber elections of 1998, which used a mix of FPTPand PR.

Cox (1997: 99) presented this M + 1 rule as a direct generalization ofDuverger’s law and tested it in various ways. Depending on the electoralsystem, M + 1 is meant here to be the number of viable candidates ORof viable lists—those winning at least one seat or coming close to win-ning. The distinction between candidates and lists is blurred at low M,where few parties expect to win more than one seat, anyway. In theNetherlands, however, where all 150 assembly seats are determined ina single nationwide district (i.e. M = 150), 151 viable candidates seemsan understatement, while 151 viable parties is clearly an overstatement.Hence the M + 1 rule is not likely to hold for viable candidates and cannotpossibly hold for viable parties in large multi-seat districts.

Reed’s argument is well grounded for SNTV, where parties are penalizedfor running too many candidates and every candidate is effectively com-peting against all others. With List PR, however, the argument might bepresented in a different way. For every seat-winning party, its viable candi-dates could well be its seat-winning candidates (Si) plus one that was closeto winning: Si + 1. With p = M1/2 seat-winning parties (Chapter 8) this

244


means a total of M + p = M + M1/2 viable candidates for the seat-winningparties. To these, one must add an estimate for viable candidates in partiesthat failed to win a single seat.

As for the number of parties, it may well be that only two parties shouldrun in a single-seat district on rational grounds. Then the average thresh-old of representation would be 50 percent. However, this is most often agross overestimate, because more parties tend to run than rational choicetheory proposes. The range of empirical estimates of the average thresholdof representation in FPTP districts extends from 35 percent (Lijphart 1994)to possibly into the low 40s. The unusually pure two-party systems in theUSA and in small island countries are exceptions. Parties beyond two maynot be ‘viable’ in the sense of having a chance to win, but they do have areal impact by lowering the threshold of inclusion for the ‘serious’ parties.Therefore, this ‘irrational’ reality cannot be ignored. Maybe such partiesthat run (including independents) could be called ‘pertinent’ even whenthey are not ‘viable’.

The minimal baseline for the number of pertinent parties (p′) is thenumber of seat-winning parties, p = M1/2. My hunch is that

p′ = M1/2 + 2M1/4,

although I cannot prove it. Here M1/2 represents the number of seat-winning parties, and 2M1/4 is an estimate of the number of further partiesenticed to run in an M-seat district. This formula would yield 3 partiesrunning when M = 1. The corresponding threshold of inclusion wouldbe 33 percent, the threshold of exclusion is definitely 50 percent, andthe arithmetic and geometric means of the two are 41.7 and 40.8 per-cent, respectively—close to the observed average. The numbers of partiesrunning in Finnish districts (Taagepera and Shugart 1989: 119) straddlethe predictions by p′ = M1/2+2M1/4: actual 7.4 versus the predicted 6.5 forM around 9; 7.9 versus 7.6 for M around 14; and 8.6 versus 9.0 for Maround 22. The formula would predict 19 pertinent electoral parties forthe Netherlands.

Returning now to the pertinent candidates, they would include theM seat-winning candidates plus one additional candidate per pertinentparty:

c = M + M1/2 + 2M1/4.

For the Netherlands (M = 150), it would suggest 169 pertinent candidates:the 150 winners, 12 near-winners from the 12 seat-winning parties and

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7 near-winners from 7 parties failing to win a single seat. A reality checkhas not been carried out.

When investigating thresholds of inclusion, such as the ones in Table15.1, one might as well plug in p′ = M1/2 + 2M1/4, rather than usingimpressionistic figures for the number of parties. The results are shownin Table 15.3. For districts with up to 5 seats, the number of pertinentparties would exceed M, while for districts of 6 or more seats it wouldfall short of M. Regardless of how p′ is chosen, d’Hondt has the highestthresholds. Sainte-Laguë has the lowest exclusion thresholds. Hare-LRhas the lowest inclusion thresholds (due to Alabama paradox) and alsothe lowest TR, taken as the geometric mean of TI and TE. At low M,arithmetic means are higher than the geometric by up to 0.15 percent-age points for d’Hondt, up to 1.2 for Sainte-Laguë, and up to 2.3 forHare-LR.

Despite the variations in Table 15.3 at given district magnitude, themeans of the inclusion and exclusion thresholds remain within ±3 per-centage points for all usual List PR formulas, even at low magnitudes. Anoverall estimate for the mean threshold of minimal representation is theaforementioned

T =75%M + 1

. [district-level approximation]

The reasons for this choice are given in Taagepera (1998b). It does notinclude the number of parties and hence could be off the mark, if

Table 15.3. Number of ‘pertinent’ electoral parties (p′) and resulting thresholdsof representation (in %), if p′ = M1/2 + 2M1/4 and TR = (TITE)1/2.

d’Hondt Sainte-Laguë Hare-LR

M p′ TI TE TR TI TE TR TI TE TR

1 3.00 33.3 50.0 40.8 33.3 50.0 40.8 33.3 50.0 40.82 3.79 20.9 33.3 26.3 17.3 33.3 24.0 13.2 33.3 21.03 4.36 15.7 25.0 19.8 12.0 25.0 17.3 7.6 25.0 13.84 4.83 12.8 20.0 16.0 9.2 20.0 13.6 5.2 20.0 10.25 5.23 10.8 16.7 13.4 7.6 16.7 11.2 3.8 16.7 8.06 5.58 9.5 14.3 11.6 6.4 13.5 9.3 3.0 13.7 6.47 5.90 8.4 12.2 10.5 5.6 11.0 7.8 2.4 11.9 5.48 6.19 7.6 11.1 9.2 5.0 10.2 7.1 2.0 10.5 4.69 6.46 6.5 10.0 8.0 4.5 8.7 6.2 1.7 9.4 4.0

10 6.72 6.4 9.1 7.6 4.0 7.5 5.5 1.5 8.5 3.620 8.70 3.6 4.8 4.2 2.1 3.2 2.6 0.6 4.4 1.650 12.39 1.6 2.0 1.8 0.9 1.1 1.0 0.2 1.8 0.5

100 16.32 0.9 1.0 0.9 0.5 0.5 0.5 0.1 0.9 0.2

Note: Cases where p′ > M are shown in bold.

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unusually many or few parties run—as seen in Table 15.1. It is approx-imately halfway between the values of TR for d’Hondt and Sainte-Laguë inTable 15.3, except at M = 1.

For single-seat districts, the formula yields T = 37.5%. It may betoo low. The empirical point of 50-50 probability for winning theseat was around 39 per cent votes in UK 1983 and as high as 49.5percent in the US elections 1970 for the House (Taagepera 1998b).Unpublished work on districts in Canada suggests a figure around 41percent.

Is T = 75%/(M + 1) theoretical or empirical? The starting point consistedof purely theoretical expressions for inclusion and exclusion thresholds,and the denominator (M + 1) harks back to the most widely occurringthreshold of exclusion. The ‘75 percent’ in the numerator, however,came from largely empirical juggling of thresholds for various seat allo-cation formulas. So the expression is an empirical one, but with roots intheory.

Nationwide Threshold of Minimal Representation and Numberof ‘Pertinent’ Electoral Parties

The above results apply to a single district. What about minimal repre-sentation, nationwide? What share of nationwide vote is likely to bring aparty one seat in the assembly when the country is divided into districtsof approximately equal magnitudes?

The crucial factor is the number of electoral districts (E ). Considerthe extreme possibilities for distribution of votes among districts. Thenationwide threshold could approach the district-level threshold, if thevotes are uniformly dispersed. At the other extreme, it could be as lowas district-level threshold divided by the number of districts, if all thenationwide votes are concentrated into one district. Thus the limits onthe nationwide threshold are

Tdistrict/E < Tnationwide < Tdistrict.

Taking the geometric mean of the extremes suggests that the nationwideaverage threshold minimal representation is the district-level thresholddivided by the square root of the number of districts (Taagepera 2002b):

T =75%

(M + 1)E 1/2. [nationwide approximation, parties]

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Here independent candidates differ from small parties, because they runin one district only, so that Tnationwide = Tdistrict/E . Hence

T =75%

[(M + 1)E ]. [nationwide, independents]

This dependence on the number of districts suggests that a small party (orindependent candidate) stands to win a seat with a lesser share of voteswith FPTP than with List PR. This apparent paradox was first pointed outby Grofman (1999). The key word is ‘nationwide’. It takes a larger shareof district votes to win by FPTP than by multi-seat PR, but a smaller shareof nationwide votes.

How should we measure the actual vote level at which a party hasa 50-50 chance of winning a seat? This empirical threshold of nation-wide minimal representation (T) can be defined as follows (Taagepera1989). At this vote share, the number of cases where parties have wonat least one seat with vote shares lower than T equals the numberof cases where they have failed to win seats with vote shares higherthan T. For most countries, this empirical threshold agrees with themodel above within a factor of 2, but UK, Spain, and Imperial Germany(1871–1917) have markedly lower thresholds than predicted (Taagepera2002b).

A lower nationwide threshold would enable more parties to surmountit. In other words, the lower the threshold, the higher the number of seat-winning parties, N0. Let us establish a more specific connection.

In terms of assembly size and district magnitude, the number of districtsis E = S/M. Plugging this value into the equation for parties above, we canexpress the nationwide threshold in terms of seat product MS:

T =75%

(MS)1/2(1 + 1M )

.

The impact of (1 + 1/M) is quite limited. This factor equals 2 when M = 1,and tends toward 1 when M becomes very large (limited by M ≤ S). Theextremes are

T =37.5%(MS)1/2

[for FPTP]

and

T =75%

(MS)1/2[for very largeM].

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The mean number of seat-winning parties was previously shown to beN0 = (MS)1/4. Hence

T =75%

N20

(1 + 1

M

) .

Reversing it yields

N0 =

( 75%T

)1/2

(1 + 1

M

)1/2 .

Accordingly, the average number of seat-winning parties ranges from N0 =(37.5%/T)1/2 for FPTP to N0 = (75%/T)1/2 when the district magnitude ishuge. This number can be determined empirically (with some practicaldifficulties, as pointed out in Chapter 8), so that this prediction can betested.

Figure 15.1 (modified from Taagepera 2002b) graphs N0 against T, bothon logarithmic scales. The predicted zone is the area between the linemarked as M = 1 and the curve marked as M = S. All actual data points

1

10

100

1001010.1

Threshold (T, in %)

Num

ber

of s

eat-

win

ning

par

ties

(N0)

2 x Average Prediction

M > 1

M = 1Legal T

Complex

T = 75%/(N02+1) [M = S]

Predicted Zone

T = 75%/2N0

2

[M = 1]

GER 1971−

GER 1920−

SPASWI

UK

JPNIRE DEN

NET 1988−

USA 1984−USA 1928−

ISR

Average Predicted/2

Figure 15.1. Nationwide number of seat-winning parties vs average threshold ofrepresentation

Reprinted, with modified labels, from Electoral Studies, R. Taagepera, ‘Nationwide Thresholdof Representation’, 383–401, © 2002, Elsevier Ltd., with permission from Elsevier.

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are within a factor of 2 of the center of this zone. Overall, they are evenlydispersed on both sides of the zone, but there are differences among thetypes of electoral systems. Agreement with model is excellent for simplemulti-seat systems, as 10 of 14 points are within the predicted zone. Forsystems with legal thresholds the number of seat-winning parties alwaysexceeds the prediction at given (legal) threshold. For the M = 1 systems,the reverse tends to be the case.

In retrospect, the derivation of the predictive model involves a flaw. Itwas assumed that the nationwide threshold could approach the district-level threshold, if the votes are uniformly dispersed among the districts.This is true only to a point. In a single district, T = 75%/(M + 1) representsthe average threshold of winning one seat with 50-50 probability. If theparty has such a vote share in E districts, it is likely to win one seat inone-half of the districts, meaning a total of E/2 seats rather than a singleone. For multi-seat PR the difference might be a minor one, but in thecase of FPTP, this would mean winning a half of all the seats available!In this light, it is surprising that the M = 1 data in Figure 15.1 fit even aswell as they do.

I have not found a way to refine the model so as to overcome this flaw.One might think of looking for nationwide thresholds of inclusion andexclusion and take their mean. These thresholds have been calculated(Taagepera 1998c), but they diverge so widely that their averages supplyno clue about the location of the vote share with 50-50 probability forlanding the first seat.

How many ‘pertinent’ parties would run nationwide? The tentativeformula p′ = M1/2 + 2M1/4 applies to districts. Extension to the nationwidelevel might involve the square root of the number of districts or theseat product, but the work remains to be done. A phantom number ofvote-getting parties (NV0) emerged in Chapter 14. One might test howextensions of the formula proposed in the previous section compare withthis phantom number. Both might be inserted into threshold formulas, tosee whether they make sense.

The Threshold of Breaking Even

The break-even point b was defined in Chapter 5 as the vote share forwhich fractional seat shares equal vote shares so that the advantage ratio(a) is 1. This point can be seen as another threshold. Attempts to builda logical model for the break-even point (Taagepera and Shugart 1989:88–91, 270) yielded b = 100%/NV, but most actual values are below this

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prediction. The flaw in the model was that it used the effective number ofelectoral parties rather than some estimate of the total number of partiesthat run in a ‘serious’ way. The model could now be revised, replacingthe effective number by the phantom number of vote-getting partiesdeduced in Chapter 14. This would connect the break-even point to theseat product, indirectly. Testing remains to be done.

However, the significance of the break-even point varies. It is clear forFPTP systems, with profiles such as shown in Figure 5.2. Here the averageproportionality profile curve intersects the horizontal line a = 1 at a sharpangle. Immediately below the break-even point, a party is heavily short-changed in terms of seats per votes, so this point matters. In contrast, theshift is more gradual and diffuse for multi-seat PR (e.g. Figure 5.4), so thatfalling somewhat below the formal break-even point is of little concern.The break-even point loses any meaning for highly dispersed profilessuch as France (Two-Rounds, Figure 5.3). Still, at least for FPTP, betterinstitutions-based prediction of the threshold of breaking even would beof interest.

The Threshold of Absolute Majority

In the calculation of inclusion and exclusion thresholds in districts, onecan go beyond minimal representation and calculate such thresholds forwinning two seats, and so on. One can also calculate the minimal voteshares at which a half of the seats could be won, and the maximal sharesat which such majority of seats could still elude a party.

Most thorough work in this direction has been carried out by RubénRuiz Rufino (2005). That study develops general equations that allow oneto calculate the inclusion and exclusion thresholds ranging from minimalrepresentation to absolute majority, for different allocation formulas. Inline with the power and generality of the equations, the number ofparameters to be fed in becomes large, and I hope a more user-friendlyversion can be worked out. Ruiz Rufino (2005) uses the effective numberof parties as an input variable, but with the help of N = (MS)1/6, purelyinstitutional expressions could be established.

Extension to nationwide vote shares is conceivable. For FPTP, it haspopped up spontaneously in the previous section. Here ‘minimal repre-sentation’ in a district implies exclusive representation in that district. Tothe extent that T = 75%/(M + 1) = 37.5% expresses the mean thresholdof minimal representation a single district, it also expresses the meanthreshold of absolute majority nationwide.

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The Paradoxical Relationships Between the Thresholds ofMinimal Representation, Breaking Even, andReaching Absolute Majority

It might seem evident that it would take fewer votes to achieve minimalrepresentation than to break even, and fewer votes to break even than toachieve absolute majority, but it can be misleading. Actual relationshipscan be tricky at district level, and even more so at the district–nationwideinterface. In one single-seat district, the minimal representation of oneseat also means overrepresentation, absolute majority, and 100 percentmajority. Relationships are less clear in multi-seat districts, and contrastsare less marked, but they still matter—and they are also harder to detect.

At district level, it may come as a surprise that not only inclusion thresh-olds but also exclusion thresholds of minimal representation mean over-representation, for the usual PR formulas. Indeed, the highest exclusionthreshold we observed was for d’Hondt: 1/(M + 1). This vote share assuresa share 1/M of the seats, meaning an advantage ratio a = (M + 1)/M =1 + (1/M), which exceeds 1. This means that, within a single district, a partywith only one seat is often overrepresented—unless we go to allocation formu-las less proportional than d’Hondt. Transfer from district to nationwidelevel is far from obvious. It was pointed out that it can take a smaller shareof nationwide votes to win an assembly seat with FPTP than with PR.

According to N0 = (MS)1/4, it is as easy to win a seat in a 625-seat assem-bly elected by FPTP than in an assembly of 25 elected by nationwide PR.There is a difference, however, in the degree of power that such minimalrepresentation involves. Having 1 seat in an assembly of 25 may make aparty a minor but serious player in cabinet formation, while having 1 seatof 625 amounts to very little. Thus the substantive meaning of minimalrepresentation depends itself on assembly size.

Legal Thresholds and their Concordance withEffective Thresholds

Some electoral systems impose legal thresholds on representation, such asrequiring 5 percent of votes before a party can participate in allocationof seats. Transition is sharp, from no seats to nearly breaking even oreven becoming overrepresented (see Figure 5.5). In contrast to the sharp‘vertical’ barrier (step function) imposed by the legal threshold, the seatproduct imposes a gradual ‘tilted’ barrier zone that no single one of the

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aforementioned types of thresholds can characterize completely. For legalthresholds, a single number mostly says it all. The tilted barrier raisedby district magnitude is all too easily assumed to be analogous to a legalthreshold, but one must exert caution.

Confusion is enhanced by failing to distinguish between district-leveland nationwide restrictions. Legal thresholds can be applied at districtlevel (Spain) or nationwide (Germany)—and the difference matters. Con-sider a party with 4.9 percent nationwide votes. If all seats are allocatedin districts of more than 10 seats, then a district-level legal thresholdof 5 percent most likely would allow it to win seats in some districts.In contrast, a nationwide 5 percent legal threshold would completelyblock it.

Previous analyses (including Taagepera and Shugart 1989; Lijphart1994) often have confused the two levels, treating nationwide legal thresh-olds as equivalent to district-level effective thresholds imposed by districtmagnitude, such as T = 75%/(M + 1). Before one juxtaposes the two, onemust correct for the effect of the number of districts (E ). As noted earlier,the effective nationwide threshold set by a district-level effective thresh-old T is around T/E 1/2—and most likely even less for FPTP.


This chapter has more questions, cautionary notes, and suggestions thanfirm answers. An operational definition of the number of parties that runseriously (or at least semi-seriously) is of interest in scholarly discussion.It impinges on various aspects of thresholds of representation. The worldtrend in democracies may be toward greater inclusion, as claimed byscholars such as Colomer (2004b) and Lijphart (1999). If so, then thethreshold of minimal nationwide representation could be a formal yard-stick of interest to political practitioners.

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16

Seat Allocation in Federal SecondChambers and the Assemblies of theEuropean Union


� Here the implications for institutional engineering are major, becausethe institutional structure of the European Union is as yet unsettled.Making use of the logical models presented and tested here could saveappreciable political wrangling and might improve the outcomes.

� The number of seats in the European Parliament seems to grope towardthe cube root of total population, which is the empirical and logicalnorm for national assemblies. This ‘cube root law of assembly sizes’could be made the official norm: The number of seats equals the cuberoot of the EU population.

� The total voting weights for qualified majority voting in the Council ofthe European Union seem to grope toward a logically founded formulafor balanced representation of total population and of member states.This formula could be made the official norm: The total of votingweights equals the sixth root of the EU population × the square root ofthe number of member states.

� Allocation of EP seats and CEU voting weights among the membersof EU has for 40 years closely approximated the one predicted by a‘minority enhancement equation’ solely on the basis of the numberand populations of member states plus the total number of seats or vot-ing weights. This logically founded formula could be made the officialnorm. The formula is more complex than for total size of assemblies,

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but it is easily workable with a computer program or even a pocketcalculator.

� All these models may be of use for some other supranational bodies andfederal second chambers.

This chapter describes a promising spin-off from the law of minorityattrition (Chapter 13), even while it is marginal for predicting party sizesas such. A predictive model is constructed and tested for allocation ofseats among member countries in the EP and the Council of the Euro-pean Union (CEU). The model is based on nothing but the constraintsimposed by the total population, the number of seats (or voting weights)in the given body and the number and populations of member countries.Applying this logically based model could save on negotiation time whennegotiators know that the eventual outcome will be close to fitting themodel, anyway. This would be the major payoff for this chapter.

The problem of seat allocation in the European bodies is part of amore general one, that of seat allocation in supranational entities suchas the United Nations (UN) as well as in national second chambers, atleast those that somehow reflect territorial subunits. It was recognizeda long time ago that allocating seats to territorial units on the basis oftheir populations represents a problem mathematically similar to that ofallocating seats to parties on the basis of votes. The latter is approximatedby the law of minority attrition. However, mathematical similarity islimited in one respect.

Small parties have no inherent right to even minimal representation. Incontrast, territorial subunits in a federal second chamber may have sucha right, simply by being a distinct subunit. The same goes for membercountries in supranational entities such as the EU and international orga-nizations. It follows that the number of constituent parts must be workedinto the minority attrition equation, along with their populations and thetotal number of seats, before this equation can be applied to allocation ofseats in such entities. Indeed, minority attrition must be reversed intominority enhancement.

As applied to seats and votes, the law of minority attrition involves thetotal number of seats in the assembly as a given. The basis for determiningthis total number should be investigated in the first place. For the first oronly chambers of national assemblies that represent individuals, the cubelaw of assembly sizes applies, with some reservations (cf. Chapter 12).To the extent that the EP can be considered the analog of national

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Seat Allocation in EU Assemblies

parliaments, one may well ask how its total size relates to the cube root ofpopulation. Both the EP and the CEU could also be argued to have morein common with the second chambers of national parliaments. So thedeterminants of the sizes of second chambers should also be considered,especially those that are somehow tied to territorial subunits.

We should know how many seats are to be allocated before tryingto allocate them. Therefore, the size question will be addressed first.Thereafter, a minority enhancement equation will be developed on thebasis of the law of minority attrition. It will be tested with EU data.Application to federal second chambers and some supranational bodiesis briefly discussed.

The Size of Subunit-Based Second Chambers

Although the second chambers do not pretend to represent the popu-lation as such, larger countries may be expected to have larger secondchambers. When the second chambers are based on territorial subunits,their size might also depend on the number of subunits. Taagepera andRecchia (2002) compiled the populations (P ) and the sizes of first andsecond chambers (F and S, respectively) for 28 contemporary countrieswhere federal or other territorial subunits form the basis of election orappointment of at least part of the second chamber—this part beingdesignated as S ′.

The sizes of assemblies and populations can take only positive values,and hence a linear relationship of logarithms is the simplest relationshipto be expected (cf. Taagepera 2008), if there is a relationship at all.For these 28 countries, the first chamber sizes (F ) roughly follow theaforementioned cube root law but fall to about one-half of that value atvery low populations—the general pattern observed in Chapter 12. Linearregression of log F on log P yields R2 = 0.81. For second chambers, thebest-fit line log S versus log P corresponds to S = 0.46P 0.304 (R2 = 0.56), orwith some rounding off, S = 0.48P 0.30. The exponent 0.30 is slightly lowerthan the 0.33 in the cube root law (Taagepera and Recchia 2002). The bestfit line of log S on log F , the size of the first chamber, corresponds toS = 1.00F 0.786 (R2 = 0.68). Thus, second chambers tend to be smaller thanfirst chambers. The values of R2 suggest that the second chamber sizesmay be affected by population through the first chamber size rather thandirectly.

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When second chambers are selected on the basis of T territorial sub-units, their sizes are subject to the following limiting constraints.

(1) If all T subunits are to be represented, the chamber must have atleast T seats. This is the lower limit on S.

(2) If people were represented as individuals, the second chamberswould be akin to first chambers and would be expected to have thesame size (F ). This is the upper limit on S.

In the absence of any further knowledge, our best guess would be thegeometric mean of the two limits, as there is no reason to stress the impactof one over the other (cf. Taagepera 2008). So the predictive model is

S = (FT)1/2.

The actual best-fit exponent was found to be 0.506, very close to theexpected 0.500. Thus the first chamber size and the number of subunitsseem, indeed, to influence the size of the second chamber to roughly thesame degree. For further testing, Taagepera and Recchia (2002) consideredonly the number of those second chamber seats allocated on the basis ofsubunits (S ′), excluding nonregional appointments. Figure 16.1 shows theactual number of such seats graphed against the expected size (FT)1/2. Weexpect S ′ = 1.00(FT)1/2 + 0. The best-fit line is S ′ = 0.96(FT)1/2 + 12, withR2 = 0.52 between S ′ and (FT)1/2. This line is seen to be extremely close toexpected one.

Note that R2 is not increased, compared to fitting with first chambersize alone. The difference is that the equation S = 1.00F 0.786 was purelypostdictive, with no explanation given for why the exponent shouldbe around 0.786 rather than something else. Here, in contrast, we havea predictive model, posited before any input of data, on very generalgrounds. It is confirmed by the agreement between the predicted andactual best-fit lines, regardless of the degree of scatter around them (cf.Taagepera 2008).

Now combine the cube root law for first or only chambers, F = P 1/3,with S = (FT)1/2. It leads to

S = P 1/6T1/2.

If a second chamber or supranational assembly is selected purely onthe basis of subunits or member states, balanced representation of totalpopulation as well as of member states may call for this total size: sixth

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Best-fit lin

e

Model

0

0

100

Num

ber

of s

ubun

it-ba

sed

seco

nd c

ham

ber

seat

s

200

300

400

100

Geometric mean of F and number of subunits

200 300

Figure 16.1. Number of subunit-based second chamber seats vs. the geometricmean of first chamber size and the number of subunits

Source: Reprinted from R. Taagepera and S. Recchia, ‘The Size of Second Chambers andEuropean Assemblies’, European Journal of Political Research, 41: 185–205, © 2002 EuropeanConsortium for Political Research, with permission from ECPR.

root of the population × square root of the number of subunits. Thismodel has not yet been fully tested.

The Sizes of the European Parliament and the CEU

The patterns observed for the first and second chambers of national par-liaments will now be applied, with obvious caution and reservations, totwo-tiered supranational assemblies. Among the various ancillary bodiesof the EU, the EP and the smaller CEU stand out. How would theirsizes look, if these were the sizes of national first and second chambers,respectively? The CEU uses two forms of voting: qualified majority voting(QMV), where larger countries have larger voting weights, and unanimityvoting, where all countries have equal votes. Only the QMV aspect will beconsidered, so that CEU size is shorthand for the total of voting weights.

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Table 16.1. The number of seats in the European Parliament—predictionby the cube root law and the actual number

Population P 1/3 Actual Actual as(million) seats % of P 1/3

1964 173 557 142 251979 277 652 410 631989 341 699 518 741994 348 703 567 811995 369 717 626 872004 454 769 785 102

Table 16.1 shows the population of the EU and the number of seatsin the EP (based on Taagepera and Hosli 2006). The cube root of thepopulation is compared to the assembly size. The initial size of the EPamounted to only one-quarter of the cube root of population represented,but it gradually approached the cube root with each expansion of theUnion and reached it by 2004. This asymptotic growth is reminiscent ofthe growth of the US House, which started in 1790 at one-third of thecube root of the population but caught up with the cube root within 40years (Taagepera and Shugart 1989: 175).

The CEU followed a rather similar pattern of asymptotically approach-ing the level of S = P 1/6T1/2, from 1958 to 1995, as seen in Table 16.2(based on data in Taagepera and Hosli 2006). However, the Treaty ofNice proposed a sharp change. The figure for 2004 in Table 16.2 refersto the number proposed in the EU Constitutional Treaty, not yet ratified.It would send the size of the CEU through the ceiling, exceeding P 1/6T1/2

by more than 300 percent.

Table 16.2. Total voting weights in the Council of the EuropeanUnion—prediction by S = P 1/6T 1/2 and actual

Population Members P 1/6T 1/2 Actual Actual as %(P , million) (T ) of P 1/6T 1/2

1958 172 6 58 17 291973 257 9 76 58 761981 288 10 81 63 741986 323 12 91 76 841995 369 15 103 87 84(2004) 454 25 140 (456) (326)

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The Minority Enhancement Equation for Seat Allocation inFederal Second Chambers and EU Bodies

Regardless of whether the total number of seats shows any regularity,those seats must be somehow allocated among the territorial subunits.Two distinct norms could be used. In national first or only chambers, thenorm is ‘a person is a person’, and subunits receive seats in proportionto their populations (with possibly a minimum of one seat per subunitstipulated). In international bodies and some federal second chambers,the prevailing norm is ‘a state is a state’, and each of them receives thesame number of seats. This is the case for the Assembly of the UN and forthe US Senate.

Some supranational bodies and federal second chambers try to accom-modate both of these conflicting norms, allocating larger states moreseats, but still short of their proportional due. This third alternative isused for the Canadian second chamber, for the EP, and also for the votingweights in the CEU. Can we make logical predictions for the outcome ofa compromise that tries to accommodate both norms, representation ofindividuals and of states? It turns out we can.

The starting point is the law of minority attrition (Chapter 13) that,among other applications, relates the seats (Si , S j ) of two parties, i and j ,to their votes (Vi , Vj ). Its first component is Si = SVn

i /�Vnk , where S is the

total number of seats (�Sk = S). Henri Theil (1969) offered formal proofthat the transformation of vote shares into seat shares must follow thisformat, because, among all functions of the form Si/S = f (Vi/Vv j ), this isthe only one that does not lead to inconsistencies in the presence of morethan two parties.

Theil (1969) also pointed out that the same general formula may beapplied to seat allocation in international institutions whenever onewishes to overrepresent smaller states. It suffices to replace votes (Vi ) bythe populations (Pi ) of the countries:

Si =SPn

i

�P nk

.

Here the summation is over N countries and an exponent smaller than1 must be used. The value n = 0 would provide each country with thesame number of seats, regardless of population size (‘a state is a state’),while n = 1 represents countries in direct proportion to their populations(‘a person is a person’).

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The value n = 0.5, halfway between these extremes, may seem a bal-anced compromise between the two norms. It has a mathematical ratio-nale, was recommended by Theil (1969) and roughly fits the distributionof CEU voting weights. However, the distribution of seats in the EP fitsonly when n is around 0.7. Why would the EU use different criteria forthe two bodies? It is that the total number of seats matters.

Indeed, if the present EU, with 25 members (as of 2006), had anassembly of only 25 seats, the only acceptable way to allocate themwould be to allocate each country one seat—which would correspond ton = 0 in the equation above. On the other hand, if the assembly werehuge, appreciable proportionality to populations of countries could beafforded, while still giving representation even to the smallest memberstates. Taagepera and Hosli (2006) present a modification of the law ofminority attrition where they express the disproportionality exponent nin terms of not only total population (P ) and total number of seats (S), butalso the number of member states (T). The model is presented in chapterappendix. The result is

n =1

log S − 1log T

1log P − 1

log T

.

This is a more complex expression than what we obtained in Chapter 15for seats and votes: n = log V/ log S. If we applied the latter to memberstates, with populations replacing votes (n = log P/ log S), the small stateswould be left with no representation, like small parties are. But here wewant to overrepresent them, population-wise, because they are distinctmembers. This is why the number of members must be brought in, and itcomplicates the expression for n. The combination of this expression withSi = SPn

i /�P nk can be called the minority enhancement equation. Once P ,

S, and T are given, the number of seats for any country i can be calculatedon the basis of its population, Pi .

Testing the Minority Enhancement Equation with theParliament and the Council of the European Union

Taagepera and Hosli (2006) tabulate the predicted and actual votingweights or seats in the CEU and EP. Table 16.3 shows the considerablechanges in these bodies, over 30 years. Yet the calculated values of expo-nent n remained quite steady: 0.47 ± 0.06 for CEU and 0.69 ± 0.03 for EP.

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Table 16.3. Characteristics of seat allocations in the Councilof the European Union and the European Parliament

CEU EP

Time period 1958–95 1964–95Number of members 6–15 SamePopulation (million) 172–387 SameSeats/voting weights 17–87 142–626Range of exponent n 0.41–0.52 0.67–0.72Shares misallocatedby predictive model 0–5.2% 2.3–5.1%

Source: As analyzed in Taagepera and Hosli (2006).

The predicted seats were rounded to integers, using the equivalent of theLargest Remainders approach, so as to preserve the total number of seats.The predictive model misallocated only 2.6 percent of the CEU votingweights, on the average, and 3.7 percent of the EP seats. Three empiricalpostdictive data fits by other authors, reviewed in Taagepera and Hosli(2006), could hardly do better than the predictive model.

Figure 16.2 shows the degree of fit in 1995, a year with average mis-allocations (2.3 percent for CEU, 3.4 for EP). The theoretical lines areshown, and they agree with data points so well that they might bemistaken for statistical best-fit lines. They are not! The lines shown arepredictions by a logical model based solely on the number of countries,total seats (or voting weight units), and country populations. The onlyconsistent deviation is an excess for the smallest member, Luxembourg—and it prevails throughout the entire period.

Data and graphs for other time periods are shown in Taagepera andHosli (2006). These are very robust results, over 30 years. The contrastbetween EP and CEU confirms that, with the same number and totalpopulation of member states, the size of the assembly makes a differ-ence: The smaller the body, the more disproportionate the representation ofpopulations is, in favor of smaller member states. No arbitrary value of thedisproportionality exponent, such as n = 0.5, can fit all sizes.

But how come that the EU institutions conform to the minorityenhancement equation when its decision-makers have not been awareof its existence? Could the negotiators or arbitrators in a supranationalorganization be mathematiciens malgré soi? To some extent, this is so,indeed.

From the very beginning, the EU seemed to respect the followingtwo ground rules. First, even the smallest member must have nonzero

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1.0

10.0

100.0

0.1 1.0 10.0 100.0

European Parliament(Theoretical Slope = 0.672)

Aut

Council of the EU(Theoretical Slope = 0.456)

Sea

ts (

Vot

ing

Wei

ghts

)

Member State Population [in Millions]

Lux

IreFin Den

Swe

Por GreBel

Net

GerUKFra

Ita

Spa

Figure 16.2. Seat and voting weight distribution in the European Parliament andthe Council of the EU in 1995—predictive model and actual values

Source: Taagepera and Hosli (2006). Reprinted from Political Studies, 54, R. Taagepera andM. Hosli, ‘National Representation in International Organizations’, 370–98, © 2006 PoliticalStudies Association, with permission from the Political Studies Association.

representation. Second, a more populous state must not have less rep-resentation than a less populous one. These ground rules are closeto two constraints of the three on which the exponent n is basedin the model (see chapter appendix). In other words, the predictivemodel is not artificial but derives largely from very simple and practicalprinciples.

So the larger states should have more representation. But how muchmore should they have, compared to the smaller? Here the modelstarts out from the norm that, if the given institution offers only oneposition, it would go to the largest member. Somehow, this highlydebatable and apparently irrelevant constraint is conducive to the sameoutcome as the complex haggling during 40 years of constructingthe EU.

The Treaty of Nice and subsequent negotiations broke, for the firsttime in EU history, the ground rule that a more populous state mustnot have less representation than a less populous one. Table 16.4 showsexamples of incongruence in seat allocations for the EP elections of 2004.The new members are shown in bold, and several of them are clearly

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Table 16.4. Incongruent seat allocations for theEuropean Parliament elections of 2004, comparedto population in 2000

Country Population Seats(million)

Slovakia 5.4 14Denmark 5.3 16Finland 5.2 16Ireland 3.8 15Lithuania 3.7 13. . . .Estonia 1.4 6Cyprus 0.8 6Luxembourg 0.4 6Malta 0.4 4

underrepresented, compared to old members with similar populations.The populations changed little, from 2000 to 2004. Rather, the EU hadimplicitly introduced second-class citizenship for new members. Alloca-tions of voting weights in the CEU underwent similar pressures.

With some of the basic ground rules bent, it is not surprising that thegap between the predictions of the minority enhancement equation andthe actual seats or voting weights has increased moderately since theTreaty of Nice—as they would for any postdictive curve fitting, too. Inparticular, the smallest member states (Cyprus, Luxembourg, and Malta)are heavily overrepresented, presumably thanks to the founding memberstatus of Luxembourg that pulls the others along (see Taagepera and Hosli2006: Figure 4).

Rather than reducing the importance of minority enhancement equa-tion as a logically based norm, the post-Nice developments show thatthe expanding EU needs to protect itself against ad hoc haggling. Allcountries have explicit seat allocation rules for parties in national parlia-ments, and federal countries have such rules for seat allocation amongfederal subunits. This seems so self-evident that some reviewers forTaagepera and Hosli (2006) found it hard to believe that EU does nothave explicit rules for seat allocation among member states. If it wantsto avoid further haggling, the EU needs firmer ground rules. The minor-ity enhancement equation offers a consistent basis on which furtherstipulations could be grafted, depending on commonly accepted specialneeds.

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Application of the Minority Enhancement Equation to FederalSecond Chambers and United Nations

A systematic analysis of federal second chambers remains to be done, soas to distinguish those that follow pure principles of either ‘a state is astate’ (USA) or ‘a person is a person’ (Austria) from those that try to followa middle course (Canada). The interesting question is, how close the lattercome to the minority enhancement equation—and could they profit fromcoming even closer.

The UN has followed a two-pronged approach. Its General Assemblytreats all countries as equals—as if it were a second chamber. In contrast,Security Council assigns more representation to its permanent members,which are among the most populous. In view of formal veto power andrealistic ability to use it, it could be argued that the USA yields power atleast commensurate with its share of the world population. In this respect,Security Council appears more akin to a national first chamber—contraryto what its relationship to the General Assembly might look at the firstglance.

One could play at determining the sizes of two assemblies that cor-respond to the world population, using the models F = P 1/3 and S =P 1/6T1/2. One could then allocate these seats according to the popula-tions of member countries of the UN, using the minority enhancementequation. This is left as an exercise for those interested.


The models presented and tested here are of direct interest for institu-tional engineering in the EU, because its institutional structure is as yetunsettled. Making use of these models could save appreciable politicalwrangling and improve the outcomes. The EU needs firmer ground rulesfor allocation of seats in the EP and voting weights in the CEU. It couldalso profit from ground rules for the sizes of these bodies. The cube rootlaw, its extension to subunit-based chambers, and the minority enhance-ment equation offer a consistent basis.

This is not a take-it-or-leave-it proposition. The models presented couldbe taken as a starting point, on which further stipulations could begrafted, depending on perceived special needs. For instance, supposeone desires even stronger overrepresentation of the smallest members,such as Luxembourg-sized countries in the EU. It could be obtained by

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adding 0.5 or 1 million to the populations of all countries. The additionwould hardly affect large or even median countries while boosting thetiniest.

All this is outside the central focus of this book—predicting party sizes.Yet it is connected methodologically. Assembly sizes affect party size dis-tribution, through the seat product MS. Analogous determinants operatein sub- and supranational bodies. The law of minority attrition enables usto connect party seat shares to their vote shares. One can easily modifyit to fit situations where even the smallest components are entitled torepresentation. Given the potential importance for institutional engineer-ing, this is a major spin-off from electoral and party studies. The reversecan also come about. The additional insights gathered from analysis ofsupranational and federal bodies might, in turn, prove useful in electoraland party studies.


Construction of the minority enhancement equation

Previous equation n = log V/ log S (Chapter 14) inspires us to look for logicalconstraints on the value of n in the case of international bodies and federalsecond chambers. The idea behind this equation is that the total numbers ofseats and votes determine the disproportionality exponent n. In the case of sub-and supranational bodies, the total number of seats (S) remains a factor, totalpopulation (P ) easily substitutes for total votes, but the number of territorial unitsto be represented (T) also matters. The way P , S, and T enter is subject to logicalconstraints. This means that the function n = f (T, S, P ) must apply to extremeor other special cases, if it is to be general. Three such logical constraints can beposited (Taagepera and Hosli 2006).

I. If the number of seats to be allocated matches the number of territorial units(S = T), then the only reasonable way to distribute them is to give each unit oneseat. Mathematically, this means f (S = T, P ) = 0. Indeed, if n = 0 is entered intoSi = SPn

i /�P nk , it becomes Si = S/T = 1, given that a0 = 1 for any finite number a.

Here the difference between party representation and territorial representationbecomes important. When seats are scarce, one could easily leave a tiny partywith no seat and provide a large party with several seats. In the case of territorialunits, however, even the tiniest member state is entitled to one seat, before eventhe largest country can receive a second one. A state is a state.

II. If the number of seats equals total population (S = P ), then every personshould get a seat, and we would have perfectly PR of populations. A person is a

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person. Mathematically, f (T, S = P ) = 1, because n = 1 in Si = SPni /�P n

k leads to Si =SPi/P and hence Si/S = Pi/P .

III. If there is only one seat to be allocated, the largest member has the strongestclaim. Equation Si = SPn

i /�P nk yields this result when n tends toward ∞. This

implies that f (T, S = 1, P ) → ∞. This constraint is more debatable than the others.Why not rotate the seat among the members, as is the case for the presidingcountry of the EU? Rotation would mean return to the norm ‘a state is a state’.Visibly, this norm applies to the EU presidency, but not to the Council of the EUand the EP. We need a third stipulation, distinct from the two previous ones, tospecify a model in three variables (S, P, and T)—and f (T, S = P ) = 1 works. But itwould be nice if it could be grounded in a different way.

The two first constraints are satisfied whenever n = f (T, S, P ) has the form

n =g(S) − g(T)g(P ) − g(T)

,

where g(x) means an identical transformation of the variables involved. Morecomplex expressions also satisfy the constraints. Occam’s razor principle holdsthat the simplest form should be chosen, unless nature imposes more complexforms. Note that n = 0.5, the value proposed by Theil (1969) implies that g(S) isthe arithmetic mean of g(T) and g(P ):

g(S) =g(T) + g(P )

2→ n = 0.5.

The third constraint is satisfied when g(x) is a function such that g(1) → ∞.Again, there are other ways to satisfy the three constraints, but this one is thesimplest. The simplest functions g(x) leading to g(1) → ∞ are g(x) = 1/(x − 1) andg(x) = 1/ log x. For reasons analogous to those presented previously in appendixto Chapter 13, the logarithmic expression is to be preferred. Empirically, g(x) =1/(x − 1) would predict values of n ranging from 0.69 to 0.95 for CEU, while theactual values of n have been under 0.6 ever since 1958. For EP, the predictionswould be above 0.96, while the actual values have remained under 0.75. Thus, thefunction g(x) = 1/(x − 1) clearly cannot explain the actual seat or voting weightallocations, while the excellent fit resulting from g(x) = 1/ log x can be seen inprevious Figure 16.2.

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17

What Can We Expect fromElectoral Laws?


� Expect electoral laws to have an effect on party system, governmentstability, and other features.

� Expect some ability to fine-tune simple electoral laws to desired goals,but be cautious.

� Do not expect any ability to tailor complex electoral laws to desiredgoals.

� Consider marginal adjustments rather than flipping to completely dif-ferent electoral laws.

� Keep the same electoral laws for at least three elections before changingthem.

What do we know about electoral systems worth conveying to polit-ical practitioners intent on creating or revising an electoral system? Iwill try to answer this question, keeping in mind that some change inthe party system is usually at least one of the goals. The next ques-tion is how can we generate further usable knowledge? I will outlinea broader agenda, going beyond macro-level models for simple leg-islative elections. This agenda involves micro-level models, more com-plex systems, and elections beyond the legislative. Chapter appendixoffers data in a form where the effect of the seat product MS has beenremoved (‘controlled for’) so that detection of other factors may beeasier.

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How Much Do We Know?

If ethnic, religious, or social groups really insist on slaughtering eachother, then no electoral system can prevent them. One cannot expect thatmuch from electoral laws. But in borderline cases, some systems may workbetter than some others. In homogeneous societies too, electoral systemscan have detectable effects on policy outputs and, over the long run, onpolitical culture. Decision-makers try to choose an electoral system thatfits the existing political culture or nudges it in a desired direction, butthey do not always succeed. Fiji is a recent example where advice bypolitical scientists clashed.

Fiji is ethnically split among original Fijians and Indo-Fijians whom theBritish colonial rulers brought in as laborers. The conflict peaked with amilitary coup in 1987. In 1996, a Constitution Review Commission aimedat consensus-building among competing ethnic groups and proposedAlternative Vote in a mix of communal and ethnically heterogeneoussingle-seat districts. It was adopted with minor changes and was used inthe elections of 1999 and 2001. Alternative Vote had been proposed byDonald Horowitz (1985, 2002, 2006), who maintained that it would leadto parties courting the second choice votes of centrist voters and henceto softening the extremist rhetoric. Fiji discarded the contrary advice byArend Lijphart (1977, 2002), who proposed a ‘consociational’ approach,with closed-list PR, group autonomy, and power sharing in multipartycoalitions.

Alternative Vote in Fiji is now considered a failure (Fraenkel andGrofman 2005, 2006a, 2006b; Stockwell 2005). Compared to multi-seatPR, single-seat districts are bound to increase the number of frustratedvoters who do not get their first preferences elected. Alternative Votecould mitigate it only if the existing political culture favors compromiserather than seeing it as more dishonorable than defeat. ContemporaryWesterners may take the existence of a culture of compromise too muchfor granted. In Fiji, disproportionality between the seat and vote sharesbecame huge. The voters’ second choices were manipulated by partyleaders, giving an edge to extremist parties at the expense of the moderateones. Like BC, AV might be a good system ‘only for honest men’ (cf. Chap-ter 3), meaning those who do not play strategic games in a divided society.

In nearby New Caledonia, ethnic strife also interrupted democratizationwith election boycotts and violent clashes from 1984 on. The existingList PR was complemented in 1998 by mandatory power sharing, decen-tralization, and improved access to voting outside the capital area. It is

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What Can We Expect from Electoral Laws?

not an unqualified success, but leaders of opposing parties have servedtogether in coalition governments and in 2004 centrist parties triumphed,by shifting the agenda away from ethnicity (Fraenkel and Grofman 2005;Maclellan 2005).

A sample of two countries is too small to draw conclusions. New Caledo-nia may have been lucky and Fiji unlucky. The consociational approachstill has to prove itself in Kosovo (Taylor 2005). But some systems maybe more failure prone than others. Single-seat districts may provide fewerviable mechanisms for consolidation of democracy (Birch 2005).

We have plenty of empirical data and precedents. Half a century ago,W.J.M. MacKenzie (1954: 54) maintained that ‘The only thing that can bepredicted with certainty about the export of elections is that an electoralsystem will not work in the same way in its new settings as in its old.’ Is itstill true? In Harry Eckstein’s terminology (1966, 1998), if institutions arenot sufficiently congruent with the existing political culture, they fail oryield unexpected results. A country can do pretty unexpected things evento straightforward electoral laws (cf. A Wuffle, quote at the start of PartIII).

Extrapolation into the future is risky even regarding the same country.The ‘cube law’ fitted the British elections up to the 1960s, but then thedisproportionality exponent shifted from 3 toward 2 and then 1.5 in the1970s (cf. Chapter 13). The importance of third parties has also increasedin the UK, without any marked change in the electoral laws.

At the level of recipes based on single country precedents, politicalscientists have little more to offer than historians or journalists. Theyhave an edge when they go comparative, detecting empirical regularitieswhile also including cautionary case studies. Even so, empirically basedregularities depend on ‘all other things being the same’. How can weknow which things must remain the same, if we do not know what causesthe observed relationship? How could we predict whether the cube rulecontinues to hold in Britain without knowing its cause?

This is where predictive models based on logical considerations (rangingfrom general to specific) offer more certainty. They help us know whichthings must be the same, for the observed regularities to hold, and towhat extent changes in inputs alter the outputs. They help but do notguarantee. The law of minority attrition explains the exponent 3 in thecube rule by interaction between the number of voters and number ofseats. But these numbers changed little in the UK at the time the exponentdropped. We might presume that the very knowledge about the seat–vote relationships enabled parties to counteract the natural tendencies

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by concentration of resources into the most promising districts. But whatis then left of the predictive ability of the minority attrition equation?

Such questions must be faced. If they cannot be immediately answered,the choice is between reverting to pure empiricism—or even to an ‘any-thing can happen’ attitude—or keeping on looking for further logicallygrounded explanations. I favor the latter, and we are making headway. Itmay take more than a few exceptions to sink a logically grounded model.When physicists noticed that energy did not seem conserved in certainsubatomic processes, they did not discard the principle of conservationof energy but posited the existence of an as yet undetectable particle, theneutrino. It took decades before more direct evidence for the existence ofthis phantom particle could be found.

Designing Electoral Laws and Waiting for aParty System to Evolve

The devil is in the detail. If you clutter electoral laws with details thatcould be avoided, then the devils of unexpected consequences will havea field day. If you keep it simple, you will have some ability to predict.If simple electoral systems produce undesirable outcomes in the givencultural context, we may at least know in retrospect what caused them,and then we can try incremental changes.

When electoral systems are made complex, any degree of rationalpredictability vanishes. Incremental adjustments to unwelcome surprisesbecome impossible when we cannot even be sure which component is atfault. Hence attempts at correction may make it worse.

For simple electoral systems, the corrective ability should not be dis-missed, nor should it be overestimated. Excessive optimism would onlylead to disappointment and complete dismissal. Even when a newlydemocratizing country chooses a simple system, inspection of previousgraphs shows that various outputs can be off by a factor of 2. We mightexpect to have 4 parties in the parliament, but can get as many as 8—oronly 2. Most political science undergraduates should be able to tell youthat few countries have more than 8 or less than 2 parties, so where ispredictability?

First, in most cases the outcome is likely to be closer to expectation.Second, gradual adjustment is possible, if you know what to adjust and byhow much to do so. If the given political culture, other institutions, andother factors combine to produce 8 parties in the assembly and one still

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wants to have 4, one might cut district magnitude by a factor of 24 = 16.This is so because of the relationship N0 = (MS)1/4 in Chapter 8 indicatesthat cutting M 16-fold would cut N0 twofold. Actually, better err on theconservative side and cut M by a factor of 8. Then allow at least threeelections to take place, and see what happens (Taagepera 2002d).

Indeed, it takes several elections with the same laws before their averagelong-term effects can be evaluated. Parties and voters need time to learnhow to use a newly adjusted system to their best advantage. If the laws arecontinuously altered, no stable system and ways to handle it can emerge.Of course, some initial choices may be so disastrous as to be given up ina hurry, but they are rare. The laws may not be that badly dysfunctional,once people learn to use them. Moreover, if you truly botched it the firsttime, what guarantees that a total flip does not lead from flaws discoveredto flaws as yet unknown? Fine-tuning may achieve the desired effectsmore safely.

Sometimes the change needed may lie outside the electoral system assuch. Party financing laws can affect the number of districts in whichparties decide to run—which, in turn, affects party votes, seats-to-votesratios, and possibly seats. Do parties strike election-time alliances but partways once in the assembly? The gut reaction might be to set higher legalthresholds for alliances, but this would complicate the electoral systemand be hard to police. In contrast, parliamentary rules that deny materialbenefits to parliamentary groupings that did not feature in elections maybe self-policing.

Simple Electoral and Party Systems

The predictions about party systems in this book often have sounded as if‘a party is a party is a party’. Parties have been treated as beings of the samekind, differing only in the number of votes and seats they command. Thisis of course far from reality. Parties differ widely in their internal structureand cohesion, among other features—and electoral systems interact withthese. Closed-list PR tends to reinforce central party leaderships, whileopen-list PR and STV enable the individual candidates to buck the leaders.Intraparty election or selection rules for leaders and for candidates ingeneral elections are another aspect of electoral systems this book hasnot dealt with. Even for national elections, after briefly describing thevariety of electoral systems, I have effectively reduced the range to ‘simpleelectoral systems’, meaning closed-list PR and FPTP as its limiting case.

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So what is my excuse for treating parties like Democritian atoms, indi-visible and lacking internal structure, and all electoral systems as simple?This question may be raised by those who argue that everything is so‘richly’ interrelated with everything else that, if one cannot investigateeverything at once, one is not entitled to investigate anything at all. Sucha claim would restrict us to holistic approaches such as religion or art onthe one hand, or to a grand linear regression equation on the other, whereall the ingredients are thrown into the same pot on an equal basis. I haveencountered such demands. But this is not how science proceeds.

Science proceeds in stages, trying to go from the more general to themore detailed, yet not starting with so broad and vague generalities thatconnections remain vacuous. Call it the middle range theory approach, ifyou will. I am well aware that most actual electoral systems are not simpleat all, that they cannot be reduced to assembly size, district magnitude,and seat allocation formula. Similarly, parties are not simple entities. ButI have put this awareness and related factual knowledge on temporaryhold, for the following reason.

If we cannot decipher the relationships among institutions, seats, votes,and parties even in simple systems, how could we expect to do so in morecomplex ones? The scientific approach is to solve problems first in simplesettings, and then gradually expand into complex. Thus, most of heattransfer is three-dimensional, but textbooks start with transfer in ideallyone-dimensional rods and then proceed to ideally two-dimensional flatplates. This was also the order in which theory was worked out in the firstplace, although all empirical experience is bound to be three-dimensional.

The approach is similar for elections and parties, except for one aspect.I could define an ideally simple electoral system, but how does one definean ideally simple, generic party, to which the models presented herewould apply foremost? If anything, it would be an ideally centralizedparty, a dictatorially run monolith. Correspondingly, the ideal electoralsystem would offer closed-list PR (or FPTP), where voters are forced tochoose between monolithic parties, not individual candidates. If we can-not make predictive sense of elections–parties interface for such a simplesetup, then how could we expect to make sense of more complex setups?

These simplifying assumptions are not normative. As founding chairof a political party in Estonia, I struggled hard for intraparty democ-racy and against the relentless hand of Michels’ iron law of oligarchy(Taagepera 2006) that pushed toward central control. Also, although Ifavor keeping electoral systems simple, I instinctively like personalizedPR and STV, despite the resulting complexities for analysis. But let us face

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it: Monolithic parties and closed-list PR (and FPTP) are easier to work intological models, compared to situations where intraparty democracy andvoters’ choice of individual candidates blur the simple picture. Puttingsuch details on hold does not mean ignoring or disliking them. They arejust relegated to the next level of analysis.

There is one caveat. Such an ordering of priorities presumes that wehave sensed correctly what comes first and is indispensable. It also pre-sumes that what comes next does not produce random noise (from theviewpoint of the presumed basic model) so huge as to drown out the pre-dictions of first-order models. This is hard to obtain under nonlaboratoryconditions.

For me, the major surprise regarding the models presented in this bookis that they actually work as well as they do. Given the dearth of trulysimple electoral systems, all these models have been tested with systemsthat present marked complexities—and yet these models hold, as averagesof many electoral systems. They seem to express, indeed, the centralfeatures of the actual systems, even when there is no inherent reason why‘ignorance-based’ models should hold upon further input of information.

It is easy to poke holes into the generalizations presented, by pointingout deviant cases, but such an attitude does not advance the orderly questfor more structured understanding, if we want to go beyond encyclope-dic compilation and cataloguing of factual knowledge. Yes, Switzerlanddiverges drastically from the prediction of the inverse square law ofcabinet duration. So what? Feathers in the wind do not disprove gravity.A fairly narrow zone, with slope 2 on log–log graph, still fits all thosesystems where cabinets depend on parliamentary confidence. Switzerlandshows that we do not yet understand everything about mean cabinetduration. It does not undo the fact that we already do understand some-thing about why cabinets last as long as they do.

Going Beyond the Simple Electoral Systems

Some aspects of simple electoral systems still need extensive testing.This applies to institutional inputs to vote shares and deviation fromPR (Chapter 14) and to the varied impacts of population size on politics(Chapter 12). The conclusion ‘This model has not yet been fully tested’also pops up in various other sections throughout the book. Why did notI carry out such testing before publishing the book? It would have takenmany more years, the more so because success in thoroughly investigating

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one aspect would bring up further questions. It is more efficient to pub-lish the many existing findings now and point out the unfinished ornew issues as challenges to anyone interested. The same applies to theweak theoretical links in well-tested models such as the third assumptionfor the minority enhancement equation (Chapter 16) or the embryonicmodel for the number of pertinent electoral parties (Chapter 15).

I now turn to aspects of the electoral–party system nexus where the sim-ple models presented here do not suffice, yet can help cast further light onmore complex situations. Although the Duvergerian macro-agenda is notcompleted, it has been investigated to the point where meaningful spin-off has become possible toward systematic investigation of more complexelectoral systems—‘second-order’ rules such as closed versus open lists,and intraparty effects of electoral rules. Much work in this direction hasalready been carried out. It is to be hoped that the mutually interlockingmodels presented in this book help advance such understanding. Anyadvances in the macro dimension also present new challenges to themicro dimension of the Duvergerian framework.

An analogous list could be presented for second-order effects of partystructures. Even if the electoral system were ideally simple, it could beexpected to have a simple effect of the kind modeled here only onsimple party systems, where all parties are monolithic. Here, I will limitmyself to the broader agenda for the study of electoral systems. Thefollowing survey owes much to the recent overview of electoral systemsby Matthew Shugart (2006). It proceeds from more general considerationstoward some details of nationwide legislative elections, and then toucheson some other levels. Among the many effects of such factors and theirinteractions, I focus on how they might affect the number and sizedistribution of parties.

The Micro Dimension of Duverger

In physics, macroscopic laws of thermodynamics were first developed,such as the ideal gas law, inducing and extrapolating from macroscopicobservation. These laws were useful in practice. Only much later didstatistical mechanics ground them in microscopic movement of particles.The relationships presented in this book are macroscopic. A major partof the micro-Duvergerian agenda would be to supply individual-levelfoundations for the system-level relationships observed in this book.

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In principle, statistical mechanics could have developed from scratch,absent nineteenth-century thermodynamics. In reality, the macroscopiclaws offered incentives, because they needed deeper explanation, and alsoa reality check on whether the micro-level calculations went in the rightdirection. The relationships presented here could play a similar role. Asobserved in appendix to Chapter 8, geometric averages of logical limitsneed not work at all, but they often do, as worldwide averages. It is up tomicro-Duvergerians to find out, why they work—and this is of course alsoa path toward explaining deviations from these averages.

The processes that lead to the Duvergerian average patterns of dis-tribution of seats need to be made more explicit. The mechanical andpsychological effects are entangled, and Benoit (2002) has argued thatthe strength of the mechanical effect has often been overstated due to‘prefiltering’ by psychological considerations. The psychological effectitself risks being a catch-all term for strategic choices of varied types, byindividual actors in individual elections. Actors include not only votersbut also party leaders and campaign contributors.

Cox (1997) achieved a major advance with his aforementioned notionof ‘strategic coordination’, which may or may not materialize so as tooffer an optimal number of candidates or lists. From his testing of theaforementioned ‘M + 1’ rule at low district magnitudes, Cox concludesthat the quality of voter information decreases with district magnitude.Blais (2000) has investigated the limits of rational choice approachesto the decision to vote or not to vote. I will not discuss here the stillbroader agenda of strategic considerations by candidates, voters, andparties which, in turn, depend on the ideological distribution of parties.

Political Culture

At given electoral system characteristics, political culture certainly shouldaffect the outputs, including the number and size of parties. Politicalculture, however, is a broad term covering various aspects which arehard to operationalize. Furthermore, political culture is a major factorin determining the type of government and electoral system a countrychooses in the first place. Consensual cultures are more likely to choosePR than winner-take-all cultures (Lijphart 1999). Thus, various aspects ofpolitical culture can act on the number and size of parties directly, bymodifying the impact of the electoral system, or indirectly, through theelectoral system itself.

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This double impact was pointed out for mean cabinet duration (Chap-ter 10). With the same number of parties, consensual polities mightbe expected to have longer lasting cabinets than the majoritarian. Butconsensual polities are also more likely to choose PR, which leads to moreparties. The two effects seem to cancel each other out. This is not a reasonfor giving up, but more refined work is needed both on measuring polit-ical culture and on connecting it to various aspects of political outputsthrough predictive models.

Ethnic and Geographical Determinants of the Size of Parties

In addition to institutions, the number of politicized social cleavagesor ‘issue dimensions’ also affects the number and size of parties. In theabsence of distinct issues, parties will not form even if the electoralsystems offer few restraints—see Vatter (2003) for a recent test. However,impressionistic estimates of the number of issue dimensions risk becometautological, as they are affected by the known number of parties. Tocounteract this risk, Ordeshook and Shvetsova (1994) introduced ethnicheterogeneity as a measurable proxy for issues. As it overlooks nonethniccleavages, it may underestimate of the number of issues, but it representsan advance toward objective measurement. The interaction of cleavagesand district magnitude has been confirmed by Ordeshook and Shvetsova(1994), Amorim Neto and Cox (1997), Cox (1997, 208–21), and Geys(2006). They find that interaction is multiplicative, while Lago Penas(2004) reports somewhat higher correlation for Spain when adding thetwo factors.

Once one agrees on which ethnic or other interest groups are distinct,their effective number can be measured. Yet ethnic heterogenity maynot increase party system fragmentation when parties are not structuredalong ethnic lines, or when various minority groups form a single party(Madrid 2005). The location of these groups also matters. A group uni-formly dispersed across the country may contribute less to heterogeneitythan does a group of equal size concentrated in a border area where itforms the majority (Mozaffar, Scarritt, and Galaich 2003). More generally,geographical location of support for different parties interacts with theeffect of electoral systems in determining their strength in the assembly(Gudgin and Taylor 1979; Johnston 1981; Eagles 1995; Park 2003), alongwith turnout differences and malapportionment (Grofman, Koetzle, and

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Brunell 1997). Maybe we would need a simple index of geographicalconcentration to express the location of minorities.

Party Financing

The way parties are financed varies—see special issue of Party Politics onparty finance, edited by Fisher and Eisenstadt (2004). The state increas-ingly finances parties, and the way it is done can affect the number andnature of parties (Burnell and Ware 1998; van Biezen 2003). If funds areallocated by seats won, it freezes out small parties underpaid in terms ofseats. If, on the contrary, funds are allocated by votes obtained, then tinyparties can survive even in FPTP districts, increasing the effective numberof electoral parties and possibly that of the legislative parties, indirectly.Hence, ‘Rules on party finance should be integrated more fully in thefuture study of the results of electoral reforms’ (Hooghe, Maddens, andNoppe 2006). They could modify the mean outcomes based on the seatproduct alone. The quantitative impact of such refinements should beworked into the predictive model.

Two-Tier PR

As one extends the study beyond the simple electoral systems, systemswith two (or more) tiers command attention in view of their wideningspread. Recall that two-tier PR systems come in two forms: parallel andcompensatory. The outcomes can be quite different, yet the two are oftenconfused. Take the example where voters cast votes in 100 FPTP districtsand also in a 100-seat nationwide district. With parallel rules, the FPTPseats may go to two major parties, while all parties win their proportionalshare in the nationwide tier. In total, third parties win about a half of theirproportional due in seats, while the two top parties are overpaid accord-ingly. With compensatory rules, in contrast, nationwide proportionality isrestored (usually subject to a legal threshold of votes), which means thatthe major parties loose whatever advantage they obtained in the FPTPdistricts.

Elklit and Roberts (1996) have stressed this ‘two-tier compensatorymember’ electoral rule as a separate category, more often called MMP. Thevolume edited by Shugart and Wattenberg (2001) updates our knowledge

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about the particularities of this approach that avoids malapportionmentproblems, yet preserves the benefits of local representation.

Several countries have recently adopted two-tier PR, either as parallelallocation (e.g. Italy and Japan) or MMP (e.g. New Zealand and Scotland),offering political scientists equivalents of crucial experiments among andwithin countries. In New Zealand, the shift from FPTP to MMP hasreduced disproportionality, as expected (Gallagher 1998), but may nothave reduced the adversarial nature of politics characteristic of FPTP(Barker and McLeavy 2000). Note that reduction in disproportionalityresults directly from a softened mechanical effect, which is instantaneous,while political style is a cultural aspect that may need more time to setin. In Japan, the shift from SNTV to FPTP and PR in parallel arguablyhas made the system more disproportional (Gallagher 1998), and thedominant Liberal Democratic Party has maintained its grip.

Italy’s shift from List PR to FPTP and PR in parallel highlights a littlenoted aspect of Duvergerian effects in FPTP districts: They tend to favorformation of two major blocks, but those blocks do not have to beunified parties. In Italy, parties form two blocks to present candidatesin the single-seat districts, while maintaining their separate identitiesthanks to the nationwide part of elections (Katz 1996). Thus Duverger’slaw is observed to work in Italy at the district level (Reed 2001), whilethe nationwide landscape remains almost as fractured as it was underList PR.

How would two tiers affect the number and size distribution of parties?Colomer’s micro-mega rule suggests that more entry points with differentrules would help small parties, but it depends on specific electoral rules atboth levels (Cox and Schoppa 2002). Moreover, it still remains debatablehow to determine an input-based effective magnitude (cf. Chapter 11) inthe face of two separate or interacting tiers, so as to establish a basis forcomparison. Frequent addition of legal thresholds in some tiers compli-cates the issue further.

In Italy, interaction between the two tiers works out in the PR tierin such a way that an increase in district magnitude actually tends toreduce the effective number of parties (Ferrara 2004). This is so becausethe voters are motivated strategically to desert the strong party they havevoted for in the FPTP district. Still, a 15-country study, which includesItaly (Moser and Scheiner 2004), finds no contamination between thetwo tiers. Nishikawa and Herron (2004) find that the overall effectivenumber of legislative parties tends to fall in between pure FPTP and PRsystems.

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Preferential-List PR

It matters more than one may think whether voters vote for parties orfor individual candidates (Grofman 1999). Shugart (2006) observes thatthe literature implicitly has equated PR with closed lists (which are alsopart of my definition of a simple electoral system), even while preferential(open) lists may be more prevalent in practice.

In fact, preferential lists can be used even in the FPTP framework,as Uruguay has done for presidential elections (Shugart 2006). Severalcandidates, possibly belonging to separate but allied parties, form openlists, where voters vote for a specific candidate. The single seat goes tothe list that achieves plurality and, within the list, to the candidate withthe most votes. It so to say combines primaries with general elections.As usually practiced, FPTP amounts to closed-list PR applied in single-seatdistricts, but it is more akin to SNTV in one respect: In both SNTV andstandard FPTP, a party is penalized for presenting an excessive number ofcandidates.

The study of preferential lists remains underdeveloped. They come in abewildering number of subtly different forms, with possibly different con-sequences. The attempt at classification of various closed list, preferential-list, quasi-list, and nonlist rules by Shugart (2006) offers a road map. Theeffect of preferential lists on the number and size distribution of partiesremains to be investigated. Defection and new party formation is the onlyrecourse for an ambitious dissenter who is ranked low in a closed list.By enabling independent-minded candidates to run and win, preferentiallists may prevent such splits. Hence they may reduce the number ofparties. On the other hand, preferential lists may loosen party disciplineto the point where the meaning of party as a unit of analysis becomesquestionable. This was certainly the effect of SNTV in Japan.

Presidential and Prime Ministerial Elections

In form, presidential elections are akin to legislative elections in a singlesingle-seat district, and the same alternatives offer themselves for choiceof electoral rules (see Chapter 3). However, deviation from proportionalityis higher, because there is no statistical evening out over many districts. Atthe same time, the stakes are higher than in a single parliamentary district,if the president has appreciable power. Therefore, some seat allocation

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rules are used or proposed that would never be considered in assemblyelections (Shugart and Taagepera 1994; Samuels and Shugart 2003).

Two-Rounds elections are more frequent in presidential than in assem-bly elections, so as to give the winner a stronger mandate than mereplurality among many candidates. Indeed, among 91 countries with directpresidential elections, Blais, Massicotte, and Dobrzynska (1997) find apreponderance of majority runoffs (54 percent), followed by plurality (22percent), and various other majority procedures (13 percent). However,when the eventual winner differs from the first round front runner, suchan inversion can actually weaken democratic governability (Pérez-Liñán2006). Presidential elections interact with assembly elections, especiallyif they precede the latter (Shugart and Carey 1992). Presidential coattailsmay boost a major party, thus reducing the effective number of partiesand altering their size distribution.

Some countries with symbolic heads of state have experimented ortoyed with the idea of direct election of prime minister. These are largelyuntested grounds in practice and theory (see Diskin and Hazan 2002).

Subnational and Supranational Elections

Elections at subnational levels vary in the number of levels and in impor-tance, which depend on the degree of formal federalization and actualdecentralization of finances (Lijphart 1999). Information on subnationalelection rules and outcomes is harder to come by than for national elec-tions. This is why they have been less studied, especially in a comparativeway, but it is becoming an active field. If the federal or lower subunitshave appreciable autonomy, then subnational elections could offer fur-ther entry points to small and new parties, unless these are blocked byelectoral and party financing rules. Thus subnational elections may ormay not increase the number of parties, compared to that in a fullyunitary country.

The only major supranational elections are those to the EP. While seatallocation to member states is centrally determined (and has been mod-eled in Chapter 16), election rules have been set by individual countries,and may differ from those used in national elections. The seat product isoften appreciably lower than in national elections, because fewer seats areat stake, for the given country. Even so, Euroelections may still advantagesmaller parties, because voters tend to consider these elections less impor-tant than the national ones. Hence some voters vote in Euroelections

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for new protest parties with whom they would not want to take theirchances in national elections. This may have an indirect effect on nationalelections, as these new or small parties achieve more visibility thanks tosuccess in Euroelections.

The Intraparty Dimension

For given vote shares, electoral rules affect not only which parties winseats but also who gets those seats within the party—see special issuesof Party Politics on Party Democracy and Direct Democracy, edited byScarrow (1999), and on Democratizing Candidate Selection, edited byPennings and Hazan (2001). In List PR, parties may wish to appeal tovarious constituencies by including women and ethnic minorities, andsome such candidates may win. In contrast, for the single candidate instandard FPTP, parties tend to prefer males of dominant ethnicity. Hencethe percentage of women tends to be higher in assemblies elected byPR (Rule 1981). PR may also promote higher intraparty turnover (Darcy,Welch, and Clark 1994; Henig and Henig 2001).

Matland and Taylor (1997) document a finer distinction: Even in multi-seat closed-list PR, parties tend to place males at the top of the list whenthey expect to win only one seat. Preferential lists may enable womento win even when the party leadership does not expect them to win.Nationwide PR might be expected to even out representation, but thisneed not be so. Geographic representation is not uniform in Israel and theNetherlands, and underrepresented regions tend to elect relatively fewerwomen (Latner and McGann 2005).

A candidate’s ability to win depends on party label and also on the‘personal vote’ his or her own image can attract. Carey and Shugart (1995)reasoned that the incentive to cultivate a personal vote should increasewith increasing district magnitude in open-list PR but decrease in closed-list PR. Indeed, the larger the district magnitude, the less incentive forpersonal activity closed lists can offer. The probability is low that personalactivity by the nth ranked candidate on the list can increase the numberof seats won by the party exactly from n − 1 to n. In open-list PR (and alsoSNTV), personal activity can put a candidate ahead of fellow candidates—and the more so when more seats are at stake. The two contrary trendsfuse at M = 1, where the single candidate is the party’s face in that district.

Two indirect tests have confirmed this conjecture. As M increases, thefrequency of initiating bills of a local character goes up for preferential-list

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PR but down for closed list (Crisp et al. 2004). So does the probability thatthe candidate is born in the district and is experienced in elected office(Shugart, Valdini, and Suominen 2005). Further distinctions betweencandidate-centered politics and localism are pointed out by Grofman(2005).

Further examples of the incidence of general election rules on intrapartypolitics are offered in Shugart (2006). Little is as yet known about them,because intraparty data are more voluminous and harder to come by thaninter-party election data.

Conclusion: Are Electoral Systems a Rosetta Stone forParts of Political Science?

Political science has been an intellectual field largely separate from poli-tics. So was physics, from civil engineering two centuries ago, and biologyfrom medicine, one century ago. Political science and politics may startto connect. It depends on how quickly political science complementspostdictive statistical methods with predictive ones.

In the study of electoral systems, we have made headway during thelast half-century and during the last decade, even while we have to gobeyond just ‘seats and votes’ (Powell 2006). We already know somethingabout electoral systems worth conveying to political practitioners. Ourquantitatively predictive ability is largely restricted to the simplest elec-toral systems where the seat product alone largely determines the numberof access points for smaller parties. Hence the advice to practitioners isto keep electoral laws simple. In more complex systems, the number ofaccess points is multiplied by ethnic and geographic variety, multilevelelections, and various second-order elections. Small party prospects maybe affected by party financing rules and presidential elections.

As advances in sciences bring new answers, they also engendernew questions. Hence the broader agenda for electoral studies thatgoes beyond the macro-Duvergerian. The study of micro-Duvergerianprocesses, complex electoral systems and intraparty impact of electoralrules are visibly at a stage where the territory is still being mapped andfurther intricacies are discovered. Here, predictive ability is spotty. In con-trast, the macro-Duvergerian agenda that focuses on the simplest electoralsystems has seen a breakthrough, since 1990, in quantitative predictionof the average impact of electoral systems on the distribution of seatsamong parties. Extension to the distribution of votes and prediction of

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disproportionality is in the process. This breakthrough is based on logicalquantitative models. To the extent that the theory of simple systems iscompleted, gradual extension to more and more complex systems canproceed.

Electoral systems are inextricably intertwined with party systems. Thenumber and strength of parties is largely measured in terms of electionresults—votes and seats. It might be more meaningful to consider cohe-sion of parties and their ability to get one’s way in negotiations, butthese are harder to measure. Thus the effective number of parties, usuallybased on election figures, remains perhaps the most widely used singleindex in political science, despite its well-known shortcomings. It pops upwhenever the party system is included as a possible factor in explainingor affecting any political phenomena.

Such penetration of other subfields made Taagepera and Shugart (1989)ask whether electoral studies could offer some branches of political sci-ence the equivalent of what Rosetta stone did for deciphering of hiero-glyphs.

Compared to other political phenomena, electoral systems deal with fairly hardnumbers: number of votes, seats, electoral districts, and so on. Thus these studiesare especially amenable to methods used in more established scientific disci-plines. . . . Votes might be to the quantitative development of political science whatmass has been for physics and money for economics: a fairly measurable basicquantity. (Taagepera and Shugart 1989: 5)

In developed sciences, quantitative expressions interlock. The same quan-tities recur in various different equations. A constant measured in onecontext is used in a different one. These numerical values are steppingstones. In comparison, quantitative knowledge in political science haslargely been fractured. The numerical values of coefficients found in aregression analysis are rarely used for further analysis. Such numericalvalues are end points, dead on arrival into printed pages.

Figuratively, quantitative relations in physics are like railroads inEurope—they interlock. Those in political science are like many railroadsin Africa—isolated tracks starting in port cities and ending in the hin-terland. Simple electoral systems are an exception. Here the product ofdistrict magnitude and assembly size leads to the number of seat-winningparties, which leads to the largest seat share, which leads to the effec-tive number of parties. Here we have the beginnings of an interlockingnetwork of equations which, through the mean duration of cabinets,promises to extend beyond the realm of electoral and party systems.

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In addition to such ‘colonization’ potential, the success of quantita-tively predictive logical models in electoral studies offers an inducementto other subfields of political studies to supplement their methodologicalapproaches. There is more to the quantitative study of politics than justregression and factor analysis on the one side and rational choice onthe other. Some other sciences have been served well by thought exper-iments based on the notions of boundary conditions and extreme cases,continuity of change between those limits, and elimination of logicalinconsistencies. Such notions can have their uses in political studies too.They will not open all doors, but this is not needed. It suffices if they opensome.

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APPENDIX

DETECTING FACTORS OTHER THAN THESEAT PRODUCT

This appendix offers data in a form where the effect of the seat product MS hasbeen removed (‘controlled for’) so that detection of other factors may becomeeasier. The predictive models based on seat product account for a fair part of theobserved variation in the number of seat-winning parties, the largest seat share,the effective number of parties, and mean duration of cabinets. The pattern is thefollowing.

The part of variation accounted for is 51 percent both for the largest seat share(Figure 8.4) and for the effective number (Figure 9.3). This means that all factorsnot correlated with the seat product account together for less than the seat productdoes alone. Hence, seat product clearly is the most important single factor. Otherfactors include features of electoral systems apart from M and S, such as the onesmentioned in Chapter 7, other institutions, cultural features, and path-dependenthistorical developments.

For mean cabinet duration, the seat product accounts for only 24 percent ofthe variation (Figure 10.2). It may thus look as if some other factor may accountfor more than does the seat product. However, the input of other factors mustlargely come through the effective number of parties, which single-handedlyaccounts for 77 percent of the variation in mean cabinet duration (Figure 10.1).Thus only 23 percent of the total variation is left for all other factors independentof N.

While assembly size is heavily determined by country population, the determi-nants of district magnitude are wide open. It is conceivable that some as yet unde-tected factors largely determine M and, quite separately, also determine the meanduration of cabinets. Even more broadly, all too many variables are interdependentrather than one-directionally dependent. Sophisticated statistical procedures canpoint out all sorts of colinearities and covariations among variables—but causallinkages are another matter.

So I have followed a different path, a more naive one, if you will, but one thathas served well in the advances of physics—advances that eventually enabled us toconstruct computers, so that even people with little mathematical sophistication

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Table A.1. Residuals of the number of seat-winning parties (N0)

Country, period and Seat product (MS)1/4 Actual N0 Residual,no. of elections (MS) R0 = N0/(MS)1/4

M = 1Germany 1871–1912, 13 396 4.5 13.6 3.02Netherlands 1888–1913, 8 100 3.2 6.5 2.03Norway 1906–18, 5 124 3.3 5.7 1.73France 1958–81, 7 470 4.7 6.7 1.43Denmark 1901–18, 7 118 3.3 4.7 1.42UK 1922–87, 19 628 5.0 6.4 1.28Australia 1901–17, 7 75 2.9 3.4 1.17New Zealand 1890–1987, 32 81 3.0 3.5 1.17Australia 1919–87, 28 106 3.2 3.7 1.16Canada 1878–1988, 31 247 4.0 4.4 1.10Italy 1895–1913, 6 508 4.8 5.1 1.06Norway 1882–1903, 8 114 3.3 2.9 0.88USA 1828–82, 28 240 3.9 3.0 0.77USA 1884–1936, 27 396 4.5 3.3 0.73Sweden 1887–1905, 8 226 3.9 2.8 0.72USA 1938–88, 26 435 4.6 2.5 0.54

M > 1Spain 1977–86, 4 2,345 7.0 12.8 1.83Ireland 1922–89, 24 525 4.8 8.2 1.71Switzerland 1919–87, 19 1,521 6.3 10.5 1.67Japan 1928–86, 22 1,920 6.6 10.0 1.52Norway 1921–49, 8 1,125 5.8 6.5 1.12Luxembourg/2 1922–51, 7 359 4.3 4.7 1.09Norway 1953–85, 9 2,707 5.9 6.3 1.07Luxembourg 1919–89, 11 751 5.2 5.5 1.06Portugal 1975–87, 7 2,814 7.3 6.9 0.95Malta 1921–45, 6 99 3.2 3.0 0.94Finland 1907–87, 30 2,800 7.2 6.8 0.94Sweden 1952–68, 6 2,059 6.7 5.7 0.85Malta 1947–87, 11 250 4.0 3.2 0.80Sweden 1908–48, 14 1,886 6.6 5.2 0.79

Note: Calculated from data in Taagepera (2002b), as graphed in Figure 8.1. ‘Luxembourg/2’ indicates electionscarried out in one-half of the country.

on their own part can instigate sophisticated statistical analyses. Once this machin-ery is available, should we abandon the simpler approaches? I do write this bookon a computer, but when conceptual thinking becomes tense, I grab for a pencil.It makes sense to use the most appropriate technology for the given purpose, notthe most advanced one in a technological sense.

This approach has led to predictive models that connect the seat product, thenumber of parties, and the mean cabinet duration. Are these connections reallycausal, so that a change in district magnitude truly would lead to a changein cabinet duration? We are reminded that maybe the best measure of overalltechnical development of a country is the per capita number of telephones, butit would be risky to put all national resources into buying telephone sets and hope

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Detecting Other Factors

that the rest would follow. I think predictions based on seat product are on safercausal grounds, but one has to maintain a healthy dose of skepticism. This dose isbetween blind acceptance and blind rejection.

Some of my colleagues may wish to follow up on this simple but time-honoredapproach of addressing only one or a few variables at a time, while carefullythinking through how (i.e. in what functional form) these variables might logicallyimpinge on the number and size distribution of parties and other features thatmay derive from it. They may discover connections to which I may be blind. Forthis purpose, the following Tables A.1–A.3 are offered. They show the residuals,that means what is left to be accounted for when the expected impact of the seatproduct has been removed (‘controlled for’).

Table A.1 shows such residuals for the number of seat-winning parties. Thecountries are listed in the order of decreasing residuals, which range from 3.0 to0.54 for single-seat systems and from 1.83 to 0.79 for multi-seat systems. A residualvalue 1.00 means that the prediction by MS fits exactly. A residual of 2 means thatthe actual number of seat-winning parties is twice the expected number, while aresidual of 0.5 means that the actual number is one-half of the expected. Onlytwo countries fall outside this range. This limited range makes detection of causal(or at least correlated) factors so much more difficult. As mentioned in Chapter 8,operational measurement of number of seat-winning parties presents difficulties,and hence a large part of the residual may be measurement error—which makesdiscovering further causal factors even harder. I offer these data nonetheless, justin case.

Table A.2 shows analogous residuals for the largest seat share. The countries arelisted in the order of decreasing residuals, which range from 1.7 to 0.58 for single-seat systems and from 1.30 to 0.67 for multi-seat systems. Here measurement erroris much smaller, which makes it more promising grounds for detecting furthercausal factors. On the other hand, the range of the residuals is even narrower thanin previous table.

Table A.3 shows the residuals for the largest seat share, the effective numberof parties, and the mean cabinet duration, all for the same data-set. (For thelargest seat share, overlap with previous table is appreciable.) The countries arelisted in the order of decreasing residuals for the effective numbers, which rangefrom 2.7 to 0.72. The residuals for mean cabinet duration have a much widerrange—from 3.1 down to 0.20—which should make detection of further factorseasier. However, the high correlation with the effective number must be kept inmind.

Here in particular, please note that the residuals refer to the theoreticallyexpected relationship, not the empirical best fit. In the case of C versus MS, R2

is 0.30 for the empirical fit but only 0.24 for the predictive model. This 24 percentis the part of variation that is not only accounted for in a statistical sense but alsoexplained in a more substantive sense. Hence, this is the part other factors are tocomplement.

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Table A.2. Residuals of the largest seat shares (s1)

Country, period and Seat product 1/(MS)1/8 Actual s1 Residual,no. of elections (MS) R1 = s1(MS)1/8

M = 1Italy 1895–1913, 6, TR 508 0.46 0.78 1.70Botswana 1965–94, 7 33 0.65 0.83 1.29Antigua 1980–89, 3 17 0.70 0.86 1.22United States 1828–1994, 84 344 0.48 0.59 1.22United Kingdom 1885–1992, 29 643 0.45 0.53 1.18Bahamas 1972–87, 4 42 0.63 0.73 1.16Canada 1878–1993, 32 247 0.50 0.58 1.14Trinidad 1961–91, 7 35 0.64 0.73 1.14St. Vincent 1974–89, 4 14 0.72 0.82 1.12Jamaica 1944–89, 11 47 0.62 0.65 1.12Mauritius 1976–95, 6 68 0.59 0.65 1.10Norway 1882–1903, 8 114 0.55 0.59 1.07Barbados 1966–91, 6 26 0.67 0.69 1.04Dominica 1975–90, 4 21 0.68 0.69 1.01Sweden 1887–1905, 8 226 0.51 0.51 1.01New Zealand 1890–1993, 34 81 0.58 0.58 1.00Belize 1979–89, 3 24 0.67 0.66 0.99Grenada 1972–90, 4 15 0.71 0.69 0.97France 1958–93, 10, TR 496 0.46 0.44 0.95Norway 1906–18, 5, TR 124 0.55 0.50 0.91Cook Islands 1965–99, 10 23 0.68 0.61 0.90St. Lucia 1974–92, 6 17 0.70 0.63 0.90Australia 1919–96, 31, AV 106 0.56 0.50 0.89Samoa 1979–2001, 7 47 0.62 0.51 0.86Denmark 1901–18, 7 117 0.55 0.46 0.84Australia 1901–17, 7 75 0.58 0.48 0.83Cuba 1901–54, 23 64 0.59 0.48 0.80St. Kitts & Nevis 1980–89, 3 10 0.75 0.51 0.68The Netherlands 1888–1913, 8, TR 100 0.56 0.34 0.61Germany 1871–1912, 13, TR 396 0.47 0.27 0.58

M > 1Spain 1977–96, 7 2,360 0.38 0.49 1.30Japan 1928–93, 24, SNTV 1,930 0.39 0.50 1.28Portugal 1975–95, 9 2,770 0.38 0.42 1.14Sweden 1908–68, 20 1,600 0.40 0.45 1.13Ireland 1922–92, 25, STV 567 0.45 0.47 1.04Norway 1921–93, 19 1,200 0.41 0.42 1.03Malta 1921–92, 18, STV 180 0.52 0.52 0.98Luxembourg 1919–94, 19 504 0.46 0.43 0.94Finland 1907–95, 32 2,860 0.37 0.33 0.89Switzerland 1919–95, 21 1,540 0.40 0.27 0.67

Note: Calculated from data in Taagepera and Ensch (2006), as graphed in Figures 8.3 and 8.4. M = 1 systemsare FPTP, unless otherwise indicated: TR = Two-Rounds; AV = alternate vote. M > 1 systems are List PR, unlessotherwise indicated: STV = single transferable vote; SNTV = single nontransferable vote. District magnitudesremained the same during the periods shown and variations in assembly size were relatively minor, except forUSA (213–437), Malta (from 10 for Government Council, 1939 and 1945, to 65) and Luxembourg (from 25 inpartial elections to 64).

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Detecting Other Factors

Table A.3. Residuals of the largest seat shares (s1), effective numbers of parties (N), andmean cabinet durations (C )—R1 = s1(MS)1/8, R N = N/(MS)1/6, RC = C (MS)1/3/42 yrs.

Country and Seat product s1 R1 N R N C RC

period (MS) (years)

M = 1Papua-NG 1977–97 108 0.40 0.71 5.98 2.74 1.65 0.19India 1977–96 542 0.55 1.21 4.11 1.44 2.4 0.47Mauritius 1976–97 68 0.62 1.06 2.71 1.34 2.1 0.20France 1959–2002, TR 508 0.44 0.97 3.43 1.21 3.1 0.59Barbados 1966–94 26 0.70 1.05 1.76 1.02 9.5 0.67Trinidad 1961–2001 36 0.75 1.17 1.82 1.00 10.0 0.79Australia 1946–96, AV 128 0.51 0.93 2.22 0.99 9.9 1.19New Zealand 1946–96 85 0.57 0.99 1.96 0.93 6.3 0.66Canada 1945–93 270 0.56 1.12 2.37 0.93 8.0 1.23Bahamas 1972–2002 42 0.73 1.17 1.68 0.90 14.9 1.23USA 1947–2000 435 0.62 1.32 2.40 0.87 7.7 1.39Jamaica 1962–89 55 0.76 1.25 1.62 0.83 9.2 0.83Botswana 1965–2004 37 0.75 1.18 1.35 0.74 39.6+ 3.14+UK 1945–97 635 0.53 1.20 2.11 0.72 8.6 1.76

M > 1Finland 1945–2003 2,940 0.27 0.73 5.03 1.33 1.5 0.51Luxembourg 1945–99 809 0.41 0.95 3.36 1.10 6.0 1.33Japan 1946–96, SNTV 1,940 0.54 1.39 3.71 1.05 3.9 1.16Norway 1945–97 1,190 0.47 1.13 3.35 1.03 4.3 1.08Ireland 1948–97, STV 538 0.48 1.06 2.84 1.00 3.8 0.74Israel 1949–96 14,400 0.38 1.25 4.55 0.92 1.75 1.01The Netherlands 1946–2002 19,600 0.34 1.18 4.65 0.90 3.3 2.13Portugal 1976–2002 2,810 0.43 1.16 3.33 0.89 3.2 1.08Costa Rica 1953–98 426 0.52 1.12 2.41 0.88 4.9 0.88Malta 1966–87, STV 294 0.53 1.08 1.99 0.77 10.6 1.68Spain 1977–2004 2,330 0.50 1.32 2.76 0.76 9.0 2.84

Note: Calculated from data in Taagepera and Sikk (2007), as graphed in Figures 9.4 and 10.2. M = 1 systems areFPTP, unless otherwise indicated: TR = Two-Rounds; AV = alternate vote. France had one PR election in 1986. M >

1 systems are List PR, unless otherwise indicated: STV = single transferable vote; SNTV = single non-transferablevote. District magnitudes remained the same during the periods shown and variations in assembly size wererelatively minor. Countries are listed in the order of decreasing residual for N (which corresponds to locations inFigure 9.4).

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306

Index

A Wuffle x, 99, 239, 271Adams, James F. xadditional member, see electoral systems,

compositeadjustment seats, see electoral systems,

compositeadvantage ratio 70–5Alabama paradox 85–7, 95, 243, 246Åland Islands 36Allende, Salvador 2Allik, Mirjam vii, 145–6, 148–51, 155,

157–8, 161alternative vote (AV) 25–7, 36, 45–6, 270

agreement with simple models 117, 125,129

Amorim Neto, Octavio 278Anckar, Carsten 188–9, 192, 195–6, 231Anckar, Dag 188Anderson, Christopher J. 108Andrews, Josephine 15Antigua, 232apparentement 24, 41, 73, 90, 127, 132approval voting 28–9assembly size 17–18, 23–4, 102, 110, 127,

184–6cube root law of 110, 188–90, 197–200,

206and openness to small parties 90–1

Australia 25, 35, 135–6sister parties 53, 125, 129

Austria 14, 64, 182, 193, 237, 266

balance in party sizes, index of 48, 56, 62,123–4, 138, 237

and effective number of parties 50–3Balinski, Michael L. 87, 95ballot structure 18, 23Banzhaf power index 56Barbados 25Barker, Fiona 280Belgium 39, 183, 237Benoit, Kenneth 17, 277

Birch, Sarah 25, 271Blais, André v, 40, 277, 282Blau, Adrian 214, 221block vote (BV), see party block voteBlondel, Jean 50–1Boix, Carles 7, 110Bolland, John 34, 50Borda, Jean Charles de 26, 29, 102Borda count (BC) 26–8, 45–6, 270Botswana 52, 125, 172, 174, 180Boucek, Françoise 56, 62Bowler, Shaun v, x, 35–6break-even point 70–3, 235, 250–1, 252Britain, see United KingdomBritish-heritage countries 15, 25, 45–6, 110,

270Brunell, Thomas L. 43, 279Burnell, Peter 279

cabinet duration vi, viii, 111, 165–75,191–2

data 289–91inverse cube law 170–2, 175inverse square law 167–70, 174–5, 275

cabinet types 56, 167Canada 135–6, 247

district level 215, 247national level 64, 135–6, 144, 237province level 104, 181, 209second chamber 261, 266

Carey, John M. v, 282–3Caribbean countries:

with high disproportionalityexponent 207, 209–12, 214, 229, 232,237

categoric ballot 17–18Caulier, Jean-François 56causal direction 7, 103, 108, 110chess rules and electoral rules 7–8Chhibber, Pradeep 154, 215Chile 2, 41, 50China 190

307

Index

Churchill, Winston 120Clark, Janet vi, 203, 283Cleavages 106closed list 18, 29, 283coalition cabinets 56, 58, 131, 202

break-up 166, 168duration 17, 167

Coleman, Stephen viiColomer, Josep M. v, vii, 7, 13–15, 22, 24,

26, 33, 102–3, 110, 132, 253micro-mega rule 83–5, 92, 94, 102, 107,

110communication channels, number of 111,

167–8, 170, 198–200compensatory seats, see electoral systems,

compensatoryconceptual anchor points 69, 208

for number of parties 91, 117, 121, 134conceptual limits 31, 162, 140

on number and size of parties 119–20,125, 135, 148, 156, 157

see also quantitatively predictive logicalmodels

conservation laws 68, 124, 135–6Costa Rica 30Cox, Gary W. v, 27, 35, 49, 94, 104–5, 149,

226, 244, 277–8Cox, Karen 280Crisp, Brian 30, 284Cross, William xCrotty, William xcube law of elections 112–13, 205, 214, 217,

218cube root law of assembly sizes, see assembly

sizecumulative voting 28Cyprus 265Czech Republic 20

Dahl, Robert A. 1, 4, 51, 187Dalton, Russell J. x, 195Dalton’s principle 77–8, 82Danish divisors 32–3, 86–8, 93, 95–6Darcy, Robert vi, 203, 283democracy and elections 1–2, 108Denmark 6, 41, 51, 183‘deterministic’ prediction ixdeviation from a norm 76deviation from PR 65–82, 113

Gallagher index (D2) 66–9, 76–82, 88–9,211, 231, 234

indices to suit any seat allocationformula 94

‘law of conservation’ 68

Lijphart index (D∞) 67, 77–9Loosemore–Hanby index (D1) 66–9,

76–82, 233master equation for indices 78prediction from institutional

inputs 231–3prediction from votes 211–13relations between indices 79–82relation to effective number of

parties 68–9d’Hondt divisors 31–2, 44

as sufficient quota 33, 94formula exponent for 93–6, 128,

130large party bias 85–9, 112modified, in Estonia 32–3openness to small parties 85, 89–90presented as Hagenbach-Bischoff 43threshold of representation 242–3,

246–7, 252Diamond, Larry v, 51Diskin, Abraham 282disproportionality exponent 112, 205–8,

213–15, 217district level alliances, see apparentementdistrict magnitude 103

definition 2, 18–19, 23determinants of 110openness to small parties 85–9unequal 37–8, 132

divisor formulas for seat allocation 31–4see also Danish; d’Hondt; Imperiali;

Jefferson; Sainte-Laguë divisorsDobrzynska, Agnieszka 282Dodd, Lawrence C. 168, 172Dogan, Mattei 166Dolez, Bernard 219, 221double ballot 40Downs, Anthony 105Droop quota 30, 33–4Drummond, Andrew J. 108Dumont, Patrick 56Dunleavy, Patrick 56, 62duration of cabinets, see cabinet durationDuverger, Maurice 101–3

mechanical effect 104, 205psychological effect, 104–5, 231two effects combined, 107, 112, 149, 231,

277, 280Duverger’s law 7, 27, 41, 192, 280

generalization as M+1 rule 105,244

Duverger’s statements (law andhypothesis) 103, 106, 107

308

Index

Duvergerian agenda 101–14degree of completion 177, 231, 234, 276,

284macro-agenda 102, 106, 109–14micro-agenda 102, 114, 141, 276–7

Eagles, Munroe 278Eckstein, Harry 5, 271effective magnitude 178–83

input based 178, 184–6output based 178–84

effective number of components 48, 55effective number of electoral parties 53–4,

113, 196, 215prediction from institutional

inputs 229–31effective number of legislative parties

(N, N2) 48–54, 56–9, 111data 289–91and index of balance 50–3and N0 and N∞ 59–60, 62–3, 97, 154,

160–4, 226and number needed for majority

coalition 58prediction from institutional

inputs 152–3, 162prediction from votes 211–12and relevant parties 63–4

effective number of parties:gap between electoral and legislative 54,

236–7power index based 56

effective number of polities 55effective thresholds, see thresholdsEisenstadt, Todd 279electoral design, see institutional

engineeringelectoral districts, definition 2, 17, 21electoral laws vs. electoral systems 22,

272–3electoral reform, see electoral systems,

choice and changeelectoral systems 23–46

aggregate 97–8, 154choice and change 13–17, 20–1, 43–5,

131, 273composite: compensatory vs.

parallel 40–1, 45–6definition 2, 5, 21openness to small parties 83–98pathologies 42–3, 46research 8–9

Elklit, Jørgen vi, 279Ellis, Andrew v, 14, 21–2, 24–5, 44–5

England, see United KingdomEngstrom, Richard L. 43Ensch, John 25, 124–7, 129–30Estonia ix–x, 16, 20, 32, 39, 274ethnic conflict and electoral systems

270–1ethnic minority representation 14, 91,

202–4, 216, 278European Parliament 43, 255–7, 266, 268

minor party vote 54n, 105, 282seat allocation 261–5size 259–60

European Union 190, 200, 255Constitutional Treaty 260, 264–5

European Union, Council of 255–7, 259–66,268

total voting weights 259–60voting weight allocation 261–5

expectation values viii, 121, 148

Farrell, David M. v, x, 21–2, 24–5, 36Fatah 1, 15federal subunits:

number and largest share 135–6representation in second chamber 266

Feld, Scott L. 26Ferrara, Frederico 40, 280Fiji 270–1financing of parties 104, 105, 279Finland 105, 132, 245

as typical d’Hondt 32, 92–3proportionality profile 73, 75, 233unusual features 37, 181, 237

first-past-the-post (FPTP) 14, 18, 24–7and deviation from PR 68–9frequency of 44–6proportionality profiles 69–71as single-seat PR 19, 23, 33, 36, 92

Fisher, Justin 279formula exponent (F ), 92–6, 128, 130

see also seat productFraenkel, Jon 270–1France 21, 193, 221

proportionality profile 72, 80two-rounds majority-plurality 25–6unusual features 55, 64, 69, 181

fraud, electoral 42French-heritage countries 45–6, 110

Galaich, Glen 53, 278Gallagher, Michael v, x, 94, 244, 280

index of deviation from PR, see deviationfrom PR

Gambetta, Diego 41

309

Index

geometric vs. arithmetic means 118, 137,218

Germany, Federal Republic 40, 64, 182, 195,253

CDU/CSU party, 50, 53, 63legal threshold 38–9, 253proportionality profile 74typical MMP 16

Germany, imperial 117, 125, 141, 144, 248Gerring, John 197Gerry, Elbridge 43gerrymander 42–3, 214

bipartisan 42–3, 214Geys, Benny 278Golder, Matt 44Gotz, Florian vi, 125–6Greece 39, 174, 182, 237Grenada 211Grofman, Bernard v, vii, x, 6, 26, 35–6,

76–7, 91, 146, 167, 188, 248, 270–1,278, 281, 284

Gudgin, Graham 278Guillory, Christine A. 108Gunther, Richard x, 51

Hagenbach-Bischoff quota 30, 33basis for d’Hondt divisors 43

Hamas 1, 29Hamilton, Alexander 33Hamilton quota 30Hanby, Victor J. 66, 242Handley, Lisa 91, 188Hare, Thomas 102Hare quota 30Hare quota and largest remainders

(Hare-LR) 30, 33, 44, 85–6, 112formula exponent 92–3, 95–6openness to small parties 85–9threshold of representation 242–3, 246

Hartmann, Christof, vi 125–6Hayes, James P. 188Hazan, Reuven Y. 282, 283Henig, R. vi, 283Henig, S. vi, 283Herron, Erik S. 280Hill, Steven 1nHooghe, Marc 279Horowitz, Donald 270Hosli, Madeleine O. 260, 262–5, 269Hsieh, John F.-S. 226

Iceland 2–3, 41, 69, 183, 193, 237ignorance-based models 110–11, 119, 141–2


Imperiali divisors 32–3, 86–8, 95–6Imperiali quotas 30, 33India 69, 209, 215

major FPTP system 44, 46province level variety 104, 181unusual features 55, 89, 126, 172

Inglehart, Ronald 174institutional constraints 4institutional engineering 9, 22, 112, 197–8,

272–3European Union, United Nations and

second chambers 255–6, 266–7first or only chambers 130–3, 155–6,

172–4, 183–4, 234, 253intra-list competition 29, 41, 283inverse square law of cabinet duration,

see cabinet durationIreland 6, 35, 137, 181, 237island countries 188, 190, 207Israel 1, 283Italy 30, 39, 41, 183, 237

number of relevant parties 63–4parallel system 40, 280two-rounds (1895–1913) 125, 144

Jackman, Robert W. 15Jamaica 180, 195Japan 69, 105, 139

Liberal Democratic Party dominance, 50,53, 139, 237

shift to two-tier PR 280typical SNTV 35, 117, 281unusual features 69, 128, 137, 195

Jefferson, Thomas 33Jefferson divisors 31Johnston, Ron J. 278Jones, Mark P. v

Kaminski, Marek M. 15Kasapovic, Mirjana viKaskla, Edgar 91, 95Katz, Richard S. v, x, 280Kendall, M. G. 205King, Gary ixKiribati 26Klingemann, Hans-Dieter 108Koetzle, William 278Kollman, Ken 154, 215Kosovo 271Krennerich, Michael vi

310

Index

Laakso, Markku 49, 60, 70, 103, 154,242

Laatsit, Mart 146, 158–9, 231Lago Penas, Ignacio 278Lange, David 16largest remainders 263

with Hare quota, see Hare quotalargest seat share (s1) 110, 122–33, 135–9

data 289–91inverse as measure of number of parties

(N∞), see number of partieslargest vote share for given largest seat

share 229–30, 235–6Latin America 207Latner, Michael 283Laurent, Annie 219, 221Laver, Michael x, 166Lebeda, Tomáš 21legal majorities 39legal thresholds, see thresholdslegitimacy 14Lijphart, Arend v, 21–2, 24, 39, 43, 49–50,

52, 54–6, 67–9, 79, 103, 108, 166–70,174–5, 178–9, 181–2, 185, 188, 192,207, 245, 253, 270, 277, 282

limited vote (LV) 29, 45–6Linz, Juan J. xlist PR 29

frequency 44–6literacy as basis for assembly size 189–90Loosemore, John 66, 242Loosemore-Hanby index for deviation from

PR, see deviation from PRLuxembourg 32, 42, 129, 237, 263, 265–6

M+1 rule 105, 244–5, 277McAllister, Ian 25, 36Macclellan, Nic 271McGann, Anthony 283MacKenzie, W. J. M. 271Mackerras Malcolm 36McLeavy, Elizabeth 280McNeill, Keith G. viiMackie, Thomas T. vi, 51, 70, 116, 122, 125,

145, 237Maddens, Bart 279Madrid, Raúl L. 278magnitude, see district magnitude; effective

magnitudeMair, Peter x, 4, 51majoritarian electoral systems 1, 24–6,

108malapportionment 24, 42–3, 65, 69

Malta 92, 134–5, 193typical STV 35unusual features 39, 51, 181, 237, 265

Massicotte, Louis 40, 282Matland, Richard E. 30, 283Mauritius 172, 174, 181mechanical effect, see DuvergerMichels, iron law of 274micro-mega rule, see ColomerMill, John Stuart 103Merill, Samuel xMexico 40, 195Mikkel, Evald 4minority attrition, law of 112–13, 201–23

derivation 216–21FPTP 204–15multi-seat districts 213–14, 219–21two-rounds 221women and ethnic minorities 202–4, 215,

222minority cabinets 56minority enhancement equation 261–8Mitchell, Paul vmixed member proportional (MMP) 16, 40,

44poportionality profile 74

Molinar, Juan 56, 62Monroe, Burt L. v, 37–8, 76–7Montero, José Ramón xMoser, Robert G. 280Mozaffar, Shaheen 2, 53, 278Mukherjee, Bumba 154multi-seat districts 18–19, 28–36

Nagayama, Masao 145Nagayama triangle 145, 148Netherlands, The 38, 64, 144, 172, 233, 283

two-rounds (1888–1913) 117as typical nationwide district 92–3,

119–20, 244–5New Caledonia 270–1New Hampshire 197New Zealand 49, 185, 207

change in electoral system vi–vii, ix, 16,280

proportionality profile 70–1, 233Nice, Treaty of, see European UnionNiemi, Richard G. 226Nishikawa, Misa 280Nohlen, Dieter vi, 91, 125–6, 207, 210–11,

230Noppe, Jo 279Norris, Pippa v

311

Index

Norway 32, 64Novák, Miroslav 21number of parties 47–64

effective, see effective number of partiesentropy-based (N1) 61, 169largest seat share based (N∞) 48, 50,

57–60, 62–3, 97, 138, 149master equation 60–1NP index 62registered 188, 192–3, 195–6relevant 48, 57, 63–4seat-winning (N0) 48, 50, 57–60, 97,

110–11, 116–22, 133–5; data 288–9serious or pertinent 226–7, 244–6, 250

number of serious candidates 105Nurmia, Matti vii

Occam’s razor 141open list 18, 29, 281, 283Ordeshook, Peter C. 278ordinal ballot 17–18‘Other parties’ 5, 122, 139, 141, 152overpayment for largest party 104

Palestine 1–2, 15, 29panachage 18, 42, 127, 181Papua-New Guinea 153, 172, 181,

197Park, Myoung Ho 278parties:

balance in size, see balance in party sizeinternal structure xmembership 188, 192–5number, see number of parties, effective

numberregional 16

party block vote (PBV) 28, 44–6party systems:

definition 5–6mapping with number and balance of

parties 50–3simple 273–5

path dependence 108, 183Pedersen, Mogens N. 67Pennings, Paul 283Pérez-Liñán, Aníbal 282physics 188

conserved quantities 135, 219general structure 120, 214, 284, 285methods 31, 272, 287sequential approach to problems vii–viii,

274, 276–7terminology 61, 121, 174

Plattner, Mark v

Plurality allocation rule 18–19, 92–3, 97–8,178–80

multi-seat plurality 19–20, 98, 178–9,182, 183, 219–20

in single-seat districts, seefirst-past-the-post

political culture 4–5, 10, 14, 131, 234consensual vs. majoritarian 108, 174–5,

277–8political engineering, see institutional

engineeringpolitical practitioners, advice to 1, 23

advantages of simple electoral laws ix, 13,101

allocating seats in second chambers andEuroparliament 255

altering assembly size 187altering the mean duration of

cabinets 165altering the number of parties 83, 115,

143, 225increasing representation of women and

minorities 201, 241measuring the number of parties and

disproportionality 47, 65simplifying complex electoral

systems 177, 269population, effect on politics 110,

187–200Portugal 32, 237Powell, G. Bingham v, 284predictive ability 10, 17, 120, 142, 212

models 112, 118quantitative prediction vi–viii, 9, 140,

211see also quantitatively predictive logical

modelspreferential voting, see open listpresidential elections 24, 125, 134, 206,

281–2primary elections 42prime ministerial elections 281–2proportional representation (PR), 7, 14,

18–19deviation from, see deviation from PR

proportionality profiles 65, 70–5, 233Przeworski, Adam 67psychological effect, see DuvergerPutnam, Robert 202

quantitatively predictive logical models 46,110, 271, 285–6

vs. directional models 127, 159, 175see also predictive ability

312

Index

quota method for seat allocation 30–3see also Droop; Hagenbach-Bischoff;

Hamilton; Hare; Imperiali quotas

R-squared, significance of 118, 126, 129,171–2

Rae, Douglas W. 103, 242ranges of variables 130, 140Rapoport, Anatol 7–8Recchia, Steven 257–9Reed, Steven R. 34, 50, 105, 120, 146, 226,

244, 280regression analysis:

logging all variables 192OLS on logged variables 117–18, 123,

125–6, 128, 153, 169, 171Overuse of vii–viii, 285

Reilly, Benjamin v, 14, 21–2, 24–6, 44–5relative majority, see pluralityrepresentation thresholds, see thresholdsresponsiveness of FPTP, see

disproportionality exponentReynolds, Andrew v–vi, 2, 14, 21–2, 24–5,

44–5Riker, William H. 103Roberts, Nigel S. 16, 279Rokkan, Stein 242Rose Amanda G. 37–8Rose, Richard vi, 51, 70, 116, 122, 125, 145,

237Rosetta stone 10, 285Ruiz Rufino, Rubén 251Rule, Wilma vi, 283Rush, Mark E, 43

St Kitts and Nevis 46, 102, 125, 197St Vincent 91Sainte-Laguë divisors 32–3, 44, 85,

112formula exponent for 92–6modified 32–3openness to small parties 85–90thresholds of representation 242–3, 246–7

Samuels, David J. 282Sartori, Giovanni 51, 56–7, 63–4Scarritt, James R. 53, 278Scarrow, Susan 283Schedler, Andreas 2Scheiner, Ethan 280Schiff, Leonard E. 121Schoppa, Leonard J. 280Schrödinger’s equation 236Schuster, Karsten 88, 95Scotland 105, 280

seat allocation formulas 18, 22–3, 112effect on openness to small parties, 85–9

seat product (MF S, MS) 92–8, 110–11,139–40, 154, 179–82

effective 131, 155seat shares of parties 144–52, 156–60seat–vote equation, 112, 204, 215, 216, 219

reversed 227second chambers:

seat allocation 261–8size 257–9

selection process constant 219, 221–3self-interest 5, 15–16, 110Shepsle, Kenneth A. 166Shugart, Matthew S. 30, 40, 94, 101–2, 106,

116, 120–1, 131, 161, 185, 209, 211,217, 276, 279, 281–5

Shvetsova, Olga 278Siaroff, Alan 51, 62Sikk, Allan 75, 152–3, 168–72, 174simple electoral systems 19–20, 103,

109–10, 178–9, 272predictability 10, 46

simple quota, see Hare quotasingle-member districts, see single-seat

districtssingle non-transferable vote (SNTV) 34–6,

44–6, 105, 117, 244single-seat districts 18–19, 24–8single transferable vote (STV) 35–6, 44–6,

117sister party problem 50, 53, 63, 125, 129,

139size and politics, see populationsmall countries, politics in 191–6Solomon Islands 197Soudriette, Richard W. 45South Korea 35, 40, 195Soviet electoral rules 15Spain 32, 38, 253, 278

deviations from predictions 38, 127, 172,181, 248

low balance in party sizes 53, 237malapportionment 55, 69

stability, political 15–16, 131, 166Steenbergen, Marco 2Stockwell, Robert F. 270Storey, Robert S. viistrategic voting and sequencing 104–5, 112,

149Strøm, Kaare 202Stuart, R. 205sub- and supranational elections 54n, 105,

282–3

313

Index

Suominen, Kati 30, 289Sweden 32, 64, 183Switzerland 32, 34, 64, 181

apparentement and panachage 41–2, 127long-lasting cabinets 169, 275small largest share 128, 141

Taiwan 35, 40Tan, Alexander C. 195Taylor, Andrew 271Taylor, Michelle 30, 283Taylor, Peter J. 278Theil, Henri 216–17, 222, 261–2, 268Thibaut, Bernhard vithought experiment 91–2, 147


threshold of majority 251–2thresholds of representation 24

confusion between district andnationwide 185, 252–3

effective, district level 38, 178effective, nationwide 247–50inclusion and exclusion 242–3, 246legal, district level 90, 144legal, nationwide 16, 38–9, 74, 127, 182

ticket splitting 75tiers, multiple 39, 44, 279–80see also electoral systems, compositeTomz, Michael ixTufte, Edward R. 4, 187, 205Turkey 195two-rounds (TR) 21, 25–7, 42, 45–6

proportionality profile 72unpredictable outcomes 117, 125, 129

two-tier, see electoral systems, composite

Ukraine 15United Kingdom (UK) 44, 91–2, 193, 215

cube law and its demise 205, 207, 214,221, 271

district level threshold ofrepresentation 247–8

Liberal Party demise 2–3proportionality profile 233typical large country FPTP 1, 15, 20, 25,

104

unusual features 180, 190, 197, 237,153

United Nations (UN) 256, 261, 266United States (USA) 44, 135, 205, 215,

266Electoral College 201, 213, 220gerrymander 42–3, 214House 86, 189, 260Large largest seat share 125, 141proportionality profile 70–1Senate 261, 266states within 33, 91, 190, 200unusually pure two-party system 104,

117, 197, 245, 247women’s representation 203–4, 222

unlimited vote (UV) 29Uruguay 41

Valdini, Melody E. 30, 284Van Biezen, Ingrid 279Van Roozendaal, Peter 170Vatter, Adrian 278Venezuela 183volatility of votes 67, 75volleyball scores 203–4, 221–2

Ware, Alan x, 279Warner, Steven 41Warwick, Paul 166wasted votes 104Wattenberg, Martin P. vi, x, 40, 195,

279Webb, Paul xWebster, Daniel 32–3Welch, Susan vi, 203, 283Weldon, Steven A. 188–9, 192–4, 196Williams, Ferd viiWittenberg, Jason ixWolinetz, Steven B. 50–1women’s representation 14, 30, 108, 283

affected by law of minorityattrition 202–4, 222–3

Young, H. Peyton 88, 95

Zambia 15Zimmermann, Joseph F. vi

314

Documents

The Logic of Simple Electoral Systems