A Dynamic Duverger’s Law

Jean Guillaume ForandDepartment of Economics, University of Waterloo

Vikram MaheshriDepartment of Economics, University of Houston

Introduction

• Duverger’s ‘law’ and ‘hypothesis’I Are predictions of static electoral models.I Are corroborated empirically with cross-sectional data on the

number of parties competing in elections.

• However, while electoral rules rarely change within countries,observed party systems are not typically stable over time.

• Political environments change over time (voter preferences,important issues, politicians’ characteristics, etc.), soadjustments to party systems should be expected.

• Question: Do different electoral rules yield systematicallydifferent party system dynamics?

Introduction

• Duverger’s ‘law’ and ‘hypothesis’I Are predictions of static electoral models.I Are corroborated empirically with cross-sectional data on the

number of parties competing in elections.

• However, while electoral rules rarely change within countries,observed party systems are not typically stable over time.

• Political environments change over time (voter preferences,important issues, politicians’ characteristics, etc.), soadjustments to party systems should be expected.

• Question: Do different electoral rules yield systematicallydifferent party system dynamics?

Introduction

• Duverger’s ‘law’ and ‘hypothesis’I Are predictions of static electoral models.I Are corroborated empirically with cross-sectional data on the

number of parties competing in elections.

• However, while electoral rules rarely change within countries,observed party systems are not typically stable over time.

• Political environments change over time (voter preferences,important issues, politicians’ characteristics, etc.), soadjustments to party systems should be expected.

• Question: Do different electoral rules yield systematicallydifferent party system dynamics?

Introduction

• This paper has two goals

1. We introduce a simple dynamic model of party formation andmaintenance.

I The model suggests that plurality rule leads to less variabilityin the number of parties over time than proportionalrepresentation.

I However, the model makes no static prediction about thenumber of parties in a given electoral system at any point intime.

2. We provide empirical evidence in favour of this prediction inelections from a panel of 44 democracies since 1945.

Introduction

• This paper has two goals

1. We introduce a simple dynamic model of party formation andmaintenance.

I The model suggests that plurality rule leads to less variabilityin the number of parties over time than proportionalrepresentation.

I However, the model makes no static prediction about thenumber of parties in a given electoral system at any point intime.

2. We provide empirical evidence in favour of this prediction inelections from a panel of 44 democracies since 1945.

Introduction

• In the model, the political environment evolves, so that acurrently minor party

I Finds it costly to compete in current elections.I But it may receive more support from future voters.

• Key strategic decision by the supporters of a currently minorparty

I Maintain their party or disband it to re-form a party underbetter electoral circumstances.

I An active party generates an option value.

• How this decision is affected by the electoral systemI Plurality rule imposes higher static costs to currently minor

parties (more incentive to exit).I Plurality rule imposes higher future barriers to entry to

currently minor parties (more incentive to stay).

• Less entry by new parties, but also less exit by establishedparties, may be expected under plurality rule.

Introduction

• In the model, the political environment evolves, so that acurrently minor party

I Finds it costly to compete in current elections.I But it may receive more support from future voters.

• Key strategic decision by the supporters of a currently minorparty

I Maintain their party or disband it to re-form a party underbetter electoral circumstances.

I An active party generates an option value.

• How this decision is affected by the electoral systemI Plurality rule imposes higher static costs to currently minor

parties (more incentive to exit).I Plurality rule imposes higher future barriers to entry to

currently minor parties (more incentive to stay).

• Less entry by new parties, but also less exit by establishedparties, may be expected under plurality rule.

Introduction

• In the model, the political environment evolves, so that acurrently minor party

I Finds it costly to compete in current elections.I But it may receive more support from future voters.

• Key strategic decision by the supporters of a currently minorparty

I Maintain their party or disband it to re-form a party underbetter electoral circumstances.

I An active party generates an option value.

• How this decision is affected by the electoral systemI Plurality rule imposes higher static costs to currently minor

parties (more incentive to exit).I Plurality rule imposes higher future barriers to entry to

currently minor parties (more incentive to stay).

• Less entry by new parties, but also less exit by establishedparties, may be expected under plurality rule.

Literature

• Duverger’s law as an electoral district-level prediction.I Most of the theoretical work is here: Feddersen (1992), Palfrey

(1989), Osborne and Slivinski (1996).I Empirical work relies on comparing the votes of runner-up and

next runner-up: Cox (1997), Fujiwara (2011).

• ‘Macro’-level studies of Duverger’s lawI Theoretically, how to address the ‘linkage problem’: Cox

(1997), Morelli (2004).I Empirical work can focus on the number of parties, but using

cross-country data: Blais and Carty (1991), Lijphart (1994),Neto and Cox (1997), Ordeshook and Svhetsova (1994),Taagepera and Shugart (1991).

• Little work on the comparative dynamics of electoral systems.I For theory: Cox (1997), Fey (1997).I For empirical: Anagol and Fujiwara (2015), Chhibber and

Kollman (1998), Gaines (1999), Reed (2001).

Model: Party Formation and Maintenance

• Elections are held at time t = 1, 2, ....

• The party that wins the election at t implements policyx t ∈ {x−1, x0, x1}, with x−1 < x0 < x1.

• Party j can be of type −1, 0 or 1.I Parties cannot commit to policies: if a party of type j is in

power at t, it implements x t = xj .I A party of type 0 is present in all elections.I Parties of type −1 and 1 are formed and maintained by two

long-lived interest groups of type −1 and 1 respectively.

• Note: the party system at any time t can feature one, two orthree parties.

Model: Party Formation and Maintenance

• Elections are held at time t = 1, 2, ....

• The party that wins the election at t implements policyx t ∈ {x−1, x0, x1}, with x−1 < x0 < x1.

• Party j can be of type −1, 0 or 1.I Parties cannot commit to policies: if a party of type j is in

power at t, it implements x t = xj .I A party of type 0 is present in all elections.I Parties of type −1 and 1 are formed and maintained by two

long-lived interest groups of type −1 and 1 respectively.

• Note: the party system at any time t can feature one, two orthree parties.

Model: Party Formation and Maintenance

• Elections are held at time t = 1, 2, ....

• The party that wins the election at t implements policyx t ∈ {x−1, x0, x1}, with x−1 < x0 < x1.

• Party j can be of type −1, 0 or 1.I Parties cannot commit to policies: if a party of type j is in

power at t, it implements x t = xj .I A party of type 0 is present in all elections.I Parties of type −1 and 1 are formed and maintained by two

long-lived interest groups of type −1 and 1 respectively.

• Note: the party system at any time t can feature one, two orthree parties.

Model: Probabilistic Elections

Preference

state st

Pr(st = s0) = q

Pr(st = s1) = 1−q2

x0

x−1

x1

s0

s1

s−1

Model: Probabilistic Elections

Preference

state st

Pr(st = s0) = q

Pr(st = s1) = 1−q2

x0

x−1

x1

s0

s1

s−1

Policy

support pt

p > p > p

p + p + p = 1

pt0 = p

pt−1 = p

pt1 = p

Model: Probabilistic Elections

Preference

state st

Pr(st = s0) = q

Pr(st = s1) = 1−q2

x0

x−1

x1

s0

s1

s−1

Policy

support pt

p > p > p

p + p + p = 1

pt0 = p

pt−1 =p+p

2

pt1 =p+p

2

Model: Probabilistic Elections

Preference

state st

Pr(st = s0) = q

Pr(st = s1) = 1−q2

x0

x−1

x1

s0

s1

s−1

Policy

support pt

p > p > p

p + p + p = 1

pt0 = p

pt−1 = p

pt1 = p

Model: Probabilistic Elections

Preference

state st

Pr(st = s0) = q

Pr(st = s1) = 1−q2

x0

x−1

x1

s0

s1

s−1

Policy

support pt

p > p > p

p + p + p = 1

pt0 = p

pt−1 = p

pt1 = p

Party

structure φtParty

support Pt

Pt0 = p

Pt−1 = p

Pt1 = p

Model: Probabilistic Elections

Preference

state st

Pr(st = s0) = q

Pr(st = s1) = 1−q2

x0

x−1

x1

s0

s1

s−1

Policy

support pt

p > p > p

p + p + p = 1

pt0 = p

pt−1 = p

pt1 = p

Party

structure φtParty

support Pt

Pt0 = p + p

Pt−1 = p

Pt1 = 0

Model: Probabilistic Elections

Preference

state st

Pr(st = s0) = q

Pr(st = s1) = 1−q2

x0

x−1

x1

s0

s1

s−1

Policy

support pt

p > p > p

p + p + p = 1

pt0 = p

pt−1 = p

pt1 = p

Party

structure φtParty

support Pt

Pt0 = p + p

Pt−1 = 0

Pt1 = p

Model: Probabilistic Elections

Preference

state st

Pr(st = s0) = q

Pr(st = s1) = 1−q2

x0

x−1

x1

s0

s1

s−1

Policy

support pt

p > p > p

p + p + p = 1

pt0 = p

pt−1 = p

pt1 = p

Party

structure φtParty

support Pt

pt0 = 1

pt−1 = 0

pt1 = 0

Model: Probabilistic Elections

• We model electoral systems as a mapping

party j ’s win probability

Party support Ptj 7−→

policy outcome x t = xj

• Plurality rule and proportional representation are representedby different ‘contest success functions’.

Model: Probabilistic Elections

Party

structure φt

x0

x−1

x1

Probability

of winning

Proportional representation

Pt0

Pt−1

Pt1

Model: Probabilistic Elections

Party

structure φt

x0

x−1

x1

Probability

of winning

Proportional representation

Pt0

Pt−1

Pt1

Model: Probabilistic Elections

Preference

state st

s0

s1

s−1

Party

structure φt

x0

x−1

x1

Probability

of winning

Plurality ruleMinority penalty α > 0

Pt0

p − α

p + α

Model: Probabilistic Elections

Preference

state st

s0

s1

s−1

Party

structure φt

x0

x−1

x1

Probability

of winning

Plurality ruleMinority penalty α > 0

Pt0

Pt−1

Pt1

Model: Probabilistic Elections

Party

structure φt−1

x0

x1

x−1

Party

structure φt

x0

x−1

x1

Probability

of winning

Plurality ruleEntry penalty β > 0

Pt0

Pt−1 + β

Pt1 − β

Model: Probabilistic Elections

Party

structure φt−1

x0

x1

x−1

Party

structure φt

x0

x−1

x1

Probability

of winning

Plurality ruleEntry penalty β > 0

Pt0

Pt−1

Pt1

Model: Payoffs and Equilibrium

• Interest group j is risk-neutral and has preferences

xj �j x0 �j x−j .

• Supporting a party is costly for activist jI Forming a new party imposes cost c > 0.I Maintaining an existing party imposes cost c ∈ (0, c).

• Interest groups have common discount factor δ ∈ (0, 1).

• We focus on Markov perfect equilibria σ = (σ−1, σ1), in whichinterest groups condition their actions at t only on thepayoff-relevant state (st , φt−1).

Model: Payoffs and Equilibrium

• Interest group j is risk-neutral and has preferences

xj �j x0 �j x−j .

• Supporting a party is costly for activist jI Forming a new party imposes cost c > 0.I Maintaining an existing party imposes cost c ∈ (0, c).

• Interest groups have common discount factor δ ∈ (0, 1).

• We focus on Markov perfect equilibria σ = (σ−1, σ1), in whichinterest groups condition their actions at t only on thepayoff-relevant state (st , φt−1).

Results for Proportional Representation

• Under proportional representation, we consider a strategyprofile σPR such that interest group j supports a party(existing or new) if and only if st 6= s−j .

• Under σPR

I Two or three parties compete in elections, depending on thepreference state.

I The party formation and maintenance decisions of party j areindependent of party structures.

Proposition 1

We identify conditions under which σPR is the unique Markovperfect equilibrium under proportional representation.

• The option value of a maintained party (indexed by c − c)must be low enough to ensure that party j is disbanded whenst = s−j .

Results for Proportional Representation

• Under proportional representation, we consider a strategyprofile σPR such that interest group j supports a party(existing or new) if and only if st 6= s−j .

• Under σPR

I Two or three parties compete in elections, depending on thepreference state.

I The party formation and maintenance decisions of party j areindependent of party structures.

Proposition 1

We identify conditions under which σPR is the unique Markovperfect equilibrium under proportional representation.

• The option value of a maintained party (indexed by c − c)must be low enough to ensure that party j is disbanded whenst = s−j .

Results for Proportional Representation

• Under proportional representation, we consider a strategyprofile σPR such that interest group j supports a party(existing or new) if and only if st 6= s−j .

• Under σPR

I Two or three parties compete in elections, depending on thepreference state.

I The party formation and maintenance decisions of party j areindependent of party structures.

Proposition 1

We identify conditions under which σPR is the unique Markovperfect equilibrium under proportional representation.

• The option value of a maintained party (indexed by c − c)must be low enough to ensure that party j is disbanded whenst = s−j .

Results for Proportional Representation

• Under proportional representation, we consider a strategyprofile σPR such that interest group j supports a party(existing or new) if and only if st 6= s−j .

• Under σPR

I Two or three parties compete in elections, depending on thepreference state.

I The party formation and maintenance decisions of party j areindependent of party structures.

Proposition 1

We identify conditions under which σPR is the unique Markovperfect equilibrium under proportional representation.

• The option value of a maintained party (indexed by c − c)must be low enough to ensure that party j is disbanded whenst = s−j .

Model: Results for Plurality Rule

• Under plurality rule, we focus on strategy profiles in whichI Interest group j forms a new party if and only if st = sj .I Interest group j supports an existing party if st ∈ {sj , s0}.

• Does interest group j support an existing party whenst = s−j?

I Yes under profile σPL with maximal participation.I No under profile σPL with minimal participation.

• Under σPL, three parties compete in all elections.

• Under σPL, two parties compete in all elections, although theiridentities change with the preference state.

• Under both σPL and σPL, the participation decisions of party jare persistent.

Model: Results for Plurality Rule

• Under plurality rule, we focus on strategy profiles in whichI Interest group j forms a new party if and only if st = sj .I Interest group j supports an existing party if st ∈ {sj , s0}.

• Does interest group j support an existing party whenst = s−j?

I Yes under profile σPL with maximal participation.I No under profile σPL with minimal participation.

• Under σPL, three parties compete in all elections.

• Under σPL, two parties compete in all elections, although theiridentities change with the preference state.

• Under both σPL and σPL, the participation decisions of party jare persistent.

Model: Results for Plurality Rule

• Under plurality rule, we focus on strategy profiles in whichI Interest group j forms a new party if and only if st = sj .I Interest group j supports an existing party if st ∈ {sj , s0}.

• Does interest group j support an existing party whenst = s−j?

I Yes under profile σPL with maximal participation.I No under profile σPL with minimal participation.

• Under σPL, three parties compete in all elections.

• Under σPL, two parties compete in all elections, although theiridentities change with the preference state.

• Under both σPL and σPL, the participation decisions of party jare persistent.

Model: Results for Plurality Rule

Proposition 2

We identify conditions under which there exist bounds α and αsuch that σPL is a Markov perfect equilibrium whenever α > α andσPL is a Markov perfect equilibrium whenever α < α.

• σPL is an equilibrium if minority penalty is sufficiently high forinterest group j to disband a party in state (s−j , {j}).

• σPL is an equilibrium if minority penalty is sufficiently low forinterest group j to maintain a party in state (s−j , {−1, 1}).

Model: Results for Plurality Rule

Proposition 2

We identify conditions under which there exist bounds α and αsuch that σPL is a Markov perfect equilibrium whenever α > α andσPL is a Markov perfect equilibrium whenever α < α.

• σPL is an equilibrium if minority penalty is sufficiently high forinterest group j to disband a party in state (s−j , {j}).

• σPL is an equilibrium if minority penalty is sufficiently low forinterest group j to maintain a party in state (s−j , {−1, 1}).

Model: Comparative Predictions

• The parameter region satisfying Proposition 2 is nested inthat satisfying Proposition 1, and it can be shown by exampleto be nonempty.

• At any time t, there can be more or less parties underplurality rule than under proportional representation.

• However, in both equilibria σPL and σPL under plurality rule,there is less expected change in the number of parties than inequilibrium σPR under proportional representation.

I No variability in the number of parties under σPL.I There is less expected party exits under σPL than under σPR ,

because there is three-party competition in s0 under σPR .

Model: Comparative Predictions

• The parameter region satisfying Proposition 2 is nested inthat satisfying Proposition 1, and it can be shown by exampleto be nonempty.

• At any time t, there can be more or less parties underplurality rule than under proportional representation.

• However, in both equilibria σPL and σPL under plurality rule,there is less expected change in the number of parties than inequilibrium σPR under proportional representation.

I No variability in the number of parties under σPL.I There is less expected party exits under σPL than under σPR ,

because there is three-party competition in s0 under σPR .

Model: Comparative Predictions

• The parameter region satisfying Proposition 2 is nested inthat satisfying Proposition 1, and it can be shown by exampleto be nonempty.

• At any time t, there can be more or less parties underplurality rule than under proportional representation.

• However, in both equilibria σPL and σPL under plurality rule,there is less expected change in the number of parties than inequilibrium σPR under proportional representation.

I No variability in the number of parties under σPL.I There is less expected party exits under σPL than under σPR ,

because there is three-party competition in s0 under σPR .

Empirical Evidence: Data

• We aim to provide a robust measure of the correlationbetween the variability in the number of parties and electoralsystems across democratic countries.

• This is a stylised fact which has yet to be established.

• We use data from Constituency-Level Elections (CLE), whichcontains (eligible) district-level electoral results for multipleelections from 44 democracies.

Empirical Evidence: Measuring Proportionality

• Few electoral systems correspond to pure plurality rule andpure proportional representation.

• Our results are robust to using three commonly used proxiesfor electoral systems

I The majoritarian dummy variable of Persson and Tabellini(2005).

I Effective district magnitude: the average number of legislatorselected per electoral district (excluding compensatory seats).

I The disproportionality index of Gallagher (1991): the sum ofsquared differences between parties’ vote and seat shares.

Empirical Evidence: Measuring Proportionality

Figure: Electoral Proportionality: Three Measures

Australia

Bots.

Can.

FranceMalaysia

Mauritius

N. Zeal.

Trin. & Tob.UK

US

Austria

Belg.

Bolivia

BulgariaC. Rica

Cyprus

Czech Rep.

Estonia

FinlandGermany

Greece

Hungary

Iceland

Ireland

Israel

ItalyLatvia

Lux.

Malta

Mexico

Neth.

Norw.

Poland

Portugal

Rom.

Russia

Slovakia

S. Africa

Spain

Sweden

Switz.

Turkey

Ven.

.1.2

.3.4

Ave

rage

Gal

lagh

er D

ispr

opor

tiona

lity

Inde

x

50 100 150Effective District Magnitude

Majoritatian Proportional

Empirical Evidence: Measuring Party Entry and Exit

• We say party j enters in district d in election e of country c if

pjdc(e−1) < 0.05 and pjdce ≥ 0.05,

and we define exit in a district similarly (our results are robustto different entry thresholds).

• Parties may not be active in all districts, but the number ofelectoral districts varies across electoral systems.

• Let σdce denote the fraction of the total seats in the nationallegislature contributed by district d .

• Let ndce denote the number of entering parties in district dduring election e in country c. The Total Entries Nce is

Nce =Dce∑d=1

ndce · σdce ,

and we define Total Exits similarly.

Empirical Evidence: Measuring Party Entry and Exit

• We say party j enters in district d in election e of country c if

pjdc(e−1) < 0.05 and pjdce ≥ 0.05,

and we define exit in a district similarly (our results are robustto different entry thresholds).

• Parties may not be active in all districts, but the number ofelectoral districts varies across electoral systems.

• Let σdce denote the fraction of the total seats in the nationallegislature contributed by district d .

• Let ndce denote the number of entering parties in district dduring election e in country c. The Total Entries Nce is

Nce =Dce∑d=1

ndce · σdce ,

and we define Total Exits similarly.

Empirical Evidence: Specification

• Our main results consist of regressing our measure of partyentry and exit on the majoritarian dummy.

• Our model supports a negative relationship.

• All continuous variables are specified in logarithms. Thismimimises concerns that more variability in the number ofparties could be mechanically driven by differences in thenumber of parties across electoral systems.

• We also control directly for the number of parties.

• This concern is especially valid since, according to the staticDuverger’s law, the number of parties decreases in thedisproportionality of electoral systems.

Empirical Evidence: Specification

• Our main results consist of regressing our measure of partyentry and exit on the majoritarian dummy.

• Our model supports a negative relationship.

• All continuous variables are specified in logarithms. Thismimimises concerns that more variability in the number ofparties could be mechanically driven by differences in thenumber of parties across electoral systems.

• We also control directly for the number of parties.

• This concern is especially valid since, according to the staticDuverger’s law, the number of parties decreases in thedisproportionality of electoral systems.

Empirical Evidence: Duverger’s Law

Table: Static Tests of Duverger’s Law

Variable (1) (2) (3) (4) (5) (6)

MajoritarianDummy

-0.23**(0.09)

-0.17*(0.10)

Effective DistrictMagnitude

0.09***(0.02)

0.05**(0.03)

DisproportionalityIndex

-2.76***(0.86)

-2.77***(0.88)

Decade, Regionaland DistrictNumber Controls

N Y N Y N Y

R2 0.06 0.23 0.10 0.23 0.31 0.42

Number ofObservations

454 454 454 454 454 454

Notes: Heteroskedasticity robust standard errors clustered by country are presented inparentheses. *** - 1%, ** - 5% and * - 10% significance level.

Empirical Evidence: Main Results

Table: Dynamic Tests of Duverger’s Law: Total Movements

Variable (1) (2) (3) (4) (5) (6)

MajoritarianDummy

-0.28**(0.11)

-0.42***(0.13)

Average DistrictMagnitude

0.09**(0.04)

0.12***(0.04)

DisproportionalityIndex

-0.93(0.60)

-0.01(0.79)

Decade,Regional,Number ofDistricts andParties Controls

N Y N Y N Y

R2 0.04 0.27 0.04 0.28 0.01 0.25

Number ofObservations

411 411 411 411 411 411

Notes: Heteroskedasticity robust standard errors clustered by country are presented inparentheses. *** - 1%, ** - 5% and * - 10% significance level.

Empirical Evidence: Robustness to Vote Thresholds

Figure: Coefficient estimates for majoritarian dummy and effectivedistrict magnitude at different vote thresholds, along with their 95%confidence intervals

-10

Dyn

amic

Duv

erge

r C

oeffi

cien

t

1 2 3 4 5 6 7 8 9 10Party Inclusion Threshold (%)

District Magnitude Persson-Tabellini

Conclusion

• We present a novel dynamic reinterpretation of Duverger’slaw.

• We construct a minimal dynamic model with a clean timeseries prediction and show that this prediction is consistentwith available evidence.

• Our ‘macro’ focus on the number of national parties, whileimportant, rules out causal analysis.

I Our paper points to the importance of studying thecomparative intertemporal properties of electoral systems.

I These questions could be posed in ways that allows causalclaims and tests.

Conclusion

• We present a novel dynamic reinterpretation of Duverger’slaw.

• We construct a minimal dynamic model with a clean timeseries prediction and show that this prediction is consistentwith available evidence.

• Our ‘macro’ focus on the number of national parties, whileimportant, rules out causal analysis.

I Our paper points to the importance of studying thecomparative intertemporal properties of electoral systems.

I These questions could be posed in ways that allows causalclaims and tests.