Citizen Candidacy, Party Formation and Duverger’s Law

February 1999

Anouk RivièreECARE - Université Libre de Bruxelles

Abstract:

This paper studies the number of candidates, their ideology and the policy outcome, in a

citizen candidate model with uncertainty and endogenous formation of parties. In a simple

citizen candidate model without parties, each ideology is represented by a candidate if the

cost of candidacy is low enough. If, in order to share the cost of candidacy, citizens are

allowed to create and become members of parties before the candidacy stage, all ideological

groups are not represented: there are generally two candidates, a centrist and an extremist.

The centrist citizens use the risk aversion of extremists, who prefer not to be represented

rather than not to have a centrist candidate: the centrists form a party whose number of

members is large enough to finance its president against one extremist, but not against two.

Therefore, the paper proposes both a first theory of party formation and an alternative

explanation for Duverger’s law.

Keywords:

Citizen candidate, party formation, party membership, risk aversion, Duverger’s law.

J.E.L. classification numbers: C72, D71, D72, D82

My acknowledgements go to Isabelle Brocas, Juan Carrillo, Micael Castanheira, Mathias Dewatripont,Guido Friebel, Marjorie Gassner, Elhanan Helpman, Samir Jahjah, Patrick Legros, Roger Myerson,Frédéric Pivetta, Andrea Prat, Gérard Roland, Yossi Spiegel, Frantisek Turnovec as well as to theparticipants to the Ph.D. Student Workshop in CERGE in May 98, to the Summerschool in PoliticalEconomy in Oberwesel in August 98 and the EEA Congress in Berlin in September 98. All remainingerrors are mine.

Address: Anouk Rivière, ECARE - CP 139, 50 Av. Fr. Roosevelt, 1050 Bruxelles, Belgium. Email:ariviere@ulb.ac.be

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0. Introduction

Since the beginning of the twentieth Century, the most widespread political regime is

the representative democracy: one or several representatives, elected by the citizens,

make the collective decisions.

The main empirical feature of elections is that the citizens who become candidates are

generally not independent: politically active citizens are organized in political parties,

whose leaders usually become candidates. Therefore, modeling elections should

explain the decisions made by citizens to remain independent citizens, to create or to

join a political party, to become candidates and eventually to vote for one or another

candidate.

To the best of my knowledge, a formal model including all these decisions has never

been proposed, probably because of the complexity of the problem. The present paper

develops such a modeling and derives theoretical results in a very simple set-up. The

primitives of the model are the citizens, their preferences and distribution, the policy

alternatives and a single representative first-past-the-post election.

I assume that the citizens and candidates are risk averse and purely ideological: they

care only about the policy, and not about who wins the election per se. 1 When elected,

the winning candidate will chose a policy. I assume that she will implement her

preferred policy: citizens and candidates are not able to credibly commit to any other

policy.

For simplicity, I restrict the ideology space to three points on a line: a left-wing, a

centrist and a right-wing ideologies.2 Moreover, there is some uncertainty: some

citizens only discover their preferences on the day of the election. Whether the median

is left-wing, centrist or right-wing is decided at that time.3 These citizens are not

active in politics.

I first develop a citizen candidate model, with very stylized assumptions. Each citizen

may become a candidate, but this incurs a fixed cost. The decision to become a

candidate is strategic, and I assume that communication is possible between the

potential candidates: the candidacy decisions are coalition-proof.4 Moreover, the

voting decisions are strategic too: voters play best response and recursively suppress

weakly dominated strategies.

2

I obtain the following results: if the cost of candidacy is very small, each ideology is

represented by a candidate in a coalition-proof political equilibrium. The number of

candidates decreases with an increase of the candidacy cost. Equilibria with two

candidates appear to be rare.

In the second part of the paper, I argue that the cost of candidacy is generally too high

to be paid by an isolated individual and I consider the possibility that citizens share

the cost of candidacy by means of political parties. Before the entry stage, there is a

party formation stage: citizens can become a president or a member of a party. If the

president becomes a candidate, the cost of candidacy will be shared equally between

all members and herself. Thus, I define political parties as cost sharing organizations.

In order to avoid coordination problems at this stage, I realistically assume that

communication is possible between potentially active citizens and demand coalition

proofness.

The results of this complete game are as follows: in subgame perfect coalition-proof

Nash equilibrium, if the cost of candidacy is not too small, not all ideological groups

are represented. Under certain expectations, there can be only a centrist candidate. But

there are generally two candidates, one centrist and one extremist. Each ideology

would like to create a party and propose a candidate but the centrists will use the risk

aversion of the extremists in order to have one extremist candidate against a centrist

instead of two extremists against a centrist. Indeed, the centrists will form a party

whose number of members is large enough to finance its president against one

extremist, but not against two. Since extremists prefer not to be represented rather

than not to have a centrist candidate to reduce the volatility of outcomes, one of the

extremist groups will be obliged to stay out of the political life.

Uncertainty is crucial in this paper, because otherwise citizens of the surely median

ideology would create a party that would always win the election, and other parties

would be useless. I think uncertainty is likely, at least at the moment parties are

created.

Thus, the present model predicts Duverger’s Law, probably the most important

informal result of political science. It asserts that plurality-like electoral systems have

a tendency toward two party systems. It is in general seen as a result of strategic entry

by parties, and strategic behavior by voters, who do not want to waste their votes. In a

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formal model with exogenously positioned candidates, Feddersen (1992) obtains

Duverger’s Law as a result of strategic and costly voting: only two candidates receive

votes.5 This simple formal model proposes a new explanation for Duverger’s Law in a

more complete set-up: it is the result of the party formation process, and more

precisely of the strategic choice by the centrist citizens of the number of members in

their party.

The present citizen candidate model without party formation should be compared with

Besley and Coate (1997) and Osborne as Slivinsky (1996), who developed the first

two citizen candidate models.

Common features of our three models are that 1) the cost of candidacy is fixed and

exogenous, 2) citizens are not able to commit to another policy than their sincerely

preferred one, 3) the results are sensitive to the size of candidacy cost, 4) they do not

in general predict a (expected) median voter result and 5) they do not in general

predict Duverger’s Law.

The main differences between my model and the two others are that in my model 1)

there are only three ideologies on a line, 2) there is uncertainty over the citizens’

preferences, and 3) the citizens are purely electoralist (while they allow for electoralist

motivations).

Concerning the strategies of citizens, my model is very far from Osborne and

Slivinsky who assume sincere voting, and closer to Besley and Coate, who consider

strategic voting. But I introduce additional refinements in the equilibrium concepts.

Except in citizen candidate models, the literature makes an arbitrary distinction

between the set of voters and the set of (potential) candidates. The first paper

considering the problem of entry was by Palfrey (1984), where a third candidate may

enter and position after the two “leader” candidates have chosen their positions in the

political spectrum. The leaders will diverge so that the entrant cannot win. Osborne

(1993) develops a model of sequential entry in which a number of potential candidates

decide whether, where and when to enter, with sincere voting. With three potential

entrants, only one becomes a candidate. Feddersen, Sened and Wright (1990) develop

a model of simultaneous entry, with strategic voting. The number of candidates is

decreasing with the candidacy cost, and all candidates hold the median position. In all

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these papers, candidates are purely electoralist, are able to commit, and there is no

uncertainty.

In all the above-mentioned models, no distinction is made between a candidate or a

party. But at the same time, some papers start to make this distinction, and they should

be compared to the present model with parties, where I define parties as cost-sharing

organizations.

The problem of membership in parties was considered by Aldrich (1983): he proposes

a model with two parties where citizens decide whether to become a member of one of

the parties. Members are price-takers, while in the present paper, each member is

decisive to the outcome. In a broader framework, Caplin and Nalebuff (1997) model

competition between a given set of institutions with endogenous membership.

Even though, to my knowledge, no paper includes a theory of party formation, a few

papers propose - often implicitly - a view on the roles of political parties. Caillaud and

Tirole (1998) consider parties as informational intermediaries who own reputation

capital and select high quality candidates, given the voters’ informational deficit:

parties regulate intra-brand competition. In Snyder (1994) parties are constituted by

candidates from different constituencies, and defend the interests of the majority

constituted by party incumbents. Austen-Smith (1984) and Krehbiel (1993) are in the

same spirit. In Alesina and Spear (1989), parties articulate intertemporal politics by

linking a junior and a senior in an overlapping generation model. In Baron (1993),

parties consist in a partitioning of the population into groups voting for one of the

three exogenously given candidates.

In all these papers, either the parties or the candidates are given exogenously.6 In the

present paper, I show that it is possible to make both parties and candidates appear

endogenously, and that the party formation process drives non trivial results.

The remainder of the paper is organized as follows: section 1 describes the world.

Section 2 develops the citizen candidate model and computes the coalition-proof

political equilibria. Section 3 considers the extended game where citizens may create

and become members of parties. The coalition-proof complete political equilibria are

computed in section 4. Section 5 considers the question of robustness and proposes

several generalizations. Section 6 concludes.

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1. Description of the world

The community is composed of N citizens, labeled j ∈ N = {1, ..., N). A representative

must be selected who will implement a policy alternative x. There is a single election.

The candidate with the highest score wins (plurality). Ties are broken by a fair coin.

Ideologies:

The ideological world is one-dimensional and formed by three ideologies: left,

represented by policy - 1; center, represented by 0 and right, represented by 1.

Citizens:

I assume that a total of 3n citizens know their ideology from the start: n are left-wing

for sure, n are centrist for sure, and n are right-wing for sure.7 The ideology of the

other N - 3n citizens is uncertain: they only acquire ideology -1, 0 or 1 the day of the

election. This assumption is made to represent uncertainty on the distribution of

voters.

The citizens are ordered in such a way that j = 1,..., n represent the surely left-wing

individuals, j = n + 1, ..., 2n the center individuals, j = 2n + 1, ..., 3n the right-wing

individuals and j = 3n + 1, ..., N the uncertain ideology individuals.

States of the world

There are three possible states of the world, depending on whether the median voter is

left-wing (-1), centrist (0) of right-wing (1). The probabilities of these states are 3/10,

4/10 and 3/10 respectively.8

Let TI = (L, C, R) denote the 3 component vector under the state of the world I, I = -1,

0, 1, where L, C and R are the number of citizens of respectively left-wing, centrist,

right-wing ideology. We assume in a general way that

T-1 = (L, C, R), with L > C + R, C ≥ R, and L + C + R = N.

T0 = (L, C, R), with L = R < C and L + C + R = N.

T1 = (L, C, R), with R > C + L, C ≥ L and L + C + R.9

Candidates:

6

Candidates are simply citizens deciding to take part in the election. The cost of

becoming a candidate is δ and is the same for everybody. This cost represents time

and money spent on election campaign.

I assume that the cost of candidacy is not too large: δ/n < 3/10.10

Candidates are purely ideological: they want to win the election for the sole purpose

of influencing the policy outcome. Thus they are not electoralist: there is no private

benefit of being elected.

No commitment:

If a type i citizen is elected, the policy outcome is i.

Utility function:

I assume that the policy implemented affects the utility of citizens in a quadratic

way11. The utility of a citizen of ideology z if policy i is given by:

Uz(i) = - (z - i)2

Default utility:

If there is no candidate, everybody gets a minimal utility, fixed here to -4. This default

utility corresponds to the minimal possible utility provided by a candidate (which

varies between -4 and 0).

2. The Citizen Candidate model

2.1. Description of the game

Timing of the game:

- t = 0: Citizen decide simultaneously whether to become candidate or not.

- t = 1/2: the state of the world is realized and observed by everybody.

- t = 1: each citizen votes for one of the candidates.

- t = 2: policy is implemented.

I seek the most relevant subgame perfect equilibria of this game.12

Entry (stage 0)

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The entry stage is a game. Citizen i’s strategy is si ∈ {0,1}, where si = 1 denotes entry

and si = 0 means staying out.

Citizens will play best response, anticipating next stages, and taking into account that

there are three states of the world, left-wing, center and right-wing, with respective

probabilities 3/10, 4/10 and 3/10.

Moreover, I assume that communication between potential candidates is possible, and

require that the entry vector s = (s1, ..., sN) given voting behaviors α at next stage is a

coalition-proof Nash equilibrium. This concept, introduced by Bernheim, Peleg and

Whinston (1987), applies in an environment where players can freely discuss their

strategies, but cannot make binding commitments. I think that it suits quite well the

situation of citizens active in politics.

Its informal idea is as follows: a vector of entry decisions is coalition-proof if, given

the strategies outside any coalition, this coalition cannot be better off by deviating in a

way such that no subcoalition would like to deviate again.

The day before the election, the state of the world is realized and the type of each

citizen becomes common knowledge. Then comes the election. Each state of the

world corresponds to a voting game. Thus stage 1 is one among three games: a game

in a left-wing world, a centrist world or a right-wing world.

Voting in a I state of the world (stage 1).

The game is a perfect information game: every citizen knows the type of all the others.

Given a candidate set C, each citizen may decide to vote for any candidate in C. Let

αIj ∈ C denote citizen j ‘s decision.13

Citizens correctly anticipate the policy that would be chosen by each candidate and

vote strategically (i.e. maximize their expected utility conditional on being pivotal).

I use the following concept of refined voting equilibrium: 1°) voting decisions are a

best response to what others do, 2°) no individual uses a weakly dominated strategy;

3°) individuals sequentially eliminate weakly dominated strategies in the set of

possible actions of the other players,14 4°) if a citizen is strategically indifferent

between all candidates, she votes for her sincerely preferred candidate; 5°) if in

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addition there are several sincerely preferred candidates, she votes for these candidates

with equal probabilities.

The concept of voting equilibrium used throughout the paper can seem very

sophisticated, but is actually not. It is clearly illustrated in the proof of lemma 1 in the

appendix. Moreover, it will appear to be consistent with sincere voting in section 2.2.

Let EI(C) denote the set of all refined voting equilibria in a I state of the world if the

set of candidates is C.

Combining the stages: definition of a coalition-proof political equilibrium

Combining the analysis of all stages, I want to define a coalition-proof political

equilibrium as a subgame perfect equilibrium with my additional refinements.

Combining the analysis of all stages, I define a coalition-proof political equilibrium as

a vector s of entry decisions and a function α describing voting behavior such that: (i)

s is a coalition-proof equilibrium of the entry game given α and (ii) for all non empty

sets C, for any I, αI(C) is a voting equilibrium.

Since each voting game possibly involves multiple equilibria, subgame-perfection

includes the idea of rational expectations: citizens at the entry stage are faced to the

problem of “guessing” which equilibrium will be selected for each possible set of

candidates (in each state of the world). These expectations are represented by a

function α(⋅).

2.2. Characterization of coalition-proof political equilibria

In this paragraph I compute the coalition-proof political equilibria. I first make the

following assumptions regarding the behavior of citizens:

Undecided passivity assumption:

The individuals whose preferences are not decided at time 0 do not participate in the

entire political process, but only in the voting stage.

9

The idea of the undecided passivity assumption is that still undecided citizens free-

ride on the political activities of the other individuals until the day of the election.

They are not active in the political life, because they do not know yet which policy

they prefer.

Normal coordination assumption:

Call α any set of rational expectations regarding the voting equilibrium in all possible

situations, C a set of citizens, Ci a set of type i citizens and P(I,J) the probability of

victory of a type J candidate if the state of the world is I.

I assume that α is such that

∀ i, C, Ci: C’ = C ∪ Ci ⇒ P(I,i)(α(C)) ≥ P(I,i)(α(C’)), ∀ I.

∀ j, C, Ci: C’ = C ∪ Ci ⇒ P(I,j)(α(C)) ≤ P(I,j)(α(C’)), ∀ I.

Basically, the normal coordination assumption says that increasing the number of

candidates of a given ideology does not improve the coordination of citizens of that

ideology with respect to the coordination of other types.15

This assumption is quite intuitive since I assume no differentiation in quality between

the candidates.16

In order to compute the different coalition-proof political equilibria. I will use the

following lemmas:

Lemma 1: In all refined voting equilibria with only candidates of different

ideologies, citizens vote for their sincerely preferred candidate.

Lemma 2: There is no coalition-proof political equilibrium with more than one

candidate of each type.

The proof of lemmas 1 and 2 are given in the Appendix. Lemma 1 is a consequence of

the refinements made upon the voting behavior, which enable an absolute majority to

insure the victory of its only candidate.

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Lemma 2 is a direct consequence of the coalition-proofness assumption and the

normal coordination assumption: Since δ > 0, not becoming candidates for any

coalition of type i individuals is always a self-enforcing deviation if some other i-type

individual becomes a candidate.

Proposition 1: In coalition-proof political equilibrium, the number of candidates

(from 0 to 3) is increasing as the cost of candidacy is decreasing. The complete

characterization is as follows:

- if δ > 4, no candidate enters. Everybody gets utility -4;

- if 2 < δ < 4, exactly one candidate enters. If her type is u, then outcome is u.

- if 1 < δ < 2, there are two types of equilibria: one candidate equilibria, with one

centrist candidate; and two candidates equilibria: with a candidate of types -1

and a candidate of type 1 (winning with ex ante equal probabilities).

- if 3/10 < δ < 1, there is one centrist candidate.

- if δ < 3/10, there is one candidate for each type.

Proof of proposition 1: see the appendix, which proposes a demonstration for the

general case with 0 < p < 1/3 (including p = 3/10).

Figure 1: Table of all coalition-proof political equilibria

-1 vs 0 vs 1 -1

-1 vs 1 1 0 0 0 φ 0 3/10 1 2 4 δ

-1 vs 1 represents an equilibrium with a left-wing and a right-wing candidates.

The basic idea of proposition 1 is as in Besley and Coate. The citizens must trade off

the cost of candidacy and moving the expected policy toward their ideology. The more

costly it is to become a candidate, the more the individuals are accepting a policy far

from their preferred one. If δ is prohibitive, there is no candidate at all. But as this cost

becomes small, each ideological group in the electorate is represented by a candidate,

who will apply her favorite policy if the state of the world appears to be her’s.

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Remarks:

1°) When δ < 3/10, figure 1 shows that there are three candidates: there is a classical

prisoners’ dilemma between the left-wing and the right-wing electorate: ex ante both

groups prefer a situation with only a centrist candidate but cannot avoid the temptation

to propose a candidate of their own.

2°) A two-candidate equilibrium is only possible for a small range of values δ

(equilibrium -1 vs 1 when 1 < δ < 2).

3°) Coalition-proofness eliminates only one pure strategy Nash equilibrium: when

4/10 < δ < 1, there is a Nash equilibrium leading to -1 vs 1, which is not coalition-

proof (and is Pareto dominated by equilibrium 0).

4°) Without coalition-proofness, equilibria with mixed strategies at the entry stage

would exist. Note that voters of one given ideology prefer equilibria where they are

able to coordinate, as is the case with pure strategies, and even more with coalition-

proofness.17

3. A Game with Endogenous Parties

In the citizen candidate model, the outcome depends on the cost of candidacy, and can

be suboptimal if this cost is too high: if δ > 4, there is no candidate and all individuals

have the minimum possible utility -4.

In the real life, independent citizens are not frequent and costs of candidacy seem to

be substantial. It then seems natural that several citizens of the same ideology would

try to regroup and share the cost of candidacy.

This section considers this possibility and introduces parties as cost sharing

organizations: before the candidacy stage, there is a simple one stage subgame, called

party formation subgame.

Each citizen may decide to create a party or to become a member of a party. I make

the following

Party membership and presidentship assumptions:

(1) any citizen has the opportunity to become a president of a party;

(2) the party president is automatically a member;

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(3) a citizen can become member of at most one party, provided that its president

share the same ideology as her’s;

(4) a member of a party is not allowed to propose her candidacy as an independent

citizen;

(5) the party president decides unilaterally if she becomes a candidate;

(6) if the president becomes a candidate, the cost of candidacy δ is shared equally

between all members and herself.

These assumptions deserve some comments. Assumption (1) seems natural in a

democracy. Assumption (3) says that parties are formed by citizens of a common

ideology, which seems realistic. Assumptions (2) and (6) define a financial egalitarism

inside a party. Assumptions (4) and (5) define the authority of the president inside a

party. They may seem a bit dictatorial, but in fact are not: since members and

president pay the same share of the cost by assumption (6) and have the same

ideologies by assumption (3), they all agree about the best candidacy decision and

eventually share the same utility.

These assumptions define parties as campaign contribution organizations. Some other

roles of parties (regarding for example free riding, coordination or even cooperation

between ideological groups) will appear endogenously as a consequence of the game,

while other important roles will obviously be neglected.

By the undecided passivity assumption, the undecided type citizens are assumed not to

play in the party formation subgame as in the candidacy stage.

Timing of the extended game:

- t = - 1: each citizen decides whether to become a party member or president.

- t = 0: each individuals decides whether to become a candidate or not.

- t = 1/2: the state of the world is realized.

- t = 1: election after realization of the state of the world.

- t = 2: policy is implemented.

I look for the subgame perfect equilibria of this game.

13

Party formation (stage -1)

In this stage, each citizen i must decide whether to become member or president of a

party.18 The strategy of citizen is mi: citizen i becoming president of a party is denoted

by mi = i; citizen i remaining independent is denoted by mi = 0; citizen i becoming

member of a party whose president is j, is denoted by mi = j ≠ i, with mj = j. Thus, by

the party membership and presidentship assumptions, mi ∈ {0, 1, ..., n} if i ∈ {1, ...,

n}, mi ∈ {0, n + 1, ..., 2n} if i ∈ {n + 1, ..., 2n}, mi ∈ {0, 2n + 1, ..., 3n} if i ∈ {2n + 1,

..., 3n}. By undecided passivity assumption, mi = 0 if i ∈ {3n + 1, ..., N}.19

As in the candidacy stage, I demand coalition-proofness in the party formation stage. I

think it is very reasonable: citizens politically active are likely to meet and discuss

with each other.

A coalition-proof party formation equilibrium is a coalition-proof Nash equilibrium

vector m given anticipated decisions s and α at the next stages.

The party formation stage gives a set of party presidents and their members. Then

comes the candidacy political game, where individuals acts as in the small game of

section 2, but the fact that party members behave according to their chart will modify

the candidacy stage (stages 1 and 2 remaining exactly as in the small game).

Candidacy game (stages 0 and next ones)

When the time of candidacy comes, m is given; the players can be partitioned in three

sets: 1°) the independent citizens, who can decide to become independent candidates

if they pay the entry cost δ, 2°) the party presidents, who can decide to become a party

candidate, all its members sharing equally the δ cost, 3°) the non president members,

who must respect their membership contracts, their strategy being restricted to si = 0.

The actions taken at the party formation stage, represented by vector m played a stage

-1, induces a vector δc (m) = (δ1, ..., δ3n) of individual cost of candidacy:

+ ∞ if i is member of some party

δi = δ/Mi if i is president of a party with Mi members

δ if i is in no party

14

The rest of the game is as in the game without parties.

I define a coalition-proof political equilibrium of the complete game with m given as

a coalition-proof political equilibrium of the small game given the vector δC(m)

instead of vector δ and expectations α(.,m) instead of α(.).

Combining the stages: coalition-proof complete political equilibrium

Combining the analysis of all stages, I want to define a coalition-proof complete

political equilibrium as a subgame perfect equilibrium with my additional

refinements. As in section 2, the subgame-perfection is complex because each

political game possibly involves multiple equilibria.

Let s(m) denote the commonly anticipated candidacy decisions when m was played at

stage -1. The function s(.) represent expectations over the candidacy stage actions.

These expectations will have to be rational.

Combining the analysis of the stages, I define a coalition-proof complete political

equilibrium (CPCPE) to be a vector m of party formation decisions, a function s(.) of

candidacy decisions and a function α(.,.) describing voting behavior such that: (i) m

is a party formation coalition-proof equilibrium given s(.), (ii) for all m, s(m) is an

equilibrium of the entry game given voting decisions α and (iii) for all non empty sets

C, all m vectors, αI(C,m) is a totally refined voting equilibrium for any i.

4) The Results

Note first that the simple concept of Nash Equilibrium at the party formation stage,

without coalition proofness, would be too weak, because it allows for coordination

failures: if every citizen expects the others not to become members, parties are useless

and no citizen creates one. If δ > 4, there exist complete political equilibria with 0, 1,

2 or 3 candidates 20, none of them being Pareto dominating. The 0 candidate result is

proven in the appendix.

15

The following three remarks are useful in understanding the mechanism of the main

results:

Remark 1: Members of parties who propose a candidate are decisive.

Indeed, if a member were not decisive, she would get out of the party in order to avoid

paying her share of the cost. Thus, active parties will be minimal coalitions.

Remark 2: The cost the president (and each member) of a party is ready to pay to

become a candidate depends on the other candidates.

This was already the case for citizens in the game without parties: the expected gain of

candidacy in term of policy depends on the set of other candidates. For example,

members of a centrist party are willing to pay up to 7/10 to present a candidate against

a right-wing candidate, while they are only willing to pay 4/10 against a right-wing

and a left-wing, because they win more often in the former situation:

EUc(C = {c,r}) = -3/10 - δc > EUc(C = {r}) = -1 if δc < 7/10

EUc(C = {l,c,r}) = -6/10 - δc > EUc(C = {r,c}) = -1 if δc < 4/10

Remark 3: The number of members necessary for a party president to become a

candidate depends on the set of other candidates.

This is a consequence of Remark 2, since having more members means each member

paying a smaller share.

Remarks 2 and 3 show that the numbers of members in the parties are the crucial

variables that will eventually decide the set of candidates.

Call Mi the number of members of the party whose president is i:

Mi = #{j mj = i} if mi = i, Mi = 0 if mi ≠ i.

Call Mi(C) the minimum number of members in the party created by i for i to become

a candidate, if all citizens in C\{i} are candidates.

16

The former remarks basically say that Mi(C) depends on C and i. Of course, if two

presidents i and j have the same ideologies, Mi(C∪ {i}) = Mj(C∪ {j}) for any C. In the

lemma 6.0 of the appendix, I characterize all possible Mi(C)s, for l, c, r respectively

left-wing, centrist and right-wing presidents.

Lemma 6.1 of the appendix computes the continuation equilibria for all possible party

formation vectors m.

Note that there exist m vectors leading to multiple equilibria. For example, consider a

situation with δ < Mc < Mc(c,l) (centrist party whose size is not sufficient to have a

candidate against an extremist), Mr(r,l) ≤ Mr < Mr(r,c) and Ml(r,l) ≤ Ml < Ml(r,c) (two

extremist parties whose sizes are sufficient to run against an opposite extremist but

not sufficient against a centrist). Such situation corresponds to cell C2.R3.L3 of

lemma 6.1 of the appendix and has two continuation equilibria: 0 and -1 vs 1. Thus,

the determination of the complete political equilibria in general depends on the

expectations s(.) for the different m vectors.

Proposition 2: Complete characterization of the equilibria:

If δ < 4/10, a situation with 3 candidates is the only CPCPE.

If 7/10 < δ, a situation with a centrist candidate is a CPCPE.

If 8/10 < δ, a situation with a centrist candidate and an extremist candidate is a

CPCPE for any set of rational expectations.

Only for some ranges of δ with δ < 21/10, a situation with 3 candidates is a

CPCPE.

← 0← -1 vs 0 vs 1

← 0 vs 1

← -1 vs 0

0 4 7 8 10 14 20 21 δ 10 10 10 10 10 10 10 10

= equilibrium for any set of rational expectations

= equilibrium for some set of rational expectations

17

= never an equilibrium

Proof of proposition 2: See the appendix (including generalization with 0 < p < 1/3).

Idea of the proof when δ < 3/10:

Parties need not exist, since any independent citizen is ready to pay the cost by her

own. There are new Nash equilibria with more or less cost sharing, but they are not

coalition-proof.

Idea of the two candidate result:

The result rests on two facts: 1°) The centrists prefer a situation with one extremist

and one centrist candidates rather than three candidates; indeed, they win more often

in the former situation; 2°) By risk aversion, to reduce volatility of outcomes,

extremists prefer not to be represented rather than not to have a centrist candidate:

left-wing prefer 0 vs 1 to -1 vs 1. Therefore, centrists form a party whose size Mc is

large enough to finance its president against one extremist, but not against two: Mc =

Mc(c,r) < Mc(c,r,l). The individual cost is too high: it would not be in the centrist

president’s best interest to become a candidate against two extremists. In such a way,

they credibly threaten not to have a candidate if a second extremist candidate enters.

And there will be only one extremist candidate against a centrist.

Thus, two assumptions are crucial to our Duvergerian result: the extremist citizens

have to be risk averse, and the probability of a centrist world may not be too small.

Otherwise, the left-wing extremists prefer -1 vs 1 rather than 0 vs 1 and the centrist

citizens are not able to make this threat not to enter.

Unfortunately, centrists are not able to apply this threat for all values of δ. If δ is too

small (< 4/10), even an independent centrist is willing to enter against two extremists.

And there also exist some small ranges of δ where the minimal numbers of members

to enter against one and against two extremists are the same.21

Idea of the one centrist candidate result:

18

A situation with only one centrist candidate is preferred by all ideologies to a situation

with three candidates, or with two extremist candidates. The citizens would like to

find sizes of parties such as to kill the prisoners’ dilemma between the left-wing and

the right-wings, and it will sometimes be possible.

Consider a situation with Mc < Mc(c,l), Mr(r,l) ≤ Mr < Mr(r,c), Ml(r,l) ≤ Ml < Ml(r,c)

(corresponding to cell C2.R3.L3 of lemma 6.1 of the appendix). Assume furthermore

that δ/Mc < 1 < δ/(Mc - 1). The continuation equilibrium is outcome 0 by coalition-

proofness.

In such situation, no coalition of extremist citizens wants to increase the size of their

party, because it would lead to continuation equilibrium -1 vs 1.

Thus, this situation is an equilibrium if no coalition of centrist members wants to

deviate, i.e. if their deviation would imply the continuation equilibrium -1 vs 1 among

the set {0, -1 vs 1) (cell C2.R3.L3 with δ/Mc > 1). And this depends on the

expectations.

Thus, 0 is an equilibrium if rational expectations are such that C(m) = {l,r} ∀ m such

that δ < Mc < Mc(c,l), Mr(r,l) ≤ Mr < Mr(r,c), Ml(r,l) ≤ Ml < Ml(r,c).

Idea of the three candidate result:

Consider a situation leading to three candidates. A situation with only one centrist

candidate would be preferred by all ideologies. If they can make a collective self-

enforcing deviation to such a situation, they will. Such deviation can happen only if

1°) outcome 0 can be reached and 2°) no left-wing (respectively right-wing) coalition

would deviate further to -1 vs 0 (respectively 0 vs 1). Such a deviation is always

possible if there exists Mc with Mc({c}) < Mc and δ/Mc < 1 for some number of

members Mc, which is guaranteed for sufficiently big δs.

Proposition 2 asserts that, when the cost of candidacy requires cost sharing via parties,

there are generally two parties eventually presenting a candidate, in sharp contrast to

the small citizen candidacy game. Thus, the paper proposes an alternative explanation

for Duverger’s law, in a stylized model with rational agents where the entire political

landscape is endogenous, and the size of the parties is crucial.

19

Note that the results in this paper predict two major parties, in the sense of two parties

presenting a serious candidate, but allow for multiple small “unimportant” parties (in

accordance with the usual interpretation of Duverger’s Law).

Note also that the equilibrium cost sharing level achieved by the parties is minimal:

citizens try to free-ride on the politically active citizens, and each member of a party

presenting a candidate is decisive.

Some hypotheses regarding the political space and the distribution of preferences are

very restrictive. In the next section, we show that the results hold in more general

conditions.

5. Robustness and Generalizations

5.1. Generalization to 0 < p < 1/3.

The core of the paper examined a very special example of distribution, with symmetry

and p = 3/10. In the appendix, I show that the spirit of proposition 4 holds for any

symmetric distribution with 0 < p < 1/3. The general result is summarized in the

Proposition 3: In a citizen candidacy model, situations with two candidates are

very rare. But if there is cost sharing via endogenous political parties, with a

symmetric distribution and p < 1/3, centrists are often able to deter simultaneous

entry of two extremist parties, leading to situations with a centrist and an

extremist parties.

Proof of proposition 3: See the appendix. As p gets closer to 0, the zones where

Duverger’s result of proposition 3 does not hold tend to the entire line, while as p gets

closer to 1/3, these zones become empty.

Note that proposition 3 does not hold if p > 1/3 (e.i. if a centrist world is less likely

than any of the extremist worlds), because under such distributions, left-wing prefer

outcome -1 vs 1 to 0 vs 1, and the centrists are not able to prevent simultaneous entry

of two extremists by the threat not to enter: there will be three candidates.

5.2. Introducing asymmetry:

In this paragraph, I consider a simple example of an asymmetric distribution to show

that the Duvergerian effect of former sections is maintained even without symmetry.

20

Proposition 4: Assume that the probabilities p, 1 - p - q and q of a left-wing,

centrist and right-wing worlds are 7/20, 8/20 and 5/20 respectively. If δ > 2, a

situation with a centrist and an extremist candidates is a CPCPE.

Proof of proposition 4: See the appendix.

Note the surprising result here: outcome 0 vs 1 is possible in equilibrium, both

candidates being strictly to the right of the expected median ideology, which is -1/10.

Again, the results could be generalized to other (p,q) couples, with the same flavor as

long as the centrist world is the most likely: 0 < q < p < 1 - p - q.

5.3. Allowing for more ideological groups.

A natural question is whether the results are specific to the three ideology case.

Unfortunately, considering more ideological groups can be problematic, because it can

bring severe multiple equilibrium problems at the voting stage, due to problems of

coordination.

A distinction should be made between the number of ideologies in the political

spectrum and the number of ideologies that the median voter can have. If the

coordination of citizens at the voting stage is not too bad, our Duvergerian result holds

with more ideological groups for very reasonable situations, when the ideologies

which the median can hold are three ideologies at the center of the spectrum.

Proposition 5: Assume there are 5 ideologies, and if the probability of positions -

2, -1, 0, 1 and 2 worlds are respectively 0, 3/10, 4/10, 3/10 and 0. If δ > 2,

situations -1 vs 0 and 0 vs 1 are CPCPE.

Proof of the proposition: see the appendix.

Introducing more ideologies in the set of potential median positions is more

problematic. Nevertheless, if the coordination of citizens at the voting stage is not too

bad, the two candidate result holds under similar distribution conditions with four

ideological groups. For example, we have the

21

Proposition 6: If there are 4 ideologies, and if the probability of a -2, -1, 0 and 1

worlds are respectively 2/10, 3/10, 3/10, 2/10, if δ > 3/4, the CPCPE have exactly 2

candidates.

Proof of proposition 6: see the appendix.

5.4. Allowing for commitment:

Through out the paper, following Besley and Coate (1997), no possibility of credible

commitment is assumed. Interestingly, the model can be generalized to a situation

with a commitment stage before the candidacy stage.22 We found that the equilibrium

with two candidates holds. The idea is that the centrist will stay out of the race and let

one of the extremist groups commit to the centrist policy: there are two extremist

candidates, one of them committed to 0.23

6. Conclusion

This paper develops a very stylized model to build a bridge between the classical

literature with exogenous parties à la Downs and the new citizen candidate literature à

la Besley and Coate, by proposing a citizen candidate model with party formation.

It proposes an alternative explanation for the well-known Duvergerian phenomenon

observed in majority like elections: the tendency towards two party system can be

explained not only by strategic considerations by exogenous potential parties and

voters, but by the formation process of political parties itself, and the strategic choice

of their sizes by the citizens.

The model relies on very restrictive assumptions, and should be seen as a first step,

showing that party formation can be modeled with non trivial results and that

considering parties or candidates is not equivalent. Section 5 shows that the spirit of

the results holds in more general conditions. Future research should try to test further

the robustness of the results with regard to the hypotheses. I could also consider other

electoral systems, mixed electoralist/ideological agents, asymmetry of information,

quality differentiation between citizens, organization inside a party...

22

Endnotes:

1. A summarizing discussion about electoralist vs ideological candidates can be found in Alesina and

Rosenthal (1995).

2. I also propose generalizations with four or five ideologies.

3. Uncertainty was first stylized in a similar way in Castanheira (1997).

4. Borrowed from Bernheim, B. Douglas, Bezalel Peleg and Michael D. Whinston (1987).

5. For a very complete overview, see Cox (1997).

6. The question of interest group formation is somewhat germane to the problem of political party

formation, and has not been yet studied either.

7. The assumption that the three groups are of the same size is inessential.

8. The results generalize to any symmetric distribution with 0 < p = q < 1/3, where p and q

respectively represent the probability of a left-wing and a right-wing world. The appendix develops

the general proofs. An asymmetric situation is considered in section 5.

9. The assumptions on L, C and R simply insure that 1°) the median voter is indeed of type i in a I

state of the world and 2°) the distribution in each state of the world is single-peaked.

10. This assumption will ensure that the entry cost is never too high if citizens may share it.

11. The qualitative results of the paper hold for any strictly concave utility function.

12. The present game differs from Besley and Coate in three ways: there is uncertainty at stage 0, the

voting equilibrium concept will be more refined, and the entry equilibrium concept will be more

refined.

13. I could easily generalize the model to allow for abstention without modifying the qualitative

results.

14. I borrow this natural concept to Moulin (1979).

15. Suppressing this assumption would not suppress any of the equilibria obtained throughout the

paper, but would allow for new equilibria with several candidates of the same ideology.

16. But in a model with differentiation, the normal coordination assumption would be restrictive.

Consider a situation with two “bad” left-wing candidates compared with a situation with these two

bad candidates and a very good left-wing candidate. The normal coordination assumption says that

coordination would be easier in the first situation, which is not obvious.

17. Symmetric equilibria, i.e. mixed strategy equilibria where citizens of same type have the same

strategy, would therefore lead to a high level of suboptimality.

18. The party formation stage could be divided into two stages, a party creation stage followed by party

membership, with similar results but heavier formalization.

19. Citizen i becoming member of a party without a president is considered as equivalent to i

remaining independent.

20. This is true even with suppression of weakly dominated strategies.

23

21. This possibility is a consequence of the discrete number of members, and could be solved by the

introduction of “fractions” of members, or “fractions” of contributions, but I don’t do it for

simplicity.

22. Developing this new model is technically very costly.

23. The extremists whose president commits to a moderate policy would like to create a second party -

extremist. This will not happen because of a free-riding problem among the extremists: they are not

ready to pay the same cost for an extra extremist candidate as for the centrist one, and members of

the extremist party expected to present a moderate candidate prefer to dissolve and let the other

party commit to the moderate policy.

24

1 A summarizing discussion about electoralist vs ideological candidates can be found in Alesina and

Rosenthal (1995).2 I also propose generalizations with four or five ideologies.3 Uncertainty was first stylized in a similar way in Castanheira (1997).4 Borrowed from Bernheim, B. Douglas, Bezalel Peleg and Michael D. Whinston (1987).5 For a very complete overview, see Cox (1997).6 The question of interest group formation is somewhat germane to the problem of political party

formation, and has not been yet studied either.7 The assumption that the three groups are of the same size is inessential.8 The results generalize to any symmetric distribution with 0 < p = q < 1/3, where p and q respectively

represent the probability of a left-wing and a right-wing world. The appendix develops the general

proofs. An asymmetric situation is considered in section 5.9 The assumptions on L, C and R simply insure that 1°) the median voter is indeed of type i in a I state

of the world and 2°) the distribution in each state of the world is single-peaked.10 This assumption will ensure that the entry cost is never too high if citizens may share it.11 The qualitative results of the paper hold for any strictly concave utility function.12 The present game differs from Besley and Coate in three ways: there is uncertainty at stage 0, the

voting equilibrium concept will be more refined, and the entry equilibrium concept will be more

refined.13 I could easily generalize the model to allow for abstention without modifying the qualitative results.14 I borrow this natural concept to Moulin (1979).15 Suppressing this assumption would not suppress any of the equilibria obtained throughout the paper,

but would allow for new equilibria with several candidates of the same ideology.16 But in a model with differentiation, the normal coordination assumption would be restrictive.

Consider a situation with two “bad” left-wing candidates compared with a situation with these two bad

candidates and a very good left-wing candidate. The normal coordination assumption says that

coordination would be easier in the first situation, which is not obvious.17 Symmetric equilibria, i.e. mixed strategy equilibria where citizens of same type have the same

strategy, would therefore lead to a high level of suboptimality.18 The party formation stage could be divided into two stages, a party creation stage followed by party

membership, with similar results but heavier formalization.19 Citizen i becoming member of a party without a president (corresponding to mi = j ≠ i while mj ≠ j)

is considered as equivalent to i remaining independent.20 This is true even with suppression of weakly dominated strategies.21 This possibility is a consequence of the discrete number of members, and could be solved by the

introduction of “fractions” of members, or “fractions” of contributions, but I don’t do it for simplicity.22 Developing this new model is technically very costly.

25

23 The extremists whose president commits to a moderate policy would like to create a second party -

extremist. This will not happen because of a free-riding problem among the extremists: they are not

ready to pay the same cost for an extra extremist candidate as for the centrist one, and members of the

extremist party expected to present a moderate candidate prefer to dissolve and let the other party

commit to the moderate policy.

References

Alesina, Alberto and Howard Rosenthal (1995): Partisan Politics, Divided Government andthe Economy, Cambridge University Press.

Alesina, Alberto (1988): “Credibility and Policy Convergence in a Two-Party System withRational Voters”, American Economic Review, Sept 1988, 796-805.

Alesina, Alberto and Stephen E. Spear (1988): “An overlapping Generations Model ofElectoral Competition”, Journal of Public Economics.

Austen-Smith, (1984): “Two-Party Competition with Many constituencies”. MathematicalSocial Sciences, 7, 177-198.

Baron, David P. (1993): “Government Formation and Endogenous Parties”, AmericanPolitical Science Review, 87 (March): 34-47.

Bernheim, B. Douglas, Bezalel Peleg and Michael D. Whinston (1987): “Coalition-ProofNash Equilibria: I. Concepts”, Journal of Economic Theory, 42, 1-12.

Besley, Timothy and Stephen Coate (1997): “An Economic Model of RepresentativeDemocracy”, Quarterly Journal of Economics, 112(1), 85-114.

Caillaud, Bernard and Jean Tirole (1997): “Parties as Political Intermediaries”, CERASworking paper n° 98-01.

Calvert, R. (1985): “Robustness of the Multidimensional Voting Model: Candidates’motivations, Uncertainty, and Convergence”. American Journal of Political Science 29, 65-95.

Caplin and Nalebuff (1997): “Competition among Institutions”. Journal of EconomicTheory 72, n°2, pp 306-342.

Cox, Gary (1990): “Multicandidate Spatial Competition”. In James M. Enelow and Melvin J.Hinich, eds, Advances in the Spatial Theory of Voting. Cambridge: Cambridge UniversityPress.

Cox, Gary (1997): Making Votes Count. Cambridge University Press.

Castanheira, Micael (1997): Voting for Losers. mimeo

Downs, Antony (1957): An Economic Theory of Democracy (New York: Harper Colllins).

Feddersen, Timothy J., Itai Sened, and Stephen G. Wright (1990): “Rational Voting andCandidate Entry under Plurality Rule”, American Journal of Political Science, XXXIV.

Feddersen, Timothy J (1992): “A Voting Model Implying Duverger’s Law and PositiveTurnout”, American Journal of Political Science, XXXVI.

Fey, Mark (1997): “Stability and Coordination in Duverger’s Law: A Formal Model ofPreelection Polls and Strategic Voting”, American Political Science Review.

Greenberg, Joseph and Kenneth Shepsle (1987): “The Effect of electoral Rewards inMultiparty Competition with Entry”, American Political Science Review.

Krehbiel, K. (1993): “Where is the Party?” British Journal of Political Science, 23.

Moulin, Hervé (1979): “Dominance Solvable Voting Schemes”, Econometrica, 47(6).

Myerson, Roger B and Robert Weber (1993): “A Theory of Voting Equilibria”, AmericanPolitical Science Review.

Palfrey, Thomas R. (1984): “Spatial Equilibrium with Entry”, Review of Economic Studies.

Osborne, Martin J. (1993): “Candidate Positioning and Entry in a Political Competition”,Games and Economic Behavior, V.

Osborne, Martin J. and Al Slivinski (1996): “A Model of Political Competition with citizencandidates”, Quarterly Journal of Economics.

Snyder Jr, James M. (1994): “Safe Seats, Marginal Seats, and Party Platforms: The Logic ofPlatform Differentiation”, Economics and Politics.

Taagepera, Rein and Matthew Soberg Shugart (1993): “Predicting the Number of Parties: AQuantitative Model of Duverger’s Mechanical Effect”, American Political Science Review.

Wittman, D. (1977): “Candidates with Policy Preferences: A Dynamic Model”. Journal ofEconomic Theory 14: 180-189.

Technical Appendix

Proof of lemma 1:The case with 1 or 2 candidates is obvious.Consider a situation with three candidates l, c and r and a left-wing state of the world.Voting for the right-wing and the left-wing candidates is a weakly dominated strategy for all centristcitizens. Voting for the right-wing (respectively left-wing) citizen is a weakly dominated strategy forall left-wing (respectively right-wing) citizens.In a left-wing state of the world, by elimination of weakly dominated strategies, the right-wingcandidate never wins: the right-wing candidate gets in expectation at most R, which is strictly smallerthan the minimum possible score of the stronger candidate among l and c, i.e. (L + C)/2:Since in a left-world L > R+C and R ≤ C, I have L > 2R > 2R - C and therefore R < (L+C)/2. But then,by iterative elimination of weakly dominated strategies, voting for the left-wing candidate becomes aweakly dominant strategy for all left-wing citizens and the left-wing candidate wins for sure. No citizenis decisive and all citizens being strategically indifferent, they vote sincerely.The proof is symmetric in a right-wing state of the world, and trivial in a centrist state of the world(since in a centrist world C > R).

1- Some refined voting equilibriaWe compute all voting equilibria for some useful sets of candidates and only give the outcome of thevoting equilibria.

Notations:Let - 1, 0 and 1 denote candidates of type left-wing, centrist and right-wing respectively.

1 candidate:He wins. His favorite policy is implemented.

2 candidates:If there are two candidates, the strategy of each voter is to vote for her sincerely preferred candidate (bysuppression of weakly dominated strategies).• -1 vs 0: O-1 = -1 , O0 = O1 = 0.• 0 vs 1: O0 = O-1 = 0, O1 = 1.• -1 vs 1; O-1 = -1, O0 = -1(1/2),1(1/2); O1 = 1.

3 candidates:• -1 vs 0 vs 1: By lemma 1, all voters vote for their sincerely preferred candidate in any state of theworld. Thus, O-1 = -1, O0 = 0 and O1 = 1.

Distributional assumption:Denote the probability of a left-wing world, a centrist world and a right-wing world by p, 1 - p - q andq respectively. Assume 1 - 2p > p = q.

The following table uses the voting equilibria in the different situations to give the expected utilities ofthe different non candidate citizens in the different possible situations (other cases are eithersymmetrical or irrelevant by lemmas):

C 0 -1 -1 vs 1 0 vs 1 -1vs0vs1EUi∈ L\C -1 0 -2 -1-3p -1 - 2pEUi∈ C\C 0 -1 -1 -p -2pEUi∈ R\C -1 -4 -2 - 1 + p -1 - 2pTable 2: Expected Utilities of non-candidates citizens with C given when p = q.

2. Coalition-Proof Political Equilibria

Proof of lemma 2:Consider any type i and any two sets of candidates C and C’ such that C includes exactly one i-typecitizen and C’ is C union one or several other type-i citizens.EUi(policy outcome under αI(C)) ≥ EUi(policy outcome under to αI(C’)),for all voting equilibria αI(.) satisfying to NCA.Therefore, as soon as δ > 0, not becoming a candidate for a group of type i individuals is a self-enforcing deviation if some other i-type individual becomes a candidate.

Proof of proposition 1:By lemma 2, there is at most one candidate of each type, and by lemma 1, voters vote for theirsincerely preferred candidate. Therefore, the expected utilities are univocally determined and wesimply indicate the expected utility associated with the set of candidates, as computed in table 2.

δ > 4:Not to enter is a strictly dominant strategy, since ∀ i ∈ {1, ..., 3n}, Eui(C = {i}) = -δ < Ui(x0) = -4.Policy outcome is x0. No coalition deviates.

2 < δ < 4:To enter is a best response if no other candidate enters sinceEUi(C = {i}) = -δ > EUi(x0) = -4 (P1.0)Not to enter is a best response for any candidate i if some other candidate j enters:- if i-type and j-types are equal:EUi(C = {i,j}) = -δ < EUi(C = {j}) = 0 (P1.1)- if i-type is -1 and j-type is 0:EUi(C = {i,j}) = -δ -1 + p < Eui(C = {j}) = -1 (P1.2)- if i-type is -1 and j-type is 1:EUi(C = {i,j}) = -δ -2 < EUi(C = {j}) = -4 (P1.3)- if i-type is 0 and j-type is 1:EUi(C = {i,j}) < -δ -1 + 2p < EUi(C = {j}) = -1 (P1.4)- other cases are symmetric.

Any situation with one candidate is coalition-proof:• centrist candidate c:No self enforcing collective deviation can happen without the candidate, since nobody wants to enter ifc is candidate by (P1.1) and (P1.2);no self-enforcing collective deviation can happen with the candidate: nobody wants to becomecandidate instead of c:∀ i ≠ c, EUi(C = {c}) ≥ -1 > -δ = Eui(C = {i})• left-wing candidate l:no self-enforcing collective deviation can happen without the candidate, since nobody wants to enter ifl is candidate;no self-enforcing collective deviation can happen with the candidate: nobody wants to becomecandidate instead of her, except a right-wing r, but l is only willing to have a self enforcing deviationwith a left-wing or a centrist:

Eul(C = {r}) = -4 < EUl(C = {l}) = -δ < Eul(C = {c}) < Eul(C = {l’})• right-wing candidate case is symmetric.

1 < δ < 2:A situation with one candidate of type 0 is a coalition-proof equilibrium.- if nobody enters, entering is a best response for a 0-type citizen c by (P1.0)- if a 0-type citizen c enters, not to enter is a best response for a 0-type citizen c’:EUc’(C = {c,c’}) = -δ < EUc’(C = {c}) = 0 (P1.5)- if a 0-type citizen c enters, not to enter is a best response for a 1 or -1-type citizen e:EUe(C = {e,c}) = -δ -1 + p < EUe(C = {c}) = -1 (P1.6)The candidate c would like to have a collective deviation such that she wouldn’t be candidate anymore,but no one wants to deviate with her:EUc’(C = {c’}) = -δ < Euc’(C = {c}) = 0EUr(C = {r}) = -δ < Eur(C = {c}) = -1.A situation with two extreme candidates is a coalition-proof equilibrium:- if there is a -1-type candidate l, entering is a best response for a 1-type candidate r:EUr(C = {l,r}) = -δ -2 > EUr(C = {l}) = -4 (P1.7)- if there is a 1-type candidate, entering is a best response for a -1-type candidate:symmetric.- if there are a -1 and a 1-type candidates, l and r, not entering is a best response for a 0-type candidatec:EUc(C = {l,c,r}) = -δ -2p < EUc(C = {l,r}) = -1 (P1.8)Everybody would prefer to have a centrist candidate, but no centrist citizen is willing to pay the cost:EUc(C = {c}) = -δ < Euc(C = {l,r}) = -1 (P1.7??)No other situation is an equilibrium:Situations without candidate, with one extreme candidate, or with several candidates of the same typeare already excluded.Situations with one candidate of each type are excluded by (P1.8)Situations with a centrist candidate and an extreme candidate are excluded by (P1.6)).

1 - 2p < δ < 1:A situation with one candidate of type 0 is a coalition-proof equilibrium.It is a Nash equilibrium by (P1.0), (P1.5) and (P1.6).The candidate c does not want to give up her position of candidate to an extremist:EUc(C = {c}) = -δ > EUc(C = {e}) = -1A situation with two extreme candidates is not a coalition-proof equilibrium:It is a Nash equilibrium by (P1.7) and (P1.8)Everybody would prefer to have a centrist candidate, and any centrist citizen is willing to pay the cost:in a situation with two extremist candidates, l and r, there is a self-enforcing collective deviation bythese two candidates and any centrist c leading to outcome 0:EUc(C = {c}) = -δ > Euc(C = {l,r}) = -1EUr(C = {c}) = EUl(C = {c}) = -1 > Eul(C = {l,r}) = -2 - δNo other situation is an equilibrium:As in the situation 1 < δ < 2.

p < δ < 1 - 2p:A situation with one centrist candidate is an equilibrium:- if nobody enters, entering is a best response for a 0-type citizen c by (P1.1)- if a 0-type citizen enters c, not to enter is a best response for a 0-type citizen by (P1.5)- if a 0-type citizen enters, not to enter is a best response for a 1 or -1-type citizen by (P1.6)This equilibrium is coalition-proof.A situation with one extreme candidate and one centrist candidate is not an equilibrium:By equation (P1.6.)

A situation with two extreme candidates is not an equilibrium:- if there are a -1 and a 1-type candidates, l and r, entering is a best response for a 0-type candidate, c:EUc(C = {l,c,r}) = -δ -2p > EUc(C = {l,r}) = -1 (P1.9)A situation with one extreme candidate is not an equilibrium:- if there is a -1-type candidate, entering is a best response for a 1-type candidate by (P1.7)

- if there is a 1-type candidate, entering is a best response for a -1-type candidate:symmetric.

δ < p:A situation with one candidate of each type is an equilibrium:- if there are a -1 and a 0-type candidate (called l and c), entering is a best response for a 1-type citizen(called r):EUr(C = {l,r,c}) = -δ -1 - 2p > EUr(C = {l,c}) = -1 - 3p (P1.10)- if there are a -1 and a 1-type candidates (called l and r), entering is a best response for a 0-typecandidate by (P1.9)- other best responses are the same for reasons of symmetry.This equilibrium is coalition-proof:The only possible collective deviation would be the two extremists getting out of candidacy, in order ofhaving only a centrist candidate. Unfortunately, such deviation is not self-enforcing, since any of themwants to deviate back:EUl(C = {c}) = -1 < Eul(C = {l,c}) = - 1 + p - δA situation with more than three candidates is not an equilibrium by lemma 1.A situation with two candidates is not an equilibrium by (P1.9) and (P1.10)A situation with zero or one candidate is not an equilibrium: easy.

�

2. Coalition-Proof Complete Political equilibriaComplete political equilibria without coalition-proofness:If δ > 4, there are complete political equilibria with 0, 1, 2 or 3 candidates.

Lemma 3:There is no coalition-proof entry equilibrium with two candidates of the same ideology.

Proof: similar to proof of lemma 2 in section 2.

Lemma 4:A i-type citizen becoming a member of a party whose creator is of i-type is not a weakly dominantstrategy.

Proof of lemma 4:Assume δ > 4, and m is as follows: all i-type citizens become a member of a party whose president isci: mi = ci ∀ i∈ I, mi = 0 otherwise. C(m) = {ci}, and outcome is i ∈ {-1,0,1}. Eui(mi = ci) = -δ/n. Thenany i’s type citizen’s best response is to deviate and get out of the party: C(m-i, i) = {ci}, outcome is thesame and Eui (mi = 0) = 0.

Proof of the 0 candidate result without coalition-proofness:Consider δ > 4. A situation without party and without candidate is a Nash equilibrium. Indeed, if mi = 0∀ I, C(m) = ∅ : no citizen wants to become a candidate since δ > 4, and no citizen deviates at the partyformation stage: C(m-i, i) = ∅ .Similarily, for δ < 4, a situation without party is a Nash equilibrium if rational expectations are suchthat C(m = 0) = C(m-i,mi) ∀ i ∈ {1, ..., 3n}, (m-i,mi) denoting m’ such that m’j = mj ∀ j ≠ i.

�Proofs of propositions 2 and 3:

Notations:• Call respectively C, L and R the set of surely centrist, left-wing and right-wing citizens.• Call c∈ C a citizen such that #{i mi = c} ≥ #{i mi = j} ∀ j ∈ C. Similarly, call l∈ L a citizen suchthat #{i mi = l} ≥ #{i mi = j} ∀ j ∈ L and call r∈ R a citizen such that #{i mi = c} ≥ #{i mi = j}∀ j ∈ R. Thus, c, l, and r are the centrist, left-wing, right-wing citizens who are presidents of the largestcentrist, left-wing and right-wing parties.• Let mS = j represent mi = j ∀ i ∈ S.Let (m -S,mS) denote any party formation strategy vector m’ such that m’j = mj ∀ j ∉ S.

Thus (m -R,mR) denotes any collective deviation from vector m by a coalition of right-wing citizens.Note that (m-i,mi) with i∈ R is a special case(unilateral deviation).

Lemma 6.0:- MI({I}) is the integer such that δ/MI < 4 < δ/(MI - 1) if it exists, I = l, c, r. MI({I}) is defined iff δ > 4.- Ml({l,c}) = Mr({r,c}) = Ml({l,c,r}) = Mr({l,c,r}) is the integer such that δ/Ml({l,c}) < p < δ/(Ml({l,c})- 1) if it exists. Ml({l,c}) = Mr({r,c}) = Ml({l,c,r}) = Mr({l,c,r}) is defined iff δ > p.- Mc({l,c}) = Mc({r,c}) is the integer such that δ/Mc({l,c}) < 1 - p < δ/(Mc({l,c}) - 1) if it exists.Mc({l,c}) = Mc({r,c}) is defined iff δ > 1 - p.- Ml({l,r}) = Mr({l,r}) is the integer such that δ/Ml({l,r}) < 2 < δ/(Ml({l,r}) - 1) if it exists. Ml({l,r}) =Mr({l,r}) is defined iff δ > 2.- Mc({c,r,l}) is the integer such that δ/Mc({l,c,r}) < 1 - 2p < δ/(Mc({l,c,r}) - 1) if it exists. Mc({c,r,l}) isdefined iff δ > 1 - 2p.Ml(C) = + ∞ if C includes 2 or more left-wing citizens.Mc(C) = + ∞ if C includes 2 or more centrist citizens.Mr(C) = + ∞ if C includes 2 or more right-wing citizens.

Proof of lemma 6.0: Bycomparison of the expected utilities for the different types of citizens as given in table 1, for thedifferent sets of candidates, and given δ.For example, Ml({r,l}) is the integer such that δ/Ml({l,r}) < 2 < δ/(Ml({l,r}) - 1), because EUl(C = {r})= -4 and EUl(C = {l,r}) = -2. A left-wing party president is therefore willing to become candidate, if aright-wing is candidate, if and only if she has to pay a candidacy cost smaller than 2, i.e. if the numberof members of her party is such that δ/Ml({l,r}) < 2.If δ < 2, any left-wing citizen is willing to be a candidate against a right-wing candidate, no party isneeded, and Ml({r,l}) is not defined.

Lemma 6.1:The continuation coalition-proof equilibria, given all possible strategy vectors m at the party formationstage and the corresponding sizes of parties Mc, Mr, and Ml, are as given in the following tables.Columnss represent variations of Mr (R1 to R4), rows variations of Ml (L1 to L4) and tables variationsof Mc (C1 to C4).Mc < Mc({c})(C1)

Mr < Mr({r})(R1)

≤ Mr < Mr({r,l})(R2)

≤ Mr < Mr({c,r})(R3)

Mr({r,c}) ≤ Mr

(R4)Ml < Ml({l})(L1)

x0 1 1 1

≤Ml < Ml({r,l})(L2)

-1 -1 or 1 1 1

≤ Ml< Ml({c,l})(L3)

-1 -1 -1 vs 1 -1 vs 1

Ml({c,l}) ≤ Ml

(L4)-1 -1 -1 vs 1 -1 vs 1

Mc({c})≤ Mc <Mc({c,r}) (C2)

Mr < Mr({r})(R1)

≤ Mr<Mr({r,l})(R2)

≤Mr< Mr({c,r})(R3)

Mr({r,c})≤Mr

(R4)Ml < Ml({l}) (L1) 0 symmetric to

R1.L2symmetric toR1.L3

1

Ml({l}) ≤ Ml <Ml({r,l}) (L2)

-1 or 0 δ/Mc > 10 if δ/Mc < 1

0 if δ/Mc < 1-1 or 0 or 1 other.

symmetric toR2.L3

1

Ml({r,l})≤Ml<Ml({c,l})(L3)

0 if δ/Mc < 1 andδ/Ml > 1-1if δ/Mc > 1 andδ/Ml < 1-1 or 0 otherwise

= R1.L3 0 if δ/Mc < 1 0 or -1 vs 1 other.

-1 vs 1

Ml({c,l}) ≤Ml (L4) -1 -1 -1 vs 1 -1 vs 1

Mc({c,l})≤ Mc <Mc({c,l,r}) (C3)

Mr < Mr({r})(R1)

≤ Mr<Mr({r,l})(R2)

≤Mr< Mr({c,r})(R3)

Mr({r,c})≤Mr

(R4)Ml <Ml({l})(L1) 0 0 0 0 vs 1Ml({l}) ≤ Ml <Ml({r,l}) (L2)

0 0 0 0 vs 1

Ml({r,l})≤ Ml

<Ml({c,l})(L3)0 0 0 (since δ/Mc < 1) 0 vs 1

Ml({c,l})≤Ml L4 -1 vs 0 -1 vs 0 -1 vs 0 -1 vs 1

Mc({c,l,r})≤ Mc

(C4)Mr < Mr({r})(R1)

≤ Mr<Mr({r,l})(R2)

≤Mr< Mr({c,r})(R3)

Mr({r,c})≤Mr

(R4)Ml <Ml({l})(L1) 0 0 0 0 vs 1Ml({l}) ≤ Ml <Ml({r,l}) (L2)

0 0 0 0 vs 1

Ml({r,l})≤ Ml

<Ml({c,l})(L3)0 0 0 0 vs 1

Ml({c,l})≤Ml L4 -1 vs 0 -1 vs 0 -1 vs 0 -1 vs 0 vs 1

Proof of lemma 6.1:By a computation similar to the computation of the coalition-proof candidacy equilibria in formersection, using table 1, with individual δis instead of a common δ.

Lemma 6.2:Some cells in the tables in lemma 6.1 are empty for some values of δ, because no integer Mi satisfy theconditions defined in lemma 6.0:If δ > 4, all cases correspond to possible Mis.If 2 < δ < 4, the continuation games are without table C1, rows L1 and columns R1.If 1 - p < δ < 2, the continuations games are without C1, L1, L2, R1 and R2.If 1 - 2p < δ < 1 - p, the continuation games are without C1, C2, L1, L2, R1 and R2.If p < δ < 1 - 2p, the continuation games are without C1, C2, C3, L1, L2, R1 and R2.If δ < p, cell C4.R4.L4 gives the continuation game equilibria.Moreover, if δ ∈ ((1 - p)(k-1), max{(1 - p)(k-1),k(1 - 2p)}), with k ≥ 2, table C3 must be suppressedtoo, since Mc({c,l}) = Mc({c;l,r}) = k.And if δ ∈ (i, max{i,(i + 1)(1 - p)}), δ/Mc < 1 is impossible in table C2, for any i integer with i ≥ 1.

Proof of lemma 6.2:Uses the fact that Mi(C)s are not defined for all δs, depending on C and I.For Mc({c,l,r}) = Mc({c,l}); it must be the case that δ > 1- p so that Mc({c,l,r}) and Mc({c,l}) are bothdefined. Therefore, Mc({c,l,r}) = Mc({c,l}) implies Mc({c,l,r}) = Mc({c,l}) ≥ 2.Mc({c,l,r}) = Mc({c,l}) = 2 and δ/2 < 1 - 2p implies (1-2p)2 > δ > 1-p.Mc({c,l,r}) = Mc({c,l}) = k is equivalent to k = Mc({c,l,r}) ≤ Mc({c,l}), therefore to δ/k < 1 - 2p andδ/(k-1) > 1 - p (> 1 - 2p) therefore to (1 - p)(k-1) < δ < k(1 - 2p) (note that this double inequalityrequires k < (1-p)/p).If δ < 1, δ/Mc < 1 is always true.If i < δ < i+1, δ/Mc < 1 in table C2 is possible iff it is not the case that Mc = i < Mc ({c,l}) = i+1 ≥ δ/(1-p) i.e. outside i < δ < (i+1)(1-p).

Lemma 6.3:A situation without candidate is never a CPCPE.

Proof of lemma 6.3:• Assume δ > 4. By subgame perfection, C = ∅ implies m such that ∀ i, Mi < Mi({i}) (case C1.R1.L1).Then, a coalition S of i-type citizens collectively deviates to mS = i, so that Mi = Mi({i}) and C = {i}(case C2.R1.L1, C1.R2.L1 or C1.R1.L2). All members of the coalition are strictly better off sinceδ/Mi{i} < 4, and no subcoalition deviates from this deviation, because it would lead to C = ∅ (back tocase C1.R1.L1).

• Assume δ < 4, Result is obvious since no continuation game as computed in lemma 3 allows foroutcome x0, as explained in lemma 4.

�Lemma 6.4:A situation with only a left-wing candidate is not a CPCPE.

Proof of lemma 6.4:• Assume δ > 4. By subgame perfection, C = {l} implies m such that Ml ≥ Ml({l}) for some left-wingcitizen l, and, Mr < Mr({r,l}), Mc < Mc({c,l}) (cases C1.R1.L2-3-4, C1.R2.L2-3-4, C2.L2.R1-2,C2L3.R1-2, C2.L4.R1-2).In any of those situations, a right-wing coalition will deviate:Consider any situation among those. Consider the set of all deviations by right-wing citizens leading toan outcome different from -1 (this set is not empty, consider for example a coalition leading to party rwith Mr({c,r,l}. the possible outcomes are 0, 1, -1 vs 1). Among them, choose one with a minimal Mr.Then such a collective deviation by right-wing citizens occurs: all members of the coalition are strictlybetter off (utility strictly larger than -4 in any case), and it is self-enforcing, any subcoalition deviatingfurther leading to the former situation (by choice of Mr minimal).• Assume 2 < δ < 4. If there is no left-wing party with at least two members, a self-enforcing deviationwith Mr = 2 leads to outcome 1.If Ml ≥ 2, a self-enforcing deviation with Mr = 2 is leads to outcome -1 vs 1.• Proof is obvious if δ < 2.

�Lemma 6.5:A situation with only a right-wing candidate is not a CPCPE.

Proof of lemma 6.5:Symmetric to proof of lemma 6.4.

Lemma 6.6:A situation with only a left-wing and a right-wing candidates is not a CPCPE.

Proof of lemma 6.6:• Assume δ > 4.Note first that the situation must be in C1.R3.L3 or C2.R3.L3 with Ml = Mr = Mr({r,l}) (by free-ridingand the fact that they prefer 0 to -1vs1).I will show that a deviation by right-wing and centrist non members citizens would happen. Consider ajoined deviation leading to situation C3R4L3 (or C4.R4.L4 if table C3 is irrelevant) with minimal Mr

and Mc. Outcome is 0 vs 1. All members of the deviation are better off since their expected utility islarger than -1, instead of -2.It then remains to note that no subcoalition wants to deviate: centrist citizens deviating would lead toC1-2.R4.L3 and -1 vs 1, right-wing citizens deviating would lead to C3.R1-3.L3 and no betteroutcome, and a mixed deviation is not wanted by right-wings.• Assume 2 < δ < 4. -1vs1 corresponds to Ml = Mr = 2 and Mc < 2. A deviation of right-wing and centrist citizens leading toMc = Mc({r,c}) > 3 and Mr = Mr({r,c}) > 7 will take place and outcome is 0 vs 1 as when δ > 4.• Assume 1 < δ < 2:Outcome -1 vs 1 implies no extremist parties (by free-riding). Centrist and right-wing will deviate toMc = Mc({c,r}) > 2 and Mr = Mr({c,r}) > 4 (situation C3-4R4L3). Outcome is 0 vs 1, all members ofthe deviation are better off, and it is self-enforcing.• Assume 1-2p < δ < 1:Situation must be in C3.R4.L4, C2.R3.L4, C2.R4.L4 or C2.R4.L3.Assume situation is in table C2. Then a coalition of extremists members will take place, lead toC2.R3.L3 and outcome 0. It is self-enforcing since left-wing prefer 0 to -1 vs 1.Assume now situation C3.R4.L4. Then a coalition of left-wing members deviates to C3.R4.L3, sincethey prefer 0 vs 1 to -1 vs 1. It is self-enforcing since any further deviation of a subcoalition would leadto 0 vs 1 or -1 vs 1.• Assume δ < 1 -2p:The only possible cells are within C4.R3-4.L3-4.

If m is inside C4.R4.L4, continuation game outcome is -1 vs 0 vs 1. If m is outside this cell, there willbe a deviation: consider situation inside C4.R3.Lj, then there will be a self-enforcing deviation of 2right-wing citizens so that Mr = 2 and situation is C4.R4.Lj.

Lemma 6.7:A situation with only a left-wing and a centrist candidate is a CPCPE iff δ does not belong to thefollowing set: [0, 1-2p] ∪ [∪ k ((1 - p)(k-1), max{(1 - p)(k-1),k(1 - 2p)}).

Proof of lemma 6.7:• Assume δ > 1 -2 p and δ ∉ [∪ k ((1 - p)(k-1), max{(1 - p)(k-1),k(1 - 2p)}).By lemma 4.2, at least cells C3-4.R3-4.L3-4 correspond to possible Mis. Consider a situation C3.Rj.L4,with j = 1 if δ > 4, j = 2 if 2 < δ < 4, j = 3 if δ < 2. Outcome is -1 vs 0.(If δ > 1-p, it corresponds to a situation with two parties: l and c are party presidents and candidates,with Ml = Ml({l,c}), Mc = Mc({c,l}), and ∀ i≠l,c, Mi = 0. If δ < 1 - p, it corresponds to Ml = 2, a left-wing candidate and an independent centrist candidate).No coalition of left-wing is wanting to deviate: they can only go to situations C3.Rj.Lj-4 and are neverbetter off.It is easy to check that no coalition including left-wing can deviate in a self-enforcing way: left-wingare only willing to take part to a coalition if it leads to {l}, no centrist nor right-wing will accept to bepart of it:EUc(C = {l,c}) > Euc (C = {l}) = -1; EUr(C = {l,c}) > Eur (C = {l}) = -4.No coalition of centrists is willing to deviate: they can only go to situations Ci.Rj.L4 for some i and arenever better off.No coalition of right-wing citizens is willing to deviate: it leads to C(m’) = {l,c} or {l,r} (C3.Rj-4.L4).And the deviation is not taking place.It remains to show that no coalition of centrist and right-wing citizens can deviate. What ever theydecide, l will be a candidate anyway (situations in L4). Therefore, centrists are never willing to deviate:C = {l,c} is their second best.• Assume 1 - 2 p < δ and δ ∈ [∪ k ((1 - p)(k-1), max{(1 - p)(k-1),k(1 - 2p)}).By lemma 4.2, M’is must be in C4.Rj.L4, j ≠ 4.This situation is not an equilibrium since a coalition of right-wing citizens will deviate to situationC4.R4.L4 with Mr = Mr({c,l,r}). Such deviation is self-enforcing.• Assume δ < 1 - 2p.then outcome is -1 vs 0 vs 1, as seen in proof of lemma 6.6.

�Lemma 6.8:A situation with only a right-wing candidate and a centrist candidate is a pure strategy coalition-proofcomplete political equilibrium iff δ does not belong to the following set: [0, 1-2p] ∪ [∪ k ((1 - p)(k-1),max{(1 - p)(k-1),k(1 - 2p)})].

Proof of lemma 6.8:Symmetric to proof of lemma 6.7.

Lemma 6.9:A situation with only a centrist candidate is a CPCPE for some set of rational expectations if δ ∉ [0,1-p].∪ ∪ i=1

∞ [i, max{i,(i+1)(1-p)}].

Proof of lemma 6.9:• Assume δ > 1 and δ ∉ ∪ i=1

∞ [i, max{i,(i+1)(1-p)}].Consider a situation with three parties: l, r and c are party presidents, with Ml({l,r}) ≤ Ml < Ml({l,c}),Mr({l,r}) ≤ Mr < Mr({r,c}), and Mc({c}) ≤ Mc < Mc({l,c}), δ/Mc < 1 < δ/(Mc -1) (inside situationC2.R3.L3) and expectations such that outcome is -1 vs 1 in any situation C2.R3.L3 with δ/Mc > 1.Centrist non-members will not participate in any coalition since they have the maximal possible utility:0.No coalition of centrist member citizens wants to deviate, since this would lead to situation C1.R3.L3or C2.R3.L3 with outcome -1 vs 1 by the expectations, and Euc(m) = - δ/Mc > -1.No right-wing coalition is deviating: This would lead to a situation in C2.L3.Rj-4, with j = 1 if δ > 4, j= 2 if 2 < δ < 4, j = 3 if δ < 2, implying maximal utility -1, that they already all have.

No coalition including right-wing citizens and left-wing is taking place: right-wing only want todeviate to get C = {r} or {c,r}, while left-wing only want to try and get {l} or {l,c}.A coalition of centrist and right-wing can lead to {c,r} (C3-4.R4.L3). This implies a larger centristparty, and we noticed above that no centrist non-member wants to deviate.Symmetric arguments apply to the other cases.• Assume 2 < δ ∈ ∪ i=1

∞ [i, max{i,(i+1)(1-p)}].A situation in C2.R1.L1 with Mc = Mc({c}) is a CPCPE under expectations such that C(m’) = -1 assoon as m’ is in C2.R1.Lj, j = 2,3,4 and C(m’) = 1 as soon as m’ is in C2.R Rj.L1, j = 2,3,4 (theseexpectations are rational by lemma 4.2).No centrist, except the one who is candidate under vector m, will be part of any deviating coalitionsince she has maximal utility (i.e. 0).A left-wing coalition could obtain a better outcome, i.e. outcome -1, but any such deviation (leading tom’) is not self-enforcing because of free-riding problems: if δ/Ml > 1, any member deviates furtherbecause EUl(C = {l}) = - δ/Ml < -1. If δ/Ml < 1, any member i deviates further because EUi(m’-i,m’i) =Ei(m’) = 0.A mixed type coalition will not happen either.• Assume 1 < δ < 2(1-p).Situation C2.R3.L3 (no party) with outcome 0 is a CPCPE for rational expectations such that outcomeis -1vs1 as soon as there is a left-wing party or a right-wing party (with at least two members).If C(m) = {c}, c cannot do anything against it. Other centrists do not want to do anything against it andno group of left-wing citizens will deviate either: if they create a party, any of them deviates further toget back to outcome 0 instead of -1 vs 1.No group of right-wing (and left-wing) citizens wants to deviate for similar reasons.• Assume 1 - p < δ < 1.Consider a situation without party and with outcome 0 (situation C2.R3.L3). It is a CPCPE, since nocentrist wants to belong to any coalition (except the future candidate), and if a coalition of extremistsdeviates, they can only get outcome -1 vs 1.• Assume 1 - 2p < δ < 1 - p.Outcome 0 requires situation in C3-4.R3.L3 but a coalition of left-wing members deviate to C3-4.R3.L4, since they prefer -1 vs 0 to outcome 0. It is self-enforcing when Ml = Ml({l,c}).• Assume p < δ < 1 - 2p.The continuation games are as in lemma 4.1, restricted to the cells in C4.R3-4.L3-4.Right-wing citizens creating a party with two members is a best response to any situation where noright-wing have done it: they prefer C4.R4.L3 (their utility being -1 + p - δ/2) to C4.R3.L3 (utility -1)and they prefer C4.R4.L4 (utility -1 -2p - δ/2) to C4.R3.L4 (utility -1 - 3p). Therefore 0 is not possiblein equilibrium.• Assume δ < p:The only continuation game outcome is -1 vs 0 vs 1 as seen before.

�

Lemma 6.10:A situation with a left-wing, a right-wing and a centrist candidates is a CPCPE for any set ofexpectations iff δ < 1 - 2p.Moreover, a situation with a left-wing, a right-wing and a centrist candidates is a CPCPE for some setof expectations if δ belongs to the following set: ∪ i=1

∞ [i, max{i,(i+1)(1-p)}].

Proof of lemma 6.10:• Assume δ > 1 and δ ∉ ∪ i=1

∞ [i, max{i,(i+1)(1-p)}]C = {l,c,r} requires that l, r and c are party presidents, with Ml({l,r,c}) ≤ Ml, Mr({l,r,c}) ≤ Mr, andMc({c,l,r}) ≤ Mc (situation C4.R4.L4).Eui(m) = -16/10 - δ/Ml if i is member of party l,Eui(m) = -16/10 - δ/Mr if i is member of party r,Eui(m) = -6/10 - δ/Mc if i is member of party c.I will prove that a coalition of members of all three parties will deviate so as to have C = {c}.Consider a collective deviation by members of the three parties leading to m’ such that Ml({l,r}) ≤ Ml <Ml({l,c}), Mr({l,r}) ≤ Mr < Mr({r,c}), and Mc({c}) ≤ Mc < Mc({l,c}), with δ/Mc < 1. Such situationcorresponds to C2.R3.L3 and C = {c} by coalition-proofness. Such a deviation is relevant by lemma4.2.Members of the deviating coalition are strictly better off, since they have the following utilities:

Eui(m’) = - 1 if i is a left-wing or a right-wing citizen, Eui(m’) = 0 if i is a centrist citizen.The core of the proof is to show that no subcoalition wants to deviate further.Centrist members of the deviation certainly won’t belong to any subcoalition (they have the maximumpossible utility).Right-wing could try to get C = {r,c} instead of C = {c}, but they are not able to do so without the helpof the centrist. More generally, no subcoalition S formed by left-wing and/or right-wing deviators istaking place: it leads to C = {l,r} or {c} (situation among C2.R3-4.L3-4).• Assume δ > 1 and δ ∈ ∪ i=1

∞ [i, max{i,(i+1)(1-p)}]A collective deviation leading to C2.Lj.Rk and outcome 0 is not always possible, since δ/Mc > 1 intable 2 (by lemma 4.2).In fact a situation in C4.R4.L4 with Mc = Mc({c,l,r}), Mr = Mr({c,l,r}) and Ml = Ml({c,l,r}) is a CPCPEif C(m) = {l,r} for any m in C2.R3.L3.In such a situation, no coalition including only one ideology will happen.No coalition including only right-wing and centrist will happen.No coalition including right-wing and left-wing will happen: it would lead to outcome 0 in table C4,and a subcoalition of one ideology (say left-wing) would deviate back to outcome -1 vs 0 (this is theprisoners’dilemma effect between left and right).I will prove that, under some expectations, no coalition of members of all three parties will deviateeither.Since a situation in table C2 implies that δ/Mc > 1 (by lemma 4.2), expectations such that outcome isnot 0 as soon as situation is in table C2 are rational. Therefore there will be no collective deviation toC2.And a deviation into C3.R2-3.L2-3 is either impossible or not self-enforcing, since left-wing woulddeviate further to C3R2-3.L4.• Assume 1 - p < δ < 1:A deviation leading to Mr = Ml = Mc = 0, leads to 0 (by coalition-proofness) and is self-enforcing(since a subdeviation would lead to -1 vs 1).• Assume 1 - 2p < δ < 1 - p:A deviation from situation C4.R4.L4 leading to outcome 0 is not self-enforcing, since a subcoalition ofleft-wing deviates further to C3-4.R3.L4 and outcome -1 vs 0 (table C2 is not relevant anymore).• Assume p < δ < 1 - 2p:Consider a situation leading to -1 vs 0 vs 1, with Mr = Mr({c,l,r}) and no centrist party. A deviation byright-wing (or left-wing) members is not wanted. A deviation by right-wing and left-wing members toC4.R3.L3 is wanted but not self-enforcing since right-wing would deviate further to 0 vs 1.• Assume δ < p:Outcome -1 vs 0 vs 1 is the only continuation game.

3. Introducing Asymmetry

Lemma 6.1 bis:The continuation coalition-proof equilibria, given all possible strategy vectors m at the party formationstage and the corresponding sizes of parties Mc, Mr, and Ml, are as given in the tables of lemma 6.1with the following modification:A table must be added between table C2 (Mc({c}) ≤ Mc < Mc({r,c}) ) and C3 (Mc({c,l}) ≤ Mc <Mc({r,c,l}) since now a value of Mc with Mc({c,r}) ≤ Mc < Mc({l,c}) must be considered.This table is denoted C2’.

Mc({c,r})≤ Mc <Mc({c,l}) (C2’)

Mr < Mr({r})(R1)

≤ Mr<Mr({r,l})(R2)

≤Mr< Mr({c,r})(R3)

Mr({r,c})≤Mr

(R4)Ml < Ml({l}) (L1) 0 0 0 0 vs 1Ml({l}) ≤ Ml <Ml({r,l}) (L2)

-1 or 0 δ/Mc > 10 if δ/Mc < 1

0 if δ/Mc < 1-1 or 0 otherwise

0 0 vs 1

Ml({r,l})≤Ml<Ml({c,l})(L3)

0 if δ/Mc < 1 andδ/Ml > 1-1if δ/Mc > 1 andδ/Ml < 1-1 or 0 otherwise

= R1.L3 0 if δ/Mc < 1 0 or -1 vs 1 other.

-1 vs 1

Ml({c,l}) ≤Ml -1 -1 -1 vs 1 -1 vs 1

Proof of proposition 4:Distribution assumption: q = 5/20, p = 7/20. Thus 1 - p - q = 4/10 as before.Proof uses lemmas 6.1 bis and is similar to proof of proposition 3

�

4. Adding IdeologiesA case with 5 ideologies and 3 states of the worldAssume there are 5 ideologies, -2, -1, 0, 1, 2, with 5n citizens who are sure of their types, n of eachtype.With probability 3/10, the state of the world is left-wing: T-1 = (3n,9n+2,3n,2n,n)With probability 3/10, the state of the world is right-wing and symmetric.With probability 4/10, the state of the world is centrist: T0 = (n,3n,10n +2,3n,n)

Proof of proposition 5:By arguments similar to the 3 ideology case - available to the author.

A case with 4 ideologies and 4 states of the worldAssume there are 4 ideologies, -2, -1, 0, 1, with 4n citizens who are sure of their types, n of each type.Ideologies -1 and 0 are called moderate, and ideologies -2 and 1 are called extremist.Let el, l, c, r respectively represent citizens of ideologies -2, -1, 0 and 1.With probability 2/10, the state of the world is extreme left-wing:T-2 = (6n + 2,3n,2n,n)With probability 2/10, the state of the world is right-wing: T1 = (n,2n,3n,6n + 2)With probability 3/10, the state of the world is left-wing: T-1 = (2n,6n+2,3n,n)With probability 3/10, the state of the world is centrist: T0 = (n,3n,6n + 2, 2n)

Proof of proposition 6:By arguments similar to the 3 ideology case - available to the author.