7/28/2019 10.1.1.177.863

1/38

Playing for Your Own Audience:

Extremism in Two Party ElectionsGbor Virg

September 2004

Abstract

This paper considers a two party election with costly voting and a single-

dimensional policy space. First, the parties choose platforms and then the votersdecide whether to vote and who to vote for. Each party has its own constituencywho does not observe the other partys action. The paper shows that in such a set-ting the parties always locate away from the median, since the voters who dislike theparties platform do not observe its policy choice and its own constituents like a pol-icy choice that caters to their taste. As the number of potential voters increases theparties adopt more extreme platforms and as it becomes infinitely large the partieslocate at the two endpoints of the policy interval, full extremism occurs. As the costof voting increases parties also tend to occupy more non-centrist locations, whileif a voter is more likely to obtain more information of the other party he does notvote for, the opposite effect appears. Making voters more symmetrically informed

about the two parties platforms increases the welfare of society, while asymmetricinformation acquisition by the voters is worse than no information acquisition at allin a private value framework.

I would like to thank Dan Bernhardt and Jozsef Molnar for their useful suggestions.University of Rochester, Department of Economics, 228 Harkness Hall, NY14627, e-mail: gvi-

rag@troi.cc.rochester.edu

7/28/2019 10.1.1.177.863

2/38

1 Introduction

An article in The Wall Street Journal1 focused on George W. Bushes strategy in the

2004 presidential elections. The following is a quotation from the Journal article: " Since

the advent of television brought presidential candidates into the voters living rooms, thegeneral-election of both major parties have been targeted toward winning swing voters

at the political center. Now, more than any modern campaign, the Bush effort, led by

White House advisor Karl Rove, downplayed that goal in favor of a drive to wring more

voters from the presidents committed core of supporters. Mr. Rove calls it a mobilization

election."

The article emphasizes that when voting is costly then it is important for a candidate

to convince his own constituentcy to vote. However, it does not take it into account that

such a policy may backfire by mobilizing the voters of the opposing party hurt by a non-

centrist platform of a candidate. To illustrate this point consider a standard model of

policy choice model with costly voting. First, the two parties choose their platforms, then

the voters decide whether to vote (incurring a cost K) or stay at home.2 Assume that a

voters loss function is the square of the distance between his ideal point and a candidates

position. In such a model one cannot explain how extremism might occur. While a more

extreme platform mobilizes the core supporters of a party, more is lost by mobilizing

the supporters of the opponent. Thus the two parties still locate at the political center.Interestingly, no voter participates in equilibrium, since the two positions are the same

and voting is costly.

One way to avoid this result is to assume that a party is able to make its core supporters

believe that he has taken an extremist platform, while making other voters believe that

he is still at the center. While a party may certainly try to follow this strategy it may

only work if voters are not rational. This paper offers a model with rational voters where

the parties take extreme positions without being able to mislead the voters. The starting

point is that if a voter is sure not to vote for a candidate, then he might not pay attention

to what this candidates platform exactly is, while he may follow more closely what "his

candidate" does when he decides whether to participate or not. An indirect evidence

1 "Bushs Unusual Campaign Plan: Focus on Base, Not Swing Vote", August 30, 20042 This setup was first considered by Ledyard (1984).

1

7/28/2019 10.1.1.177.863

3/38

supports this starting assumption. According to a New York Times article3 Fox Newss

news coverage achieved a higher rating than all of the three big network channels (ABC,

NBC, CBS) for the first time during the 2004 Republican National Convention. A report

issued by the Reuters4 attribute it to the fact that conservative voters are more likely to

watch Fox News than the general electorate and that they are also more likely to watch

the Republican Convention than others: "Fox benefited from the fact that its audience is

"ideologically very much aligned with the Republican Party"5 and apparently turned out

in large numbers to watch the GOP convention than the Democrats, said independent

network news analyst Andrew Tyndall." The same report observes that CNN with a more

liberal audience compared to Fox News did much better even in absolute terms than Fox

News during the Democratic National Convention: "In the 10 p.m. hour Thursday, which

included President Bushs speech, ..., Fox averaged 7.3 million viewers... Atlanta-basedCNN, ... which won the cable-ratings battle during the Democratic convention, attracted

just 2.7 million viewers in the 10 p.m. hour on Thursday." If we assume that ones decision

of which channel to watch is independent of what campaign event is exactly covered then

the data supports our main assumption that voters are observing their own party more

closely.

Under this modelling assumption a party may take an extreme position, since it knows

that the voters of the other party do not observe that position, while its core supporters like

this deviation from the political center. When voters are more polarized those tendencies

for extremism are even stronger since there are less voters in the center, so an extreme

position carries less punishment from voters in the center.

The most surprising result of this paper is that in a large electorate the two parties

locate at the two extremes of the policy space. The reason is that in such a large election

only a small fraction of the voters participate, otherwise the probability of being pivotal

would converge to 0, and abstaining would be the best strategy. But a more extreme type

has more incentive to vote if losses are quadratic (in the distance between the implemented

3 Bill Carter: "Networks Left to Reflect on Weeks Poor Ratings," September 3, 20044 Steve Gorman: "Bush, Republicans Outpoll Kerry, Democrats on TV", available at

http://story.news.yahoo.com/news?tmpl=story2&u=/nm/20040904/tv_nm/television_convention_dc5 The article quotes research data published by the Pew Research Center this June: "Twenty-five

percent of Republicans say they watch CNN regularly, while 41 percent are Fox watchers, according tothe study. Meanwhile, 44 percent of Democrats watch CNN, while 29 percent watch Fox."

2

7/28/2019 10.1.1.177.863

4/38

policy and a voters ideal point).6 Then only the most extreme voters participate in large

elections and a party has every incentive to cater to their needs by moving to the extremes.

The model may not be the most appropriate to apply to large elections since - similarly

to the literature initiated by Ferejohn and Fiorina (1974) - the model predicts that the

participation rate converges to zero if the number of voters become infinitely large. Then

our result should be only interpreted as a possibility: If there are no counterbalancing

effects the two parties locate at the very extremes of the policy space, depending on who

participates in the elections. If the voters who show up are mostly extremist then the

model predicts that the parties take extreme positions as well.

In such a setting welfare losses readily arise. It is important to note that rational

voters do not take it on face value what their own party claims about the position of

the opponent, since it is always in a partys interest to picture the opponent as the mostextreme. One instance when this problem does not arise is when the two candidates

participate in debates and their positions can be compared. However, it is not clear that

voters who are relatively far from the center actually watch those debates or take the time

to study the program of the party they do not vote for. If that is indeed the case then

the conclusions of the model become relevant in two party elections.

There are three more comparative statics results worth mentioning. First, as the

participation costs increase less voters participate, only the more extreme ones and thus

the parties take more extreme positions. Second, if we allow a voter to obtain with

probability p the action of the other party as well, then we show that the parties move

toward the center, and when p = 1 (perfect information is acquired about the platforms)

the parties locate near the median. Third, if we let moderate voters (located between t

and 1 t) to observe the platforms of both parties then the parties move their platformcloser to the center. More interestingly, the parties locate in such a way so that these

moderate (and symmetrically informed) voters never vote in equilibrium.

It might be argued that it is welfare improving to have news sources that are each

individually unbiased compared to a situation where the different biases of the different

news sources counterbalance each other. Indeed, if each voter subscribes to at most one

6 The assumption of a convex loss function may also capture that voters at the extremes are morelikely to participate, because they have a lower cost of participation or equivalently they care more aboutwhat policy is implemented.

3

7/28/2019 10.1.1.177.863

5/38

newspaper and that newspaper adequately covers only the position of one of the candidates

then it creates a situation where voters are asymmetrically informed. Note again, that if

it is well known that this newspaper supports one of the candidates then a rational reader

does not pay attention to what it says about the other candidate, since it always has the

incentive to claim that the other candidate is an extremist. Analyzing how media bias,

costly voting and information acquisition may relate to political extremism is a topic of

future research.

2 Literature review

The seminal works on voting are Downs (1957) and Hotelling (1929), who consider spatial

voting models in a two-party system and establish the result office oriented candidates

move to the center of the political spectrum. Ferejohn and Fiorina (1974) analyze large

voting games with two parties, fixed platforms and costly voting. They show that the

participation rate is converging to zero as the electorate becomes very large, the paradox

of voting. The paper most closely related to this is Ledyard (1984), who analyzes a game

of costly voting with strategic choice of platforms where the voters perfectly observe

the actions of both parties and the cost of voting is stochastic. Under the assumption

of concave utility functions (an assumption we maintain throughout the analysis) and

other technical assumptions he shows that parties choose the same platform and no voterparticipates. As a special case of our model when voters have perfect information about

the two platforms we obtain similar results. On the other hand, as we move further from

this perfect information case in our model these results no longer hold.

Borgers (2004) analyzes costly voting with two candidates and concludes that compul-

sory participation yields inferior welfare consequences compared to the voluntary voting

case. The reason is that by making voting compulsory the probability that a certain voter

is pivotal decreases and thus his utility from voting goes down as well. In our model there

is an extra effect that may make compulsory voting beneficial. If voting is voluntary then

extreme types vote only and the two parties move away from the median. If voting was

compulsory then both parties locate in the center, since now they have to convince all

voters to vote for them. If the electorate is large and voting costs are small then we can

show that compulsory voting is welfare increasing.

4

7/28/2019 10.1.1.177.863

6/38

Turner and Weninger (2001) conducts an empirical analysis7 and shows that in the

case of industry lobbying firms more moderate preferences are less likely to participate in

a public meeting. This result is somewhat similar to the finding of the this model that

more extreme voters are more likely to vote.

Palfrey and Rosenthal (1983) analyze two party elections, where voting costs are known

and the candidates have fixed positions (like in Borgers (2004)). They find multiple

equilibria some of them are inefficient. Our model finds only inefficient equilibria in

the sense that if voters observe the positions of the parties imperfectly then the parties

locate away from the median imposing welfare costs on the society. The inefficiency we

concentrate on derives from the locational choices of the parties, while in Palfrey and

Rosenthal (1983) this issue did not arise.

This paper is also related to the literature on political extremism in two-party elections.Several papers show that if the two parties are at least partially policy motivated then

they do not necessarily locate in the political center, if there is uncertainty about which

voters will particiapte in the voting. Some examples are Calvert (1985), Roemer (1997)

and Wittman (1983). A more recent paper is Ghosh (2002) who studies the behavior of

policy motivated parties who choose their own candidates. In this case political extremism

occurs in equilibrium as well. The starting point of this paper is rather different. We

assume that parties are purely office motivated and assume that voters do not observe

policy positions perfectly in an asymmetric manner, i.e. voters tend to observe the action

of their own party better. Then the factor that induces extremism is informational rather

than preference determined.

3 The model

There are two parties, L and R who compete for votes. There are n bidders who decide

whether to participate or not and who to vote for. Bidders are ex-ante identical and

their types are independently and uniformly distributed on the [0, 1] interval. First, the

two parties make their decisions about their locations on the same unit interval, i.e. vL,

vR [0, 1]. After that the voters on the left (yL 0.5) observe party Ls decision andthey decide whether to vote or not and who to vote for. Voters on the other side observe

7 Using the theoretical model of Osborne, Rosenthal, Turner (2000).

5

7/28/2019 10.1.1.177.863

7/38

only the action of party R before voting. The party with the more votes win, in case of tie

they employ a symmetric tie-breaking rule. The two parties are committed to implement

their announced platform.

We assume that the parties maximize the probability with which they win. Equiv-

alently we can assume that they maximize their vote share or the expected number of

votes they receive. Voters care only about the policy outcome, v. The utility of a voter

with type y is

(v y)2 K

if he votes and

(v y)2

if he does not. The expected utility of type yL from voting is

((vR yL)2 (vL yL)2)1

2Pr(piv)K,

while the expected utility from abstaing is normalized 0. In the above formula Pr(piv)

denotes the probability that (given the strategies of the other players) a player is pivotal,

i.e. without his vote there would be a tie or his own party, party L would be one vote down.

We need to multiply this probability by 12

because a voter can change the probability of

his party winning only by 50% in those cases. Note, that the actual policy of his own

party, vL appears in the formula, while vR denotes his belief of what the other party does.We analyze pure strategy, symmetric perferct Bayesian equilibria. The parties and

the voters follow pure strategy. Symmetry amounts to the following: i) The two parties

occupy a position at the same distance from the median:

vL + vR = 1.

ii) Voters with the same type use the same strategy (i.e. who to vote for if at all). iii)

A voter with type y votes for the opposite party as type 1

y, if that type participates,

otherwise he abstains as well.

Finally, we require that voters play a Bayesian Nash-equilibrium in the voting phase

of the game.

6

7/28/2019 10.1.1.177.863

8/38

4 Preliminary analysis

In this Section we discuss some features of the model that makes it easier to follow the

formal analysis in the next section. We make the assumption that voters who observe party

Ls decision (those with low types) do not infer anything from that about the location ofparty R. 8 Note, that party L would never locate at vL > 0.5, because high types do not

observe this choice, so they do not respond, while low types punish such behavior. Thus

in a symmetric equilibrium a voter is always closer to his own party (except the median

who is at the same distance), and he only has to decide between voting for his own party

or abstaining. The median voter never participates in a symmetric equilibrium, since he

is indifferent between the two platforms and voting is costly. Before moving on to the

formal analysis a remark about the parties incentives is in order. A party cannot change

the behavor of the constituent of the other party. The only thing it can change is the

amount of participation his own constituents engage in9. So, the strategic component

between the two parties is completely elimiated in the sense that no matter what the

opponent actually does the same decision is optimal for a given party. Thus a party

chooses its location to maximize the number of votes it can get. But that decision rule

maximizes its vote share and probability of winning as well. A party cannot induce too

high level of participation, since then voters would not be pivotal often enough. As a

consequence, no matter how well a party is located compared to its opponent (or rathercompared to the belief about that location) a party cannot induce more than a certain

share of the voters to participate. In a sense a too successful candidate is a victim of his

own succes, the more voters vote for him the less likely that a certain voter is pivotal, so

the less incentive an extra voter has to participate. This problem cannot be eliminated

by moving closer to where the voters are, i.e. moving to the center.

8 Otherwise party L would always have an incentive to signal that his opponent is an extremist. If weallowed such inference of the voters then many equilibria would arise, because if a party deviates thenits constituency might think that the opponent is at the middle and so they do not vote for their party.

Also, these equilibria do not seem reasonable, since a given party always want to signal the same thingabout its opponent, a not very satisfactory case for signaling.

9 Stricly speaking by locating at some unfavorable location the party can induce his own constituentsto vote for the oher party, but since it is clearly suboptimal to do this we can rule this case out.

7

7/28/2019 10.1.1.177.863

9/38

5 Analysis

We will concentrate on pure strategy, symmetric equilibria. Let us analyze the decision

problem of the voters on the left side of the unit interval, the voters of the other party

face similar problems. Unless otherwise mentioned the probability for being pivotal is fora voter who considers voting for party L. Let vL ( 1/2) be the position that party Loccupies and let the voters believe that party R occupies position vR ( 1/2). A voterto the left of the median does not vote for R, because it is better for him to abstain.

Then the only decision to make is whether to vote or not to vote. The next Lemma

shows that extreme types have more incentives to vote under the assumed (convex) utility

specification.

Lemma 1 If y0

L < yL 1/2 and yL votes for L then y0

L votes for L as well.

Proof. The utility from participating is the product of the probability of being pivotal

and the gain that is made conditional on being pivotal. The probability component is the

same for different types of the same player, only the gain component is different:

g = (vR yL)2 (vL yL)2 = (v2R v2L) 2yL(vR vL)< (vR y0L)2 (vL y0L)2 = (v2R v2L) 2y0L(vR vL) = g0.

So, if type yL has an incentive to vote then so does type y0L.

Thus in any such equilibrium the types who vote can be characterized by a cutoff

value, y such that types less than y vote for L, while types between y and 1/2 do not

vote. A similar rule applies for types above 1/2.

Let us study first the case where there is only one voter to gain some intuition. In

this case

Pr(piv | vL, vR) = 1

regardless of the cutpoints, (vL, vR). First, we show that the parties do not move to the

position of the median voter if there is positive participation costs. Indeed, suppose that

vL = vR = 0.5

was true in an equilibrium. Then voing is not profitable for any type, since the two

positions are the same and voting costs preclude voting then. If K is not too big then

8

7/28/2019 10.1.1.177.863

10/38

there is a profitable deviation for both parties. Party L can move as close to 0 as needed

to induce some participation from the extreme types. 10 The reason is that when a party

moves to the extreme it is noticed only by its own constituents, so costs of extremism are

not imposed, since voters on the other side do not observe this deviation and so they are

not motivated to vote.

Let us find the equilibrium in this simple one-voter game.

Lemma 2 In any symmetric, pure strategy equilibrium if n = 1 andK is not too big, vL

and vR are such that the left and righ cutoff values satisfy the equations that

yL = vL and yR = vR.

Proof. First, we can show that full participation (i.e. yL = yR = 1/2) cannot be an

equilibrium, since in a symmetric equilibrium type 1/2 does not gain from participation,

so he always abstains. Also, if K is not too big, it is not an equilibrium if no type

participates. Suppose that vL 6= yL < 1/2. Then if party L changes his location to v0L in a

way that it is closer to yL, then type yL achieves higher utility from participating, while

the utility from non-participation remains the same. Then that type strictly prefers to

participate. But then some type higher than type y0L will satisfy the indifference condition

[(vR y0L)2 (v0L y0L)2] = 2K. (1)

This means that party L can increase the number of his voters to include type y0L, which

means that he will receive more votes, so the original location was not optimal.

Then equation (1) and the above Lemma implies that

(vR vL)2 = 2K.

In a symmetric equilibrium

vR = 1 vL.

Then using the last two equations yields

vL =1

2K

2and vR =

1 +

2K

2.

10 This is possible if(0.5 0)2 = 1/4 > 2K.

We will make this assumption throughout, to rule out equilibria when noone participates.

9

7/28/2019 10.1.1.177.863

11/38

For concreteness let K = 0.02. Then

vL = 0.4 and vR = 0.6.

To obtain a comparative statics as the number of voter changes we turn now to a two player

game. The only difference in the derivation involves the calculation of the probability of

being pivotal. A voter (on the left) is pivotal if and only if the other voter does not vote

for party L, which is with probability 1 yL:

Pr(piv | L, yL, yR) = 1 yL.

Lemma 3 In any symmetric, pure strategy equilibrium if n = 2 andK is not too big, vL

and vR are such that the left and righ cutoff values satisfy the equations that

yL = vL < 1/2 and yR = vR > 1/2.

Proof. First, we can show that full participation (i.e. yL = yR = 1/2) cannot be an

equilibrium, since in a symmetric equilibrium type 1/2 does not gain from participation, so

he always abstains. Also, if K is not too big, in equilibrium some types must participate.

The utility of type yL from participation if the (left) cutoff is also yL is

[(vR

yL)2

(vL

yL)

2](1

yL).

Note, that this expression is strictly decreasing in yL, so the indifference condition

[(vR yL)2 (vL yL)2](1 yL) = 2K

has a unique solution. This means that given the beliefs about yR and the observation of

vL the cutoff type, yL is uniquely determined.

Suppose that vL 6= yL < 1/2. Then if party L changes his location to v0L in a way

that it is closer to yL, then type yL achieves higher utility from participating (if the other

player employs cutoff startegy yL), while the utility from non-participation remains the

same. Then that type strictly prefers to participate. But then some type higher than

type y0L will satisfy the indifference condition

[(vR y0L)2 (v0L y0L)2](1 y0L) = 2K. (2)

10

7/28/2019 10.1.1.177.863

12/38

This means that party L can increase his share of votes, which means that he will receive

more votes, so the original location was not optimal.

Then equation (2) and the above Lemma implies that

(vR vL)2

(1 vL) = 2K.In a symmetric equilibium

vR = 1 vL.

Combining the last two equations yields:

(1 2vL)2(1 vL) = 2K.

It is easy to show that vL is lower now than in the case with only one voter. For example

when K = 0.02 then we have now

vL = 0.38.

This suggests, that the more voters there are the more extreme the parties political

position become. The intuition is that with more voters each possible voter is less likely

to participate (since he is pivotal less often). But then only extremists participate, so the

parties need to cater to their tastes.

With more than two voters calculations become more involved. The main difficulty

is that there might be complemetarities between the strategies of the different voters, i.e.

if the others participate more then a voter might have more incentive to do so. These

complementarities may give rise to multiple equilibria and it is also not uniquely pinned

down what happens in the voting stage of the game. For this reason we introduce the

next assumption, which helps avoiding the problems of multiple equilibria in the voting

phase of the game. We assume that voters coordinate and vote in a manner that the

maximum amount of participation is reached for the given observations and beliefs:

Assumption: If the constituency of L observes policy position vL and believes that

voters on the right side use cutoff strategy yR and party R locates at vR then all types

less than yL participate, such that i) yL = 0.5, if it is a Nash-equilibrium for all types to

participate, i.e.

[(vR 0.5)2 (vL 0.5)2]1

2Pr(piv | 0.5, yR) K.

11

7/28/2019 10.1.1.177.863

13/38

ii) or if the last condition does not hold then

yL = maxz

[(vR z)2 (vL z)2]1

2Pr(piv | z, yR) = K

or iii) yL = 0 if even this equation cannot be satisfied. For voters on the right side of the

interval the appropriate corresponding assumptions are made.

We can easily show that there exists a cutoff value, yL that satisfies this assumption.

While this seems a strong restriction we can show that even this does not lead to high

participation levels ifn becomes large, so the assumption can be viewed as a technical one

that simplifies the analysis. The reason we need the above assumption is to prevent voters

from coordinating to a different voting equilibrium when the party changes its location

slightly. Since it is actually the only Pareto efficient outcome for the voters (and the

party prefers it obviously as well) in the voting stage (conditional on vL and their beliefsabout vR, yR) this assumption seems a reasonable starting point. The main bite of it is

indicated by the following Lemma that is exactly the same as the previous ones, but now

using also the assumption of maximum participation:

Lemma 4 In any symmetric, pure strategy equilibrium that satisfies the assumption of

maximum participation, vL andvR are such that the left and righ cutoffvalues satisfy the

equations that

yL = vL and yR = vR.

Proof. We start with the same steps as before applied to the equilibrium that has the

highest participation in the voting stage. Note, that in a symmetric equilbrium type 0.5

never participates, so we use the second part of the assumption here.11 Again, we can show

that as we move the locational choice closer to the original cutoff value it becomes true

that if everyone else uses cutoff level yL then type yL has strict incentive to participate:

[(vR

yL

)2

(v

L yL

)2]1

2Pr(piv | y

L, y

R) > K

Then by continuity there there is y0L > yL such that

[(vR y0L)2 (vL y0L)2]1

2Pr(piv | y0L, yR) = K.

11 Case iii) when no type participates is ruled out if K is not too large.

12

7/28/2019 10.1.1.177.863

14/38

But then by our assumption at least this new, higher level of participation is induced

when party L deviates to v0L, which makes it a profitable deviation.

We now prove that in general n voter games an equilibrium exists under mild restric-

tions. The main difficulty lies in establishing that the choice of the left cutoff values are

strategic substitutes in the sense that if the other voters employ a higher cutoff value,

yL then a given voter has less chance for being pivotal and thus less incentive to partici-

pate. This feature eliminates the problems of multiple equilibria in the voting phase and

guarantees that an equilibrium indeed exists:

Theorem 1 If n is even or n is odd and it is large enough then there exists a unique

equilibrium in the class of symmetric, pure strategy equilibria satisfying the assumption of

maximum participation.

Proof. Step 1 : First, we construct a candidate for such an equilibrium and in Step 2

we verify that it is indeed such an equilibrium. We will assume throughout that K is not

too big, so that there is no equilibrium in which no types participate. Let party L and

its constituency believe that party R uses strategy vR. Then to have a candidate for an

equilibrium it must hold that yR = vR, so this is the belief we assign to party L and its

voters. We are looking for a strategy, vL that satisfies two conditions:

i) Type yL = vL is indifferent between participating and abstaining if the other voters

use cutoff level yL. Formally,

[(vR vL)2 (vL vL)2]1

2Pr(piv | yL = vL, yR = vR) = K.

ii) Symmetry is satisfied:

vL = 1 vR.

Putting these conditions together yields

T(vL) = (1 2vL)2 Pr(piv | vL, 1 vL) = 2K. (3)

To prove that there exists a solution to this equation note that

T(0.5) = 0 < 2K

and

T(0) = 1 > 2K.

13

7/28/2019 10.1.1.177.863

15/38

Moreover, we can show that there is a unique solution of the above indifference condition.

To see this it is sufficient to prove that for all vL

Pr(piv | vL, 1 vL)vL

< 0.

We can verify this inequality by exploiting the fact that if participation rates are higher

then more other voters participate, which makes it less likely that one is pivotal. The

details are in the Appendix.

Step 2: Now, we prove that the solution of equation (3) is an equilibrium satisfying

the condition of maximum participation. We need to verify two conditions:

a) There is not a cutoff level, y0L > vL, s.t.

[(vR

vL)2

(vL

y0L)

2]1

2

Pr(piv | y0L, vR)

K.

b) There is not a strategy v0L, and cutoff strategy y0L > vL such that

L[(vR y0L)2 (v0L y0L)2]1

2Pr(piv | y0L, vR) K.

Part a) rules out that the voters can coordinate on an equilibrium (in the voting stage)

that achieves higher participation levels. Part b) rules out that party L can achieve higher

turnout by deviating from vL. The following claim helps establishing the Theorem:

Claim: If for all y0L

> yL = vL = 1

yR = 1

vR

Pr(piv | y0L, vR)

y 0L 0

then conditions a) and b) are satisfied applied to the candidate equilibrium found in Step

1.

Proof: By definition of the solution found in Step 1 :

[(vR vL)2 (vL vL)2]1

2Pr(piv | vL, vR) = K.

If y0L > vL then

[(vR vL)2 (vL vL)2] > [(vR y0L)2 (vL y0L)2].

Also, by assumption

Pr(piv | vL, vR) Pr(piv | y0L, vR).

14

7/28/2019 10.1.1.177.863

16/38

Then

[(vR vL)2 (vL y0L)2]1

2Pr(piv | y0L, vR) < K

follows, which establishes part a). To verify condition b) we prove that if all the other

voters use cutoff strategy y0L and party L uses strategy v0L, then it does not worth it even

for type yL to participate. But then certainly a higher type, y0L strictly prefers abstaining

as well. To see this consider the problem of such a type yL. His utility from participation

is

[(vR yL)2 (v0L yL)2]1

2Pr(piv | y0L, vR).

Since vL = yL it follows that

[(vR yL)2 (vL yL)2] > [(vR yL)2 (v0L yL)2].

By assumption

Pr(piv | vL, vR) Pr(piv | y0L, vR).

Thus

K =

[(vR yL)2 (vL yL)2]1

2Pr(piv | vL, vR) >

[(vR

yL)

2

(v0L

yL)

2]1

2

Pr(piv | y0L, vR),

which concludes the proof of the claim.

Finally, we show that the assumption of the previous claim is valid:

Claim: For any n that is even or odd and is large enough n for all y0L > yL = 1 yRPr(piv | y0L, vR)

y 0L 0.

Proof: The probability of being pivotal if the other voters use cutpoint strategies

characterized by (y0L, yR) is

Pr(piv | y0L, yR) = Pr(piv | y0L, yR, 0 voted) Pr(0 voted | y

0L, yR) + ... +

+Pr(piv | y0L, yR, n 1 other voted)Pr(n 1 other voted | y0L, yR).

Let

qi = Pr(piv | y0L, yR, i other voters voted)Pr(i other voted | y

0L, yR)

15

7/28/2019 10.1.1.177.863

17/38

and

pi = Pr(i other voted | y0L, yR).

Then

Pr(piv | y0L, yR) = p0q0 + ... + pn1qn1

and

Pr(piv | y0L, vR)

y 0L= (q0

p0y 0L

+ ... + qn1pn1

y 0L) + (p0

q0y 0L

+ ... + pn1qn1

y 0L).

In the Appendix we show that

A = (q0p0y 0L

+ ... + qn1pn1

y 0L) < 0.

Showing that

B = (p0 q0y 0L

+ ... + pn1qn1y 0L

) < 0

is more straightforward. If k other voters voted and k is even then a voter is pivotal if

and only if there is tie among the others. For k = 0 this has probability 1, i.e.

q0 = 1.

For the case when k > 0 the probability that some other voter votes for party L icondi-

tional on voting at all is

=

y0Ly0L + 1 yR .

Then the probability of being pivotal is

qk =k!

(k/2)!(k/2)!k/2(1 )k/2,

which decreases in y0L under our assumption that y0L > 1yR and thus > 1/2. A similar

calculation shows that the same result holds when k is odd. Then we have that

q0y 0L

= 0 andqky 0L

< 0 for k > 0,

which implies that B < 0 must hold.

As we argued earlier the more voters there are the smaller chance for any given voter

to be pivotal given a cutoff value, vL. This suggests then that as the number of voters

increases the participation rate must go down. The following Theorem states this result

formally:

16

7/28/2019 10.1.1.177.863

18/38

Theorem 2 In the unique pure strategy, symmetric equilibrium that satisfies the assump-

tion of maximum participation, the participation rate, vL is decreasing in n.

Proof. Suppose that the solution of equation (3) is indeed an equilibrium (the

conditions of the previous Theorem are sufficient for this to happen). If it was true that

Pr n(piv | vL, 1 vL) > Pr n+1(piv | vL, 1 vL), (4)

for all n then it would follow that

vL(n) > vL(n + 1).

To verify condition (4) we just need to note that in a symmetric case (i.e. when yL+ yR =

vL + (1 vL) = 1) it holds thatq0 > q1 = q2 > q3 = q4 > .... = qk > ...

and that those values do not depend on n. On the other hand as n goes up then (holding

vL constant) it is more likely that more voters participate. More precisely we can show

that for all k npn0 + ... + p

nk > p

n+10 + ... + p

n+1k .

Thus there is a first order stochastic dominance relationship between the two probability

distributions and since the sequence of qs is descending (and does not depend on the

number of voters) this last inequality implies that

Pr n = pn0q0 + ... + p

nn1qn1 + 0qn >

Pr n+1 = pn+10 q0 + ... + p

n+1n qn.

Now, we characterize the equilibrium when n becomes arbitrarily large. The following

Theorem shows that in the limit only types at the extremes participate:

Theorem 3 In any pure strategy equilibrium as n we have

yL 0, yR 1.

17

7/28/2019 10.1.1.177.863

19/38

Proof. Step 1: In equilibrium the probability of being pivotal cannot converge to 0,

since then there would be no incentive to participate. Then it is sufficient to show that if

the result of the above Theorem was false then the probability of being pivotal converged

to 0. It is easy to see that if exactly one of the two cutoffs converged to the extreme

then the probability of being pivotal certainly coverges to 0. So, we will suppose below

that neither of them converges to the extremes. Moreover, yL and yR cannot converge to

asymmetric values (i.e. yL + yR = 1 must hold), otherwise the probability of being pivotal

would converge to zero as well.

Step 2: Then take a sequence of equilibria. We can find a subsequence, such that

ynL byL > 0.Then for any k

Pr(number of other votes to L k) 0.

By Step 1 there is symmetry between the two cutoff levels, thus the following holds:

Pr(number of other votes to R k) 0.

Step 3: If 2k other voters voted12 then a voter is pivotal if there is a tie if he did not

vote. By symmetry (from Step 1 ) the probability of this is

Pr k(piv) =

(2k)!k!k!

22k =

1

2

3

4

5

6 ...

2k

1

2k .

It is easy to show that

Pr k(piv) 0

as k .Step 4: It holds that

Pr(piv) =

Pr(number of votes to each

k)Pr(piv | number

k) +

+Pr(number of votes to each > k)Pr(piv | number > k).

By Steps 2 and 3 as n we can find sufficiently big values for k, such that

Pr(number of votes to each k) 012 The case of odd number of voters can be handled similarly.

18

7/28/2019 10.1.1.177.863

20/38

and

Pr(piv | number of votes to each > k) 0.

Step 5: Then Using the last step we conclude that the probabilit of being pivotal

converges to 0 in contradiction to Step 1.

The result that only very few voters participate in large voting games with costly

voting is standard, see e.g. Borgers (2004). In addition to reproducing this feature, our

model predicts that exactly the most extreme types participate in the limit. Putting

together the last Theorem and Lemma and using that in the limit any equilibrium is

close to a symmetric one (since yL + yR 1) yields the following suprising result:

Proposition 1 In any pure strategy equilibrium that satisfies the assumption of maximum

participation, as n becomes arbitrarily large the two parties locate at the extremes:

vL 0, vR 1.

Since in the limit only the most extreme types participate, catering to them means that

the parties move also to the extremes. Thus if the constituents of the other party do not

pay attention then the extreme voters will hijack the elections, inducing high welfare losses.

Recent years have seen a polarization of the US media, new television channels and radio

programs appeared that now more perfectly serve their audience with the appropriate

slant in their coverage. This phenomenon may decrease the publics knowledge about

the viewpoint of the other party. So, rather paradoxically the appearance of more news

sources may lead to an electorate that is less precisely informed. Although the politicians

themselves may present what they claim is the opinion of the other candidate, rational

voters do not take these sources at face value, since they are avare that the politicians

always have the incentives to make the voters believe that the opponent party is an

extremist.

An appealing feature of the equilibrium is that in equilibrium voters know the positionof the party they do not vote for. Consequently, if they were given the opportunity to

acquire information about the platform of that party it would not yield any benefit to

them, so they would not pursue that option even if there is a very small cost attached to

it.

19

7/28/2019 10.1.1.177.863

21/38

As the cost of voting, K increases the policies become more extreme:

vLK

< 0.

The intuition is that as the cost goes up only more extreme voters participate and the

parties trying to convince these voters on the margin to participate adopt more extreme

platforms. This leads us to the observation that a higher level of participation is desirable

for non-extreme voters (and the public in general), because in that case the marginal voter

who participates is closer to the median and thus the parties adopt centrist platforms.

This result is in contrast with the findings of Borgers (2004) who shows that in a two-

party model with fixed party platforms a higher participation probability has negative

externality on the other voters, since it decreases their probability of being pivotal. Then

making voting compulsory reduces welfare. In our model this effect is opposed by theone that higher participation rates induce the parties to choose more moderate policies.

Indeed, suppose that K is very small and n goes to infinity. If voting is compulsory

then the parties locate in the median but the voters all have to incur the voting costs.

Nonetheless, if these costs are small then it is not a big loss for them. On the other hand,

if voting is voluntary then the parties choose extreme platforms and although the voting

cost, K is avoided it does not make up for the loss induced by the implementation of an

extreme platform. Compulsory voting is superior in this case.

6 Robustness of the results

In this Section I propose some modifications of the above game to assess the consequences

of more symmetric information structures. First, suppose that each voter now observes

the policy of the other party with probability p > 0. When p = 1 we are back to the

standard model with perfect information and positive cost of voting. Ledyard (1984)

analyzed this game and proved that in every equilibrium both parties locate near the

median and no voter participates. This suggests that as p increases our extremism result

is weakened and the participation level goes down.

We illustrate these results in a simple one-voter game, but all of the results hold for

the general n-voter case, if p is small enough.13 Assume that each party maximizes its

13 The calculations for that case appear in Appendix 2.

20

7/28/2019 10.1.1.177.863

22/38

probability of winning the elections.14 The case when p is high induces an equilibrium

with no participation and centrist platforms, the results of Ledyard (1984). Focusing on

the case when p is not too high we obtain the following result:

Theorem 4 In a one-voter model if p p =12K

1+2K then in the unique symmetric, purestrategy equilibrium with positive participation the following results hold:

vLp

> 0

andyLp

< 0.

Proof. See Appendix.

Making voters more informed in the sense that p goes up has a positive effect on

welfare, since it induces parties to move closer to the median, thereby reducing the effi-

ciency loss induced by extreme policies adopted. It is very important that it is not more

knowledge itself, but the reduction of the asymmetry in a voters information about the

two parties that contributes to this favorable outcome. Maybe more interestingly, making

voters more informed about their own parties leads to a reduction in welfare.

Let us consider one more modification of the baseline model. Assume that p = 0,

but that voters located on interval [t, 1 t] are moderates, who observe the positions of

both parties. First, we consider the model with only one voter. Let (x, 1 x) denote thelocation of the two parties in the baseline model, where no voter observes the platforms

of both parties, i.e. x satisfies

(1 2x)2 = 2K.

There are three cases, depending on the proportion of moderate voters:

Case 1, t x: In this case there are not enough moderates to change the votingoutcome. In the equilibrium of the baseline model (with locations (x, 1x)) no moderatevoter votes and one can easily show it remains an equilibrium even in the modified game.

The key observation is that a party cannot increase the number of its voters in a way

that was not available in the baseline model. So, if there are not enough moderate voters,

who observe both platforms, then they are ompletely neglected, since extremist voters are

more likely to vote.

14 We would obtain qualitatively similar results if a party maximized its share of the votes.

21

7/28/2019 10.1.1.177.863

23/38

Case 2, x > t > 12q

14 2K: In this case one can check that it is not an equilibrium

that both voters locate at the median, (1/2, 1/2). The equilibrium of the baseline model,

(x, 1 x) is not an equilibrium either, since now some moderate voters vote and actuallythey are the voters on the margin who the parties strive to convince to vote or to abstain

(e.g. party L tries to convince "marginal voters" on the left to vote and "marginal voters"

on the right to abstain). Since, the marginal voters observe now both platforms there is

an incentive for both parties to move close to the median. This argument bites, as long

as the type on the left, who is indifferent between participating and abstaining, vL is a

moderate. Then it follows, that

vL t

must hold. But if

vL < t

in a symmetric equilibrium then it implies that

vL < x

and also that the marginal voter of the opponent is an extremist. In that case it is a

good idea to take a slightly more extreme position for party L, since that does not induce

more participation of voters on the right hand side, while it increases participation among

voters on the left hand side.15 So, it cannot be an equilibrium either. Thus it must hold

in equilibrium that

vL = t, (5)

which means that the parties move away from the median exactly so much that no mod-

erate voters participate, but all extremists do. Moving more to the extreme would start

mobilizing some of the (moderate) voters of the other party (R). Moving towards the cen-

ter would decrease the amount of votes party L receives, without inducing more abstantion

among voters of party R, since in order to increase abstantion among those voters, someextremists must be convinced now to abstain, which is not feasible, because they do not

15 The reason it increases participation among voters of party L is that in this case y > vL and thusby moving closer to vL party L can increase the willingness of this type to participate. Since there is nothreat of increased participation among voters of party R, so the argument of the baseline model appliesto show that decreasing the distance between y and vL increases the winning probability of party L.

22

7/28/2019 10.1.1.177.863

24/38

observe the position of party L. Then the only symmetric equilibrium, (y, 1 y) is suchthat condition (5) holds, i.e.

(1 y t)2 (y t)2 = 2K

or

y =1

2 K

1 2t > x.

For concreteness take K = 0.02. Then x = 0.4 holds as we have shown in the baseline case.

Now, let t = 0.3, that is there are enough moderate voters to change the equilibrium of the

game. Then the last equation implies that y = 0.45, so parties move closer to the political

center than in the baseline case where noone observes the platforms of both parties and

party L locates at x = 0.4. One can show that the more voters there are who observe

both platforms, the less extreme positions will be taken by the two parties:

y

t= 2K

(1 2t)2 < 0.

Case 3: t 12q

14 2K: In that case there are so many voters observing both

platforms that it becomes an equilibrium of the voting game that both parties locate

at the median and all voters abstain. The proof that it becomes an equilibrium is very

simple: If the other party occupies that centrist position then party L by moving away

from the center induces type 1 t, a voter of party R to participate before it can inducehis most loyal voter, type 0 to do so. But then the mobilization effort backfires and it

is better not to induce any participation at all. Locating at (1/2, 1/2) is not the only

equilibrium in this case, but it is more important for our purposes that if t is close enough

to 0 (i.e. most voters observe both platforms) then all equilibria will converge to one

where both parties are close to the median and no voter paticipates.16

Let us consider the general n-voter case. Let xn denote the (unique) equilibrium when

no voters observe the platforms of both parties:

(1 2xn)2 Pr(piv | xn, 1 xn) = 2K.

The results below can be proven by extending the above argument:

16 Other equilibria are in the form considered in case 2. The difference is that in case 3 there are fullabstantion equilibria as well.

23

7/28/2019 10.1.1.177.863

25/38

Theorem 5 If there are n voters, p = 0 then the following results hold: i) If t xnthen the equilibrium of the baseline model remains an equilibrium of the modified model,

moreover this is the unique pure strategy, symmetric equilibrium satisfying the assumption

of maximum participation. ii) If xn > t >12

q14

2K then there is a unique symmetric,

pure strategy equilibrium satisfying the assumption of maximum participation, (y, 1 y)and it holds that vL = t, i.e. exactly the extreme voters participate. Moreover, the more

moderate voters there are, the more centrist positions the parties take

y

t< 0.

iii) If t min(12q

14 2K, x) then it becomes an equilibrium that both parties locate at

1/2 and no voter participates.

An interesting conclusion of the model is that moderate voters, who observe both

platforms never participate in equilibrium, since a party makes it sure that the moderate

voters of the other party reach a minimum utility that induces them not to participate.

One can say then that parties move to the extremes to please their only own base only so

much so that they do not mobilize the moderate (and informed) voters of the other party.

Finally, consider a model more in the spirit of probabilistic voting. Suppose that the

ideal points of the bidders are still distributed uniformly, but the cost of voting is not

constant, but a random variable uniformly distributed on [c, c], independently of the idealpoint of the voter. Let c (0, 1/2) and c be large enough, so that for no voter withthis cost it is ever optimal to participate. The other assumptions are as in the baseline

model. One can again show that extreme types are more likely to vote and there is a

highest type, xL 1/2 such that only voters with ideal points less than this value votewith positive probability for parrty L. A similar cutoffvalue exists among voters of party

R. In the characterization of the equilibrium we take the strategy of part R as given and

characterize the best reply of party L.

Theorem 6 For any n that is even or odd and large enough, under the assumption of

maximum participation there exists a unique symmetric equilibrium of the voting game,

characterized by equation

(1 2vL)(1 4vL)1

2Pr(piv | xL = 1 2vL, xR = 2vL) = c. (6)

24

7/28/2019 10.1.1.177.863

26/38

Proof. See Appendix 3.

Similar results can be established as in the baseline case. In particular, the parties

do not locate at the median in equilibrium. Also, as the number of voters increase the

parties locate closer to the extremes of the unit interval and from (6)

vLc

< 0

follows. Intuitively, the higher voting costs are the more extreme the participating voters

are, and the parties have incentives to move closer to the participating voters, yielding a

result of heightened extremism.

7 Conclusion

We considered a model of a costly voting with two parties, who choose their political

platforms strategically. The main departure from the previous literature is that a partys

constituency has better information about this party than about the opponent. Then each

partys action is primarily observed by its own voters, which induces them to locate away

from the median to motivate these voters to participate in the election. As the electorate

becomes larger only more extreme voters participate and the parties to please them occupy

more extreme positions. Also, as the cost of voting increases or the probability that voters

observe the platform of the party they do not vote for decreases the same effect arises.The welfare implications of such extremism are negative if voters are symmetrically

distributed and our metric for welfare is the average loss a voter faces by not being

able to determine the policy himself. In this setup one can show that contrary to some

previous results mandatory voting might yield higher welfare than voluntary, since in

the mandatory system the parties face the usual logic of voting, which induces them to

move to the center. An interesting implication of the model is that making voters more

informed in an asymmetric manner can reduce welfare, so symmetric ignorance might be

preferable in such a private value framework.

An important future research avenue is to analyze how the bias in different media

sources can contribute to this extremism result by providing less reliable information

about the position of one candidate than about the other.

25

7/28/2019 10.1.1.177.863

27/38

Appendix

Proof. Proof that Pr(piv|vL,1vL)vL < 0: In a symmetric case (i.e. when yL + yR = vL +

(1 vL) = 1) it holds that

q0 > q1 = q2 > q3 = q4 > .... = qk > ....

On the other hand as vL goes up then it is more likely that more voters participate. Let

v0L > vL and let pk (p0k) denote the probability that k other voter participates if the cutoff

value is vL (v0L). We can establish that for all k n 2

p0 + ... + pk > p00 + ... + p

0k.

Thus there is a first order stochastic dominance relationship between the two probability

distributions and since the sequence of qs is descending this last inequality implies that

Pr = p0q0 + ... + pn1qn1 >

Pr = p00q0 + ... + p0n1qn1.

Proof. Proof that A < 0: First note that

p0 + ... + pn1 = 1,

sop0y 0L

+ ... +pn1

y 0L= 0. (7)

The probability that k other voted is

pk =(n 1)!

k!(n 1 k)!(yR y0L)

n1k(1 yR + y0L)k.

Let

r = (yR y0

L), s = 1 r = (1 yR + y0

L).Then

pky 0L

=1

pk(kr (n 1 k)s) = 1

pk(k (n 1)s)

Thenpky 0L

0 k k.

26

7/28/2019 10.1.1.177.863

28/38

If k is even then as we have shown above

qk =k!

(k/2)!(k/2)!k/2(1 )k/2,

while for k

1 odd:

qk1 =(k 1)!

(k/2 1)!(k/2)!k/21(1 )k/2.

Thenqk+2

qk= 4(1 )k + 1

k + 2< 1

and similarlyqk+1qk1

= 4(1 )k + 1k + 2

< 1.

Also under the assumption that y

0

L > 1 yR and thus > 1/2,qk+1

qk= 2(1 )k + 1

k + 2< 1,

butqk

qk1= 2 1.

Let k be the largest k such thatpky 0L

0.

First, assume that k is even and that n 1 is odd. Let

A = H+ G,

such that

H = (q0p0y 0L

+ q2p2y 0L

+ ... + qkpky 0L

) + (qk+2pk+2

y 0L+ ... + qn1

pn1y 0L

)

and

G = (q1 p1

y 0L+ q3 p

3

y 0L+ ... + qk1p

k1

y 0L) + (qk+1p

k+1

y 0L+ ... + qn2p

n2

y 0L).

Also, let

= (p0y 0L

+p2y 0L

+ ... +pky 0L

) + (pk+2

y 0L+ ... +

pn1y 0L

) = 1 + ( 1)

27

7/28/2019 10.1.1.177.863

29/38

and

= (p1y 0L

+p3y 0L

+ ... +pk1

y 0L) + (

pk+1y 0L

+ ... +pn2

y 0L) = 1 + ( 1).

Since

q0 > q2 > ...qk > qk+2... > qn1

andpky 0L

0 for k k and pky 0L

> 0 for k > k

it follows that

H qk(p0y 0L

+p2y 0L

+ ... +pky 0L

) + qk+2(pk+2

y 0L+ ... +

pn1y 0L

) =

= qk1 + qk+2( 1) = (qk qk+2)1 + qk+2.

Similarly,

G (qk1 qk+1)1 + qk+1.

Using the two upper bunds from above and that + = 0 it follows that

A = H+ G (qk qk+2)1 + (qk1 qk+1)1 + (qk+2 qk+1).

After using the above results about variables {qk} it is now sufficient to show that 0to obtain that the upper bound on the right hand side is negative and thus conclude

the proof. Recall that r = (yR y0L), s = 1 r = (1 yR + y0L). For convenience leten = n 1 denote the number of other bidders. Then the following formula follows aftersimple calculations:

=

= enrhn1 + en!2!(en 2)!(2srhn2 (en 2)s2rhn3) + ....+

+

en!

k!(en k)!(ksk1rhnk (

en k)skrhnk1) =

= en[rhn1 (en 1)!1!(en 2)!srhn2 + (en 1)!2!(en 3)!s2rhn3 ....

(en 1)!(k 1)!(en k)!sk1rhnk + (en 1)!k!(en k 1)!skrhnk1 + ...]

= en(r s)hn1 < 0.28

7/28/2019 10.1.1.177.863

30/38

The last inequality follows because we assumed that en = n1 was odd. Ifn was odd, butlarge enough then it is easy to show that yR 1 must hold in our candidate equilibrium.Then party L cannot induce non-extreme types (y0L separated fom 0 in the limit) to

participate, since then the probability for its constituents to be pivotal would converge

to 0, which cannot be the case. So, r 1 and s 0 for large enough n for any possibledeviation, which guarantees that < 0 holds in this case as well. Finally, we check the

case when k 1 is even. Let H, coresspond to the even components and G, to theodd ones as before. Following identical steps, but now using that k, the critical value is

odd we obtain the following upper bound for A :

A (qk1 qk+1)1 + (qk qk+2)1 + (qk+1 qk+2).

After observing thatqk+1 > qk > qk+2

we can follow the same argument as in the other case to complete the proof.

Proof of Theorem 4:

Proof. Suppose that in the equilibrium the parties take positions (vL, vR). Then in

this simple one-voter case the following cutoff values apply in equilibrium17:

yL =v2R v2L 2K

2(vR

vL

)

and

yR =v2R v2L + 2K

2(vR vL).

Suppose that party L deviates to position v0L. Then the cutoff value for his constituents

become

y0L =v2R (v0L)2 2K

2(vR v0L). (8)

The cutoffvalue for the other voters if they do not observe this deviation (with probability

1p) remains yR. The cutoff value for those who observe it changes to

z0R =v2R (v0L)2 + 2K

2(vR vL). (9)

17 Note, that in equilibrium the voters are correct about the positions, so it does not cahnge theirdecision if they actually observe the position of the other party or not.

29

7/28/2019 10.1.1.177.863

31/38

The probability for party L to win is

Pr(L wins) = y0L +1

2[p(z0R y0L) + (1p)(yR y0L)] =

y0L + pz0R

2+

(1p)yR2

.

Then party L has to maximize y0L + pz0R with respect to v

0L. Using equations (8) and (9)

and that in equilibrium vR + vL = 1 the solution to this problem is characterized by

(1 2vL)2 = 2K1p1 + p

, (10)

which implies thatvLp

> 0.

Then substituting vL from equation (10) and using that vR = 1 vL implies

y

0

L = yL =

v2R

(vL)2

2K

2(vR vL) =vR + vL

2 K

vR vL ==

1

2 K

1 2vL=

1

2 K

1 2vL.

ThenyLp

=yLvL

vLp

< 0.

We can check that under our condition that p p it holds that yL 0 in equilibrium, butif p becomes larger then the above equations would yield a negative participation rate,

yL, so in that case there would not be any symmetric equilibrium with positive turnout.

8 Appendix 2

We analyze symmetric, pure strategy equilibria in the case when p > 0 as well. The

following assumption is made as an equilibrium selection device:

Assumption (Help the deviator): Suppose that in the candidate equilibrium party L

locates at vL, party R locates at vR and the voters choose cutoffstrategies (yL, yR). Then if

party L deviates to policy position v0L then the voters all choose the Bayesian equilibrium

in the voting stage such that it maximizes the vote share of the deviator.

We call this assumption the Assumption of Helping the Deviator. While with a positive

p it is no longer true that this is the only Pareto-optimal equilibrium for the voters who

observe v0L, we can employ a forward induction type of argument to justify it. Party L

30

7/28/2019 10.1.1.177.863

32/38

presumably deviated to increase his vote share, so it might be reasonable to require that

whenever possible the voters should coordinate to satisfy the partys needs.

Let us derive now a necessary condition for such a symmetric equilibrium under our

last assumption. So, consider a candidate equilibrium with vR = 1

vL and yR = 1

yL.

Since in a pure strategy equilibrium beliefs are correct for sure, so the cutoff strategies

used do not depend on whether one observes the action of the other party. Suppose that

party L deviates to position v0L. Then its constituents would choose position y0L. Other

voters use strategy z0R if they observe v0L and stick to strategy yR if they do not. It is easy

to see that if v0L is close to vL then there exist (unique) values (y0L, z

0R) close to (y

0L, yR)

such that these strategies together with yL constitue an equilibrium in the voting game,

that does not necessarily satisfy the Assumption of Helping the Deviator. If such an

equilibrium increases the expected vote share of party L above 1/2 (the vote share inthe symmetric situation) then the Assumption of Helping the Deviator imply that voters

coordinate on an equilibrium that gives at least this big expected vote share for L, so we

found a profitable deviation.

Thus, any voting equilibrium nearby must give expected vote share that is not more

than 1/2. Moreover, one can prove that such a nearby equilibrium values are differentiable

functions of the position of party L. Then if this equilibrium would bring an expected vote

share strictly less than 1/2 then choosing the opposite deviation would yield an expected

vote share above 1/2, a contradiction. So, in that neighbourhood the expected vote share

must have a 0 derivative with respect to vL. This condition can be formalized as follows:

[p(1 z0R) + (1p)(1 yR)]v 0L

=y 0Lv 0L

.

Thus

pz 0Rv 0L

+y 0Lv 0L

= 0. (11)

By definition of an equilibrium it follows that

Pr(piv | L, v0L)[(vR y0L)2 (v0L y0L)2] == Pr(piv | R, v0L)[(z

0R v0L)2 (z0R vR)2] = 2K.

Since

Pr(piv | L, v0L) = Pr(piv | R, v0L)

31

7/28/2019 10.1.1.177.863

33/38

locally, since expected vote shares remain 1/2 for each, it follows that in this neigbourhood:

[(vR y0L)2 (v0L y0L)2] = [(z0R v0L)2 (z0R vR)2].

After some algebra we obtain

y0L + z0R = vR + v

0L.

Thusz 0Rv 0L

+y 0Lv 0L

= 1. (12)

Using equations (11) and (12) we obtain

z 0Rv 0L

=1

1p (13)

and

y 0Lv 0L

= p1p . (14)

After differentiating equation

Pr(piv | L, v0L)[(vR y0L)2 (v0L y0L)2] = 2K, (15)

evaluating it at v0L = vL = 1 vR and using formulas (13) and (14) we obtain

Pr 0(piv | L, vL)(12vL)(12yL)+Pr(piv | L, vL)[2vRp

1p2(vLyL)2vLp

1p ] = 0.

Let H(y) be the probability of being pivotal if fraction y votes for L, fraction y votes for

R and (1 2y) abstains.18 By equation (14) it holds that this fraction behaves in such away that

y

v 0L=

p1p .

With this new notation we can write

Pr(piv | L, vL) = H(yL)

andPr 0(piv | L, vL) = H

0(yL) p

1p .

18 This can be written as (assuming n is even)

H(y) = y0+0(1 2y)n + y1(1 2y)n1 n!(n 1)! + n(n 1)y

2(1 2y)n2 + ... + n!(n/2)!(n/2)!

yn.

32

7/28/2019 10.1.1.177.863

34/38

Then our necessary condition for having a symmetric equilibrium becomes:

pH0(yL)(1 2vL)(1 2yL) = H(yL)[2(1p)(vL yL) 2p(1 2vL)]. (16)

From this it is obvious that if p = 0 then vL = yL, but when p > 0 then

vL yL > 0

must hold. Equation (15) can be rewritten as

H(yL)(1 2vL)(1 2yL) = 2K. (17)

Using this equation and equation (16) yields

pH0(yL)2K = H2(yL)[(1p)(12K

(1 2yL)H(yL) 2yL) 2p

2K

H(yL)(1 2yL)]. (18)

Note that at p = 0 this simplifies to

H(yL)(1 2yL)2 = H(yL)(1 2vL)2 = 2K,

our previous equation for the baseline model. Let

(yL) = H2(yL)[(1p)(1

2K

(1 2yL)H(yL) 2yL) 2p

2K

H(yL)(1 2yL)].

It is easy to show that

0(yL) < 0.

For p = 0 condition (18) has only one solution then. Using continuity arguments it follows

that if p is close enough to 0 then it is still true. Moreover,

p< 0

implies that for small p (where the left hand side of (18) is nearly constant in yL) it holds

thatyLp

< 0

and using equation (17) and the fact that H0

(yL) < 0 implies thatvLp

> 0.

Since second order conditions held strictly for the p = 0 case a continuity argument implies

that ifp is small enough then party L does not have a profitable non-local deviation either,

i.e. the (unique) solution of (18) is indeed an equilibrium of our game.

33

7/28/2019 10.1.1.177.863

35/38

9 Appendix 3

Proof of Theorem 6:

Proof. Fix vR and analyze the optimal choice for party L. The definition ofxL implies

that this type with cost level c is indifferent between voting and abstaining:

((vR xL)2 (vL xL)2)1

2Pr(piv | vL, vR) = c.

19 (19)

Let P be the probability of voting for the constituent of party L and introduce the

variable Pr(piv | P) to denote the probability that a voter of L is pivotal taking vR as

given. Formally,

P =

xL

Z0((vR x)2 (vL x)2)12 Pr(piv | P) c

c cdx. (20)

At the optimal choice (vL) ofL the first order condition implies that

P

vL= 0. (21)

Note, that at such a pointPr(piv | P)

vL= 0. (22)

Using equations (19), (20) and (22) equation (21) becomes

xLZ0

(vL x)dx = 0

or

vL =xL2

.

Using this last equation and symmetry, vR = 1 vL and xR = 1 xL implies togetherwith equation (19) that in equilibrium

(1 2vL)(1 4vL)1

2Pr(piv | xL = 1 2vL, xR = 2vL) = c.

19 Implicitly we use the assumption of maximum participation here to rule out cases in which voters ofparty L coordinate on abstention, which is always an equilibrium in the voting phase of the game.

34

7/28/2019 10.1.1.177.863

36/38

This equation has a unique solution in vL under our assumptions, so there is one candidate

equilibrium.20 We only need to show that the global second order conditions are satisfied

for party L to conclude the proof. Let

eP = P(c c).Differentiating (20) yields

ePvL

=

xLZ0

(vL x)Pr(piv | P)dx +ePvL

Pr(piv | P)

PK,

where K is a positive number. Then

ePvL

=

Pr(piv | P)

xL

Z0

(vL x)dx

1 Pr(piv|P)P K.

If we start at a candidate equilibrium and want to induce a higher participation of the

voters of L then we have shown in the proof of existence in the baseline model that

Pr(piv | P)

P< 0,

which implies that

P

vL

sgn=

xLZ0

(vL x)dx = xL(xL2 vL),

20 To show this it is sufficient to have that

Pr(piv | xL = 2vL, xR = 1 2vL, vL, vR = 1 vL)vL

< 0.

This is equivalent to (as seen in the proof of the baseline case):

P(xL = 2vL, xR = 1 2vL, vL, vR = 1 vL)vL

> 0.

After writing P up we have:

P =

2vLZ0

c( vR+vL2xvR+vL2xL

1)c c dx =

2vLZ0

c( 12x14vL

1)c c dx,

which after differentiation yields the result.

35

7/28/2019 10.1.1.177.863

37/38

where the signsgn= means that the two expressions have the same sign. Then it is sufficient

to prove that if vL is increased above its level at the candidate equilibrium then

vL >xL2

holds. (A similar argument will work when we decrease the level of vL.) Let us start nowfrom the candidate equilibrium where

Pr(piv | P)

vL= 0

and thus (because of (19))

((vR xL)2 (vL xL)2)vL

= 0.

Using that

vL =xL2 , vR = 1

xL2

we obtainxLvL

=xL vLvR vL

=xL/2

1 xL< 2.

Thus if we increase vL then we locally have

vL >xL2

and thusP

vL< 0.

So, P is decreasing in vL to the right of the level of vL of the candidate equilibrium.

Suppose that for some larger vLP

vL= 0.

There

vL =xL2

and thus (using that xL < 1/2 < vR)

xLvL =

xL

vLvR vL =

xL/2

vR xL/2 < 2.But this means again that

P

vL< 0

must hold to the right of such vL, and consequently the global second order conditions

are satisfied.

36

7/28/2019 10.1.1.177.863

38/38

References

[1] Borgers, T. (2004): "Costly Voting," American Economic Review, 94(1), pp. 57-66

[2] Calvert, R. L. (1985): "Robustness of the Multidimensional Voting Model: Candidate

Motivation, Uncertainty and Convergence," American Journal of Political Science,

29, pp. 69-95

[3] Downs, A. (1957): An Economic Theory of Democracy, New York, Harper and Row

[4] Ferejohn, J. and M. Fiorina (1974): "The Paradox of Not Voting: A Decision Theo-

retic Analysis," American Political Science Review, 68(2) pp. 525-536

[5] Ghosh, P.(2002): "Electoral Competition, Moderating Institutions and Political Ex-

tremism," Mimeo, University of British Columbia

[6] Hotelling, H. (1929): "Stability in Competition," Economic Journal, 39, pp, 41-57

[7] Ledyard, J. O. (1984): "The Pure Theory of Large Two-Candidate Elections," Public

Choice, 44(1), pp. 7-41

[8] Osborne, M., Roshental, J. and M. Turner (2000): "Meetings With Costly Partici-

pation", American Economic Review, 90(4), pp. 927-43

[9] Palfrey, T. and H. Rosenthal (1983): "A Strategic Calculus of Voting," Public Choice,

41(1), pp. 7-53

[10] Palfrey, T. and H. Rosenthal (1985): "Voter Participation and Strategic Uncertainty,"

American Political Science Review, 79(1), pp. 62-78

[11] Turner, M. and Q. Weninger (2001): "Meetings With Costly Participation: An Em-

pirical Analysis," Review of Economic Studies, forthcoming

[12] Roemer, J. (1997): "Political-economic Equilibrium When Parties Represent Con-

stituencies: The Unidimensional Case," Social Choice and Welfare, 14, pp. 479-502

[13] Wittman, D. (1983): "Candidate Motivation: A Synthesis of Alternatives," American

Political Science Review, 77, pp. 142-157