Economics Working Paper Series 

 
2019/015 

 
 

Technological change, campaign spending and 
Polarization 

 
 

Pau Balart, Agustin Casas and  Orestis Troumpounis 
 
 

The Department of Economics 
Lancaster University Management School 

Lancaster LA1 4YX 
UK 

 
 
 
 
 
 
 
 

© Authors 
All rights reserved. Short sections of text, not to exceed 

two paragraphs, may be quoted without explicit permission, 
provided that full acknowledgement is given. 

 
 

LUMS home page: http://www.lancaster.ac.uk/lums/ 



Technological change, campaign spending and

polarization∗

Pau Balart† Agustin Casas‡ Orestis Troumpounis§

December 12, 2018

Abstract

We focus on changes in technology and campaign management to study the

documented simultaneous increase in campaign spending and polarization. In our

model, some voters are ideological while others are impressionable. If the distribu-

tion of voters between types is endogenous and depends on parties’ platform choices,

our results show that a) an increase in the effectiveness of electoral advertising or

a decrease in the electorate’s political awareness, surely increases polarization and

may also increase campaign spending, while b) a decrease in the cost of advertising

does not affect neither polarization nor spending.

Keywords: electoral competition, campaign spending, impressionable voters, semiorder

lexicographic preferences

JEL codes: D72.

∗We appreciate valuable feedback from participants in the following events: 2017 Contests: Theory
and Evidence Conference, 2018 European Winter Meeting of the Econometric Society, Petralia Sottana
2018: Applied Economics Workshop, 2018 Political Economy Workshop Rotterdam, 2018 Madrid Po-
litical Economy Workshop, 3d Grode Workshop: Game Theory. Financial support from the Agencia
Estatal de Investigacion (AEI) and the European Regional Development Funds (ERDF) through grants
ECO2017-86305-C4-1-R (Balart) and ECO2017-85763-R (Casas) is acknowledged. Balart also acknowl-
edges financial support by the Fundación Ramón Areces.
†Department of Business Economics, Universitat de les Illes Balears, Carretera de Valldemossa, km

7.5 (Edifici Jovellanos), 07022 Palma de Mallorca, Spain. pau.balart@uib.cat
‡CUNEF, C/Leonardo Prieto Castro, 2. Madrid, 28040, Spain. acasas@u.northwestern.edu
§Department of Economics, Lancaster University, Bailrigg, Lancaster, LA1 4YX, United Kingdom

and Department of Economics, University of Padova, via del Santo 33, 35123 Padova, Italy. troumpou-
nis@gmail.com

1



1 Introduction

A well documented fact in US politics is the simultaneous increase of campaign spending

and polarization. Updated measures indicate that polarization in 2015 was at the highest

level since the era of Reconstruction, with this trend being well documented since earlier

work by Poole and Rosenthal (1984).1 Also campaign spending, with the exception of

the 2016 presidential election, has been steadily increasing since 1960 with a noteworthy

increase in the 2008 and 2012 presidential elections.2 In this paper, we provide a new the-

oretical explanation for the occurrence of these phenomena by linking campaign spending

and polarization with recent technological advances and changes in the management of

electoral campaigns.

Focusing on technology is relevant because while it is still gaining importance (e.g.,

Obama’s campaigns, the use of big data, and social media (Nickerson and Rogers, 2014)),

research so far has not drawn conclusive evidence on its effects on electoral competition

(Herrera, Levine, and Martinelli, 2008). Dating back to the introduction of nationwide

TV in the 1960s party centered non-professional campaigns were gradually abandoned.

Instead parties had to rely on professional campaign management due to the emergence

of multiple channels and 24/7 news coverage and the internet revolution in the mid-1990s

(Norris, 2000; Sabato, 1981). During recent years, the importance of technology is at its

peak and it is widely acknowledged that campaigns involve highly sophisticated tools.

The main question we aim to address is: What is the effect of such changes on electoral

competition?

In line with changes in technology and campaign management, we present a theory

where technology affects a) the effectiveness of electoral advertising –e.g., due to the

professionalization of campaign management and advanced targeting technologies, b) the

costs of electoral advertising –e.g., due to the possibility of reaching larger masses at a

low marginal cost, and c) the electorate’s political awareness –e.g., due to the presence of

various sources of information. To sum up our main results, an increase in the effectiveness

of electoral advertising or a decrease in the electorate’s awareness increase polarization

and may also increase campaign spending. On the contrary, changes in the marginal cost

of advertising cannot explain changes neither in polarization nor campaign spending.

We analyze the effects of technological changes on campaign spending and polarization

by combining the two seminal models of Downs (1957) and Tullock (1980). In particular,

we assume that two office motivated parties first choose their electoral platforms and

then decide upon the optimal level of costly electoral advertising. Voters can be either

1See http://pooleandrosenthal.com/political_polarization_2015.htm.
2See https://www.fec.gov/data/elections/president/2016/.

2

http://pooleandrosenthal.com/political_polarization_2015.htm
https://www.fec.gov/data/elections/president/2016/


impressionable or ideological as in the seminal papers by Baron (1994); Grossman and

Helpman (1996). As common in this literature, ideological voters support the party that

proposes the platform closest to their bliss point. Hence, parties compete for a share of

ideological voters as if they were competing in a Downsian model of electoral competi-

tion. On the contrary, impressionable voters are swayed towards one party or the other

through costly electoral advertising. Given each party’s electoral advertising, the effec-

tiveness of the latter determines the fraction of impressionable voters that supports each

of them. Differing from previous literature, we endogenize the division of voters across

impressionable and ideological. Such division depends on the differentiation between

the proposed platforms, with the fraction of ideological voters increasing in polarization.

This assumption captures the idea that the more diverse platforms are, more voters vote

based on their ideological preferences since platforms become “salient”. On the contrary,

when parties’ platforms are similar, voters may have a “hard” time or little interest in

distinguishing them, and turn to electoral advertising that determines (probabilistically)

their voting behavior. Alas intuitive, one way to formalize this behavior is that voters’

preferences are described by a lexicographic semiorder (see Luce 1956; Tversky 1969;

Rubinstein 1988; Leland 1994; Manzini and Mariotti 2012), consistent with the notion of

the “just noticeable difference”.3 With such preferences, each individual chooses which

party to support on the basis of platforms, but only if those are different “enough” (i.e.,

above a certain threshold). If the platforms are not sufficiently different, then the voter

is influenced exclusively by parties’ electoral advertising.

The above sketched model encompasses the effect of technological changes on elec-

toral competition through three distinct and non-mutually exclusive channels. The first

two reflect the way campaigns for impressionable -Tullock- voters are conducted and how

technology and changes in campaign management affect a) the effectiveness, and b) the

cost of electoral advertising. One could reasonably argue that recent technological ad-

vances have increased the effectiveness of electoral advertising since campaigns can be

well targeted, and have decreased the marginal cost of advertising given the possibility

of reaching large masses. The third channel captures how technology affects electoral

competition through the electorate’s political awareness and the endogenous division of

voters to impressionable and ideological. More precisely we focus on the conversion rate

at which impressionable voters become ideological as polarization increases. Although

not straightforward how this the conversion rate has evolved, we show how it affects

electoral competition.

Our setup proves relatively tractable. In contrast to previous contributions with en-

3In experimental psychology, the Weber-Fechner law remarks that the “just noticeable difference” is
not necessarily influenced by the physiological but rather by psychological factors. The law states that
the just-noticeable difference is increasing in the absolute level of the subtracts (Falmagne, 2002).

3



dogenous platforms and electoral advertising – that we later discuss, our model: i) has

a unique Nash equilibrium in pure strategies when parties are symmetric, and ii) can

be solved allowing heterogeneity in parties’ marginal cost of campaigning. But most

importantly, in equilibrium, several comparative statics arise regarding the effect of tech-

nological advances on electoral competition.

Consider first an increase in the effectiveness of electoral advertising. Since every dol-

lar spent on campaigns leads to higher returns, parties have incentives to symmetrically

increase their campaign spending (spending effect). To mitigate such increase in electoral

advertising, parties have incentives to polarize their platforms and reduce the number of

impressionable voters and hence their spending (polarization effect). If the spending effect

dominates, campaign spending and polarization increase simultaneously and can explain

the observed trends in US politics. If the polarization effect dominates, an increase in the

electoral effectiveness is overcompensated by an increase in polarization and campaign

spending decreases. Actually, the effectiveness of electoral advertising proves crucial if

one wants to explain the simultaneous increase in campaign spending and polarization in

terms of campaign management. A decrease in the marginal costs of running a campaign

does not affect neither polarization nor total spending. While a decrease in the marginal

cost of advertising leads to more advertising, the lower marginal cost of the latter leaves

campaign spending and polarization unaffected. Finally, the conversion rate at which

impressionable voters become ideological when polarization increases affects both cam-

paign spending and polarization in a similar manner as a change in the effectiveness. If

one wants to explain the simultaneous increase in polarization and spending exclusively

through this channel, then the conversion rate has been decreasing, meaning that voters

do not respond much to changes in platforms for example due “media malaise”, possibly

associated with mistrust of politicians and disenchantment with politics (Norris, 2000;

Newton, 1999).

1.1 Related Literature

In terms of results, our work complements existing models of platform choice with en-

dogenous advertising. The closest paper in terms of research question and methodology

is by Herrera et al. (2008) who explicitly model changes in the targeting effectiveness

and its effect on polarization and spending. Contrary to us, they show that an improved

campaign technology reduces polarization. This is becasue while in our model polar-

ization softens competition in the electoral advertising stage (as the endogenous valence

literature), in their model it has the exact opposite effect. Prummer (2018) focuses on

changes in targeting technology and fragmentation of media networks as determinants of

polarization. Moving away from a targeting story, Rivas (2017) provides an alternative

4



justification for the simultaneous increase in polarization and campaign spending in a

model where the latter is financed through lobbies.

The structure of our model, where parties first choose platforms and then spend-

ing, resembles existing models of endogenous valence (e.g., Ashworth and Bueno de

Mesquita (2009); Zakharov (2009); Carrillo and Castanheira (2008); Iaryczower and Mat-

tozzi (2013) among others). In that literature, voters typically have additive separable

preferences over platforms and valence (i.e., electoral advertising). In the closest work to

ours (Ashworth and Bueno de Mesquita, 2009; Zakharov, 2009), platform diversification

softens the competition in the valence accumulation stage.4 These dynamics are exactly

the ones presented in our model through the endogenous division of voters to ideological

and impressionable. Our model, however, proves more tractable and permits the analysis

of the effects of technological changes on electoral competition. In contrast to Ashworth

and Bueno de Mesquita (2009), our model under symmetry admits a pure strategy equi-

librium in the campaign stage setting and does not require that voters’ ideologies are

uniformly distributed. A non-uniform distribution is also permitted in Zakharov (2009)

but only when focusing on local Nash equilibria. In contrast, we are able to characterize

Nash equilibria in pure strategies for a general distribution of voters’ ideology (symmetric

and log-concave in this paper) and perform relevant comparative statics. Also in con-

trast to the previous models, we are also able to solve the model and obtain results when

parties have heterogeneous campaign costs, for example due to an incumbency advantage

(Meirowitz, 2008; Pastine and Pastine, 2012).

Summing up, one could think of our model as one of endogenous valence where

rather than additive separable preferences over platforms and campaigns, individuals have

semiorder lexicographic preferences (see Luce 1956; Tversky 1969; Manzini and Mariotti

2012; Rubinstein 1988; Leland 1994) and is in line with salience models where decision

makers overweight attributes that exhibit greater differences in the available choice set

(Bordalo, Gennaioli, and Shleifer 2012, 2013a,b, 2015; Bushong, Rabin, and Schwartzstein

2017; Kőszegi and Szeidl 2012; Spiegler 2014). Semiorder lexicographic preferences can

be seen as a particular case of salience in which: a) only the difference in one attribute

(policy platforms) is relevant to assign the weight of each dimension, and b) individual

weights on each attribute are discrete and take value 0 or 1. In aggregate terms, our as-

sumption on the endogenous division of voters across types has a similar effect to the one

of salience at the individual level: the smaller the distance between platforms, the smaller

is their weight on the final outcome of the electoral competition. Callander and Wilson

(2006) and Nunnari and Zápal (2017) introduce related context-dependent preferences in

4The opposite effect may occur when parties have ideological motives (Epstein and Nitzan, 2004;
Cardona, De Freitas, and Rub́ı-Barceló, 2018) or there is uncertainty regarding the valence investment
(Carrillo and Castanheira, 2008).

5



political economy models. In the former, the utility of voting depends not only on the

direct benefits of turnout but also on the context, i.e., the candidates’ polarization. In

the latter, the authors provide a model of “focusing”, where voters attention is captured

more on the issues that candidates’ proposals differ more. Similarly, Amorós and Puy

(2013); Aragonés, Castanheira, and Giani (2015); Denter (2017) focus on electoral com-

petition models when parties have the ability to affect the relative salience of different

issues via their strategic actions (e.g., allocation of time or effort). In our model, parties’

strategic actions affect the salience of platforms versus advertising, and hence voters in

the society split among those casting votes either in an informed or uninformed manner.

Therefore, although very different in nature, our model links with recent literature where

some voters may be partially informed regarding parties’ policy proposals (Aragonès and

Xefteris, 2017; Eguia and Nicolò, 2018).

Finally, our model contributes to the contest theory literature (see Corchón (2007);

Konrad (2009); Serena and Corchón (2018) for surveys) since parties compete for a share

of impressionable voters as if they were competing in a Tullock contest (with the “noise”

of the latter capturing the effectiveness of electoral advertising). Notice that the value

of the “prize” (of the contest) allocated based on electoral advertising is endogenous and

depends on platform selection. Consequently, by fixing closer platforms parties not only

attract more voters from their competitor as in any Downsian model, but also increase

the share of Tullock voters for which rent-dissipation arises (Tullock, 1980; Nitzan, 1994).

In terms of results, the value of the prize depending on platforms provides a result in

contrast to most of the contest literature where the relationship of electoral advertising

and the noise of the contest success is monotonic.

2 Model

Let two political parties i ∈ {L,R} first propose (and commit to) platforms xi in the
policy space X = [0, 1] and then choose the level of campaign advertising ei ≥ 0. Without
loss of generality, we assume xL ≤ xR. Let Si(xL,xR,eL,eR) be the vote share for party
i and ci(ei) = µiei the cost of advertising, with µi > 0 denoting the constant marginal

cost of advertising. Without loss of generality, we assume µL ≤ µR. Parties’ are office
motivated with payoffs Πi = Si(xL,xR,eL,eR) − ci(ei).5

Voters have a preferred policy x drawn from distribution G(x) with corresponding

density g(x) symmetric and log-concave (i.e., (ln g(x))′′ ≤ 0), with full support in X.6

5Here each party’s objective is to maximize its vote share net of the campaign costs. Alternatively,
Si(xL,xR,eL,eR) can also be interpreted as the probability of winning by assuming parties’ uncertainty
on voters’ ideology as in Aragonès and Xefteris (2017).

6Under symmetric campaign costs, our results can be extended to asymmetric distributions of ideol-

6



Independent of their ideal policy some voters are ideological and some are impressionable.

The ideological citizens vote sincerely for the party whose proposed platform is closer to

them and split their vote if the proposed platforms coincide (à la Downs ). The utility of

a voter with ideology x that votes for party i is ux(i) = −|x−xi|.7

The impressionable citizens’ vote depends only on electoral advertising. In partic-

ular, we assume that given parties’ advertising, the probability that an impressionable

citizen votes for party i is determined à la Tullock and hence equal to e
η
i /(e

η
L + e

η
R),

where parameter η > 0 captures the effectiveness of electoral advertising.8 If η → 0,
impressionable voters split equally across the two parties. However, as η increases, the

allocation of impressionable voters across parties becomes more responsive to electoral

advertising. Impressionable voters voting on the basis of persuasive electoral advertising

is a standard assumption in this literature (see for example the seminal papers by Baron

(1994); Grossman and Helpman (1996) and a large literature thereafter). The specific

proposed function is the seminal contest success function (CSF) introduced by Tullock

(1980). This function is extensively used in the literature and apart from tractability also

satisfies relevant axiomatic properties (Skaperdas, 1996) and can be micro-founded in a

reasonable manner (also in our setting, see Section 5.4 of the Appendix).9

The endogenous division of voters across ideological and impressionable depends on

the level of polarization. Let y = xR − xL ∈ X be the platforms’ polarization and
F(y) a continuous cumulative distribution function, log-concave (i.e., (ln F(y))′′ ≤ 0),
with corresponding density f(y) with full support in X. The share of ideological voters is

F(y), and therefore the share of impressionable voters is 1 −F(y).
It is informative to briefly stress how the above presentation of our model in aggregate

terms reflects voters’ individual behavior. Consider voters with semiorder lexicographic

preferences. Let the first attribute reflect policies and the second attribute electoral ad-

vertising. Each voter draws a preferred policy x from G(x) and a level of “sensitivity”

φ from Fφ(y) = Pr(φ ≤ y), denoted F(y) for simplicity. The value of φ determines if
the voter is voting based on policies or advertising. If polarization is greater than φ,

the voter is ideological. If polarization is less than φ the voter is impressionable. That

is, as platforms diverge, an individual voter is ideological with ex-ante probability F(y),

ogy (available upon request).
7The assumption of a particular distance function is made without loss of generality.
8We assume η > 0 to focus on cases in which our model differs from a Downsian model. If η = 0,

equilibrium spending is always zero and platform choice coincides with that of the standard Downsian
model.

9The campaign stage for impressionable voters is resolved via Tullock’s ratio-form CSF that facilitates
the comparative statics of our model. In the symmetric case, our results would have the same qualitative
features if we considered the difference-form CSF proposed by Alcalde and Dahm (2007), the tractable
noise CSF proposed by Amegashie (2006) or the relative-difference CSF by Beviá and Corchón (2015)
under the parameter restrictions proposed by Balart, Chowdhury, and Troumpounis (2017).

7



and impressionable with ex-ante probability 1−F(y). Hence, despite the stark partition
of voters into ideological and impressionable presented in the main text, an individual

voter’s expected behavior a priori depends both on platforms and advertising. To avoid

additional notation we focus on the aggregate version of the model throughout the pa-

per. The microfoundations of individuals’ behavior (in terms of semiorder lexicographic

preferences as above and in terms of a salience model) and the effect of advertising on

individual voting behavior (and the rise of the Tullock CSF) are detailed in Section 5.4

of the Appendix.

The timing of the game is as follows: At t = 1, the political parties simultaneously

choose the political platforms that maximizes their payoff. At t = 2, having observed the

platforms choices and the share of impressionable voters determined by the polarization,

parties choose the advertising levels. Finally, at t = 3, voters vote. Given the nature of

our game, we focus on subgame perfect Nash equilibria in pure strategies.

3 Results

Given the described game, let x̄ = xL+xR
2

be the indifferent ideological voter for xL 6= xR.
Ideological voters with x ≤ x̄ vote for L, while the remaining ones vote for R. Thus,
party L obtains a share SLIdl = G(x̄) of the ideological voters and party R, S

R
Idl = 1 −

G(x̄). If xL = xR, then S
R
Idl = S

L
Idl =

1
2
. Given that the individual probability that an

impressionable citizen votes for party i is
e
η
i

e
η
L

+e
η
R

, the expected share of impressionable

votes to party i is SiImp =
e
η
i

e
η
L

+e
η
R

for party i. Hence, the expected vote share obtained by

the parties can be then written as a weighted average of the previous two:

Si(xL,xR,eL,eR) =F(y)S
i
Idl(xL,xR) + (1 −F(y))S

i
Imp(eL,eR). (1)

This expression highlights the effect of platform choices in our game. First, platform

choice affects how ideological voters split between the parties (via SiIdl(xL,xR)). Second,

platform choice affects the ideological-impressionable composition of the electorate (via

F(y)). As common in Downsian type models, converging towards the opponent is ben-

eficial due to the relocation of the indifferent voter. However, in our model platform

convergence results into an increase in the share of impressionable voters, and hence a

tougher competition in the (costly) advertising stage. The above trade off makes plat-

forms’ choice a non-trivial task.

8



3.1 Symmetric parties

For illustrative purposes, and to highlight our main results in the simplest framework, we

first pay attention to parties having identical marginal costs, i.e., µA = µB = µ. Recall

that voters’ behavior is essentially parametric and hence the last stage in our backward

induction reasoning is the choice of advertising. Equilibrium advertising can be solved as

effort in a Tullock contest with symmetric players, in which the prize of winning equals

the share of impressionable voters. The equilibrium in this stage is described in the

lemma below

Lemma 1. For all η ≤ 2 there exists a unique Nash equilibrium in the campaign stage
and advertising is given by e∗i (xL,xR) = (1 −F(xR −xL))

η
4µ

, for all i.

All proofs appear in the Appendix.

Our first Lemma draws from previous results in the contest theory literature. It

characterizes the equilibrium advertising levels while stating a condition on the cam-

paigns’ effectiveness η such that an equilibrium in pure strategies exists. If the campaigns

are not effective “enough” (i.e., η ≤ 2), an equilibrium in pure strategies exists and is
unique with advertising being: a) increasing in the campaign effectiveness η, b) decreas-

ing in the marginal cost µ, and c) decreasing in the platforms’ polarization y (recall

that y = xR − xL).10 Note that the symmetric spending in equilibrium implies that in
equilibrium impressionable voters split between the two parties (i.e., SLImp = S

R
Imp = 0.5).

Anticipating the advertising levels in the second stage, the political parties’ maxi-

mization problem in the first stage is to choose the platform that maximizes their payoff.

For instance, for party L, the payoff at t = 1 is ΠL (xL,xR,e
∗
L(xL,xR),e

∗
R(xL,xR)), which

can be written as:

SL(xL,xR)−cL(e∗L(xL,xR)) = F(xR−xL)S
L
Idl(xL,xR)+(1−F(xR−xL))S

L
Imp−µe

∗
L(xL,xR)

where SLImp =
1
2

and SLIdl = G(x̄) if xL 6= xR or S
L
Idl =

1
2

if xL = xR.

The trade-off parties face is now evident. Consider that for a given set of platforms

(xL,xR), the leftist party chooses to propose a less extreme platform. On the one hand,

the indifferent voter is more to the right, which has a positive effect on SLIdl as in a standard

Downsian model. On the other hand, it converts some ideological voters to impressionable

ones, which by Lemma 1 increases the spending on advertising in the second stage of game.

10If campaigns are effective “enough” (i.e., η > 2), there is no equilibrium in pure strategies in that
stage and all mixed-strategy equilibria are payoff equivalent (Alcalde and Dahm, 2010) with parties’
expected payoffs in that stage zero and E(e∗i (xL,xR)) = (1 −F(xR −xL))

1
2µ

). For recent advances on

the properties of such mixed equilibria and relevant literature refer to Ewerhart (2015).

9



Similar to Tirole (1988) and Ashworth and Bueno de Mesquita (2009) divergence is a tool

of softening competition in the vertical dimension (the advertising stage in our model).

The importance of each of the two forces present in the trade-off is determined by

the rate at which impressionable voters become ideological as polarization increases (i.e.,
f(y)
F(y)

) and the mass of voters around the indifferent voter g(x̄). The relative importance

of these two forces is moderated by the campaign effectiveness η. We characterize the

equilibrium of the platform stage in the following proposition.

Proposition 1. Let η ≤ 2 and ȳ implicitly defined by f(ȳ)
F(ȳ)

= 2
η
g( 1

2
). For any µ > 0 there

exists a unique subgame perfect Nash equilibrium.

For F(0) > 0, the following equilibrium types arise:

• (Convergent equilibrium) x∗L = x
∗
R =

1
2
, if

f(0)
F(0)
≤ 2

η
g( 1

2
),

• (Interior equilibrium) x∗L =
1
2
− ȳ

2
, and x∗R =

1
2

+ ȳ
2
, if

f(1)
F(1)

< 2
η
g( 1

2
) <

f(0)
F(0)

• (Extremism equilibrium) x∗L = 0 and x
∗
R = 1, if

2
η
g( 1

2
) ≤ f(1)

F(1)
.

If F(0) = 0, the extremism equilibrium arises if and only if 2
η
g( 1

2
) ≤ f(1)

F(1)
and the

interior equilibrium arises otherwise.

Electoral advertising for each of the above SPNE is uniquely characterized in Lemma 1.

First, notice that in contrast to previous literature with similar dynamics –where

equilibrium platforms require mixed strategies –Proposition (1) shows that there exists a

unique pure strategy SPNE.11 In this unique equilibrium, the level of polarization (y∗) can

be zero (convergent equilibrium), one (extremism equilibrium) or ȳ (interior equilibrium).

The emerging type of equilibrium depends on: a) the concentration of voters around

the median g( 1
2
), b) the rate at which impressionable voters become ideological when

polarization increases
f(y)
F(y)

, and c) the effectiveness of electoral campaigns η.

Starting with the effectiveness of the electoral campaigns, a large value of η makes

the competition for impressionable votes tougher, which exacerbates advertising costs in

the second stage (Lemma 1). Therefore, a high value of η provides incentives to polarize

platforms in the first stage in order to reduce the number of impressionable voters. If

F(0) > 0 (i.e., there exists a share of ideological voters under convergence), as η increases,

indeed we may move across types of equilibria with convergence occurring for a larger set

11Note here that we have restricted attention to η ≤ 2 due to the mixed strategies in the campaign
stage for η > 2. However, the unique pure strategy equilibrium characterized in the platform substage
–which is the main difference to Ashworth and Bueno de Mesquita 2009– is also an equilibrium for η > 2
with equilibrium platforms the same as the ones characterized for η = 2. This is due to the payoff
equivalence result described in the previous footnote.

10



of parameters. But also platforms become less polarized in the interior equilibrium (i.e., ȳ

is decreasing in η). Similarly, when many voters are concentrated around the median (i.e.,

a high value of g( 1
2
)), there are strong incentives to propose moderate platforms. Thus, a

strong presence of moderate voters leads to equilibria of “low” polarization (again, either

across equilibria types or within the interior equilibrium). If F(0) = 0 (i.e., no ideological

voters when parties fully converge), nothing changes except that a convergent equilibrium

never arises.

The conversion rate at which impressionable voters become ideological when polar-

ization increases (i.e, the reverse hazard rate
f(y)
F(y)

) also helps understanding our result.

This rate captures the incentives of increasing polarization as a way of reducing electoral

advertising. By log-concavity of F(y), the rate is monotonically decreasing and hence

takes its maximum value at y = 0 and its minimum value at y = 1. If the maximum value

of the conversion rate is small “enough” (i.e.,
f(0)
F(0)
≤ 2

η
g( 1

2
)) the original Downsian result

of platform convergence emerges. Despite full convergence, the conversion rate is so low

that increasing polarization does not increase the share of ideological voters enough to

diminish electoral advertising in a profitable fashion. Analogously, if its minimum value

is large “enough” (i.e., 2
η
g( 1

2
) ≤ f(1)

F(1)
), polarization is very effective in restraining electoral

advertising and extremism emerges. A distributional change in the function determining

the distribution of voters across types gives interesting comparative statics.

Notation 1. Let ρ parametrize the sensitivity of the conversion rate due to inputs other

than polarization (e.g., awareness or interest in politics).

The conversion rate f(y; ρ)/F(y; ρ) satisfies the monotone likelihood ratio property (MLRP)

in y if for any ρ1 > ρ0 it holds that

f(y; ρ1)

F(y; ρ1)
>
f(y; ρ0)

F(y; ρ0)

where F(y; ρ) and f(y; ρ) are differentiable in ρ.

Any increase in ρ makes the conversion rate of impressionable voters to ideological

more responsive to changes in polarization. This may move platforms across types of

equilibria (favoring more polarization), while also at any interior equilibrium, polariza-

tion is increasing in ρ (it follows from applying implicit differentiation to the interior

equilibrium condition
f(ȳ)
F(ȳ)

= 2
η
g( 1

2
)).

Finally, note that platforms’ characterization does not depend on the costs of cam-

paigning µ in any manner. Given that this cost is symmetric for the two parties, increasing

or decreasing it would only rescale the equilibrium levels of advertising ei (from Lemma

1) but will not modify the actual level of campaign spending µei and hence polarization.

11



Effects on Campaign Spending

A technological change that increases the campaign effectiveness has an ambiguous effect

on campaign spending. This is apparent when we look at the relevant expression:

∂µe∗i (x
∗
L,x

∗
R)

∂η
=

Spending effect (+)︷ ︸︸ ︷
1 −F(y∗)

4

Polarization effect (-)︷ ︸︸ ︷
−f(y∗)

∂y∗

∂η

η

4
(2)

On the one hand, ceteris paribus an increase in η increases advertising (Lemma 1). On the

other hand, it also increases in equilibrium level of polarization y∗ (Proposition 1), which

in turn, decreases the share of impressionable voters and so the levels of advertising

(Lemma 1). We call the former the spending effect, while we label the latter as the

polarization effect.

At the fully divergent and fully convergent equilibria there is no polarization effect,
∂y∗

∂η
= 0, and spending either increases monotonically with η due to the spending effect

or is unaffected. At the interior equilibrium however, the polarization effect takes place

and mitigates the spending effect. If polarization increases disproportionally with η,

the polarization effect may even overturn the spending effect, and hence observe η and

campaign spending move in opposite directions. In Lemma (2) below, we provide the

conditions for a simultaneous increase in campaign spending and polarization, and we

use an example to illustrate it.

Lemma 2. In any interior equilibrium, a technological change that increases the cam-

paign effectiveness η – and hence polarization – also increases campaign spending due

to the spending effect dominating the polarization effect if and only if effectiveness is

low “enough”. Formally, in any interior equilibrium
∂µe∗i (x

∗
L
,x∗
R

)

∂η
≥ 0 if and only if

η ≤ 2g( 1
2
)

1−F(ȳ)
f(ȳ)

[
F(ȳ)
f(ȳ)

]′
.

Example 1: F(y) is uniformly distributed over (0, 1] with a mass at zero

F(0) = 1
10

. The “conversion rate” now – from impressionable to ideological voters –

is proportional to polarization:
f(y)
F(y)

=
9
10

1
10

+ 9
10
y
. From Proposition (1), we are at a conver-

gent equilibrium for η ≤ 2
9
g( 1

2
), at an interior equilibrium for 2

9
g( 1

2
) < η < 20

9
g( 1

2
) and at

an extremism equilibrium for η ≥ 20
9
g( 1

2
). These conditions highlight how the concentra-

tion of voters around the median gives rise to different equilibrium types. If for example

g( 1
2
) is large then extremism can be excluded as an outcome for any level of campaign

effectiveness. In the interior equilibrium polarization given by y∗ = ȳ = η
2g( 1

2
)
− 1

9
and

it is straightforward to see that polarization is increasing in η. Using the condition in

Lemma (2) we can also get the non-monotone comparative statics on campaign spending

12



Figure 1: Comparative statics on η for campaign spending and polarization. Uniform
distribution of F(y) with F(0) = 1

10
and g(1/2) = 1/2.

in the interior equilibrium and show that campaign spending is increasing in the campaign

effectiveness for η low enough (i.e., η ≤ 10
9
g( 1

2
)) and decreasing otherwise.

In Figure 1 we graphically represent the comparative statics of changes of η on polar-

ization and campaign spending also assuming that g( 1
2
) = 1

2
. Consider first the equilib-

rium levels of polarization (solid line). For η lower than 1
9

or greater than 10
9

the convergent

and extremist equilibria respectively arise. For 1
9
< η < 10

9
the interior equilibrium arises

and polarization is strictly increasing in η. Let’s now turn to campaign spending (dashed

line). For η ≤ 1
9
, polarization is constant and equal to zero and campaign spending is

monotonically increasing in η (as in any standard Tullock contest). If η ≥ 10
9

the ex-

tremism equilibrium arises and due to all voters being ideological campaign spending is

zero.12 In the interior equilibrium interval (i.e., η ∈ ( 1
9
, 10

9
)), the non-monotonicity arises.

Campaign spending increases until reaching η = 5
9

due to the spending effect being larger

than the polarization one. On the contrary, spending decreases for values of η larger than
5
9

due to the polarization effect overcoming the spending effect and till extremism arises.

Polarization and campaign spending are also affected by the rate at which impression-

able voters become ideological as polarization increases parametrized by ρ. A technolog-

12 Zero spending under extremism arises because of the simplification of no mass of impressionable
voters under maximal platform separation. One could trivially extend our model by including a mass of
impressionable voters 0 < δ < 1 even under extreme polarization. In that case, in the extreme equilibrium
campaign spending is positive and strictly increasing in η. Our characterization in Proposition 1 would
remain unaffected and advertising levels in Lemma 1 will be simply rescaled but not qualitatively affected.

13



ical change that increases ρ has an ambiguous effect on campaign spending. As above,

the condition comes from looking at the derivative of campaign spending with respect to

ρ.

∂µe∗i (x
∗
L,x

∗
R)

∂ρ
=

Spending effect (+)︷ ︸︸ ︷
−
∂F (y∗)

∂ρ

η

4

Polarization effect (-)︷ ︸︸ ︷
−
∂F (y∗)

∂y∗
∂y∗

∂ρ

η

4

On the one hand, an increase in ρ increases the “stock” of impressionable voters

1−F(y) (this follows directly from MLRP which implies first order stochastic dominance).
The spending effect then suggests that, for a given level of polarization y, an increase in ρ

increases advertising (Lemma 1). On the other hand, an increase in ρ makes voters more

responsive to polarization, by affecting the “conversion” rate
f(y)
F(y)

, and therefore provides

incentives to increase polarization (Proposition 1). But this increased polarization in

turn, decreases the share of impressionable voters and so decreases the levels of advertising

(Lemma 1). As before, we label this latter effect as the polarization effect.

At the convergent equilibria there is no polarization effect since ∂y
∗

∂ρ
= 0 and spending

increases monotonically with ρ. That is, an increase in ρ keeping polarization constant

would increase the number of impressionable voters and their weight in the parties’ maxi-

mization problem and parties would have higher incentives to increase advertising. In the

interior equilibrium the polarization effect kicks in and the net effect on spending depends

on the magnitude of these two effects. Recall that changes in ρ enter in the spending

effect due to changes in “stock”, while they enter in the polarization effect due to changes

in the “conversion”. Finally, polarization is constant in ρ once the extremism equilibrium

is reached and spending is constant and equal to zero since there are no impressionable

voters. The following lemma summarizes the above for the interior equilibrium.

Lemma 3. In any interior equilibrium, a technological change that increases ρ – and

hence polarization – also increases campaign spending due to the spending effect domi-

nating the polarization effect if and only if the effect of ρ on the “stock” of impressionable

voters is “large” enough. Formally, in any interior equilibrium
∂µe∗i (x

∗
L
,x∗
R

)

∂ρ
≥ 0 if and only

if −∂F(ȳ)
∂ρ
≥ f(ȳ)∂ȳ

∂ρ
.

Finally, notice that in the symmetric case, the marginal cost of advertising plays no

role in equilibrium: neither the platforms nor the campaign spending depend on µ. When

we introduce a cost-asymmetry between the parties, the equilibrium changes, favoring the

party with the lower marginal cost.

14



3.2 Asymmetric parties

In contrast to previous models with endogenous valence, we can incorporate cost asym-

metries and obtain results of similar intuition. In this section we assume, without loss of

generality, that µL < µR. We first characterize campaign spending in the second stage

and then show how equilibrium platforms are affected by the asymmetry. Following Baik

(1994) and Nti (1999) the following Lemma arises.

Lemma 4. Let η̄ implicitly defined by µ
η̄
L + µ

η̄
R = η̄µ

η̄
R. For all η ≤ η̄ there exists a

unique Nash equilibrium in the campaign stage and advertising is given by e∗i (xL,xR) =

(1 −F(y)) η
µi

µ
η
L
µ
η
R

(µ
η
L

+µ
η
R

)2
for all i.

Lemma (4) shows that, in equilibrium, parties choose different levels of advertising

(e∗i (xL,xR)) although spend equal amounts (µie
∗
i (xL,xR)).

13 The share of impressionable

voters is not any longer equally split across parties, giving an advantage to the party

with the lower marginal cost. This generates an asymmetry in parties’ incentives to use

polarization as a device to reduce campaign spending and eliminates the interior equilibria

where x∗L + x
∗
R = 1. In any interior equilibrium parties propose asymmetric platforms.

The convergent and extremism equilibria described previously also arise. We present the

conditions for the rise of each type of equilibrium in the following propositions.

Proposition 2. (Convergent equilibrium) For any η ≤ η̄, µL < µR and F(0) > 0, there
exists a unique SPNE with x∗L = x

∗
R =

1
2

if and only if
f(0)
F(0)
≤ g( 1

2
)
/(

µ
η
L
µ
η
R

(µ
η
L

+µ
η
R

)2
2η+

µ
η
R
−µη

L

µ
η
L

+µ
η
R

)
.

Campaign spending for the convergent equilibrium is uniquely characterized in Lemma 4.

As in the symmetric case, the convergent equilibrium arises when a lot of voters are

concentrated around the median (i.e., high g( 1
2
)) and/or the conversion rate at y = 0

(i.e., at its maximal level
f(0)
F(0)

) is low. In the asymmetric case, the size of the asymmetry

is also a determinant of platform convergence. If the asymmetry converges to zero, then

the inequality characterizing a convergent equilibrium converges to the one of Proposition

1. As the asymmetry however increases, platform convergence becomes less likely (the

denominator at the right hand side of the inequality is increasing in the asymmetry).

This is because, as the asymmetry increases, the (symmetric) convergent equilibrium

becomes less attractive for the disadvantaged party that loses the competition for im-

pressionable voters and hence has incentives to diversify and propose distinct platforms.

This potentially leads to an interior equilibrium.

13The condition on η that guarantees equilibrium in pure strategies is more restrictive than in the
symmetric case, since η̄ is lower or equal than 2. For η ∈ (η̄, 2) the equilibrium is characterized by Wang
(2010) and Ewerhart (2017), while payoff equivalence by Alcalde and Dahm (2010) can be used to solve
the platform stage for η > 2.

15



An equilibrium is interior if at least one of the parties chooses an interior platform.

That is, a platform between the median and the corner (either 0 or 1, depending on the

party). Taking this definition into account, Proposition 3 describes the necessary and

sufficient conditions for the existence and uniqueness of an interior equilibrium, which

depend on the extent of the parties’ cost asymmetries. Then, within the scope of the

proposition, the corollary below characterizes the interior equilibria.

Proposition 3. (Interior Equilibrium) For any η ≤ η̄ and µL < µR, let x̄∗ = G−1(
µ
η
R

µ
η
L

+µ
η
R

)

and ȳ be implicitly defined by
f(ȳ)
F(ȳ)

=
g(x̄∗)

2η

(µ
η
L

+µ
η
R

)2

µ
η
L
µ
η
R

, there exists a unique interior SPNE if

and only if one of the following conditions is satisfied

F(2x̄∗ − 1)
f(2x̄∗ − 1)

g(x̄∗) < 2η
µ
η
Lµ

η
R

(µ
η
L + µ

η
R)

2
<
F(1)

f(1)
g(0.5) +

µ
η
R −µ

η
L

µ
η
R + µ

η
L

if
µ
η
R

µ
η
L + µ

η
R

< G(
3

4
) (3)

F( 1
2
)

f( 1
2
)
g(

1

2
) − 2G(

3

4
) +

2µ
η
R

µ
η
R + µ

η
L

< 2η
µ
η
Lµ

η
R

(µ
η
L + µ

η
R)

2
if

µ
η
R

µ
η
L + µ

η
R

≥ G(
3

4
) (4)

Corollary 1. (Characterization of Interior Equilibria) There are two types of Interior

equilibria. Proposition 3 describes the necessary and sufficient conditions for existence

and uniqueness. Their characterization is as follows:

• (Interior/Interior) x∗L = x̄
∗ − ȳ

2
and x∗R = x̄

∗ + ȳ
2

if and only if Condition 3 and

2η
µ
η
L
µ
η
R

(µ
η
L

+µ
η
R

)2
<

F(2−2x̄∗)
f(2−2x̄∗) g(x̄

∗) are satisfied.

• (Interior/Corner) x∗L < 0.5 and x
∗
R = 1, otherwise.

Campaign spending for each of the above SPNE is uniquely characterized in Lemma 4

All interior equilibria are asymmetric. It can be that both parties propose interior

platforms (Interior/Interior) or that the cost advantaged party proposes an interior plat-

form while the disadvantaged party proposes an extreme platform (Interior/Corner).

The cost asymmetry plays an important role in determining the type but also polariza-

tion level at the interior equilibrium. The cost disadvantaged party R has more incentives

than the advantaged party to reduce the share of (costly) impressionable voters. Con-

sequently, R has more incentives to separate its platform from L. Thus, in any interior

equilibrium platforms are shifted towards the cost-disadvantaged party R. That is, in

the (Interior/Interior) equilibrium the point around which parties propose equidistant

platforms is to the right (i.e., x̄∗ > 0.5), while in the (Interior/Corner) it is the disadvan-

taged party that proposes an extreme platform. Note that in general, if the asymmetry is

low “enough” (i.e., (3) is satisfied) the disadvantaged party may also propose an interior

platform. If this asymmetry however is high “enough” (i.e., (4) is satisfied) only the (In-

terior/Corner) equilibrium arises, the disadvantaged party would like to separate more

but cannot due to having reached the corner. The disadvantaged party best responds

16



to that by proposing an interior platform. In that sense, “large” asymmetries never give

rise to extremism equilibria either as the following proposition summarizes.

Proposition 4. (Extremism Equilibrium) For any η ≤ η̄ and µL < µR, there exists a
unique SPNE with x∗L = 0 and x

∗
R = 1 if and only if g(0.5)

/(
2η

µ
η
L
µ
η
R

(µ
η
L

+µ
η
R

)2
− µ

η
R
−µη

L

µ
η
L

+µ
η
R

)
≤ f(1)

F(1)

and campaign spending as characterized in Lemma 4.

If the asymmetry converges to zero, the relevant inequality converges to the one of

Proposition 1. As the asymmetry however increases, extremism becomes less likely (the

denominator on the left hand side of the inequality is decreasing in the asymmetry).

This is because, the advantaged party has no incentives to polarize as it would reduce

the share of impressionable voters for whom it enjoys a great advantage. Also, as in the

symmetric case, extremism is more likely when the rate at which impressionable voters

become ideological is great enough under maximal platform separation (i.e.,
f(1)
F(1)

is high).

As with the symmetric case, the inequalities in all equilibrium conditions also depend

on the effectiveness of campaigns, the mass of ideological voters at the median and con-

version rates at different points. While we have covered the most relevant ones, especially

what concerns the cost asymmetries, in general, the incentives to polarize (or not) and

move across equilibria types remain similar to the symmetric case. Proceeding with ex-

amples, similar intuition to the results of the symmetric case arise. The simultaneous

increase in polarization and campaign spending observed can be reconciled also in the

presence of an asymmetry.

Example 2: F(y) and G(y) uniformly distributed in X. In Figure 2 Panel (a)

we plot the case of “low” asymmetry (µR/µL = 1.2) while in Panel (b) we plot the

case of “high” asymmetry (µR/µL = 2). For both levels of asymmetry, we are at a

convergent equilibrium only when η = 0. That is, when impressionable voters are split

equally across the two parties regardless of electoral advertising parties do not have

incentives to differentiate their platforms since the asymmetry vanishes. As long as η > 0

we illustrate the divergent and asymmetric (Interior/Interior) equilibrium platforms as

characterized in Corollary 1. Note that in both panels, and as described in our results, the

disadvantaged party R is proposing a relatively more extreme platform compared to the

advantaged party L. In the example of “low” asymmetry, platforms diverge monotonically

as η increases (as in the symmetric case) and hence polarization is also increasing. In

the example of “high” asymmetry, and despite polarization being again increasing in the

effectiveness of electoral advertising, the advantaged party does not respond to increases

in η in a monotonic manner. Finally, when it comes to campaign spending, as the upper

right panel shows, we may again encounter a situation of a non-monotonic relationship

17



between campaign spending and η (as in Figure 1 and the symmetric case). But perhaps

more importantly, our results can again sustain the simultaneous increase in polarization

and campaign spending due to technological changes even in the presence of asymmetries

for a wide range of parameters.

Panel (a): “low” asymmetry: µR/µL = 1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0.2

0.4

0.6

0.8

1

xR

xL

η

P
la
tf
o
r
m
s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0

0.2

0.4

0.6

0.8

1

η

P
o
la
r
iz
a
ti
o
n

0

5 · 10−2

0.1

0.15

0.2

S
p
e
n
d
in
g

Panel (b): “high” asymmetry: µR/µL = 2

0 0.2 0.4 0.6 0.8 1 1.2

0.2

0.4

0.6

0.8

1

xR

xL

η

P
la
tf
o
r
m
s

0 0.2 0.4 0.6 0.8 1 1.2
0

0.2

0.4

0.6

0.8

1

η

P
o
la
r
iz
a
ti
o
n

0

5 · 10−2

0.1

0.15

0.2

S
p
e
n
d
in
g

Figure 2: Solid lines depict equilibrium platforms (x∗
L
,x∗
R

) in the left panels and polarization (x∗
R
−x∗

L
) in the right

panels. Dotted lines depict campaign spending. Graphs are plotted on the interval of η that guarantees an equilibrium in
pure strategies in the advertising stage.

4 Conclusion

Citing Herrera et al. (2008), “commentators have suggested that the reason for both the

increased polarization and campaign spending is that skilled political operatives using so-

phisticated statistical tools and purchasing advertising in local markets are better able to

18



target particular voters” (see for example, NBC 2017). However, existing results so far

were linking such technological advances with a reduction in polarization, and therefore

favored alternative channels that may drive polarization such as more volatile preferences.

Our results in contrast, are the first to justify the simultaneous increase in polarization

and campaign spending due to recent technological changes and better targeting of the

electoral campaigns.

Given that one would naturally expect further advances in campaign technology two

natural questions arise: a) should we expect a further increase in polarization?, and b)

what about campaign spending? Our theory would say yes, further advances in targeting

will lead to further polarization. Ways to go against this trend would require policies that

improve the awareness of the electorate and a shift of voters’ attention from persuasive

campaigns to political platforms. With regards to campaign spending, recall that our

theory does not provide a monotone comparative static. As the targeting technology im-

proves, parties have incentives to increase their campaign spending, but at the same time

to polarize (which reduces campaign spending). Hence, our results are not incompati-

ble with the observations of the 2016 presidential US election where campaign spending

dropped. Furthermore, following this presidential campaign, the diffusion of plausible

but false information received the name of “fake news”. This notion, widely used during

and after the campaigns, has direct implications on the electorate’s awareness: for any

given “real” platforms, the stock of ideological voters decreases with the amount of fake

news. In terms of our model, fake news would increase the reversed hazard rate, causing

an increase in polarization as well.

While one of our contributing messages could be the importance of the electorate’s

awareness as a way to affect campaign spending and polarization, political pundits have

paid special attention to caps on campaign spending. In the context of symmetric costs,

our model provides very clear and intuitive implications regarding the effects of this policy.

If a cap is below the equilibrium campaign spending, it will completely shut down the

polarization effect in our model. That is, when a cap is binding, the parties will not induce

an increase in campaign spending by moving their platform towards their competitor.

This eliminates one of the elements of the parties’ trade-off in the choice of platform

location. As a consequence, platform convergence is the only possible outcome. In other

words, any cap on campaign spending smaller or equal to the equilibrium spending, will

also induce convergence at the median voter. Thus our model predicts an important

impact of campaign caps on party polarization. The same intuition can be sustained

under asymmetric costs if the inverse hazard rate is great enough under full convergence.

19



5 Appendix - Proofs

This section is organized as follows. In Subsection 5.1 we show the pure strategy equi-

librium in advertising for asymmetric costs. Additionally, we discuss the extension for

η > η̄ in the symmetric case. In Subsection 5.2 we write and solve the platform selec-

tion problem for the general case with asymmetric costs. Thus, taking into account the

Kuhn-Tucker conditions, we solve this maximization problem and we show the first and

second order conditions. Lemmi A.1, A.2 and Remark 1 in that subsection provide the

conditions for existence and uniqueness of platforms that maximize the Lagrangian for

the general case. Next, we show uniqueness and existence of a unique (pure) Subgame

Perfect Nash Equilibrium, and we characterize the equilibrium platforms for the sym-

metric case (Proposition 1). Later on, we do the same for asymmetric costs. In this case,

we show separately the convergent equilibrium (Proposition 2), the interior equilibrium

(Proposition 3 and Corollary 1) and the extremism one (Proposition 4).

5.1 Advertising

We begin proving the more general Lemma 4, i.e., µL 6= µR, and after the proof we discuss
the implications of symmetry, i.e., Lemma 1.

Proof. For any given pair (xL,xR), parties simultaneously choose ei that maximizes (1−
F(y))

e
η
i

e
η
L

+e
η
R
−µiei(xL,xR). Hence, from the FOC, we obtain the first necessary condition

for an interior PSNE: ei =
1−F(y)
µi

η
µ
η
L
µ
η
R

(µ
η
L

+µ
η
R

)2
. The equilibrium payoffs can be written as

(1 −F(y))[
µ
η
i

µ
η
L + µ

η
R

−η
µ
η
Lµ

η
R

(µ
η
L + µ

η
R)

2
],

which are positive if and only if (µ
η
L +µ

η
R)µ

η
i −ηµ

η
Lµ

η
R ≥ 0. Therefore, the second necessary

condition for PSNE is that η ≤ η̄ : µη̄R +µ
η̄
L = η̄µ

η̄
R. Moreover, for all η lower or equal than

η̄, the SOC holds with strict inequality, assuring not only existence but also uniqueness

of the equilibrium.

The symmetric case. Notice that η̄ ≤ 2 holds with strict equality if and only if
µL = µR. In the symmetric case, for all η ≤ 2, the conditions on the existence and
uniqueness of PSNE are satisfied, and therefore ei =

1−F(y)
4µi

η. For η > 2, Alcalde and

Dahm (2010) show that “the contest possesses an all-pay auction equilibrium” (Theorem

3.2 in their paper). In a symmetric contest, this theorem implies full dissipation, which

means that the expected payoff would be 0 for parties L and R, and the corresponding

expected bids would be 1
2

1−F(y)
µi

η.

20



5.2 Polarization and Spending

The objective function, existence and uniqueness

Throughout this section let y = xR − xL, x̄ = xR+xL2 , and S
L
Idl(xL,xR) = G(x̄) = 1 −

SRIdl(xL,xR) for xL 6= xR and S
L
Idl(xL,xR) = S

R
Idl(xL,xR) =

1
2

otherwise. By backward

induction and using the equilibrium expressions of the of advertising subgame, we can

write the first-stage payoff for the political parties as follows

Πi(xL,xR) = F(y)S
i
Idl(xL,xR) + (1 −F(y))

µ
η
−i

µ
η
R + µ

η
L

− (1 −F(y))η
µ
η
Rµ

η
L

(µ
η
R + µ

η
L)

2
.

Lemma A.1. In any equilibrium xL ≤ 1/2 and xR ≥ 1/2.

Proof. Remind that we have assumed without loss of generality that xL ≤ xR. First
consider a divergent equilibrium, i.e., xL 6= xR. We proceed by contradiction. Assume
an equilibrium such that 1/2 < x̃L < x̃R. Then party R is strictly better of by deviating

to xR = 2x̃L − x̃R, which maintains unchanged the proportion of each type of voter and
the share of impressionable votes and strictly increases the share of ideological votes of

party R. Analogously we can show that party L always has a profitable deviation to

xL = 2x̃R − x̃L when x̃L < x̃R < 1/2
Now consider a convergent equilibrium xL = xR. Given the assumption of xL ≤ xR

two necessary conditions for a convergent equilibrium at xL = xR = x are:

limxL→x− ΠL(xL,x) ≤ ΠL(x,x)
limxR→x+ ΠR(x,xR) ≤ ΠR(x,x)

For x > 1
2
, limxL→x−G(xL,x) >

1
2

= G(x,x), which implies that limxL→x−ΠL(xL,x) >

ΠL(x,x) and violates the equilibrium condition for party L. A similar argument applies

for party R if x < 1
2
.

Remind that when xL = xR 6= 1/2, a discontinuity can arise in the objective function,
due to the discontinuity of SiIdl(x). However, the previous lemma restricts convergence to

the case xL = xR = 0.5, in which S
i
Idl(x), and consequently Πi(xL,xR), are continuous.

Thus, by including the equilibrium constraints (xL ≤ 1/2 and xR ≥ 1/2) in the maxi-
mization problem, the Lagrangians below are continuous, and provided that the second

order conditions are met, will be solved by the equilibrium platforms (due to the pres-

ence of equilibrium constraints solving the Lagrangian is a necessary but not a sufficient

condition for equilibrium, as we show in Remark 1 below).

For i ∈{L,R}, let λi ≥ 0 be the multipliers associated with the feasibility constraints

21



and νi ≥ 0 with the equilibrium constraints. The Lagrangians are:

LL = ΠL(xL,xR) −λL(−xL − 0) −νL(xL −
1

2
) (5)

LR = ΠR(xL,xR) −λR(xR − 1 ) −νR(
1

2
−xR). (6)

The first order conditions (FOC from now on) are:

∂LL
∂xL

=
∂ΠL(xL,xR)

∂xL
+ λL −νL = 0, (7)

∂LR
∂xR

=
∂ΠR(xL,xR)

∂xR
−λR + νR = 0. (8)

Where,

Π′L ≡
∂ΠL(xL,xR)

∂xL
= F(y)

g(x̄)

2
−f(y)G(x̄) + f(y)

µ
η
R

µ
η
R + µ

η
L

−f(y)η
µ
η
Rµ

η
L

(µ
η
R + µ

η
L)

2
, (9)

Π′R ≡
∂ΠR(xL,xR)

∂xR
= −F(y)

g(x̄)

2
+ f(y)[1 −G(x̄)] −f(y)

µ
η
L

µ
η
R + µ

η
L

+ f(y)η
µ
η
Rµ

η
L

(µ
η
R + µ

η
L)

2
(10)

Lemma A.2. Let F(x) be log-concave and g(x) be symmetric and log-concave. The

objective functions are strictly quasiconcave in the policy space when xL ≤ 12 ≤ xR, hence
the second order conditions (SOC) are satisfied.

The constrained optimization problem includes two linear constraints for each party,

thus focusing on the quasiconcavity of the payoff functions suffices.

Proof. Without loss of generality, consider party L. Under the conditions of Lemma

(A.1), continuity of ΠL is assured. Then, let us modify Equation (9) by dividing it over

the densities f(y) and g(x̄):

Π̃′L ≡
Π′L

f(y)g(x̄)
=

F(ȳ)

2f(ȳ)
−

1

g(x̄)

[
G(x̄) −

µ
η
R

µ
η
R + µ

η
L

+ η
µ
η
Rµ

η
L

(µ
η
R + µ

η
L)

2

]
(11)

Let Π̃L be the primitive of Π̃
′
L. Π̃L is strictly quasiconcave if and only if Π̃

′
L(x)(x

′−x) > 0
whenever Π̃L(x

′) > Π̃L(x). Since strict quasiconcavity is determined by the sign of Π̃
′
L(x),

which is the same of the sign of Π′L(x) (because f(y)g(x̄) is strictly positive), the strict

quasiconcavity of Π̃L(x) guarantees the strict quasiconcavity of ΠL(x).

Therefore, by showing the strict concavity of Π̃L (i.e., Π̃
′′
L =

∂
Π′
L

f(y)g(x̄)

∂xL
< 0), we will be

proving that ΠL(xL,xR) is strictly quasiconcave too. Hence Π̃
′′
L is

∂
Π′
L

f(y)g(x̄)

∂xL
= −

1

2

[
F(ȳ)

f(ȳ)

]′
−

1

2g(x̄)2

{
g(x̄)2 −g′(x̄)

[
G(x̄) −

µ
η
R

µ
η
R + µ

η
L

+ η
µ
η
Rµ

η
L

(µ
η
R + µ

η
L)

2

]}

22



By log-concavity of F(y), the term −[ F(ȳ)
2f(ȳ)

]′ is negative, so we can focus on the nega-

tivity of H = −
{
g(x̄)2 −g′(x̄)

[
G(x̄) − µ

η
R

µ
η
R

+µ
η
L

+ η
µ
η
R
µ
η
L

(µ
η
R

+µ
η
L

)2

]}
in the expression above to

guarantee strict concavity of Π̃L(x) (and hence strict quasi-concavity of ΠL(x)). Let us

consider two cases.

• If g′(x̄) ≥ 0, then −g′(x̄)[ µ
η
R

µ
η
R

+µ
η
L
−η µ

η
R
µ
η
L

(µ
η
R

+µ
η
L

)2
]) is negative (strictly negative for g′(x̄) >

0) because the term in brackets is always positive given η < η̄ (see proofs of Lemmi

1 and 4). Log-concavity of g(x) implies log-concavity of G(x), thus −[g(x̄)2 −
g′(x̄)G(x̄)] is negative (strictly negative for g′(x̄) = 0). Hence H is strictly negative

and ΠL(x) strictly quasi-concave.

• If g′(x̄) < 0, suppose there exists x̂L : G( x̂L+xR2 ) =
µ
η
R

µ
η
R

+µ
η
L
−η µ

η
R
µ
η
L

(µ
η
R

+µ
η
L

)2
. Since G(x)

is increasing in x, for xL > x̂L, H would be strictly negative and ΠL(x) strictly

quasi-concave. For xL ≤ x̂L, G(x̄) ≤
µ
η
R

µ
η
R

+µ
η
L
− η µ

η
R
µ
η
L

(µ
η
R

+µ
η
L

)2
implies that

∂ΠL(xL,xR)
∂xL

is

strictly positive which directly implies that ΠL(xL,xR) is strictly quasiconcave for

xL ∈ [0, x̂L].

We can proceed similarly to show that ΠR(xL,xR) is also strictly quasiconcave (in

that case we use that g(x) implies that 1 −G(x) is log-concave).

Remark 1. Let (xL,xR) be a solution to the Lagrangians. Then, in the platforms stage,

there is a divergent equilibrium (xL 6= xR) only when νL = νR = 0 and there is a
convergent one (xL = xR) only when νL ≥ 0 and νR ≥ 0.

Proof. For the divergent equilibria, let xL 6= xR be an equilibrium with νR = 0. Suppose
νL > 0. Then xL =

1
2
< xR and Π

′
L(

1
2
,xR) > 0. Thus, party L has incentives to deviate

to a platform strictly larger than 1
2

which violates Lemma (A.1). Then xL 6= xR with
νR = 0 < νL cannot be an equilibrium. Hence, νL must be 0 for this type of equilibrium

to exist. The same holds true for νL = 0 < νR.

For the convergent equilibrium, Lemma (A.1) implies xL = xR =
1
2

is the only can-

didate to equilibrium. Hence, at the solution to the Lagrangian, νL ≥ 0 and νR ≥ 0 is
true. Notice that even if Π′L(

1
2
, 1

2
) ≥ 0 and Π′R(

1
2
, 1

2
) ≤ 0, like in a standard Dawnsian

game, they do not have incentives to deviate. Suppose L deviates to a platform strictly

larger than 1
2
, then it would become the Right party, which implies that if would have

incentives to converge to 1
2
.

Proof of Proposition 1

Proof. Lemma A.1 shows that we can constraint the analysis to the case with xL ≤ 12 ≤ xR
without loss of generality. Lemma A.2 shows strict quasiconcavity of the maximization

23



problem. Hence, the first order conditions are necessary and sufficient to characterize the

equilibrium. With µL = µR equations 9 and 10 become:

Π′L ≡
∂ΠL(xL,xR)

∂xL
= F(y)

g(x̄)

2
−f(y)G(x̄) + f(y)

1

2
−f(y)η

1

4
, (12)

Π′R ≡
∂ΠR(xL,xR)

∂xR
= −F(y)

g(x̄)

2
+ f(y)[1 −G(x̄)] −f(y)

1

2
+ f(y)η

1

4
(13)

Remark 1 shows that convergence in the symmetric case implies Π′L ≥ 0 and Π
′
R ≤ 0.

If these conditions are met, xL =
1
2

= xR implies y = 0. Notice that if F(0) = 0, then

there is no convergent equilibrium. When F(0) > 0, both conditions can be re-written

as
f(0)
F(0)
≤ 2

η
g( 1

2
). Similarly, from Remark 1 we have full polarization when Π′L ≤ 0 and

Π′R ≥ 0, evaluated at xL = 0, xR = 1. Using that xL = 0 and xR = 1 imply y = 1,
both conditions above can be re-written as

f(1)
F(1)
≥ 2

η
g( 1

2
). And when there is a divergent

equilibrium (without full polarization), we obtain what we call the interior/interior equi-

librium when Π′L = Π
′
R = 0. This equality implies that polarization is implicitly defined

by
f(ȳ)
F(ȳ)

= 2
η
g( 1

2
), when there is no convergent or extremism equilibrium, i.e., when the

two inequalities above do not hold:

f(1)

F(1)
<

2

η
g(

1

2
) <

f(0)

F(0)
.

Proof of Proposition 2

Proof. Because of Lemma (A.2) and Remark (1), solving the Lagrangian in the case

λL = λR = 0 and νL ≥ 0 νR ≥ 0 suffices for a convergent equilibrium. By Lemma A.1,
a convergent equilibrium can only take place at xL = xR =

1
2
, which implies λi = 0 for

i = L,R. Given that νL and νR are positive, the Kuhn-Tucker conditions for a convergent

equilibrium imply
∂ΠL(x,x)
∂xL

≥ 0 and ∂ΠR(x,x)
∂xR

≤ 0 and using that G(0.5) = 1
2

they can be

written for i = L,R and −i 6= i as:

F(0)

f(0)
g(0.5) ≥ 2η

µ
η
Lµ

η
R

(µ
η
L + µ

η
R)

2
+
µ
η
i −µ

η
−i

µ
η
L + µ

η
R

.

Given that µR ≥ µL, if the equation for R is satisfied, it will also be so for L. Finally,
note that by substituting µL = µR we obtain the convergent condition for the proof of

Proposition (1). For F(0) = 0, there is not convergent equilibrium.

24



Proof of Proposition 3 and Corollary 1

Below we provide the conditions for the existence (Proposition 3) and the characterization

(Corollary 1) of the divergent equilibrium in which both parties play interior platforms.

Following the classification of Corollary 1 we prove the interior/interior equilibrium in

Lemma A.3 and the interior/corner equilibrium in Lemma A.4.

Lemma A.3. Let x̄∗ = G−1(
µ
η
R

µ
η
L

+µ
η
R

) and y∗ = ȳ be implicitly defined by
f(ȳ)
F(ȳ)

=
g(x̄∗)

2η

(µ
η
L

+µ
η
R

)2

µ
η
L
µ
η
R

.

Hence, there is a unique divergent interior equilibrium x∗L = x̄
∗ − ȳ

2
, x∗R = x̄

∗ + ȳ
2

if and

only if
F(2x̄∗ − 1)
f(2x̄∗ − 1)

g(x̄∗) < 2η
µ
η
Lµ

η
R

(µ
η
L + µ

η
R)

2
<
F(2 − 2x̄∗)
f(2 − 2x̄∗)

g(x̄∗)

which only holds for
µ
η
R

µ
η
R

+µ
η
L
< G( 3

4
).

Proof. Because of Lemma (A.2) and Remark (1), solving the Lagrangian in the case

λi = νi = 0 for all i suffices for a divergent interior equilibrium. Using x̄ = (xL + xR)/2

and y = xR −xL, the pair (x∗L,x
∗
R) is uniquely defined by the pair (x̄

∗,y∗) with y∗ = ȳ.

We begin by proving that there is a unique (x̄∗, ȳ) that solves the FOCs when λi = 0 = νi

for all i. From ∂LL
∂xL

+ ∂LR
∂xR

= 0 and G(x) strictly increasing in x, we obtain the unique x̄∗

that solves the FOCs:

G(x̄∗) =
µ
η
R

µ
η
L + µ

η
R

⇐⇒ x̄∗ = G−1(
µ
η
R

µ
η
L + µ

η
R

).

Plugging x̄∗ in 7 or 8, we obtain
f(ȳ)
F(ȳ)

=
g(x̄∗)

2η

(µ
η
L

+µ
η
R

)2

µ
η
L
µ
η
R

. Since the conversion rate is

increasing and g(x̄∗) can be treated as a constant, there is a unique polarization level ȳ

that solves the FOCs. Hence, we obtain the unique

x∗L = x̄
∗ −

ȳ

2

and

x∗R = x̄
∗ +

ȳ

2

that solve the FOCs. Note that for µL = µR we immediately obtain that G(x̄
∗) = 1

2
and

by symmetry o G(x) that x̄∗ = 1
2

Finally, we have to check that the above solutions, x∗L and x
∗
R lie within the corre-

sponding policy sub-space x∗L ∈ (0,
1
2
) and x∗R ∈ (

1
2
, 1). First, G(x) is increasing, so its

inverse is as well. Hence, from x∗L ∈ (0,
1
2
), we obtain that ȳ > 2G−1(

µ
η
R

µ
η
R

+µ
η
L

) − 1, and

25



from x∗R ∈ (
1
2
, 1), we obtain that ȳ < 2 − 2G−1( µ

η
R

µ
η
R

+µ
η
L

). Hence, it must be the case that

ȳ ∈
(

2G−1(
µ
η
R

µ
η
R + µ

η
L

) − 1, 2 − 2G−1(
µ
η
R

µ
η
R + µ

η
L

)

)
⇐⇒ ȳ ∈ (2x̄∗ − 1, 2 − 2x̄∗) , (14)

which only holds if
µ
η
R

µ
η
R

+µ
η
L
< G( 3

4
). Also, in Equation 14, we can use the definition of ȳ and

that the rate
F(x)
f(x)

is increasing and invertible to obtain the conditions of the equilibrium

in terms of the conversion rate:

F

(
2G−1(

µ
η
R

µ
η
R

+µ
η
L

) − 1
)

f

(
2G−1(

µ
η
R

µ
η
R

+µ
η
L

) − 1
) < F(ȳ)

f(ȳ)

1

g

(
G−1(

µ
η
R

µ
η
R

+µ
η
L

)

) < F
(

2 − 2G−1(
µ
η
R

µ
η
R

+µ
η
L

)

)
f

(
2 − 2G−1(

µ
η
R

µ
η
R

+µ
η
L

)

)

⇐⇒

F(2x̄∗ − 1)
f(2x̄∗ − 1)

<
F(ȳ)

f(ȳ)

1

g(x̄∗)
<
F(2 − 2x̄∗)
f(2 − 2x̄∗)

Note that for the symmetric cost case, µL = µR, the above simplifies to
F(0)
f(0)

g( 1
2
) <

η
2
<

F(1)
f(1)

g( 1
2
)

Lemma A.4. (0 < x∗L ≤
1
2
,x∗R = 1) Let x̄

∗ = G−1(
µ
η
R

µ
η
L

+µ
η
R

) and ȳ be implicitly defined

by
f(ȳ)
F(ȳ)

=
g(x̄∗)

2η

(µ
η
L

+µ
η
R

)2

µ
η
L
µ
η
R

. Let x∗L be implicitly defined by 2G(
1+x∗

L

2
) =

F(1−x∗
L

)

f(1−x∗
L

)
g(

1+x∗
L

2
) +

2
µ
η
R

µ
η
R

+µ
η
L
− 2η µ

η
R
µ
η
L

(µ
η
R

+µ
η
L

)2
.

For
µ
η
R

µ
η
R

+µ
η
L
< G( 3

4
), there is a unique equilibrium (x∗L, 1) if and only if

F(2 − 2x̄∗)
f(2 − 2x̄∗)

g(x̄∗) < 2η
µ
η
Lµ

η
R

(µ
η
L + µ

η
R)

2
<
F(1)

f(1)
g(0.5) +

µ
η
R −µ

η
L

µ
η
R + µ

η
L

For
µ
η
R

µ
η
R

+µ
η
L
≥ G( 3

4
), there is a unique equilibrium (x∗L, 1) if and only if

F( 1
2
)

f( 1
2
)
g(

3

4
) − 2G(

3

4
) +

2µ
η
R

µ
η
R + µ

η
L

< 2η
µ
η
Lµ

η
R

(µ
η
L + µ

η
R)

2

Proof. Because of Lemma (A.2) and Remark (1), solving the Lagrangian in the case

where νL = νR = λL = 0 and λR ≥ 0 is sufficient to find an equilibrium where x∗L ≤
1
2

and x∗R = 1. Let x̄ =
1+xL

2
and y = 1 − xL, then taking into account the conditions on

26



the Lagrange-multipliers:

∂ΠL(xL,xR = 1)

∂xL
+
∂ΠR(xL,xR = 1)

∂xR
= λR ≥0

µ
η
R

µ
η
R + µ

η
L

≥G(
xL + 1

2
)

2G−1(
µ
η
R

µ
η
R + µ

η
L

) ≥xL + 1 (15)

Lemma A.1 and Equation (15) imply that xL ≤ min{2G−1(
µ
η
R

µ
η
R

+µ
η
L

) − 1, 1
2
}. Taking into

account that 2G−1(
µ
η
R

µ
η
R

+µ
η
L

) − 1 < 1
2
⇐⇒ µ

η
R

µ
η
R

+µ
η
L
< G( 3

4
), we solve the following:

• If µ
η
R

µ
η
R

+µ
η
L
< G( 3

4
), then

∂ΠL(xL,1)
∂xL

|xL=2x̄∗−1 ≤ 0 must hold in equilibrium. Plugging in

xL = 2x̄
∗ − 1 and xR = 1 and using G(x̄∗) =

µ
η
R

µ
η
R

+µ
η
L

in Equation (7) we obtain

F(2 − 2x̄∗)
f(2 − 2x̄∗)

g(x̄∗) ≤ 2η
µ
η
Lµ

η
R

(µ
η
L + µ

η
R)

2
(16)

• If µ
η
R

µ
η
R

+µ
η
L
≥ G( 3

4
), then

∂ΠL(xL,1)
∂xL

|xL=0.5 ≤ 0 must hold in equilibrium. Using xL = 0.5
in Equation (7) we obtain

F(0.5)

f(0.5)
−

2

g( 3
4
)
G(

3

4
) +

2

g( 3
4
)

µ
η
R

µ
η
R + µ

η
L

−
2

g( 3
4
)
η

µ
η
Rµ

η
L

(µ
η
R + µ

η
L)

2
≤ 0

⇐⇒
F(0.5)

f(0.5)
g(

3

4
) − 2G(

3

4
) +

2µ
η
R

µ
η
R + µ

η
L

≤ 2η
µ
η
Lµ

η
R

(µ
η
L + µ

η
R)

2
(17)

Finally, the fully divergent equilibrium is excluded when we assure xL > 0, i.e.,

when 2η
µ
η
L
µ
η
R

(µ
η
L

+µ
η
R

)2
<

F(1)
f(1)

g(0.5) +
µ
η
R
−µη

L

µ
η
R

+µ
η
L

(see Proposition 4 below). Notice that when
µ
η
R

µ
η
R

+µ
η
L
≥ G( 3

4
), the latter inequality always holds.14

Proof of Proposition 4

Proof. Because of Lemma (A.2) and Remark (1), solving the Lagrangian in the case

where νi = 0 and λi ≥ 0 for all i is sufficient to find an equilibrium with xL = 0 and

14Recall that η ≤ η̄ implies that 2η µ
η
L
µ
η
R

(µ
η
L
+µ

η
R
)2

< 2
µ
η
L
+µ

η
R

µ
η
R

µ
η
L
µ
η
R

(µ
η
L
+µ

η
R
)2

= 2
µ
η
L

µ
η
L
+µ

η
R

. Also 2
µ
η
L

µ
η
L
+µ

η
R
≤

µ
η
R
−µη

L

µ
η
R
+µ

η
L
⇐⇒ µηR ≥ 3µ

η
L. Then, single-peakness and symmetry of g(x) imply G(

3
4
) > 3

4
. Finally

µ
η
R ≥ 3µ

η
L follows directly from

µ
η
R

µ
η
R
+µ

η
L
≥ G( 3

4
)

27



xR = 1. Hence, from Equation 7 and 8,

λL ≥ 0 iff
F(1)

f(1)
g(0.5) +

µ
η
−i −µ

η
i

µ
η
L + µ

η
R

≤ 2η
µ
η
Lµ

η
R

(µ
η
L + µ

η
R)

2
.

As µR ≥ µL, the condition for party L is the sufficient one. Last,
F(1)
f(1)

g(0.5) > 0, so a

necessary condition for a fully divergent equilibrium is

2ηµ
η
Lµ

η
R ≥ µ

2η
R −µ

2η
L = (µ

η
R −µ

η
L)(µ

η
R + µ

η
L) ≥ (µ

η
R −µ

η
L)ηµ

η
R,

where the last inequality follows from η ≤ η̄. And the expression above simplifies to
3µ

η
L ≥ µ

η
R.

5.3 Comparative statics

Proof of Lemma (2).

Proof. By Proposition (1), the interior equilibrium arises if η ∈ [2g( 1
2
)
F(0)
f(0)

, 2g( 1
2
)
F(1)
f(1)

].

Using the implicit function theorem we can write: ∂ȳ
∂η

= 1
2g(0.5)[F(ȳ)f(ȳ) ]

′ . Then,

∂µe∗i (x
∗
L,x

∗
R)

∂η
= −f(ȳ)

∂ȳ

∂η

η

4
+

1 −F(ȳ)
4

= −
f(ȳ)

2g(0.5)
[
F(ȳ)
f(ȳ)

]′ η4 + 1 −F(ȳ)4 .
Hence,

∂µe∗i (x
∗
L,x

∗
R)

∂η
≥ 0 ⇐⇒ 1 −F(ȳ) ≥

f(ȳ)

2g(0.5)
[
F(ȳ)
f(ȳ)

]′η ⇐⇒ η ≤ 1 −F(ȳ)f(ȳ) 2g(0.5)
[
F(ȳ)

f(ȳ)

]′

where
[
F(ȳ)
f(ȳ)

]′
is a positive number by log-concavity of F(y).

Proof of Lemma (3)

Proof. By Proposition (1), the interior equilibrium arises if η ∈ [2g( 1
2
)
F(0)
f(0)

, 2g( 1
2
)
F(1)
f(1)

].

Then,
∂µe∗i (x

∗
L,x

∗
R)

∂ρ
= −

∂F (y∗)

∂ρ

η

4
−
∂F (y∗)

∂y∗
∂y∗

∂ρ

η

4

and hence
∂µe∗i (x

∗
L
,x∗
R

)

∂ρ
≥ 0 if and only if −∂F(ȳ)

∂ρ
≥ f(ȳ)∂ȳ

∂ρ
.

28



5.4 Voters’ behavior

In this section we formalize and derive two main assumptions of the model. Namely, a)

the endogenous division of voters across ideological and impressionable (via semiorder lex-

icographic preferences or a model of salience), and b) the impressionable voters’ behavior

that results in a Tullock contest during the campaign stage.

In their “strict” formulation, lexicographic preferences require a tie in the first at-

tribute to compare alternatives over a second attribute. In their “weak” formulation (or

semiorder, (Tversky, 1969)), small differences between alternatives in the first attribute

also lead to indifference in that attribute. In other words, small differences on the first

attribute are disregarded (Fishburn, 1974). This is the exact intuition we presented in

our model where as polarization decreases and platforms look more alike, individuals

turn their attention to campaigns instead of platforms. In particular, semiorder pref-

erences with binary choices (i.e., two platforms in our setup) are unrestrictive in the

sense of Manzini and Mariotti (2012), and the use of semiorder lexicographic preferences

microfounds the aggregate behavior presented in the main text.

Assume a population of measure one, in which voters have semiorder lexicographic

preferences and are heterogeneous in two dimensions. First, they draw an ideal policy

x ∈ [0, 1] from G(x). Second, voters also draw a level of sensitivity, φ, towards differences
in the ideology space, from Fφ(y) = Pr(φ < y). Hence, φ is the minimal distance between

the two platforms that a voter considers to be “relevant” or “distinguishable”.15 When

the distance between two platforms is less than φ the voter treats them as identical and

moves to the second attribute, electoral advertising, ei. Although x is ex-post irrelevant

for impressionable voters, all individuals are identified by the pair (x,φ).

The above features can be represented in an analytical manner by adapting the

semiorder lexicographic structure proposed by Luce (1978). Considering voter (x,φ),

we can write the voter’s evaluation of party i as:

ϑx,φ(i) = − | x−xi | Υ(φ ≤ xR −xL) + ti(eR,eL,θi)[1 − Υ(φ ≤ xR −xL)] (18)

where Υ(φ ≤ xR−xL) is an indicator function taking value 1 when a voter is ideological,
i.e., φ ≤ xR −xL = y, and 0 when a voter is impressionable, i.e., φ > xR −xL = y, as in
the main text. Without loss of generality − | x−xi | is the utility derived from voting
for party i ∈{L,R} according to ideology; and ti(eR,eL,θi) is the impressionable voter’s
utility from voting for party i ∈{L,R} according to advertisements.

Let θi in ti(eR,eL,θ
i) be a random variable (from the candidates’ points of view) that

15In terms of the experimental literature in human perception (or psychophysics), φ can be interpreted
as the just-noticeable difference.

29



captures how much of a party’s advertisement “leaks” to voters in the following way

ti(eR,eL,θ
i) = log(e

η
i ) + θ

i. Similarly to multinomial applications in industrial organiza-

tion (Nevo, 2000) and political economy (Casas, Fawaz, and Trindade, 2016), assume θi

to be drawn i.i.d. from an type I extreme-value distribution. As in McFadden (1974),

the probability Pr(tL > tR) – which in our case is the probability that an impressionable

voter votes for L – becomes the contest success function Pr(tL > tR) =
e
η
L

e
η
L

+e
η
R

. With

a continuum of voters, we interpret this probability as the share of impressionable vot-

ers that vote for L. On the same lines, (Jia, 2008; Jia, Skaperdas, and Vaidya, 2013)

show that for ti(eR,eL,θ
i) = eiθ

i, if θL and θR follow independent inverse exponential

distributions with parameter η > 0, the probability also becomes Pr(tL > tR) =
e
η
L

e
η
L

+e
η
R

.16

For simplicity, let Fφ(y) be written as F(y), which can be used to denote the propor-

tion of ideological voters given a level of polarization y. Platform preferences and ideol-

ogy sensitivity are assumed to be independent. Consequently, for a given pair of policy

platforms xL and xR, the votes of ideological and impressionable voters can be indepen-

dently aggregated (integrating over φ). By integrating over θL and θR under the above

distributional assumptions, the impressionable vote share of party i is SiImp =
e
η
i

e
η
L

+e
η
R

.

For ideologogical voters, by integraing over x one immediately obtains that for party L,

SLIdl = G(x̄) if xL 6= xR and S
L
Idl =

1
2

if xL = xR. By taking into account the above,

parties’ vote shares can be immediately written as in Equation (1).

Salience and attention

Our model can also be interpreted as an extreme case of (Bordalo et al. 2012, 2013a,b,

2015), in which “salient thinkers” give more weight to attributes that exhibit greater

heterogeneity in the available choice set.

In the papers on salience and attention, the salient attribute is the one in which the

differences are more pronounced. The less salient attribute receives less weight. For

instance, in Bordalo et al. (2015) they look at price and quality: if quality is the salient

attribute, the utility from consuming good k is qk −ωpk with ω exogenously determined
and in [0, 1]. Instead of comparing attributes, we compare the platforms differentiation

with a baseline level of polarization, φ, which is exogenously drawn from Fφ(y). Thus, for

φ smaller than the equilibrium polarization, voters take platforms as the salient attribute.

In particular, for φ ≤ y, ω = 0. And for φ ≥ y, 1 −ω = 0. Thus, one can consider the
rank-based weighting salience function proposed by Bordalo et al. 2012, 2013b, 2015,

where:

ϑx,φ(i) = −(| x−xi |)ω + ti(eL,eR,θi)(1 −ω)

The heterogeneity of “salient thinkers” is given by Fφ(y), and everything else is exactly

16The assumption that party shocks are identical to all individuals is made without loss of generality,
as long as they are independent of ideology.

30



as in the previous section.

31



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36


	BTC.pdf
	Introduction
	Related Literature

	Model
	Results
	Symmetric parties
	Asymmetric parties

	Conclusion
	Appendix - Proofs
	Advertising
	Polarization and Spending
	Comparative statics 
	Voters' behavior