#  Echo Chambers:
Voter-to-Voter Communication and Political Competition222This paper is based
on a chapter of my PhD dissertation at the University of Warwick. I am
grateful to Mirko Draca, Walter Ferrarese, Carlo Perroni, Herakles
Polemarchakis, Simone Tonin. I am also grateful to seminar participants at the
Erasmus Political Economy Workshop and at the Workshop for Woman in Political
Economy.

Monica A. Giovanniello${}^{\hskip 1.4457pt}$444University of the Balearic
Islands, Departament d’Economia Aplicada, Palma de Mallorca, 07122, Spain;
e-maiul<EMAIL_ADDRESS>tel: +34 971 173437

> Abstract
>

>> I study how strategic communication among voters shapes both political
outcomes and parties’ advertising strategies in a model of informative
campaign advertising. Two main results are derived. First, echo chambers arise
endogenously. Surprisingly, small ideological distance between voters is not
sufficient to guarantee that a chamber is created, biases’ direction plays a
crucial role. Second, when voters’ network entails a significant waste of
information, parties target their advertising only to the voters leaning
toward their opponent’s rather than to their own supporters.

>>

>>
>
>>

>> Keywords: Echo Chambers; Social Networks; Information Aggregation;
Political Advertising.

>>

>> JEL CLASSIFICATION: D72, D83, M37

##  1 Introduction

One of the main characteristics of the new information society is how it
horizontalizes information: the users of information are not just passive
recipients, as was the case for traditional media (radio, television,
newspaper), but are now able to spread information themselves, and in this way
shape public opinion. By mediating the exposure to information, these social
media technologies have opened up an unexpected role for voters – they now not
only cast their votes but also have a greatly enhanced role in spreading
information that can subsequently shape public opinion.111Donald Trump’s main
channel of communication was social media. Brad Parscale, digital director of
Trump’s campaign, stated that Facebook helped generate $250 million in on-line
fund-raising: “Facebook and Twitter were the reason we won this thing”.
Another example is the growth of the Five Star movement in Italy, which
emerged from a blog by the comedian Beppe Grillo. The Five Star movement
organizes face-to-face interactions via websites such as MeetUp.com and
solicits supporter views on its central blog. Similarly, the UK’s English
Defense League (EDL) emerged directly from a Facebook group centred on local
race-related political developments in Luton. See Schönwälder and Bartlett
(2016) for a summary of these and other similar political movements.

Earlier studies focusing on the consumption of “top-down” news (i.e. print
newspapers, TV, on-line news websites) found evidence of information
segregation in on-line news consumption (Gentzkow and Shapiro,
2011).222Previous works focused on the media bias in traditional news formats,
such as high circulation newspapers (Mullainathan and Shleifer, 2005; and
Gentzkow and Shapiro, 2006). A notable empirical exploration of the
determinants of newspaper media bias is Gentzkow and Shapiro (2010).

Nowadays this paradigm has changed: voters are not simply exposed to
information; instead they interact and exchange information with deliberately
selected sources and receivers. Bakshy et al. (2015) find that individuals’
sharing choices plays a stronger role than the Facebook content recommendation
algorithm in limiting exposure to cross-cutting content, voters network
reinforces patterns of segregation. Thus, any political information that
originates from traditional outlets is mediated by social media, which creates
scope for strategic information sharing, hiding, or misrepresenting.333 _”We
just put information into the bloodstream to the internet and then watch it
grow, give it a little push every now and again over time to watch it take
shape. And so this stuff infiltrates the on-line community and expands …,”_
Mark Turnbull, managing director of Cambridge Analytica and SCL Elections (The
Guardian, E. Graham-Harrison and C. Cadwalladr, 21/03/2018).

This paper provides a first theoretical investigation of how social network
interactions and network structure influence the diffusion of politically-
relevant information, and thus political outcomes.444Several empirical studies
show that interactions within intimate social networks increases the
likelihood of political participation (McClurg, 2003; Plutzer and Zipp, 1996;
Rainie and Smith, 2012), and study the effect of social media on voters’
exposure to political information (Bakshy et al., 2015; Boutyline and Willer,
2017; DiFonzo, 2017; Garrett, 2009). I develop a two-level game that embeds:
_(i)_ a model of political competition where parties compete in campaign
advertising, and _(ii)_ a model of personal influence in a peer-to-peer
information network where voters can strategically communicate with each other
in order to affect the policy outcome. I focus on political competition
between two policy-motivated parties. Each party selects, given its candidate
type (policy position), the level of informative advertising in order to
persuade voters. Parties’ advertisement is costly, and it is assumed to be
truthfully informative.555Parties invest in campaign to create advertisement
able to capture the attention of voters and with which voter can emphasize.
While voters communication is ’cheap’ as they can tweet or share messages
whose content has been already created. Voters have preferences over policies,
but do not know ex-ante the ideological position of the parties’ candidates.
Voters can obtain information about parties’ candidate types either by direct
exposure to the parties’ advertisements or by strategically communicating with
other voters within their network. Thus, voters update their beliefs and cast
their vote sincerely. The party that obtains a simple majority of votes wins
the election and enacts the policy corresponding to its candidate ideology.

Two results are put forward. First, echo chambers arise endogenously. When
voters can costlessly and strategically communicate with their friends they
chose to share valuable information only with like-minded peers and do not
share any valuable information otherwise. As result, information travels only
in ideologically-aligned groups, this is what I mean by echo chambers. I show
that echo chambers arise if and only if voters are leaning toward the same
party (i.e. voters are biased in the same direction), and their ideological
distance is not too big. In other words, I show that small ideological
distance between voters is not sufficient to guarantee that a chamber is
created, biases’ direction plays a crucial role. Thus, voters do not share
valuable information neither with ideologically close voters who are biased
toward the opposite party, nor with voters biased toward their favoured party
if the ideological distance is not small enough.

Second, I analyse the effect of echo chambers, via social network’s structure,
on parties advertising strategies. Echo chambers, i.e. an enclosed system in
which information is credibly shared, can act as a hurdle to the spread
truthful information among voters and restrict access to sources of reliable
information. If voters network is characterized by few links and/or the
probability that like-minded voters interact is low the network acts as a
filter, parties advertisement is not echoed among voters and valuable
information is lost in the communication stage. When this is the case parties
maximize their probability of winning by targeting those voters who, if
uninformed, would vote for their opponent rather than targeting their own
supporters (whom if uninformed would cast their vote for them). If, instead,
the voters’ network is characterized by many links and/or by an high
probability that like-minded voters interact the echo chambers act as
information diffusion device. Parties maximize their probability of winning by
randomly advertising their candidates, in this way they reach voters biased
toward both parties, who will echo the advertisement within their network.

Two recent contributions examine the implications of information segregation
in networks. Bloch et al. (2018) study under which conditions rational agents
have incentive to spread rumours in social networks. They consider a model in
which bias and unbiased agents can spread possibly false information to affect
the voting outcome. They find that, for high priors beliefs about the desired
state of the world, the social network acts like a filter: agents block
messages received by their biased neighbours, in this way, the circulating
messages, even if incorrect, may convey useful information to take the right
collective decision. Jann and Schottmüller (2016), analyse a cheap talk game
with multiple senders and receivers, and find that players self-select
themselves into like-minded and homogeneous groups. Interestingly they find
that, when players have different preferences and information, players self-
segregation is Pareto improving.

However, differently from the current paper, neither study the mechanism
behind the rise of an echo chamber, as I do here. I consider a situation in
which voters with similar and dissimilar preferences freely communicate with
each other and given the underlying social network structure, information
travels from many-to-many without any stigma. I show that echo chambers are a
product of voters communication strategies rather than environments
predetermined by exogenous factors. To the best of my knowledge, this paper is
the first theoretical contribution that formalizes how network structure
affect endogenous echo chambers formation, and their effect on both the
information diffusion and parties advertisement strategies.

This paper is also related to the theoretical literature on communication
flows (Crawford and Sobel, 1982). A well-established result in this literature
is that truthful information transmission is possible whenever the sender and
the receiver have similar preferences, i.e. the bias gap is not too large
(Austen-Smith, 1992). In contrast, I show that, although it is necessary that
information is transmitted among voters with similar preferences, this is not
sufficient to create an echo chamber, voters must also be biased toward the
same direction (same party).

Another related contribution is Galeotti and Mattozzi (2011) who investigate
the effects of social learning on political outcomes in a model of truthful
interpersonal communication and informative campaign advertising. They find
that non-strategic communication networks with a richer structure lead to
political polarization. In contrast with their model, I focus on the effect of
strategic communication transmission among voters on the political outcome. I
find that, when voters can falsify or hide information to shape public opinion
and affect the outcome of the election, information diffusion and aggregation
depend on the structure of the communication network which, in turn, affect
the size of echo chambers that can be exploited by parties. This different
information environment highlights the importance of the voters’ network
structure and its effect on the political outcome.

The remainder of the paper is organized as follows. Section 2 describes the
model. Section 3 develops the benchmark model in which the only source of
voters information is the parties’ advertisement. Section 4 characterizes the
equilibrium of the communication game. Section 5 provides the characterization
of the political equilibrium. In Section 6 the main assumptions of the paper
are discussed. The last section concludes. All proofs are in Appendix.

##  2 The Model

Parties. The policy space is one-dimensional. There are two policy-motivated
parties, $J\in\\{L,R\\}$, that compete in an election. Each party is
represented by a candidate who can be either a moderate ($m$) or an extremist
($e$). A candidate type, $t_{J}\in T=\\{e,m\\}$, where $0<m<1/2$ and $e=m/2$,
is independently and exogenously assigned to each party.666The assumption
$e=m/2$ is made only to simplify computation exposition, as letting $e<m$ does
not change the quality of my results.,777I relax this assumption later in
Section 6.1 allowing the parties to strategically select also the candidate
type. With probability $\sigma_{J}$ the candidate of party $J$ is a moderate
and with probability $1-\sigma_{J}$ he is an extremist. The probability
distributions $\sigma_{J}:T\rightarrow[0,1]$, for $J\in\\{L,R\\}$, are common
knowledge. A state of the world is, then, defined by the profile of candidate
types for the two parties, $\theta=(t_{L},t_{R})\in T\times T$.

Abusing of notation, I denote by $t_{L}$ the ideology of the candidate of
party $L$ and by $1-t_{R}$ the ideology of party $R$’s candidate. Parties have
distance preferences over the policy space and their ideological bliss point
coincide with the ideological bliss point of their extremist candidate, _i.e._
party $L$ bliss point is $i_{L}=e$ and $R$’s bliss point is $i_{R}=1-e$.

Parties privately observe their candidate type and simultaneously choose a
level of informative campaign advertising, $x_{J}(t_{J})\in[0,1]$.888The level
of informative advertisement, $x_{J}(t_{J})$, can be seen as the ability of
the party’s message to capture the attention of voters. The advertisements
have the following two characteristics: parties can only advertise their own
candidate, which implies there is no negative advertising; parties cannot lie
about the ideological position of their candidates. The advertising is
truthful, but costly. If party $J$ advertises its candidate with intensity
$x_{J}(t_{J})$, each voter obtains perfect information about party $J$’s
candidate type with probability $x_{J}(t_{J})$. The party that wins the
electoral competition enacts the policy preferred by its candidate.

Then, party $L$ chooses to advertise its candidate $t_{L}$ with intensity
$x_{L}(t_{L}):T\rightarrow[0,1]$ to maximize its expected payoff:

$\displaystyle U_{L}(x\mid t_{L})$ $\displaystyle=$
$\displaystyle\sigma_{R}(m)[\pi_{L}(x\mid\theta)(i_{L}-t_{L})+(1-\pi_{L}(x))(i_{L}-(1-m))]$
$\displaystyle+$
$\displaystyle\sigma_{R}(e)[\pi_{L}(x\mid\theta)(i_{L}-t_{L})+(1-\pi_{L}(x))(i_{L}-(1-e))]-cx_{L}(t_{L})$

where $\sigma_{R}(t_{R})$ is the probability that party $R$ has a candidate of
type $t_{R}$, $\pi_{L}(x\mid\theta)$ is the probability that party $L$ wins
the election, given the state of the world $\theta$, with
$x=[x_{L}(t_{L}),x_{R}(t_{R})]$ being the advertising levels chosen by the
parties, and $c>0$ is the advertising marginal cost. Substituting $i_{L}$ into
(2), after some manipulation the expected payoff of party $L$ can be expressed
in a more parsimonious way:

$U_{L}(x\mid
t_{L})=\sum_{t_{R}\in\\{e,m\\}}\sigma_{R}(t_{R})[\pi_{L}(x\mid\theta)(1-t_{R}-t_{L})+(e-(1-t_{R}))]-cx_{L}(t_{L}),$
(1)

and party $R$’s expected payoff is

$U_{R}(x\mid
t_{R})=\sum_{t_{L}\in\\{e,m\\}}\sigma_{L}(t_{L})[\pi_{R}(x\mid\theta)(1-t_{R}-t_{L})+(t_{L}-(1-e))]-cx_{R}(t_{R}),$
(2)

where $1-e$ is the ideological bliss point of party $R$. Notice that the
utilities of the two parties are increasing in the ideological distance
between their candidates, as $-\pi_{J}(t_{J}\mid\theta)(t_{L}-(1-t_{R}))>0$,
and decreasing in the ideological distance between the opponent candidate and
the party’s bliss point, as $e-(1-t_{R})<0$ (resp. $t_{L}-(1-e)<0$) for party
$L$ ($R$). This implies that a party always prefer to lose against a moderate
candidate and win with an extremist one.

Voters. There is a continuum of voters of unit measure which have distance
preferences over candidates’ positions,

$u_{i}(t)=\begin{cases}\quad-\mid i-t_{L}\mid$ \quad\quad\quad if $L$ wins the
election, $\\\ -\mid i-(1-t_{R})\mid$ \quad\,\ if $R$ wins the election,
$\end{cases}$ (3)

where $i\in(0,1)$ denotes a voter’s ideological bliss point. There are two
types of voters, partisans and independent. To distinguish among different
types of voter I introduce the following definition:

###### Definition 1

Denote by $j=\\{R,L\\}$ the favourite party of voter $i$. The preferences of a
partisan voter over the candidates type of the two parties are
$e_{j}\succ_{i}m_{j}\succ_{i}m_{j^{-}}\succ_{i}e_{j^{-}}$ with $j^{-}\neq j$,
i.e. she always votes for her favourite party regardless the candidates type
of the two parties. The preferences of an independent voter over the candidate
types of the two parties are
$m_{j}\succ_{i}m_{j^{-}}\succ_{i}e_{j}\succ_{i}e_{j^{-}}$, i.e. she always
prefer to vote for the moderate candidate of the party she likes less over an
extremist candidate of her favourite party.

Using this voters’ type definition and their preferences it is easy to show
that ex-ante (before the advertising takes place) the ideological bliss points
of the partisans’ group are distributed on the intervals $(0,1/2-m/4)$ and
$(1/2+m/4,1)$ for the left and right partisans respectively. While
independents’ voters bliss point are distributed on the interval
$[1/2-m/4,1/2+m/4]$. To avoid trivial results I assume the two partisans
groups have the same size and that the identity of the median independent
voter is ex ante uncertain. The independent voters’ ideological bliss points
are uniformly distributed in the interval $[\mu-\tau,\mu+\tau]$, where $\mu$
is drawn from a uniform distribution with support $[1/2-m/4,1/2+m/4]$,
$\tau>0$, and $m<1/4-\tau/2$.

Figure 1: Parties, candidates, and voters ideologies

Information Diffusion. Voters can obtain information about the candidates type
either by direct exposure to the parties’ advertisement or by strategically
communicating with other voters.

Parties, after observing their candidate type, select the level of informative
advertising, $x_{J}(t_{J})$. Then, each each voter is directly exposed to the
information about $t_{J}$ with probability, $x_{J}(t_{J})$.

Each voter is exogenously matched with $z+k>0$ voters, where $k\geq 1$ is the
number of her sendersand $z>0$ is the number of her receivers. I assume that
there is no preference uncertainty within the personal network of each voter,
_i.e._ each voter can perfectly observe the ideological bliss point of all her
$z+k$ peers, but voters can not observe their peers’ networks nor their voting
decisions.999It is, in general, possible to distinguish two types of online
networks: those based on common interests, e.g. Twitter, and those based on
off-line relations, e.g. Facebook (Pew Research Center, Survey March 7 – April
4 2016, “Social Media Update”). In both type of networks voters have a good
approximation of the ideological position of their peers. Although, as
expected in the second type of networks the level of homophily within users’
networks is reduced, this does not prevent information diffusion. Many
empirical studies find networks based on intimate social relations are likely
to stimulate interest in politics (Mac Carron et al., 2016; Goncalves et al.,
2011). Thus, once the advertisement has taken place, all voters simultaneously
receive $k$ and send $z$ pairs of private messages about the candidate type of
each party $(M_{L}(t_{L}),M_{R}(t_{R}))$, where
$M_{J}(t_{J})\in\\{e,m,\emptyset\\}$ is a message about the candidate type
$t_{J}$ of party $J\in\\{L,R\\}$.101010By assuming simultaneous communication
among voters I take into account the life-span of new information on social
media. As pointed out by Hodas and Lerman (2012), on social media new pieces
of information ”compete in time”, as the recommendation algorithms moves them
to less visible queues, ”and space”, as users divide their limited attention
among their peers. They show that in order to maximize the public users-voters
tend to retweet information immediately after it has been tweeted for the
first time, i.e. when it is most visible. In fact, given the limited
attentions of the their peers users-voters visibility is crucial and so it is
the timing of publication/retweet. The sender in each of the $k$ matched pair
aims to maximize the probability that his favourite candidate wins the
election. In other words, when sending the message pair to each of her
receiver, the sender behaves as if she believes the receiver’s vote would be
the pivotal one. Then, the sender’s intra-stage utility from each possible
pair of messages is

$\displaystyle u_{s}(M_{L}(t_{L}),M_{R}(t_{R})\mid I^{s},x,r)$
$\displaystyle=$ $\displaystyle\Pr\left(v_{r}=1\mid
I^{r},x,r\right)\mathbb{E}[t_{L}-s\mid I^{s},x]$ (4) $\displaystyle\ \ \ +$
$\displaystyle\Pr\left(v_{r}=0\mid I^{r},x,r\right)\mathbb{E}[s-(1-t_{R})\mid
I^{s},x],$

where $s$ denotes the bliss point of the sender in each matched pair, $I^{s}$
is the information set of the sender and $\Pr\left(v_{r}=1\mid
I^{r},x,r\right)$ is the probability that the receiver with ideological bliss
point $r$ votes for the party $L$, given her ex-post information set $I^{r}$.

Figure 2: Left: Voter $i$ personal network. Right: Voter $i$ information
sources

After the communication stage, the information set of voter $i$ becomes
$I^{i}=$ $(t_{L},$ $\overline{M}_{L}(t_{L}),$
$t_{R},\overline{M}_{R}(t_{R}))$, where $t_{J}$ denotes the information about
the candidate type of party $J$ acquired through direct exposure to the
parties’ advertisement, and $\overline{M}_{J}(t_{J})$ denotes the vector of
messages received from the $k$ partners about the candidate type of party $J$.
Let $\rho(t_{L},t_{R}\mid I^{i},x,k)$ denote the posterior belief of voter $i$
about the state of the world, $\theta=(t_{L},t_{R})$, where voters’ prior
beliefs about the true state of the world are consistent with $\sigma$. The
posterior belief, whenever possible, is derived using Bayes’ rule:

$\displaystyle\rho(t_{L},t_{R}\mid I^{i},x,k)$ $\displaystyle=$
$\displaystyle\frac{\Pr(t_{L},t_{R})\Pr(I^{i}\mid
t_{L},t_{R},x,k)}{\sum_{t^{\prime}_{L},t^{\prime}_{R}\in
T}\Pr(t^{\prime}_{L},t^{\prime}_{R})\Pr(I^{i}\mid
t^{\prime}_{L},t^{\prime}_{R},x,k)}.$ (5)

The indifferent independent voter. After updating their beliefs, voters cast
their vote sincerely. Voter $i$’s utility function is

$u_{i}(v_{i}\mid\theta,I,k)=\sum_{t_{L}\in T}\sum_{t_{L}\in
T}\Big{(}v_{i}\rho(t_{L},t_{R}\mid\theta,I,k)(t_{L}-i)+(1-v_{i})\rho(t_{L},t_{R}\mid\theta,I,k)(i-(1-t_{R}))\Big{)}$
(6)

where $v_{i}\in\\{0,1\\}$ is $i$’s voting decision and $v_{i}=1$ (resp.
$v_{i}=0$) denotes the decision of voting for party $L$ ($R$). Note that
voter’s utility does not depend on her message strategy.111111 Studies show
that citizens share political news mostly motivated by the gratification of
information seeking or for socializing rather than to political mobilize other
voters (Lee and Ma (2012), Kaye and Johnson (2004)). The communication intra-
stage utility can be taken into account by adding to the voters utility
function a gratification payoff $\Phi w_{i}$, where $w_{i}\in\\{0,1\\}$ and
$w_{i}=1$ denotes the voter’s decision of sending $z$ private message pairs
that maximize the probability of obtaining his favoured outcome, and $\Phi>0$
is the resulting gratification.  Through strategic communication the partisan
voters aim to maximize the probability that their party’s candidate wins the
election, while the independent voters aim to maximize the probability that a
moderate candidate (preferably of their favoured party) wins the election.

Thus, given an information set $I$, the voter votes for party $L$ if and only
if

$u_{i}(v_{i}=1\mid\theta,I,k)\geq u_{i}(v_{i}=0\mid\theta,I,k).$ (7)

It follows that a voter with ideology $i$ and information set $I$ votes for
party $L$ if and only if $i<i^{*}(I)$, where $i^{*}(I)$ solves the equation
(7) with equality, _i.e._ it is the bliss point of the indifferent independent
voter with information set $I$. The identity of the indifferent voter for any
information set $I$, is given by121212See Appendix for derivation.

$i^{*}(I)=\frac{1}{2}+\frac{1}{2}\displaystyle\sum_{t_{L},t_{R}\in
T}\left(\rho(t_{L},t_{R}\mid\theta,I,k)\,t_{L}-\rho(t_{L},t_{R}\mid\theta,I,k)\,t_{R}\right).$
(8)

Party $L$’s vote share is

$\mu^{*}(x\mid\theta,x,k)=\sum_{I}i^{*}(I)\Pr(I\mid\theta,x,k),$

then party $L$’s candidate wins the election with the following probability:

$\pi_{L}(x\mid\theta,k)=\begin{cases}0\hskip 113.81102pt\quad$ if
$\quad\mu^{*}(x\mid\theta,k)<\frac{1}{2}-m\\\
\dfrac{\mu^{*}(x\mid\theta,k)+m-\frac{1}{2}}{2m}\qquad\,$ if
$\quad\mu^{*}(x\mid\theta,k)\in\Big{(}\frac{1}{2}-m,\frac{1}{2}+m\Big{)}\\\
1\hskip 113.81102pt\quad$ if
$\quad\mu^{*}(x\mid\theta,k)>\frac{1}{2}+m.\end{cases}$ (9)

Timing. I study a symmetric five-stage Bayesian game. In the first stage
parties, given their candidate type, select a level of informative campaign
advertising, $x_{J}(t_{J})$. In the second stage each voter observe the
candidate type of each party $J$ with probability $x_{J}(t_{J})$, and she is
matched with $k+z>0$ other voters. In the third stage (communication stage)
voters may be exposed to political information throughout strategic
communication: each voter receives a pair of private messages,
$(M_{L}(t_{L}),M_{R}(t_{R}))$, from her $k$ peers about the candidate type of
the two parties. In the fourth stage, voters update their belief and sincerely
cast their vote. Finally, the candidate that obtains a simple majority of
votes wins the election and enacts the policy corresponding to his ideology.

Figure 3: Timing of the game

I analyse symmetric political equilibria, that is I assume the two parties
have the same probability of competing in the election with a moderate
candidate, $\sigma_{L}=\sigma_{R}=\sigma$.

###### Definition 2

A symmetric political equilibrium consists of: (i) symmetric parties’
advertising strategies $x^{*}=(x^{*}_{L}(t_{L}),x^{*}_{R}(t_{R}))$ ; (ii) a
pair of message strategy for each sender in each matched pair
$M_{J}(t_{J}):T\rightarrow M$, for $J\in\\{L,R\\}$; (iii) voters’ belief
functions $\rho(t_{L},t_{R}\mid I^{i},x,k)$, (iv) indifferent independent
voters type $i^{*}(\cdot)$ such that:

  * 1.

$(x^{*}_{L}(t_{L}),x^{*}_{R}(t_{R}))$ are mutual best responses given the
subsequent communication equilibrium and voting behaviour;

  * 2.

$\rho^{*}(\cdot)$ is consistent and sequentially rational with the probability
of the true state of the world $\sigma$, the optimal advertising strategies of
the parties $x^{*}$, and the senders’ optimal message strategies
$(M_{L}(t_{L}),M_{R}(t_{R}))$.

##  3 Benchmark Model

In this section I analyse parties advertisement strategies when voters can not
communicate with each other, that is $k=0$. Thus, voters’ information set is
reduced to $I_{0}=(t_{L},t_{R})$, which may contain only the information
gathered from parties advertisement.131313A seminal contribution on
informative advertisement is Grossman and Shapiro (1984). They analyse a model
in which consumers choose among products of which they have seen the
advertisement. As firms use advertising to create surplus they over-advertise,
this way reducing profits. The main difference between the Grossman and
Shapiro’s model and my model is that voters may vote also for candidate they
have not seen the advertisement. In other words, seeing and not seeing
advertisement carries information as voters form expectations about them at
the time of voting. This implies that Grossman and Shapiro’s firms and the
parties in this paper have different incentives. The firms over-advertise
(even hurting their own profits) as it is the only way to create surplus,
while the parties in this model obtains votes even when voters are uninformed.

Before proceeding with the analysis it is worth notice thata party that
advertises an extremist incurs two costs: a direct cost of advertisement
$c>0$, and an indirect cost due to disclosing harmful information. By
advertising an extremist the party lowers the probability of winning the
election as the independent voters, who are decisive for the outcome of the
election, always prefer to vote for a moderate candidate rather than
extremist.

###### Lemma 1

Parties never advertise an extremist candidate.

Proof: See Appendix.

Thus, voters’ information about the candidate of each party $J$ gathered from
advertisement can be summarized with $t_{J}=\\{m,\emptyset\\},\ \forall
J\in\\{L,R\\}$, and a voter is said to be perfectly informed about the
candidate type of party $J$ whenever he observes advertisement $t_{J}=m$. An
voter uninformed about the candidate type of party $J$, i.e. a voter who does
not receive the advertisement, knows that with probability
$\sigma(1-x_{J}(m))/1-\sigma x_{J}(m)$ the $J$’s candidate is a moderate and
with probability $1-\sigma(1-x_{J}(m))/1-\sigma x_{J}(m)$ he is an extremist.

Working backwards, I first analyze the voting decisions. Once the
advertisement has taken place the possible information sets hold by the voters
are $\\{(m,m);(m,\emptyset);$ $(\emptyset,m);(\emptyset,\emptyset)\\}$.

First, consider the case in which a voter holds an information set is
$I_{0}^{i}=(\emptyset,m)$. The voter casts her votes for party $L$ whenever
the equation (7) holds, i.e. whenever

$\displaystyle\rho(m,m\mid I_{0}^{i},x)(m-i)+\rho(e,m\mid
I_{0}^{i},x)\left(e-i\right)\geq i-(1-m),$

where $\rho(e,m\mid I_{0}^{i},x)=1-\rho(m,m\mid I_{0}^{i},x)$ with
$\rho(m,m\mid I_{0}^{i},x)=\frac{\sigma(1-x_{L}(m))}{1-\sigma x_{L}(m)}$, and
$e=m/2$. Which implies that a voter with information set
$I_{0}^{i}=(\emptyset,m)$ votes for party $L$ if and only if her bliss point
is

$i\leq\alpha^{0}_{l}\equiv\frac{1}{2}-\frac{m}{4}\left(1-\frac{\sigma(1-x_{L}(m))}{1-\sigma
x_{L}(m)}\right).$

Thus, a voter informed only about the candidate of party $R$ casts her votes
for party $L$ if and only if her bliss point is $i\in(0,\alpha^{0}_{l})$. By
the symmetry of the game it follows that a voter informed only about the
candidate of party $L$ ($I_{0}^{i}=(m,\emptyset)$) casts her votes for the
party $R$ if and only if her bliss point is $i\in(\alpha^{0}_{r},1)$, where
$\alpha^{0}_{r}\equiv\frac{1}{2}+\frac{m}{4}\rho(m,e\mid I_{0}^{i},x)$. By
following the same steps above it is easy to show that, whenever a voter hold
information set $I_{0}^{i}=(\emptyset,\emptyset)$ or $I_{0}^{i}=(m,m)$, she
casts her vote for party $L$ if and only if she is $L$-leaning, _i.e._ $i\leq
1/2$, and votes for party $R$ otherwise.

Using Definition 1 it is easy to distinguish also the ex-post groups-types of
voters, _i.e._ whether, after the advertising, the voter with bliss point $i$
is a partisan or an independent. Once the parties advertise their candidates,
if a voter’s ideological bliss point is such that either
$i\in(0,\alpha^{0}_{l})$ or $i\in(\alpha^{0}_{r},1)$ she is ex-post partisan,
as her preferences are
$e_{j}\succ_{i}m_{j}\succ_{i}m_{j^{-}}\succ_{i}e_{j^{-}}$.If, instead, ex-post
the voter’s bliss point is $i\in(\alpha^{0}_{l},\alpha^{0}_{r})$ she is ex-
post independent, as her preferences are
$m_{j}\succ_{i}m_{j^{-}}\succ_{i}e_{j}\succ_{i}e_{j^{-}}$. By symmetry, the
interval defined by the two cut-off points always contains the centre of the
ideological space (i.e. $\alpha^{0}_{l}<1/2<\alpha^{0}_{r}$), voters with
ideological bliss point on the same side of the center point are said to be
biased toward the same party.

###### Remark 1

There exist two ideological cut-off points, $\alpha^{0}_{l}$ and
$\alpha^{0}_{r}$ which together with the central point define the ex-post
group-type to which each voter belongs once the advertising has taken place:

  * •

if $i\in(0,\alpha^{0}_{l})$ (resp. $i\in(\alpha^{0}_{r},1)$) the voter is an
ex-post $L$-leaning ($R$-leaning) partisan;

  * •

if $i\in(\alpha^{0}_{l},1/2)$ (resp. $i\in(1/2,\alpha^{0}_{r})$) the voter is
an ex-post $L$-leaning ($R$-leaning) independent.

From Lemma 1 follows that parties may advertise their candidate only if they
run for the election with a moderate one. Given the symmetry of the game, it
is sufficient to focus only to the case in which party $L$ has a moderate
candidate.

In what follows I consider the case in which parties have access to two
advertising technologies: i) a random technology, which allow each voter to be
exposed at the advertisement with the same positive probability, ii) and group
targeting technology which direct the advertisement exclusively to the voters
who are leaning toward a given party (either only $L$-leaning or only
$R$-leaning voters).

The benchmark model captures the minimum cost under which parties are willing
to advertise their candidates when the two technologies are available, but
voters can not communicate among each other.

#### Random Advertising.

First I analyse the case in which party $L$ can only randomly advertise its
candidate.

Suppose that party $R$ has an extremist candidate. Whenever this is the case
what matters is the distribution of indifferent voters, which in turn depends
on the information set held. Given that party $L$ has a moderate candidate and
it randomly advertises him with intensity $x_{L}(m)$, the proportion of
uninformed voters is $1-x_{L}(m)$. While no voter receive any information from
party $R$ as $x_{R}(e)=0$. Then party $L$ votes share, given any possible
information set $I_{0}$, is

$\displaystyle\mu^{0}(x\mid\theta)$ $\displaystyle=$
$\displaystyle\sum_{I_{0}}i^{*}(I_{0})\Pr(I_{0}\mid\theta,x)=\frac{1}{2}(1-x_{L}(m))+\left(\frac{1}{2}+\frac{m}{4}\rho(m,e\mid
I,x)\right)x_{L}(m)$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{m}{4}(1-\rho(m,m\mid I,x))\,x_{L}(m),$

which implies that party $L$ wins the election with probability

$\pi^{0}(m,e\mid x)=\frac{1}{2}+\frac{1}{8}\,(1-\rho(m,m\mid I,x))\,x_{L}.$

If both parties have a moderate candidate and both randomly advertise them
each party wins the election with probability $\pi(m,m\mid x)=1/2$. Thus,
party $L$’s expected utility from randomly advertising a moderate candidate,
when $k=0$, is

$\displaystyle U^{0}_{L}(x_{L}\mid\theta)$ $\displaystyle=$
$\displaystyle\sigma\Big{[}\frac{1}{2}(1-2m)-\frac{2-3m}{2}\Big{]}$
$\displaystyle+$ $\displaystyle(1-\sigma)\Big{[}4+\frac{2-3m}{16}\rho(m,e\mid
I^{0},x)x_{L}(m)-(1-m)\Big{]}-cx_{L}(m).$

which is linear in $x_{L}(m)$. Then, it follows that party $L$ advertises its
candidate if and only if $U^{0}_{L}(x_{L}\mid m,t_{R})>U^{0}_{L}(x_{L}=0\mid
m,t_{R})$, where

$\displaystyle U^{0}_{L}(x_{L}=0\mid m,t_{R})$ $\displaystyle=$
$\displaystyle\sigma\Big{[}\pi^{0}_{R}(m,m\mid
x_{L}=0,x_{R})(1-2m)-\frac{2-3m}{2}\Big{]}$ $\displaystyle+$
$\displaystyle(1-\sigma)\Big{[}\frac{1}{2}\left(1-\frac{3}{2}m\right)-(1-m)\Big{]}-cx_{L}(m)$

Thus, whenever the cost of advertisement is such that

$c\leq c^{0}\equiv\frac{1}{16}(1-\sigma)(2-3m)(1-\rho(m,m\mid I^{0},x))$ (10)

party $L$ advertise with intensity $\bar{x}_{L}^{0}(m)=1$, and never advertise
a moderate candidate otherwise.

This implies that, if $c<c^{0}$, a voter uniformed about the candidate of
party $J$ would anticipate that the party is running with an extremist
candidate. Thus, if both parties have a moderate or an extremist candidate,
each wins the election with probability $\pi^{0}(m,m\mid x)=\pi^{0}(e,e\mid
x)=1/2$, while if only one party has a moderate candidate it wins the election
with probability $\pi^{0}(m,e\mid x)=\pi^{0}(e,m\mid x)=1$.141414This result
depends on the specific functional form of the cost function. Suppose the cost
function is quadratic, $c(x)=c\frac{x^{2}}{2}$. Then parties advertise a
moderate candidate if and only if $c\leq
c^{00}\equiv\frac{1}{8}\left((2-3m)(1-\sigma)\rho(m,e\mid
I^{0},x)+2(1-2m)(1-\rho(m,e\mid I^{0},x))\sigma\right)$ and
$\bar{\bar{x}}^{0}_{J}(m)=\frac{(2-3m)(1-\sigma)\rho(m,e\mid I^{0},x)}{16c}$.
From which follow that uninformed voters can not anticipate with certainty the
candidate type when $c<c^{00}$.

#### Targeted Advertising.

Suppose parties can target their advertisement to only one ideological group
of voters, i.e.  either to the $L$-leaning or to the $R$-leaning, and all
voters that belong to the chosen group receive the advertisement with
probability one, while all the voters that belong to the other group do not
receive it. Thus, each voter that belongs to the targeted group $j$ will
receive the advertisement with probability $x_{Jj}^{d}(t_{J})=1$, where $d$
denotes that party $J$ is using a targeting technology which directs the
information exclusively to the $j$-leaning voters.

In what follows I consider only the case in which $c>c^{0}$, as whenever
$c\leq c^{0}$ party advertise their candidate to the entire population,
$\bar{x}^{0}(m)=1$, which clearly dominates any targeting strategy.

Whenever $c>c^{0}$ parties prefer to do not randomly advertise their
candidate. To simplify the argument, for the moment suppose that only party
$L$ can deviate by targeting its advertisement. If party $L$ targets
$L$-leaning voters (its own supporters): all left leaning voters are informed
about the candidate’s type of their favoured party, and $L$ wins the election
with probability $1/2$, regardless of the candidate type of party $R$. By not
advertising party $L$ obtains the same probability of winning the election but
at a zero cost, it follows that party $L$ never targets its advertisement to
its own supporters.

If, instead party $L$ targets the voters leaning toward its opponent
(henceforth opponent’s supporters) it wins the electoral competition with
probability $\pi_{L}(m,x_{R}(t_{R})\mid x^{\tau})=1$, as I assumed that party
$R$ has no access to targeting technology for now and $x_{R}(t_{R})=0$ for all
$t_{R}$ when $c>c^{0}$. Then, party $L$ target its advertisement to the
opponent supporters, $x_{JJ^{-}}^{\tau}(t_{J})=1$, whenever its utility from
targeting its advertisement,
$U_{L}^{\tau^{0}}(x_{L}^{\tau}(m),x_{R}(t_{R})\mid\theta)$, is higher than the
utility it receives when it does not advertise its candidate,
$U^{0}_{L}(x_{L}(m)=0,x_{R}(t_{R})\mid m,t_{R})$. Thus, party $L$ targets its
advertisement to the opponent supporters whenever

$c\leq\frac{(2-3m)(1-\sigma)}{4}.$ (11)

By symmetry it follows that also party $R$ has an incentive target its
advertisement to $L$-leaning voters. This implies that, if a party runs with a
moderate, has access to targeting technologies, and
$c\leq\frac{(2-3m)(1-\sigma)}{4}$, it wins the election with with probability
$1/2$ if both run with a moderate candidate, and with probability one if the
opponent has an extremist. Then, it is easy to show that if

$c<c^{\tau}\equiv\frac{(2-3m-m\sigma)}{4},$ (12)

where $c^{\tau}>c^{0}$, parties advertise a moderate candidate by targeting
their advertisement to voters leaning for their opponent.

###### Theorem 1

In a symmetric pure strategy equilibrium without voters communication, parties
advertise a moderate candidate with intensity $x^{0}(m)^{*}=1$ if and only if
the unit cost of advertisement $c$ is such that

$c\leq c^{0}\equiv\frac{1}{16}(1-\sigma)(2-3m)(1-\rho(m,m\mid I^{0},x)),$

target their advertisement to the opponent supporters,
$x_{JJ^{-}}^{\tau}(t_{J})=1$ , if and only if

$c<c^{\tau}\equiv\frac{(2-3m-m\sigma)}{4},$

where $c^{0}<c<c^{\tau}$, and never advertise a moderate candidate otherwise.

When voter’s communication does not takes place independent voters may cast
their vote for their less favoured party if and only if they are hit by its
advertisement. Then, in equilibrium, if the cost of the advertisement is low,
parties advertise with intensity $x^{*}(m)=1$ a moderate candidate. In other
words, parties inform the entire population of their candidate type if the
advertising technology is efficient enough. If, instead, the cost of
advertisement is high parties prefer to target their advertisement to the
opponents’ supporter and let their own supporter uninformed, this way to
guarantee the victory of their moderate whenever it runs against an extremist
and win with probability 1/2 otherwise.151515Policy motivated parties always
prefer to lose against a moderate and win against an extremist.

##  4 Echo Chambers

In this section I introduce the communication game assuming that each voter is
linked to $k\geq 1$ senders and $z\geq 1$ receivers. In the communication
game, then, information travels from many to many, given the underlying social
network structure, and I show that echo chambers arise as a product of voters’
communication strategies.

Once the advertisement has taken place, each voter receives $k\geq 1$ (resp.
sends $z\geq 1$) private messages about the candidate’s type of both parties
($M_{L}(t_{L}),M_{R}(t_{R})$) from each of her $k$ senders (resp. to her $z$
receivers).

Working backwards I start analysing the voters decision for each possible
information set, in this way I can characterize the ex-post communication
groups (henceforth simply ex-post groups) of partisan and independent voters.

In the last stage voter $i$’s decision is

$v_{i}=\begin{cases}\ 1$\quad\quad if $\mathbb{E}\left[t_{L}-i\mid
I^{i},x,k\right]\geq\mathbb{E}\left[i-(1-t_{R})\mid I^{i},x,k\right],\\\ \
0$\quad\quad otherwise, $\end{cases}$ (13)

where $x$ is the equilibrium advertising strategy vector of the two parties,
and $I^{i}=(t_{L},M_{L}(t_{L}),t_{R},M_{R}(t_{R}))$ voter’s $i$ information
set.

For each voter’s personal network $k$, subsequent communication strategies,
and advertising strategies, voters can be divided into ex-post partisans and
ex-post independents.

###### Lemma 2

There exist two ideological cut-off points,

$\alpha_{l}=\dfrac{1}{2}-\dfrac{m}{4}\left[1-\rho(m,m\mid
I,x,k)\right]\quad\text{and}\quad\alpha_{r}=\dfrac{1}{2}+\dfrac{m}{4}\left[1-\rho(m,m\mid
I,x,k)\right],$

which together with the central point define the ex-post group-type to which
each voter belongs once both the advertising has taken place and each voter
has received the $k$ message pairs:

  * •

if $i\in(0,\alpha_{l})$ (resp. $i\in(\alpha_{r},1)$) the voter is an ex-post
left-leaning (right-leaning) partisan;

  * •

if $i\in(\alpha_{l},1/2)$ (resp. $i\in(1/2,\alpha_{r})$) the voter is an ex-
post left-leaning (right-leaning) independent.

Proof: See Appendix.

Whenever voters’ ideological bliss point $i$ is such that
$i\in(\alpha_{l},\alpha_{r})$, voters are ex-post independents; otherwise they
are ex-post partisans (see Appendix). More specifically, if a voter’s
ideological bliss point $i$ is such that: $i\in(0,\alpha_{l})$ (_resp._
$i\in(\alpha_{r},1)$ ) the voter is an ex-post $L$-leaning (_resp._
$R$-leaning) partisan; $i\in(\alpha_{l},\alpha_{r})$ the voter is an ex-post
independent voter, and she is said to be $L$-leaning (_resp._ $R$-leaning)
whenever $i\in(\alpha_{l},1/2)$ (_resp._ $i\in(1/2,\alpha_{r})$.

Before proceeding with the analysis it is worth noticing that from Lemma 1
follows that none of the voters can credibly convey a message $M_{J}(e)$,
given that voters have zero probability of receiving information $t_{J}=e$
from parties advertising. Then, it must that in equilibrium the sender is
indifferent between $M_{J}(e)$ and $M_{J}(\emptyset)$ as the two messages can
convey the same information. Thus, in what follows, without loss of
generality, I assume that a sender indifferent between two pairs of message
always prefers to send a truthful one, and this allows me to restrict the
message’s space to $M_{J}(t_{J})=\\{m,\emptyset\\},\ \forall J\in\\{L,R\\}$.

In the next sections I first derive the properties of a pairwise communication
equilibrium (Section 3.1), and then I characterizes the echo chambers
equilibrium (Section 3.2).

###  4.1 Pairwise Communication

To derive the main result of the paper it is useful to analyse the basic
network structure to highlight the role played by the direction of the
ideological biases in the communication game. I assume that each voter is
matched with only one sender (_i.e._ , $k=1$), and, abusing of notation, I
denote by $s$ and $r$ the ideological bliss points of the sender and the
receiver, respectively.

In the last stage the receiver’s voting decision depends on her information
$I^{r}$, that is on the information she may receive from each party
advertisement ($t_{L},t_{R}$) and/or from the sender
$(M_{L}(t_{L}),M_{R}(t_{R}))$, the parties’ advertising strategy
$x=\left(x_{L},x_{R}\right)$, and the ideological bliss point of her sender
$s$. Thus, the receiver votes for party $L$ if and only if

$\mathbb{E}\left[t_{L}-r\mid
I^{r},x,s\right]\geq\mathbb{E}\left[r-(1-t_{R})\mid I^{r},x,s\right].$

In turn, this implies that the informational incentive of the sender depends
not only on the information he holds, but also on both the receiver’s
information and receiver’s ideological bliss point. Recall that the sender’s
intra-stage utility from each possible pair of messages is

$\displaystyle u_{s}(M_{L}(t_{L}),M_{R}(t_{R})\mid I^{s},x,r)$
$\displaystyle=$ $\displaystyle\Pr\left(v_{r}=1\mid
I^{r},x,r\right)\mathbb{E}[t_{L}-s\mid I^{s},x]$ (14) $\displaystyle\ \ \ +$
$\displaystyle\Pr\left(v_{r}=0\mid I^{r},x,r\right)\mathbb{E}[s-(1-t_{R})\mid
I^{s},x].$

Assuming that an indifferent sender sends a truthful messages, truthful
revelation is incentive-compatible for the sender if and only if

$u_{s}(M_{L}(t_{L}),M_{R}(t_{R})\mid\theta,I^{s},x,r)\ \geq\
u_{s}(M_{L}(\hat{t}_{L}),M_{R}(\hat{t}_{R})\mid\theta,I^{s},x,r).$ (15)

I state the following result.

###### Lemma 3

In the pairwise cheap-talk game truthful revelation is possible if and only if

  * (i)

the receiver is an ex-post partisan;

  * (ii)

the $J$-leaning sender knows his favoured party runs with a moderate, i.e.
$t_{J}=m$, and is uninformed about the candidate type of the party she likes
less ($t_{J^{-}}=\emptyset$);

  * (iii)

both the sender and the receiver belong to the same ex-post group type
(partisan/independent) and are on the same side of the ideological center
point.

Proof: See Appendix.

The intuition behind point (i) of Lemma 3 is trivial. The sender always sends
a truthful message to an ex-post partisan receiver (_i.e._ , either
$r<\alpha_{l}$ or $r>\alpha_{r}$) as she always vote for her favoured party
regardless the information held.

In what follows I briefly analyse the incentive of the sender in the
communication game when matched with an ex-post independent receiver.

First consider case in which the sender is partially informed about the
candidate type of his favoured party, _i.e._ he only knows that his favoured
party has a moderate candidate and has no information about the candidate’s
type of the opponent. In this case, by lying to an ex-post independent
receiver the sender may harm the probability that his favoured party wins the
election with a moderate candidate. By sending a truthful message the sender
never harms the probability that his favoured party wins the election and he
neither harms the probability that a moderate is elected, given that an ex-
post independent voter always prefer to vote for a moderate. Thus, whenever a
sender is partially informed about the candidate type of his favoured party
always sends a truthful message (Lemma 3 (ii)).

To shortly illustrate the core argument of point (iii) of Lemma 3 suppose the
sender is left leaning, and that he knows that the candidate of party $R$ is a
moderate, but has no information about the candidate type of his favoured
party, $L$.

An ex-post partisan sender knows that by truthfully sharing his information
with an ex-post independent receiver always increases the probability that
party $R$ wins the election, while he would prefer the left party to win the
election (even with an extremist). Thus, an ex-post partisan sender does not
truthfully transmit his information to an ex-post independent receiver, even
if his receiver is biased toward the party the sender prefers.161616It is easy
to show that the same reasoning applies to the case in which the ex-post
partisan sender is uninformed. If perfectly informed about both candidates’
type the ex-post partisan sender truthfully share his information with voters
leaning toward the party he prefers, and lies otherwise.

If the sender is an ex-post independent there are two possible scenarios: 1)
sender and receiver are biased toward the same party (_i.e._ the ideological
bliss points of both voters are on the same side of the central ideology); 2)
the two voters are biased toward different parties. In the first case, the
sender knows that by truthfully revealing her information he does not harm the
probability of his favourite moderate to being elected. In fact, a $L$-leaning
receiver votes for party $R$ if and only if he knows the $R$’s candidate is a
moderate and has no information about the $L$’s candidate type (like the
sender would have done). Thus, when the two voters are ex-post independent and
their biases are aligned the sender, by truthfully sharing his information,
increases the probability of a moderate being elected without harming the
probability of election of a moderate of her favoured party $L$.

In the second case truthful revelation by the sender harms the probability
that his favourite party with a moderate candidate wins the election: the
$R$-leaning receiver uses the sender’s information to vote for the sender’s
opponent party, regardless of the information he holds about the sender’s
favourite party’s candidate ($L$).

Thus, while an ex-post partisan receiver always obtains truthful information
in the pairwise communication game, an ex-post independent receiver obtains
truthful information in two cases: 1) both the sender and the receiver have
aligned preferences (they are biased toward the same party) and belong to the
same ex-post group type (partisans/independents); 2) the sender is informed
about the party’s candidate type he prefers and has no information about the
candidate type of the opponent.

###### Lemma 4

Whenever voters in the matched pair have divergent preferences and/or do not
belong to the same group type, the sender’s truthful messages are not credible
in equilibrium.

###  4.2 Social Networks and Echo Chambers

In this section I characterize communication equilibrium in the social
network. The structure of this equilibrium takes a specific form, which I
refer to as echo chambers.

Each voter personal network is characterized by $k\geq 1$ senders, by $z>0$
receivers, and by the probability, $\beta\in(0,1)$, that both $i$ and her
$z+k$ peers are leaning toward the same party (degree of homophily). Voters’
networks are acyclic and the richness of the network, i.e. the number of links
of each voter ($k$ and $z$), and the degree of homophily, $\beta$, are common
knowledge. All voters simultaneously receive $k$ and send $z$ pairs of private
messages about the candidate type of each party $(M_{L}(t_{L}),M_{R}(t_{R}))$
from and to their peers.171717Allowing the information to travel for more than
one step complicate the analysis, but it does not affect the qualitative
features of my results. In each matched pair of voters an ex-post independent
receiver discards the message pairs received from both ex-post partisan
senders and from ex-post independent senders if they are leaning for her less
favoured party. Thus, the receiver behaves as if she never received such
messages. In instead the voters receives credible message pairs then she
updates her believes and truthfully shares his information with her next-step
receiver if and only if either they are both ex-post independent with aligned
preferences or the receiver is an ex-post partisan. This implies that truthful
and credible information travels along paths in which the Lemma 3 and 4 hold
in each node and it is blocked otherwise.

Now I am ready to state my main result:

###### Theorem 2

In any pairwise $k$-player cheap-talk game there are two ideological cut-off
points, $0<q_{l}(k,\beta,x_{L})<1/2<q_{r}(k,\beta,x_{R})<1$, such that
truthful revelation in each matched pair is possible and credible if and only
if the bliss points of both the sender and the receiver belong to the interval
defined by those cut-off points and their preferences are aligned (_i.e._ both
are biased for the same party).

Proof: See Appendix.

As voter $i$ has $k+1$ sources of information, the parties’ advertisement and
the $k$ message pairs he receives from their network’s peers, the sender’s
informational incentives depends not only on the information he gathers trough
exposure to the parties’ advertisement and the receiver’s ideological bliss
point, but also on the information that each of his receivers gather from the
other $k-1$ senders.

Given that within the sampled network of each voter there is no preference
uncertainty and the communication among voters has the form of private
messages, each voter evaluates the $k$ messages pairs received one by one. It
follows that if a voter receives at least one credible and informative message
about the candidate’s type of party $J$ then he knows that the $J$’s candidate
is a moderate with probability one. Conversely, if the receiver observes $k$
credible but uninformative messages about party $J$ candidate type then he
knows that the candidate of party $J$ is a moderate with probability
$\sigma(1-x_{J}(m))^{k}/(1-x_{J}(m))^{k}\sigma+1-\sigma$ and an extremist
otherwise. Given the voters’ personal network structure there exist two
ideological cutoffs,

$q_{l}(k,\beta,x_{L})=\frac{1}{2}-\frac{m}{4}\frac{1-\sigma}{1-\sigma+\sigma(1-x^{*}_{L}(m))^{\beta
k+1}},$

and

$q_{r}(k,\beta,x_{R})=\frac{1}{2}+\frac{m}{4}\frac{1-\sigma}{1-\sigma+\sigma(1-x^{*}_{R}(m))^{\beta
k+1}}$

that along side with the ideological center point define what I call ’echo
chambers’: an enclosed system in which information is truthfully and credibly
shared and outside which no credible information is transmitted.

A voter whose ideological bliss point is between the cut-offs
$(q_{l}(k,\beta,x_{R}),1/2)$ (resp. $(1/2,q_{r}(k,\beta,x_{R}))$) truthfully
and credibly transmit information only to her peers leaning toward party $L$
($R$), and do not convey any information otherwise.181818It can be shown that
this result will also hold if network affiliation were costly. To see this,
consider the case in which an ex-post independent voter has to decide if and
which network to join at a cost $\alpha>0$. From Theorem 1 it follows that he
rely on the information received from voters with aligned biases that belong
to his group type and he disregards all the information received by others.
Then, if the degree of homophily is high enough, an independent voter would be
willing to incur a cost to join a network, which, in turn, reinforces the echo
chambers effect. Note also that an ex-post partisan would never pay to buy
information that she is not willing to use.

It is easy to prove that the two cut-offs $q_{l}(k,\beta,x_{L})$ and
$q_{r}(k,\beta,x_{R})$ are, respectively, decreasing and increasing in the
degree of homophily $\beta$, in the sources of receiver information $k$, and
in the advertising level $x_{J}(t_{J})$, which imply that as the number of
reliable sources of information increases the echo chambers become wider, i.e.
the distance $\mid q_{j}(k,\beta,x)-1/2\mid$ for each $j\in\\{l,r\\}$
increases.

##  5 Echo Chambers and Parties’ Advertising Strategies

I assume parties can advertise their candidate either by using a random
technology, which allow each voter to be exposed at the advertisement with the
same positive probability, or by relying on a group targeting technology,
which direct the advertisement only to voters who are either $L$-leaning or
$R$-leaning. I compare the maximum unit cost that a party is willing to pay to
advertise its candidate under both technologies and I show how those choices
are affected by the echo chambers.

I show that if the voters’ network is characterized by many (resp. few) source
of credible information, _i.e._ $k\beta$ is large (small), parties exploit the
echo chambers by advertising their candidate randomly (to targeted groups).

###  5.1 Random advertising

Suppose party $L$ has a moderate candidate and it randomly advertises him.
Then, the proportion of voter informed about $L$’s candidate type is

$\gamma_{L}(x_{L},k,\beta)=1-\big{(}1-x_{L}(m)\big{)}^{\beta k+1},$

which is increasing in the number of information sources of the voters $k$, in
the degree of homophily in the network $\beta$, and in the level of
advertising $x_{J}(m)$.

When both parties have a moderate candidate, given they play symmetric
strategies, each party wins the election with probability $\pi(m,m\mid
x,k,\beta)=1/2$. If party $R$ has an extremist candidate what matter is the
distribution of indifferent voters which depends on the information set held.
Then, for any possible information set $I$, the votes share of party $L$ is

$\mu(x\mid\theta,k,\beta)=\frac{1}{2}+\frac{m}{4}\rho(m,e\mid
I,k,\beta)\gamma_{L}(x_{L},k,\beta),$

which is increasing in $\gamma_{L}(x_{L},k,\beta)$. Thus, when the voters’
personal network features a high degree of homophily ($\beta$) and/or the
network richness ($k$) is high, the echo chamber are wide and advertisement is
echoed among voters.191919Wider echo chambers refers to the two cut-offs in
Theorem 2, as $\beta$ or $k$ or both increase the two cut-offs
$(q_{l}(k,\beta,x_{L}),q_{r}(k,\beta,x_{R}))$ moves toward the extremes of the
ideological line making the echo chambers wider, i.e. credible information
travels among an higher number of voters. Conversely, if the voters’ personal
network features a low degree of homophily and/or a small number of peers, the
echo chambers are small preventing the diffusion of information among voters.
As result, many voters remain uninformed and they will vote for their favoured
party.

###### Theorem 3

In a symmetric pure strategy equilibrium there exist critical levels of
richness of the network $k$ and degree of homophily $\beta$, such that party
$J$ randomly advertises a moderate candidate if and only if

$c(k,\beta)<c^{*}(k,\beta)\equiv\frac{(2-3m)(1-\sigma)(\beta
k+1)(1-\rho(m,m\mid I,\beta,k))}{16},$

and never advertise a moderate candidate otherwise.

Proof: See Appendix.

###  5.2 Targeted Advertising

The new informational environment has made available a large amount of voters’
personal data to political actors: voters/users to join social media create
profiles in which they specify personal information like gender, age,
education, jobs, and beliefs. Access to this data virtually allows parties to
target their advertisement and manipulate the flow of information diffusion
exploiting the presence of echo chambers. In this section I find the
conditions under which parties would rely on a group targeting advertisement
technology. I assume that if a party targets its advertisement to voters
biased toward party $J$ then all those voters receive the advertisement with
probability one.

###### Theorem 4

In a symmetric pure strategy equilibrium parties never target their
advertisements to their own supporters, and there exist critical levels of
richness of the network, $k$, and degree of homophily, $\beta$, such that
party $L$ advertises a moderate candidate exclusively to the opponent’s
supporters if and only if the cost of advertisement is such that

$c(k,\beta)<\hat{\bar{c}}\equiv\frac{(2-3m-\sigma m)}{4}.$

Proof: See Appendix.

Given that parties never advertise an extremist, to see the intuitions behind
Theorem 3, as before, it is sufficient to focus on the case in which the
candidate of party $L$ is a moderate.

First, I focus on the first part of the statement: parties never exclusively
target the advertisement on their own supporters. From Theorem 2 follows that
the information does not travel between left and right leaning voters even if
they belong to the group of ex-post independent voters. Thus, if party $L$
targets the advertisement to its own supporters all the right leaning voters
remain uninformed about its own candidate, and consequently they vote for
party $R$. Thus, party $L$ wins the election with probability $1/2$,
regardless of the candidate type of its opponent. If party $L$ randomly
advertise its candidate whenever it faces an extremist opponent it wins the
electoral competition with probability $\pi_{L}(m,e\mid x)=1/2+(1-\rho(m,m\mid
I,\beta,k))(1-(1-x_{L})^{(\beta k+1)}>1/2$, while if it faces a moderate
advertised randomly, then it wins with probability $1/2$. In other words, by
targeting the advertisement to its own supporters, a party wastes its
persuasion power on those voters who would have voted for it even if they were
not hit by the advertisement.

The second part of Theorem 4 links the network structure, which affects
information diffusion, to parties’ choice of advertisement technology, i.e.
under which conditions parties prefer to target their advertisement to the
voters leaning toward their opponent (opponent’s supporters) rather than
randomly advertise their candidate to all voters. Consider the case in which
party $L$ has a moderate candidate and it targets the advertisement to the
opponent’s supporters, while party $R$ randomly advertise its candidate
whenever it has a moderate. By targeting its advertisement to $R$’s supporters
all the ex-post left independent voters will be uninformed about their
favoured party’s candidate ($t_{L}=\emptyset$), while every ex-post right
independent biased voter will hold the information $t_{L}=m$.

It follows that $R$’s candidate is an extremist, party $L$ wins the electoral
competition with probability one. As both ex-post $R$-leaning independent
voters - with information $(m,M_{L}(m),\emptyset,M_{R}(\emptyset))$ \- and the
uninformed ex-post $L$-leaning independent voters vote for party $L$. If,
instead, party $R$ has a moderate and advertise him randomly then a proportion
of ex-post independent voter $\gamma_{R}(x_{R},k,\beta)=1-(1-x_{R}(m))^{\beta
k+1}$ vote for party $R$. As a consequence, by targeting the opponent
supporters ($x_{LR}^{d}(m)$), party $L$ wins the election with probability
$\pi^{d}_{L}(m,m\mid x_{L}^{d},x_{R})=1/2-((1-\rho(m,m\mid
I,\beta,k))(1-2(1-x_{R})))/16<1/2=\pi_{L}(m,m\mid x)$, where $\pi_{L}(m,m\mid
x)$ is the probability that party $L$ wins the election if it would have
randomly advertised a moderate.

Theorem 3 and 4 highlight the relationship between the advertisement’s cost
and the network structure. The wider (resp. small) the echo chambers are the
lowest (higher) is the parties’ incentive to target the supporters of their
opponent.

It is easy to show that $\hat{\bar{c}}(k,\beta)<c^{*}(k,\beta)$ for
$k\beta\geq\bar{k\beta}$. When voters’ network is characterized by high degree
of homophily and/or the richness of the network it acts as an information
diffusion device.202020Where
$\bar{k\beta}\equiv\frac{(2-3m)\left((2+(1-\rho(m,m\mid
I,\beta,k))\sigma)+(1+\rho(m,m\mid
I,\beta,k))\right)-4m\sigma)}{(2-3m)(1-\rho(m,m\mid I,\beta,k))(1-\sigma)}$
(see Appendix for derivation). In turn this implies that by randomly
advertising their candidate parties can reach many echo chambers, which spread
the information, and allow parties to maximize the probability of winning the
election for any given level of advertisement $x_{j}(m)$. On the contrary,
when the voters’ network is characterized by low homophily and/or the richness
of the network is small, $k\beta<\bar{k\beta}$, the network’s structure
entails a significant waste of information. This implies that the proportion
of voters informed about the $R$’s candidate type,
$\gamma_{R}(x_{R},k,\beta)$, is small and party $L$’s probability to win the
election by targeting the $R$-leaning voters is high.

My results are robust to the introduction of heterogeneity between the two ex-
ante bias groups of voters. Let the degrees of homophily within the left and
right group of independent voters be, respectively, $\beta_{l}<\beta_{r}$.
From Theorem 2 follows that the $R$-leaning voters have wider echo chambers
($|1/2-q_{l}(\beta_{l},k)|<|1/2-q_{r}(\beta_{r},k)|$) which implies, ceteris
paribus, that $R$-leaning ex-post independents would have access to more
credible sources of information than the ex-post $L$-leaning independents
have.

Thus, party $L$ has two advantages: first, the ex-post left partisan group
will be wider than the ex-post right partisans one; second, party $L$ needs
less advertising than the right party in order to persuade the same proportion
of $R$-leaning ex-post independent voters with respect to that party $R$ needs
to persuade the $L$-leaning ex-post independent voters. This setting mimics a
scenario in which the two parties incur different costs to randomly advertise
their candidate. From Theorem 3 and 4 follows that party $L$ will randomly
advertise its moderate candidate and exploit the echo chambers without
jeopardizing the support of its own supporters.

##  6 Discussion

The preceding analysis aimed to disentangle the mechanisms by which
communication through social media networks shapes electoral competition. I
designed a model to highlight the structure of the problem in its simplest
possible form, but the main results do not hinge on some of the specific
assumptions I made. For example, both the symmetry in candidates types and in
the voters’ network structure are convenient but not essential. I have also
abstracted from negative advertising: allowing parties to spread truthful
negative advertising about the candidate’s type of their opponent translate in
a model with a larger set of signals, but the structure of the problem remains
unchanged.212121Although part of the political content on social media is
negative information about competitors (produced by candidates rather than
parties’ elites), it has been proven by Hill (1989) and Merritt (1984) that
negative advertising elicits negative effects for both the target and the
sponsor. Ansolabehere and Iyengar (1996) show that negative advertising
mobilizes partisans, but depresses turnout among independents.

Another important assumptions in the model is that voters network each voter
knows exactly the ideology of the other voters in her network. This implies
that, in equilibrium, a voter will be fully convinced that a candidate is
moderate as soon as she either receives party’s advertisement or a single
cheap-talk message from someone who belongs to her echo chamber. That is
definitely not true once one consider the possibility of preference
uncertainty in the voters’ network. In this case voters would need to look at
the distribution of messages, as one message is not going to be sufficient,
and the parties might need to invest more on advertisement to convince a large
enough mass of voters.

###  6.1 Strategic candidate’s selection

In what follows I briefly analyse the case in which parties can strategically
select their candidate. I show that if advertisement technology is inefficient
parties select an extremist candidate (and never advertise him), otherwise
parties randomize between the two types of candidate and advertise only a
moderate one. To simplify the argument I assume that parties may only randomly
advertise their candidates. Let $s_{J}=(\sigma_{J}(t_{J}),x_{J}(t_{J}))$ be
the strategy of party $J$, where $\sigma_{J}(t_{J})$ is now the probability
that party $J$ selects a candidate of type $t_{J}\in\\{e,m\\}$.222222The
Theorem 5 is an easy corollary of the Proposition 1 of Galeotti and Mattozzi
(2011), in which $\overline{c}(k,\beta)=\frac{(2-7m)(1+\beta
k)}{16(1-k\beta(1-\underline{p})}$ and $\underline{p}$ solves
$\frac{16c}{2-3m}=(1+\beta k)\underline{p}^{\beta k},$ with
$\underline{p}=1-x$.

###### Theorem 5

For every $k$ and $\beta$ there exist a critical level of
$\overline{c}(k,\beta)$ such that in equilibrium:

  * i)

If $c\geq\overline{c}(k,\beta)$ both parties always run with an extremist and
do not advertise him.

  * ii)

If $c<\overline{c}(k,\beta)$ parties randomize between moderate and extremist
and $\sigma^{*}$ and $x^{*}=x^{*}(m)$ solve

$\displaystyle(1-\sigma)(\beta k+1)\left(1-\rho(m,m\mid I,k,\beta)(1-x)^{\beta
k}\right)=\frac{16c}{2-3m}$ (16) $\displaystyle 1-\rho(m,m\mid
I,k,\beta)\left(1-(1-x)^{\beta k+1}\right)=\frac{4m+16cx^{*}}{2-3m}.$ (17)

First consider the symmetric equilibria in pure strategy. Suppose parties
always select a moderate, $t^{*}=m$. Note that in any pure strategy equilibria
it must be that $x^{*}(t^{*})=0$, as voters will anticipate parties
candidate’s choice. Then, if a moderate is selected in a pure-strategy
equilibrium we have that $x^{*}(m)=0$, which implies that the parties have
incentive to deviate and select an extremist. In fact, by selecting an
extremist a party increases its utility, as it is policy motivated, without
harming the probability of winning the electoral competition as voters, which
in equilibrium are uninformed, would believe that both parties are running
with a moderate.

Now consider the case in which parties always select an extremist, $t^{*}=e$.
If a party deviates by selecting a moderate and advertising him there are two
effects to consider. First, there is a positive effect on the party’s expected
utility: by advertising a moderate the party increases its probability of
winning the elections. As independent voters prefers to vote a moderate
candidate, informing them of the deviation does increase the party’s
probability of winning the election. Second, there is a negative effect on
party’s expected utility, the advertisement is costly. Thus, if the cost of
advertisement is too high $c\geq\overline{c}(k,\beta)$, i.e. the marginal cost
of the deviation (the cost of advertisement) is higher than its marginal
benefit (the increase in the probability of winning the election), parties
always select an extremist candidate and do not advertise him.

When the advertisement technology is efficient, choosing a moderate increases
the probability of winning the election and compensate the cost of
advertisement. But, as parties are policy motivated, they would still prefer
to win with an extremist candidate rather than with a moderate.232323This also
implies that, ceteris paribus, as more centrist is the moderate candidate’s
position the less often a moderate candidate is selected by the party. Then,
whenever the cost of the advertisement is not too high parties randomly select
a candidate type and advertise only the moderate one.

##  7 Conclusion

Recent technological changes have lead to the rapid development of
communication networks through social media, and its political implications
are far from being completely understood. This paper poses the basis to
understand how social interaction among voters with heterogeneous preferences
influence the information diffusion and shapes the political outcome. I build
a formal model that embeds _(i)_ a model of campaign competition, where
parties compete in campaign advertising, and _(ii)_ a model of personal
influence, where voters, with possibly heterogeneous preferences, can
strategically communicate with each other to affect the political outcome.

The first result shows that whenever voters can strategically communicate with
each other echo chambers arise endogenously, voters share valuable information
only with like-minded peers. In other words, a chamber arises if and only if
the senders and the receiver are biased toward the same party, i.e. small
ideological distance between senders and receivers is not enough to guarantee
truthful communication, as one would have expected. Although small ideological
distance between the sender and the receiver is necessary for truthful
communication it is not sufficient, voters need to be leaning toward the same
party to create an echo chamber. The second result suggests that both the
richness of the network and the degree of homophily play an important role in
determining the advertising strategies of the parties. Specifically, whenever
one of the two or both are low, the parties are likely to target their
advertising campaign to voters ideological biased toward their opponent. This
is in sharp contrast with the literature with truthful communication, which
suggests that parties seek to target their ideologically closer voters.

From a more general perspective, my results suggest that the role of social
networks in political competition should not be underestimated, and neither
should the role-played by the voters within their network. Understanding how
social networks shape voting sharing strategies is crucial to stem the
phenomenon of “fake news” and eventually identifying the incentives of the
nodes that create and spread them.

## Appendix

This Appendix contains proof of all results stated in the text.

#### The indifferent voter.

Given an information set $I$, a voter $i$ is indifferent between voting for
the two parties if and only if

$u_{i}(v_{i}=1\mid\theta,I,k)=u_{i}(v_{i}=0\mid\theta,I,k)$ (18)

where

$\displaystyle u_{i}(v_{i}=1\mid\theta,I,k)$ $\displaystyle=$
$\displaystyle\sum_{t_{L},t_{R}\in T}\rho(t_{L},t_{R}\mid I,x,k)(t_{L}-i)$
(19) $\displaystyle u_{i}(v_{i}=0\mid\theta,I,k)$ $\displaystyle=$
$\displaystyle\sum_{t_{L},t_{R}\in T}\rho(t_{L},t_{R}\mid I,x,k)(i-(1-t_{R}))$
(20)

To ease the exposition I denote the candidate type of party $J$ as
$t_{J}=\\{e_{J},m_{J}\\}$. Given that $\sum_{t_{L},t_{R}\in
T}\rho(t_{L},t_{R}\mid\theta,I,k)=1$, the (19) becomes

$\displaystyle u_{i}(v_{i}=1\mid I,x,k)$ $\displaystyle=$
$\displaystyle\big{(}\rho(m,m\mid I,x,k)+\rho(m,e\mid I,x,k)\big{)}m_{L}$
$\displaystyle+$ $\displaystyle\big{(}\rho(e,m\mid I,x,k)+\rho(e,e\mid
I,x,k)\big{)}e_{L}-i$

and (20) becomes

$\displaystyle u_{i}(v_{i}=0\mid\theta,I,k)$ $\displaystyle=$
$\displaystyle-\big{(}\rho(m,m\mid I,x,k)+\rho(e,m\mid I,x,k)\big{)}m_{R}$
$\displaystyle-$ $\displaystyle\big{(}\rho(m,e\mid I,x,k)+\rho(e,e\mid
I,x,k)\big{)}e_{R}+i-1$

Thus, by substituting (19) and (20) into (18), after some manipulation,

$\displaystyle 1-2i$ $\displaystyle=$ $\displaystyle\rho(m,m\mid
I,x,k)(m_{R}-m_{L})+\rho(e,e\mid I,x,k)(e_{R}-e_{L})$ $\displaystyle+$
$\displaystyle\rho(m,e\mid I,x,k)(e_{R}-m_{L})+\rho(e,m\mid
I,x,k)(m_{R}-e_{L})$

from which it is immediate to obtain (8). $\blacksquare$

#### Sketch of proof of Lemma 1.

To simplify the argument consider the case in which party $L$’s candidate is
an extremist advertised with intensity $x_{L}(e)>0$, and there is only one
voter $i$. When voter $i$ is informed only about the party $L$’s candidate,
i.e.  her information set is $t_{L}=m$ and $t_{R}=\emptyset$, she will vote
for the left party if and only if her bliss point is such that
$i<\frac{1}{2}-\frac{m}{4}\rho(e,m\mid(e,\emptyset),x)\equiv\underline{i}$.
If, instead voter $i$ is uniformed she votes for the party $L$ if and only if
her bliss point is such that $i<\frac{1}{2}\equiv\overline{i}$. Given that
$\overline{i}>\underline{i}$, party $L$ by advertising its extremist
candidate, not only pays a cost $c>0$, but it also lowers his-own probability
of winning the election. $\blacksquare$

#### Sketch of proof of Lemma 2.

An ex-post partisan voter is a $J$-leaning voter who, regardless the
information she holds on the candidates’ type, always votes for the party she
is ex-ante biased for, i.e. $J$. An ex-post $J$-leaning independent voter is a
voter who votes for the opponent party, $J^{-}$, whenever he knows that
$J^{-}$’s candidate is a moderate and has no information about the candidate
of the party he is ex-ante biased for.

Consider a right-leaning voter $i\in(1/2,1)$ and $\rho(m,m\mid\theta,I,x,k)$
be the voter $i$ posterior beliefs that both candidate are moderate given his
information set $I$, the parties strategies $x$, and his network $k$. From
Definition 1 follows that a right leaning voter is an ex-post right-leaning
independent voter if and only if she votes for the left party whenever his
information set $(t_{L},M_{L}(t_{L}),t_{R},M_{R}(t_{R}))$ is either
$(m,\emptyset,\emptyset,\emptyset)$ or $(\emptyset,m,\emptyset,\emptyset)$,
that is the following inequality holds

$u_{i}(v_{i}=1\mid\theta,I,x,k)\geq u_{i}(v_{i}=0\mid\theta,I,x,k),$ (21)

where $v_{i}\in\\{0,1\\}$ is $i$’ voting decision and $v_{i}=1$ denotes the
decision of voting for party $L$. The inequality (21) holds whenever

$i\leq\dfrac{1}{2}+\dfrac{m}{4}\left[1-\rho(m,m\mid\theta,I,x,k)\right]\equiv\alpha_{r}.$

Thus, all voters whose bliss point are such that $i\in(1/2,\alpha_{r})$ are
ex-post right-leaning independent voters. Given the symmetry of the game by
the same argument it is possible to show that all voters whose bliss point is
such that $i\in(\alpha_{l},1/2)$ are ex-post left-leaning independent voters,
where $\alpha_{l}=\dfrac{1}{2}-\dfrac{m}{4}\left[1-\rho(m,m\mid
I,x,k)\right]$. Putting together the above results, a voter is ex-post
independent whenever his ideological bliss point $i$ belongs to the interval
$[\alpha_{l},\alpha_{r}]$, and he is an ex-post partisan otherwise.
$\blacksquare$

#### Proof of Lemma 3.

The proof of point (i) of Lemma 1 is trivial. To prove the remaining two
points I divide the proof in two parts. I first analyse the voting decision of
a receiver who believes the sender shares only truthful messages. Then, I
prove the result (ii) and (iii) of Lemma 3.

Part _(1)_ : If the ex-post independent receiver knows his favoured party has
a moderate candidate then, regardless of the information he holds on the
opponent’s candidate, she casts her vote for her favoured party.

An ex-post independent receiver credibly and partially informed only about
$L$’s candidate type,i.e. who holds an information
$I^{r}=\\{(m,M^{s}_{L}(\emptyset),\emptyset,M^{s}_{R}(\emptyset));(m,M^{s}_{L}(m),$
$\emptyset,M^{s}_{R}(\emptyset));(\emptyset,M^{s}_{L}(m),\emptyset,M^{s}_{R}(\emptyset))\\}$,
votes for party $L$ if and only if $u_{r}(1\mid I^{r},x,s)>u_{r}(0\mid
I^{r},x,s)$, which holds for $r<\dfrac{1}{2}+\dfrac{m}{4}\left[1-\rho(m,m\mid
I^{r},x,s)\right]$. Mutatis mutandis, an ex-post independent receiver votes
for party $R$ when credibly and partially informed only about $R$’s candidate
type if and only if $r>\dfrac{1}{2}-\dfrac{m}{4}\left[1-\rho(m,m\mid
I^{r},x,s)\right]$.

The uninformed ex-post independent receiver
$I^{r}=(\emptyset,M_{L}^{s}(\emptyset),$ $\emptyset,M_{R}^{s}(\emptyset))$
always casts his vote for her favoured party.

Part _(2)_ : The sender behaves as if his receiver is pivotal. Then, truthful
revelation is incentive compatible if and only if

$u_{s}((M_{L}(t_{L}),M_{R}(t_{R}))\mid I^{s},x,r)\geq
u_{s}(\hat{M}_{L}(t_{L}),\hat{M}_{R}(t_{R})\mid I^{s},x,r).$

Assume that the sender is matched with an ex-post left-leaning independent
receiver _i.e._
$r\in\left(\frac{1}{2}-\frac{m}{4}\frac{1-\sigma_{L}}{\sigma_{L}(1-x_{L})^{k+1}+1-\sigma_{L}},\frac{1}{2}\right)$.

First, consider a perfectly informed sender, $I^{s}=(m,m)$. The sender’s
expected payoffs from every possible message pair $M_{L}(t_{L}),M_{R}(t_{R})$
are:

$\displaystyle u_{s}(m,m\mid I^{s},x,r)$ $\displaystyle=$ $\displaystyle
u_{s}((m,\emptyset)\mid I^{s},x,r)=m-s,$ $\displaystyle u_{s}(\emptyset,m\mid
I^{s},x,r)$ $\displaystyle=$ $\displaystyle x_{L}(m-s)(1-x_{L})(-1+m+s),$
$\displaystyle u_{s}(\emptyset,\emptyset\mid I^{s},x,r)$ $\displaystyle=$
$\displaystyle(1-x_{R}(1-x_{L}))(m-s)+(x_{R}(1-x_{L}))(-1+m+s).$

Then, the sender truthfully shares his information if and only if
$u_{s}(m,m\mid\theta,I^{s},x,r)\geq u_{s}(\emptyset,m\mid\theta,I^{s},x,r)$
and $u_{s}(m,m\mid\theta,I^{s},x,r)\geq
u_{s}(\emptyset,\emptyset\mid\theta,I^{s},x,r)$, that hold if and only if
$s\leq 1/2$. A perfectly informed sender truthfully reveal his information if
and only if the receiver is biased toward his favoured party, regardless
whether the receiver is an ex-post partisan or an ex-post independent voter.
Clearly, the same is true whenever the sender is only informed about the
party’s candidate type he prefers.

When the sender holds the information set $I^{s}=(\emptyset,m)$, the sender’s
payoff from each possible message pair are:

$\displaystyle u_{s}(\emptyset,m\mid\theta,I^{s},x,r)$ $\displaystyle=$
$\displaystyle\rho(e,m\mid\theta,I^{s},x,r)(-1+m+s)$ $\displaystyle+$
$\displaystyle\rho(m,m\mid\theta,I^{s},x,r)[x_{L}(m-s)+(1-x_{L})(-1+m+s)];$
$\displaystyle u_{s}(m,\emptyset\mid\theta,I^{s},x,r)$ $\displaystyle=$
$\displaystyle u_{s}(m,m\mid I^{s},x,r)$ $\displaystyle=$
$\displaystyle\rho(m,m\mid\theta,I^{s},x,r)(m-s)+\rho(e,m\mid\theta,I^{s},x,r)(m/2-s);$
$\displaystyle u_{s}(\emptyset,\emptyset\mid\theta,I^{s},x,r)$
$\displaystyle=$ $\displaystyle\rho(m,m\mid\theta,I^{s},x,r)[x_{L}(m-s)$
$\displaystyle+$
$\displaystyle((1-x_{L})x_{R})(-1+m+s)+(1-x_{R}(1-x_{L}))(m-s)]$
$\displaystyle+$
$\displaystyle\rho(e,m\mid\theta,I^{s},x,r)[(1-x_{R})(m/2-s)+x_{R}(-1+m+s)].$

The sender never lies by revealing false information about the candidate’s
type of the left party. When the sender discloses false information about the
left candidate $(m,M_{R}(t_{R}))$ then the sender knows the receiver uses his
message to cast his vote for the left party and with probability
$(1-\sigma_{L})$ will take the wrong decision by voting for an extremist
candidate. Thus, the only two message pairs to consider are $(\emptyset,m)$
and $(\emptyset,\emptyset)$. It follows that the sender truthfully shares his
information if and only if $u_{s}(\emptyset,\emptyset\mid\theta,I^{s},x,r)\geq
u_{s}(\emptyset,m\mid I^{s},x,r)$ which holds if and only if
$s\geq\frac{1}{2}-\frac{m}{4}\frac{1-\sigma_{L}}{\sigma_{L}(1-x_{L})^{2}+1-\sigma_{L}}$.

An ex-post partisan sender informed only about the candidate of his favoured
party always truthfully shares his information. On the contrary, if an ex-post
partisan sender is informed only about the candidate of the party he likes
less, he never truthfully shares his information. This proves Lemma 1 _(ii)_.

A $J$-leaning ex-post independent sender always truthfully shares his
information, regardless the information hold, if also the receiver is
$J$-leaning, i.e. the preferences of the voters in the matched pair are
aligned (Lemma 3 (iii)) .

When the sender has no information about neither of the candidate type of the
two party (he is uninformed) his expected payoffs from each messages pair are:

$\displaystyle u_{s}(m,m\mid\theta,I^{s},x,r)$ $\displaystyle=$ $\displaystyle
u_{s}(m,\emptyset\mid\theta,I^{s},x,r)=$ $\displaystyle=$
$\displaystyle(m-s)(\rho(m,m\mid\theta,I^{s},x,r)+\rho(m,e\mid\theta,I^{s},x,r))$
$\displaystyle+$
$\displaystyle(m/2-s)(\rho(e,m\mid\theta,I^{s},x,r)+\rho(e,e\mid\theta,I^{s},x,r));$
$\displaystyle u_{s}(\emptyset,m\mid\theta,I^{s},x,r)$ $\displaystyle=$
$\displaystyle\rho(m,m\mid\theta,I^{s},x,r)(x_{L}(m-s)+(1-x_{L})(-1+m+s))$
$\displaystyle+$
$\displaystyle\rho(m,e\mid\theta,I^{s},x,r)(x_{L}(m-s)+(1-x_{L})(-1+m/2+s))$
$\displaystyle+$
$\displaystyle\rho(e,m\mid\theta,I^{s},x,r)(-1+m+s)+\rho(e,e\mid\theta,I^{s},x,r)(-1+m/2+s);$
$\displaystyle u_{s}(\emptyset,\emptyset\mid\theta,I^{s},x,r)$
$\displaystyle=$
$\displaystyle\rho(e,e\mid\theta,I^{s},x,r)(m/2-s)+\rho(m,e\mid\theta,I^{s},x,r)(m-s)$
$\displaystyle+$
$\displaystyle\rho(m,m\mid\theta,I^{s},x,r)\left((1-x_{R}(1-x_{L}))(m-s)+x_{R}(1-x_{L})(-1+m+s)\right)$
$\displaystyle+$
$\displaystyle\rho(e,m\mid\theta,I^{s},x,r)(x_{R}(-1+m+s)+(1-x_{R})(m/2-s)).$

The sender truthfully share his information if and only if the following two
conditions hold:

  * i)

the sender is not an ex-post partisan voter leaning for the party from which
he did not receive any information,
$s\geq\frac{1}{2}-\frac{m}{4}\frac{1-\sigma_{L}}{\sigma_{L}(1-x_{L})^{2}+1-\sigma_{L}}$,

  * ii)

the sender’s preferences are aligned with his receiver’s preferences $s<1/2$.

$\blacksquare$

#### Proof of Lemma 4.

This result follows from sequential rationality and consistency requirements.
$\blacksquare$

#### Proof of Theorem 2.

To show that a senders never deviate from the message strategies described in
the two-players game, even with richer the communication network, it is
sufficient to show that there is not profitable deviation. Let $(k-1)$ senders
follow the equilibrium strategies of the two-player cheap talk game described
before, and analyse the incentive to deviate of the $k^{th}$ sender, with
information $I^{s}$.242424This is without loss of generality, given that
within the sampled network of each voter there is no preference uncertainty
and the communication among voters has the form of private messages. Thus,
each voter evaluates the messages she receives one by one and can disregard
all message pairs received from senders biased for the party she likes less.
Denote by $\beta\in(0,1)$ the probability that in each of the $k$ matched pair
both voters have aligned preferences, they both prefer the same party.

To simplify the exposure I analyse the case in which the sender is matched
with a left leaning receiver.

The expected payoffs of a perfectly informed sender, $I^{s}=(m,m)$, from every
messages pair are:

$\displaystyle u_{s}(m,m\mid I^{s},x,k,\beta)$ $\displaystyle=$ $\displaystyle
u_{s}(m,\emptyset\mid I^{s},x,k,\beta)=m-s$ $\displaystyle
u_{s}(\emptyset,m\mid I^{s},x,k,\beta)$ $\displaystyle=$ $\displaystyle
x_{L}^{\beta(k-1)+1}(m-s)+(1-x_{L})^{\beta(k-1)+1}(-1+m+s)$ $\displaystyle
u_{s}(\emptyset,\emptyset\mid I^{s},x,k,\beta)$ $\displaystyle=$
$\displaystyle\left((1-x_{R})(1-x_{L})\right)^{\beta(k-1)+1}(m-s)$
$\displaystyle+$
$\displaystyle\left((x_{R}(1-x_{L}))\right)^{\beta(k-1)+1}(-1+m+s)$

and a perfectly informed sender sends a truthful message to a left biased
receiver if and only if $u_{s}(m,m\mid I^{s},x,k,\beta)\geq
u_{s}(\emptyset,m\mid I^{s},x,k,\beta)$ and $u_{s}(m,m\mid
I^{s},x,k,\beta)\geq u_{s}(\emptyset,\emptyset\mid I^{s},x,k,\beta)$. The two
inequalities hold if and only if $s\leq 1/2$. A perfectly informed sender has
no incentive to deviate from the equilibrium strategy of two-players
communication game: he truthfully shares his information whenever the his
preferences are aligned with the receiver, regardless of the ex-post group
type (partisan/independent) each of them belongs.

The expected payoffs of a sender only informed about the candidate type of the
receiver’s less favoured party, $I^{s}=(\emptyset,m)$, from each messages pair
are:

$\displaystyle u_{s}(\emptyset,m\mid I^{s},x,k,\beta)$ $\displaystyle=$
$\displaystyle\rho(m,m\mid
I^{s},k,\beta)\left(x_{L}^{\beta(k-1)+1}(m-s)+(1-x_{L})^{\beta(k-1)+1}(-1+m+s)\right)$
$\displaystyle+$ $\displaystyle\left(1-\rho(m,m\mid
I^{s},k,\beta)\right)(-1+m+s)$ $\displaystyle u_{s}(\emptyset,\emptyset\mid
I^{s},x,k,\beta)$ $\displaystyle=$ $\displaystyle\rho(m,m\mid
I^{s},k,\beta)\left(x_{L}^{\beta(k-1)+1}(m-s)+((1-x_{L})x_{R})^{\beta(k-1)+1}(-1+m+s)\right.$
$\displaystyle+$
$\displaystyle\left.((1-x_{R})(1-x_{L}))^{\beta(k-1)+1}(m/2-s)\right)$
$\displaystyle+$ $\displaystyle(1-\rho(m,m\mid
I^{s},k,\beta))\left(x_{R}^{\beta(k-1)+1}(-1+m+s)+(1-x_{R})^{\beta(k-1)+1}(m/2-s)\right)$

A sender truthfully shares his information if and only if
$u_{s}(\emptyset,m\mid I^{s},x,k,\beta)\geq u_{s}(\emptyset,\emptyset\mid
I^{s},x,k,\beta)$, that is if and only if
$s\geq\frac{1}{2}-\frac{m}{4}\frac{1-\sigma_{L}}{\sigma_{L}(1-x_{L})^{\beta
k+1}+1-\sigma_{L}}\equiv\overline{s}_{l}$. Thus, a sender partially informed
about the receiver’s less favoured candidate never deviates from the
equilibrium strategy prescribed in the two-player cheap talk game.

The expected payoffs of an uniformed sender, $I^{s}=(\emptyset,\emptyset)$,
from each messages pair are:

$\displaystyle u_{s}(m,m\mid I^{s},x,k,\beta)$ $\displaystyle=$ $\displaystyle
u_{s}((m,\emptyset)\mid I^{s},x,\theta)$ $\displaystyle=$
$\displaystyle(\rho(m,m\mid I^{s},k,\beta)+\rho(m,e\mid I^{s},k,\beta))(m-s)$
$\displaystyle+$ $\displaystyle(\rho(e,m\mid I^{s},k,\beta)+\rho(e,e\mid
I^{s},k,\beta))(m/2-s),$ $\displaystyle u_{s}(\emptyset,m\mid
I^{s},x,k,\beta)$ $\displaystyle=$ $\displaystyle\rho(m,m\mid
I^{s},k,\beta)(x_{L}^{\beta(k-1)+1}(m-s)+(1-x_{L})^{\beta(k-1)+1}(-1+m+s))$
$\displaystyle+$ $\displaystyle\rho(m,e\mid
I^{s},k,\beta)\left(x_{L}^{\beta(k-1)+1}(m-s)+(1-x_{L})^{\beta(k-1)+1}(-1+m/2+s)\right)$
$\displaystyle+$ $\displaystyle\rho(e,m\mid
I^{s},k,\beta)(-1+m+s)+\rho(e,e\mid I^{s},k,\beta)(-1+m/2+s),$ $\displaystyle
u_{s}(\emptyset,\emptyset\mid I^{s},x,k,\beta)$ $\displaystyle=$
$\displaystyle\rho(m,m\mid
I^{s},k,\beta)\left(\left(((1-x_{L})(1-x_{R}))^{\beta(k-1)+1}+x_{L}^{\beta(k-1)+1}\right)(m-s)\right.$
$\displaystyle+$
$\displaystyle\left.(x_{R}(1-x_{L}))^{\beta(k-1)+1}(-1+m+s)\right)$
$\displaystyle+$ $\displaystyle\rho(e,m\mid
I^{s},k,\beta)\left(x_{R}^{\beta(k-1)+1}(-1+m+s)+(1-x_{R})^{\beta(k-1)+1}(m/2-s)\right)$
$\displaystyle+$ $\displaystyle\rho(e,e\mid I^{s},k,\beta)(m/2-s)+\rho(m,e\mid
I^{s},k,\beta)(m-s).$

An uniformed sender truthfully share his information rather than lie in favour
of the receiver’s favourite candidate whenever
$s\geq\frac{1}{2}-\frac{m}{4}\frac{1-\sigma_{L}}{\sigma_{L}(1-x_{L})^{\beta
k+1}+1-\sigma_{L}}\equiv\overline{s}_{l}$, and the sender’s preferences are
aligned with his receiver’s preferences $s<1/2$. Again, the result of Lemma 3
apply to the $k$-pairwise communication game in each matched pair.

From sequential rationality and consistency requirements that Lemma 4 holds
also for the k-player cheap talk game.

Then, if a voter receives at least one credible and informative message about
the candidate’s type of party $J$, that is $M_{J}(t_{J})=m$, he knows that the
$J$’s candidate is a moderate with probability one. Conversely, if the
receiver observes $k$ credible but uninformative messages about party $J$
candidate type, $M_{J}(t_{J})=\emptyset$, then he knows that the candidate of
party $J$ is a moderate with probability
$\sigma_{J}(m)(1-x_{J}(m))^{k}/\sigma_{J}(m)(1-x_{J}(m))^{k}+1-\sigma_{J}(m)<1$.

Consider a voter whose bliss point is $i>1/2$, then $i$ is an ex-post
independent voter if and only if the following inequality holds

$u_{i}(v_{i}=1\mid\tilde{I},x,k)\geq u_{i}(v_{i}=0\mid\tilde{I},x,k),$

where $\tilde{I}=(t_{L},M_{L}(t_{L}),t_{R},M_{R}(t_{R}))$ is either
$(m,M_{L}(t_{L}),\emptyset,\emptyset)$ or $(t_{L},m,\emptyset,\emptyset)$ for
at least one credible message $M_{L}(t_{L})$ and all $k$ credible messages
$M_{R}(t_{R})$, which holds whenever

$i\leq\frac{1}{2}+\frac{m}{4}\frac{1-\sigma_{R}}{1-\sigma_{R}+\sigma_{R}(1-x^{*}_{R}(m))^{\beta
k+1}}\equiv q_{r}(k,\beta,x_{R}).$

By symmetry, a voter biased toward the left party is an ex-post independent
voter if

$i\geq\frac{1}{2}-\frac{m}{4}\frac{1-\sigma_{L}}{1-\sigma_{L}+\sigma_{L}(1-x^{*}_{L}(m))^{\beta
k+1}}\equiv q_{l}(k,\beta,x_{L}).$

It is immediate to notice that the two cut-offs that define the ideology of
ex-post independent voters $(q_{l}(k,\beta,x_{L}),q_{r}(k,\beta,x_{R}))$
coincide with the cutoffs $\overline{s}_{l}$ and $\overline{s}_{r}$, where
$\overline{s}_{r}=\frac{1}{2}+\frac{m}{4}\frac{1-\sigma_{R}}{1-\sigma_{R}+\sigma_{R}(1-x^{*}_{R}(m))^{\beta
k+1}}$ given the symmetry in the communication game.

This prove my result. Voters, whose ideological bliss point is between this
two cut-off points truthfully and credibly transmit information whenever their
preferences are aligned. $\blacksquare$

In what follows, to ease the proof of both Theorems 3 and 4, I assume party
$L$ has a moderate candidate $t_{L}=m$.

#### Proof of Theorem 3.

Parties can only randomly advertise their candidate. If both parties have a
moderate candidate, in the symmetric equilibrium, each wins the election with
probability $1/2$. If party $R$ has an extremist $t_{R}=e$ what matter is the
distribution of indifferent voters for each possible information set,
$I=\\{(m,m,\emptyset,\emptyset),(\emptyset,m,\emptyset,\emptyset),$
$(m,\emptyset,$ $\emptyset,$ $\emptyset),$ $(\emptyset,$ $\emptyset,$
$\emptyset,\emptyset)\\}$, hold by voters. The indifferent voter for each
information is

$i^{*}(I)=\frac{1}{2}+\frac{1}{2}\displaystyle\left(\sum_{t_{L},t_{R}\in
T}\rho(t_{L},t_{R}\mid\theta,I,x,k,\beta)\,t_{L}-\rho(t_{L},t_{R}\mid\theta,I,x,k,\beta)\,t_{R}\right),$

which implies

$\pi(m,e\mid x)=\frac{4+\rho(m,e\mid I,x,k,\beta)(1-(1-x_{L})^{\beta
k+1})}{8}.$

Party $L$’s expected utility from randomly advertising a moderate candidate is

$\displaystyle U_{L}(x_{L}\mid\theta,k,\beta)$ $\displaystyle=$
$\displaystyle\sigma_{R}\Big{[}\frac{1}{2}(1-2m)-\frac{2-3m}{2}\Big{]}$
$\displaystyle+$
$\displaystyle(1-\sigma_{R})\Big{[}4+\frac{2-3m}{16}\rho(m,e\mid
I,x,k,\beta)(1-(1-x_{L})^{\beta k+1})-(1-m)\Big{]}-cx_{L}$

and the optimal level of advertisement $x^{*}_{L}(m)\in(0,1)$ solves

$(1-\sigma_{R}(m))(\beta k+1)\rho(m,e\mid I,\beta,k)(1-x_{L})^{\beta
k}=\frac{16c}{2-3m}.$ (22)

if $(1-\sigma_{R})(\beta k+1)\rho(m,e\mid\theta,I,x,k,\beta)(2-3m)/16>c$ party
$L$ will never randomly advertise a moderate candidate and $x_{L}=0$,
otherwise it will advertise a moderate if and only if the expected utility
form advertising him with intensity $x^{*}_{L}$,
$U^{\emptyset}_{L}(x_{L}^{*}\mid\theta,k,\beta)$, is higher than the expected
utility of not advertising him,
$U^{\emptyset}_{L}(\tilde{x}\mid\theta,k,\beta)$, where $\tilde{x}_{L}(m)=0$.
The expected payoff of the left party when she does observe that its candidate
is a moderate, and decides to not advertise is

$\displaystyle U^{\emptyset}_{L}(\tilde{x}\mid\theta,k,\beta)$
$\displaystyle=$
$\displaystyle\sigma_{R}\Big{[}\Big{(}\frac{1}{2}-\frac{\rho(e,m\mid
I,\beta,k)(1-(1-x_{R})^{\beta k+1})}{8}\Big{)}(1-2m)-\frac{2-3m}{2}\Big{]}$
$\displaystyle+$
$\displaystyle(1-\sigma_{R})\Big{[}\frac{1}{2}\frac{2-3m}{2}+(1-m)\Big{]}.$

Thus $U_{L}(x_{L}\mid\theta,k,\beta)\geq
U^{\emptyset}_{L}(\tilde{x}\mid\theta,k,\beta)$ if and only if

$cx_{L}<\dfrac{\rho(e,m\mid I,\beta,k)(2-3m+\sigma_{R}m)(1-(1-x_{L})^{(\beta
k+1)})}{16}.$ (23)

Thus, $x^{*}_{L}(m)$ solves

$(1-\sigma_{R}(m))(\beta k+1)\rho(m,e\mid I,\beta,k)(1-x_{L})^{\beta
k}=\frac{16c}{2-3m},$

and party randomly advertises a moderate candidate whenever

$c<c^{*}(k,\beta)\leq\frac{(1-\sigma_{R}(m))(\beta k+1)\rho(m,e\mid
I,\beta,k)(2-3m)}{16}$

and never advertises a moderate candidate otherwise. $\blacksquare$

#### Proof of Theorem 4

Each party, when using a targeting technology, can target their advertisement
only to the $L$-leaning or $R$-leaning voters. If a party targets its
advertisement to $J$-leaning voters then all of them receive the advertisement
with probability one while no $J^{-}$-leaning voter receives the
advertisement.

Let $x_{Jj}^{d}(t_{J})=1$ denote the advertising strategy of a party $J$ who
target ($d$) its advertisement to voters leaning toward party $j=\\{L,R\\}$.

To simplify the argument assume that party $R$ has no access to the targeting
technology and it randomly advertises a moderate candidate if
$c<c^{*}(k,\beta)$.

First I show that parties never target their advertisement to their own
supporters. Consider the case in which party $L$ targets its own supporters,
then all $L$-leaning voters, both partisans and independents, are informed
about the candidate’s type of their favoured party and, regardless of the
candidate type and advertising strategy of party $R$, party $L$ wins the
election with probability $1/2$ it would win if it does not advertise its
candidate.

It follows that if $c>c^{*}(k,\beta)$, which implies party $R$ never
advertises a moderate candidate, party $L$ prefers to do not advertise a
moderate candidate either, as it will win the electoral competition with
probability $1/2$ without paying any advertisement cost. If, instead,
$c<c^{*}(k,\beta)$, party $L$ by randomly advertising a moderate candidate
wins the electoral competition with probability $\pi_{L}(m,m\mid x)=1/2$ when
it faces a moderate opponent, and with probability $\pi_{L}(m,e\mid
x)=1/2+\rho(m,e\mid I,\beta,k)(1-(1-x_{L})^{(\beta k+1)}>1/2$ when it faces an
extremist opponent. Then, party $L$ target the advertisement to its own
supporter rather than randomly advertise its candidate if and only if
$U_{L}(x_{LR}^{d}(m)\mid\theta,\beta,k)\geq U_{L}$
$(x_{L}^{*}\mid\theta,\beta,k)$, i.e. if and only if

$\displaystyle(1$ $\displaystyle-$
$\displaystyle\sigma_{R})\Big{[}\Big{(}\frac{\rho(m,e\mid
I,\beta,k)(1-(1-x_{R})^{\beta
k+1})}{8}-\frac{1}{2}\Big{)}\Big{(}\frac{2-3m}{2}\Big{)}\Big{]}$
$\displaystyle+$ $\displaystyle\sigma_{R}\Big{(}\frac{\rho(m,e\mid
I,\beta,k)(1-2(1-x_{R})^{\beta k+1}}{8}\Big{)}(1-2m)>0,$

which never holds for any $x_{R}\in(0,1)$. Thus, in a symmetric pure strategy
equilibrium parties never target their advertisements to their own supporters.

Now consider the case in which party $L$ targets the advertisement to voters
leaning for the opponent.

If $c>c^{*}(k,\beta)$ parties never randomly advertise their candidate.
Suppose party $R$ does not advertise its candidate and only party $L$ has
access to targeting technology. Party $L$ by targeting its advertisement to
the opponent’s supporter wins the elections with probability one. It is
trivial to show that party $L$ deviate from $x_{L}(m)=0$ whenever the cost of
the advertisement is such that $c^{*}(k,\beta)<c<\frac{1-2m}{2}$, which is the
case only when the credible source of voters information are not too many,
i.e. $\beta k+1<\frac{16c}{(2-3m)(1-\sigma)(1-\rho(m,m\mid x,I,k,\beta)}$. By
symmetry it follows that also party $R$ has an incentive deviate by targeting
$L$-leaning voters. This would imply that each party would win the election
with probability $1/2$, if both run with a moderate candidate, and with
probability one if their opponent has an extremist. It is easy to show that if
$c^{*}(k,\beta)<c<\frac{1-2m}{2}$ it does not exist any pure strategy
symmetric equilibrium.252525Parties will mix between the two strategy with
probability $\zeta=\frac{1-m+2c}{1-2m}$.

If, instead, $c<c^{*}(k,\beta)$ parties randomly advertise their moderate
candidate. Suppose again only party $L$ has access to targeting technology. By
targeting the opponent’s supporters party $L$ wins the election with
probability $\pi_{L}^{R}(m,e\mid x^{d}_{LR},x_{R}^{*})=1$ when the opponent
runs with an extremist, and with probability $\pi_{L}(m,m\mid
x^{d}_{LR},x_{R}^{*})=1/2-(\rho(e\mid
I,\beta,k)(1-2(1-x_{R}))/16<1/2=\pi_{L}(m,m\mid x_{L}^{*},x_{R}^{*})$ if the
opponent runs with a moderate. Then, party $L$ targets the advertisement to
the $R$-leaning voters if and only if
$U_{L}(x_{LR}^{d}(m)\mid\theta,\beta,k)\geq U_{L}(x_{L}\mid\theta,\beta,k)$,
i.e. if and only if

$\displaystyle\sigma_{R}\Big{[}\frac{1}{2}(1-2m)-\frac{2-3m}{2}\Big{]}+(1-\sigma_{R})\Big{[}\frac{2-3m}{2}-(1-m)\Big{]}-c\geq$
$\displaystyle\sigma_{R}\Big{[}\frac{1}{2}(1-2m)-\frac{2-3m}{2}\Big{]}+(1-\sigma_{R})\Big{[}\frac{1}{2}\frac{2-3m}{2}-(1-m)\Big{]},$

where recall $x^{d}_{LR}=1$, and which implies

$c<\hat{c}\equiv\frac{(2-3m)(1-\sigma_{R})}{4}.$

The party $R$ also has incentive to target $L$-leaning voters whenever
$c<\hat{c}$. This implies that, as in the benchmark model (Section 3) the cost
of advertisement must be

$c<\hat{\bar{c}}\equiv\frac{(2-3m-\sigma m)}{4}.$ (24)

Looking at the cost thresholds it is easy to show that
$c^{*}(\beta,k)\geq\hat{\bar{c}}$ whenever voters’ personal network is
characterized by high degree of homophily and/or the richness of the network,
i.e. if

$k\beta\geq\bar{k\beta}\equiv\frac{(2-3m)\left((2+(1-\rho(m,m\mid
I,\beta,k))\sigma)+(1+\rho(m,m\mid
I,\beta,k))\right)-4m\sigma)}{(2-3m)(1-\rho(m,m\mid I,\beta,k))(1-\sigma)}.$

Whenever this is the case the network it acts as an information diffusion
device and by randomly advertising their candidate parties can exploit the
echo chambers. If, instead, voters’ personal network is characterized by few
sources of credible information $c^{*}(\beta,k)<\hat{c}$, and parties have
higher incentive in targeting the opponent’s supporters as the voters’
personal network entails a significant waste of information. $\blacksquare$

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