# Utilitarian Beliefs in Social Networks: Explaining the Emergence of Hatred

Houda Nait El Barj, Stanford University
Théophile Sautory, U.C. Berkeley

(July 2021)

###### Abstract

We study the dynamics of opinions in a setting where a leader has a payoff
that depends on agents’ beliefs and where agents derive psychological utility
from their beliefs. Agents sample a signal that maximises their utility and
then communicate with each other through a network formed by disjoint social
groups. The leader has a choice to target a finite set of social groups with a
specific signal to influence their beliefs and maximise his returns.
Heterogeneity in agents’ preferences allows us to analyse the evolution of
opinions as a dynamical system with asymmetric forces. We apply our model to
explain the emergence of hatred and the spread of racism in a society. We show
that when information is restricted, the equilibrium level of hatred is
determined solely by the belief of the most extremist agent in the group
regardless of the inherent structure of the network. On the contrary, when
information is dense, the space is completely polarised in equilibrium with
the presence of multiple “local truths” which oscillate in periodic cycles. We
find that when preferences are uniformly distributed, the equilibrium level of
hatred depends solely on the value of the practical punishment associated with
holding a hate belief. Our finding suggests that an optimal policy to reduce
hatred should focus on increasing the cost associated with holding a racist
belief.

_“If societies cannot enforce the separability of the human mind into
sensitivity for what is science and for what is not science, then we can
predict that as humans evolve, the taste for truth will disappear
incrementally as it yields less pleasure” - Friedrich Nietzsche, Gay Science
1882._

##  1 Introduction and motivation

Not all beliefs are born equal. Traditional models of learning often assume
that the only obstacles to the truth are agents’ cognitive capacity and the
information structure. However, in some cases, the main obstacle preventing an
agent from learning the truth could be her own self. Take the example of an
employee who is being told repeatedly by a colleague that he is doing a bad
job. Our employee may choose to ignore what his colleague is telling him
because it hurts his self-image. In fact, his colleague is sending him a
truthful signal about his work performance. After a certain period of time,
the employee gets fired for bad performance. If the truth did not hurt the
employee, he might have been more attentive to the signals sent by his
colleague. This would have potentially allowed him to improve his work
performance and avoid getting fired.

In general, knowing the truth allows agents to take optimal actions in the
real world. This is the main motivation for humans to learn about the world in
which they live in. In certain cases, knowing the truth also yields some
psychological utility, and as such this can alter the extent to which agents
learn. Knowledge of a certain topic can create a trade-off between the
psychological and the practical utility associated with it. When learning the
truth yields psychological discomfort, we can observe that agents “choose
their own version of reality”.

In a February 2021 article, the New York Times interviewed a number of experts
in different fields on how the Biden administration can solve what they call
our new “reality crisis”: a world where an increasing number of citizens chose
to create “their own version of reality” from hoaxes and conspiracies.111“How
the Biden Administration Can Help Solve Our Reality Crisis These steps,
experts say, could prod more people to abandon the scourge of hoaxes and
lies.” New York Times, 02/02/2021 Examples of conspiracy theories relate to
whether the Covid Vaccine will implant a tracking microchip or whether Satanic
cannibalistic pedophiles run a global child sex traffic. Conspiracy theories
are gaining in popularity and can have practical consequences for the
democracy and social stability of a country.

Indeed, various theories propogated the idea that COVID-19 was manufactured in
a Chinese lab fueling anti-Asian racism globally. Xenophobic and violent acts
targetted at the Chinese community spurged after COVID-19: as of October 2020
more than 2,800 incidents were officially reported in the USA.222As reported
by STOP AAPI Hate. Similarly, the United Kingdom disclosed an increase of 21%
in anti-asian hate crimes as compared to the previous year suggesting the
pattern is not unique to the
US.333https://www.theguardian.com/world/2020/may/13/anti-asian-hate-crimes-
up-21-in-uk-during-coronavirus-crisis In parallel, the Black Lives Matter
movement raised awareness on the ubiquitous violence and racism that impacts
the global Black community.

The rise in hate crimes has opened up a social and political debate about the
roots and mechanisms of hatred. In our paper, we show that hatred can be
exactly understood as a belief that yields utility. Starting with René Girard,
the anthropological litterature argues that in order for someone to turn
hateful towards a social group, they must be able to blame the scapegoat for
some event they suffered from. As such, degrees of hatred can be equated to
strength of beliefs: “The more I am convinced that COVID-19 has been created
by the Chinese, the more likely I am to act upon it by comitting a racist
act”. Besides, hatred can also result from network effects, where certain
individuals turn hateful the more they are surrounded with hateful people to
feel socially integrated to their circle.

In fact, hate crimes and scapegoating are not a new phenomenon. Armenians were
blamed for the decline of the Ottoman Empire, Jews were blamed for the
austerity in Germany following World War I, Tutsis were blamed by the Hutus
for Rwanda’s economic crisis in the 80s…444Moise, Jean. 2014. “The Rwandan
Genocide: The True Motivations for Mass Killings”Not only have scapegoats
emerged to take on the blame for economic and social downturns, but various
social groups have also been designated as the cause of the appearance and
spread of pandemics. Jews were blamed for the Plague in 1347, AIDS was blamed
onto the gay community and the Chikungunya disease was blamed on the Comorean
immigrants. One could think that strong institutions, social progress, and
increasing education levels would eradicate such irrational behaviours. Yet
the Coronavirus pandemic brought scapegoating and racist acts to modern
evidence. Even in democratic and well-educated nations, the Chinese have been
blamed by some of the local population for the virus. A natural question that
arises then is how does hatred emerge and how can we reduce it?

This paper aims to provide a general model of utilitarian beliefs that can be
applied to answer this question. We study the mechanisms through which beliefs
that yield psychological utility emerge and spread in a social network. The
recent psychological litterature has revealed the key role of social groups
and cognitive dissonance in shaping agents’ choices of beliefs. We base our
model on psychological evidence for these forces in order to better capture
and explain the dynamics of utalitarian beliefs. Our framework is based on
Yariv (2002) belief utility model where we assume that a given belief yields
both instrumental and psychological utility. Thus, when agents face a choice
of signals to sample to learn about a state of the world, they sample
whichever brings them the highest utility. In our setting, agents have
hetereogenous preferences: the relative importance of the psychological
utility in determining their choice of belief vary. This reflects the fact
that people value the truth differently. We can think of the value of the
truth as being pinned down by some psychological cost. Some people will want
to learn the truth no matter what, whereas some other people will avoid
learning the truth to keep a good self image. We analyse a society composed of
disjoint social groups where agents want to learn a state of the world. Their
belief determines their action and the associated pay-off but also the
psychological pleasure they feel. As in DeGroot (1974), beliefs evolve as
agents communicate with other agents in their group. Through their
communication, they update their beliefs until a consensus is reached. In our
framework, the consensus reached in a social group will depend on the topology
of the network, the signals agents received as well as their underlying
preferences.

We then apply our model to explain the dynamics of hatred. We review the
psychological and anthropological litterature related to hatred. We identify
two forces: the desire to blame and the desire to belong, that are essential
in the emergence of hatred. Our general model of utilitarian beliefs can be
conveniently applied to the case of hatred, as it encompasses these two
forces. In our setting, a leader has a political interest in blaming the cause
of a downturn or pandemic onto a social minority group. The majority of agents
suffered from this event. They hold a belief on how likely the scapegoat is
responsible for it. This belief then determines whether or not they commit a
racist act but also yields them psychological utility. If they commit a racist
act, agents receive a punishment. Individual’s preferences, the structure of
social groups -their sizes, connectedness and segregation as well as the
access to information all determine the level of hatred. Our model thus
provides a rich framework through which we can study the dynamics of hatred.

We show that when information is restricted, the level of hatred is determined
solely by the belief of the most extremist agent in the network. However, in
the presence of free and diverse source of information, the space becomes
polarised with the local consensus varying accross groups. These consensus are
driven by network effects, where some individuals end up chosing a hate belief
only after their neighbour adopts it. On the contrary we find that when
information is dense, the equilibrium level of hatred oscillates in periodic
cycles and the space is completely polarised. This gives a greater incentive
to the leader to spread hatred. In such societies, agents can create their own
information circles to hold a belief that yields them higher utility without
inccuring the cost of social dissonance. We show that when agents have uniform
preferences, , the equilibrium level of hatred depends uniquely on the
practical punishment associated with holding such a belief. As the number of
agents in the population grows, the population tends to satisfy these
assumptions.

The primary contribution of this paper is to provide a framework guided by
psychological evidence to study the forces that shape the formation of
utilitarian beliefs. In contrast to prior work, our model studies the dynamics
of beliefs with a network fragmented into social groups and from which agents
yield utility. This allows us to study the impact of both preferences and
topological structures of communities on the evolution of opinions. To the
best of our knowledge, we are not aware of any prior work that provides such a
framework. This paper also makes a contribution to the theoretial literature
on the analysis of dynamical systems. An advantage of our framework is that we
can define our system under asymetric forces resulting in the characterisation
of periodic equilibria. In such a setting, our model describes a population
with a polarised basis of agents, and a set of oscillating members whose
beliefs evolve. Grounded on psychological and anthropological evidence, our
model can be applied to study the dynamics of hatred within societies. We hope
that our conclusions can help inform the design of policies to limit racism.

The remainder of the paper proceeds as follows. Section 2 presents the model.
Section 3 exposes the psychological litterature on which our model is based,
reviews the related economic litterature and lays the groundwork for our
application to hate dynamics. Consequently, Section 4 applies the model to
study the diffusion of hatred following a major disastrous event. Section 5
concludes. The Appendix contains the proofs for all the results.

##  2 Model

Consider a world with two possible states $\Theta=\\{0,$$1\\}$. Society is
composed of $N$ agents. Let $\mathcal{G}=\\{G_{1},G_{2},...,G_{k},G_{k+1}\\}$
be the set of all $K+1$ fixed social groups.555In our paper, we are not
interested in how social groups are defined: they could represent different
political affiliations, demographics or preferences. We impose that agents
belong to a single social group, i.e., for all $j,k\in\\{1,2,..,K+1\\}$ such
that$j\neq k,\hskip 10.00002pt$$\hskip 10.00002ptG_{j}\cap G_{k}=\emptyset$.
Agents evaluate the probability of the true state. We let
$\mu_{t,i}=p_{t,i}(\theta=1)$ be the belief of agent $i$ in time $t$. At the
beginning of period $0,$ $\mu_{0,i}=0$ $\forall i$. This means that initially,
all agents believe the true state to be $\theta=0$ with certainty.

Rounds of communication take place at each period through which agents discuss
and update their beliefs. We will explain how beliefs are updated later in
this section. For now, we want to introduce the concept of “credibility” and
define how agents interact. Let $\phi_{i,j}$ represent the credibility of
agent$j$ for agent $i$, this represents how much agent $i$ listens and trusts
agent$j$, where $\phi_{i,j}$ $\leq 1$. If $\phi_{i,j}=1$, then agent $i$
believe agent $j$ to tell the truth, and at the other extreme, $\phi_{i,j}=0$
implies that agent $i$ does not listen to agent $j$. Define $g$ to be the
function that maps an invidiual $i$ to his social group $g$ in society,
$g:N\to\mathcal{G}$. We assume that communication happens only within agents
of the same social group, and that each agent listens to any agent of its
group, including herself, thus imposing:

$\phi_{i,j}=0\iff g(i)\neq g(j)$

There is a leader, who is not part of any social group. He has a payoff that
depends on agents’ beliefs. In particular, he maximises his payoff when all
agents believe the true state to be $\theta=1$ (i.e. $\mu_{i}=1$, $\forall
i$). To influence agents’ belief and increase his return, the leader can send
them a signal suggesting that the true state is $\theta=1$ . Denote the signal
sent by the leader as $s^{L}=1$. The leader can choose which social groups to
target with such signals. All agents which are part of social groups not
targetted by the leader stick to their initial belief. This means that the
leader can affect only the beliefs of the social groups he explicitly targets.

Similarly to how agents hold grant certain credibility for each other, agents
also grant credibility to the leader. We denote by $\phi_{i}^{L}$ the
credibility of the leader for agent $i$. While the credibility of agents are
fixed, we let the credibility of the leader vary. In particular, we assume
that the more social groups the leader targets with his signals, the less
credible he is. 666This has a natural interpretation: when targetting a single
social group, a leader can adjust his communication to the exact preferences
of that social group in order to persuade them. As the number of social groups
he targets increases, his speech becomes less specific and de facto less
effective. Let $\mathcal{S}\subset\mathcal{G}$ be the set of social groups
targetted by the leader. We then have that:

$\begin{cases}\phi_{i,\mathcal{S}}^{L}>\phi_{i,\mathcal{S}^{\prime}}^{L}&\iff|\mathcal{S}|<|\mathcal{S}^{\prime}|\qquad\forall
i,\mathcal{S},\mathcal{S}^{\prime}\hskip
10.00002ptg(i)\in\mathcal{S},g(i)\in\mathcal{S}^{\prime}\\\
\phi_{i,\mathcal{S}}^{L}=0&\iff g(i)\notin\mathcal{S}\end{cases}$

Consequently, the return of the leader will depend on the set of social groups
he targets. Let $R_{t}:\mathcal{S\to\mathbb{R}}$ be the function that gives
the return of the leader in a period $t$:

$R_{T}(\mathcal{S})=\sum_{i\in\mathcal{S}}\mu_{T,i}-|\mathcal{S}|c$

where $c$is a cost of communication. The leader maximisation problem is thus
to find a set of groups $S^{*}$ that will maximise his returns.

We now explain how agents form and update their beliefs. At the beginning of
each period $t$, agents are exposed to a single or to multiple signals,
$s_{t}\in(0,1]$ and can choose whether or not to sample one of them. 777We
assume that a signal exactly equal to $0$ is impossible whereas we allow for a
signal to be exactly equal to $1$ since such a signal is sent by the leader
which we endow with communicative and manipulative skills.Each agent weigh a
signal $s_{t}^{C}$ from source $C$ by the the trust or accuracy they assign to
the source. We denote by $\lambda_{i}^{C}$ be the trust agent $i$ gives to
source $C$. Let $\hat{\mu}_{t,i}$ be agent $i$ chosen belief in period $t$,
after she decided which signal to sample (if any). Agent $i$’s chosen belief
fully determines her action in period $t$. We let $a_{t,i}$ denote the action
of agent $i$. $a_{t,i}\in\\{0,1\\}$ where $\hat{\mu}_{t,i}=p(a_{t,i}=1)$.

An agent chooses her belief $\hat{\mu}_{t,i}$ to maximise her per-period
utility:

$U(\mu_{t,i})=u(\mu_{t,i})+\sigma_{i}v(\mu_{t,i})$

Assume no discounting such that each util is felt in the present period and
vanishes afterwards. In this setting, the first component $u$ is the
instrumental utility that the agents gets from the actions implied by her
belief. The instrumental utility function is the following :

$u(\mu_{t,i})=\begin{cases}\chi&a_{t,i}=0\\\ -\chi&a_{t,i}=1\end{cases}$

where $\chi>0$. The second component of our utility, $v,$ is the psychological
utility the agent gets from her belief. $\sigma$ characterises how much an
agent cares about the psychological pleasure derived from his belief relative
to its practical consequences materialised by $\chi$. In our set up, we want
the psychological utility to be maximal when agents believe the true state of
the world to be $\theta=1$ at antipodes with their instrumental utility,
creating a dilemma. An agent’s belief depend on the signal they chose to
sample. As such, and to provide to provide a psychological utility in the
range $[-1,1]$, we define $v$:

$v(\mu_{t,i})=2\text{
$\lambda_{t,i}^{C}$$\left(s_{t,i}^{C}-\frac{1}{2}\right)$}$

Note that $v$is not defined when the agent chooses not to sample a signal
(i.e. when $s_{t,i}=\emptyset$ ). In such a case, we set $v(\mu_{t,i})=0$.

Once agent have chosen their belief, rounds of communications happen within
each social group. We represent our social network as a directed weighted
graph $G(N,E(G))$ where each connected component corresponds to a social
group. Vertices correspond to agents and edges corresponds to interactions
between agents. The weight on an edge $\overrightarrow{ij}$ captures the
influence of agent $i$ on agent$j$, which corresponds to $\phi_{j,i}$ in our
model and is assumed to be fixed accross periods. 888Fixing the credibility of
agents can be interpreted as agents having bounded rationality. Indeed,
rational agents should update the credibility or accuracy they give to a
certain source when new information is revealed to them. Note that
$\phi_{i,j}\neq\phi_{j,i}$ is allowed since our model allows for asymetric
influence or credibility accross agents. Let $\Phi$ be the $n\times n$ non-
negative stochastic matrix of interactions and $\boldsymbol{\mu}_{t,k}$ be the
row vector of beliefs of agents in period $t$ after $k$ rounds of
communication. The dynamics are described by:

$\mu_{t,k}=\mu_{t,k-1}\Phi$

We then have that $\mu_{t,k+r}=\mu_{t,k}\Phi^{(r)}$ where $\Phi$ can be
thought of the transition matrix of a Markov chain. We assume that $\Phi$ is
irreducible and aperiodic which allow us to apply the martingale convergence
theorem to the beliefs.999See Kemeny and Snell (1960) Irreducibility
corresponds to the existence of a path connecting any two vertices in the
graph and implies aperiodicity when $\phi_{i,i}>0$$\forall i$ (all agents
value their previous belief). Define $\omega^{*}$ as the left row eigenvector
corresponding to the unique eigenvalue equal to 1 for $\Phi$. After enough
rounds of communication in period $t$, beliefs converge :

$c_{t}=\omega^{*}\hat{\mu}_{t}$

Let’s now illustrate the dynamics of our model with a simple example.

######  Example.

A society with two social groups $G_{1}$ and $G_{2}$. Only $G_{1}$ is targeted
by the leader.

There are $3$ agents in group $G_{1}$. Since $G_{2}$ is not targeted by the
leader, all agents in this social group will stick to their prior by
construction, so we are only interested in the dynamics of the group $G_{1}$.

Let $\chi=0.4$. At the start of period $0$, they all have a belief of exactly
$0$. Now, at the begining of period $0$, they are exposed to only one signal
$s^{L}=1$. With each agent is associated a triplet $\\{\sigma,\phi^{L}\\}$ .
We have for agent $1$: $\\{\sigma_{1}=0.1,\phi_{1}^{L}=0.2\\}$, for agent $2$:
$\\{\sigma_{2}=0.9,\phi_{2}^{L}=0.5\\}$ and for agent $3$:
$\\{\sigma_{3}=0.9,\phi_{3}^{L}=0.8\\}$.

We characterise the matrix of interactions $\Phi$ in this example to be :

$\Phi=\left(\begin{array}[]{ccc}0.1&0.4&0.5\\\ 0.1&0.6&0.3\\\
0.3&0.5&0.2\end{array}\right)$

Let’s analyse the belief decision of agent 1.

If he samples no signal, i.e. $s_{0}=\emptyset$, then he stays at his prior
which is $\mu_{o}=0$ and so his utility is:

$\displaystyle U(\hat{\mu}_{0}|s_{o}=\emptyset)=$
$\displaystyle-\chi\cdot\mu_{0}+\chi\cdot(1-\mu_{0})+\sigma_{1}\cdot 0$
$\displaystyle=$ $\displaystyle 0\cdot(-0.4)+0.4\cdot 1+0.1\cdot 0$
$\displaystyle=$ $\displaystyle 0.4$

If he samples $s_{0}=1$, then his chosen belief is
$\hat{\mu}_{o}=\phi_{1}^{L}\cdot s^{L}+(1-\phi_{1}^{L})\cdot\mu_{o}=0.2\cdot
1+0.8\cdot 0=0.2$ and so his utility is

$\displaystyle U(\hat{\mu}_{0}|s_{o}=1)=$
$\displaystyle-\chi\cdot\mu_{0}+\chi\cdot(1-\mu_{0})+\sigma_{1}\cdot 2\text{
$\phi_{1}^{L}$$\left(s^{L}-\frac{1}{2}\right)$}$ $\displaystyle=$
$\displaystyle 0.2\cdot(-0.4)+0.4\cdot 0.8+0.1\cdot 0.1\cdot 1$
$\displaystyle=$ $\displaystyle 0.25$

In this case, for agent 1,
$U(\hat{\mu}_{0}|s_{o}=\emptyset)>U(\hat{\mu}_{0}|s_{o}=1)$, and as such, he
will chose not to sample the signal and stick to his prior. Thus, his chosen
belief at the begining of period $0$, is $\hat{\mu}_{0,1}=0$. Applying the
same logic to agent $2$ and $3$, we get $\hat{\mu}_{0,2}=0.5$ and
$\hat{\mu}_{0,3}=0.8$. Thus, our row vector of initial chosen belief in period
$0$ is :

$\hat{\mu}_{0}=(\begin{array}[]{ccc}0&0.5&0.8\end{array})$

After one round of communication, the row vector of beliefs is given by :

$\displaystyle\mu_{0,1}=$ $\displaystyle\hat{\mu}_{0}\cdot\Phi$
$\displaystyle=$
$\displaystyle(\begin{array}[]{ccc}0&0.5&0.8\end{array})\cdot\left(\begin{array}[]{ccc}0.1&0.4&0.5\\\
0.1&0.6&0.3\\\ 0.3&0.5&0.2\end{array}\right)$ $\displaystyle=$
$\displaystyle(\begin{array}[]{ccc}0.29&0.70&0.30\end{array})$

After $6$ rounds of communication, beliefs eventually converge,
$\mu_{0,6}=(\begin{array}[]{ccc}0.51&0.51&0.51\end{array})$. Thus,
$c_{0}=0.51$ and the period $0$ ends. Period $1$ begins, and all agents of
$G_{1}$ initially hold a belief $\mu_{1}=0.51$. Then, the same steps described
for period $0$ happen and similarly for all future periods.

##  3 Psychological evidence and related litterature

In our model, we want to capture the idea that the dynamics of learning are
singular in presence of utilitarian beliefs. In particular they depend on how
much weight agents place on the psychological component of their belief.
Furthermore, as humans are social beings who communicate with one another, the
dynamics of opinions should be studied from a social network framework. This
becomes even more crucial in the presence of utilitarian beliefs. Indeed, we
will show in Section $4$ that network effects can create incentives for
individuals to adopt certain beliefs that they would not have adopted with
different neighbours. Consequently, in this section, we want to provide
psychological evidence for the forces that shape the dynamics of utilitarian
beliefs. We review the psychological litterature associated with scapegoating
to show how hatred can be understood as a belief that yields utility. This
will allow us to apply our model to the dynamics of racism in Section $4$.

In his seminal book “Le bouc émisssaire”, French anthropologist René Girard
establishes a theory where scapegoating is defined as a collective phenomenon
that emerges when individuals have been experiencing an endemic and
unconscious violence. By being able to gather “all against one”, the scapegoat
allows for violence to be released and expressed in society. Under his theory,
the choice of the scapegoat is arbitrary, as it only serves the purpose of an
external outlet through which the majority can get deliverance from their
internal anxiety. Consequently, moments of scapegoating in History should be
recurring, as they represent a collective crisis that rises in intensity until
the internalized violence culminates and must be released. While the
designation of the scapegoat is independent of any cause, the targetted group
must fit certain criteria. First, the scapegoat must be a minority group that
can clearly distinguished from the majority. Second, the majority must believe
that the scapegoat is somewhat responsible for the misfortune they are
accusing them of. Finally, the scapegoat must present some extreme quality
such as wealth, beauty, vice, or weakness.

René Girard’s theory is closely related to Freud’s notion of projection which
he defines as the “self reproach repressed by erecting defensive symptom of
distrust of other people. In this way, the subject withdraws his
acknowledgement of the self-reproach”(Freud, Further Remarks on the Neuro-
Psychoses Of Defence 1896b) . In Freudian psychoanalytic theory, projection is
an unconscious defense mechanism through which agents get relief by displacing
their reproach onto someone else and can do so “without any consideration for
reality”. Doing so allows the agent to reduce cognitive dissonance that could
result from discovering unflattering aspects of one’s image or reality such as
the notion of guilt (Freud, Fragment of An Analysis Of A Case Of Hysteria
1905).

Building up on Freud’s ideas, Dollard et al. (1939) theorize the emergence of
scapegoats as the result of society’s frustration following a specific event.
Individuals cannot target their violence at the original source (either
because it is absent or non accessible). Consequently, the aggression is
displaced towards an easy target, which usually happens to be a minority
group. Providing evidence for the frustration-aggression hypothesis of
Dollard, Hovland and Sears (1940) show that incidence of lynchings of Black
people in Southern states is positively correlated with bad economic
indicators. Amongst other works, Douglas (1995) characterizes scapegoating as
a strategy used by agents to minimize feelings of guilt over the outcome of a
negative event for which they could be responsible. Related to this theory,
Rothschild et al. (2012) posit that a scapegoat allows for the majority to
keep their moral integrity by displacing guilt. In three studies, they show
that a negative outcome that could be linked to one’s own actions or whose
source is unknown increase scapegoating. In their theory, two fundamental
forces lead to scapegoating : the need to perceive oneself as morally valuable
and the need to perceive oneself as having control over one’s
environment.101010Glick (2002) and Glick (2005) also develop a theory where
scapegoating is attractive as it allows individuals to make sense of negative
outcomes without a clear cause in a simple way, which restores the desire for
control over their environment, as they can now simply punish the scapegoat.

Common to all these theories is the desire to blame a social group to get a
sense of relief from the potential responsibility agents could have in their
own misfortune. The need for blame as a fundamental process in social
cognition has been documented in the psychology litterature. Beardsley (1970)
characterizes blame as “a power and poignancy for human life unparalleled by
other moral concepts”. While blame allows agent to project guilt onto a chosen
scapegoat, they must be able to justify that the victim deserves her
mistreatment. (McKenna, 2012). Cikara, Botvinick and Fiske (2019) show that
the more an individual perceive themselves as threatened by the minority, the
more pleasure they get from punishing them.

Malle, Guglielmo and Monroe (2014), propose a theory of blame in which an
ordering of criteria (event detection, agent causality, intentionality,
obligation, reasons and capacity) defines degrees of blame. The human desire
for meaning, and avoidance of uncertainty as well as avoidance of the
responsibility for a negative outcome all justify the desire for blame (Malle
and Knobe 1997, Wong and Weiner 1981, Hilton 2007). Experimental evidence for
blame has been provided in Gurdal, Miller and Rustichini (2013) where
principals blame agents for the negative outcome of a lottery even though they
are aware it is a random event.

Another essential force of scapegoating is operated via the collective power
of social groups versus the targeted minority. An essential criterion is the
ability of the blamers to self-dissociate from the blamed, which must hence
stem from categorization of social groups into in-groups (the blamers) and the
outgroups (the scapegoat).111111 See L. Z. Tiedens & C. W. Leach (Eds.).
Studies in emotion and social interaction. The social life of emotions (p.
314–335).Tajfel et al. (1971) show that more defined social categorization
allows for increased discriminative behaviors with more “ingroup favoritism”
and more “outgroup prejudice”. Tajfel (1981) and Chen and Li (2009) show that
agents usually seek to maximize the difference between salient in-groups and
outgroups. Categorization of society into social groups operates as the
necessary structure through which differential treatments of oneself and the
other can happen, so long as they can be differentiated. Therefore, the more
distinct and distanced the scapegoat in society relative to other groups, the
easier it is for agents to direct hate and blame at them.

At the heart of social groups lies the human need for belonging. From an
evolutionary perspective, belonging could represent a survival advantage since
groups can better hunt, protect themselves from predators, and find potential
mates to reproduce. (Mangel and Clark 1985). Baumesteir and Leary (1995)
define the need to belong as an “evolutionary selection that guide individual
human beings into social groups and lasting relationships. These mechanisms
would presumably include a tendency to orient towards other members of the
species, a tendency to experience affective distress when deprived of social
contact or relationships, and a tendency to feel pleasure or positive affect
from social contact and relatedness”. Baumesteir and Wotman (1992), show that
belonging to a social group is usually associated with positive emotions such
as happiness and safety, while Baumesteir and Tice (1990), Leary and Tambor
(1993), Argyle (1987) and Myers (1992) all show that social exclusion, and a
sense of isolation as well as being deprived of relationships is associated
with higher depression, anxiety and grief. Kiecolt-Glaser et al. (1984) found
that loneliness was associated with decrease in immunocompetence and increase
in urinary cortisol respectively. Lynch (1977) summarises evidence from many
studies to conclude that “U.S. mortality rates for all causes of death are
consistently higher for divorced, single, and widowed individuals" than for
married individuals”.

Social groups define their own social norms, which are a set of implicit rules
that categorize what is acceptable within the group. Deviation from one’s
group social norms has been shown to lead to communication to produce
conformity and eventually loss of social status (Festinger 1950, and Schachter
1951). Since belonging is a fundamental need for humans, the above imply that
agents would not in general not want to deviate from their group’s accepted
beliefs at a given time.

The structure of social groups has a dual impact on the formation of hatred:
at the scapegoat level, it allows individual to dissociate from that
particular group and to direct the hatred at them. Further, when social groups
are rigid and static, agents feel more pressured to adhere to the average
group belief as they would not be integrated in society otherwise. If a belief
forms in a given group and the agent would not initially choose to believe in
it, she might feel pressured to do so in order to be integrated in her own
social group. However, this pressure lessens when the agent has the
possibility to belong to a different social group if she expresses different
views than his initial group majority. Therefore, in our model, agents’ desire
to belonging is an essential force in the formation of the hate belief that is
shaped by the topology of social groups.

The other essential force that we will capture is the desire for self-esteem.
This implies that agents are more inclined into believing a specific group is
responsible for the bad economic environment when they have themselves
suffered from it. Agents blame the scapegoat following an unconscious desire
to vengeance, holding them responsible for a global event that causes them
distress.121212Galofré-Vilà et al. (2017) find that between 1930 and 1933,
German districts most impacted by radical austerity measured are associated
with higher vote shares for the Nazi party.
https://www.nber.org/system/files/working_papers/w24106/revisions/w24106.rev0.pdf
Even if they do not feel attacked or threatened by the scapegoat, holding them
responsible allows hatred to operate as a mean of projection of distress into
a clear recipient object. This mechanism provides agents with relief and a
sense of certainty.

To our knowledge, the economic literature on scapegoating is limited to the
work of Glaeser (2005). He uses a model of supply and demand to derive an
equilibrium level of hatred in an economy where factors such as desire for
vengeance, time spent listening to atrocities stories about the minority as
well as costs from reduced economic interactions impact the demand side while
the desire to complement policies from a political candidate, financial
resources of actors and homogenous party platforms determine the supply side.
Our work is closely related to Glaeser (2005) since we both characterise
hatred as being initially sparked by a leader who has some political gain in
blaming a minority and where agents have some utility in feeling hateful.
However, our model uses a new framework of non-Bayesian learning where beliefs
enter the utility of agents. This allows us to study how both the interactions
within social groups and selfish preferences impact the level of hatred.
Furthermore, in our model, we are able to characterise how some individuals
are incited to hatred only because of their neighbours being hateful. In
Glaeser (2005) agents become haters as a function of the amount of time spent
hearing horrible stories, whereas in our model, hatred is a belief that varies
with the information access and the preferences of each agent. In our setting,
the macroscopical level of hatred is determined globally by the structure of
social groups. On the contrary to Glaeser (2005), limiting factors take
account of the propensity through which individuals update their belief with
the available information and through communication.

Our work is also closely related to the economic literature on psychological
expected utility whereby beliefs about a given state impact the utility
directly. Laying the grounds for our model are Yariv, (2001); Koszegi, (2003),
Oster, Shoulson and Dorsey (2014) and Caplin and Leahy (2001). In those
models, agents actions depend on their beliefs and are the result of a
utility-maximisation problem. These beliefs yield ego-utility which interferes
with the optimal actions being taken. On the other hand, our paper also
relates to the literature on non-bayesian updating. The closest works to ours
in the questions asked are Golub and Jackson (2010), Acemoglu et al. (2008)
and De Marzo, Vayanos, and Zwiebel (2003). Other relevant paper to our work
are Banerjee (1992); Acemoglu and Ozdaglar (2010), Bikhchandani, Hirshleifer,
and Welch (1992); Ellison and Fudenberg (1993), Acemoglu Chernozhukov, and
Yildiz (2016), Banerjee and Fudenberg (2004) Mossel, Sly and Tamuz (2015),
Molavi, Tahbaz-Salehi and Jadbabaie (2018) and Reshidi (2020).

##  4 Application of our model to the dynamics of hatred

We can now aply our model to understand how hatred emerges and spread within a
society.

Consider a society initially prosperous, until a major harsh economic or
social event happens that touches a significant share of the
population.131313Examples of such events can be found in history. For
instance, following World War I, Germany was held responsible for the damages
caused and had to repay an amount of 67.8 billion goldmarks, which led to a
period of austerity that touched in particular middle and lower classes.
Another modern example could be the pandemic of the Coronavirus that led to
more than 300,000 deaths in the USA.This cause-event is disastrous at a large
scale: not only does it impact the life of many agents but the impact itself
is consequential for each agent.

Following such a large-scale event, no clear responsible can be unequivocally
identified. This represents a threat for the political leader141414In our
model we assume the existence of a unique leader interested in spreading
hatred. Extensions with multiple competing political parties are left for
future research. to be blamed. Targetting the blame at a specific group
represents a political advantage where he can win some electorate. As argued
in section $3$, the chosen scapegoat group must meet two criteria : they must
be a minority to provide a political advantage to the leader (otherwise he
would loose more electorate than he can potentially win) and the majority
should be able to make a link between the current event and the scapegoat
group.151515For instance, in the 1920s in Germany, the majority of middle and
lower classes lived under dire economic conditions due to the austerity
imposed while Jews were usually in the upper class and hence very wealthy.
Thus Hitler could introduce the idea that since Jews were untouched by
austerity they must have wanted it, and are hence responsible for Germany’s
misery. In a similar way, Trump introduces the idea the Chinese people are
responsible for the Coronavirus pandemic since the virus originated in China.

Consider a world with two possible states , $\Theta=\\{0,$$1\\}$ with
$\theta=1$ corresponding to the event “The scapegoat is responsible for the
event and should be punished” and $\theta=0$ corresponding to the event “The
scapegoat is not responsible”. Let $\mu_{t,i}=p_{t,i}(\theta=1)$ be the belief
of agent $i$ in time $t$. Initially, at the onstart of period $0$,
$\mu_{0,i}=0$ $\forall i$, essentially representing the fact that racism is
not “innate” since we are interested in studying its emergence and dynamics.

There are $K+1$ social groups in our society, assume that the group $G_{K+1}$
is the scapegoat. It is a minority group associated with the global event.
Then, the leader has a choice of which subset
$\mathcal{S\subseteq}\mathcal{\mathcal{G}\setminus}G_{k+1}$ to target with
hateful signals about the scapegoat to incite hatred.

###  4.1 A dictator world

First, let’s study the dynamics of hatred in a dictator world, where
information is restricted to the one provided by the leader.

Consider a group in $\mathcal{\mathcal{G}\setminus}G_{k+1}$ and assume that
they are targetted by the leader. We know that initially all agents start with
$\mu_{0,i}=0$, and at each period they receive a signal
$s_{t,i}=s^{L}=1,\forall i,t$ and $\lambda_{t,i}=\phi_{i}^{L},\forall t$. Thus
the choice of agents when receiving this signal is either to sample it, or
ignore it and stick to their prior. Since initially $\mu_{i,-1}=0$ $\forall
i$, agents must gain a lot of blaming-utility to choose to adopt the leader’s
belief. There are huge initial costs of holding a hate belief. 161616An
interesting empirical question would be whether $\phi^{L}$ and $\sigma$ are
independent. Is the credibility or accuracy an agent assigns to another
dependent on how much they want to believe in their words. Or said
differently, do we tend to take as accurate statements that we desire to
believe in? Thus after being exposed to the leader in period $0$, the beliefs
of agents are :

$\begin{cases}\hat{\mu}_{0,i}=\phi_{i,L}&\text{if }\sigma_{i}>2X\\\
\hat{\mu}_{0,i}=0&\text{if }\sigma_{i}\leq 2X\end{cases}$

Then, after choosing their belief, they communicate, and they do so at each
new period. We then have the following result:

######  Proposition 1.

Equilibrium level of hatred

At finite horizons, the network topology matters, in particular, within one
round of communication we can guarantee a minimum consensus belief at under
the following condition on the network topology. We have
$\omega_{j}^{*}\leq\omega_{k}^{*}$ for every $\sigma_{j}\geq\sigma_{k}$ and
$\phi_{j}^{L}<\phi_{k}^{L}$ . On the contrary, at infinite horizon, the
network topology is irrelevant. Provided there is a single agent with
$\sigma_{i}>2X$, then
$lim_{k\to\infty}c_{k}=\max\\{\phi_{i}^{L}\\}_{i:\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{q})}{\phi_{i}^{L}},\text{for
some q}}$. In particular, if there a single agent $k$ with $\sigma_{k}>2X$ and
$\phi_{k}^{L}=\max\\{\phi_{i}^{L}\\}_{i\in N}$ then
$lim_{k\to\infty}c_{k}=\phi_{k}^{L}$ regardless of the network topology.

This proposition reveals an important dynamic of a utilitarian belief. At the
finite horizons, for a given sequence of initially chosen beliefs, as
determined by $\sigma$, agents with lowest beliefs on hate should be those who
have the highest social influence on the network. Under the dynamics of our
system, the finite horizon consensus belief is a convex combination of the
leader’s credibility for agents. Some agents are forced into hatred due to the
communication round and convergence of opinions without it ever being the
result of the influence of the leader on them. Those agents can be interpreted
as individuals who do not have much incentive to develop hatred and the only
way their individual belief is contributing into the final hate consensus
level is via them adopting their group belief. On the reverse, agents with
high incentive to hatred make an individual choice to become hateful and their
adopting of the leader’s belief has a direct impact on the final belief. But
at infinite horizon, the steady-state level of hatred is independent of the
network topology and at equilibirum, the leader targets all agents in the
majority group, and they all have belief
$\mu_{\infty}=\max\\{\phi_{i}^{L}\\}_{i:\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{q})}{\phi_{i}^{L}},\text{for
some q}}$ and commit a racist act with probability $\mu_{\infty}$. The
intuition for infinite horizon equilibrium is the follow. At the beginning of
each period, agents can adopt the leader’s belief weighted by his credibility
whenever it yields enough utility. Utility is increasing in the probability
that the true state is $\theta=1$, where more agents adopt the leader’s belief
giving other agents an incentive to adopt it as it establishes social norms in
their society. Said differently, the people with the highest utility to blame
will be the first to adopt the leader’s belief. Even though preferences of
agents and the punishment for holding a hate belief never change, as the
consensus increases more agent have an incentive to adopt the leader’s belief
as it becomes the norm. This is exactly because of the group effect that we
described in section $3$.

Given these dynamics, we are now interested in the decision of the leader. Who
will he target with hateful messages to spark hatred?

Consequently, the leader has a choice of choosing which groups in
$S=\\{G_{1},G_{2},...,G_{k}\\}$ to target with hateful messages in order to
maximise his returns as defined in Section $2$.

Assume that the leader cares about the level of hatred in a future period $r$
(for example $r$ could be the election period) but sends hateful messages in
all periods $0$ to $r$.

We can thus redefine our leader maximisation problem using our assumption of
distinct groups :

$\displaystyle\max_{i}\sum_{i\in S}\mu_{i}-|S|c\hskip 10.00002pt$
$\displaystyle\iff\max_{G\in S}c_{k,G}|G|-|S|c\hskip 10.00002pt\hskip
10.00002pt$ subject to
$\begin{cases}\phi_{i,\mathcal{S}}^{L}>\phi_{i,\mathcal{S}^{\prime}}^{L}\hskip
10.00002pt\iff|\mathcal{S}|<|\mathcal{S}^{\prime}|&\hskip 10.00002pt\hskip
10.00002pt\forall i,\mathcal{S},\mathcal{S}^{\prime}\hskip
10.00002ptg(i)\in\mathcal{S},g(i)\in\mathcal{S}^{\prime}\\\
\phi_{i,\mathcal{S}}^{L}=0\hskip 10.00002pt\hskip 10.00002pt\iff
g(i)\notin\mathcal{S}\end{cases}$

where $g$ is the mapping function defined in section $2.$

In particular, we let
$\phi_{i,S}^{L}=\max\\{0,\phi_{i}^{L}-\frac{N_{s}}{K}\\}$ where
$N_{s}=|\cup_{G\in S}G|$ and $K=|G_{1}\cup...\cup G_{k}|$. Then his return
$R:2^{S}\to\mathbb{R}$, is given by
$R(S^{{}^{\prime}})=c_{k,G}|G|-|S^{{}^{\prime}}|c$ for $G\in S^{{}^{\prime}}$
and $S^{{}^{\prime}}\subseteq S$. In the below, we will assume that $c=0$,
representing the fact that nowadays targetting individuals with messages can
be done at virtually no cost via social platforms such as Twitter or Facebook.
This assumption does not alter the qualitative or comparative nature of our
results. Adding a cost reduces the incentive for the leader homogenously
accross all groups. We will be looking at a period $r$where all groups are in
equilibrium consensus as defined in the above section.

######  Proposition 2.

In a dictator world, where the leader can have full control of the signals
sent to individuals, if $\max_{i\in G}\phi_{i}^{L}$ is equal in all groups,
then the choice of the leader is simply determined by the size of the groups.
Let $S^{*}$ denote the set of social groups maximising the leader’s returns,
which may not be unique, then $S^{*}$ is chosen such that
$|S^{*}|=\phi^{L,\max}\cdot\frac{K}{2}$. Under heterogenous preferences, and
equal group sizes, $|g|=N\hskip 10.00002pt\forall g,$ then the leader targets
all group $g$ with $\phi_{g}>\frac{N}{K}$. Finally, if social groups are not
heterogenous in their preferences and sizes, then the leader return maximising
set must balance off adverse effects of group sizes. In fact if $S^{*}$ is a
return-maximising set of groups, then: all omitted groups $G_{m}$ $\notin
S^{*}$ are such that $\phi_{m}<\frac{|G_{m}|}{K}+2\sum_{s\in
S^{*}}\frac{|g_{s}|}{K}$ and all included groups $G_{s}$ $\in S^{*}$ verify
$\phi_{s}>\frac{|G_{s}|}{K}+2\sum_{l\in S^{*},l\neq s}\frac{|g_{l}|}{K}$.

Proposition $2$ calls attention to an important fact: the size of a group is
at the same time a curse and an advantange for the leader. The bigger the
group, the greater the number of people that turn hateful if this group is
targetted by the leader, which increases his returns. However, the bigger the
group, the more it represents an absolute advantage to the leader so the more
he has to adapt his communication to fit the preferences of this group.
Consequently he is less credible for all groups, thus reducing back his
returns. As such, when groups all have a similar extreme individual (i.e. as
an individual whose credibility for the leader is maximal and similar in all
groups) then the choice of the leader simply comes back to selecting social
groups such that his choice is half the population size hence balancing the
trade off. However, when groups are very heterogenous in the agent who has
maximum credibility, then the leader has to select which groups to target
individually where both $(\phi_{G}^{L,\max},|G|)$ matter. Intuitively, he
wants to target groups with the highest possible $\phi_{G}^{L,\max}$ but with
a size $|G|$ enhancing the absolute effect of $\phi_{G}^{L,\max}$ (more
hateful people) instead of having more effect on reducing
$\phi_{G^{\prime}}^{L,\max}$ for all other groups $G^{{}^{\prime}}\neq G$
(leader is less credible for everyone because of high size group). Thus, even
when the cost of communication is $0$, to optimise his returns and maximise
the level of hatred in society, the leader should not target all groups. The
relative sizes of groups and their preferences will be an important factor in
determining whether or not they are being targetted by hateful messages from
the leader to spark hatred.

###  4.2 A world with diverse information

We now focus on a modern world of free diverse information where agents are
not only targetted by signals from the leader but they are also exposed to a
continuum of signals on $(0,1]$.171717We assume that $s_{k}$ can be equal to
$1$ exactly since it is the signal sent from the leader who has some
manipulative power whereas a signal can never be exactly 0. Since there is a
continuum of signals on $(0,1]$ we will simplify our choice space and assume
that the agent has at the begining of each period $k$ a choice to sample a
signal $s_{k}\in\\{s^{0},1,\emptyset\\}$ where $s^{0}\thickapprox 0$ is a
signal sent from a policymaker whose interest is to fight hatred and
$s_{k}=1=s^{L}$ is the signal sent from the leader. Indeed, since the agent
choses which signal to sample based on her realised utility, she is always
comparing levels of utility. We assume that this comparison ends up being a
binary comparison between a signal of almost $0$ and a signal of $1$.
181818This is not exactly correct, as signals of different magnitude can be
weighted differently (reflecting accuracy or trust of each source) and so the
comparison is not always transitive. However, in the below, we will assume
that we can always find two signals of very close magnitude coming from
infinite sources which are hence assigned different level of trust, such that
our comparison can be transitive and reduce to a choice between a signal of
magnitude almost $0$ and a signal of magnitude 1. If the agent does not sample
a signal, she sticks to her prior which is last period’s consensus.

In the presence of multiple signals and a leader targetting at each period all
agents to promote hatred, one might wonder when does the policymaker have the
most return in treating agents with information to reduce their hate belief.
In this set up, agents sample a signal if and only if it provides the most
utility. Thus, signals from the policymaker might be never sampled and
represent a pure economic cost. In the next proposition, we show when the
policymaker has the highest return in sending truthful information to agents.

######  Proposition 3.

Suppose the policymaker had the option to randomly send truthful information
($s^{P}\thickapprox 0)$ to agents in order to reduce their hate belief. When
agents yield ego utility from their belief and sample a signal in order to
maximise their utility, then the policymaker has the most return in sending
truthful signals when $\sigma$ is distributed on $\mathcal{U}(0,1)$, where
$\mathcal{U}$ is the uniform distribution. 191919E. Weinan, Li, Tiejun and
Vanden-Eijnden. 2019. “Applied Stochastic Analysis”._Graduate Studies In
Mathematics. American Mathematical Society._ has been a useful resource to
derive this result.

The result of proposition $3$ is very intuititive. The choice of signal
sampled by the agent depends on the value of their $\sigma$. If the
policymaker treats randomly agents with information, she wants the effect of
her treatment to be the greatest. This happens exactly when $\sigma$ is
uniformly distributed. Indeed, if $\sigma$ were deterministic, the policymaker
would be better off learning about the types of agents beforehands and
selecting the agents whose $\sigma$ imply they will sample her signal. Yet,
when she treats randomly agents (because the cost of learning their type is
too high for instance) she maximises the effect of her treatment when $\sigma$
is random as well. Practically, the policy maker has the most return in
sending informative signals randomly when an event touches a large population
in a similar manner (a pandemic, a large economic crisis…).

Recall that agents weigh the signal from the leader by $\phi_{i}^{L}.$ In the
below we will assume that $\lambda_{i}^{P}$, the accuracy or trust the agent
has in the policymaker is $\lambda_{i}^{P}=1-\phi_{i}^{L}.$ Since we restrict
signals to be diametrically opposite, they likely come from antithetic
sources. As such, agents will trust one or the other depending on their
identity and preferences. Hence at the beginning of each period, agents choose
to sample or not a signal such that :

$s_{k,i}=\operatorname*{arg\,max}_{\hat{\mu}_{k,i}}U(\hat{\mu}_{k,i})\hskip
10.00002pt\hskip 10.00002pt\hskip 10.00002pt\text{subject to $\hskip
10.00002pt\hskip
10.00002pt$$\hat{\mu}_{k,i}$=$\lambda_{k,i}^{C}$$s_{k,i}^{C}$+(1-$\lambda_{k,i}^{C}$)$c_{k-1}$}\hskip
10.00002pt\hskip 10.00002pt\forall k,\hskip
10.00002pts_{k,i}^{C}\in\\{s_{0},1,\emptyset\\}$

Studying the dynamics of hatred, we get the following result which describes
how society becomes more polarised as the consensus belief increases. The
society becomes split into followers (agents who always sample the signal of
the leader and follow him) and resistants (agents who follow the policynaker)
and the agents who are indifferent decreases.

######  Proposition 4.

Under the presence of both a leader and a policymaker targetting all agents
with signals respectively $s^{P}\thickapprox 0$ and $\mu^{L}=1$, when agents
choose their belief in order to maximise their utility, then as the consensus
group belief increases, the network becomes more polarised. Eventually, when
$c_{k}>\frac{1}{2}$, the society is completely polarised with agents either
following the leader by sampling his belief or resisting him by sampling the
policymaker belief and these form two distinct groups.

When $c_{k}<\frac{1}{2}$ , as the consensus increases, more people are
incentivised to sample the signal from the policymaker and similarly more
people are incentivised to sample the signal from the leader. Equivalently, as
the consensus increases, less people have an incentive to stick to the
consensus. This is reflected by different effects on each group.

Initially, when the consensus is low, a large group of people would rather
stick to the consensus without explicitly taking an individual risk in either
updating their belief towards the policymaker or towards the leader. Sampling
the leader’s signal leads to a higher hate belief and implies higher risk of
being punished by acting upon it. On the reverse, sampling the policymaker
signal implies a lower hate belief and hence imposes a psychological cost on
the agent (who can no longer blame as much on the scapegoat). Therefore,
sticking to the consensus is the conservative option and balances out the two
risks. Said differently, it is less costly for agents to adopt either version
of reality passively rather than explicitly chosing it. However, as the
consensus increases, this group of people becomes smaller as they have more
incentive to adopt either version of reality. The consensus increasing has
different implication on the group of people who are inclined to sample the
signal of the leader vs those who sample the policymaker’s.

As the consensus increases, agents with low utility of blaming perceive a
higher risk of being punished by adopting a high hate belief if they stick to
the average group belief and as such make an explicit decision to chose a
lower hate belief by sampling the policymaker’s signal. On the reverse, agents
with higher utility of blaming are more inclined into explicitly adopting a
higher hate belief by sampling the signal of the policymaker since the average
group belief is already high anyways. (i.e. they are incentivised to be more
hateful when other people around them are already somehow hateful).

These two effects are opposite on each group and can be described of as risk-
aversion (or empathy) versus ego utility effect.

Eventually, when $c_{k}>\frac{1}{2}$, the society becomes completely polarised
with agents either explicitly adopting the signal of the leader or adopting
the the signal of the policymaker. On the one hand agents with high utility of
blaming will always follow the leader (followers) and agents with low utility
of blaming (or equivalently agents who weigh more the practical implications
of hatred -lack of empathy, loss of opportunities in life..) will resist the
leader (resistants) but no one is indifferent.

Given these asymetric dynamics, we can study the equilibrium level of hatred.
In the next proposition, we show that we can characterise it under different
settings.

######  Proposition 5.

Let $F$ be the function describing the dynamics of the consensus, i.e.
$c_{k}=F(c_{k-1})$ and $c^{*}$ denote its equilibrium if it exists. We then
have the following results:

1\. When $\phi^{L}$ and $\sigma$ are independently distributed on
$\mathcal{U}(0,1)$, then $c^{*}=1-\chi$, as $N\to\infty$.

2\. Without any assumption on the distribution of $\phi^{L}$ and $\sigma$,
then $F$ is piecewise linear function defined on $[0,1]$ composed of $M+1$
disjoint intervals $I_{1},...,I_{M}$, where $M$ is the number of
discontinuities of $F$ over $[0,1]$. Then, $F$ admits a stable fixed point
$c^{*}$ if $c^{*}$ is the fixed point of the first visited interval such that
$F(I_{m})\subseteq I_{m}$.

3\. When $F$ does not admit a stable fixed point $c^{*}$ in the first interval
where $F(I_{m})\subseteq I_{m}$ and without any assumption on the distribution
of $\phi^{L}$ and $\sigma$, then the consensus belief is eventually periodic.

The results of proposition $5$ are striking. Recall that $\sigma$ describes
how much an agent weighs the psychological utility yield from his beliefs
relative to its practical utility. As such, $\sigma$ pins down which agents
sample the leader or the policymaker’s signal or none. Which agents belong to
which category further depends on the general level of hatred at each period,
and hence varies with time. Consequently, when we do not assume any
distribution for $\phi^{L}$ and $\sigma$, the equilibrium level of hatred will
depend on the relative proportions of these three groups, which vary accross
periods. Either these proportions balance out exactly, and $F$ admits a stable
fixed point, or the consensus belief will be eventually periodic. In such a
case, an exact equilibrium is not guaranteed, and the level of hatred varies
periodically. In this equilibrium state, the population is split into three
distinct groups: the resistants, the followers, and the agents who oscillate
between beliefs.Regardless of the bounds of the periodic oscillations, the
value of the punishment $X$ plays a crucial role in the equilibrium level of
hatred. As $X$ increases, the number of agents who always sample the signal
from the policy-maker (leader) cannot decrease (increase). This suggests that
directing policy towards increasing the practical cost associated with holding
a hate belief has potential to reduce each agent’s hatred level.

In fact, in a society where agents’ credibility for the leader and their
desire to blame are uniformly distributed, then the equilbrium level of hatred
is solely a function of the punishment associated with holding a hate belief.
This means that if the cause event touched agents homogeneously, then the best
way to reduce the level of hatred, is to increase the cost associated with it.
This observation complements the findings from Proposition $3$. When $\sigma$
is uniformly distributed, the policymaker has the highest returns in proposing
general information campaigns, rather than focusing on specific agents in the
population. This is because treating agents randomly, even if they are not the
most hateful, will reduce the general level of hatred via network feedback
effects. In large populations, the equilibrium level of hatred becomes
completely pinned down by the punishment value. This is because the types of
agent as defined by their value of $\\{\sigma,\phi^{L}\\}$ determine their
actions. When $\\{\sigma,\phi^{L}\\}$ are distributed uniformly and
independently, the asymetric forces guiding agents’ choices eventually balance
out in the system. Then, the only threshold determining the direction of their
force (i.e. whether they lean towards the policymaker or the leader), is only
a function of the punishment $\chi$. Consequently, to reduce the level of
hatred, governments should increase the cost associated with it. This can be
understood either as legal punishment for hate crimes, social cost associated
with being racist…This suggests an important direction for future policy.

##  5 Conclusion

Humans optimal decision-making process is often associated with desires to
reduce suffering and maximise happiness. As such, it makes sense to interpret
beliefs within the same framework when they yield utility to the agent who
holds them. We propose a framework to study utilitarian beliefs within social
networks. We believe that it is important to understand how such beliefs must
be analysed differently. They result from the utility-maximising choice of the
agent, where heterogeneity in preferences implies different individual
decision-making processes. Studying these asymmetric forces within network
effects is essential to better understand how these beliefs evolve. We based
our model on psychological and anthropological evidence for such forces and
applied it to study the dynamics of hatred, -which can be represented as a
utilitarian belief. We find that when preferences are uniformly distributed,
the equilibrium level of hatred depends solely on the value of the practical
punishment associated with holding a hate belief. Our finding suggests that
misinformation campaigns are inefficient when agents derive utility from their
beliefs. An optimal policy should instead focus on increasing the cost of
holding a racist belief.

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##  Appendix A Appendix : Proofs

### Proof of Proposition 1

###### Proof.

First, let’s prove the finite horizon case.

In period $0$, the consensus is $c_{0}=\omega_{i}^{*}\widehat{\mu}_{0,i}$,
where $\hat{\mu}_{0}$ is either $0$ or $\phi_{i}^{L}$. Agents start period $0$
with no hatred, i.e. $\mu_{-1,i}=0$ for all $i$ and are xposed to
$s_{0}=\mu_{L}=1$. They can choose to adopt it weighted by $\phi_{i}^{L}$,
where $\phi_{i}^{L}>0$ in which case $\widehat{\mu}_{0,i}=\phi_{i}^{L}>0$ or
stick to his prior in which case $\hat{\mu}_{0,i}=$$\mu_{-1,i}=0$.

Utility if she adopts the leader belief

$\displaystyle\mathbb{E}(U|x_{0}=\mu_{L})$
$\displaystyle=\phi_{i}^{L}(-X)+(1-\phi_{i}^{L})X+\sigma_{i}\phi_{i}^{L}$
$\displaystyle=-2X\phi_{i}^{L}+X+\sigma_{i}\phi_{i}^{L}$

Utility if she sticks to his prior

$\displaystyle\mathbb{E}(U|x_{0}=\emptyset)$
$\displaystyle=\mu_{-1}(-X)+(1-\mu_{-1})X$ $\displaystyle=X$

Thus an agent adopts the leader belief in period 0 if and only if
$\mathbb{E}(U|x_{0}=\mu_{L})>\mathbb{E}(U|x_{0}=\emptyset)$ which happens when
$\sigma_{i}>2X$

Consequently after their utility maximisation choice, agents have either
$\widehat{\mu}_{0,i}=\phi_{i}^{L}$ if $\sigma_{i}>2X$ or
$\widehat{\mu}_{0,i}=\mu_{-1}=0$ if $\sigma_{i}\leq 2X$ .

Then
$c_{0}=\sum_{i|\sigma_{i}>2X}\omega_{i}^{*}\phi_{i}^{L}+\sum_{i|\sigma_{i}\leq
2X}\omega_{i}^{*}\cdot 0=\sum_{i|\sigma_{i}>2X}\omega_{i}^{*}\phi_{i}^{L}$ .
It follows that
$\min_{\omega^{*}}c_{0}\iff\omega_{j}^{*}\leq\omega_{k}^{*}\text{ for
every}\sigma_{j}\geq\sigma_{k}\text{ and }\phi_{j}^{L}<\phi_{k}^{L}.$

Let’s now prove the infinite case now.

Let’s evaluate the utility at any period $k>0.$ In any other period, the agent
is evaluating the utility if he adopts the leader belief

$\displaystyle\mathbb{E}(U|x_{1}=\mu_{L})$
$\displaystyle=\phi_{i}^{L}(-X)+(1-\phi_{i}^{L})X+\sigma_{i}\phi_{i}^{L}$
$\displaystyle=-2X\phi_{i}^{L}+X+\sigma_{i}\phi_{i}^{L}$

vs the utility if he sticks to his prior:

$\displaystyle\mathbb{E}(U|x_{2}=\emptyset)$
$\displaystyle=c_{k-1}(-X)+(1-c_{k-1})X$ $\displaystyle=-2c_{k-1}X+X$

Indeed,at round $k$, an agent adopts the leader belief if
$\phi_{i}^{L}>\mu_{k-1}$ and
$\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{k-1})}{\phi_{i}^{L}}$or sticks to last
period consensus belief $c_{k-1}$.

We can rewrite the consensus in period $k$ as :

$c_{k}=\sum_{\left\\{i:\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{k-1})}{\phi_{i}^{L}},\phi_{i}^{L}>c_{k-1}\right\\}}w_{i}\phi_{i}^{L}+\sum_{\left\\{i:\sigma_{i}<\frac{2X(\phi_{i}^{L}-c_{k-1})}{\phi_{i}^{L}}\right\\}}w_{i}c_{k-1}+\sum_{\left\\{i:\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{k-1})}{\phi_{i}^{L}},\phi_{i}^{L}<c_{k-1}\right\\}}w_{i}c_{k-1}$

$c_{k}=\sum_{\left\\{i:\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{k-1})}{\phi_{i}^{L}},\phi_{i}^{L}>c_{k-1}\right\\}}w_{i}\phi_{i}^{L}+c_{k-1}\left(\sum_{\left\\{i:\sigma_{i}<\frac{2X(\phi_{i}^{L}-c_{k-1})}{\phi_{i}^{L}}\right\\}}w_{i}+\sum_{\left\\{i:\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{k-1})}{\phi_{i}^{L}},\phi_{i}^{L}<c_{k-1}\right\\}}w_{i}\right)$

Let’s prove the first result. First note that as long as there is an agent for
whom
$\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{k-1})}{\phi_{i}^{L}},\phi_{i}^{L}>c_{k-1}$
then $c_{k}>c_{k-1}$ .

If there is at least one agent $j$ with $\sigma_{j}>2X$, he adopts
$\phi_{j}^{L}$ in period 0, then $c_{0}>0$ and $c_{k}$ is updated dynamically
as more agents have an incentive to switch to hatred. Indeed, in period $k$,
the incentive to switch to hatred is
$\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{k-1})}{\phi_{i}^{L}}$ whereas, in period
$k+1$ it is $\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{k})}{\phi_{i}^{L}}$ .

Let the consensus becomes stable from period $K^{*}$ onwards. Then, in period
$K^{*}$ all agents have $\mu_{k^{*}}=c.$ Assume for the sake of contradiction
that
$c<\max\\{\phi_{i}^{L}\\}_{i:\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{q})}{\phi_{i}^{L}},\text{for
some q}}$. Let agent $m$ be the agent such that
$\phi_{m}^{L}=\max\\{\phi_{i}^{L}\\}_{i:\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{q})}{\phi_{i}^{L}},\text{for
some q}}$ but $\phi_{m}^{L}>c$.

Then in period $K^{*},$ agent $m$ choose $\phi_{m}^{L}$ which contradicts our
initial assumption.

Thus,
$lim_{k\to\infty}c_{k}=\max\\{\phi_{i}^{L}\\}_{i:\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{q})}{\phi_{i}^{L}},\text{for
some q}}$.

To prove the first result, let $j$ be an agent such that $\sigma_{i}>2X$ and
$\phi_{j}^{L}=\max\\{\phi_{i}^{L}\\}_{i\in N}$.

Then by the above, if agent $j$ has incentive to switch in period $0$ to
$\phi_{j}^{L}$, then at each period he chooses $\phi_{j}^{L}$. By definition,
at stationary state, $\mu_{i}=c\hskip 10.00002pt\forall i$ .

Again by contradiction, at stationary state, agent $c=\phi_{j}^{L}$.

Thus $\lim_{k\to\infty}c_{k}=\max\\{\phi_{i}^{L}\\}_{i\in N}$ ∎

### Proof of Proposition 2

Let $S$ be the set of groups from which the leader can choose.
$S=\\{G_{1},G_{2},...,G_{k}\\}$.

We define his return $R:2^{S}\to\mathbb{R}$ and for a given set
$S^{{}^{\prime}}\subseteq S$, we have :

$R(S^{{}^{\prime}})=\sum_{g\in S^{{}^{\prime}}}(c_{r,g}|g|-(r+1)c|g|)$

Assuming $c=0$,

$R(S^{{}^{\prime}})=\sum_{g\in S^{{}^{\prime}}}c_{r,g}|g|$

Since $r$is a period of equilibrium, then we have showed above that
$c_{r,g}=\max_{i\in
g}\\{\phi_{i}^{L}\\}_{i:\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{q})}{\phi_{i}^{L}},\text{for
some q}}$. In large groups, we can assume that the condition
$i:\sigma_{i}>\frac{2X(\phi_{i}^{L}-c_{q})}{\phi_{i}^{L}},\text{for some q}$
is satisfied, and hence $c_{r,g}=\max_{i\in g}\phi_{i}^{L}=\phi_{g}$.

Then we defined $\phi_{i,S}^{L}=\max(0,\phi_{i}^{L}-\frac{|\cup_{g\in
S}g|}{K})$, so $\phi_{g,S}^{L}=\max(0,\phi_{g}^{L}-\frac{|\cup_{g\in
S}g|}{K})$.

Since we already defined $\phi_{G}=\max_{i\in g}\phi_{i}^{L}$, then we will
let $\phi_{g,S}^{L}>0,$ hence $\phi_{g,S}^{L}=\phi_{g}^{L}-\frac{|\cup_{g\in
S}g|}{K}.$

Then

$\displaystyle R(S^{{}^{\prime}})=$ $\displaystyle\sum_{g\in
S^{{}^{\prime}}}(\phi_{g}-\frac{|\cup_{g\in S^{{}^{\prime}}}g|}{K})|g|$
$\displaystyle=$ $\displaystyle\sum_{g\in
S^{{}^{\prime}}}(\phi_{g}|g|-\frac{|\cup_{g\in
S^{{}^{\prime}}}g|\cdot|g|}{K})$

Since we defined social groups such that $g_{i}\cap g_{j}=\emptyset\hskip
10.00002pt\forall i\neq j$ then,

$\displaystyle R(S^{{}^{\prime}})=$ $\displaystyle\sum_{g\in
S^{{}^{\prime}}}(\phi_{g}|g|-\frac{|g|^{2}}{K})$

In the case with homogenous preferences, $\phi_{G}=\phi\hskip
10.00002pt\forall g$

$\displaystyle R(S^{{}^{\prime}})=$ $\displaystyle\sum_{g\in
S^{{}^{\prime}}}(\phi\cdot|g|-\frac{|g|^{2}}{K})$

Then the return becomes only a function of the total size of the groups chosen
in a given set $S^{{}^{\prime}}$. Let $N=\sum_{g\in S^{{}^{\prime}}}|g|$ for a
given set $S^{{}^{\prime}}$.

We can rewrite

$\displaystyle R(N)=$ $\displaystyle\phi N-\frac{N^{2}}{K}$

Then we see that the return maximising set $S^{{}^{\prime}}$ is characterised
by $N=\phi\frac{K}{2}$

In the case with non-homegenous preferences but equal size groups, we can
rewrite

$\displaystyle R(S^{{}^{\prime}})=$ $\displaystyle\sum_{g\in
S^{{}^{\prime}}}(\phi_{G}\cdot M\cdot N-\frac{M^{2}N^{2}}{K})$
$\displaystyle=$ $\displaystyle MN\sum_{g\in
S^{{}^{\prime}}}(\phi_{G}-\frac{MN}{K})$

where $M=|S^{{}^{\prime}}|$ and $N=|g|,\hskip 10.00002pt\forall g$

Then notice that adding a group increases return by $\phi_{G}-\frac{N}{K}$.
Consequently, the leader will target all groups $g$ with
$\phi_{g}>\frac{N}{K}$.

Finally, under non-homegenous preferences and unequal group sizes, then, we
can rewrite

$\displaystyle R(S^{{}^{\prime}})=$ $\displaystyle\sum_{g\in
S^{{}^{\prime}}}(\phi_{g}|g|-\frac{|g|^{2}}{K})$ $\displaystyle=$
$\displaystyle\sum_{g\in S^{{}^{\prime}}}\phi_{g}|g|-\sum_{g\in
S^{{}^{\prime}}}\frac{|g^{2}|}{K}-2\frac{\sum_{j=1}^{M}\sum_{i=1}^{j-1}|g_{i}|\cdot|g_{j}|}{K}$

If $S^{{}^{\prime}}$ is a return-maximising set, then if $g_{m}$ is an omitted
group in equilibrium, we must have

$R(S^{{}^{\prime}}\cup
g_{m})-R(S^{{}^{\prime}})<0\iff\phi_{g_{m}}|g_{m}|-\frac{|g_{m}|^{2}}{K}-2\sum_{l\in
S^{{}^{\prime}}}\frac{|g_{m}|\cdot|g_{l}|}{K}<0$

Thus all omited groups $g_{m}\notin S^{{}^{\prime}}$ are such that
$\phi_{g_{m}}<\frac{|g_{m}|}{K}+2\sum_{l\in
S^{{}^{\prime}}}\frac{|g_{l}|}{K}$.

And on the contrary, all included groups, $g_{s}\in S^{{}^{\prime}}$ are such
that $\phi_{g_{s}}<\frac{|g_{s}|}{K}+2\sum_{l\in S^{{}^{\prime}},l\neq
s}\frac{|g_{l}|}{K}$

### Proof of Proposition 3

Define the outcome $x_{j}$ to be the event that when the policy maker sends a
signal $x_{k}=0$ to agent $j$, the consensus in our network $c_{k}$ ends up
being lower versus when they do not.

We have,

$\displaystyle p_{j}=$ $\displaystyle\mathbb{P}(X=x_{j})$ $\displaystyle=$
$\displaystyle\mathbb{P}(c_{k}|\text{ agent $j$ gets signal $x_{k}=0$
}<c_{k}|\text{ agent $j$ does not get signal }x_{k}=0)$ $\displaystyle=$
$\displaystyle\mathbb{P}(\text{agent j samples signal $x_{k}=0$})$
$\displaystyle=$ $\displaystyle\mathbb{P}(U_{k,j}(\text{agent j samples signal
$x_{k}=0$})>U_{k,j}(\text{agent j does not sample signal $x_{k}=0$}))$
$\displaystyle=$ $\displaystyle\begin{cases}\mathbb{P}(\sigma_{j}\leq
2Xc_{k-1})&\mbox{if }c_{k-1}\leq\frac{1}{2}\\\
\mathbb{P}\left(\sigma_{j}<2X(\phi_{j}^{L}+c_{k-1}-2\phi_{j}^{L}c_{k-1})\right)&\mbox{if
}c_{k-1}>\frac{1}{2}\end{cases}$

We can then define our Shannon entropy to provide the expected information
content of $X$:

$H(X)=\mathbb{E}(I(X))=\mathbb{E}(-log(\mathbb{P}(x_{j}))$

We then get:

$H(X)=-\sum_{j=1}^{n}p_{j}log(p_{j})$

Let’s prove that $H$ is maximized when $X$ and hence $\sigma$ is uniformely
distributed.

We define the function $f$ on the range (0, 1] as follows:

$f(x)=x\log(x)$

We can then write:

$H(X)=-\sum_{j=1}^{n}f(p_{j})$

The second derivative of $f$ is:

$f^{{}^{\prime\prime}}(x)=\frac{1}{x}$

which is strictly positive on the (0, 1]. Hence $f$ is strictly convex on (0,
1], and the sum $\sum_{j=1}^{n}f(p_{j})$ is a stricly convex function. Thus,
$H$ is strictly concave. From the convexity of $f$, we make use of the
Jensen’s inequality applied to the random variable $\mathbb{P}(X)$, to write
the following:

$\displaystyle
f\left(\mathbb{E}(\mathbb{P}(X))\right)\leq\mathbb{E}\left(f(\mathbb{P}(X))\right)\implies$
$\displaystyle
f\left(\frac{\sum_{j=1}^{n}p_{j}}{n}\right)\leq\frac{1}{n}\sum_{j=1}^{n}f(p_{j})$
$\displaystyle\implies$ $\displaystyle\frac{1}{n}\sum_{j=1}^{n}f(p_{j})\geq
f\left(\frac{1}{n}\right)$ $\displaystyle\implies$ $\displaystyle-H(X)\geq
n\frac{1}{n}\log\left(\frac{1}{n}\right)$ $\displaystyle\implies$
$\displaystyle H(X)\leq\log(n)$

For a uniform distribution, where $p_{j}=\frac{1}{n},\forall j$, the Shannon
entropy is:

$H(X)=-\sum_{j=1}^{n}p_{j}log(p_{j})=-n\frac{1}{n}\log\left(\frac{1}{n}\right)=\log(n)$

which achieves the upper bound limit.

Finally, as $H$ is a striclty concave function, the uniform distribution is
the unique distribution which maximizes the Shannon entropy.

Intuitively this mean that information has highest power when $\sigma$ is
uniformely distributed. Since a signal $x=0$ can only reduce the consensus,
then the effect of information is strictlly positive. Therefore, maximising
the expected information content of our information release is equivalent to
maximising the return of information.

### Proof of Proposition 4

Fix a period $k$ and remember that now the agent has a choice to sample three
signals $s_{k}\in\\{s^{P},s^{L},\emptyset\\}$ where $s^{P}\thickapprox 0$ is
the signal from the policymaker, $s^{L}=\mu^{L}=1$ is the signal from the
leader. Let’s evaluate the utility under each sampling possibility.

  * •

Utility if the agent samples the signal from the policy-maker

The belief of the agent becomes
$\hat{\mu}_{k,i}=\lambda_{i}^{P}s^{P}+(1-\lambda_{i}^{P})c_{k-1}\thickapprox(1-\lambda_{i}^{P})c_{k-1}$
since $s_{k}\thickapprox 0$ and the expected utility $U$ is:

$\displaystyle E(U(\hat{\mu}_{k,i}|s=s_{k}^{P}=0))=$ $\displaystyle
X(1-(1-\lambda_{i}^{P})c_{k-1})-X(1-\lambda_{i}^{P})c_{k-1}-\sigma_{i}\lambda_{i}^{P}$
$\displaystyle=$ $\displaystyle
X-2X(1-\lambda_{i}^{P})c_{k-1}-\sigma_{i}\lambda_{i}^{P}$

  * •

Utility if the agent samples the signal from the leader

The belief of the agent becomes
$\hat{\mu}_{k,i}=\phi_{i}^{L}+(1-\phi_{i}^{L})c_{k-1}$ and the expected
utility $U$ is:

$\displaystyle E(U(\hat{\mu}_{k,i}|s=s_{k}^{L}=1))=$ $\displaystyle
X(1-(\phi_{i}^{L}+(1-\phi_{i}^{L})c_{k-1}))-X(\phi_{i}^{L}+(1-\phi_{i}^{L})c_{k-1})+\sigma_{i}\phi_{i}^{L}$
$\displaystyle=$ $\displaystyle
X-2X\phi_{i}^{L}-2X(1-\phi_{i}^{L})c_{k-1}+\sigma_{i}\phi_{i}^{L}$

  * •

Utility if the agent sticks to its prior

The belief of the agent becomes $\hat{\mu}_{k,i}=c_{k-1}$ and the expected
utility $U$ is:

$\displaystyle E(U(\hat{\mu}_{k,i}|s_{k}=\emptyset))=$ $\displaystyle
X(1-c_{k-1})-Xc_{k-1}$ $\displaystyle=$ $\displaystyle X-2Xc_{k-1}$

The agent will sample from the policy-maker or the leader iff it provides the
highest utility of the three methods, assume that
$\lambda_{i}^{P}=1-\phi_{i}^{L}$.

Then the agent prefers sampling the signal of the leader over sticking to the
consensus when

$\displaystyle E(U(\hat{\mu}_{k,i}|s=s_{k}=1))>$ $\displaystyle
E(U(\hat{\mu}_{k,i}|s=s_{k}=\emptyset))$ $\displaystyle\iff
X-2X\phi_{i}^{L}-2X(1-\phi_{i}^{L})c_{k-1}+\sigma_{i}\phi_{i}^{L}>$
$\displaystyle X-2Xc_{k-1}$ $\displaystyle\iff\sigma_{i}>$ $\displaystyle
2X(1-c_{k-1})$

Then the agent prefers sampling the signal of the leader over the policymaker
when

$\displaystyle E(U(\hat{\mu}_{k,i}|s=s_{k}=1))>$ $\displaystyle
E(U(\hat{\mu}_{k,i}|s=s_{k}=0))$ $\displaystyle\iff\sigma_{i}>$
$\displaystyle\frac{2X(\phi_{i}^{L}+c_{k-1}(1-2\phi_{i}^{L}))}{1}$
$\displaystyle\sigma_{i}>$ $\displaystyle
2X(c_{k-1}+\phi_{i}^{L}-2\phi_{i}^{L}c_{k-1})$

Finally, the agent prefers sticking to the consensus over sampling the
policymaker when:

$\displaystyle E(U(\hat{\mu}_{k,i}|s=s_{k}=\emptyset))>$ $\displaystyle
E(U(\hat{\mu}_{k,i}|s=0))$ $\displaystyle\iff X-2Xc_{k-1}>$ $\displaystyle
X-2X(1-\lambda_{i}^{P})c_{k-1}-\sigma_{i}\lambda_{i}^{P}$
$\displaystyle\iff\sigma_{i}>$ $\displaystyle 2Xc_{k-1}$

We can then define :

$\displaystyle\alpha$ $\displaystyle=2X(1-c_{k-1})$ $\displaystyle\beta$
$\displaystyle=2X(c_{k-1}+\phi_{i}^{L}-2\phi_{i}^{L}c_{k-1})$
$\displaystyle\gamma$ $\displaystyle=2Xc_{k-1}$

We will then rank $\alpha,\beta,\lambda$ on $[0,1]$

  1. 1.

Rank $\alpha,\beta$

$\displaystyle\alpha<\beta\iff$ $\displaystyle
1-c_{k-1}<c_{k-1}+\phi_{i}^{L}-2\phi_{i}^{L}c_{k-1}$ $\displaystyle\iff$
$\displaystyle 1-2c_{k-1}<\phi_{i}^{L}(1-2c_{k-1})$ $\displaystyle\iff$
$\displaystyle\phi_{i}^{L}>1,c_{k-1}<\frac{1}{2}\quad\mbox{or}\quad\phi_{i}^{L}<1,c_{k-1}>\frac{1}{2}$
$\displaystyle\iff$ $\displaystyle
c_{k-1}>\frac{1}{2}\quad\mbox{as}\quad\phi_{i}^{L}<1\forall i.$

  2. 2.

Rank $\alpha,\gamma$

$\alpha<\gamma\iff 1-c_{k-1}<c_{k-1}\iff c_{k-1}>\frac{1}{2}$

  3. 3.

Rank $\beta,\gamma$

$\beta<\gamma\iff c_{k-1}+\phi_{i}^{L}-2\phi_{i}^{L}c_{k-1}<c_{k-1}\iff
c_{k-1}>\frac{1}{2}$

We then have two cases

Case 1 $c_{k-1}<\frac{1}{2}$

Then we have $\begin{cases}\alpha>\beta\\\ \alpha>\gamma\\\
\beta>\gamma\end{cases}$

Hence we have $\alpha>\beta>\gamma$

Let $\sigma_{i}\in[0,1]$ where $0\leq\gamma<\beta<\alpha\leq 1$. We evaluate
the preferences on each interval. Define $\mathcal{C=}\\{L,P,C\\}$ to be the
set of choices for the agent where he can of sampling the leader signal, the
policmaker’s and sticking to the cosnensus. Per above each is associate with
an expected utility level so we can establish preferences.

In $[0,\gamma]$, we have $C\gtrsim L,P\gtrsim L$ and $P\gtrsim C$, hence by
transitivity, the agent prefers to sample the policymaker’s signal.

In $(\gamma,\beta]$, we have $C\gtrsim L,P\gtrsim L$ and $C\gtrsim P$, hence
by transitivity, the agent prefers to stick to the consensus.

In $(\beta,\alpha]$, we have $C\gtrsim L,L\gtrsim P$ and $C\gtrsim P$, hence
by transitivity, the agent prefers to stick to the consensus.

In $(\alpha,1]$, we have $L\gtrsim C,L\gtrsim P$ and $C\gtrsim P$, hence by
transitivity, the agent prefers to sample the leader.’s signal.

Therefore we get a complete mapping on $[0,1]$ for the choices of sampling by
the agent :

$\begin{cases}\text{Agent samples the policymaker
signal}&\iff\sigma_{i}\in[0,2Xc_{k-1}]\\\ \text{Agent sticks to the
consensus}&\iff\sigma_{i}\in(2Xc_{k-1},2X(1-c_{k-1})]\\\ \text{Agent samples
the leader signal }&\iff\sigma_{i}\in(2X(1-c_{k-1}),1]\end{cases}$

Consequently we see that as $c_{k-1}\uparrow$, less people choose to stick to
the consensus.

Case 2 $c_{k-1}>\frac{1}{2}$

Then we have $\begin{cases}\alpha<\beta\\\ \alpha<\gamma\\\
\beta<\gamma\end{cases}$

Hence we have $\alpha<\beta<\gamma$

In $[0,\alpha]$, we have $C\gtrsim L,P\gtrsim L$ and $P\gtrsim C$, hence by
transitivity, the agent prefers to sample the policymaker’s signal.

In $(\alpha,\beta]$, we have $L\gtrsim C,P\gtrsim L$ and $P\gtrsim C$, hence
by transitivity, the agent prefers to sample the policymaker’s signal.

In $(\beta,\gamma]$, we have $L\gtrsim C,L\gtrsim P$ and $P\gtrsim C$, hence
by transitivity, the agent prefers to sample the leader’s signal.

In $(\gamma,1]$, we have $L\gtrsim C,L\gtrsim P$ and $C\gtrsim P$, hence by
transitivity, the agent prefers to sample the leader’s signal.

Therefore we get a complete mapping on $[0,1]$ for the choices of sampling by
the agent :

$\begin{cases}\text{Agent samples the policymaker
signal}&\iff\sigma_{i}\in[0,2X(c_{k-1}+\phi_{i}^{L}-2\phi_{i}^{L}c_{k-1})]\\\
\text{Agent sticks to the leader signal
}&\iff\sigma_{i}\in(2X(c_{k-1}+\phi_{i}^{L}-2\phi_{i}^{L}c_{k-1}),1]\end{cases}$

Consequently when $c_{k-1}>\frac{1}{2}$, the choice space becomes completely
polarised, and agents adopt either the policymaker’s signal or the leader
signal.

### Proof of Proposition 5

Part 1. _When the punishment value $X$ is such that $X\geq\frac{1}{2}$, agents
never sample the leader’s signal and $\forall k\in\mathbb{N}$ the consensus
belief $c_{k}=0$._

###### Proof.

Let’s prove that if $X\geq\frac{1}{2}$, then $\forall k\in\mathbb{N}$ the
consensus belief remains $c_{k}=0$.

We will prove the proposition by induction.

Let $X\geq\frac{1}{2}$ and suppose $c_{k-1}=0$, $k-1\in\mathbb{N}$. We will
show that no agent will sample from the leader nor the policy-maker. Thus, the
consensus belief will remain $c_{k}=0$.

From the Proof of Proposition 4, with $c_{k-1}=0$, we write that agents will
sample the leader’s signal:

$\iff\sigma_{i}>2X(1-c_{k-1})=2X\geq 1.$

We also note that agents will sample the policy-makers’s signal:

$\iff\sigma_{i}\leq 2Xc_{k-1}=0.$

Recall that:

$\forall i\in n,\sigma_{i}\in(0,1).$

Thus, no agent sample from the leader nor the policy-maker at timepoint $k$.
Each agent will stick to the consensus and hold a belief
$\mu_{i,k}=c_{k-1}=0$.

Therefore,

$c_{k}=\sum_{i=1}^{n}w_{i}\mu_{i,k}=0.$

We proved that:

$\forall k\in\mathbb{N},X\geq\frac{1}{2},c_{k-1}=0\implies c_{k}=0.$

We note that the consensus belief at timepoint $k=0$ is such that $c_{k}=0$.
Thus $c_{1}=0$ and we proved by induction that:

$X\geq\frac{1}{2}\implies\forall k\in\mathbb{N},c_{k}=0.$

∎

Part 2. _For $X<\frac{1}{2}$, if we assume that $\sigma_{i}$ and
$\phi_{i}^{L}$ are realizations of i.i.d uniformly distributed random
variables $\Sigma$ and $\Phi^{L}$ over $(0,1)$ respectively, and let the
number of agents $n$ tend to infinity, then the sequence of consensus belief
$c_{k}=f(c_{k-1})\forall k\in\mathbb{N^{*}}$ starting at $c_{0}=0$ is
monotonically increasing and converges to $c_{\infty}=1-X$._

###### Proof.

We will follow three steps to prove Proposition 2: 1) we will show that by
assuming $\sigma_{i}$ and $\phi_{i}^{L}$ are realizations of i.i.d uniformly
distributed random variables $\Sigma$ and $\Phi^{L}$ over $(0,1)$
respectively, and letting the number of agents $n$ tend to infinity, we can
express the consensus sequence $c_{k}=f(c_{k-1})\forall k\in\mathbb{N}$ as a
quadratic and linear function of $c_{k-1}$ for $c_{k-1}\in[0,\frac{1}{2}]$ and
$c_{k-1}\in(\frac{1}{2},1]$ respectively. 2) These expressions will allow us
to prove that for $X<\frac{1}{2}$, the consensus belief sequence is
monotonically increasing on $[0,1-X)$. 3) We will then be able to conclude on
the convergence of the sequence to $c_{\infty}=1-X$.

We will first use the results from Proof of Proposition 4 to express for all
$k$ in $\mathbb{N}$ the sequence $c_{k}=f(c_{k-1})$ as a quadratic and linear
function of $c_{k-1}$ for $c_{k-1}\in[0,\frac{1}{2}]$ and
$c_{k-1}\in(\frac{1}{2},1]$ respectively.

Recall that the consensus belief is defined as:

$c_{k}=\sum_{i=1}^{n}w_{i}\mu_{i,k},\forall k\in\mathbb{N},$

where $\mu_{i,k}$ represents the hatred belief of agent $i$ at timepoint $k$
and $w_{i}$ is fixed $\forall i$ such that $\forall i\in n,w_{i}>0\mbox{ and
}\sum_{i=1}^{n}w_{i}=1$.

To explore the dynamics of the consensus belief sequence, we will express
$c_{k}$ as a function of the parameters $\sigma_{i},\phi_{i}^{L}$ for agents
$i$, the constant $X$, and the previous consensus belief $c_{k-1}$.

The Proof of Proposition 4 allows us to express $f(c_{k-1})$ over the domain
$c_{k-1}\in[0,\frac{1}{2}]$ as:

$f(c_{k-1})=\sum_{i,\sigma_{i}\leq
2Xc_{k-1}}\left[w_{i}\phi_{i}^{L}c_{k-1}\right]+\sum_{i,2Xc_{k-1}<\sigma_{i}\leq
2X(1-c_{k-1})}\left[w_{i}c_{k-1}\right]+\sum_{i,\sigma_{i}>2X(1-c_{k-1})}\left[w_{i}\left(\phi_{i}^{L}+(1-\phi_{i}^{L})c_{k-1}\right)\right].$
(1)

Similarly, over the domain $c_{k-1}\in(\frac{1}{2},1]$, we can write:

$f(c_{k-1})=\sum_{i,\sigma_{i}\leq
2X(\phi_{i}^{L}+c_{k-1}-2\phi_{i}^{L}c_{k-1})}\left[w_{i}\phi_{i}^{L}c_{k-1}\right]+\sum_{i,\sigma_{i}>2X(\phi_{i}^{L}+c_{k-1}-2\phi_{i}^{L}c_{k-1})}\left[w_{i}\left(\phi_{i}^{L}+(1-\phi_{i}^{L})c_{k-1}\right)\right].$
(2)

We will now apply the assumption that $\sigma_{i}$ and $\phi_{i}^{L}$ are
realizations of i.i.d uniformly distributed random variables $\Sigma$ and
$\Phi^{L}$ respectively, to simplify Equations (1,2).

From this assumption, $c_{k}$ is the sum of i.i.d random variables and is
itself a random variable. We can thus apply the weak law of large numbers as
the number of agents $n\rightarrow\infty$. From now on, we let
$n\rightarrow\infty$. We can write: $c_{k}=\mathbb{E}(c_{k})$. We will now
show that this allows us to express the function $f$ over the domain
$[0,\frac{1}{2}]$ and $(\frac{1}{2},1]$ as a quadratic and linear function of
$c_{k-1}$ respectively.

We will start by taking the expected values of $c_{k}$ as defined in Equations
(1,2) respectively, and simplify the expression for $f$ over $[0,1]$. In order
to compute these expected values, we will remove the dependencies on
$\sigma_{i}$ of the sum terms in Equation (1,2) and express $f$ as a single
sum over all agents $i\in N$. To do so, we let, $H$ be the Heaviside function
defined over $\mathbb{R}$ such that:

$H(x):=\begin{cases}0&x\leq 0\\\ 1&x>0\end{cases}.$

For clarity, we will compute the expected values of Equation (1) and Equation
(2) separately.

We now consider $f$ in the region $c_{k-1}\in[0,\frac{1}{2}]$ defined by
Equation (1) and make use of the Heaviside function $H$ to compute the
expected value for $c_{k}.$ Starting from Equation (1), we write:

$\displaystyle f(c_{k-1})=$
$\displaystyle\mathbb{E}\left[\sum_{i=1}^{n}w_{i}\left[\phi_{i}^{L}c_{k-1}+c_{k-1}(1-\phi_{i}^{L})H\left(\sigma_{i}-2Xc_{k-1}\right)+\phi_{i}^{L}(1-c_{k-1})H\left(\sigma_{i}-2X(1-c_{k-1})\right)\right]\right]$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}w_{i}\mathbb{E}\left[\Phi^{L}c_{k-1}+c_{k-1}(1-\Phi^{L})H\left(\Sigma-2Xc_{k-1}\right)+\Phi^{L}(1-c_{k-1})H\left(\Sigma-2X(1-c_{k-1})\right)\right]$
(3) $\displaystyle=$
$\displaystyle\mathbb{E}\left[\Phi^{L}c_{k-1}+c_{k-1}(1-\Phi^{L})H\left(\Sigma-2Xc_{k-1}\right)+\Phi^{L}(1-c_{k-1})H\left(\Sigma-2X(1-c_{k-1})\right)\right].$

To simplify Equation (3), we note that the expected value of $H(\Sigma-a)$ for
$a\in\mathbb{R}$ is equal to the probability of the event $\Sigma>a$. When
$a\in[0,1]$ and $\Sigma\sim\mathcal{U}(0,1)$ :

$\mathbb{E}\left[H(\Sigma-a)\right]=\mathbb{P}(\Sigma>a)=1-\mathbb{P}(\Sigma\leq
a)=1-a$

We are now ready to express $f(c_{k-1})$ in the range
$c_{k-1}\in[0,\frac{1}{2}]$ as a quadratic function of $c_{k-1}$ :

$\displaystyle f(c_{k-1})=$ $\displaystyle
c_{k-1}\left(\frac{1}{2}+(1-\frac{1}{2})(1-2Xc_{k-1})-\frac{1}{2}\left(1-2X(1-c_{k-1})\right)\right)+\frac{1}{2}\left(1-2X(1-c_{k-1})\right)$
(4) $\displaystyle=$
$\displaystyle-2Xc_{k-1}^{2}+(2X+\frac{1}{2})c_{k-1}+\frac{1}{2}-X.$

Equation (4) will allow us to study the dynamics of the consensus belief
$c_{k}$ as $k$ increases in the range $c_{k-1}\in[0,\frac{1}{2}]$ .

The next step of the analysis is to simplify the expression for $f$ as defined
over the range $c_{k-1}\in(\frac{1}{2},1]$ by Equation (2). We will then be
ready to explore the dynamics of the consensus belief in the range
$c_{k-1}\in[0,1]$.

We will now express $f$ as defined in Equation (2) as a linear function of
$c_{k-1}$ by using the Heaviside function and computing the expected value of
$c_{k}$.

$\displaystyle f(c_{k-1})=$
$\displaystyle\mathbb{E}\left[\sum_{i=1}^{n}w_{i}\left[\phi_{i}^{L}c_{k-1}+\left(\phi_{i}^{L}+(1-2\phi_{i}^{L})c_{k-1}\right)H\left(\sigma_{i}-2X(\phi_{i}^{L}+c_{k-1}-2\phi_{i}^{L}c_{k-1})\right)\right]\right]$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}w_{i}\mathbb{E}\left[\Phi^{L}c_{k-1}+\left(\Phi^{L}+(1-2\Phi^{L})c_{k-1}\right)H\left(\Sigma-2X(\Phi^{L}+c_{k-1}-2\Phi^{L}c_{k-1})\right)\right]$
(5) $\displaystyle=$
$\displaystyle\mathbb{E}\left[\Phi^{L}c_{k-1}+\left(\Phi^{L}+(1-2\Phi^{L})c_{k-1}\right)H\left(\Sigma-2X(\Phi^{L}+c_{k-1}-2\Phi^{L}c_{k-1})\right)\right].$

We must now compute the expected value expressed by Equation (5).
Specifically, we must calculate
$\mathbb{E}\left[H\left(\Sigma-2X(\Phi^{L}+c_{k-1}-2\Phi^{L}c_{k-1})\right)\right]$.
To do so, we let $T$ be the random variable defined as:

$T=2X(c_{k-1}+\Phi^{L}(1-2c_{k-1})).$

We will now prove that $T$ is a uniformly distributed random variable over
$[2X(1-c_{k-1}),2Xc_{k-1}]$, which will allow us to prove that

$\forall X\in[0,\frac{1}{2}),\forall
c_{k-1}\in(\frac{1}{2},1],\mathbb{E}\left[H\left(\Sigma-2X(\Phi^{L}+c_{k-1}-2\Phi^{L}c_{k-1})\right)\right]=1-X.$

We first note that:

$X=0,\Sigma\sim\mathcal{U}(0,1)\implies\mathbb{E}\left[H\left(\Sigma-2X(\Phi^{L}+c_{k-1}-2\Phi^{L}c_{k-1})\right)\right]=\mathbb{E}\left[H\left(\Sigma-0)\right)\right]=1=1-X.$

Then, we note that:

$X\in(0,\frac{1}{2}),c_{k-1}\in(\frac{1}{2},1]\implies\frac{dT}{d\Phi^{L}}=2X(1-2c_{k-1})<0.$

Thus $T$ is defined by a monotonically decreasing function of $\Phi^{L}$.
$\Phi^{L}\sim\mathcal{U}(0,1)$, therefore $T$ is uniformly distributed over
its range. The boundaries of $T$ are:

$\displaystyle T(\Phi^{L}=0)$ $\displaystyle=2Xc_{k-1}.$ $\displaystyle
T(\Phi^{L}=1)$ $\displaystyle=2X(1-c_{k-1}).$

We note that:

$X\in(0,\frac{1}{2}),c_{k-1}>\frac{1}{2}\implies 2Xc_{k-1}>2X(1-c_{k-1}).$

Therefore, $T\sim\mathcal{U}(2X(1-c_{k-1}),2Xc_{k-1})$. By letting $f_{T}(y)$
be the density function of the random variable $T$ we write:

$f_{T}(y)=\begin{cases}\frac{1}{2Xc_{k-1}-2X(1-c_{k-1})}=\frac{1}{2X(2c_{k-1}-1)},&\mbox{if
}y\in[2X(1-c_{k-1}),2Xc_{k-1}],\\\ 0,&\mbox{otherwise.}\end{cases}$

We are now ready to compute
$\mathbb{E}\left[H\left(\Sigma-2X(\Phi^{L}+c_{k-1}-2\Phi^{L}c_{k-1})\right)\right]$.
We let $F_{\Sigma}$ be the cumulative distribution function of the random
variable $\Sigma\sim\mathcal{U}(0,1)$ and write:

$\displaystyle\mathbb{E}\left[H(\Sigma-T)\right]=$ $\displaystyle
1-\mathbb{P}(\Sigma\leq T)$ (6) $\displaystyle=$ $\displaystyle
1-\int_{2X(1-c_{k-1})}^{2Xc_{k-1}}F_{\Sigma}(y)f_{T}(y)\mathrm{{d}y}$
$\displaystyle=$ $\displaystyle
1-\frac{1}{2X(2c_{k-1}-1)}\int_{2X(1-c_{k-1})}^{2Xc_{k-1}}F_{\Sigma}(y)\mbox{dy}.$

To calculate the integral in Equation (6) we note that

$X\in[0,\frac{1}{2}),c_{k-1}\in[\frac{1}{2},1]\implies[2X(1-c_{k-1}),2Xc_{k-1}]\subseteq[0,1].$

Thus:

$\displaystyle\mathbb{E}\left[H(\Sigma-T)\right]=$ $\displaystyle
1-\frac{1}{2X(2c_{k-1}-1)}\int_{2X(1-c_{k-1})}^{2Xc_{k-1}}y\mbox{dy}$
$\displaystyle=$ $\displaystyle
1-\frac{1}{2X(2c_{k-1}-1)}\cdot\frac{(2X)^{2}\cdot(c_{k-1}^{2}-(1-c_{k-1})^{2})}{2}$
$\displaystyle=$ $\displaystyle 1-\frac{2X(2c_{k-1}-1)}{2(2c_{k-1}-1)}$
$\displaystyle=$ $\displaystyle 1-X.$

Therefore we proved that:

$\forall X\in[0,\frac{1}{2}),\forall
c_{k-1}\in(\frac{1}{2},1],\mathbb{E}\left[H\left(\Sigma-2X(\Phi^{L}+c_{k-1}-2\Phi^{L}c_{k-1})\right)\right]=1-X.$

We can now simplify the expression for $f$ defined in Equation (5) for
$c_{k-1}\in(\frac{1}{2},1]$ as a linear function of $c_{k-1}$:

$\displaystyle f(c_{k-1})=$
$\displaystyle\mathbb{E}\left[\Phi^{L}c_{k-1}+\left(\Phi^{L}+(1-2\Phi^{L})c_{k-1}\right)H\left(\Sigma-2X(\Phi^{L}+c_{k-1}-2\Phi^{L}c_{k-1})\right)\right]$
(7) $\displaystyle=$
$\displaystyle\frac{1}{2}c_{k-1}+(\frac{1}{2}+(1-2\frac{1}{2})c_{k-1})(1-X)$
$\displaystyle=$ $\displaystyle\frac{1}{2}c_{k-1}+\frac{1-X}{2}.$

Equation (7) will allow us to study the dynamics of the consensus belief
$c_{k}$ in the range $c_{k-1}\in(\frac{1}{2},1]$.

The sequence of consensus belief $c_{k}$ is now defined by Equation (4) and
Equation (7) on $c_{k-1}\in[0,\frac{1}{2}]$ and $c_{k-1}\in(\frac{1}{2},1]$
respectively. This will allow us to enter the second step of the proof and
show that the sequence $c_{k}=f(c_{k-1})$ starting at $c_{0}=0$ is
monotonically increasing on $c_{k-1}\in[0,1-X).$ We will then prove that the
$\lim_{k\rightarrow\infty}c_{k}=1-X$.

To prove that $c_{k}$ is monotonically increasing with $k$ on the region
$c_{k-1}\in[0,1-X)$, we will first prove that the consensus belief increases
montonically up to a value $c_{k^{*}}$ at a period $k^{*}\in\mathbb{N}$ such
that $c_{k^{*}}>\frac{1}{2}$. At that timepoint, it will enter in the region
where the sequence is defined by Equation (7). We will then prove that it
increases monotonically and converges to $c_{\infty}=1-X$.

Let’s prove that $\forall X\in[0,\frac{1}{2})$, the consensus belief sequence
is monotonically increasing in the range $c_{k-1}\in[0,\frac{1}{2}]$, and
reaches a value $c_{k^{*}}$ at a period $k^{*}\in\mathbb{N}$ such that
$c_{k^{*}}>\frac{1}{2}$.

From Equation (4), we note that:

$\displaystyle c_{k}-c_{k-1}$
$\displaystyle=-2Xc_{k-1}^{2}+(2X+\frac{1}{2})c_{k-1}+\frac{1}{2}-X-c_{k-1}$
(8) $\displaystyle=-2Xc_{k-1}^{2}+(2X-\frac{1}{2})c_{k-1}+\frac{1}{2}-X.$

We will now prove that:

$X\in[0,\frac{1}{2}),c_{k-1}\in[0,\frac{1}{2}]\implies c_{k}>c_{k-1}.$

First, we observe that for $X=0$:

$\forall c_{k-1}\in[0,\frac{1}{2}],X=0\implies
c_{k}-c_{k-1}=\frac{1}{2}(1-c_{k-1})>0.$

Thus, for $X=0$, the sequence for $c_{k}$ is increasing for
$c_{k-1}\in[0,\frac{1}{2}]$. Let’s now prove it is also the case for
$X\in(0,\frac{1}{2})$.

We note that on $c_{k-1}\in[0,\frac{1}{2}]$:

$c_{k}-c_{k-1}=0\iff-2Xc_{k-1}^{2}+(2X-\frac{1}{2})c_{k-1}+\frac{1}{2}-X=0.$
(9)

We can now show that the quadratic function defined by Equation (9) is
striclty positive for $c_{k-1}\in[0,\frac{1}{2}]$, and thus that the consensus
sequence $c_{k}$ is monotonically increasing for $c_{k-1}\in[0,\frac{1}{2}]$.

When $X\in(0,\frac{1}{2})$, the discriminant of Equation (9) is
$\Delta=-4X^{2}+2X+\frac{1}{4}$, and $\Delta\in(\frac{1}{4},\frac{1}{2}]$.
Equation (9) admits real solutions $c_{1},c_{2}$ of the form

$c_{1}=\frac{(2X-\frac{1}{2})-\sqrt{-4X^{2}+2X+\frac{1}{4}}}{4X}\mbox{ and
}c_{2}=\frac{(2X-\frac{1}{2})+\sqrt{-4X^{2}+2X+\frac{1}{4}}}{4X}.$

By substituting the extreme values for $\Delta,$ we observe that $c_{1}<0$ and
$c_{2}>\frac{1}{2}$. Therefore we observe that for
$c_{k-1}\in[0,\frac{1}{2}]$, $c_{k}-c_{k-1}$ is defined by a quadratic
function with a negative coefficient for the 2nd order polynomial term, whose
roots are $c_{1}<0$ and $c_{2}>\frac{1}{2}$. Therefore:

$X\in(0,\frac{1}{2}),c_{k-1}\in[0,\frac{1}{2}]\implies c_{k}-c_{k-1}>0.$

We proved that:

$X\in[0,\frac{1}{2}),c_{k-1}\in[0,\frac{1}{2}]\implies c_{k}>c_{k-1}.$

The sequence $c_{k}$ is thus monotonically incerasing for
$c_{k-1}\in[0,\frac{1}{2}]$.

Let’s now show that the sequence of consensus belief $c_{k}$ will reach a
value $c_{k^{*}}$ at period $k^{*}$ such that $c_{k^{*}}>\frac{1}{2}$ with
$c_{k^{*}-1}\leq\frac{1}{2}$. This will allow us to show that the sequence
enters the region $c_{k-1}>\frac{1}{2}$, in which it will converge to
$c_{\infty}=1-X$.

First, for $X=0$, we notice that

$c_{k}=\frac{1}{2}-X+\frac{1}{2}c_{k-1}=\frac{1}{2}(1+c_{k-1})\implies
c_{1}=\frac{1}{2}\implies c_{2}>\frac{1}{2}.$

Thus for $X=0$, the sequence of consensus belief $c_{k}$ reaches a value
$c_{k^{*}}$ at period $k^{*}\in\mathbb{N}$ such that $c_{k^{*}}>\frac{1}{2}$.

Let’s prove that the statement holds $\forall X\in(0,\frac{1}{2})$.

For $c_{k-1}\in[0,\frac{1}{2}]$, $f$ is a polynomial function of $c_{k-1}$
with a negative coefficient for the 2nd order polynomial term. Therefore, $f$
is monotonically increasing on the region
$(-\infty,\operatorname*{arg\,max}_{c_{k-1}}f(c_{k-1}))$ . We notice that:

$X\in(0,\frac{1}{2})\implies\operatorname*{arg\,max}_{c_{k-1}}f(c_{k-1})=\frac{1}{2}+\frac{1}{8X}>\frac{1}{2}.$

Thus $\forall X\in(0,\frac{1}{2})$, $f$ is monotonically increasing on
$c_{k-1}\in(-\infty,\frac{1}{2}+\frac{1}{8X})$, and specifically in
$[0,\frac{1}{2}]$. Thus:

$\forall
X\in(0,\frac{1}{2}),c_{k-1}\in[0,\frac{1}{2}]\implies\operatorname*{max}f(c_{k})=f(\frac{1}{2})=\frac{3}{4}-\frac{X}{2}>\frac{1}{2}.$
(10)

Therefore, for $c_{k}$ to remain smaller than $\frac{1}{2}$ for all
$k\in\mathbb{N}$, the sequence must converge to a point
$c_{k-1}\in[0,\frac{1}{2}]$. Recall that we proved Equation (8) admits no
solution in $c_{k-1}\in[0,\frac{1}{2}]$. Therefore, for
$c_{k-1}\in[0,\frac{1}{2}]$ there exist no fixed point of $f$, and the
consensus belief cannot converge to a value $\in[0,\frac{1}{2}]$. Thus,
because the sequence of consensus beliefs $c_{k}$ is monotonically increasing
for $c_{k-1}\in[0,\frac{1}{2}]$, and that for $c_{k-1}\in[0,\frac{1}{2}]$,
$\operatorname*{max}f(c_{k})>\frac{1}{2}$, there exists a period
$k^{*}\in\mathbb{N}$ such that $c_{k^{*}}>\frac{1}{2}$.

We will now prove that the consensus belief then increases monotonically on
$c_{k-1}\in(\frac{1}{2},1-X)$ and converges to $c_{\infty}=1-X$.

We will start by proving that the sequence of consensus belief is
monotonically increasing for $c_{k-1}\in(\frac{1}{2},1-X)$.

From Equation (2), we note that:

$c_{k}-c_{k-1}=\frac{1}{2}c_{k-1}+\frac{1-X}{2}-c_{k-1}=\frac{1-X}{2}-\frac{1}{2}c_{k-1}.$
(11)

Thus,

$c_{k}-c_{k-1}>0\iff\frac{1-X}{2}-\frac{1}{2}c_{k-1}>0\iff c_{k-1}<1-X.$ (12)

Therefore, we conclude by induction that $\forall k\in\mathbb{N},k\geq
k^{*}\implies c_{k}<1-X$ and the sequence of consensus belief $c_{k}$ is
monotonically increasing on $c_{k-1}\in(\frac{1}{2},1-X]$.

We will now complete the proof by showing that $c_{k}$ is bounded above by
$1-X$, which allows us to prove the third step described, namely that
$\lim_{k\rightarrow\infty}c_{k}=1-X$.

We use the result from Equation (12) to prove by induction that:

$\forall k\in\mathbb{N},c_{k-1}\in(\frac{1}{2},1],c_{k-1}<1-X\implies
c_{k}<1-X.$

Suppose that $c_{k-1}<1-X$, then:

$c_{k}<\frac{1}{2}(1-X)+\frac{1-X}{2}\implies c_{k}<1-X.$

We now verify that the relation holds for $c_{k^{*}}$, the first term such
that $c_{k}>\frac{1}{2}$. From Equation (10), we note that $c_{k^{*}}$ is
bounded such that:

$c_{k^{*}}\leq\frac{3}{4}-\frac{X}{2}.$

We observe that:

$\frac{3}{4}-\frac{X}{2}<1-X\iff X<\frac{1}{2}.$

Thus:

$X\in[0,\frac{1}{2})\implies c_{k^{*}}<1-X.$

The consensus sequence is thus bounded above by $1-X$. Therefore, it is
monotonically incereasing and bounded above, which implies its limit
$\lim_{k\rightarrow\infty}c_{k}$ exists. Let $c_{\infty}$ be this limit. We
can express $c_{\infty}$ as:

$\lim_{k\rightarrow\infty}c_{k}=\lim_{k\rightarrow\infty}c_{k-1}\implies\frac{1}{2}c_{\infty}+\frac{1-X}{2}=c_{\infty}\implies
c_{\infty}=1-X.$

We proved that, $\forall X\in(0,\frac{1}{2})$, by assuming $\sigma_{i}$ and
$\phi_{i}^{L}$ are realizations of i.i.d uniformly distributed random
variables $\Sigma$ and $\Phi^{L}$ over $(0,1)$ respectively, and letting the
number of agents $n$ tend to infinity, the consensus belief starting at
$c_{0}=0$ is monotonically increasing with $k\in\mathbb{N}$ and converges to
$c_{\infty}=1-X$. ∎

Part 3._f is a piecewise linear function of $M+1\in\mathbb{N}$ intervals
$I_{1},...,I_{M+1}$, whose range are defined by the $M$ discontinuities of $f$
over $[0,1]$. The discontinuites repesent the unique threshold values of
$c_{k-1}$ for which at least one agent $i$ changes the signal it samples. The
M+1 pairwise disjoint intervals are such that $\cup_{i=1}^{M+1}I_{i}=[0,1]$.
Some of these interval $I^{*}\subset I$ admit fixed points. If the consensus
belief converges to a fixed point, then it will converge to the fixed point of
the first interval it visits which belongs to I*._

###### Proof.

We are interested in providing explicit equations for the fixed points $c$ of
each of the linear functions which define $f$ in its continuous intervals. The
discontinuities of $f$ repesent the unique threshold values of $c_{k-1}$ for
which at least one agent $i$ changes opinion. To provide an explicit equation
for each fixed point, we will first proceed in three steps: 1) group the
agents based on their change in signal sampling as the consensus increases,
and define $t_{i}$ as the threshold value of $c_{k-1}$ for agent $i$ for such
a change to occur, 2) order the agents in each of these respective groups such
that $i<j\implies t_{i}<t_{j}$, 3) express the equation for the fixed point
$c$ of each of these intervals. We will then prove that intervals $I^{*}$ of
$[0,1]$ which admit a fixed point are such that $f(I^{*})\subset I^{*}$. This
will allow us to prove that, if the consensus belief converges to a fixed
point $c$, then the fixed point is that of the linear function defining $f$ in
the first visited interval $I^{*}$.

We will start by providing explicit equations for the fixed points of the
linear functions which define each interval of $f$.

From Proof of Proposition 5 part 2, we know that over $c_{k-1}\in[0,1]$, $f$
is defined as a piecewise linear function expressed by Equations (1,2) for
$c_{k-1}\in[0,\frac{1}{2}]$ and $c_{k-1}\in(\frac{1}{2},1]$ respectively. We
will thus proceed with our three step analysis for the range
$c_{k-1}\in[0,\frac{1}{2}]$ and $c_{k-1}\in(\frac{1}{2},1]$ separately.

Let’s proceed with the three defined steps for the region
$c_{k-1}\in[0,\frac{1}{2}]$.

1) We will show that we can group all agents in $i$ into three sub-groups as
$c_{k-1}$ increases in $[0,\frac{1}{2}]$: a) agents who always sample the
leader’s signal, b) agents who stick to the consensus before always sampling
from the policy-maker when a threshold value for $c_{k-1}$ is reached, c)
agents who stick to the consensus before always sampling from the leader when
a threshold value for $c_{k-1}$ is reached.

From Proof of Proposition 4, we recall that:

$\begin{cases}0\leq\sigma_{i}\leq 2Xc_{k-1},&\mbox{the agent samples the
policy-maker,}\\\ 2Xc_{k-1}<\sigma_{i}\leq 2X(1-c_{k-1}),&\mbox{the agent
sticks to the consensus,}\\\ 2X(1-c_{k-1})<\sigma_{i}\leq 1,&\mbox{the agent
samples the leader.}\end{cases}$

To show that we can group the agents in a), b), and c), we first show that at
timepoint $k=1$, agents have either stuck to the consensus or sampled from the
leader:

$\forall i\in N,\sigma_{i}>0\mbox{ and }c_{0}=0\implies\sigma_{i}>2Xc_{0},$

and no agent sampled from the policy-maker. Hence at period $k=1$, agents
stuck to the consensus or sampled from the leader. This will allow us to group
the agents in the sub-groups as defined in a), b) and c).

a) Some agents will always sample from the leader’s signal as $c_{k-1}$
increases in $[0,\frac{1}{2}]$. Indeed, an agent $i$ will sample the leader at
period 1 if $\sigma_{i}>2X$. We notice that the threshold $2X(1-c_{k-1})$
decreases as $c_{k-1}$ increases in $\mathbb{R}$ and escpecially on
$[0,\frac{1}{2}]$. Thus, all agents $i$ such that $\sigma_{i}>2X$ will sample
from the leader for all $c_{k-1}\in[0,\frac{1}{2}]$ . Let
$\mathcal{A}=\\{i|\sigma_{i}>2X\\}$. Without loss of generality, we relabel
the agents in $\mathcal{A}$ from $1$ to $|\mathcal{A}|=A$.

b) We will now show that some agents will change from sticking to the
consensus to sampling the policy-maker as $c_{k-1}$ increases from $0$ to
$\frac{1}{2}$. By observing that:

$\forall c_{k-1}\in[0,\frac{1}{2}],2X(1-c_{k-1})\in[X,2X],$

we state that:

$\forall c_{k-1}\in[0,\frac{1}{2}],\forall i,\sigma_{i}\leq
X\implies\sigma_{i}\leq 2X(1-c_{k-1}).$

Therefore, for $c_{k-1}\in[0,\frac{1}{2}]$, agents such that $\sigma_{i}\leq
X$ will never sample the leader. In the limit
$c_{k-1}\rightarrow\frac{1}{2},2Xc_{k-1}=2X(1-c_{k-1})$ and these agents will
therefore change from sticking to the consenus to sampling from the polic-
maker as $c_{k-1}$ increases in $[0,\frac{1}{2}]$. For each of these agents
$i$, the change will occur at a threshold value $t_{i}$ for $c_{k-1}$, such
$t_{i}=\frac{\sigma_{i}}{2X}$. Let $\mathcal{B}$ be the set of cardinality $B$
of agents $i$ such that $\sigma_{i}\leq X$. Without loss of generality, we
label the agents in $\mathcal{B}$ from $A+1$ to $A+B$.

c) We will now show that some agents will change from sticking to the
consensus to sampling the leader as $c_{k-1}$ increases from $0$ to
$\frac{1}{2}$. By observing that:

$\forall c\in[0,\frac{1}{2}],2Xc\in[0,X],$

we state that:

$\forall c_{k-1}\in[0,\frac{1}{2}],\forall
i,\sigma_{i}>X\implies\sigma_{i}>2Xc_{k-1}.$

Therefore, for $c_{k-1}\in[0,\frac{1}{2}]$, agents such that $\sigma_{i}>X$
will never sample from the policy-maker. Using the same argument as in point
b), as $c_{k-1}$ increases in $[0,\frac{1}{2}]$, these agents will transfer
from sticking to the consensus to sampling from the leader. For each of these
agents $i$, the change will occur at a threshold
$t_{i}=1-\frac{\sigma_{i}}{2X}$. Let $\mathcal{D}$ be the set of cardinality
$D$ of agents such that $\sigma_{i}>X$. Without loss of generality, we label
the agents in $\mathcal{D}$ from $A+B+1$ to $A+B+D$.

Now that the agents who exhibit the same behaviour as $c_{k-1}$ increase in
$[0,\frac{1}{2}]$ are grouped, we are ready to reorder these agents within
each group. This will allow us to provide explicit expressions for the fixed
points $c$ of each of the linear functions which define the intervals of $f$.

2) Without loss of generality, we re-order the agents in $\mathcal{B}\mbox{
and }\mathcal{D}$, such that within each group $i<j\implies t_{i}\leq t_{j}$.

We will now express the explicit equations for the fixed points of the linear
functions defining $f$ for $c_{k-1}\in[0,\frac{1}{2}]$.

3) Let $\mathcal{E}$ be the set of agents who change opinion as $c_{k-1}$
increases in $[0,\frac{1}{2}]$ : $\mathcal{E}:=\mathcal{B}\cup\mathcal{D}$. We
order the set of threshold values $t_{i}$ for all $i$ in $\mathcal{E}$, such
that $i<j\implies t_{i}\leq t_{j}$. We have $E=|\mathcal{E}|=B+D$. Let
$I_{1},...,I_{B+D+1}$ be the $E$ pairwise disjoint intervals such that
$\cup_{i=1}^{B+D+1}I_{i}=[0,\frac{1}{2}]$, where $t_{1},...,t_{B+D}$ are the
discontinuities of the function, $t_{0}=0$ and $t_{B+D+1}=\frac{1}{2}$. We can
now define the equation for the consensus update in the interval
$I_{m}=(t_{m},t_{m+1}]$, where $1\leq m\leq B+D$. We let $b$ be the number of
agents $i$ in $\mathcal{B}$ such that $t_{i}\leq t_{m}$, and $d$ the number of
agents $i$ in $\mathcal{D}$ such that $t_{i}\leq t_{m}$. We define
$f_{m}:=f(c_{k-1}),\forall c_{k-1}\in I_{m}$. According to Equation (1) we can
express that:

$\displaystyle f_{m}(c_{k-1})$
$\displaystyle=\sum_{i=1}^{A}w_{i}(\phi_{i}^{L}+(1-\phi_{i}^{L})c_{k-1})+\sum_{i=A+1}^{A+b}w_{i}\phi_{i}c_{k-1}+\sum_{i=A+b+1}^{A+B}w_{i}c_{k-1}$
$\displaystyle+\sum_{i=A+B+1}^{A+B+d}w_{i}(\phi_{i}^{L}+(1-\phi_{i}^{L})c_{k-1})+\sum_{i=A+B+d+1}^{A+B+D}w_{i}c_{k-1}.$

We can now write an explicit equation for the fixed poins of the linear
intervals of $f$ for $c_{k-1}\in[0,\frac{1}{2}]$. A point $c$ must be such
that $c\in I_{m}$ and $f_{m}(c)=c$. We express the latter condition as:

$c=\frac{\sum_{i=1}^{A}w_{i}\phi_{i}^{L}+\sum_{i=A+B+1}^{A+B+d}w_{i}\phi_{i}^{L}}{1-\left[\sum_{i=1}^{A}w_{i}(1-\phi_{i}^{L})+\sum_{i=A+1}^{A+b}w_{i}\phi_{i}+\sum_{i=A+b+1}^{A+B}w_{i}+\sum_{i=A+B+1}^{A+B+d}w_{i}(1-\phi_{i}^{L})+\sum_{i=A+B+d+1}^{A+B+D}w_{i}\right]}.$
(13)

We expressed the equation and conditions for a fixed point to exist in the
region $c_{k-1}\in[0,\frac{1}{2}]$. We will now repeat the anlysis on the
range $c_{k-1}\in(\frac{1}{2},1]$ before showing that if these fixed points
exist for $f$ over $c_{k-1}\in[0,1]$, then they are stable and the sequence of
consensus belief converges to the fixed point $c$ of the first visited
interval which admits a fixed point.

Let’s repeat the analysis for $f$ defined on $c_{k-1}\in(\frac{1}{2},1]$ by
Equation (7).

1) We will start by showing that we can split the agents into four distinct
sub-groups as $c_{k-1}$ increases over $(\frac{1}{2},1]$: a) agents who will
always sample from the policy-maker, b) agents who will always sample from the
leader, c) agents who will transfer from sampling the leader to sampling the
policy-maker when a threshold value for $c_{k-1}$ is reached, and d) agents
who will transfer from sampling the policy-maker to sampling the leader when a
threshold value for $c_{k-1}$ is reached.

We will now understand the behaviour of agents within
$c_{k-1}\in(\frac{1}{2},1]$, to subsequently group them.

From Proof of Proposition 4, we can write:

$\begin{cases}0\leq\sigma_{i}\leq
2X(\phi_{i}^{L}+c_{k-1}-2\phi_{i}^{L}c_{k-1}),&\mbox{the agent samples the
policy-maker},\\\ 2X(\phi_{i}^{L}+c_{k-1}-2\phi_{i}^{L}c)<\sigma_{i}\leq
1,&\mbox{the agent samples the leader.}\end{cases}$

Agents will change opinion if the order of $\sigma_{i}$ and
$2X(\phi_{i}^{L}+c_{k-1}-2\phi_{i}^{L}c_{k-1})$ changes. To understand the
behaviour of agents, we explore the variation of
$2X(\phi_{i}^{L}+c_{k-1}-2\phi_{i}^{L}c_{k-1})$ with $c_{k-1}$ in the range
$(\frac{1}{2},1]$.

$\displaystyle\frac{\mathrm{d(2X(\phi_{i}^{L}+c_{k-1}-2\phi_{i}^{L}c_{k-1}))}}{\mathrm{dc_{k-1}}}=$
$\displaystyle 2X(1-2\phi_{i}^{L}).$ (14)

The sign of $2X(1-2\phi_{i}^{L})$ varies on $\phi_{i}^{L}$, hence we split the
agents between $\phi_{i}^{L}\leq\frac{1}{2}$ and $\phi_{i}^{L}>\frac{1}{2}$ to
analyse their respective decision making behaviour.

We will now explore the behaviour of each agent $i$as $c_{k-1}\mbox{ increases
in }(\frac{1}{2},1]$ .

We first consider agents $i$ such that $\phi_{i}^{L}\leq\frac{1}{2}$ and
observe that:

$\forall c_{k-1}\in(\frac{1}{2},1],\forall
i,\phi_{i}^{L}\leq\frac{1}{2}\implies\frac{\mathrm{d(2X(\phi_{i}^{L}+c_{k-1}-2\phi_{i}^{L}c_{k-1}))}}{\mathrm{dc_{k-1}}}\geq
0.$

We note that the thresholds at the boundaries of the interval
$c_{k-1}\in(\frac{1}{2},1]$ for agents to change are respectively:

$\begin{cases}2X(\phi_{i}^{L}+\frac{1}{2}-2\frac{1}{2}\phi_{i}^{L})=X,&\mbox{if
}c_{k-1}=\frac{1}{2},\\\
2X(\phi_{i}^{L}+1-2\phi_{i}^{L})=2X(1-\phi_{i}^{L}),&\mbox{if
}c_{k-1}=1.\end{cases}$ (15)

Thus, by using the bounds from Equation (15), we state that the threshold
value of $\sigma$ for an agent to sample the leader increase linearly from $X$
to $2X(1-\phi_{i}^{L})$ a $c_{k-1}$ increases from $\frac{1}{2}$ to $1$. So we
note that:

$\forall c_{k-}\in(\frac{1}{2},1],\forall
i,\phi_{i}^{L}\leq\frac{1}{2},\begin{cases}\sigma_{i}\leq
X,&\implies\text{\mbox{agent $i$ always samples the policy-maker, }}\\\
\sigma_{i}>2X(1-\phi_{i}^{L}),&\implies\text{\mbox{agent $i$ always samples
the leader, }}\\\ X<\sigma_{i}\leq 2X(1-\phi_{i}^{L}),&\implies\mbox{agent $i$
samples the leader, then the policy-maker.}\end{cases}$

We now consider agents $i$ such that $\phi_{i}^{L}>\frac{1}{2}$. We note that:

$\forall c_{k-1}\in(\frac{1}{2},1],\forall
i,\phi_{i}^{L}>\frac{1}{2}\implies\frac{\mathrm{d(2X(\phi_{i}^{L}+c_{k-1}-2\phi_{i}^{L}c_{k-1}))}}{\mathrm{dc}_{k-1}}<0.$

Thus the threshold value of $\sigma_{i}$ for an agent to sample the leader
decreases linearly from $X$ to $2X(1-\phi_{i}^{L})$ as $c_{k-1}$ increases
from $\frac{1}{2}$ to $1$. Therefore, we note that:

$\forall c_{k-1}\in(\frac{1}{2},1],\forall
i,\phi_{i}^{L}>\frac{1}{2},\begin{cases}\sigma_{i}\leq
2X(1-\phi_{i}^{L})&\implies\text{\mbox{agent $i$ always samples the policy-
maker. }}\\\ \sigma_{i}>X&\implies\text{\mbox{agent $i$ always samples the
leader. }}\\\ 2X(1-\phi_{i}^{L})<\sigma_{i}\leq X&\implies\mbox{agent $i$
samples the policy-maker, then the leader.}\end{cases}$

We are now ready to split the population into the four distinct sets defined
in a), b), c) and d). From these groups, we will be able to write down the
expressions for fixed points of the affine functions which define $f$ in the
region $c_{k-1}\in(\frac{1}{2},1]$.

Let’s group the agents.

a) Agents who will always sample from the policy-maker. We showed that agents
$i$ such that

$\begin{cases}\phi_{i}^{L}\leq\frac{1}{2},&\sigma_{i}\leq X,\\\
\phi_{i}^{L}>\frac{1}{2},&\sigma_{i}\leq 2X(1-\phi_{i}^{L}).\end{cases}$

always sample from the policy-maker. We denote $\mathcal{R}$ the set of such
agents, with $R=|\mathcal{R}|$. Without loss of generality, we label the
agents in $\mathcal{R}$ from $1$ to $R$.

b) Agents who will always sample from the leader. We showed that agents $i$
such that

$\begin{cases}\phi_{i}^{L}\leq\frac{1}{2},&\sigma_{i}>2X(1-\phi_{i}^{L}),\\\
\phi_{i}^{L}>\frac{1}{2},&\sigma_{i}>X.\end{cases}$

always sample from the leader. We denote $\mathcal{U}$ the set of such agents,
with $U=|\mathcal{U}|$. Without loss of generality, we label the agents in
$\mathcal{U}$ from $R+1$ to $R+U$.

c) Agents who will transfer from sampling the leader to sampling the policy-
maker when a threshold value for $c_{k-1}$ is reached. We showed these agents
$i$ are such that

$\phi_{i}^{L}\leq\frac{1}{2},X<\sigma_{i}\leq 2X(1-\phi_{i}^{L}).$

Let $\mathcal{W}$ be the set of such agents $i$, with $W=|\mathcal{W}|$. These
agents will change opinon when threshold values $t_{i}$ of $c_{k-1}$ is
reached such that the order between $\sigma_{i}$ and
$2X(\phi_{i}^{L}+c_{k-1}-2\phi_{i}^{L}c_{k-1})$ changes. Without loss of
generality, we label the agents in $\mathcal{W}$ from $R+U+1$ to $R+U+W$. We
define $\forall
i\in\mathcal{W},t_{i}=\frac{\frac{\sigma_{i}}{2X}-\phi_{i}^{L}}{1-2\phi_{i}^{L}}$.

d) Agents who will transfer from sampling the policy-maker to sampling the
leader when a threshold value for $c_{k-1}$ is reached. We showed these agents
$i$ are such that

$\phi_{i}^{L}>\frac{1}{2},2X(1-\phi_{i}^{L})<\sigma_{i}\leq X.$

Let $\mathcal{Y}$ be the set of such agents $i$, with $Y=|\mathcal{Y}|$. These
agents will change opinon when threshold values $t_{i}$ of $c_{k-1}$ is
reached such that the order between $\sigma_{i}$ and
$2X(\phi_{i}^{L}+c_{k-1}-2\phi_{i}^{L}c_{k-1})$ changes. Without loss of
generality, we label the agents in $\mathcal{Y}$ from $R+U+W+1$ to $R+U+W+Y$.
We define $\forall
i\in\mathcal{Y},t_{i}=\frac{\frac{\sigma_{i}}{2X}-\phi_{i}^{L}}{1-2\phi_{i}^{L}}$.

We grouped agents into four distinct sub-groups. We are now ready to reorder
the agents within the groups and express the fixed points of each of the
linear functions defining $f$ for $c_{k-1}\in(\frac{1}{2},1]$.

2) Without loss of generality, we re-order the agents in $\mathcal{W}\mbox{
and }\mathcal{Y}$, such that within each group $i<j\implies t_{i}\leq t_{j}$.

We will now express the explicit equations for the fixed points of the linear
functions defining $f$ for $c_{k-1}\in(\frac{1}{2},1]$.

3) Let $\mathcal{Z}$ be the set of agents who change opinion as $c_{k-1}$
increases in $(\frac{1}{2},1]$: $\mathcal{Z}:=\mathcal{W}\cup\mathcal{Y}$. We
order the set of threshold values $t_{i}$ for all $i$ in $\mathcal{Z}$, such
that $i<j\implies t_{i}\leq t_{j}$. We have $Z=|\mathcal{Z}|=W+Y$. Let
$I_{B+D+2},...,I_{B+D+W+Y+2}$ be the $Z$ pairwise disjoint intervals such that
$\cup_{i=B+D+2}^{B+D+W+Y+2}I_{i}=(\frac{1}{2},1]$, where
$t_{B+D+2},...,t_{B+D+W+Y+1}$ are the discontinuities of $f$ on
$c_{k-1}\in(\frac{1}{2},1]$, and $t_{B+D+W+Y+1}=1$. We can now define the
equation for the consensus update in the interval $I_{m}=(t_{m},t_{m+1}]$,
where $B+D+2\leq m<B+D+W+Y+2$. We let $w$ be the number of agents $i$ in
$\mathcal{W}$ such that $t_{i}\leq t_{m}$, and $y$ the number of agents $i$ in
$\mathcal{Y}$ such that $t_{i}\leq t_{m}$. We define
$f_{m}:=f(c_{k-1}),\forall c_{k-1}\in I_{m}$. According to Equation (2) we can
express that:

$\displaystyle f_{m}(c_{k-1})=$
$\displaystyle\sum_{i=1}^{R}w_{i}\phi_{i}^{L}c_{k-1}+\sum_{i=R+1}^{R+U}w_{i}(\phi_{i}^{L}+(1-\phi_{i}^{L})c_{k-1})+\sum_{i=R+U+1}^{R+U+w}w_{i}\phi_{i}^{L}c_{k-1}+\sum_{i=R+U+w+1}^{R+U+W}w_{i}(\phi_{i}^{L}+(1-\phi_{i}^{L})c_{k-1})$
$\displaystyle+\sum_{i=R+U+W+1}^{R+U+W+y}w_{i}(\phi_{i}^{L}+(1-\phi_{i}^{L})c_{k-1})+\sum_{i=R+U+W+y+1}^{R+U+W+Y}w_{i}\phi_{i}^{L}c_{k-1}$

To be a fixed point in the interval $m$, a point $c$ must be such that $c\in
I_{m}$ and $f_{m}(c)=c$.

$c=\dfrac{\sum\limits_{R+1}^{R+U}w_{i}\phi_{i}^{L}+\sum\limits_{P+U+w+1}^{P+U+W+y}w_{i}\phi_{i}^{L}}{1-\left[\sum\limits_{i=1}^{R}w_{i}\phi_{i}^{L}+\sum\limits_{i=R+1}^{R+U}w_{i}(1-\phi_{i}^{L})+\sum\limits_{R+U+1}^{P+U+w}w_{i}\phi_{i}^{L}+\sum\limits_{R+U+w+1}^{R+U+W+y}w_{i}(1-\phi_{i}^{L})+\sum\limits_{R+U+W+y+1}^{R+U+W+Y}w_{i}\phi_{i}^{L}\right]}$
(16)

We expressed the equation and conditions for a fixed point to exist in the
region $c_{k-1}\in[0,1]$. We are now ready to study the dynamics of these
fixed points, and conclude on the convergence of the sequence of consensus
belief.

We have shown how to compute the fixed ponits for the affine function defined
within each interval separating discontinuities. We will now prove that such
fixed points which are in $f$ are stable.

With $M+1$ as the number of distinct linear intervals of $f$ for
$c_{k-1}\in[0,1]$, $\forall m\in[1,...,M+1]$, we define the slope of the
affine function in $I_{m}$ as $\beta_{m}$. If we define $n_{1m},n_{2m},n_{3m}$
as the sets of indices of agents who choose to sample the policy-maker, stick
to the consensus, and sample the leader respectively in each interval $I_{m}$,
we can write:

$\forall m,\beta_{m}=\sum_{i\in n_{1m}}w_{i}\phi_{i}^{L}+\sum_{i\in
n_{2m}}w_{i}+\sum_{i\in n_{3m}}w_{i}(1-\phi_{i}^{L}).$

We note that:

$\forall i\in n,\phi_{i}^{L}\in(0,1),w_{i}\in(0,1),\sum_{i=1}^{n}w_{i}=1.$

Thus with $n_{1m}\cup n_{2m}\neq\emptyset$,

$0<\sum_{i\in n_{1m}}w_{i}\phi_{i}^{L}+\sum_{i\in n_{2m}}w_{i}+\sum_{i\in
n_{3m}}w_{i}(1-\phi_{i}^{L})<\sum_{i\in n_{1m}}w_{i}+\sum_{i\in
n_{2m}}w_{i}+\sum_{i\in n_{3m}}w_{i}=1$

Thus, for any continuous interval $I_{m}$ of the function, if a fixed point
$c\in I_{m}$, $|\beta_{m}|<1$ and $F(I_{m})\subseteq I_{m}$. Therefore, all
fixed point defined over $[0,1]$ are stable.

Thus, if a fixed point $c$ exists in $c_{k-1}\in[0,1]$, the fixed point is a
solution of Equation (13) or Equation (16) if $c$ is such that
$c\leq\frac{1}{2}$ or $c>\frac{1}{2}$ respectively. We have proved that if the
consensus belief visits an interval $I_{m}$ such that $f$ admits a fixed point
on $I_{m}$ , then the consensus belief $c_{k}$ will converge to that fixed
point as $k\rightarrow\infty$. The consensus belief can therefore admit a
fixed point and converge to a single value. ∎

Part 4. _Let $I_{v}$ be the set of intervals visited by the consensus belief
sequence $c_{k}=f(c_{k-1})$ as $k\rightarrow\infty$. Let $x\in\mathbb{N^{*}}$.
Let $D_{x}$ be the set of discontinuities of the $x$th iterate of $f$ over
$c_{k-1}\in[0,1]$, defined as $f^{x}$. If $\nexists I^{*}\in
I_{v},f(I^{*})\subset I^{*},$ and $\exists x^{*}\in\text{$\mathbb{N}$},\forall
x\geq x^{*},\forall c_{k-1}\in[0,1],f^{x}(c_{k-1})\in D_{x-1}\implies
c_{k-1}\in D_{x-1}$, then the consensus belief sequence is eventually
periodic. Once the consensus belief is periodic, the agents are split into
three distinct sub-groups: a) agents who will always sample the policy-maker,
b) agents who will always sample the leader, c) agents who will periodically
oscillate between a combination of sampling the leader, the policy-maker, or
sticking to the consensus. _

###### Proof.

We will prove Proposition 4 by showing that if the discontinuities in the
iterates $f^{x}$ of function $f$ are finite as $x\rightarrow\infty$, then $f$
is a piecewise linear function whose iterates converge to a piecewise step
function. The discontinuities of the piecewise step function are a subset of
the discontinuities of all the iterates of $f$. If the consensus belief visits
no interval of $f$ admitting a fixed point, then the sequence of consensus
belief is eventually periodic.

We sart by expressing a condition for the disconinuities in the iterates
$f^{x}$ of $f$ to be finite as $x\rightarrow\infty$. This will allow us to
prove that $\lim_{x\rightarrow\infty}f^{x}$ is a stepwise function with a
fintite number of discontinuities, and thus show that the sequence of
consensus belief is eventually periodic.

Let $x\in\mathbb{N}$ and $f^{x}$ be the $x$th iterate of $f$. We define
$D_{x}$ as the set of discontinuities of $f^{x}$ over $c_{k-1}\in[0,1]$.
Assume that: