###### Abstract

In a stylized voting model, we establish that increasing the share of critical
thinkers – individuals who are aware of the ambivalent nature of a certain
issue – in the population increases the efficiency of surveys (elections) but
might increase surveys’ bias. In an incentivized online social media
experiment on a representative US population ($N=706$), we show that different
digital storytelling formats – different designs to present the same set of
facts – affect the intensity at which individuals become critical thinkers.
Intermediate-length designs (Facebook posts) are most effective at triggering
individuals into critical thinking. Individuals with a high need for cognition
mostly drive the differential effects of the treatments.

##  1 Introduction

Individuals often hold conflicting attitudes towards complex issues, known as
attitudinal ambivalence (Kaplan (1972)). Such attitudes are particularly
prevalent when an issue presents both positive and negative aspects and when
the available hard evidence is inconclusive. For example, the debate on
whether society should implement strict digital policies to protect users’
privacy at the expense of less innovation may be perceived as a dilemma by
individuals since the arguments available do not support unambiguously one
side or the other. This dilemma of balancing privacy and innovation can lead
to ambivalent attitudes as different facts and arguments can support both
sides. Such dilemma situations, with digital content produced through
generative Large Language Models and Artificial Intelligence hardly easily
verifiable with clear-cut evidence, are likely to become prevalent on digital
platforms.

When faced with complexity and ambivalence, individuals may respond through
critical thinking (Kahneman (2011)).222Individuals may also respond based on
emotional bias and motivated reasoning (Kunda (1990), Bénabou* (2015));
however, in this paper, we focus on critical thinking. Critical thinking goes
beyond simple cost-benefit analysis and involves a metacognitive process of
becoming aware of one’s ambivalent attitude toward a dilemma and evaluating
the various perspectives that underlie each attitude (Halpern (2013)). Through
this measurable process (List (2022)), individuals can overcome their
ambivalence and pass from holding raw preference to forming stable preference.
For decision-makers relying on surveys and elections, stable preferences
represent more reliable data than raw preferences.

Consider public figures or organizations whose social image or economic
returns depend on the approval of their public stance on a particular issue.
These principals must predict the public stance their audience expects them to
take because a public endorsement is a reputational commitment and a “focusing
event” that nudges agents to think critically about their raw preferences and
establish a stable preference, which might go against the principal’s stance.
Hence, the principal should estimate agents’ stable preferences before taking
a stance, minimizing the risk of lasting backlash. Suppose such an estimate is
based on a poll. Then, its quality hinges on agents reporting their stable
preference, for which it is necessary that they have formed them in the first
place.333One might conceive a strategic voting setting where agents misreport
their stable preference even after they have formed one. We think this concern
is second-order for the application we have in mind, so we assume that forming
and reporting a stable preference to go hand in hand.

Eliciting stable preferences might also be important for an institutional
principal, i.e., a policy-maker responsible for designing an economic policy
on a societal issue that presents a binary dilemma (for instance, egalitarian
versus freedom). The principal can choose from a larger (continuum) set of
policies. The optimal policy is a function of the distribution of stables
preferences, i.e., the share of individuals that would prefer one alternative
over another at the end of the critical thinking process. For concreteness,
think of a situation where the optimal amount of a social welfare program
depends on the share of citizens who support an egalitarian society against a
free-market one. The policy-maker must adopt a policy aligned with one of two
opposing worldviews. At most, they would know their stable preference on the
issue but risk imposing it on the remaining population. In this
interpretation, the policy-maker uses the citizens’ stable preferences
distribution as a normative criterion. Hence, here also, it is in the
institutional principal’s interest to anticipate (and incentivize) the
formation of agents’ stable preferences before making her decision.

In summary, we posit a principal who cares about the distribution of stable
preferences either because she fears her action will cause a backlash if they
are too distant from the target or because she is using such distribution “for
lack of anything better” as a proper normative criterion for social
aggregation of preferences. Elections would be efficient if all individuals
reported their stable preference at the poll, allowing a perfect tracking of
the relevant unknown. However, individuals find their stable preference only
at the end of a critical thinking process which not every agent is able or
willing to undertake. Also, eliciting such a process requires novel
experimental methods since standard survey methods fail to classify the types
of agents.

In such an ambivalent environment, hard information is insufficient to resolve
agents’ ambivalent attitudes, hence, underlying the role of “storytelling
formats” and the cognitive styles of agents in forming stable preferences. In
an ambivalent environment, where there is a lack of consensus on important
issues, the role of media is not only to provide objective information but
also to push individuals to realize the ambivalent nature of the issue. These
“focus events” are not based on what is traditionally known as “information”
but rather on stylized and partial facts communicated through various media
formats such as Facebook posts, news articles, and tweets. These facts, while
potentially irrelevant to the judgment of value, can still impact the speed at
which individuals recognize the ambivalence of the issue.

This paper refers to such a “ storytelling format” as the packaging (UX
design) of information, which is crucial in triggering individuals toward
critical thinking. Related to Aragones et al. (2005), they perform ’fact-free
self-learning’. A well-reasoned essay may trigger individuals to become aware
of the issue’s complexity, while a sequence of bombarding facts may induce the
same outcome. The quantity and quality of information presented and the
individual’s cognitive style play a crucial role in this process. In our
stylized digital economy, the media is a nudging device, triggering
individuals to transition into critical thinking. This is because once an
individual realizes they are ambivalent about an issue, no “objective fact”
can drive their preference. Instead, critical thinking, in which different
worldviews are weighted, is necessary to form a stable preference. Therefore,
we will adopt the terminology “stories” and “storytelling formats” to refer to
“media content” and “media format,” respectively. How a problem is presented,
from a shallow tweet-storm to a well-reasoned newspaper article, can affect
how quickly individuals realize ambivalence about an issue.

##### Aims of the paper.

In this paper, we study the role of storytelling formats in shaping this
process, with the idea that realizing the ambivalent nature of an issue (i.e.,
starting the process) is an event whose likelihood depends on the storytelling
format associated with the informational environment. Hence, the question we
address is the following: do storytelling formats affect the efficacy of polls
at eliciting stable preferences by impacting an individual’s critical thinking
process? Besides, can agents’ cognitive styles explain this effect? To address
this question, we need a theoretical model to formalize the awareness of
ambivalence within a population affects the predictive power of elections and
an empirical demonstration that different storytelling formats push
individuals into critical thinking at different rates. This paper accomplishes
both tasks and provides an affirmative answer: storytelling formats matter for
the efficiency of elections in theory and practice.

##### Theoretical setting.

We construct a simple but novel theoretical model in which the intensity,
$\lambda$, at which ace individuals in a (large) population enter critical
thinking is a relevant welfare measure. The principal uses the outcome of
elections to guess the distribution of stable preferences inside the
population. The raw preference (or stereotype preference), $x_{S}$, is drawn
from a perturbation of the distribution of stable preferences – the object the
principal wants to elicit. Although the reported preference, $x_{A}$, is
stochastic (given the stable preference) even during the critical thinking
phase, allowing for “reasoned indecisiveness,”444The baseline model, which can
be thought of as a simplified version of a fully identified three-state model,
is presented in the Appendix and delivers no significant qualitative
difference in the results, wherein individuals complete their critical
thinking process and finally discover (and report) their stable preference.
removing the stereotype component orthogonal to the stable preference improves
the “quality” of the reported preference. For this reason, the average
reported preference becomes more informative about the average stable
preference the more agents successfully perform critical thinking. Since such
informativeness coincides with the welfare of a principal that “plays” the
estimator obtained in the poll, we posit that $\lambda$ is indeed a welfare
measure.555Comparative statics are ambiguous if we consider an institutional
principal that has to act according to the election outcome and, therefore,
must consider election bias.

##### Social Media Experiment.

Based on this sharp theoretical result, we conduct an online experiment to
demonstrate that storytelling formats affect $\lambda$ and, therefore, the
efficiency of elections. To do so, we present a representative sample with a
“relatively new” ambivalent issue, namely, digital privacy. Accordingly, the
debate is whether we should remunerate users for data sharing. We randomly
assign subjects to one of four storytelling format treatments: twitter,
facebook, newspaper, or partisan twitter. We maintain the same content (i.e.,
we give agents the same selection of information) but alter its storytelling
format. We elicit pre- and post-treatment awareness of attitudinal ambivalence
in two different – although both incentivized – ways and test whether the
storytelling format alters the likelihood of the agent moving into critical
thinking. We find that intermediate-length stories (the facebook treatment)
are significantly more effective at nudging individuals into critical
thinking. We then investigate whether individuals’ cognitive styles can (at
least partially) explain this heterogeneous response to the storytelling
format. To this end, we split our sample along two cognitive traits that we
also elicit throughout the experiment using standard measures: need for
cognition (Cacioppo and Petty (1982)) and cognitive flexibility (Martin and
Rubin (1995)). We find that most of the heterogeneity in the response is
driven by individuals with a high need for cognition. At the same time, the
cognitive flexibility score is irrelevant in determining responsiveness to
critical thinking.

##### Literature Review.

##### Critical thinking and non-dogmatism.

In its normative interpretation, our behavioral model relates to Millner
(2020). In his model, non-dogmatic social planners are insecure in their
future time preferences and entertain the possibility of endorsing different
ones. Millner’s work is reminiscent of our normative interpretation in which
the planner takes the distribution of stable preferences as the “truth” when
it comes to a dilemma and is only concerned that an election would give a
partisan view because not everyone experiences moral uncertainty.

##### Simplistic Rhetoric.

Research has shown that political discourse has become increasingly simplified
over time (Jordan et al. (2019)) and that this trend intensifies during
election periods (Tetlock (1981), Thoemmes and Conway III (2007), Conway III
et al. (2012)). Our model suggests that this simplification can lead to
awareness for individuals. Additionally, our model also suggests that
simplification can lead to a skewing of preferences among individuals in a
stereotypical phase of the process, where one argument may be seen as more
persuasive due to overconfidence Ortoleva and Snowberg (2015) and the belief
that one’s experiences are more informative about policy than they are. This
highlights the importance of considering the effects of simplistic rhetoric in
political discourse and the importance of individuals being aware of their
ambivalence towards an issue.

##### Predictive Power of Elections.

As demonstrated in the seminal work of Feddersen and Pesendorfer (1997), there
are instances where many voters effectively aggregate information, resulting
in an equilibrium outcome that is fully information-equivalent. However,
preference heterogeneity can impede a voting procedure from effectively
aggregating individual voters’ information (Kim and Fey (2007); Gul and
Pesendorfer (2009); Bhattacharya, 2013; Acharya, 2016; Ali et al., 2018). This
literature highlights that when voters have divergent preferences and
incomplete information about the state of nature, they may collectively choose
an outcome that is less favorable for society or preferred by a minority. The
central concern in these papers is strategic voting, an issue not present in
our analysis. In our model, every citizen votes for their current preference.
Still, the extent to which this preference accurately reflects the payoff-
relevant stable preference depends on the citizen’s cognitive state, which is
influenced by politics. Political can be seen as a method of making citizens
view their stable preference as private information that an election aims to
uncover.

##### Social Media and Welfare.

Our paper contributes to expanding an already vast literature that has been
focusing more and more on how a specific format over the last decade (i.e.,
social media could shape agents’ political behaviors such as voting).
Gorodnichenko et al. (2021) empirically showed that using bots in social media
has a role in influencing public opinion. Munir (2018) provided evidence of
the impact of social media in shaping the voting behavior of Scottish youth
during the Scottish Independence Referendum of 2014. Falck et al. (2014)
studied the impact of more general information disseminated over the Internet
on voting behavior and found that it influenced it.

##### Structure of the paper.

Section 2 describes the behavioral model and its main positive and normative
results. Section 3 elaborates on the experimental design. Section 4 elaborates
on the empirical results. Section 5 concludes and discusses possible caveats
and extensions of the model and experiment.

##  2 Theoretical Setting

We consider a stylized political choice setting. The utility is the distance
between the political action and a target determined by the distribution of
the preferences individuals hold if they had completed a critical thinking
process. This parameter is unknown at the onset and can only be estimated
using the outcome of a poll held at some time $t$, when (a part of) citizens
might still not have completed their process.

The main aim of this section is to establish – Proposition 1 – that the
intensity at which citizens complete their critical process is a relevant
welfare measure: whatever the time of elections is, higher intensity increases
the information content of elections. This gives our exercise of estimating
the intensity associated with different propaganda formats – the objective in
the experiment presented in Section 3 – to have normative content. However, we
also show that the main message relies on assuming that we have a principal
who can freely manipulate the outcomes of the poll when taking her action
(which we refer to as $P$ositive principal). If the election outcome
constrained her action, as is most likely the case for an $I$nstitutional
principal, then a bias-precision trade-off emerges, which makes the
comparative statics ambiguous.

We present the two welfare benchmarks (corresponding to the two types of
principals discussed in the introduction) and voters’ critical thinking
process separately. We then put them together to derive our main result and
some important caveats that shed light on the assumptions needed to get the
unambiguous comparative statics.

###  2.1 Principal: Two Welfare Benchmarks

The relevant unknown is the distribution of stable preferences in a large
(continuum) population, namely the share $p\in\left[0,1\right]$ of individuals
who would prefer outcome $1$ to outcome $0$ after they complete their critical
thinking process. Welfare realizes the distance between the social action $a$
and its target $p$666Although the space of preferences — individuals’
resolution of the moral dilemma — is binary, the policy space is continuous.
This corresponds to a situation where the planner can fine-tune the policy to
the distribution of individuals’ preferences. the example in the introduction
of choosing the size of the welfare program based on the share of people that
hold an egalitarian (rather than a free-market) view fits this story. A
different specification would $a^{\star}=\mathbb{I}[p>\frac{1}{2}]$ (binary
action space) delivers similar insight but is less tractable. :

$W\left(a,p\right)=-\left(a-p\right)^{2}$

Ex-ante, $p$ is unknown and drawn from a normal distribution
$p\sim\mathcal{N}(\mu,\sigma)$; absent the information from the election, the
principal would then choose $a=\mu$ and get value $-\sigma{{}^{2}}$.777The
normality assumption gives tractable conditional expectations and closed-form
welfare. It is inconsistent with the compact support $[0,1]$. The analysis
with ex-ante uniform $p$ (and $p_{S}$) is algebraically more involved but does
not change the qualitative results. For tractability, we keep the normal
setup, implicitly assuming that $\sigma$ is “small enough” that the mass
outside $[0,1]$ is negligible. Before choosing $a\in\left[0,1\right]$, the
principal observes the proportion $\bar{p}$ of agents that report to prefer
alternative $1$. We call $\bar{p}$ the election outcome. and consider two
types of principals that differ in the use they can make of this information.

##### Positive Principal.

A $P$ositive principal, for which the election outcome is not binding, namely
who can choose any $a\in[0,1]$ no matter the realization of $\bar{p}$. The
Positive principal uses the election outcome and his knowledge of the critical
thinking process inside the population (next Section) to estimate $p$. His
optimal action is the conditional expectation

$\displaystyle a^{\star}=\hat{p}\coloneqq\mathbb{E}[p|\bar{p}]$

that achieves value

$W_{P}=-\mathbb{E}\left[\left(\hat{p}-p\right)^{2}\right],$ (1)

equal to the dispersion of the conditional mean $\hat{p}$ around $p$.888The
expectation operator $\mathbb{E}$ integrates under the joint distribution of
$p,\bar{p},\hat{p}$, which depend on the voting behavior and citizens’
critical thinking process and that we derive in the next section. Connecting
to the discussion in the introduction, one can think of such principals as
public figures (e.g., multinational firms or social influencers with
reputational concerns) that need to take a stance on an ambivalent issue. They
privately run a poll (say, by asking a polling agency) and use its outcome as
they wish to fine-tune their statement. Payoff depends on the (distribution
of) stable preferences because the statement acts as a “focusing event” that
pushes the relevant population into critical thinking: the preferences
individuals judge the principal on are (potentially) different from those they
report at the poll.

##### Institutional Principal.

An $I$nstitutional principal, whose action is constrained to be $a=\bar{p}$,
which achieves value

$W_{I}=-\mathbb{E}\left[\left(\bar{p}-p\right)^{2}\right]$ (2)

One can think of such principals as democratic institutions that must comply
with the election outcome (say, by empowering a parliament whose composition
is proportional to $\bar{p}$).999In this context, the interpretation of $p$
differs. Rather than focusing on the potential backlash from stable
preferences, we envision an institutional principal considering $p$ as a
normative criterion for aggregating social preferences about an ambivalent
issue. Essentially, the distribution of preferences of individuals who have
undergone the critical thinking process determines the “right thing to do”.
Via the standard decomposition, we get that

$W_{I}=W_{P}-B,$ (3)

where

$B=\mathbb{E}\left[\left(\bar{p}-\hat{p}\right)^{2}\right]>0$

is the bias of election, representing how the average reported preference
differs systematically from the stable ones. A principal $P$ who can correct
for such social tendencies only suffers from the dispersion of the estimator
$\hat{p}$ around the parameter $p$, whereas the principal $I$ is also
concerned by the bias of the election since they cannot correct them.

###  2.2 Agent: Cognitive and Voting Processes

Each individual is characterized by a stable preference

$y\sim Ber(p)$

where $p$ is the welfare relevant unknown that the principal wants to match.
For example, an individual with $y=1$ has a stable preference for alternative
$1$. If asked at a poll, however, individuals do not necessarily report their
stable preference. This is because the stable preference is “discovered” at
the end of a critical thinking process that individuals undergo.

##### The Cognitive Process.

Agents transition through two critical thinking _states_
$\left\\{S,A\right\\},$ where $S$ means Stereotype and $A$ means Awareness. We
assume that the critical thinking process follows a simple dynamic in
continuous time: all individuals start at $t=0$ in state $S$ and,
independently of $y$ (and other voting parameters), they transition to the
absorbing state $A$ with intensity $\lambda\in\left(0,\infty\right)$.
Therefore, at time $t$ there will be a fraction

$\eta_{S}=\exp\left\\{-\lambda t\right\\}$

of agents that are still $S$tereotypes and $\eta_{A}=1-\eta_{S}$ that
transitioned to Awareness.101010The assumption that $A$ is an absorbing state,
with no transitions from $A$ to $S$, captures the idea that awareness is an
irreversible process. A straightforward extension of the model prevents a
scenario where all individuals eventually reach state $A$: a constant fraction
$\nu<\lambda$ exits the economy and re-enters in the awareness state $S$.
Qualitative results would be unchanged as the associated share of stereotypes:
$\eta_{S}(t)=\frac{\nu}{\lambda}+\exp\left(-\lambda
t\right)\left(1-\frac{\nu}{\lambda}\right)$ would still be decreasing in
$\lambda$,$t$. The parameter $\lambda$ is key for our analysis. It represents
the intensity with which individuals realize the issue at hand is ambivalent.
We estimate its value by estimating in our experiment for different propaganda
formats: the idea is that the way news is presented has a role in determining
the speed at which individuals move into $A$ – and that such difference
interacts with other cognitive abilities.

##### Voting Behavior.

We denote $x$ the preference that individuals report at the polls and assume
it depends as follows on the stable preference $y$ and on the stage of the
critical thinking process $\\{S,A\\}$. Before realizing the issue is
ambivalent, the reported preference $x_{S}$ is

$x_{S}\left|y\right.=\begin{cases}Ber\left(p_{S}\right)&\text{w.p. }\beta\\\
y&\text{w.p. }1-\beta\end{cases}$

In words, $x_{S}$ is equal to the stable preference with probability
$\beta\in[0,1]$, while complementary probability is drawn from a distribution
of stereotypical preferences $p_{S}\sim\mathcal{N}(\mu,\sigma)$, independent
of $p$. Since we still have a parameter $\beta$ that drives the correlation
between average stereotypes and stable preferences, the independence
assumption is innocuous. It only requires the formation of stereotypical
preferences involving factors not solely related to $p$.111111The identical
distribution of $p$,$p_{S}$ is instead for tractability alone. Most
derivations in the Appendix utilize non-identically distributed normal
variables ($\mu_{p}$, $\sigma_{p}$, $\mu_{p_{S}}$, $\sigma_{p_{S}}$).
Specifically, the condition $\mu_{p}\neq\mu_{p_{S}}$ illustrates a scenario in
which the principal is aware that stereotypical preferences exhibit systematic
bias, potentially due to the ease of presenting superficially persuasive
arguments in favor of one alternative. This, along with the relaxation of
other symmetry assumptions inherent in our model, is further explored in
Section [to be added]. For example, if $\beta=\eta_{S}=1$, corresponding to a
poll held at $t=0$ and where stereotypes are independent of stable
preferences, then the election outcome $\bar{p}=p_{S}$ is uninformative about
$p$.

The preference reported by individuals in $A$ loses its dependence on the
nuisance parameter $p_{S}$ and becomes a function of the stable preference
alone,

$x_{A}\left|y\right.=\begin{cases}y&\text{w.p. }\xi\\\ 1-y&\text{w.p.
}1-\xi\end{cases}$

The parameter $\xi\in[\frac{1}{2},1]$ allows for a situation where citizens
realize the issue is ambivalent but still have not found their stable
preference. We think of our two-stage critical thinking process as a reduced
form of a fully identified three-stage process – detailed in the Appendix –
where $A$ is an intermediate stage where agents have realized the ambivalent
nature of the issue but have not formed their stable preference yet – i.e.,
they are in a phase of normative uncertainty. In this interpretation, the case
$\xi=1$ corresponds to a situation where individuals discover their stable
preference immediately after realizing the ambivalence of the issue, while
$\xi=\frac{1}{2}$ a situation of permanent indecisiveness of $A$ individuals.
Notice that if all individuals were in $A$ state (i.e., a $t\to\infty$ poll),
then the election outcome would be $\bar{p}=\xi\cdot p+(1-\xi)\cdot(1-p)$,
which whenever $\xi>\frac{1}{2}$ is a strictly monotonic (hence invertible)
function of $p$. Since the election outcome contains all information about
$p$, then $\hat{p}=p$, and the $P$ositive principal always chooses the correct
action.121212Institutional principal still needs to consider the attenuation
bias driven by $A$’s indecisiveness. In general (i.e. for interior shares
$\eta$), the election outcome is given by:

$\overline{p}=\eta_{S}\left(\beta
p_{S}+\left(1-\beta\right)p\right)+\eta_{A}\left(\xi\cdot
p+(1-\xi)\cdot(1-p))\right)$ (4)

and the parameter $\xi$ affects the $P$ositive welfare too. We can now use the
(joint) normality assumption to write $\bar{p}$ as well as the conditional
expectation $\hat{p}$ as a linear function of the fundamental unknowns
$p,p_{S}$, that is

$\bar{p}=\alpha_{0}+\alpha_{1}\cdot p+\alpha_{2}\cdot p_{S}$
$\hat{p}=\gamma_{0}+\gamma_{1}\cdot p+\gamma_{2}\cdot p_{S}$

where the loadings $\boldsymbol{\alpha,\gamma}$ are functions of the
structural parameters $\vartheta=[\beta,\xi,\boldsymbol{\mu,\sigma}]$ and the
critical thinking process statistic $\eta$ as detailed in the Appendix.
Moreover, once we specify the joint normal expectation operator, we can
compute (the evolution of) both positive and institutional welfare
LABEL:eq:PWelf-LABEL:eq:NWelf in closed form and arrive at our main result.

###### Proposition 1

$i)$ For all values of the structural parameters $\vartheta$, $W_{P}$ is
increasing in $t$ and $\lambda$.

$ii)$ $W^{I}$ has non-trivial comparative statics in $\lambda,t$. If
$\beta<1-\xi$, then it is monotonically increasing; if
$\beta>\frac{\left(1-\xi\right)\left(\left(1-2\mu\right)^{2}+4\sigma^{2}\right)}{2\sigma^{2}}$
then it is monotonically decreasing; else it grows local to $t=0$ (resp.
$\lambda=0$) up to a finite time $t^{\star}$ (finite intensity
$\lambda^{\star}$) then eventually decreases.

Figure [to be added] gives a graphical representation of the results collected
in Proposition 1, which we now discuss. Point $i$ establishes that if the
principal knows the value of the structural parameters $\vartheta$ and can
utilize the outcome of elections without constraints, then the faster
individuals move into critical thinking (the higher $\lambda$), the higher the
efficiency of elections. Indeed in all the plots of the figure, we observe
that positive welfare $W^{P}$ is increasing in time.131313As the proof relies
on $W^{P}$ being decreasing in the share of stereotypes eta s, the same graph
would be obtained if we fix the time and let $\lambda$ vary. The bottom-right
panel explains the dynamics for high and low $\lambda$. The reason behind this
result is simple to grasp: as fewer and fewer individuals are $S$, the
election outcome $\bar{p}$ becomes less and less dependent on the nuisance
unknown $p_{S}$ which confounds the inference of the welfare relevant unknown
$p$.141414Indeed, this result does not require the normality assumption but
can be directly deduced by the expression of $\bar{p}$. This is an important
result for our analysis as it establishes that $\lambda$ is a welfare measure
in a well-definite sense in our setting.

However, in point $ii)$, we also hint at a potential limitation of such a
result in case the principal is constrained to act according to the election
outcome due to the (potentially perverse) effect that moving into critical
thinking has on the bias of the election. The most paradoxical result – the
condition that if $\beta$ is large enough, institutional welfare is actually
decreasing in $\lambda$ – has a natural explanation. When $\beta$ is large,
then stereotypes are strong predictors of the stable preference (in the
extreme where $\beta=1$, all stereotypes vote $y$ despite failing to realize
the ambivalent nature of the issue),151515In our setting there is no intrinsic
social value for being critical thinkers so if all agents get their stable
preference right we have perfect elections. However, a related phenomenon
studied by Bernheim et al. (2021), “mindset flexibility” might have social
benefits beyond increasing the accuracy of elections. The challenge for us is
to derive $\lambda$ as a welfare measure even without a direct beneficial
effect from critical thinking. hence moving in the $A$wareness state
indecisiveness and associated attenuation bias – case $\xi<1$ – pushes
$\bar{p}$ away from $p$ and thus reduces efficiency. This seems – at least to
us – a pathological case since it requires. but is useful to highlight the
potential role of the bias. For this reason, we further investigate conditions
under which the two rules coincide, that is, whether there is a level of
$\eta$ such that “by divine coincidence” the loadings
$\boldsymbol{\alpha}=\boldsymbol{\gamma}$ so the positive and institutional
principal have the same action rule – and hence the same value at potential
limitations of this interpretation; we indeed show that if $\xi=1$ or
$\beta<\frac{1}{2}$, the condition for $W^{I}$ monotonically increasing is
vacuously satisfied.

Solving the system of equations $\alpha\equiv\gamma$ gives a share of
stereotypes $\eta^{\star}$ such that the two coincide. Hence, there exists an
interior time where the average reported preference is unbiased for $p$.
Formalizing this result we obtain:

###### Proposition 2

If there is no bias in the stereotype pool and $\beta$ is large enough, i.e.
if

$\mu_{p}=\mu_{p_{S}}and\frac{1-\beta}{\beta}<\frac{\sigma_{p_{S}}^{2}}{\sigma_{p}^{2}}$

, then there exists a finite time $t^{\star}$ such that
$B\left(t^{\star}\right)=0$. If, in addition,

$\xi=1thent^{\star}=-\frac{1}{\lambda}\log\left(\frac{\sigma_{x}^{2}}{\beta\left(\sigma_{x}^{2}+\sigma_{y}^{2}\right)}\right)$

with immediate comparative statics.

Figure 1: $\beta<1-\xi_{NU}$ $\Rightarrow$ $\eta^{\star}=0$ $\Rightarrow$
Inst. Welfare is decreasing Figure 2: $\eta^{\star}\in(0,1)$ $\Rightarrow$
Inst. Welfare has interior maximum, after the zero-bias time  $t^{\star}$
Figure 3: $\eta^{\star}=1$ $\Rightarrow$ Inst. Welfare is increasing Figure 4:
$\lambda$ is a welfare measure.

###  2.3 Discussion of The Model Primitives

##### Implicit Assumptions.

We have presented a relatively parsimonious model of voting while undergoing a
critical thinking process (from $S$tereotypes to $A$ware citizens) of
discovery of stable preferences. Its fundamental functioning is easily
explained. There is a nuisance parameter $p_{s}$ in the sterotypes preferences
that confounds the election outcome because it adds a component that is
orthogonal to the stable preference. As more citizens become critical thinkers
the election outcome is less influenced by $p_{s}$ and the principal gets a
better estimate of the relevant parameter $p$. The share of critical thinkers
increases with time and with the intensity of the critical thinking process,
and this is the channel through which $\lambda$ impacts the efficiency of
elections.

The two parameters $\beta,\xi$ are meant to capture the inherent quality of
the preference reported in the two stages of the critical thinking process.
High-$\beta$ environments represent situations where, despite not realizing
the ambivalent nature of the issue, stereotypes get their stable preference
right with high probability; it also makes the independence assumption between
$p$ and $p_{s}$ immaterial. We think of $\xi$ instead as a reduced form
parameter for a three-stages critical thinking process in which agents first
realize ambivalence, and then discover the stable preference; $\xi$ is
inversely proportional to the length of this second transition. The
flexibility added by those two parameters does not alter the qualitative
evolution of welfare for a $P$ositive principal, but through their impact on
the (evolution of) bias they are consequential for an $I$nstitutional one,
possibly yielding to perplexing comparative statics. We have always maintained
an implicit symmetry assumption, since all structural parameters are not
allowed to depend on the stable preference $y$. A relaxation of this
assumption would require to model
$\beta_{i}=\mathbb{P}\left[x_{S}=y|y=i\right],\xi_{i}=\mathbb{P}\left[x_{A}=y|y=i\right]$
with a different specification for the residual uncertainty in the preference
of stereotypes. Insofar as overconfidence can be interpreted as individuals’
resistance to critical thinking, evidence in Ortoleva and Snowberg (2015) also
questions the fact that the intensity $\lambda$ is independent of $y$: if
stable preference predict cognitive traits associated with critical thinking
(or the impact of different storytelling formats), then the $A$ware pool would
be selected based on $y$, which constitutes an additional source of bias.

Extensions that allow these empirical regularities to manifest would not alter
the main message of our theoretical model: if there are more $A$ware voters,
polls contain more information about the distribution of stable preferences.
This is all that matters for a principal that can “filter out” all systematic
tendencies in voting — including the asymmetries in stereotype reporting and
critical thinking transition —, while a principal that cares about getting the
election outcome as close as possible to $p$ needs to trade off accuracy with
election bias.

##### Bias in Stereotypes and Critical Thinkers’ Dilemma

An immediate extension of our model is to allow for the presence of bias in
the stereotype pool, i.e. to let $\mu_{s}\neq\mu$, corresponding to a
situation where the principal knows that a specific opinion is prevalent
before individuals realize the ambivalent nature of an issue. This possibility
— which seems compelling whenever one of the positions is more prone to be
defended by means of superficial arguments (nationalism) — means the
$I$nstitutional principal additionally benefits from increasing the intensity
$\lambda$ (or simply “letting time pass”) as having a larger share of $A$
voters would mechanically remove this type of systematic bias.161616The
evolution of welfare for the $P$ositive principal would instead be unaffected
by this extension, as she could “clear out” all systematic noise in the poll.
Importantly, if this such bias became apparent to $A$ citizens too, this might
affect their voting rule. In particular, think about critical thinkers who
have not yet discovered their stable preference; their problem is particularly
interesting. Because they have “lost” their stereotype preference and not
formed a stable one yet, they most likely to abstain during an election and be
more sensitive to costly voting (an important margin, see Cantoni and Pons).
If they were aware of a systematic bias in stereotypes (which seems realistic,
since they just escaped that state) they might decide to use their vote to
“compensate” for such bias.

Finally, by adding a penalty for waiting (discounting the utility from taking
an action later), we can use our setting to discuss the optimal timing of
elections; even the $P$ositive principal wouldn’t postpone her decision until
$t=\infty$ where she would obtain a precise estimate of $p$. Studying how the
timing of the optimal election varies with the intensity $\lambda$ (and other
structural parameters) amounts to analyzing the problem of a principal that
controls the type and duration of storytelling she wants to subject her agents
before administering a poll to maximize its accuracy.

As for the impact of information, one implication is how different people
react moving into and out of critical thinking, adding the amount of
information, presenting a set of facts as bullet points, or re-elaborating
those facts in a more structured piece. The challenge is that we do not
observe people in critical thinking. Hence, how can we identify whether
individuals are in critical thinking, and if so, how can we classify them as
$S$ or $A$ throughout the different timing of exposure to storytelling
formats? We propose an experimental design and classification strategy to
approximate the observation of such a critical thinking process.

##  3 Experimental Setting

We have formalized an ambivalent environment in which $\lambda$, interpreted
as the speed of the critical thinking process, determines the efficiency of
surveys and elections. We now empirically investigate whether different
storytelling formats are associated with different $\lambda$s. To do so, we
estimate the $\lambda$ associated with different storytelling format
treatments in an online incentivized experiment.171717The Princeton
Institutional Review Board approved the experiment. See the Appendix for the
detailed Princeton IRB approval. Figure 5 below provides an overview of the
experimental design and its primary elicitations, which we will elaborate on
in subsequent sections.

In a nutshell, in our experiment, we expose subject participants to different
storytelling formats and elicit their pre- and post-treatment stages in the
critical thinking process associated with an ambivalent issue. We would then
compare how the likelihood of transitioning from $S$ to $A$ varies across
formats, testing whether they are statistically different. Throughout the
experiment, we also collect data about participants’ cognitive styles – using
standard measures from the psychological literature. This allows us to test
whether the effectiveness of certain storytelling formats channels through
identifiable cognitive traits. We used incentivized elicitations for the key
individual variables —pre- and post-awareness states—and implemented anti-
cheating policies and attention screeners to ensure optimal data collection
quality.

We gathered 900 participants from a representative US population using
Prolific, a data collection platform increasingly favored by economists due to
its high data quality. Following meticulous screening for attention, cheating,
and quality, as outlined in the upcoming sections, our final sample size
consisted of $N=706$. Participants received a fixed payment of $2 and a bonus
payment of up to $5, resulting in an average payment of approximately $6.

$\bullet\,\,$ Short & Crude$\bullet\,\,$ Medium & Reasoned$\bullet\,\,$ Long &
Reasoned$\bullet\,\,$ Short, Bias & Crude randomized3-step test4
formatsCooling Period2-step testElicitation ofEx-Ante
TypesTreatmentElicitation of Cognitive StylesElicitation of Ex-Post Types$ : %
correct answers$ : attention check$ : % AI grade Demographics Ex-Ante
Preferences Knowledge Test Listing Reasons Issue Familiarity Cooling Period
Need For Cognition Cognitive Flexibility Essay Writing Ex-Post Preferences
Internal Uncertainty News Habits

Figure 5: Experiment Design

###  3.1 Classification of Critical Thinking States

We now present – and describe the rationale for – the two different strategies
we employ to classify participants into the critical thinking states
$\\{S,A\\}$ pre and post treatments. Table 1 summarizes both strategies.

##### Pre-treatment classification strategy.

We use a three-pronged test to design our pre-treatment classification
strategy as $\\{S,A\\}$. This test is based on the following heuristic
conditions critical thinkers must satisfy: i) they must have basic knowledge
of the issue at hand; ii) they must have thought about the issue before; iii)
they must be aware that there exist both pros and cons for the issue. All
i)-iii) characteristics are needed to be a critical thinker about an issue to
avoid misclassification (as it could be by only using iii)).

To elicit condition i), we rely on an assessment designed by Pew Research
(Vogels and Anderson (2019)) and launched on a representative US population,
referred to as knowledge test in Figure 5. In our experiment, to pass the
knowledge test, participants must score at least as high or above the score of
the nationally-representative US population found by Pew Research.181818It
consists of 10 questions. See the Appendix for the wording details. To elicit
condition ii), we ask participants to report whether they have thought about
the issue before coming to our experiment. To elicit condition iii), we ask
them to provide evidence by providing two reasons that support their
preference for the digital privacy issue and two that go against their
preference. This task is instrumental in more accurately targeting the pre-
treatment awareness state of individuals. We refer to this task as Listing
reasons is Figure 5. If (and only if) subjects are already beyond their raw
preference stage, we can provide a complete classification.

We cannot rely on the same three-pronged tests to provide the post-treatment
classification of participants in terms of $\\{S,A\\}$. Since the pre-
treatment classification test includes condition iii) and our storytelling
format treatments expose subjects to a series of pros and cons about the issue
(see the next section), relying on the same condition here can misidentify
critical thinking as memory effects. Indeed, subjects might not be critical
thinkers, having accepted the issue as ambivalent by default, but happen to
remember their list of pros and cons reported pre-treatment. Therefore, we
need a different post-treatment classification strategy.

##### Post-treatment classification strategy.

We classify participants’ post-treatment critical thinking state as follows.
We require participants to write an incentivized essay discussing their
preferences on the issue at hand. Subjects are instructed to present the issue
and articulate their argumentative stance.191919See Appendix to see the
specific instructions to participants. Their payment is based on the quality
of their essay as measured by a Large Language Model (generative AI) powered
software, Grammarly.

Although Grammarly is efficient in assessing the overall quality of writing
(at the time of our experiment, still powered by models similar to GPT-3), it
lacks the capacity to capture the nuances of critical thinking, especially in
terms of discerning whether the writer demonstrates an awareness of the
ambivalence surrounding the issue. To address this limitation, we ask
cognitive psychologists with Ph.D. degrees to provide a professional
assessment of the essays. These experts are randomly assigned participants’
essays and are asked to evaluate whether the essay reflects a state of
awareness or not, assigning a pass or fail grade accordingly. While
participants receive payment based on the AI’s evaluation, our analysis
focuses on the cognitive psychologists’ assessment, with AI’s scores serving
as a robustness check (see Section 4.3).

Treatment | $A$ | $S$
---|---|---
|  Knowledge Test Score $>$ $\tau_{KTS}$ |
before | Issue Familiarity = 1 | Else
|  Reasons List $>$ $\tau_{RL}$ |
after | Psychologists Grade = Pass | Else
Table 1: Classification strategy before/after treatment

###  3.2 Measuring Cognitive Styles

Since we are interested in explaining possible drivers of our results, through
the experiment, we measure participants’ cognitive styles and correlate those
with the treatments effectiveness. We measure cognitive styles along three
metrics. The first two are standard in the psychology literature: the Need for
Cognitive Scale (NCS) and the Cognitive Flexibility Scale (CFS).

NCS measures a participant’s willingness to think deeply. It was proposed by
neuroscientists, and cognitive psychologists Cacioppo and Petty (1982) and
have become a gold standard in cognitive psychology. It comprises a series of
six questions that each receive a score between 1 and 5. We compare the
aggregate score to the sample average to classify the participants into a high
or low need for cognition.202020No average for the US population is available
for this score, despite being used widely across the social sciences and
psychology. In addition, it was originally developed as a 34-question version,
but the authors developed a shorter, more efficient 18-question version to
elicit other psychological characteristics during the same laboratory session.
Since then, it has been considered the benchmark scale widely used in
cognitive and social sciences. An even shorter 6-question version has been
tested and validated, allowing it to be implemented in a field survey
experiment in which the participant’s attention is even more scarce. We will
use this later.

CFS measures an agent’s ability to switch between thoughts and courses of
action. It was proposed by cognitive psychologist Martin and Rubin (1995) and
is a standard scale in cognitive psychology. It comprises a series of six
questions that each receive a score between 1 and 6. We compare the aggregate
score to the US population’s average to classify participants into high or low
cognitive flexibility.212121The average is provided by the authors: 55.

Finally, we use the AI generated score of an essay, unrelated to the core
issue of our experiment, measuring individual’s abilities to coherently
present an argument.

###  3.3 Description of Treatments

Participants are randomly assigned to one of four treatments. These treatments
contain the same content (i.e., the same selection of facts about the digital
issue, but they differ in the semantic style and graphic design, as elaborated
in the introduction) and last the same amount of time.

To recap, such formats range from the semantically crudest presentation of
facts to the most refined presentation. The twitter treatment presents them
most crudely through a “tweet” format. The facebook treatment uses the format
of “facebook posts.” The newspaper treatment presents them in the most refined
way through “newspaper articles.” The partisan Twitter treatment uses only a
partisan Twitter format (either only pros or cons)222222In the appendix we
detail and provide examples of each treatment..

Before the treatment starts, participants are explicitly informed that despite
their high resemblance to real news, the tweets and Facebook posts are fake.
At the end of the experiment, participants were debriefed and reminded that
the tweets and Facebook posts were fake, following common practice in
behavioral and experimental economics and as per our Institutional Review
Board (IRB) approval.

###  3.4 Incentive Mechanisms and Quality Screening

##### Incentive mechanism.

In the experiment, participants receive two types of payments. First, they
receive a fixed reward of $2 for completing the experiment fully by having
answered the comprehension questions correctly, guaranteed. Second, they
receive a bonus payment, at most $6, as described below.

Most of the participants’ bonus payments (up to 5$; participants’ performance
in the writing exercise, which captures their critical thinking process,
determines their bonus). We ask the participants to write two short essays
during this study that will be graded from 0 to 100 points by the artificial
intelligence (AI)-powered software, Grammarly.232323We, the authors, confirm
to have neither professional ties nor a business contract with this company.
See the Appendix for a summary of how this AI works. We divide the bonus
payment into two parts.

The biggest part (from $0 to $5) is proportional to the weighted average score
in the essay writing task; the second essay receives more weight (2/3) because
it requires more writing (400 characters as opposed to 200 characters). The
score can vary from 0 to 100 points, and the reward will be proportional to
the score. If the participants get a score of 0, they win $0. If they get a
score of 50, then they win $2.50. If they get a score of 100, then they win
$5. An essay that receives a low score from the AI can still earn a high score
regarding critical thinking and awareness, despite the writer’s difficulty
with English. In the instructions to the psychologist graders, we define and
exemplify what we mean by an “ambivalent issue,” “realizing that the issue is
ambivalent,” and “critical thinking”. We also run robust checks with
philosophers.

To be eligible for the remaining bonus payment (up to $1), participants must
receive at least an average score of 50/100 in the essay exercise in addition
to the bonus from the writing essay. Such a requirement ensures that
participants do the exercise seriously; cheaters and agents that are
inconsistent in their preferences are not eligible for this bonus payment. The
participant’s performance in the knowledge test determines this additional
bonus. The test consists of 10 questions, and each participant receives $0.10
for each question answered correctly.

##### Monitoring Algorithms for Cheating Behavior.

We implement three attention screeners as it is standard in online
experimental economics. The core of our experiment is for participants to
write an original essay by themselves. We need the subjects to avoid accessing
external information during the writing task. As such, we implement two
algorithms to monitor cheating behavior.

Before starting their experiment and on par with the IRB, we inform
participants that they must not access external information during the
experiment, particularly during the knowledge test and essay exercise. In
addition, their essay must be original. Failing to do so would be considered
“cheating behavior.” As such, they would be red-flagged and prevented from
receiving anything other than the fixed payment. We excluded such participants
from our data analysis.242424We provide both algorithms as open source in our
GitHub.

The first algorithm tracks the number of times that the participants open a
new tab on their computer during the essay exercise and how much time they
spend on our essay writing web-page252525For legal privacy purposes, we did
not access the content of the opened tab but gathered only the following
information: ’participant $i$ has opened a new tab during the essay, $n$
number of times, for such and such period $t$.. The second algorithm checks
whether the participants copy-paste external information by comparing the
number of written characters and the number of keyboard clicks. If the number
of keyboard clicks is strictly inferior to the number of written characters,
it implies that the participants have copy-pasted external information. This
second algorithm cannot distinguish between original external
information262626In the situation in which some participants had already
written on the topic or a relevant topic and saved it on their computer before
coming to the experiment. and plagiarism. Therefore, we use a feature in the
AI software to check for plagiarism after the participants have finished the
experiment.

##  4 Main Results

###  4.1 Storytelling formats Affect Critical Thinking

We now test whether storytelling formats have a role in the critical thinking
process of individuals. To this end, we compute for each treatment
$i=\\{newspaper,twitter,facebook\\}$, the frequency $\hat{\lambda}_{i}$ with
which agents subject to format $i$ transition from critical thinking state $S$
to critical thinking state $A$. Formally,

$\displaystyle\hat{\lambda}_{i}=\frac{\\#(S\rightarrow
A)_{i}}{\\#(S\rightarrow A)_{i}+\\#(S\rightarrow S)_{i}}$

We use the estimated intensities to perform a difference-in-means test of the
null hypotheses $\lambda_{i}=\lambda_{j}$ for all possible combinations of
treatments $\\{i,j\\}$.

Table 2 collects point estimates and confidence intervals. From this table, we
observe that the only significant difference is between Facebook and Twitter,
where the former performs better in transitioning subjects from critical
thinking state $S$ to $A$. Through this significant result, we establish that
the format impacts the critical thinking process. By being exposed to a
different way of presenting the same basic information, individuals realize
the ambivalent nature of the issue at hand differently. In section –, we
perform robustness checks of this result (different thresholds, etc., metrics
for success) to understand potential drivers of this effect.

Treatment | newspaper | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 1.332 | -0.865
|  | (0.054) | (0.054)
twitter | $\cdot$ | $\cdot$ |  -2.249**
|  |  | (0.053)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.1$, ** $p<0.05$, *** $p<0.01$ |
Table 2: t-ratio difference-in-means

A possible explanation for the observed difference in the impact of the
storytelling formats on critical thinking, as highlighted in Table 2, is that
the treatment twitter may rely on a format that is too simplistic or naive to
nudge users towards critical thinking effectively. This explanation can be
further elaborated as follows.

First, while Twitter imposes a character limit on its content, forcing users
to use concise language and simplifying complex ideas, Facebook allows for
longer and more detailed post272727This experiment was designed and launched
before Musk Twitter’s area, which led to the increase of tweets lengths for
Blue Twitter users, which now can be considered as our Facebook treatment..
This difference in content structure could impact how individuals process
information and engage in critical thinking.

Second, the fast-paced nature of Twitter feeds, and the emphasis on real-time
information sharing could discourage users from pausing, reflecting, and
analyzing the content they consume. This constant influx of new information
might contribute to a shallower engagement with the material, reducing the
likelihood of critical thinking.

Third, Twitter’s focus on short, attention-grabbing headlines and sound bites
may encourage users to form quick opinions based on surface-level information
rather than delving deeper into the nuances of an issue. This aspect of the
platform’s design might hinder the development of well-informed perspectives
and critical thinking.

Fourth, the prevalence of echo chambers on Twitter, where users primarily
follow and interact with those who share their views, could further contribute
to the observed limitations of the Twitter format in promoting critical
thinking. This selective exposure to information might reinforce pre-existing
beliefs and discourage users from challenging their assumptions.

Fifth, another factor to consider is the nature of user engagement on these
platforms. Facebook is known for fostering more personal connections and
allowing in-depth conversations, while Twitter primarily emphasizes short,
quick information exchanges. This contrast in user engagement could contribute
to the observed difference in the effectiveness of the storytelling formats on
critical thinking.

Finally, the role of media consumption habits might be influential in
explaining the difference in critical thinking outcomes. Users of Facebook may
be more inclined to read longer posts and engage in reflective thinking,
whereas Twitter users might be more accustomed to quickly skimming through
bite-sized information. As a result, individuals’ media consumption habits
could shape their receptiveness to the storytelling formats on these
platforms, ultimately affecting their critical thinking process.

###  4.2 Heterogeneity in Cognitive Styles

We explore whether the cognitive traits we elicited explain the differential
effect by conducting a split-sample difference in means. We test whether
$\lambda_{i}=\lambda_{j}$ by partitioning our sample into high or low
individuals in our cognitive metrics. The idea is that a more in-depth (like
the journal article) might be more effective for individuals more prone to
think deeply.The efficacy of the Facebook treatment was driven by its
differential impact on High Need for Cognition agents.

Treatment | NP | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 0.764 |  -2.238*
|  | (0.070) | (0.079)
twitter | $\cdot$ | $\cdot$ |  -3.087**
|  |  | (0.075)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 3: t-ratio for High Need for Cognition.

These results suggest that subjects who are the most affected by the
storytelling format are those who exhibit a high need for cognition. For them,
treatment facebook seems to provide the right format to maximally capture
their attention to present an issue so that it successfully nudges them into
performing the critical thinking process.

One possible explanation for the results observed in Table 3 could be rooted
in the characteristics of individuals with a High Need for Cognition. These
individuals typically exhibit a greater tendency to engage in effortful
cognitive activities and prefer more complex information processing (Cacioppo
and Petty (1982)). Consequently, the Facebook format might provide a more
stimulating environment for critical thinking by offering a richer and more
nuanced presentation of information than the Twitter format.

Furthermore, it has been suggested that individuals with a higher need for
cognition are more likely to seek out, attend to, and remember information
consistent with their attitudes and beliefs (Hass & Linder, 1981). As a
result, the Facebook format could be more effective in capturing their
attention and motivating them to evaluate the content critically. This might
explain why the Facebook treatment significantly impacts transitioning
subjects from critical thinking state S to A among those with a High Need for
Cognition.

Future research could explore the specific features of the Facebook format
that contribute to its efficacy in promoting critical thinking among
individuals with a high need for cognition. For instance, it would be
interesting to investigate the role of multimedia elements, interactivity, and
the integration of diverse information sources in fostering an environment
conducive to critical thinking.

###  4.3 Robustness Analysis

We address two potential challenges to ensure the robustness of our findings –
threshold sensitivity and writing similarity checks – that we present now.

##### Threshold sensitivity.

Our conclusions should remain consistent regardless of the specific values of
the threshold used in characteristics i) and iii) of the three-pronged test we
use to classify participants pre-treatment. Recall that i) refers to the
digital knowledge test and iii) refers to the reasons listing exercise.

Regarding i), in the baseline, we require participants to score at least 7
correct answers out of 10 questions. Comparing to the original setting
provided by Pew Research, our threshold is much more demanding. The quiz by
Pew Research was launched on a large US representative sample of 4,272 adults
living in the United States. The median number of correct answers was four.
Only 20% of adults answered seven or more questions correctly, and just 2% got
all 10 questions correct. Despite this difference, still, we are interested to
check whether our treatment effectiveness depends on scoring higher or lower
than scoring 7 out of 10.

Regarding iii), in the baseline, we require participants to be capable of
listing at least one reason for one side (pro or con) and two reasons for the
other side (pro or con). We are interested to check whether our treatment
effectiveness depend on the capacity of participants to list more than one
reason for each side.

##### Writing similarity.

Our findings should not be influenced by the similarity in length between the
essay task and any specific treatment, particularly the Facebook treatment. By
comparing outcomes across different essay lengths or imposing length
constraints, we can verify that the observed effects are not artifacts of such
similarities, ensuring the robustness of our results.

Overall, our robustness analysis confirm that the treatment effectiveness does
not depend neither on the threshold sensitivity test nor on the writing
similarity test.282828We provide the analysis in the appendix.

##  5 Conclusion

In this paper, we built a simple but flexible model to measure the gain in
election efficiency by becoming aware that the issue is ambivalent ($A$). We
experimented and determined that the format in which news is presented affects
a person’s transition into $A$. This effect is driven by individuals with a
high need for cognition (the flexibility scale is insignificant). Realizing
the ambivalent nature of an issue is an essential step in discovering one’s
stable preference since it improves the “quality” of one’s preference from raw
to stable. As such, critical thinking is _also_ good for the efficiency of
elections.

What is broadly referred to as a storytelling format (e.g., newspapers,
television, social media, social echo chambers) might impact the probability
of realizing ambivalence. Beyond “informing” and “persuading,” it also affects
an individual’s critical thinking process. Additionally, the format in which
news is presented—many short messages vs. more coherent but greedy attention
discourse—matters. In particular, unexplored (by us) physiological drivers
were correlated with standard metrics of cognition/flexibility.

Stable preferences $y$ are _not observable_ , and the model is not (fully)
identified. We rely on a reduced form for an identified model with three
cognitive stages $S\to A\to T$ and a final transition to a stable preference
(resolving awareness) with qualitatively similar results. First, $=A$ realized
the issue was ambivalent but still did not find our $y$. Second, how did
voters in $A$ vote? Strategic voting in the presence of stereotype bias, “I
still have not resolved my awareness about [topic], but I see much prejudice
in favor of position $0$, so I vote $1$ to compensate.” Additionally, stable
preference $y$ is independent of other individual types
($\beta,\xi_{A},\lambda\dots$). Prejudices often coincide with a stable
preference if the latter is $1$. $\beta_{1}>\beta_{0}$. Prejudices might
correlate with the likelihood of becoming aware (Ortoleva and Snowberg
(2015)).

Regarding the internal validity of our experiment, we recognize that
classifying individuals’ critical thinking states is inherently challenging.
We devised different classification rules for pre- and post-treatment to avoid
mistaking memory for critical thinking. Second, we used a noisy measure to
look at the difference across treatments. Regarding the external validity of
our experiment, readers should refrain from interpreting our experiment as a
comparison of social media, concluding that “Facebook is better” but rather
“the format matters.” In this interpretation, the whole class of social media
becomes a storytelling format: one is exposed to a greater number of views,
but they are possibly superficial. Does it help to become aware of the issue’s
ambivalence relative to one’s life experience or the in-depth study of a topic
(more personal and reasoned but time-consuming and unlikely to occur)?

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[sections] [sections]l1

##  Appendix A Proofs of The Main Model

###  A.1 Preliminary Results on $\hat{p}$ and alike

The following three steps explicitly show how to analyse the evolution of the
cognitive state process over time for each agent and how this relates to the
parameters of the model.

1) $\mu_{S}=\exp\left\\{-\lambda_{1}t\right\\}$ and $\mu_{C}=1-\mu_{S}$
represent the masses. $\lambda_{1}$ represents the intensity with which agents
pass from the cognitive state $S$ to the cognitive state $A$ over time.
Moreover, we define the unknown parameter $\bar{p}$ as function of $\mu$ and
$p$

$\displaystyle\overline{p}\left(\mu,p\right)$
$\displaystyle=\mu_{S}\left(\mathbb{E}\left[x_{S}\left|p\right.\right]\right)+\mu_{C}\left(\mathbb{E}\left[x_{C}\left|p\right.\right]\right)$
$\displaystyle=\mu_{S}\left(\beta
p_{S}+\left(1-\beta\right)p\right)+\mu_{C}\left(\xi_{C}p+\left(1-\xi_{C}\right)\left(1-p\right)\right)$
$\displaystyle=\mu_{S}\left(\beta
p_{S}+\left(1-\beta\right)p\right)+\mu_{C}\left(1-p-\xi_{C}\left(1-2p\right)\right)$

Thus

$\overline{p}\left(\mu,p\right)=\mu_{S}\left(\beta
p_{S}+\left(1-\beta\right)p\right)+\mu_{C}\left(1-p-\xi_{C}\left(1-2p\right)\right)$
(5)

From which we can derive the expression for $p$ as a function of $p_{S}$

$\displaystyle\bar{p}$ $\displaystyle=\mu_{S}\beta
p_{S}+\mu_{S}\left(1-\beta\right)p+\mu_{C}-\mu_{C}p-\mu_{C}\xi_{C}+2\mu_{C}p\xi_{C}$
$\displaystyle\bar{p}$ $\displaystyle=\mu_{S}\beta
p_{S}+\mu_{C}-\mu_{C}\xi_{C}+p\left[\mu_{S}\left(1-\beta\right)-\mu_{C}\left(1-2\xi_{C}\right)\right]$
$\displaystyle=\mu_{S}\beta
p_{S}+\mu_{C}-\mu_{C}\xi_{C}+p\left[\mu_{S}\left(1-\beta\right)-\mu_{C}\left(1-2\xi_{C}\right)\right]$

Thus $p$ is defined as

$p=\frac{\bar{p}-\mu_{S}\beta
p_{S}-\mu_{C}\left(1-\xi_{C}\right)}{\mu_{S}\left(1-\beta\right)-\mu_{C}\left(1-2\xi_{C}\right)}$
(6)

Finally, we can define the parameter $\hat{p}$ that is defined as the
expectation of $p$ conditioning on $\bar{p}$

$\hat{p}\&=\frac{\bar{p}-\left[\mu_{S}\beta\mathbb{E}\left[p_{S}|\bar{p}\right]+\mu_{C}\left(1-\xi_{C}\right)\right]}{\mu_{S}\left(1-\beta\right)+\mu_{C}\left(2\xi_{C}-1\right)}$
(7)

Thus $\bar{p}$ is defined as

$\hat{p}=\mathbb{E}\left[p|\alpha_{1}p+\alpha_{2}p_{S}=\bar{p}\right]$ (8)

$\displaystyle\hat{p}$
$\displaystyle=\frac{\beta^{2}\mu_{S}^{2}\mu_{X}\sigma_{Y}^{2}+\sigma_{X}^{2}\left(1-\bar{p}-\xi_{C}+\mu_{S}\left(-1+\beta\mu_{Y}+\xi_{C}\right)\right)\left(1-2\xi_{C}+\mu_{S}\left(-2+\beta+2\xi_{C}\right)\right)}{\beta^{2}\mu_{S}^{2}\sigma_{Y}^{2}+\sigma_{X}^{2}\left(1-2\xi_{C}+\mu_{S}\left(-2+\beta+2\xi_{C}\right)\right)^{2}}$
$\displaystyle=\frac{\beta^{2}\mu_{S}^{2}\mu_{X}\sigma_{Y}^{2}+\sigma_{X}^{2}\left(\bar{p}-\left[\mu_{S}\beta\mu_{y}+\left(1-\mu_{S}\right)\left(1-\xi_{C}\right)\right]\right)\left(1-2\xi_{C}+\mu_{S}\left(-2+\beta+2\xi_{C}\right)\right)}{\beta^{2}\mu_{S}^{2}\sigma_{Y}^{2}+\sigma_{X}^{2}\left(1-2\xi_{C}+\mu_{S}\left(-2+\beta+2\xi_{C}\right)\right)^{2}}$

2) It is worth noting that the NWF can be expressed as the sum of the PWF and
a biased term due to the elections. Indeed,

$\displaystyle NWF$
$\displaystyle=-\mathbb{E}\left[(p-\bar{p})^{2}\right]=-\mathbb{E}\left[\left(p-\hat{p}+\hat{p}-\bar{p}\right)^{2}\right]$
$\displaystyle=-\mathbb{E}\left[(p-\hat{p})^{2}+2(p-\hat{p})(\hat{p}-\bar{p})+(\hat{p}-\bar{p})^{2}\right]$
$\displaystyle=-\left[\mathbb{E}\left[(p-\hat{p})^{2}\right]+2\mathbb{E}\left[(p-\hat{p})(\hat{p}-\bar{p})\right]+\mathbb{E}\left[(\hat{p}-\bar{p})^{2}\right]\right]$
$\displaystyle=-\left[\mathbb{E}\left[(p-\hat{p})^{2}\right]+2(\hat{p}-\bar{p})\underbrace{\mathbb{E}\left[(p-\hat{p})\right]}_{0}+\mathbb{E}\left[(\hat{p}-\bar{p})^{2}\right]\right]$
$\displaystyle=-\left[\underbrace{\mathbb{E}\left[(p-\hat{p})^{2}\right]}_{\text{Precision
of
elections}}+\underbrace{\mathbb{E}\left[(\hat{p}-\bar{p})^{2}\right]}_{\text{Bias
of elections}}\right]$

Thus it can be rewritten as

$NWF=PWF+Bias$

At this stage, we define the two welfare functions given the distributions of
the parameters

3)What the theoretical analysis wants to show is the evolution of the welfare
functions over time and the main differences between the evolution of the PWF
and the NWF. In particular, in order to study the evolution, we take the first
derivative of the two functions with respect to $\mu_{S}$. It is necessary and
sufficient to show the sign of this derivative in order to have an all rounded
understanding of the evolution of the two functions. Indeed, $\mu_{S}$ as
defined above depends negatively on $t$ and $\lambda_{1}$. Hence, once we
define the relation between the functions and $\mu_{S},$ we immediately get to
know the relation between the functions and the time/lambda. Thus let’s start
from showing the behavior of the PWF

$\displaystyle\frac{\partial PWF}{\partial\mu_{S}}$ $\displaystyle=$
$\displaystyle\frac{2\beta^{2}\mu_{S}\sigma^{2}\left(-1+2\xi_{C}\right)\left[1-2\xi_{C}+\mu_{S}\left(-2+\beta+2\xi_{C}\right)\right]}{\left\\{2\mu_{S}^{2}\left[\beta^{2}+2\beta\left(-1+\xi_{C}\right)+2\left(-1+\xi_{C}\right)^{2}\right]+\left(1-2\xi_{C}\right)^{2}-2\mu_{S}\left(-1+2\xi_{C}\right)\left(-2+\beta+2\xi_{C}\right)\right\\}^{2}}$
$\displaystyle\propto
1-2\xi_{C}+\mu_{S}\left(\beta-2\left(1-\xi_{C}\right)\right)$

Since almost everything is bigger or equal than $0$, if we want to study the
sign of the above formula, then we just have to analyse the sign of the
following term

$\displaystyle 1-2\xi_{C}+\mu_{S}\left(-2+\beta+2\xi_{C}\right)$
$\displaystyle<0$ $\displaystyle 0<\mu_{S}$
$\displaystyle<\underbrace{\frac{2\xi_{C}-1}{-2+\beta+2\xi_{C}}}_{\geq 1?}$

###### Proposition 3

$W_{P}$ is increasing in $t$ and $\lambda$ if

$\frac{2\xi_{C}-1}{-2+\beta+2\xi_{C}}>1\iff 1>\beta$

Let’s study the right hand side of the inequality

$\displaystyle 2\xi_{C}-1$ $\displaystyle\geq-2+\beta+2\xi_{C}$
$\displaystyle\beta$ $\displaystyle\leq 1$

Therefore, we can conclude that PWF is decreasing in $\mu_{S}$ for each time
$t$, because

$0<\mu_{S}<\frac{2\xi_{C}-1}{-2+\beta+2\xi_{C}},\;\;\;\forall\mu_{S}\in[0,1]$

In other words, the PWF is an increasing function of both $t$ and
$\lambda_{1}$.

###  A.2 Proof of Proposition 1

###### Proof 1

Notice preliminary that using the chain rule the following result is valid for
both welfare functions

$\frac{\text{d}W}{\text{d}\lambda}=\frac{\text{d}W}{\text{d}\eta}\cdot\underbrace{\frac{\text{d}\eta}{\text{d}\lambda}}_{<0}\implies\frac{\text{d}W}{\text{d}\lambda}\propto-\frac{\text{d}W}{\text{d}\eta}$

and therefore welfare moves in $\lambda$ (and $t$) contrary to how it moves in
the share of stereotypes. First, for positive welfare,
$\frac{\text{d}W}{\text{d}\eta}$ is always negative for the following
computations

$\begin{split}\frac{\partial PWF}{\partial\mu_{S}}=\\\
\frac{2\beta^{2}\mu_{S}\sigma^{2}\left(-1+2\xi_{C}\right)\left[1-2\xi_{C}+\mu_{S}\left(-2+\beta+2\xi_{C}\right)\right]}{\left\\{2\mu_{S}^{2}\left[\beta^{2}+2\beta\left(-1+\xi_{C}\right)+2\left(-1+\xi_{C}\right)^{2}\right]+\left(1-2\xi_{C}\right)^{2}-2\mu_{S}\left(-1+2\xi_{C}\right)\left(-2+\beta+2\xi_{C}\right)\right\\}^{2}}\\\
\propto
1-2\xi_{C}+\mu_{S}\left(\beta-2\left(1-\xi_{C}\right)\right)\end{split}$

Since almost everything is bigger or equal than $0$, if we want to study the
sign of the above formula, then we just have to analyse the sign of the
following term

$\displaystyle 1-2\xi_{C}+\mu_{S}\left(-2+\beta+2\xi_{C}\right)$
$\displaystyle<0$ $\displaystyle 0<\mu_{S}$
$\displaystyle<\underbrace{\frac{2\xi_{C}-1}{-2+\beta+2\xi_{C}}}_{\geq 1?}$

Let’s study the right hand side of the inequality

$\displaystyle 2\xi_{C}-1$ $\displaystyle\geq-2+\beta+2\xi_{C}$
$\displaystyle\beta$ $\displaystyle\leq 1$

Therefore, we can conclude that PWF is decreasing in $\mu_{S}$ for each time
$t$, because

$0<\mu_{S}<\frac{2\xi_{C}-1}{-2+\beta+2\xi_{C}},\;\;\;\forall\mu_{S}\in[0,1]$

Furthermore, $\frac{\text{d}W}{\text{d}\eta}$ has a nontrivial solution. That
is by studying the sign of the derivative of welfare elections with respect to
$\mu_{S}$ we obtain

$\frac{\partial
NWF}{\partial\mu_{S}}=-\left[4\beta^{2}\mu_{S}\sigma^{2}+4\beta\left(-1+\mu_{S}\right)\sigma^{2}\left(-1+\xi_{C}\right)+4\beta\mu_{S}\sigma^{2}\left(-1+\xi_{C}\right)+2\left(-1+\mu_{S}\right)\left[\left(1-2\mu\right)^{2}+4\sigma^{2}\right]\left(-1+\xi_{C}\right)^{2}\right]$

The sign of the term in brackets is

$4\beta^{2}\mu_{S}\sigma^{2}+4\beta\sigma^{2}\left(-1+\xi_{C}\right)\left(-1+2\mu_{S}\right)+2\left(-1+\mu_{S}\right)\left[\left(1-2\mu\right)^{2}+4\sigma^{2}\right]\left(-1+\xi_{C}\right)^{2}>0$

$\mu_{S}\left[4\beta^{2}\sigma^{2}+8\beta\sigma^{2}\left(-1+\xi_{C}\right)+2\left(-1+\xi_{C}\right)^{2}\left[\left(1-2\mu\right)^{2}+4\sigma^{2}\right]\right]>4\beta\sigma^{2}\left(-1+\xi_{C}\right)+2\left(-1+\xi_{C}\right)^{2}\left[\left(1-2\mu\right)^{2}+4\sigma^{2}\right]$

$\mu_{S}>\underbrace{\frac{4\beta\sigma^{2}\left(-1+\xi_{C}\right)+2\left(-1+\xi_{C}\right)^{2}\left[\left(1-2\mu\right)^{2}+4\sigma^{2}\right]}{4\beta^{2}\sigma^{2}+8\beta\sigma^{2}\left(-1+\xi_{C}\right)+2\left(-1+\xi_{C}\right)^{2}\left[\left(1-2\mu\right)^{2}+4\sigma^{2}\right]}}_{\text{Threshold}<1?}$

Saying that the threshold is less than one also means that $\frac{\partial
NWF}{\partial\mu_{S}}<0\iff\mu_{S}>\text{Threshold}.$ We want to study this
threshold. The conditions given in the text correspond to such threshold being
below $0$ and above, $1$, respectively.

$\begin{cases}COND1\rightarrow\text{Threshold}<0&\text{Always decreases in
time}\\\ COND2\rightarrow\text{Threshold}>1&\text{Always increase in time}\\\
COND3\rightarrow\text{Threshold}\in\left(0,1\right)&\text{Increases first,
decreases later}\end{cases}$

COND1 occurs according to the following expression

$\beta>\frac{\left(1-\xi_{C}\right)\left(\left(1-2\mu\right)^{2}+4\sigma^{2}\right)}{2\sigma^{2}}$

then welfare is always decreasing.
For COND2 to occur the numerator of the threshold must be higher than the
denominator. Hence, since there are only two terms differing between numerator
and denominator, the following must be true

$\displaystyle 4\beta\sigma^{2}\left(-1+\xi_{C}\right)$
$\displaystyle>4\beta^{2}\sigma^{2}+8\beta\sigma^{2}\left(-1+\xi_{C}\right)$
$\displaystyle 4\beta\sigma^{2}\left(-1+\xi_{C}\right)$
$\displaystyle>4\beta\sigma^{2}\left(\beta+2\xi_{C}-2\right)$
$\displaystyle\beta$ $\displaystyle<1-\xi_{C}$

then welfare is always increasing.
Finally, COND3 can be discussed intuitively. Since $\mu_{S}$ is monotonically
decreasing in time, there must be by continuity a $t^{max}$ such that

$\begin{cases}\frac{\partial W^{N}}{\partial\mu_{S}}<0&\text{for }t<t^{max}\\\
\frac{\partial W^{N}}{\partial\mu_{S}}<0&\text{for }t>t^{max}\end{cases}$

therefore, $t^{max}$ is a maximum interior of $W^{N}$ when Threshold
$\in\left(0,1\right)$

###  A.3 Proof of Proposition 2

###### Proof 2

Firstly, define the parameters associated with $\bar{p}$

$\bar{p}=\mu_{T}p+\mu_{S}\left(\beta
p_{S}+\left(1-\beta\right)p\right)+\mu_{C}\left[1-p+\xi_{C}\left(2p-1\right)\right]$

where

$\displaystyle\alpha_{0}$ $\displaystyle=\mu_{C}\left(1-\xi_{C}\right)$
$\displaystyle\alpha_{1}$
$\displaystyle=1-\beta\mu_{S}-2\mu_{C}\left(1-\xi_{C}\right)$
$\displaystyle\alpha_{2}$ $\displaystyle=\beta\mu_{S}$

Then $\hat{p}$ is given by

$\hat{p}=\frac{\frac{\bar{p}-\left[\alpha_{0}+\alpha_{2}\mu_{y}\right]}{\alpha_{1}}\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}\mu_{x}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$

where

$\displaystyle\gamma_{0}(t,\lambda)$
$\displaystyle=\frac{\alpha_{2}^{2}\sigma_{y}^{2}\mu_{x}-\alpha_{2}\alpha_{1}\sigma_{x}^{2}\mu_{y}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$
$\displaystyle\gamma_{1}\left(t,\lambda\right)$
$\displaystyle=\frac{\alpha_{1}^{2}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$
$\displaystyle\gamma_{2}(t,\lambda)$
$\displaystyle=\frac{\alpha_{2}\alpha_{1}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$

The bias is zero if and only if the following system has a solution

$\begin{cases}\alpha_{1}=\gamma_{1}\\\ \alpha_{2}=\gamma_{2}\end{cases}$

that is

$\begin{cases}\alpha_{1}=\frac{\alpha_{1}^{2}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}\\\
\alpha_{2}=\frac{\alpha_{2}\alpha_{1}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}\end{cases}$

It is immediate to check that
$\gamma_{1}\left(t,\lambda\right)=\alpha_{1}\left(t,\lambda\right)\iff\gamma_{2}\left(t,\lambda\right)=\alpha_{2}\left(t,\lambda\right)$,
so we actually have a single equation and we need to claim that exists a time
such that

$\displaystyle\frac{\alpha_{1}^{2}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$
$\displaystyle=\alpha_{1}\iff\frac{\left(1-\beta\mu_{S}-2\mu_{C}\left(1-\xi_{C}\right)\right)^{2}\sigma_{x}^{2}}{\left(1-\beta\mu_{S}-2\mu_{C}\left(1-\xi_{C}\right)\right)^{2}\sigma_{x}^{2}+\left(\beta\mu_{S}\right)^{2}\sigma_{y}^{2}}=\left(1-\beta\mu_{S}-2\mu_{C}\left(1-\xi_{C}\right)\right)$
$\displaystyle\iff\left(1-\beta\mu_{S}-2\mu_{C}\left(1-\xi_{C}\right)\right)\sigma_{x}^{2}=\left(1-\beta\mu_{S}-2\mu_{C}\left(1-\xi_{C}\right)\right)^{2}\sigma_{x}^{2}+\left(\beta\mu_{S}\right)^{2}\sigma_{y}^{2}$
$\displaystyle\iff\left(1-\beta\mu_{S}-2\mu_{C}\left(1-\xi_{C}\right)\right)\sigma_{x}^{2}\left(\beta\mu_{S}+2\mu_{C}\left(1-\xi_{C}\right)\right)=\left(\beta\mu_{S}\right)^{2}\sigma_{y}^{2}$
$\displaystyle\iff\frac{\left(1-\beta\mu_{S}-2\mu_{C}\left(1-\xi_{C}\right)\right)\left(\beta\mu_{S}+2\mu_{C}\left(1-\xi_{C}\right)\right)}{\left(\beta\mu_{S}\right)^{2}}=\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$
$\displaystyle\iff\frac{\left(1-\beta\mu_{S}-2\left(1-\mu_{S}\right)\left(1-\xi_{C}\right)\right)\left(\beta\mu_{S}+2\left(1-\mu_{S}\right)\left(1-\xi_{C}\right)\right)}{\left(\beta\mu_{S}\right)^{2}}$

When $\mu_{S}=0$ there cannot be the zero-bias time, because as
$t\rightarrow\infty$ this explodes (? can we show this is always increasing in
$\mu_{S}$) because in the limit there is always bias. On the other hand, there
could be a zero-bias time that coincides with $t^{\star}=0$. Indeed, when
$\mu_{S}=1$

$\frac{1-\beta}{\beta}=\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$

An even more special case is when $\xi_{C}=1$. Indeed,

$\displaystyle\frac{\left(1-\beta\mu_{S}\right)^{2}\sigma_{x}^{2}}{\left(1-\beta\mu_{S}\right)^{2}\sigma_{x}^{2}+\left(\beta\mu_{S}\right)^{2}\sigma_{y}^{2}}$
$\displaystyle=\left(1-\beta\mu_{S}\right)$
$\displaystyle\left(1-\beta\mu_{S}\right)\sigma_{x}^{2}$
$\displaystyle=\left(1-\beta\mu_{S}\right)^{2}\sigma_{x}^{2}+\left(\beta\mu_{S}\right)^{2}\sigma_{y}^{2}$
$\displaystyle\left(\frac{1-\beta\mu_{S}}{\beta\mu_{S}}\right)$
$\displaystyle=\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$

Substituting the expression of $\mu_{S}$as a function of $t$ and $\lambda$

$\frac{\sigma_{x}^{2}}{\beta\left(\sigma_{x}^{2}+\sigma_{y}^{2}\right)}=e^{-t\lambda_{1}}$

that becomes

$t^{\star}=-\frac{1}{\lambda_{1}}\log\left(\frac{\sigma_{x}^{2}}{\beta\left(\sigma_{x}^{2}+\sigma_{y}^{2}\right)}\right)$

where the argument of the log must be smaller than 1

$\frac{\sigma_{x}^{2}}{\beta\left(\sigma_{x}^{2}+\sigma_{y}^{2}\right)}<1$

that is

$\frac{\left(1-\beta\right)}{\beta}<\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$

##  Appendix B Experimental Design Details

###  B.1 Treatments Details and Examples

In the newspaper treatment, the participants are exposed to two news articles:
one that is for and one that is against the issue. In the facebook treatment,
participants were exposed to six Facebook posts: two for and two against an
issue as well as two irrelevant posts. In treatment twitter, participants are
exposed to twenty-four tweets: ten for digital privacy, ten against digital
privacy, and four irrelevant tweets. Each tweet has an average length of 40
characters, corresponding to 20 words.292929This corresponds to the average
length of tweets on twitter.com, see the Appendix for details.. We give
participants 5 seconds to read each tweet before the next one automatically
pops until the last one, which is in line with the average reading speed in
the US population. In the partisan twitter treatment, participants are exposed
to 13 tweets: 10 for and 3 irrelevant ones or 10 against and 3 irrelevant
ones. Within each treatment, tweets, Facebook posts, and news articles arrive
in a random order sequentially (one by screen) and remain on screen for a
given fixed amount of time (the participant cannot move to the next screen by
him or herself). Each participant is randomly assigned to one of the
treatments.

see the online appendix about participant’s experimental instructions.

###  B.2 Detailed Data Collection

##### Preventing duplicates.

Submissions to studies on Prolific are guaranteed to be unique by the
firm303030See Prolific unique submission guarantee policy here.. Our system is
set up such that each participant can have only one submission per study on
Prolific. That is, each participant will be listed in your dashboard only
once, and can only be paid once. On our side, we also prevent participants to
take up several times our experiment in two steps. First, we enable the
functionality “Prevent Ballot Box Stuffing” which permits to…Second we check
participant ID and delete the second submission from the data set of the same
ID if we find any.

##### Drop-out rates.

Here put the drop out (or in the main text).

##### High vs low-quality submissions.

Participants joining the Prolific pool receive a rate based on the quality of
their engagement with the studies. If they are rejected from a study then they
receive a malus. If they receive too much malus, then they are removed by the
pool from the company313131See Prolific pool removal Policy here.. Based on
this long term contract, participants are incentivized to pay attention and
follow the expectations of each study. Hence, a good research behavior has
emerged on Prolific according to which, participants themselves can vol
voluntarily withdraw their submissions if they feel they did a mistake such as
rushing too much, letting the survey opened for a long period of time without
engaging with it, and so on323232See Prolific update regarding this behavior
here.. According to these standards, we kept submissions rejections as low as
possible, following standard in online experimental economics. Participants
who fail at least one fair attention check are rejected and not paid.
Following Prolific standards, participants who are statistical outliers (3
standard deviations below the mean) are excluded from the good complete data
set.

##### Payments and communication.

We make sure to review participants’ submissions within within 24-48 hours
after they have completed the study. This means that within this time frame,
if we accept their submission, they receive their fixed and bonus payment.
Otherwise, we reject their submissions and send to them a personalized
e-mail(333333Partially-anonymized through Prolific messaging app which put the
researcher’s name visible to the participants and only the participants ID
visible to the researcher.), detailing the reason of the rejection, leaving
participants the opportunity to contact us afterwards if they firmly believe
the decision to be unfair (motivate their perspective). Participants can also
contact us at any time if they encounter problems with our study or just have
questions about it.

###  B.3 Detailed Elicitations

####  B.3.1 Political Preferences

##### Ex-ante and ex-post political preferences.

Before the treatment, we survey participants’ preferences (self-reported) on
political issues: guns, crime, climate, welfare, and digital privacy issues.
We use the standard congressional metrics, including digital issues. We elicit
more than only digital preferences to ensure that participants do not guess at
this stage which preferences we focus on in the remaining of the experiment
(treatment and critical thinking essay), to minimize their social desirability
bias. After the treatment on digital privacy, we survey again participants to
elicit their preferences about digital privacy. We use the following scale.

  1. 1.

On the issue of gun regulation, do you support or oppose the following
proposal?

  2. 2.

On the issue of environmental policies, do you support or oppose the following
proposal?

  3. 3.

On the issue of crime policies, do you support or oppose the following
proposal?

  4. 4.

On the issue of digital policies, do you support or oppose each of the
following proposals?

####  B.3.2 Digital Knowledge Test

see the online appendix about participant’s experimental instructions.

####  B.3.3 Issue Familiarity

  1. 1.

In the remainder of the experiment, we will focus on the following political
issue. Please state again your preference.

  2. 2.

Have you thought deeply about this issue before participating in this study?
[Yes/No]

####  B.3.4 Listing Reasons

If $y$es to the previous question, then participants see this question:

You answered “Yes” to the previous question. you will be asked now to provide,
at most, two reasons which justify your position and two reasons which justify
the opposite position. If you do not know any reasons, please select “I am
unable to list any logical reason at the moment”. you do not need to agree
with these reasons: they just need to be a logical justification for or
against your position. your payment WILL NOT depend on your answer to this
question. However, your honest answer is of paramount importance for the
success of this study.

  1. 1.

Reasons which justify your position

     * •

Reason 1: [write text here]

     * •

Reason 2: [write text here]

     * •

I am unable to list any logical reason at the moment

  2. 2.

Reasons which oppose your position

     * •

Reason 1: [write text here]

     * •

Reason 2: [write text here]

     * •

I am unable to list any logical reason at the moment

####  B.3.5 Internal Uncertainty

How certain are you of your preference regarding the digital privacy issue? By
“Certain”, we mean that you feel confident enough to vote for your political
preference if asked to you in a real life political committee. Select among
the following options:

  * •

Completely Uncertain

  * •

Rather Uncertainty

  * •

Rather Certain

  * •

Completely Certain

####  B.3.6 Need for Cognition

For each sentence below, please select how uncharacteristic or characteristic
this is for you personally.

####  B.3.7 Cognitive Flexibility

####  B.3.8 Habits of News Consumptopn

see the online appendix about participant’s experimental instructions.

###  B.4 Detailed Description of Graders’ Instructions

We recruited 20 psychologists (doctoral level or above) specializing in
cognitive psychology at Princeton University. Each grader was randomly
assigned a “grading treatment” (i.e., a set of essays to grade). Such a set of
essays was randomly built, containing essays from all four treatments.
Additionally, graders were not informed about which treatment the subjects
were assigned. Psychologists must grade a very short paragraph (around 300
words or fewer) as follows. The grading consists of giving a passing grade if
psychologists judge that the participant “realizes that the issue is
ambivalent,” a failing grade otherwise. What may happen is to confound high
cognitive sophistication (i.e., the ability to write well-written essays in
English), facilitated by the fact that they read some arguments right before
this essay exercise with their self-reasoning skill “realizing that the issue
is ambivalent”, which is the variable that we want to elicit. This is a
specific case that is still challenging for AI-based grading software and the
main reason why human expertise is uniquely useful.

We define “realizing that the issue is ambivalent” as the awareness of an
individual to recognize that there can be perfectly logical but opposite
arguments in favor of and against the same issue that renders the decision-
making process complex. Such attitudinal ambivalence leads to temporarily
conflicting preferences; namely, one preference for the issue at hand and one
preference against the issue at hand. There are different ways of measuring
this “awareness,” as documented in the social psychology and cognitive
psychology literature. In our study, we capture this awareness by observing
individuals reasoning and elaborating in a personal way on the pros and cons
of the same issue in a textual format.

Each grader was paid a fixed fee of $50 for each grading session. Each grader
could participate up to three times in our experiment, and no grader could be
assigned twice to the same grading treatment. For robustness, each essay was
corrected three times by different psychologists. Despite “triple-eliciting”
such grades, this metric can still be prone to measurement error. Accordingly,
we suggest interpreting the estimated _levels_ of $\lambda$ with caution.
However, our focus is on the difference between the treatments. Therefore,
such measurement error does not affect this difference.

see the online appendix about participant’s experimental instructions.

###  B.5 Heterogeneous Critical Thinking Classification

####  B.5.1 Critical Thinking Classification Results

Table 4 shows the classification results of of individuals as Stereotype and
Aware.

Treatment | $S_{0}\rightarrow S_{1}$ | $S_{0}\rightarrow A_{1}$ | $A_{0}\rightarrow A_{1}$
---|---|---|---
newspaper | 111 | 49 | 12
twitter | 135 | 43 | 15
facebook | 111 | 60 | 11
$N$ | 357 | 152 | 38
Table 4: Table 2: Classification results before/after treatment

####  B.5.2 Awareness With Cognitive Styles Heterogeneity with Unanimity

Awareness With Cognitive Flexibility, with Unanimity

Treatment | NP | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 1.301 | 1.661
|  | (0.091) | (0.095)
twitter | $\cdot$ | $\cdot$ | 0.422
|  |  | (0.093)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 5: t-ratio for High Need for Cognition with Unanimity Treatment | NP | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 0.320 | -0.388
|  | (0.066) | (0.065)
twitter | $\cdot$ | $\cdot$ | -0.965
|  |  | (0.064)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 6: Table 6: t-ratio for Low Need for Cognition with Unanimity

Awareness With Need for Cognition, with Unanimity

Treatment | NP | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 1.061 | -2.238*
|  | (0.084) | (0.084)
twitter | $\cdot$ | $\cdot$ | -1.300
|  |  | (0.083)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 7: Table 6: t-ratio for High Need for Cognition with Unanimity Treatment | NP | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 0.455 | 0.705
|  | (0.069) | (0.069)
twitter | $\cdot$ | $\cdot$ | 0.258
|  |  | (0.069)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 8: Table 6: t-ratio for Low Need for Cognition with Unanimity

####  B.5.3 Awareness With Cognitive Styles Heterogeneity with Majority

Awareness With Cognitive Flexibility, with Majority

Treatment | NP | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 0.658 | -0.924
|  | (0.076) | (0.087)
twitter | $\cdot$ | $\cdot$ | -1.609
|  |  | (0.081)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 9: Table 6: t-ratio for High Flexibility Treatment | NP | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 1.132 | -0.388
|  | (0.062) | (0.064)
twitter | $\cdot$ | $\cdot$ | -1.564
|  |  | (0.061)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 10: Table 7: t-ratio for Low Flexibility

Awareness With Need for Cognition, with Majority

Treatment | newspaper | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 0.764 | -2.238*
|  | (0.070) | (0.079)
twitter | $\cdot$ | $\cdot$ | -3.087**
|  |  | (0.075)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 11: Table 4: t-ratio for high need for cognition Treatment | newspaper | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 1.094 | 0.703
|  | (0.066) | (0.067)
twitter | $\cdot$ | $\cdot$ | -0.396
|  |  | (0.063)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 12: Table 5: t-ratio for low need for cognition

###  B.6 Threshold changes

Treatment | newspaper | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 1.278 | -0.923
|  | (0.054) | (0.054)
twitter | $\cdot$ | $\cdot$ |  -2.262*
|  |  | (0.053)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 13: t-ratio difference-in-means with threshold of KTS = 8 and Reason Counter = 2 Treatment | newspaper | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 1.222 | -0.799
|  | (0.054) | (0.054)
twitter | $\cdot$ | $\cdot$ | -2.067*
|  |  | (0.053)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 14: t-ratio difference-in-means with threshold of KTS = 7 and Reason Counter = 3 Treatment | newspaper | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 1.136 | -0.703
|  | (0.049) | (0.052)
twitter | $\cdot$ | $\cdot$ | -1.985*
|  |  | (0.049)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 15: t-ratio difference-in-means with threshold of KTS = 7 and Reason Counter = 2 Treatment | newspaper | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 1.428 | -0.507
|  | (0.049) | (0.052)
twitter | $\cdot$ | $\cdot$ | -1.985*
|  |  | (0.049)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 16: t-ratio difference-in-means with threshold of KTS = 6 and Reason
Counter = 3

###  B.7 AI Digital Grade

The last robustness check that we perform is to split the sample conditioning
on the the grade of the essay on the digital topic evaluated by the algorithm
of the AI.Indeed, the essay is written after undertaking the experiment and it
might influence the writing quality of the essay. In particular, our reasoning
is that if no difference in proportion is statistically significant, this
means that there is no systematic difference between those who were treated
through Facebook and those through Twitter, and indeed from Table 17 this is
the case.

Treatment | newspaper | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 1.716 | 0.234
|  | (0.078) | (0.081)
twitter | $\cdot$ | $\cdot$ | -1.565
|  |  | (0.074)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 17: t-ratio with High AI Digital Grades Treatment | newspaper | twitter | facebook
---|---|---|---
newspaper | $\cdot$ | 0.098 | -1.093
|  | (0.061) | (0.067)
twitter | $\cdot$ | $\cdot$ | -1.207
|  |  | (0.065)
facebook | $\cdot$ | $\cdot$ | $\cdot$
Standard errors in parentheses |
* $p<0.05$, ** $p<0.01$, *** $p<0.001$ |
Table 18: t-ratio with Low AI Digital Grades

##  Appendix C Model With Three-State Critical Thinking

We propose an additional model where agents can be at three different states
of critical thinking: not engaging with critical thinking, performing critical
thinking (either in its first or second state), and having finished performing
critical thinking. In our two-stage model, we considered performing critical
thinking as having finished performing it. In this scenario, we propose a
three-stage (not fully identified) model that considers the three stages
distinctively.

In this economy, the object of interest is the distribution of stable
preferences over a binary policy space in a large population. Each individual
$j$ inside the population is characterized by a three-dimensional type

$\left(x_{j},y_{j},i_{j}\right)\in\mathcal{J}\coloneqq\left\\{0,1\right\\}\times\left\\{0,1\right\\}\times\left\\{0,1\right\\}$

where $x_{j}$, represents the stereotypical preference individual $j$ would
self-report when presented with an ambivalent issue for the first time – that
is, by definition, before undergoing a critical thinking phase; $y_{j}$
differs potentially from $x_{j}$ as it represents the stable preference that
$j$ holds after completing their period of critical thinking; the cognitive
type $i_{j}$ refers to the cognitive type $i_{j}$, interacting with the
format, determines how easily individual $j$ moves into (and out of) critical
thinking.

Individuals go through a three-step process of “critical thinking” as they
form their preferences. The process begins with a “stereotypical-self” state,
followed by a period of critical thinking, and ultimately leading to a
“stable-self” state. We assume that this process is irreversible and that once
individuals reach a stable-self state, they no longer question their
preferences. There is no additional “information” that has to come and change
their worldview: the process of critical thinking provides a final and stable
answer to ambivalent issues. When asked to report their preferences on a
policy issue, individuals in either their stereotypical self or stable self
state will vote according to their respective preferences, $x_{j},y_{j}$,
respectively. Those who are still in the critical thinking phase will abstain
from voting.

The transition between the different phases is determined by an individual’s
cognitive style and the characteristics of the storytelling format. Hence, the
storytelling format is instrumental in the agent’s transition from a
stereotypical state to the stable one. By constructing our model, this
transition is captured by the critical thinking phase. An economy of stable
preferences is preferable from efficiency and welfare perspectives to an
economy of stereotypical preferences. We formally present such an economy
below.

###  C.1 Model Identification

Using reported preferences of individuals that do the $S(Stereotype)\to
T(Type)$ transition (i.e. we observe ex ante $x_{S}$ then $y$), we get

$\mathbb{E}\left[x_{S}\left|y=1\right.\right]=\left(1-\beta\right)+\beta
p_{S}$ $\mathbb{E}\left[x_{S}\left|y=0\right.\right]=\beta p_{S}$

which gives the estimators

$\hat{\beta}=1-\left(\bar{x}_{S\left|1\right.}-\bar{x}_{S\left|0\right.}\right)$

and

$\hat{p}_{S}=\frac{\bar{x}_{S\left|0\right.}}{\hat{\beta}}$

clearly $\hat{p}=\bar{y}$. Finally, using the reported preferences of
individuals that do the $A\to T(Type)$ transition we can estimate $\xi_{A}$ as

$\mathbb{E}\left[x_{A}\left|y=1\right.\right]=\xi_{A}$
$\mathbb{E}\left[x_{A}\left|y=0\right.\right]=1-\xi_{A}$

so $\hat{\xi}_{A}=\bar{x}_{NU\left|1\right.}$ or
$\hat{\hat{\xi}}_{A}=1-\bar{x}_{NU\left|0\right.}$. Notice that we can test
the assumed symmetry by testing that $\hat{\xi}_{A}=\hat{\hat{\xi}}_{A}$.
Since in our dataset we have few agents that start in $A$ this test has almost
no power.

###  C.2 General Results

The basic decomposition

$\displaystyle W_{E}$ $\displaystyle=W_{P}+Bias$
$\displaystyle-\mathbb{E}\left[\left(p-\bar{p}\right)^{2}\right]$
$\displaystyle=-\left(\mathbb{E}\left[\left(p-\hat{p}\right)^{2}\right]+\mathbb{E}\left[\left(\hat{p}-\bar{p}\right)^{2}\right]\right)$

is still clearly valid. However, $\bar{p}$ is now given by

$\displaystyle\bar{p}$ $\displaystyle=\mu_{T}p+\mu_{S}\left(\beta
p_{S}+\left(1-\beta\right)p\right)+\mu_{A}\left[1-p+\xi_{A}\left(2p-1\right)\right]$
$\displaystyle=\alpha_{0}\left(t,\lambda\right)+\alpha_{1}\left(t,\lambda\right)p+\alpha_{2}\left(t,\lambda\right)p_{S}$

with

$\displaystyle\alpha_{0}$ $\displaystyle=\mu_{A}\left(1-\xi_{A}\right)$
$\displaystyle\alpha_{1}$
$\displaystyle=1-\beta\mu_{S}-2\mu_{A}\left(1-\xi_{A}\right)$
$\displaystyle\alpha_{2}$ $\displaystyle=\beta\mu_{S}$

where $\hat{p}$ (that was wrong in the previous file since for non-normal
random variables we do not know the expectation of $p$ given the convex
combination $\beta p_{s}+\left(1-\beta\right)p$) is given by

$\hat{p}=\frac{\frac{\bar{p}-\left[\alpha_{0}+\alpha_{2}\mu_{y}\right]}{\alpha_{1}}\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}\mu_{x}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}=\tilde{\alpha_{0}}\left(t,\lambda\right)+\tilde{\alpha_{1}}\left(t,\lambda\right)p+\tilde{\alpha_{2}}\left(t,\lambda\right)p_{S}$
$\frac{\frac{\alpha_{0}\left(t,\lambda\right)+\alpha_{1}\left(t,\lambda\right)p+\alpha_{2}\left(t,\lambda\right)p_{S}-\left[\alpha_{0}+\alpha_{2}\mu_{y}\right]}{\alpha_{1}}\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}\mu_{x}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$

so

$\tilde{\alpha_{1}}\left(t,\lambda\right)=\frac{\alpha_{1}^{2}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$
$\tilde{\alpha_{2}}\left(t,\lambda\right)=\frac{\alpha_{2}\alpha_{1}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$

when is it

$\displaystyle\tilde{\alpha_{1}}\left(t,\lambda\right)$
$\displaystyle=\alpha_{1}\left(t,\lambda\right)\iff\frac{\alpha_{1}^{2}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}=\alpha_{1}\iff$
Same $\sigma$
$\displaystyle=\alpha_{1}\left(1-\alpha_{1}\right)=\alpha_{2}^{2}\iff\left(1-\beta\mu_{S}\right)\left(\beta\mu_{S}\right)=\left(\beta\mu_{S}\right)^{2}$

It is immediate to check that
$\tilde{\alpha_{1}}\left(t,\lambda\right)=\alpha_{1}\left(t,\lambda\right)\iff\tilde{\alpha_{2}}\left(t,\lambda\right)=\alpha_{2}\left(t,\lambda\right)$
so we actually have a single equation and we need to claim that $\exists$ time
such that

$\displaystyle\frac{\alpha_{1}^{2}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$
$\displaystyle=\alpha_{1}\iff\frac{\left(1-\beta\mu_{S}-2\mu_{A}\left(1-\xi_{A}\right)\right)^{2}\sigma_{x}^{2}}{\left(1-\beta\mu_{S}-2\mu_{A}\left(1-\xi_{A}\right)\right)^{2}\sigma_{x}^{2}+\left(\beta\mu_{S}\right)^{2}\sigma_{y}^{2}}=\left(1-\beta\mu_{S}-2\mu_{A}\left(1-\xi_{A}\right)\right)$
$\displaystyle\iff\left(1-\beta\mu_{S}-2\mu_{A}\left(1-\xi_{A}\right)\right)\sigma_{x}^{2}=\left(1-\beta\mu_{S}-2\mu_{A}\left(1-\xi_{A}\right)\right)^{2}\sigma_{x}^{2}+\left(\beta\mu_{S}\right)^{2}\sigma_{y}^{2}$
$\displaystyle\iff\left(1-\beta\mu_{S}-2\mu_{A}\left(1-\xi_{A}\right)\right)\sigma_{x}^{2}\left(\beta\mu_{S}+2\mu_{A}\left(1-\xi_{A}\right)\right)=\left(\beta\mu_{S}\right)^{2}\sigma_{y}^{2}$
$\displaystyle\iff\frac{\left(1-\beta\mu_{S}-2\mu_{A}\left(1-\xi_{A}\right)\right)\left(\beta\mu_{S}+2\mu_{A}\left(1-\xi_{A}\right)\right)}{\left(\beta\mu_{S}\right)^{2}}=\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$

now substituting $\mu_{S},\mu_{A}$ we have the LHS is increasing to $\infty$
in $t$, therefore there is a unique solution provided that it starts below
$\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$, that is if
$\frac{1-\beta}{\beta}<\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$ ($\beta$ is
large enough)

as $\mu_{S}\to 0$, this explodes (there is always bias in the limit), while at
the beginning there is zero bias iff

$\frac{1-\beta}{\beta}=\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$

[example, $\sigma_{x}^{2}=\frac{1}{2},\sigma_{y}^{2}=\frac{1}{6}$
$\beta=\frac{3}{4}\implies\frac{1-\beta}{\beta}=\frac{1}{3}$ ]

this is the zero-bias time. Even more special cases $\xi_{A}=1$

$\frac{\left(1-\beta\mu_{S}\right)^{2}\sigma_{x}^{2}}{\left(1-\beta\mu_{S}\right)^{2}\sigma_{x}^{2}+\left(\beta\mu_{S}\right)^{2}\sigma_{y}^{2}}=\left(1-\beta\mu_{S}\right)$
$\left(1-\beta\mu_{S}\right)\sigma_{x}^{2}=\left(1-\beta\mu_{S}\right)^{2}\sigma_{x}^{2}+\left(\beta\mu_{S}\right)^{2}\sigma_{y}^{2}$
$\left(\frac{1-\beta\mu_{S}}{\beta\mu_{S}}\right)=\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$

Since the LHS is decreasing in $\mu_{S}$ and the RHS is increasing, then there
is at most one solution. It has none if

$\frac{1-\beta}{\beta}>\frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}$

Furthermore we get

$\displaystyle W_{P}$
$\displaystyle=-\mathbb{E}\left[\left(p-\hat{p}\right)^{2}\right]=-\frac{\alpha_{2}^{2}\sigma_{y}^{2}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$
$\displaystyle\text{If }\sigma_{x}^{2}=\sigma_{y}^{2}=\sigma^{2}$
$\displaystyle=-\frac{\alpha_{2}^{2}}{\alpha_{1}^{2}+\alpha_{2}^{2}}\sigma^{2}$
$\displaystyle\xi_{A}=1$
$\displaystyle=\frac{\left(\beta\mu_{S}\right)^{2}}{2\beta\mu_{S}\left[1-\beta\mu_{S}\right]+1}\sigma^{2}??$

#### Aside: No Bias

Condition for no bias is that coefficients in $\bar{p}$ are the same as in
$\hat{p}$ that is,

$If\exists t:B\left(t\right)=0$ $\alpha_{0}+\alpha_{1}p+\alpha_{2}p_{S}$
$\frac{\frac{\alpha_{1}p+\alpha_{2}p_{S}-\alpha_{2}\mu_{y}}{\alpha_{1}}\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}\mu_{x}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}=\frac{\alpha_{2}^{2}\sigma_{y}^{2}\mu_{x}-\alpha_{2}\alpha_{1}\sigma_{x}^{2}\mu_{y}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}+\frac{\alpha_{1}^{2}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}p+\frac{\alpha_{2}\alpha_{1}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}p_{S}$
$\alpha_{0}=\frac{\alpha_{2}^{2}\sigma_{y}^{2}\mu_{x}-\alpha_{2}\alpha_{1}\sigma_{x}^{2}\mu_{y}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}=\mu\frac{\alpha_{2}^{2}\sigma_{y}^{2}-\alpha_{2}\alpha_{1}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$
$\alpha_{1}=\frac{\alpha_{1}^{2}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$
$\alpha_{2}=\frac{\alpha_{2}\alpha_{1}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$

notice that if $\beta=\frac{1}{2}$ then at $t=0$ we have a solution iff
$\sigma$ are the same at $t=0$,

$\displaystyle\alpha_{0}$ $\displaystyle=0$ $\displaystyle\alpha_{1}$
$\displaystyle=\frac{1}{2}$ $\displaystyle\alpha_{2}$
$\displaystyle=\frac{1}{2}$
$\alpha_{0}=\frac{\alpha_{2}^{2}\sigma_{y}^{2}\mu_{x}-\alpha_{2}\alpha_{1}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$
$\alpha_{1}=\frac{\alpha_{1}^{2}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$
$\alpha_{2}=\frac{\alpha_{2}\alpha_{1}\sigma_{x}^{2}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$

and

$\mathbb{E}\left[\hat{p}|p\right]=\frac{\mathbb{E}\left[\frac{\alpha_{1}p+\alpha_{2}p_{S}-\alpha_{2}\mu_{y}}{\alpha_{1}}\right]\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}\mu_{x}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}=\frac{\alpha_{1}^{2}\sigma_{x}^{2}p+\alpha_{2}^{2}\sigma_{y}^{2}\mu_{x}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}$

and

$\mathbb{E}\left[\hat{p}\right]=\frac{\alpha_{1}^{2}\sigma_{x}^{2}\mu_{x}+\alpha_{2}^{2}\sigma_{y}^{2}\mu_{x}}{\alpha_{1}^{2}\sigma_{x}^{2}+\alpha_{2}^{2}\sigma_{y}^{2}}=\mu_{x}$

#### Welfare Expressions

The general formula is in the mathematica file, under the restriction
$\mu_{x}=\mu_{y}$ and $\sigma_{x}=\sigma_{y}$ we get

$W_{E}=-\left[\left(\alpha_{0}-\left(1-\alpha_{1}-\alpha_{2}\right)\mu\right)^{2}+\left(\left(1-\alpha_{1}\right)^{2}+\alpha_{2}^{2}\right)\sigma^{2}\right]$

We have welfare at $t=0$, where $\mu_{S}=1$. Namely

$W_{E}=-\beta^{2}\left(\underbrace{\left(\mu_{x}-\mu_{y}\right)^{2}}_{\text{Prior
Bias}}+\sigma_{x}^{2}+\sigma_{y}^{2}\right)$
$W_{P}=-\beta^{2}\frac{\sigma_{x}^{2}\sigma_{y}^{2}}{\sigma_{x}^{2}\left(1-\beta\right)^{2}+\sigma_{y}^{2}\beta^{2}}$

Then,

$\displaystyle\frac{W_{E}}{W_{P}}$
$\displaystyle=\frac{\left(\mu_{x}-\mu_{y}\right)^{2}+\sigma_{x}^{2}+\sigma_{y}^{2}}{\frac{\sigma_{x}^{2}\sigma_{y}^{2}}{\sigma_{x}^{2}\left(1-\beta\right)^{2}+\sigma_{y}^{2}\beta^{2}}}$
Assume equal $\sigma$
$\displaystyle=\frac{\left(\mu_{x}-\mu_{y}\right)^{2}+2\sigma^{2}}{\frac{\sigma^{2}}{\left(1-2\beta+2\beta^{2}\right)}}=\underbrace{\frac{\left(\mu_{x}-\mu_{y}\right)^{2}}{\frac{\sigma^{2}}{\left(1-2\beta+2\beta^{2}\right)}}}_{>0}+2\left(1-2\beta+2\beta^{2}\right)^{2}>2\left(\frac{1}{2}\right)=1$

so if $\mu_{x}=\mu_{y}$ (no prior bias), then
$W_{E}\left(0\right)=W_{P}\left(0\right)$ iff $\sigma_{x}=\sigma_{y}$.

#### Results

$W_{E}>W_{P}$ this is because the bias/variance decomposition

$\displaystyle W$
$\displaystyle=-\mathbb{E}\left[\left(p-\bar{p}\right)^{2}\right]=-\mathbb{E}\left[\left(\left(1-\mu_{T}-\mu_{A}\left[2\xi_{A}-1\right]-\mu_{S}\left(1-\beta\right)\right)p+\beta\mu_{S}p_{S}+\mu_{A}\left(1-\xi_{A}\right)\right)^{2}\right]$
$\displaystyle=-\mathbb{E}\left[\left(p-\hat{p}+\hat{p}-\bar{p}\right)^{2}\right]=-\left(\mathbb{E}\left[\left(p-\hat{p}\right)^{2}\right]+\mathbb{E}\left[\left(\hat{p}-\bar{p}\right)^{2}\right]+\xcancel{2\mathbb{E}\left[\left(p-\hat{p}\right)\left(\hat{p}-\bar{p}\right)\right]}\right)$
$\displaystyle=-\left(\underbrace{\mathbb{E}\left[\left(p-\hat{p}\right)^{2}\right]}_{\text{Precision
of
election}}+\underbrace{\mathbb{E}\left[\left(\hat{p}-\bar{p}\right)^{2}\right]}_{\text{Bias
of elections}}\right)$

finally holds, the election have a bias.

The full characterization of the derivative (assuming equal $\mu$ and
$\sigma$)

$\frac{\text{d}}{\text{d}t}W_{E}|_{t=0}=-4\beta\lambda_{1}\sigma^{2}\left(1-\beta-\xi_{A}\right)$

Instead assuming only equal $\mu$ we have

$\frac{\text{d}}{\text{d}t}W_{E}|_{t=0}=-2\beta\lambda_{1}\left(\beta\sigma_{y}^{2}-\sigma_{x}^{2}\left(2\left(1-\xi_{A}\right)-\beta\right)\right)$

so

$\frac{\text{d}}{\text{d}t}W_{E}|_{t=0}>0\iff 1-\beta<\xi_{A}$

or in general

$\frac{\beta}{2\left(1-\xi_{A}\right)-\beta}<\frac{\sigma_{x}^{2}}{\sigma_{y}^{2}}$

a sensible condition. Also, $\lambda_{1}$ magnifies either the positive or the
negative change local to $0$ and in particular if $1-\beta>\xi_{A}$ then more
$\lambda_{1}$ is bad for welfare local to $t=0$. To the contrary,

$\frac{\text{d}}{\text{d}t}W_{P}|_{t=0}=\frac{2\left(1-\beta\right)\beta^{2}\lambda_{1}\sigma^{2}\left(2\xi_{A}-1\right)}{\text{sthg}^{2}}>0$

and the welfare of the unconstrained principal is [but this is just a
conjecture not falsified by Math plots] always increasing in both
$\lambda_{1},t$.

Conjecture

$W_{P}$ is increasing in $t$ (and $\lambda_{1}$)— We show that

$\displaystyle\frac{\text{d}}{\text{d}t}W_{P}$
$\displaystyle\propto-\left[\underbrace{2\left(1-\xi_{A}\right)\mu_{S}}_{+}\underbrace{\frac{\text{d}}{\text{d}t}\mu_{A}}_{?}+\underbrace{\left(1-2\left(1-\xi_{A}\right)\mu_{A}\right)}_{+}\underbrace{\frac{\text{d}}{\text{d}t}\mu_{S}}_{-}\right]$
$\displaystyle=\underbrace{\exp\left\\{-\left(\lambda_{1}+\lambda_{2}\right)t\right\\}}_{+}\lambda_{1}\left(2\left(1-\xi_{A}\right)-\exp\left\\{\lambda_{2}t\right\\}\right)2\left(1-\xi_{A}\right)-\exp\left\\{\lambda_{2}t\right\\}<2\left(1-\xi_{A}\right)-1$
$\displaystyle=1-2\xi_{A}<0$

when computed in

$\frac{\text{d}}{\text{d}\lambda_{1}}W_{P}\propto-\left[\underbrace{2\left(1-\xi_{A}\right)\mu_{S}}_{+}\underbrace{\frac{\text{d}}{\text{d}\lambda_{1}}\mu_{A}}_{?}+\underbrace{\left(1-2\left(1-\xi_{A}\right)\mu_{A}\right)}_{+}\underbrace{\frac{\text{d}}{\text{d}\lambda_{1}}\mu_{S}}_{-}\right]$

which has the same sign as

$\frac{\text{d}}{\text{d}\lambda_{1}}W_{P}\propto-\exp\left\\{\lambda_{2}t\right\\}\lambda_{2}^{2}t+2\lambda_{2}\left[\left(\exp\left\\{\lambda_{2}t\right\\}\lambda_{1}t\right)+\left(1-\xi_{A}\right)\exp\left\\{\left(\lambda_{2}-\lambda_{1}\right)t\right\\}-\left(1-\xi_{A}\right)\left(1+\lambda_{1}t\right)\right]-\lambda_{1}^{2}t\left(\exp\left\\{\lambda_{2}t\right\\}-2\left(1-\xi_{A}\right)\right)$

Furthermore, analyse $\xi_{A}$:

$\displaystyle\xi_{A}$
$\displaystyle=-\exp\left\\{\lambda_{2}t\right\\}\lambda_{2}^{2}t+2\lambda_{2}\left[\left(\exp\left\\{\lambda_{2}t\right\\}\lambda_{1}t\right)\right]-\lambda_{1}^{2}t\left(\exp\left\\{\lambda_{2}t\right\\}\right)$
$\displaystyle=-t\exp\left\\{\lambda_{2}t\right\\}\left(\lambda_{2}-\lambda_{1}\right)^{2}+2\lambda_{2}\left(1-\xi_{A}\right)\left[\exp\left\\{\left(\lambda_{2}-\lambda_{1}\right)t\right\\}-\left(1+\lambda_{1}t\right)\right]+2\lambda_{1}^{2}t\left(1-\xi_{A}\right)$
$\displaystyle=\underbrace{-t\exp\left\\{\lambda_{2}t\right\\}\left(\lambda_{2}-\lambda_{1}\right)^{2}}_{\textit{negative}}+2\lambda_{2}\left(1-\xi_{A}\right)\left[\exp\left\\{\left(\lambda_{2}-\lambda_{1}\right)t\right\\}-\left(1+\lambda_{1}t\right)+2\lambda_{1}^{2}t\right]$

Now if the second addendum is negative then we are done; so assume it is
positive, that is

$\displaystyle\exp\left\\{\left(\lambda_{2}-\lambda_{1}\right)t\right\\}-\left(1+\lambda_{1}t\right)+2\lambda_{1}^{2}t>0$

then the sum is smaller than

$\displaystyle\underbrace{-t\exp\left\\{\lambda_{2}t\right\\}\left(\lambda_{2}-\lambda_{1}\right)^{2}}_{<0}+\lambda_{2}\left[\exp\left\\{\left(\lambda_{2}-\lambda_{1}\right)t\right\\}\left(1+\lambda_{1}t\right)+2\lambda_{1}^{2}t\right]$
$\displaystyle=\lambda_{1}^{2}t-\exp\left\\{\lambda_{2}t\right\\}\left(\lambda_{2}-\lambda_{1}\right)^{2}t+\lambda_{2}\left(-1+\exp\left\\{\left(\lambda_{2}-\lambda_{1}\right)t\right\\}-\lambda_{1}t\right)$
$\displaystyle=\lambda_{1}^{2}t+\lambda_{2}\exp\left\\{\left(\lambda_{2}-\lambda_{1}\right)t\right\\}-\left[\exp\left\\{\lambda_{2}t\right\\}\left(\lambda_{2}-\lambda_{1}\right)^{2}t+\lambda_{2}\left(1+\lambda_{1}t\right)\right]<0$

so it remains to show that this is always negative; if $\lambda_{1}\approx 0$

$\displaystyle-\lambda_{2}\left(1-\exp\left\\{\lambda_{2}t\right\\}\left(1-\lambda_{2}t\right)\right)<-\lambda_{2}^{2}t<0$

$W_{E}$ has interesting comparative statistics due to the interaction with
bias. In particular, it seems that for $\beta>\text{stgh}$, then [if there is
no prior bias, $\mu_{x}=\mu_{y}$] there is a time $t$ such that
$Bias\left(t\right)=0$ because the evolution of $\mu_{S},\mu_{A}$ is such that
$\alpha^{E}=\alpha^{P}$. This seems interesting, possibly a result to put in a
proposition.

Based on our model, we can draw three main results and one additional
interesting result.

##### Inefficiency of twitter economy and non-monotonicity in election times.

The first result relates to the political institutions of the digital economy.
From our model is that a Twitter-Facebook economy where everyone can speak
their mind is not necessarily good: indeed we want only those that went
through critical thinking to vote. Following this point, the naturally arising
question, _when do we want to hold elections_? Our model clearly implies non-
monotonicity in time for election periods.

##### Typology of voting-users and adverse selection.

The second result relates to the typology of voting-users. The “ _clients” of
news outlets_ , in a micro-foundation of the $\lambda$ functions are either
low $i$ partisans (that look at it for fun) or frustrated critical thinking
voting-users that look for some facts (positive predictions). On a related but
different point, we can identify the _adverse selection in the vote-force_
(under some conditions the strengths of the stereotype pool weakens), and how
the format amplifies / reduces this issue (always true that it is better if
only types vote, at least in the symmetric case).

##### Partisan format and compensation effect.

The third and most intriguing result relates to the format. We can study the
_impact of different storytelling formats_ (more in depth, helps the high $i$,
but how it correlates with $\alpha$): more in depth, but keep it somehow
primitive. In particular and more interestingly, we can allow for
_asymmetries_ : either there is a “better” policy (say $\beta=1$, so upon
reflecting everyone agrees $1$ is right), or stereotypes of one side are less
likely to enter critical thinking (evidence that conservatives are
overconfident), how does this change the outcome, as well as the incentives
for the critical thinking agents (that may vote for those that are less
confident because of the bias in the type pool). The problem of asymmetries is
that a partisan format, or is the fact that one stereotype is more attractive
than the other to make the problem of agents in critical thinking more
problematic: remember they are smart but unwise, so they cannot ignore the
fact of a stereotyped partisan pool, either because stereotypes are more
resistant, or because the shifts the stereotypes. Hence, we propose to explain
such a situation by an effect that we label the “Compensation Effect”: When
you perceive the device to be partisan in one direction, you vote in the
opposite direction when in critical thinking.

##### The benefits of making voting costly.

A resulting and potentially controversial consequence of such an asymmetry is
that voting costs in this situation may be positive because they can also
exclude the strategic types that recognize the stereotype pool is partisan and
cannot morally abstain or vote against their type. They can use the excuse not
to go to vote.

###  C.3 Proofs

##  Appendix D Experiment With A Three Cognitive-State Model

In the experiment, we gathered data to decompose the critical thinking process
into three stages: ${S,A,T}$. Here, $S$ remains unchanged. Now, $A$ denotes an
intermediate, transitory stage during which agents experience internal
uncertainty regarding the formation of their stable preferences. Finally, $T$
denotes the stage in which agents have completed the critical thinking process
and have formed their stable preferences.

$\bullet\,\,$ Short & Crude$\bullet\,\,$ Medium & Reasoned$\bullet\,\,$ Long &
Reasoned$\bullet\,\,$ Short, Bias & Crude
randomized$Part1$$Part2$$Part3$$Part4$$Part5$$Part6$4-step test4 storytelling
formatsCooling Period3-step testElicitation ofEx-Ante
TypesTreatmentElicitation of Cognitive StylesElicitation of Ex-Post
TypesIncentivizedIncentivized Demographics Ex-Ante Preferences Knowledge Test
Listing Reasons Issue Familiarity Internal Uncertainty Need For Cognition
Cognitive Flexibility Essay Writing Ex-Post Preferences Internal Uncertainty
News Habits

Table 19 shows the classification strategy of participants as Stereotype,
Aware, and Type.

Treatment | $T$ | $A$ | $S$
---|---|---|---
|  Knowledge Test Score $>$ $\tau_{KTS}$ |  |
before | Issue Familiarity = 1 |  Knowledge Test Score $>$ $\tau_{KTS}$ |
| Internal Uncertainty $\neq 0$ |  |
|  Reasons List $>$ $\tau_{RL}$ |  |
|  | Issue Familiarity = 1 |
after | Psychologists Grade = Pass | Else |
Table 19: Classification strategy before/after treatment

The analysis presents the frequencies of the three states of participants
before and after the treatment

| $S_{1}$ | $A_{1}$ | $T_{1}$
---|---|---|---
$S_{0}$ | 475 | 153 | 39
$A_{0}$ | 0 | 21 | 2
$T_{0}$ | 0 | 0 | 30
Table 20: Table: Frequencies before/after treatment