# News Media as Suppliers of Narratives
(and Information)††thanks: Spiegler acknowledges financial support from
Leverhulme Trust grant RPG-2023-120. We thank Alex Clyde, Daniel Gottlieb,
Shira Dvir-Gvirtsman, and the seminar audience at the Hebrew University for
helpful comments.

Kfir Eliaz and Ran Spiegler Eliaz: School of Economics, Tel-Aviv University
and David Eccles School of Business, University of Utah. E-mail:
<EMAIL_ADDRESS>Spiegler: School of Economics, Tel-Aviv University and
Economics Dept., University College London. E-mail<EMAIL_ADDRESS>
###### Abstract

We present a model of news media that shape consumer beliefs by providing
information (signals about an exogenous state) and narratives (models of what
determines outcomes). To amplify consumers’ engagement, media maximize
consumers’ anticipatory utility. Focusing on a class of separable consumer
preferences, we show that a monopolistic media platform facing homogenous
consumers provides a false “empowering” narrative coupled with an
optimistically biased signal. Consumer heterogeneity gives rise to a novel
menu-design problem due to a “data externality” among consumers. The optimal
menu features multiple narratives and creates polarized beliefs. These effects
also arise in a competitive media market model.

##  1 Introduction

Standard models of news media regard them as suppliers of information,
providing noisy signals of an underlying state of Nature. A complementary
view, which is absent from standard models, is that news media are a vehicle
for spreading narratives, as reflected in the following quotes by prominent
journalists:

> “It’s all storytelling, you know. That’s what journalism is all about.” (Tom
> Brokaw)
>
> “We’re supposed to be tellers of tales as well as purveyors of facts. When
> we don’t live up to that responsibility, we don’t get read.” (William
> Blundell)

“Stories” or “narratives” are of course loaded terms with rich meanings in the
context of news reporting. We conceive of narratives as models or frames that
condition media consumers’ thinking about the significance of reported
information. For example, while many exogenous variables can be reported, the
media often selects only some of them as relevant for outcomes of interest and
therefore worthy of reporting. Another example is the shaping of popular
perceptions about the role of personal agency and external factors in life
outcomes. In particular, when reporting about discrimination, the media can
peddle a narrative that focuses on the role of personal effort in achieving
material success. Alternatively, it can offer a narrative that attributes
economic outcomes solely to discrimination; or a complex narrative that
incorporates both factors.111This example echoes Glenn Loury’s (2020)
distinction between “development” and “bias” narratives.

A related example involves narratives about financial investments. A popular
narrative aimed at retail investors is that the value of their portfolio
depends on how they manage it (how active are they? how do they allocate the
portfolio between types of risk?). Such a narrative is misleading because it
neglects the possibility that market fundamentals (interest rates, business
sectors’ risk or growth potential) are already reflected in security prices
and thus affect investment returns. Thus, when financial-news media report
about market fundamentals (e.g., that certain sectors are experiencing
growth), the simplistic narrative may lead media consumers to draw wrong
conclusions from this information (e.g., that they should invest in a growth
sector).

This paper presents a model of news media (in a broad sense that includes
content platforms) that is based on a fusion of the two views: The media
provides information about exogenous states as well as a narrative, which is a
model of the determination of outcomes as a function of states and actions.
Media consumers use the narrative to interpret statistical regularities,
forming beliefs about the mapping from states and actions to outcomes. A false
narrative is a misspecified model, which can therefore induce distorted
beliefs.

The fusion of the information-based and narrative-based views enables us to
offer a new model of media bias. There is a common intuition that this
phenomenon is driven in large part by consumer demand (Gentzkow and Shapiro
(2010) back this intuition with empirical evidence). Yet, the standard model
of consumer behavior assumes that demand for information is purely
instrumental. Expected-utility maximizers weakly prefer more informative
signals. Therefore, unless there are frictions on the supply side that prevent
media from providing complete and objective information, the market will
provide it. Even if consumers have heterogeneous preferences, they all want
more informative news.

Studies across several disciplines (psychology, political science and
communication) have shown evidence that consumer demand for news media
reflects non-instrumental attitudes to beliefs (e.g., Hart et al. (2009), Van
der Meer et al. (2020), Taber and Lodge (2006)). These findings have inspired
models of media bias in which beliefs enter directly into consumers’ utility
function (see Prat and Strömberg (2013) and Gentzkow et al. (2015) for
surveys). We propose a model of demand for news in which consumers have
intrinsic preferences over their posterior beliefs, which they arrive at via
Bayesian updating of prior beliefs shaped by the narrative they adopt. To our
knowledge, this integration of non-instrumental and Bayesian aspects of
consumer beliefs is new to the literature on media bias. We discuss the
relation to relevant literature in Section 6.

One can distinguish between two types of intrinsic preferences over posterior
beliefs. Retrospective preferences rank beliefs about past outcomes — e.g.,
wanting to believe that one is the just side in a dispute (see Chopra et al.
(2023)). Prospective preferences rank beliefs about future outcomes, taking
into account private or public actions in response to beliefs. We focus on the
latter type: Consumers approach news media with the desire to maximize their
anticipatory utility — i.e., the expected indirect utility from their
posterior beliefs. According to this view, demand for news is driven by
consumers’ pursuit of optimistic posterior beliefs. By “optimism”, we do not
necessarily mean that the media paint a rosy picture of reality, but rather
that it gives hope that if the right decisions are made, future prospects will
be good. Real-life manifestations of this idea include patriotic coverage of
international conflicts or sporting events that amplify the prospect of
victory;222In discussing the popularity of patriotic coverage of the war in
Afganistan and Iraq, a New York Times story (Ruthenberg (2003)) quotes MSNBC’s
president Erik Sorenson: “After Sept. 11 the country wants more optimism and
benefit of the doubt…It’s about being positive as opposed to being negative.”
business-news media conveying the impression that retail financial investors
can beat the market; and reports of police brutality or climate change which
send a message that inflates the ability of policy reforms (“defunding the
police”, switching to green energy) to improve social welfare.

However, under the conventional assumption that news media only supply
information, prospective preferences over (Bayesian) posterior beliefs cannot
give rise to media bias. The reason is that expected anticipatory utility is
the upper envelope of linear functions of posterior beliefs (hence convex in
these beliefs). As a result, when the media caters to consumer demand by
offering a signal function that maximizes consumers’ anticipatory utility, it
will weakly prefer full information provision. Thus, even when we allow for
prospective non-instrumental demand for information, the standard view of the
media as information providers cannot generate media bias.

This is where our view of media as joint providers of narratives and
information enters. We show that this more comprehensive approach provides a
non-trivial model of media bias, such that distortion of the truth consists of
biased/inaccurate reports together with false narratives. Moreover, there is a
synergy between these two instruments: They complement each other in producing
the hopeful beliefs that consumers seek.

Overview of model and results

In the basic version of our model, a representative consumer takes an action
after observing a signal about a state of Nature. There is an objective
stochastic mapping from states and actions to outcomes. The consumer is
endowed with a vNM utility function which is separable in the action. A
monopolistic media outlet commits ex-ante to a “media strategy”, which
consists of: $(i)$ a Blackwell experiment (namely, a stochastic mapping from
states to signals), and $(ii)$ a narrative, which selects a subset of the
outcome’s true causes.

There are four feasible narratives. The true narrative acknowledges both
states and actions as causes. The “empowering” narrative postulates that
actions are the sole cause of outcomes (in terms of the discrimination example
above, this narrative says that only personal effort determines economic
outcomes, thus suppressing the role of discrimination). The “fatalistic”
narrative postulates that only the state matters for the outcome (e.g., it
says that only the external force of discrimination determines outcomes, thus
suppressing the role of personal agency). Finally, the “denial” narrative
removes both the action and the state as causes, thus implicitly attributing
outcomes to unspecified other factors.

Given an empirical long-run distribution over states, actions and outcomes, a
narrative produces a subjective (and possibly distorted) belief by “fitting”
the narrative to this distribution. For example, the empowering narrative
interprets the empirical correlation between actions and outcomes as a causal
quantity — i.e., it attributes the variation in outcomes entirely to variation
in actions. Once the consumer adopts a narrative, his strategy (a stochastic
mapping from signals to actions) prescribes actions that maximize expected
utility with respect to the narrative-induced belief. In equilibrium, this
strategy is consistent with the empirical long-run distribution. The need for
an equilibrium definition of consumer response to a given narrative is typical
of models of decision making under misspecified models (e.g., Esponda and
Pouzo (2016), Spiegler (2016), Eliaz and Spiegler (2020)). The reason is that
changes in long-run behavior can lead to changes in the consumer’s perceived
mapping from actions to outcomes.

The media’s problem is to find a strategy and an equilibrium (induced by the
strategy) that maximize the consumer’s ex-ante expected anticipatory utility.
The rationale for maximizing anticipatory utility is that a consumer’s
engagement with the media increases with the amount of optimism he can derive
from its consumption. The better the media performs in generating optimistic
beliefs, the higher the demand for it. A crucial feature of the problem is
that the media takes into account equilibrium effects when designing its
strategy. This is similar in spirit to information-design problems (e.g.,
Bergemann and Morris (2019)). However, in standard models equilibrium effects
arise in multi-agent settings with payoff externalities. In contrast, in our
model equilibrium effects arise because of misspecified beliefs induced by
false narratives.

This account of news media raises a number of questions: Will the media
provide accurate, unbiased information? If not, what is the structure of media
inaccuracy/bias, and which narratives will it peddle? Our analysis of the
baseline model in Section 3 addresses these questions. We begin with a full
characterization of the optimal media strategy in a specification of our model
— inspired by the above discrimination and financial-investment scenarios —
which serves as a running example in our paper. The optimal strategy consists
of the empowering narrative and a signal with an optimistic bias (i.e., always
correctly reporting good news and sometimes misrepresenting bad news). The
magnitude of the bias is tailored to consumer preferences.

We then show that this combination is a robust feature: For any action-
separable utility function, if a media strategy outperforms the true-
narrative, perfect-information benchmark, then it must involve the empowering
narrative. Also, it must provide information that induces different behavior
from the benchmark (as long as the benchmark leads to state-contingent
actions). Thus, there is a synergy between false narratives and biased
information, which is essential for the media’s mission to maximize consumers’
anticipatory utility. This result also demonstrates the value of our model in
making specific predictions about the structure of media narratives and media
bias.

Section 4 introduces preference heterogeneity among consumers in the context
of our running example. This naturally calls for a model in which consumers
can choose between multiple narrative-information combinations. We now
envisage our monopolistic media provider as a gatekeeper or a platform that
restricts the entry of these combinations. Effectively, this means that the
media chooses a menu of media strategies, aiming to maximize aggregate
anticipatory utility.333For tractability, we restrict media strategies to
report good news in the good state, which is an endogenous feature of the
optimal strategy in the baseline model. From the menu, each consumer chooses
the narrative-information pair that maximizes his own anticipatory utility,
evaluated according to the equilibrium joint distribution over states, actions
and outcomes that arises from the aggregate behavior of all consumer types.

At first glance, it may appear that incentive-compatibility should be moot in
this model, because media and consumers have a common objective. However, this
is not the case because of the “data externality” that exists among consumer
types. Although they are separate individuals with idiosyncratic preferences,
they all rely on the same aggregate data to form beliefs given the narratives
they adopt. Consequently, changes in the behavior of one segment of the
consumer population can change how another segment evaluates narrative-
information pairs. Dealing with this externality in the context of a menu
design problem is a methodological novelty of our paper, and one of our
motivations for introducing heterogeneity in the first place.

The data externality turns out to have a significant effect on the optimal
menu, compared with the representative-consumer case. The contrast is
particularly stark when consumer types are uniformly distributed. Instead of
biased, partially informative signals that are finely tailored to consumer
types, now none of the consumers receive $any$ information. Nevertheless, the
population is split into two “camps” with starkly different beliefs, driven by
the different narratives they adopt. One segment opts for the empowering
narrative and takes one constant action, while the other segment opts for the
denial narrative and always takes the opposite action. Thus, our model shows
how a heterogeneous population of consumers trying to make sense of the same
aggregate data can end up holding highly polarized beliefs based on no
information, simply because of the narrative-peddling aspect of media
strategy.

We then explore the role of market structure by examining a “perfect
competition” version of the heterogeneous-consumers model. Each media provider
is “small” in the sense that it takes the joint distribution over states,
actions and outcomes as given, without taking into account how its media
strategy affects this distribution via its effect on consumers’ beliefs and
actions. A “competitive equilibrium” is a profile of media strategies, one for
each consumer type, such that: $(i)$ the strategy associated with a type
maximizes his anticipatory utility given the aggregate distribution; and
$(ii)$ this distribution arises from each type best-replying to the belief
induced by the media strategy he adopts. We show that in the essentially
unique equilibrium, only the true and fatalistic narratives prevail, where the
former narrative is coupled with complete information. When consumer types are
uniformly distributed, the segment of consumers who act on informative signals
is larger than in the monopolistic case. Thus, while perfect competition
under-performs relative to monopoly in terms of consumers’ anticipatory
utility (because media firms fail to incorporate the data externality), it
provides more accurate information, even though it does not eradicate wrong
beliefs due to false narratives.

In Section 5 we return to the baseline model and extend it in two directions.
Our first extension mixes the consumer population with rational consumers, who
know the true model and whose demand for information is conventionally
instrumental. This extension introduces a screening problem along an
unconventional dimension: consumers’ willingness to adopt a false hopeful
narrative. We show that it is optimal for the media to provide a singleton
menu. When there are few rational consumers, this menu consists of the
empowering narrative and a biased signal (albeit less biased than in the
baseline model). When there are few non-rational consumers, the menu consists
of the fatalistic narrative and a fully informative signal.

Next, we consider different separable specifications of the consumer’s utility
function. When it is separable in the state, the only false narrative that can
outperform the true narrative is the fatalistic narrative. We illustrate this
finding with an example in which actions have objective “unintended
consequences” that the false narrative neglects. When the utility function is
separable in the outcome, the true narrative, coupled with complete
information, is optimal.

The extensions cement the main insight of our paper: When media demand is
driven by motivated reasoning, peddling false narratives is a key feature of
media bias.

##  2 A Model

We begin by introducing the primitives of our model. There are four relevant
variables: a state of Nature $t$, an action $a$ taken by a representative
consumer, a signal $s$ that the consumer observes before taking the action (he
can only condition his action on $s$), and an outcome $y$. All variables take
finitely many values. The consumer’s vNM utility takes the form:

$u(t,a,y)=v(t,y)-c(a)$

The objective data-generating process is a probability distribution $p$
defined over the four variables, which can be factorized as follows:

$p(t,s,a,y)=p(t)p(s\mid t)p(a\mid s)p(y\mid t,a)$ (1)

The first and last terms on the R.H.S are exogenously given; they describe the
prior distribution of the state of Nature, and the outcome distribution
conditional on the state and the consumer’s action, respectively. Note that
the signal has no direct effect on the outcome. The term $p(s\mid t)$
describes the signal distribution conditional on the state. This distribution
is determined ex-ante by a monopolistic media outlet. Finally, the term
$p(a\mid s)$ represents the consumer’s strategy (i.e., his action distribution
conditional on the signal). The strategy’s endogenous determination will be
described below.

The causal structure underlying this data-generating process can be described
by the following directed acyclic graph (DAG), denoted $N^{\ast}$:

$\begin{array}[]{ccc}t&\rightarrow&s\\\ \downarrow&&\downarrow\\\
y&\leftarrow&a\end{array}$

In this graphical representation, borrowed from the Statistics/AI literature
on probabilistic graphical models (Pearl (2009)), a node represents a
variable, and an arrow represents a direct causal relation. For example, the
link $s\rightarrow a$ means that $s$ is a direct cause of $a$.

Let us now describe the interaction between the media and the representative
consumer. The media moves first, committing ex-ante to a pair $(I,N)$, where:
$I$ is a signal function, which is a Blackwell experiment assigning a
distribution over signals to each state (this is the conditional probability
distribution $(p(s\mid t))_{t,s}$ described above); and $N$ is a narrative,
which is a subset of the two direct causes of $y$.

In particular, the true narrative $N^{\ast}$ acknowledges both $t$ and $a$ as
the direct causes of $y$. The “empowering” narrative $N^{a}$ postulates that
$a$ is the sole direct cause of $y$. The “fatalistic” narrative $N^{t}$
postulates that $t$ is the sole direct cause of $y$. Finally, the “denial”
narrative $N^{\emptyset}$ postulates that neither $a$ nor $t$ are direct
causes of $y$ (thus implicitly attributing the outcome to other, unspecified
exogenous factors). The three false narratives ($N^{a},N^{t},N^{\emptyset}$)
can be represented by DAGs that omit at least one of the causal links into $y$
(for example, $N^{a}$ omits the link $t\rightarrow y$).

Given an objective joint distribution $p$ with full support and the pair
$(I,N)$, the consumer forms the following conditional belief over $t$ and $y$
given the signal realization $s$ and the action $a$:

$\tilde{p}(t,y\mid s,a)=p_{I}(t\mid s)p_{N}(y\mid t,a)$ (2)

where $p_{I}(t\mid s)$ is the objective posterior probability of $t$
conditional on $s$, which is induced by the signal function $I$ and given by
Bayes’ rule; and $p_{N}(y\mid t,a)$ is the perceived probability of $y$
conditional on $t$ and $a$, which is induced by the narrative $N$.
Specifically,

$\displaystyle p_{N^{\ast}}(y$ $\displaystyle\mid$ $\displaystyle t,a)=p(y\mid
t,a)\qquad\qquad p_{N^{a}}(y\mid t,a)=p(y\mid a)$ $\displaystyle p_{N^{t}}(y$
$\displaystyle\mid$ $\displaystyle t,a)=p(y\mid t)\qquad\qquad\quad
p_{N^{\emptyset}}(y\mid t,a)=p(y)$

The interpretation is that $p$ represents a long-run empirical distribution
(reflecting the decisions of other consumers who faced the same problem); and
the narrative $N$ makes sense of this distribution by imposing a particular
explanation for what causes variation in outcomes. The belief $p_{N}(y\mid
t,a)$ is a systematic, narrative-based distortion of the objective conditional
outcome distribution.

Thus, the media affects the consumer’s beliefs via two channels: $(i)$ the
signal function given by $I$, which determines the reporting of current
events; and $(ii)$ the narrative $N$, which provides a perspective — based on
an interpretation of historical regularities — for drawing implications from
the signal. The consumer’s beliefs about $t$ are induced by $I$, whereas his
beliefs about how $y$ depends on $t$ and $a$ are induced by $N$ (and the long-
run objective distribution $p$).

Importantly, when the narrative $N$ is false, $p_{N}(y\mid t,a)$ is not
invariant to the long-run consumer average behavior given by $(p(a\mid
s))_{a,s}$. To see why, elaborate $p_{N}(y\mid t,a)$ for each of the false
narratives:

$p_{N^{a}}(y\mid t,a)=\sum_{t^{\prime}}p(t^{\prime}\mid a)p(y\mid
t^{\prime},a)=\sum_{s^{\prime},t^{\prime}}p(s^{\prime}\mid a)p(t^{\prime}\mid
s^{\prime})p(y\mid t^{\prime},a)$ (3) $p_{N^{t}}(y\mid
t,a)=\sum_{a^{\prime}}p(a^{\prime}\mid t)p(y\mid
t,a^{\prime})=\sum_{s^{\prime},a^{\prime}}p(s^{\prime}\mid t)p(a^{\prime}\mid
s^{\prime})p(y\mid t,a^{\prime})$ (4) $\displaystyle p_{N^{\emptyset}}(y$
$\displaystyle\mid$ $\displaystyle
t,a)=\sum_{t^{\prime}}p(t^{\prime})\sum_{a^{\prime}}p(a^{\prime}\mid
t^{\prime})p(y\mid t^{\prime},a^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{t^{\prime}}p(t^{\prime})\sum_{s^{\prime},a^{\prime}}p(s^{\prime}\mid
t^{\prime})p(a^{\prime}\mid s^{\prime})p(y\mid t^{\prime},a^{\prime})\vskip
12.0pt plus 4.0pt minus 4.0pt$

It is evident that the terms $p(s^{\prime}\mid a)$ and $p(a^{\prime}\mid
s^{\prime})$ involve the consumer’s strategy. In other words, long-run
consumer behavior affects narrative-based perception of the mapping from
actions to consequences (given a signal), which in turn affects the consumer’s
subjectively optimal decisions. Thus, if we view the long-run distribution $p$
as a steady state, we need an equilibrium notion of the consumer’s subjective
optimization.

######  Definition 1 (Equilibrium)

Given $(I,N)$, a strategy $(p(a\mid s))_{a,s}$ is an $\varepsilon$-equilibrium
if, whenever $p(a\mid s)>\varepsilon$, $a$ maximizes

$V_{I,N}(s,a)=\sum_{t,y}p_{I}(t\mid s)p_{N}(y\mid t,a)u(t,a,y)$ (6)

A strategy is an equilibrium if it is a limit of a sequence of
$\varepsilon$-equilibria, where $\varepsilon\rightarrow 0$.

This is essentially the definition of personal equilibrium in Spiegler (2016),
and it coincides with Berk-Nash equilibrium (Esponda and Pouzo (2016)) when
the consumer’s subjective model is given by the DAG that $N$ induced. The role
of trembles in this definition is merely to avoid conditioning on null events;
they play no meaningful role in our analysis.

We assume that the media chooses $(I,N)$ ex-ante to maximize

$U(I,N)=\sum_{t}p(t)\sum_{s}p(s\mid t)\max_{a}V_{I,N}(s,a)$ (7)

subject to the constraint that $(p(a\mid s))_{a,s}$ is an equilibrium. The
media’s objective function is the consumer’s expected anticipatory utility.
The idea behind this objective function is that anticipatory utility drives
the consumer’s demand for news media. The higher his anticipatory utility, the
greater his media engagement. Our task is to characterize the media’s optimal
strategy.

In solving its problem, the media takes into account the consumer’s
equilibrium response to the media strategy. This naturally raises the question
of whether the media knowingly anticipates equilibrium effects. One
interpretation is that the media is not aware of them a priori. Instead, it
reacts to past data about consumer engagement, possibly using algorithmic
learning. The equilibrium effects that shape consumers’ media engagement will
be reflected in the learning process.

The necessity of false narratives for media bias

Suppose that the media is restricted to providing the true narrative
$N^{\ast}$. This reduces the model to standard information provision by a
sender who can commit ex-ante to a Blackwell experiment. The sender faces a
Bayesian receiver whose indirect utility from a posterior belief $\mu$ over
$t$ is

$\max_{a}\sum_{t}\mu(t)\sum_{y}p(y\mid t,a)u(t,a,y)$

Since this indirect utility is a maximum over functions that are linear in
$\mu$, it is convex in $\mu$. Therefore, it is (weakly) optimal for the sender
to commit to a fully informative signal — i.e., $p(s=t\mid t)=1$ for every
$t$. It follows that in our model, given the media’s objective of maximizing
the consumer’s ex-ante anticipatory utility, the media has no strict incentive
to provide partial or biased information unless it also peddles a false
narrative.

Thus, when the media only supplies information, it treats the consumer as if
he conventionally maximizes his material expected utility. In other words, it
does not matter whether the consumer tries to maximize his actual welfare or
merely his anticipated welfare. Throughout the paper, we refer to the maximal
anticipatory utility attained by the true narrative and complete information
as the rational-expectations benchmark.

Comment on the interpretation of $a$ and $y$

According to one interpretation of our model, $a$ represents a private action
that an individual media consumer takes, and $y$ is a personal outcome of his
choice. For example, $a$ can represent the agent’s career decision or a
dietary choice, in which case $y$ represents his earnings or health outcome,
respectively. The data that the consumer relies on to form beliefs (via the
factorization according to $N$) is aggregate, reflecting the historical
choices and outcomes of other consumers.

An alternative interpretation is that $a$ represents a public choice (such as
economic or foreign policy), and $y$ represents a public outcome (economic
growth, national security). According to this interpretation, the media
consumer is a representative voter, and the probability $p(a\mid s)$ is the
frequency with which society selects a political leadership that implements
$a$. This is a reduced-form representation of a democratic process, such that
society’s choice matches what the representative voter deems optimal.

Our model departs from the canonical information-design framework (see
Bergemann and Morris (2019)), since it allows the designer to influence the
subjective model that the receiver holds. Nevertheless, the assumption that
the consumer always correctly perceives $p(t,s,a)$ ensures that the standard
revelation principle in the information-design literature can be adapted to
the present setting.

######  Remark 1 (A revelation principle)

Without loss of optimality, we can restrict attention to signal functions that
assign a distribution over recommended actions to each state, and to
equilibria in which $a=s$ with probability one for each $s$.

The proof of this remark follows the footsteps of Theorem 1 in Bergemann and
Morris (2016) — adapted to the single-player setting — and is therefore
omitted. The proof involves manipulating the signal function given by
$(p(s\mid t))_{t,s}$ and the consumer’s strategy given by $(p(a\mid
s))_{a,s}$. In general, when the consumer forms beliefs according to a
misspecified model $N$, such changes may affect $p_{N}(y\mid t,a)$, which
could violate the revelation principle. The reason the principle does hold in
our setting is that the manipulation of $(p(s\mid t))_{t,s}$ and $(p(a\mid
s))_{a,s}$ in the proof leaves $(p(t,a))_{t,a}$ unchanged. As evident from
(3)-(2), this means that $p_{N}(t,a)$ and $p_{N}(y\mid t,a)$ also remain
unchanged, regardless of how $t$ and $a$ are jointly distributed with $s$.
This enables the standard proof to go through. The revelation principle will
simplify our analysis in the sequel.

##  3 Analysis

In this section we analyze the media’s optimal strategy. We begin with a
specification that serves as a running example in the paper. We then show that
the qualitative features of the optimal media strategy in our example are
robust.

###  3.1 An Example: “The American Dream”

In this example, all variables take values in $\\{0,1\\}$. The exogenous
components of the data-generating process are:

$p(t=1)=\frac{1}{2}\qquad\qquad p(y=1\mid t,a)=\frac{1}{2}a\left(2-t\right)$

The consumer’s payoff function is

$u(a,t,y)=ty-ca$

where $c\in(0,\frac{1}{2})$. (We will later handle the case of
$c>\frac{1}{2}$.) The action $a$ represents a private decision whether to
initiate a costly economic activity. The outcome $y$ indicates whether the
activity is successful. The state $t$ represents the return from a successful
outcome. High returns are associated with lower chances of a successful
outcome.

For a specific example, the consumer is a retail financial investor choosing
whether to engage in active or passive portfolio management. The realization
$y=1$ represents an increase in the portfolio’s value. The state $t$
represents an effective discount factor that reflects economic fundamentals
such as the interest rate for alternative investments, the expected timing of
a successful outcome, or macroeconomic uncertainty. The negative correlation
between $t$ and $y$ has a natural interpretation in this context. Market
fundamentals tend to be reflected in current security prices. For instance, a
low-risk environment is reflected in high current prices, lowering the
prospects of an increase in the portfolio’s value.

An alternative story is that the consumer is a college student who decides how
seriously to take his studies. A successful outcome means graduating (rather
than dropping out). The realization $t=1$ represents a college wage premium.
The negative correlation between $t$ and $y$ is due to the fact that as
returns to high education rise, colleges respond by becoming more demanding,
such that graduating becomes less likely.

Under both stories, the media provides information about the fundamentals
represented by $t$, as well as a narrative about what drives the outcome $y$.
These stories are meant to capture the general idea that media coverage
affects whether a person attains material success or failure is attributed to
either internal factors that the person controls or to external factors (see
Iyengar (1990) for evidence on this in the context of escaping poverty). Our
task is to characterize the optimal media strategy, considering each of the
feasible narratives.

Rational-expectations benchmark

Suppose the media offers the true narrative $N^{\ast}$. As we saw, it is
optimal to couple this narrative with a fully informative signal. When $t=0$,
the consumer knows that $ty=0$, and therefore plays $a=0$. When $t=1$, he
knows that $p(y=1\mid t=1,a=1)=\frac{1}{2}$. Since $c<\frac{1}{2}$, the
consumer plays $a=1$. It follows that the rational-expectations benchmark in
this example is $\frac{1}{4}-\frac{1}{2}c$.

Narratives that omit the link $a\rightarrow y$

Under the narratives $N^{t}$ and $N^{\emptyset}$, the consumer believes that
his action has no effect on $y$, and therefore prefers to take the costless
action $a=0$. In any equilibrium, $a=0$ with certainty for every $t$. However,
since $y=0$ whenever $a=0$, it follows that $p(y=1\mid t)=p(y=1)=0$ for every
$t$. Therefore, the consumer’s anticipatory utility is necessarily zero, which
is below the rational-expectations benchmark. It follows that the media will
necessarily offer a narrative that retains the link $a\rightarrow y$.

The empowering narrative

Under the narrative $N^{a}$,

$p_{N^{a}}(ty=1\mid a,s)=p(t=1\mid s)p(y=1\mid a)$

Since $p(y=1\mid a=0)=0$, the consumer’s subjective payoff from $a=0$ is zero
for every signal he receives. Let us calculate the consumer’s subjective
expected payoff from $a=1$. Denote $q_{t}=p(s=1\mid t)$. By the revelation
principle, we can restrict attention to binary signals and an equilibrium in
which the consumer always plays $a=s$. Then,

$p(y=1\mid a=1)=\frac{1}{2}+\frac{1}{2}p(t=0\mid
a=1)=\frac{1}{2}+\frac{1}{2}\cdot\frac{q_{0}}{q_{1}+q_{0}}$

Therefore,

$p_{N^{a}}(ty=1\mid
s=1,a=1)=\frac{q_{1}}{q_{1}+q_{0}}\cdot\left(\frac{1}{2}+\frac{1}{2}\cdot\frac{q_{0}}{q_{1}+q_{0}}\right)$
(8)

and

$p_{N^{a}}(ty=1\mid
s=0,a=1)=\frac{1-q_{1}}{2-q_{1}-q_{0}}\cdot\left(\frac{1}{2}+\frac{1}{2}\cdot\frac{q_{0}}{q_{1}+q_{0}}\right)$
(9)

In order for the consumer’s strategy to be an equilibrium, we need (8) and (9)
to be weakly above and below $c$, respectively. If these constraints hold, the
consumer’s anticipatory utility is $p(s=1)\cdot[p_{N^{a}}(ty=1\mid
s=1,a=1)-c]$, given by

$\frac{q_{1}+q_{0}}{2}\cdot\left[\frac{q_{1}}{q_{1}+q_{0}}\cdot\left(\frac{1}{2}+\frac{1}{2}\cdot\frac{q_{0}}{q_{1}+q_{0}}\right)-c\right]$
(10)

Observe that when the media offers a fully informative signal ($q_{1}=1$,
$q_{0}=0$), this expression coincides with the payoff from $N^{\ast}$. Thus,
if the false narrative $N^{a}$ is to outperform the true narrative, it must be
coupled with incomplete information. We now proceed to calculate the optimal
$I=(q_{0},q_{1})$ that accompanies $N^{a}$.

###### Claim 1

When $c<\frac{1}{2}$, it is optimal to set $q_{1}=1$.

Thus, if the optimal signal function has a bias, it must be an optimistic one,
as the media always reports good news ($s=1$) when the state is good ($t=1$).
The proof of this claim (like all proofs in this paper) is in the Appendix.
The claim implies the following simplified expression for the consumer’s
anticipatory utility:

$\frac{1}{2}\left[\frac{1}{2}+\frac{1}{2}\cdot\frac{q_{0}}{1+q_{0}}-c(1+q_{0})\right]$
(11)

Note that $q_{1}=1$ also implies that (9) is below $c$, such that playing
$a=0$ when $s=0$ is optimal for the consumer. It is now straightforward to
derive the optimal value of $q_{0}$:

$q_{0}=\min\left\\{1,\sqrt{\frac{1}{2c}}-1\right\\}$ (12)

Plugging (12) in (11), the consumer’s ex-ante anticipatory payoff is

$\begin{array}[]{ccc}\frac{1}{2}-\sqrt{\frac{c}{2}}&if&c\in\left[\frac{1}{8},\frac{1}{2}\right)\\\
\frac{3}{8}-c&if&c\in\left(0,\frac{1}{8}\right)\end{array}$

which exceeds the rational-expectations benchmark.

Thus, when $c<\frac{1}{2}$, the optimal media strategy involves the narrative
$N^{a}$ coupled with positively biased information: always sending a good
signal in the good state, and sending it with positive probability in the bad
state.

In terms of the interpretation we offered for this example, the false
narrative $N^{a}$ attributes a successful investment outcome entirely to the
consumer’s actions, without taking into account the role of the fundamentals.
The accompanying signal function has an optimistic bias in the direction of
claiming that returns are high even when they are not. Thus, on one hand the
media exaggerates the attractiveness of the investment environment, while on
the other hand it suppresses — via the empowering narrative — the negative
effect that good fundamentals have on the chances of good investment outcomes.
Thus, we find it apt to refer to the media as peddling “the American dream” in
this example.

Biased information is necessary for $N^{a}$ to beat the rational-expectations
benchmark. One way to see why is to imagine that the media provides perfect
information. This means that $t$ and $s$ are perfectly correlated ($s\equiv
t$). By the revelation principle, $a$ and $s$ are perfectly correlated
($a\equiv s$). Therefore, $a$ and $t$ are perfectly correlated. But this means
that omitting $t$ as an explanatory variable for $y$ does not lead to
erroneous beliefs: $p(y\mid a)$ coincides with $p(y\mid t,a)$. In turn, this
means that the consumer effectively has rational expectations and perfectly
monitors $t$, which is precisely the rational-expectations benchmark.
Therefore, imperfect information is necessary for $N^{a}$ to enhance the
consumer’s anticipatory utility.

So far, we assumed that $c<\frac{1}{2}$. It is easy to see that when
$c\geq\frac{1}{2}$, none of the feasible narratives can generate positive
utility. As we already saw, the narratives $N^{t}$ and $N^{\emptyset}$
generate zero utility for every $c$. It is also immediate from the expressions
for the anticipatory utility induced by $N^{\ast}$ and $N^{a}$ that when
$c\geq\frac{1}{2}$, these narratives cannot generate positive payoffs.

###  3.2 A Characterization Result

The investment-narrative example has two noteworthy features. First, the
empowering narrative emerges as optimal. Second, it is coupled with biased
information that impacts consumer behavior. We now show that both features
hold generally under the model of Section 2.

###### Proposition 1

If the media can outperform the rational-expectations benchmark, then $N^{a}$
is part of an optimal strategy.

Thus, the empowering narrative $N^{a}$ is an essential feature of a media
strategy that outperforms the rational-expectations benchmark. The logic
behind the result is as follows. Because $u$ is action-separable, a false
narrative can have an effect on ex-ante anticipatory utility only when it
distorts the joint distribution of $(t,y)$. By definition, the fatalistic
narrative $N^{t}$ cannot do that. In principle, the denial narrative
$N^{\emptyset}$ can attain such a distortion. However, this effect can be
replicated by $N^{a}$ coupled with no information.

The next result addresses the consumer behavior that the optimal media
strategy induces. We say that the payoff function and the exogenous data-
generating process form a regular environment if, under the true narrative and
complete information, the consumer has a unique best-reply which is a one-to-
one function of the state. That is, in regular environments different states
prescribe different unique actions under rational expectations.

###### Proposition 2

Suppose the environment is regular. If the optimal media strategy outperforms
the rational-expectations benchmark, then its induced conditional distribution
$(p(a\mid t))_{t,a}$ is different from that benchmark.

Thus, when the media deviates from the rational-expectations benchmark, it
necessarily induces changes in consumer behavior. Since regularity assumes a
unique optimal action in each state (under rational expectations), this means
that the outcome induced by the media’s strategy departs from what a
paternalistic social planner (aiming to maximize consumers’ material payoffs)
would prescribe.

Note that our result does not claim that the media necessarily departs from
fully informative signals. We cannot rule out the possibility that the media
sends a fully informative signal in every state and that the consumer’s
subjective best-reply involves mixing, which will be sustained in equilibrium
thanks to the false narrative $N^{a}$.

Regularity plays a key role in the result. To see why, consider the payoff
specification of Section 3.1, and let the data-generating process satisfy
$p(t=1)=\frac{1}{2}$ and $p(y=1\mid t,a)=1-t$ for every $t,a$. Under rational
expectations, the consumer’s optimal action is $a=0$ for every $t$, and the
rational-expectations payoff is $0$ (because $a=0$ and $ty=0$ with probability
one). Using similar arguments as in Section 3.1, it can be shown that it is
optimal for the media to provide $N^{\emptyset}$ (or, equivalently, $N^{a}$)
and no information. The consumer responds by playing $a=0$. His anticipatory
payoff is $\frac{1}{4}$, beating the rational-expectations benchmark, although
the behavior is the same. Thus, without regularity, it is possible for the
media strategy to outperform the benchmark without any effect on consumer
behavior.

##  4 Heterogeneous Consumers

In this section we extend the model by introducing consumer preference
heterogeneity. Accordingly, the supply side consists of multiple media
strategies that consumers can choose from, each according to his preferences.
We analyze two market structures. In Section 4.1, we consider a monopolistic
media platform acting as a gatekeeper that restricts the entry of media
providers (each represented by a distinct media strategy). The monopolist’s
objective is to maximize consumers’ aggregate anticipatory utility —
reflecting the continued assumption that this corresponds to maximizing their
platform engagement. In Section 4.2, we remove the gatekeeper and analyze a
“perfectly competitive” media market, in which each provider targets a
particular consumer type and tries to maximize his anticipatory utility.

This extension introduces a methodological innovation. While each consumer
type maximizes his own anticipatory utility, this utility — shaped by the
narrative he adopts — is evaluated according to the joint distribution over
actions and outcomes, which reflects the aggregate behavior of all consumers.
In other words, when consumers adopt a false narrative, they are subjected to
a “data externality” from other consumers. This externality changes the
formulation and analysis of the monopolistic and competitive models of the
media market. A key difference between the two market structures is that the
monopolist is an “externality maker” (who internalizes the data externality)
while competitive media providers are “externality takers”. This leads to
qualitatively different characterizations of media strategies that emerge
under these market structures.

###  4.1 Monopoly

In this version of the model, a monopolistic media platform commits ex-ante to
a menu $M$ of pairs $(I,N)$. The set of consumer types is $C=[0,1]$. Types are
distributed according to a continuous and strictly increasing $cdf$ $F$. Let
$u_{c}$ be type $c$’s payoff function. Each type $c$ selects a pair
$(I_{c},N_{c})\in M$ and a signal-dependent action $a_{c}(s)$ to maximize his
ex-ante anticipatory utility. The platform’s objective is to maximize
consumers’ aggregate ex-ante anticipatory utility.

This design problem is a novel type of “second degree” discrimination, which
arises because the platform cannot prevent consumers from freely choosing
their favorite media strategy from the menu. At first glance, this seems to be
a trivial problem, since there is no conflict of interest between the two
parties: Both the consumer and the monopolist are guided by maximizing
consumer anticipatory utility. However, consumers’ choices exert a non-
standard externality on one another: Their narrative-shaped beliefs are all
based on the same aggregate distribution $p$, which reflects the individual
choices of all consumers. This novel interdependence among consumers is what
makes the menu-design problem non-trivial.

The menu design problem

To formally describe the design problem, we begin with how consumers evaluate
alternatives. Fix some profile of consumer types’ media-strategy choices and
signal-dependent actions, $(I_{c},N_{c},(a_{c}(s)))_{c\in C}.$ Aggregate
consumer behavior is given by $(p(a\mid t))_{a,t},$ where

$p(a\mid t)=\int_{c}\sum_{s}p_{I_{c}}(s\mid t)\cdot 1[a_{c}(s)=a]dF(c)$ (13)

and where $\left(p_{I_{c}}(s\mid t)\right)_{s,t}$ is the Blackwell experiment
given by $I_{c}.$ Denote $\mathbf{a}\equiv(a(s))_{s}$. Given $(p(a\mid t))$,
consumer type $c$’s ex-ante evaluation of any $(I,N,\mathbf{a})$ is:

$U_{c}(I,N,\mathbf{a})=\sum_{s}p_{I}(s)\sum_{t,y}p_{I}(t\mid s)p_{N}(y\mid
t,a(s))u_{c}(t,a(s),y)$ (14)

In this formula, $p_{I}(s)$ and $p_{I}(t\mid s)$ are induced by the objective
prior probability $p(t)$ and the Blackwell experiment given by $I$. The
conditional probability $p_{N}(y\mid t,a)$ is as defined in Section 2, based
on $p(t,a,y)=p(t)p(a\mid t)p(y\mid t,a)$, with $p(a\mid t)$ representing
aggregate consumer behavior as in (13). Thus, the anticipatory payoff that
some type $c$ gets from his choice of triplet $(I_{c},N_{c},(a_{c}(s)))$ is
affected by the choices of triplets made by all the other types since these
determine the joint aggregate distribution $p(t,a,y).$

In actuality the media platform offers a menu of $(I,N)$ pairs, and consumer
types select items from this menu together with signal-dependent actions. An
optimal menu maximizes consumers’ aggregate anticipatory utility such that
consumers’ choices and actions satisfy some constraints. An equivalent and
more convenient way to describe the media platform is that it chooses a
profile of triplets $(I_{c},N_{c},\mathbf{a}_{c})_{c\in C}$ to maximize

$\int_{c}U_{c}(I_{c},N_{c},\mathbf{a}_{c})dF(c)$

subject to the constraints that for every $c\in C$: $(i)$ the triplet
$(I_{c},N_{c},\mathbf{a}_{c})$ maximizes $U_{c}$ over the set
$\\{I_{c},N_{c},\mathbf{a}_{c}\\}_{c\in C}$; and $(ii)$ $\mathbf{a}_{c}$
maximizes $U_{c}(I_{c},N_{c},\mathbf{a})$ given $(I_{c},N_{c})$.

To see the “data externality” that the menu design problem reflects, suppose
some consumer types change their choice of triplet. If this change involves
different signal-dependent actions, it can affect the aggregate distribution
$p(a\mid t),$ which in turn may affect the anticipatory payoff of types who
did not change their choice.

The “American dream” example revisited

Complete characterization of this menu-design problem is beyond the scope of
this paper. Here we make do with applying it to the “American dream” example
of Section 3.1, extending it by introducing consumer heterogeneity.
Specifically, we identify consumer types with the cost parameter
$c$.444Although we only analyze the menu design for this example, we chose to
present the general problem first, because we believe this makes its logic
more transparent.

In addition, we restrict the domain of feasible information strategies:
Signals are binary, $s\in\\{0,1\\}$, and the set of feasible signal functions
satisfy $\Pr(s=1\mid t=1)=1$. This restriction entailed no loss of generality
in the representative-consumer case of Section 3.1. This is no longer the case
here; we impose the restriction for tractability, as it lowers the
dimensionality of media strategies. The restriction also means that we cannot
apply the revelation principle. Accordingly, we will not take it for granted
that consumers’ actions mimic the signal they receive.

Thus, in what follows, each signal function $I$ is identified with $q$, which
is the probability of submitting $s=1$ when $t=0$. The probability of $t=1$
conditional on $s$ is thus

$p_{q}(t=1\mid s)=\frac{s}{1+q}$ (15)

In particular, when the consumer observes the signal $s=0$, he infers that
$t=0$ and therefore $ty=0$ with probability one. Hence, we can take it for
granted that all consumer types play $a=0$ and earn zero payoffs when
receiving the signal $s=0$.

It follows that $U_{c}(q,N,\mathbf{a})$ can be simplified into

$\displaystyle U_{c}(q,N,\mathbf{a})$ $\displaystyle=$ $\displaystyle
p_{q}(s=1)\cdot[p_{q}(t=1|s=1)p_{N}(y=1|t=1,a(s=1))-ca(s=1)]$ $\displaystyle=$
$\displaystyle\frac{1+q}{2}\cdot\left[\frac{1}{1+q}p_{N}(y=1|t=1,a(s=1))-ca(s=1)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}p_{N}(y=1|t=1,a(s=1))-\frac{c(1+q)}{2}a(s=1)\vskip
12.0pt plus 4.0pt minus 4.0pt$

Likewise, consumers’ aggregate state-dependent behavior can be simplified into

$p(a=1\mid t=1)=\int_{0}^{1}a_{c}(1)dF(c)\qquad\qquad p(a=1\mid
t=0)=\int_{0}^{1}q_{c}a_{c}(1)dF(c)$

We can now restate the platform’s problem: Choose a profile
$(q_{c},N_{c},a_{c}(1))_{c\in C}$ that maximizes

$\int_{0}^{1}U_{c}(q_{c},N_{c},a_{c}(1))dF(c)$

subject to the constraints that for every $c$,
$U_{c}(q_{c},N_{c},a_{c}(1))\geq
U_{c}(q_{c^{\prime}},N_{c^{\prime}},a_{c^{\prime}}(1))$ for every
$c^{\prime}\in C$; and that $a_{c}(1)$ maximizes $U_{c}$ given
$(q_{c},N_{c})$. The latter constraint can be written as follows:

$\begin{array}[]{cl}&\frac{1}{1+q}\cdot p_{N_{c}}(y=1\mid
t=1,a=a_{c}(1))-ca_{c}(1)\\\ \geq&\frac{1}{1+q}\cdot p_{N_{c}}(y=1\mid
t=1,a=1-a_{c}(1))-c(1-a_{c}(1))\end{array}$

The rest of this sub-section is devoted to analyzing this menu design problem.

###### Proposition 3

The media maximizes its objective function with a menu that has the following
features:
$(i)$ The menu includes exactly one pair $(q^{a},N^{a})$; furthermore,
$q^{a}>0$, and there is $c^{\ast}\in(0,1)$ such that all consumer types in
$[0,c^{\ast}]$ choose $(q^{a},N^{a})$ and play $a=s$.
$(ii)$ The menu includes exactly one narrative
$N\in\\{N^{t},N^{\emptyset}\\}$, coupled with an arbitrary $q$; there is
$c^{\ast\ast}\in[c^{\ast},1)$ such that all consumer types in
$[c^{\ast\ast},1]$ choose $(q,N)$ and play $a=0$ with probability one.
$(iii)$ If $c^{\ast\ast}>c^{\ast}$, then the menu also includes exactly one
pair $(q^{\ast},N^{\ast})$; furthermore, $q^{\ast}<q^{a}$, and all consumer
types in $(c^{\ast},c^{\ast\ast})$ choose $(q^{\ast},N^{\ast})$ and play
$a=s$.

There are a few noteworthy differences from the homogenous case. First, under
homogeneity, a single narrative ($N^{a}$) serves all consumers. The
differentiation between consumer populations (characterized by distinct $c$)
is done through the signal function. In contrast, differentiation between
types in the heterogeneous case is carried out by offering a menu of
narratives. Each of the narratives that keep the link $a\rightarrow y$ is
coupled with a specific signal function. The reason is that given our
restricted domain of signal functions, different media strategies that share
the same narrative are Blackwell-ordered — and therefore unambiguously ranked
in terms of the anticipatory utility they confer. As a result, no consumer
will select dominated media strategies.

More specifically, the menu includes the narrative $N^{a}$, which is coupled
with biased information toward $a=1$; the narrative $N^{\ast}$ (which need not
be in the menu) has a smaller, potentially zero bias in that direction; while
the other narratives generate the action $a=0$. Thus, we have a proliferation
of narratives, which lead to polarized beliefs and polarized behavior.

Second, in the homogenous case, market coverage is partial: Consumer types
$c>\frac{1}{2}$ receive zero payoffs; they are effectively unserved. In
contrast, in the heterogeneous case they earn positive anticipatory payoffs,
thanks to the narratives $N^{t}$ or $N^{\emptyset}$. This is made possible by
the externality between types: Types with high $c$ “free ride” on low-$c$
types, who play $a=1$ with positive probability.

The following result completes the characterization of the optimal menu when
$c$ is uniformly distributed.

###### Proposition 4

When $c\sim U[0,1]$, the optimal menu consists of two media strategies:
$(q=1,N^{a})$ and $(q=1,N^{\emptyset})$. Consumers with $c<\frac{3}{11}$
choose the former pair and always play $a=1$; whereas consumers with
$c>\frac{3}{11}$ choose the latter pair and always play $a=0$.

Thus, under uniformly distributed types, the media never provides $any$
information to any consumer. Consumer behavior is highly polarized: Consumers
with high $c$ always play $a=0$ whereas consumers with low $c$ always play
$a=1$. What generates this polarization is the different narratives that the
two consumer segments adopt: Low-$c$ consumers opt for the empowering
narrative while high-$c$ consumers opt for the denial narrative.

###  4.2 Perfect Competition

Let us now consider a competitive media market, in which every media firm is
small and therefore cannot affect aggregate consumer behavior.

######  Definition 2 (Competitive equilibrium)

A profile $(I_{c},N_{c},\mathbf{a}_{c})_{c\in C}$ is a competitive equilibrium
if for every $c\in C$: $(i)$ $(I_{c},N_{c},a_{c})$ maximizes $U_{c}$ over
$\\{(I_{c},N_{c},\mathbf{a}_{c})\\}_{c\in C}$; and $(ii)$ $\mathbf{a}_{c}$
maximizes $U_{c}$ given $(I_{c},N_{c})$; where $U_{c}$ is defined as in (14).

Unlike the monopoly case, here each media strategy targets a consumer type and
maximizes his anticipatory utility. Media suppliers do not internalize the
data externality between types, because they take the aggregate consumer
behavior implicit in $p$ as given.

As in the previous sub-section, let us apply this definition to the “American
dream” example. As before, consumer types are identified by their cost
parameter. Unlike the previous sub-section, here we need not restrict the set
of feasible signal functions, except the purely expositional restriction to
binary signals that take the values $0$ or $1$.

###### Proposition 5

There is an essentially unique competitive equilibrium. Specifically, there is
$\bar{c}\in(0,1)$ given by $2\bar{c}+F(\bar{c})=1$, such that: $(i)$ for every
$c<\bar{c}$, $I_{c}$ is the fully informative signal function and
$N_{c}=N^{\ast}$; and $(ii)$ for every $c>\bar{c}$, $N_{c}=N^{t}$.

By essential uniqueness, we mean that there could be other media strategies
that implement the same beliefs and actions. For example, when a consumer
chooses $N^{\emptyset}$, the exact signal function is irrelevant for his
beliefs and actions. Also, we could replace $N^{\ast}$ with $N^{a}$ in the
characterization, and consumers’ beliefs would be identical.

It follows that for consumer types with low $c$, perfect competition leads to
an unambiguous improvement in the informativeness of news media compared with
monopoly. To see why, note that under monopoly, a positive measure of consumer
type with $c$ close to $0$ receive biased information (which is coupled with
the narrative $N^{a}$), whereas under competition they have rational
expectations and full information. When $c\sim U[0,1]$, we have
$\bar{c}=\frac{1}{3}$, which is above the cutoff $c^{\ast}=\frac{3}{11}$ of
the monopoly case — i.e., competition improves informativeness for all
consumer types.

##  5 Variations

This section returns to the homogenous-consumer case and explores variations
on the basic model.

###  5.1 Introducing Rational Consumers

So far, we assumed that consumers’ demand for news is entirely non-
instrumental — rather, they use news media to cultivate desirable beliefs.
This gives a role to false narratives as a vehicle for sustaining such
beliefs. Consumers are not dogmatic: They are willing to accept any narrative,
and their sole criterion for selecting a narrative is the anticipatory utility
it induces.

In this sub-section we introduce instrumental motives into the basic model.
There are two ways to do this: assuming that consumers are homogenous but they
mix instrumental and non-instrumental motives; or assuming that the consumer
population consists of conventionally rational types, in addition to the type
described by the basic model. We follow the second route.

Let us return to the basic setting in which $u(t,a,y)=ty-ca$, where all
consumers have the same $c\in(0,\frac{1}{2})$, and introduce a new
heterogeneity. A fraction $\lambda$ of the consumer population have
traditionally instrumental demand for information — that is, they aim to
maximize their objective expected material payoff, rather than their
anticipatory payoff. Furthermore, these consumers have rational expectations:
They know the true model $N^{\ast}$ and are therefore immune to narrative
peddling by the media. We refer to these consumers as rational, and to the
remaining consumers (who behave as in previous sections) as non-rational.

The way rational consumers evaluate a media strategy $(I,N)$ is thus quite
simple, because it is equivalent to the way non-rational consumers evaluate
the strategy $(I,N^{\ast})$. The reason is that under the true model
$N^{\ast}$, the distinction between anticipatory utility and objective
expected material utility disappears: $U(I,N^{\ast})$ is the ex-ante expected
material payoff when the consumer subjectively best-replies to the beliefs
induced by $I$.

We handle the new heterogeneity as we handled the heterogeneity in $c$ in
Section 4. In particular, we adopt the simplifying assumption that $I$ must
involve two signals, $0$ and $1$, such that in state $t=1$ the signal is $s=1$
with probability one. In this setting, this entails no loss of generality but
facilitates exposition by emphasizing the methodological connection to the
model of Section 4.1. Thus, each $I$ is identified with the probability $q$ of
$s=1$ in $t=0$. The media’s problem is to choose a menu of media strategies,
$\\{(q_{r},N_{r}),(q_{nr},N_{nr})\\}$ to maximize

$\lambda\cdot U_{r}(q_{r},N_{r})+(1-\lambda)\cdot U_{nr}(q_{nr},N_{nr})$

subject to the constraint that $(q_{i},N_{i})$ maximizes $U_{i}$ for every
type $i\in\\{r,nr\\}$, given the aggregate distribution induced by the
strategy each type plays; and subject to the constraint that each type $i$’s
strategy is a best-reply given the beliefs induced by the type, the pair
$(q_{i},N_{i})$ he chooses, and the objective distribution.

###### Proposition 6

There is an optimal menu that consists of a single pair $(q,N)$.

Thus, the media’s problem of screening consumers according to their
rationality is degenerate: The media can offer the same strategy to all
consumers. The result holds more generally when $u(t,a,y)=v(t,y)-ca$, when
$a,t\in\\{0,1\\}$, $v$ is an arbitrary function, and $c\in(0,\frac{1}{2})$.

For brevity, we do not provide a detailed derivation of the optimal media
strategy in this environment for all values of $\lambda$. We make do with
illustrating it for extreme values of $\lambda$.

###### Proposition 7

When $\lambda$ is sufficiently close to $0$, the optimal media strategy is
$(\min\left\\{1,\sqrt{\frac{1-\lambda}{2c}}-1\right\\},N^{a})$. When $\lambda$
is sufficiently close to $1$, the optimal media strategy is $(0,N^{t})$.

Thus, introducing a small group of rational consumers into a population of
non-rational consumers increases the informativeness of the signal that the
media provides, while still keeping the empowering narrative. In contrast,
introducing a small group of non-rational consumers into a population of
rational consumers causes the media to offer the fatalistic narrative while
continuing to give full information.

###  5.2 Other Separable Utility Specifications

In this sub-section we examine alternative specifications of $u(t,a,y)$. All
definitions are adapted straightforwardly.

####  5.2.1 An Example: “Whac-a-Mole”

Impose the following structure on the data-generating process:

$p(t=1)=\frac{1}{2}\qquad\qquad p(y=1\mid t,a)=\beta(1-a)+(1-\beta)t\vskip
6.0pt plus 2.0pt minus 2.0pt$

where $\beta\in(\frac{1}{3},1)$. The consumer’s payoff function is
$u(a,t,y)=\mathbf{1}[a=y]$.

We adopt the following interpretation for this specification. The action $a$
represents a public decision how to allocate a scarce resource between two
sectors or locations. For example, the dilemma is whether to allocate policing
effort to one area of criminal activity or another. The state $t$ indicates
which sector is more dangerous. The outcome $y$ indicates which sector ends up
being active. Public policy is successful if it allocates the policing effort
to the relevant sector. However, criminal activity exhibits a “whac-a-mole”
property: When the government cracks down on one area of activity, criminals
partly divert their activity to the other area. This explains the negative
correlation between $a$ and $y$. In this context, the media reports on the
dangers posed by various sectors, and conveys a narrative about what
ultimately determines the active sector. Consumer choice represents support
for a certain public policy (e.g., voting for a political party that runs on
this policy).

As before, let us begin our quest for optimal media strategies with the case
in which the narrative is $N^{\ast}$. As usual, we can assume that the media
provides full information. When $t=1$, the consumer’s payoff from $a=1$ is
$1-\beta$, and the payoff from $a=0$ is $0$. Therefore, the consumer plays
$a=1$ when $t=1$, and his payoff is $1-\beta$. The case of $t=0$ is handled
symmetrically: the consumer plays $a=0$, and earns a payoff of $1-\beta$. It
follows that the consumer’s ex-ante anticipatory utility is $1-\beta$. Thus,
when the media conveys the true narrative and fully informs the consumer about
$t$, the consumer correctly identifies the dangerous sector and plays $a=t$.
At the same time, the consumer correctly takes the whac-a-mole effect into
account.

We will later see that the narratives $N^{a}$ and $N^{\emptyset}$ (which omit
the link $t\rightarrow y$) are weakly inferior to $N^{\ast}$. Therefore, let
us focus on the narrative $N^{t}$. We apply the revelation principle and take
it for granted that $a=s$ in equilibrium. By definition,

$p_{N^{t}}(y=1\mid s,a)=\sum_{t}p(t\mid s)p(y=1\mid t)$

where

$p(t=1\mid s=1)=\frac{q_{1}}{q_{0}+q_{1}}\qquad\qquad p(t=1\mid
s=0)=\frac{1-q_{1}}{2-q_{0}-q_{1}}\vskip 6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle p(y$ $\displaystyle=$ $\displaystyle 1\mid t=1)=\sum_{s}p(s\mid
t=1)p(y=1\mid a=s,t=1)$ $\displaystyle=$ $\displaystyle
q_{1}\cdot(1-\beta)+(1-q_{1})\cdot 1=1-\beta q_{1}$

and

$\displaystyle p(y$ $\displaystyle=$ $\displaystyle 1\mid t=0)=\sum_{s}p(s\mid
t=0)p(y=1\mid a=s,t=0)$ $\displaystyle=$ $\displaystyle q_{0}\cdot
0+(1-q_{0})\cdot\beta=\beta(1-q_{0})$

It follows that the consumer’s payoff from playing $a=1$ when $s=1$ is

$U_{N^{t}}(s=1)=\frac{q_{1}}{q_{0}+q_{1}}\cdot(1-\beta
q_{1})+\frac{q_{0}}{q_{0}+q_{1}}\cdot\beta(1-q_{0})$

Likewise, the consumer’s payoff from playing $a=0$ when $s=0$ is

$U_{N^{t}}(s=0)=1-\left[\frac{1-q_{1}}{2-q_{0}-q_{1}}\cdot(1-\beta
q_{1})+\frac{1-q_{0}}{2-q_{0}-q_{1}}\cdot\beta(1-q_{0})\right]$

In order for this strategy to be an equilibrium, we need both expressions to
be weakly above $\frac{1}{2}$. We will confirm this below. The strategy $a=s$
induces the following ex-ante anticipatory utility:

$\frac{q_{0}+q_{1}}{2}\cdot
U_{N^{t}}(s=1)+\left(1-\frac{q_{0}+q_{1}}{2}\right)\cdot U_{N^{t}}(s=0)$

This expression reduces to

$1+\frac{1}{2}\cdot\left[(2q_{1}-1)(1-\beta
q_{1})-q_{1}\right]+\frac{1}{2}\cdot\left[\beta(2q_{0}-1)(1-q_{0})-q_{0}\right]$

If the media employs a fully informative signal (i.e., $q_{1}=1$, $q_{0}=0$),
this expression is equal to $1-\beta$, which is the maximal payoff from the
true narrative $N^{\ast}$. It follows that as in the example of Section 3.1,
the false narrative $N^{t}$ can only be optimal when accompanied by
imperfectly informative signals. The optimal signal function is

$q_{1}=\frac{1}{4}+\frac{1}{4\beta}\qquad\qquad
q_{0}=\frac{3}{4}-\frac{1}{4\beta}$

Note that the optimal signal treats the two states symmetrically (since
$q_{0}+q_{1}=1$).

Plugging these values of $q_{0}$ and $q_{1}$, we can confirm that
$U_{N^{t}}(s)>\frac{1}{2}$ for every $s$. The consumer’s ex-ante anticipatory
payoff is

$\frac{(1+\beta)^{2}}{8\beta}$

which is greater than $1-\beta$.

The false narrative $N^{t}$ that emerges from this exercise neglects the
effect of $a$ on $y$, and thus effectively pretends that the whac-a-mole
effect does not exist. This enables the consumer to be more optimistic about
the success of policies, but only when the narrative is accompanied by
imprecise information.

####  5.2.2 A Characterization Result

The following result shows that the optimality of the narrative $N^{t}$ in the
whac-a-mole example is not a coincidence.

###### Proposition 8

Suppose that $u(t,a,y)=v(a,y)+w(t)$. If the media can outperform the rational-
expectations benchmark, then $N^{t}$ is part of an optimal strategy.

Thus, when $u$ is separable in $t$, the fatalistic narrative is optimal. It is
the analogue of the result that $N^{a}$ is optimal when $u$ is separable in
$a$ (Proposition 1). It can also be shown that in regular environments, this
narrative can outperform the rational-expectations benchmark only if it leads
to different behavior than the benchmark. The proof is the same as that of
Proposition 2.

Finally, consider utility functions that are separable in $y$. This turns out
to be a degenerate case, in the sense that it does not give rise to false
narratives.

###### Proposition 9

Suppose that $u(t,a,y)=v(t,a)+w(y)$. The benchmark media strategy is optimal.

This completes the characterization of utility functions that are separable in
at least one of the variables.

##  6 Discussion of Related Literature

This paper belongs to a research program on the role of causal narratives in
economic and political interactions. Eliaz and Spiegler (2020) presented a
modeling framework that formalizes causal narratives as directed acyclic
graphs (building on Spiegler (2016)), where agents’ adoption of narratives is
based on the anticipatory utility they generate. Eliaz and Spiegler (2020) and
Eliaz et al. (2022) applied this framework to political competition. The
present paper brings the modeling approach to the market for news, focusing on
the role of media as suppliers of narratives. Methodologically, its main
contributions are: $(i)$ modeling the media’s joint provision of narratives
and information; $(ii)$ the novel screening problem that arises under consumer
heterogeneity (in preferences or in rationality), due to the “data
externality” between consumer types; and $(iii)$ a new conception of a
competitive media market.555Recent empirical and experimental approaches to
causal economic narratives include Ash et al. (2021), Andre et al. (2022),
Charles and Kendall (2022), Macaulay and Song (2023) and Ambuehl and Thysen
(2023).

In terms of economic substance, our paper is part of the literature on media
bias. This phenomenon has been extensively studied from various points of
view. Prat and Strömberg (2013) and Gentzkow et al. (2015) provide
comprehensive reviews of this literature. Our paper contributes to a
theoretical strand in this literature that tries to explain media bias as a
demand-based phenomenon arising from non-instrumental aspects of consumers’
attitude to information. The basic idea in this literature is that consumer
derive intrinsic utility from beliefs or from the news they consume,
independently of their effect on decisions. This idea draws on findings in
disciplines outside economics. For example, a meta-study by Hart et al. (2009)
finds that when participants are faced with a choice between information that
supports their prior beliefs and information that may challenge it, they
exhibit a preference for the former. Within the context of news media, Van der
Meer et al. (2020) find evidence that participants are more likely to view
news that confirm their prior beliefs than news that oppose them.

Mullainathan and Shleifer (2005) attempt to model this phenomenon. They
formalize both states of Nature and news as points along an interval. When a
consumer confronts news, he incurs a cost that increases in the distance
between the news and the mean of his prior belief. Media’s strategic choices
are thus reduced to a Hotelling-style model, where the consumer’s
psychological cost is analogous to a transportation cost in the standard
Hotelling model.

Gentzkow et al. (2015) present a model in which consumers’ utility has two
additively separable components. The first component is a standard material
expected-utility term that employs the consumer’s posterior beliefs, which are
obtained conventionally via Bayesian updating. This component treats beliefs
in the usual instrumental manner. The second component is a function of the
consumer’s prior belief and the distribution of signals, such that if the
prior leans in the direction of one state, then the function increases in the
frequency of the signal whose label coincides with that state’s label. This
captures the idea that people like consuming news that support their prior
beliefs. Note that this non-standard component does not reflect any belief
updating. In particular, if the media always sends a signal that coincides
with the state the consumer deems more likely (such that effectively the
signal is entirely uninformative), the non-instrumental term reaches its
maximal possible level given the consumer’s prior belief.

Thus, both Mullainathan and Shleifer (2005) and Gentzkow et al. (2015) assume
that the hedonic effect of news is orthogonal to Bayesian belief updating.
This dissociation between the Bayesian and hedonic aspects of belief formation
is one limitation of existing models we are aware of. Moreover, even when a
consumer behaves as if he has a disutility from a clash between news and his
prior beliefs, this need not be a primitive of the consumer’s preferences, but
rather a reflection of the expected change in his belief as a result of his
exposure to news. Thus, it is possible that what the consumer ultimately cares
about his posterior beliefs.

Against this background, our model introduces two innovations. To our
knowledge, it is the first model of news media as suppliers of narratives in
addition to information. It also appears to be the first model in which the
hedonic aspect of media consumers’ beliefs is fully integrated with Bayesian
updating of these beliefs. Consumers’ intrinsic utility from beliefs is a
function of Bayesian posteriors induced by the information the media provides
and the narrative it peddles. Eliaz and Spiegler (2006) is a precedent for
this aspect of our model. In that paper, we presented of demand for
information — represented by prior-dependent preferences over Blackwell
experiments — which is driven by maximization of expected utility from
(correctly specified) Bayesian posterior beliefs. Since that model allowed for
non-convex utility from beliefs, it could accommodate demand for information
that is non-increasing in Blackwell informativeness. Lipnowski and Mathevet
(2018) examined optimal information provision for agents with such
preferences.

Our assumption of Bayesian updating rules out non-Bayesian responses to
information due to motivated reasoning. For example, Taber and Lodge (2006)
show that when subjects are confronted with information that questions their
prior beliefs, they try to discredit it. Thaler (2023) is an experimental
study of the supply of information to agents who exhibit motivated reasoning
(defined as non-Bayesian updating that is affected by the valence of beliefs).
In a similar vein, Furthermore,

The idea that misspecified models can be used to manipulate agents’ beliefs
has been studied in other contexts. Eliaz et al. (2021a) analyzed a cheap-talk
model in which the sender provides not only information but also statistical
data (or, equivalently, a model) that enables the receiver to interpret the
information. Eliaz et al. (2021b) characterized the maximal distortion of
perceived correlation between two variables that a causal model can generate
in Gaussian environments. Schwartzstein and Sunderam (2021) and Aina (2023)
studied persuasion problems in which the sender proposes models, formalized as
likelihood functions, and the receiver chooses among them according to how
well they fit historical data. Finally, our paper is related to a small
literature on strategic communication with agents whose inference from signals
departs from the standard Bayesian, rational-expectations model (e.g.,
Hagenbach and Koessler (2020), Levy et al. (2022), de Clippel and Zhang
(2022)).

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Appendix: Proofs

Claim 1

Let us begin by showing that we can set $q_{1}=1$. First, note that we can
rewrite (10) as

$\frac{1}{2}\left[\frac{q_{1}}{2}+\frac{1}{2}\cdot\frac{q_{1}q_{0}}{q_{1}+q_{0}}-c(q_{1}+q_{0})\right]$

The second and third terms inside the brackets are invariant to permuting
$q_{0}$ and $q_{1}$, whereas the first term is increasing in $q_{1}$ and
invariant to $q_{0}$. Therefore, it is optimal to set $q_{1}\geq q_{0}$.

Second, note that we can rewrite (10) as

$\frac{q_{1}+q_{0}}{2}\cdot\left[\frac{1}{1+\frac{q_{0}}{q_{1}}}\cdot\left(\frac{1}{2}+\frac{1}{2}\cdot\frac{\frac{q_{0}}{q_{1}}}{1+\frac{q_{0}}{q_{1}}}\right)-c\right]$

Thus, the expression inside the square brackets only depends on the ratio
$q_{0}/q_{1}$, while the term outside them increases in both $q_{0}$ and
$q_{1}$. It follows that $q_{1}=1\geq q_{0}$ in optimum. $\blacksquare$

Proposition 1

Denote $\min_{a}c(a)=c^{\ast}$. Consider the narrative $N^{t}$. In this case,
the consumer believes that $a$ has no causal effect on $y$. Therefore, for
every $s$, he will only mix over actions that minimize $c$. Then, in
equilibrium,

$\begin{array}[]{ccc}U(I,N^{t})&=&\sum_{s}p(s)\sum_{a}p(a\mid
s)\sum_{t}p(t\mid s)\sum_{y}p(y\mid t)v(t,y)-c^{\ast}\\\
&=&\sum_{s}p(s)\sum_{t}p(t\mid
s)\sum_{y}\left(\sum_{a^{\prime}}p(a^{\prime}\mid t)p(y\mid
t,a^{\prime})\right)v(t,y)-c^{\ast}\\\ &=&\sum_{t}p(t)\sum_{s}p(s\mid
t)\sum_{y}\left(\sum_{a^{\prime}}p(a^{\prime}\mid t)p(y\mid
t,a^{\prime})\right)v(t,y)-c^{\ast}\\\
&=&\sum_{t}p(t)\sum_{a^{\prime}}p(a^{\prime}\mid t)\sum_{y}p(y\mid
t,a^{\prime})v(t,y)-c^{\ast}\end{array}$

Since $c(a^{\prime})=c^{\ast}$ whenever $p(a^{\prime}\mid t)>0$, the last
expression can be rewritten as

$\sum_{t}p(t)\sum_{a^{\prime}}p(a^{\prime}\mid t)\left[\sum_{y}p(y\mid
t,a^{\prime})v(t,y)-c(a^{\prime})\right]$

which is by definition weakly below

$\sum_{t}p(t)\max_{a}\left[\sum_{y}p(y\mid t,a)v(t,y)-c(a)\right]$

The final expression is the rational-expectations benchmark. Therefore,
$N^{t}$ cannot be part of a media strategy that outperforms it.

Now consider the narrative $N^{\emptyset}$. As in the previous case, the
consumer believes that $a$ has no effect on $y$. Therefore, for every $s$,
$p(a\mid s)>0$ only if $a$ minimizes $c$. It follows that the consumer’s ex-
ante anticipatory utility is

$\begin{array}[]{ccl}U(I,N^{\emptyset})&=&\sum_{s}p(s)\sum_{t}p(t\mid
s)\sum_{y}p(y)v(t,y)-c^{\ast}\\\ &=&\sum_{t}p(t)\sum_{y}p(y)v(t,y)-c^{\ast}\\\
&=&\sum_{t}p(t)\sum_{y}\left(\sum_{a^{\prime}}p(a^{\prime})p(y\mid
a^{\prime})\right)v(t,y)-c^{\ast}\\\
&=&\sum_{t}p(t)\sum_{a^{\prime}}p(a^{\prime})\sum_{y}\left[p(y\mid
a^{\prime})v(t,y)-c(a^{\prime})\right]\\\
&\leq&\sum_{t}p(t)\max_{a}\left[\sum_{y}p(y\mid
a)v(t,y)-c(a)\right]\end{array}$

The final expression is the ex-ante anticipatory utility induced by the
narrative $N^{a}$ coupled with no information. It follows that the maximal
anticipatory utility from $N^{\emptyset}$ can be replicated by the narrative
$N^{a}$ (coupled with fully uninformative signals). $\blacksquare$

Proposition 2

Assume the contrary — i.e., suppose there is a media strategy that induces the
same $(p(a\mid t))_{t,a}$ as in the rational-expectations benchmark, yet
outperforms it.

We first show that $N^{a}$ is the only narrative that can be part of the
strategy. The proof of Proposition 1 showed that $N^{t}$ can never outperform
the benchmark; and $N^{\ast}$ cannot do so by definition. Now consider
$N^{\emptyset}$. Under this narrative, the consumer will assign probability
one to $\arg\min_{a}c(a)$ for every $t$. By assumption, this is also the
consumer’s behavior under rational expectations, but this contradicts the
definition of regularity. This leaves $N^{a}$ as the only possible narrative.

By regularity, $p(a\mid t)$ assigns probability one to a distinct action for
each $t$. Let $t(a)$ be the unique state for which $a$ is played under $p$.
Since $t=t(a)$ whenever $p(t,s,a)>0$, it follows that

$p(y\mid a)=p(y\mid t,a)$

for every $(t,a)$ in the support of $p$. Consequently,

$p_{N^{a}}(t,y\mid s,a)=p(t,y\mid s,a)$

and therefore, the consumer’s anticipatory utility under $p$ and $N^{a}$ is
equal to the rational-expectations benchmark, a contradiction. $\blacksquare$

Proposition 3

The proof proceeds stepwise.

Step 1: Without loss of generality, each narrative is coupled with a unique
$q$.

Assume the contrary — i.e., $M$ contains two pairs $(q,N)$ and
$(q^{\prime},N)$ with $q^{\prime}<q$. This means that the signal function
given by $q^{\prime}$ Blackwell-dominates the signal function given by $q$
(recall that $\Pr(s=1\mid t=1)=1$ under both functions). Any consumer type $c$
who compares the two pairs will weakly prefer $(q^{\prime},N)$. The reason is
that consumers take the objective distribution $p$ as given. Since both pairs
share the same narrative $N$, they both induce the same $p_{N}(y\mid t,a)$.
This reduces the comparison between the pairs to a standard comparison between
signal functions by an expected-utility maximizer.

Consider consumer types $c$ who choose $(q,N)$ from $M$. They must be
indifferent between this pair and $(q^{\prime},N)$. Except for a zero-measure
set of types, this can only be the case if the consumers take a constant
action given each of the pairs (if their subjectively optimal action were
state-contingent, then $q$ and $q^{\prime}$ would induce different ex-ante
expected utility). Moreover, this must be the same constant action since
otherwise, only a particular (zero measure) type would be ex-ante indifferent
between $(q,N)$ and $(q^{\prime},N)$. Now suppose we remove $(q,N)$ from the
menu. Then, since $(q^{\prime},N)$ was optimal for these consumer types under
$M$, this will continue to be the case and all these consumers will therefore
choose $(q^{\prime},N)$. This will have no effect on the aggregate consumer
strategy because $(q^{\prime},N)$ and $(q,N)$ induce the same choices by
consumers who chose $(q,N)$ from $M$. Therefore, the switch by these consumers
from $(q,N)$ to $(q^{\prime},N)$ has no effect on how other consumer types
evaluate any media strategy. It follows that without loss of generality, we
can remove $(q,N)$ from the menu. $\square$

Step 2: Under the optimal menu, a positive measure of consumer types play
$a=1$ with positive probability.

Assume that under the optimal menu, all consumer types play $a=0$ with
certainty. Then, regardless of the media strategy they choose, their
anticipatory utility is $0$. This is obviously the case for consumer types who
choose $N^{\ast}$ or $N^{a}$, because these narratives induce the correct
belief that $a=0$ causes $y=0$ with certainty.

As to types who choose $N^{t}$, they estimate the conditional probability

$p(y=1\mid t)=\sum_{a}p(a=1\mid t)\cdot\frac{2-t}{2}=0$

for every $t$. Therefore, these types earn zero anticipatory utility as well.

Finally, types who choose $N^{\emptyset}$ form the correct belief that
$p(y=1)=0$ (because $a=0$ with probability one by assumption, and $p(y=1\mid
a=0)=0$). It follows that all types earn zero anticipatory utility. However,
if the monopolist offers the singleton menu consisting of the media strategy
$(0,N^{\ast})$, every type $c<\frac{1}{2}$ will earn
$\frac{1}{4}-\frac{c}{2}>0$, a contradiction.

Step 3: Interval structure of types’ choices

By Step 1, we can assume that for each feasible narrative $N$ there is at most
one $q$ such that $(q,N)\in M$. Because the narratives $N^{t}$ and
$N^{\emptyset}$ omit $a$ as a cause of $y$, any consumer who chooses a media
strategy that includes one of these narratives will always play $a=0$.
Furthermore, whenever a consumer chooses a media strategy that includes
$N^{\ast}$ or $N^{a}$, he will play $a=0$ in response to $s=0$, since $t=0$
with certainty conditional on $s=0$. Therefore, by Step 2, $M$ must include a
pair $(q,N)$ such that $N\in\\{N^{\ast},N^{a}\\}$, and there is a positive
measure of consumer types who select this pair and play $a=1$ in response to
$s=1$.

How consumer types rank $(q^{\ast},N^{\ast})$ and $(q^{a},N^{a})$ when playing
$a=s$ in response to both pairs

Suppose that $M$ includes both $(q^{\ast},N^{\ast})$ and $(q^{a},N^{a})$ such
that for each of these pairs, there is a positive measure of consumer types
who choose it and play $a=1$ in response to $s=1$. The ex-ante anticipatory
utility that these pairs induce for a consumer of type $c$ is:

$\displaystyle U_{c}(q^{\ast},N^{\ast})$ $\displaystyle=$
$\displaystyle\frac{1}{2}p(y=1\mid t=1,a=1)-\frac{1+q^{\ast}}{2}c\vskip 6.0pt
plus 2.0pt minus 2.0pt$ $\displaystyle=$
$\displaystyle\frac{1}{2}\cdot\frac{1}{2}[2-1]-\frac{1+q^{\ast}}{2}c=\frac{1}{4}-\frac{1+q^{\ast}}{2}c$

and

$\displaystyle U_{c}(q^{a},N^{a})$ $\displaystyle=$
$\displaystyle\frac{1}{2}p(y=1\mid a=1)-\frac{1+q^{a}}{2}c\vskip 6.0pt plus
2.0pt minus 2.0pt$ $\displaystyle=$
$\displaystyle\frac{1}{2}\cdot\frac{1}{2}[2-p(t=1\mid
a=1)]-\frac{1+q^{a}}{2}c$

It is immediate that $U_{c}(q^{\ast},N^{\ast})>U_{c}(q^{a},N^{a})$ only if
$q^{\ast}<q^{a}$. Therefore, $q^{a}>0$ in this case. Consequently, if
$U_{c}(q^{\ast},N^{\ast})>U_{c}(q^{a},N^{a})$, then
$U_{c^{\prime}}(q^{\ast},N^{\ast})>U_{c^{\prime}}(q^{a},N^{a})$ for every
$c^{\prime}>c$. It follows that if both $(q^{\ast},N^{\ast})$ and
$(q^{a},N^{a})$ are in $M$ and induce $a=1$ in response to $s=1$, then the set
of types who choose $(q^{\ast},N^{\ast})$ lies above the set of types who
choose $(q^{a},N^{a})$.

Showing that $M$ includes $(q^{a},N^{a})$ without loss of generality

Suppose that $M$ does not include $(q^{a},N^{a})$. Then, $M$ includes a pair
$(q^{\ast},N^{\ast})$ such that a positive measure of consumer types choose
this pair and play $a=1$ in response to $s=1$. (The reason is that consumers
always play $a=0$ in response to $N^{t}$ or $N^{\emptyset}$, as well as in
response to $N^{\ast}$ when $s=0$.) Now add $(q^{\ast},N^{a})$ to the menu. It
is evident that $U_{c}(q^{\ast},N^{a})\geq U_{c}(q^{\ast},N^{\ast})$ for every
$c$. Therefore, if playing $a=s$ is optimal given $(q^{\ast},N^{\ast})$, then
it is also optimal given $(q^{\ast},N^{a})$.

Moreover, if types who previously chose $(q^{\ast},N^{\ast})$ and played $a=s$
switch to $(q^{\ast},N^{a})$ and thus continue to play $a=s$, this switch does
not change the joint aggregate distribution $p(t,a)$ because
$(q^{\ast},N^{a})$ and $(q^{\ast},N^{\ast})$ share the same signal function
and induce the same consumer strategy.

Finally, consider types who previously chose a media strategy that induces
$a=0$ for all $s$ now switch to $(q^{\ast},N^{a})$. By revealed preferences,
the switch improves their own anticipatory utility, hence they must play $a=s$
(because if they play $a=0$, their anticipatory utility is $0$). At the same
time, the switch does not affect $p(t=1\mid a=1)$ because this probability is
equal to

$\frac{\frac{1}{2}\left(m(q^{\ast},N^{a})+m(q^{\ast},N^{\ast})\right)}{\frac{1}{2}\left(m(q^{\ast},N^{a})+m(q^{\ast},N^{\ast})\right)+\frac{1}{2}q^{\ast}\left(m(q^{\ast},N^{a})+m(q^{\ast},N^{\ast})\right)}=\frac{1}{1+q^{\ast}}$

where $m(q^{\ast},N^{a})+m(q^{\ast},N^{\ast})$ is the total mass of types who
choose either $(q^{\ast},N^{\ast})$ or $(q^{\ast},N^{a})$ (which is precisely
the mass of types who choose $a=s$). As a result, the switch does not affect
$U_{c}(q^{\ast},N^{a})$ for any $c$. By definition, it also does not affect
$U_{c}(q^{\ast},N^{\ast})$.

It follows that we can sustain an equilibrium with weakly higher aggregate
anticipatory utility when $(q^{\ast},N^{a})$ is added to the menu. Thus, the
menu will contain some pair $(q^{a},N^{a})$ that induces $a=1$ in response to
$s=1$.

The set of types who choose $(q^{a},N^{a})$ and play $a=s$

We now establish that there is $c^{\ast}>0$ such that all types in
$[0,c^{\ast})$ choose $(q^{a},N^{a})$ and play $a=s$. To see why, suppose
first that $(q^{\ast},N^{\ast})$ is in $M$ and that there is a positive
measure of consumers who select this pair and play $a=s$. Then, as we showed,
the set of types who select $(q^{a},N^{a})$ over $(q^{\ast},N^{\ast})$ and
play $a=s$ lies to the left of the set of types who choose
$(q^{\ast},N^{\ast})$ and play $a=s$.

Now suppose $(q^{a},N^{a})$ is the only media strategy in $M$ that induces
$a=s$. If type $c$ prefers $(q^{a},N^{a})$ to a media strategy that induces
him to always play $a=0$, then so does every $c^{\prime}<c$.

Thus, the set of types who prefer $(q^{a},N^{a})$ and play $a=1$ in response
to $s=1$ is at the low end of $[0,1]$.

The set of types who always play $a=0$

Suppose first that $M$ includes $(q^{\ast},N^{\ast})$ such that a positive
measure of types choose this pair and play $a=s$. The payoff from this choice
is $\frac{1}{4}-\frac{1}{2}c$, which is negative for $c>\frac{1}{2}$. Thus,
such types will respond to the same pair by always playing $a=0$.

Now suppose that the only media strategy that induces $a=1$ with positive
probability is $(q^{a},N^{a})$. But then,

$U_{c}(q^{a},N^{a})=\frac{1}{2}\cdot\frac{1}{2}[2-\frac{1}{1+q^{a}}]-\frac{1+q^{a}}{2}c$

which is negative for $c\approx 1$. Hence, such types will respond to
$(q^{a},N^{a})$ by always playing $a=0$.

Clearly, every type $c$ who chooses $N^{t}$ or $N^{\emptyset}$ always plays
$a=0$.

Finally, if type $c$ prefers a media strategy that induces him to always play
$a=0$, then so does type $c^{\prime}>c$. Therefore, there must be
$c^{\ast\ast}<1$ such that all types $c>c^{\ast\ast}$ choose a media strategy
that induces $a=0$ for all $s=0$.

To conclude this step, there are two cutoffs, $0<c^{\ast}\leq c^{\ast\ast}<1$,
such that: $(i)$ all types below $c^{\ast}$ choose $(q^{a},N^{a})$ and play
$a=s$; $(ii)$ all types between $c^{\ast}$ and $c^{\ast\ast}$ choose
$(q^{\ast},N^{\ast})$ and play $a=s$; and $(iii)$ all types above
$c^{\ast\ast}$ play $a=0$ with probability one. $\square$

Step 4: The menu contains exactly one of the narratives $N^{t}$ or
$N^{\emptyset}\vskip 3.0pt plus 1.0pt minus 1.0pt$

First, observe that the menu need not include both $N^{t}$ and
$N^{\emptyset}$. The reason is that both narratives induce $a=0$ with
probability one, such that they only potentially differ in the subjective ex-
ante expected value of $ty$ that they induce. In particular, they exert the
same externality on other types. Therefore, the menu will include only the one
that yields the higher payoff.

Second, suppose that $M$ contains neither $N^{t}$ nor $N^{\emptyset}$. Then,
types above $c^{\ast\ast}$ will select the narratives $N^{\ast}$ or $N^{a}$
and always play $a=0$, thus obtaining a payoff of $0$. Suppose we add $N^{t}$
or $N^{\emptyset}$ coupled with no information. By Step 2, $a=1$ with positive
probability, and therefore, both narratives will induce strictly positive
payoff for types above $c^{\ast\ast}$. We need to examine the possibility that
lower types will switch from $(q^{a},N^{a})$ or $(q^{\ast},N^{\ast})$ to the
new media strategy. However, if a type $c<c^{\ast\ast}$ deviates in this
direction, then so does every $c^{\prime}\in(c,c^{\ast\ast})$. Consider two
cases.

Case $1$: $N^{\ast}$ is not in $M$. In this case, the deviation does not
change $p(t=1\mid a=1)$ because this quantity is not affected by increasing
the share of consumers who always play $a=0$, and the set of consumers who
play $a=s$ all induce the same $\Pr(a\mid t)$. Therefore, it does not affect
$U_{c}(q^{a},N^{a})$ for any $c$. By revealed preference, the deviation
improves the ex-ante payoff of the deviating types. It follows that there is
an unambiguous increase in aggregate consumer payoffs.

Case $1$: $N^{\ast}$ is in $M$. In this case, the set of deviating types is
some interval $[c^{\ast\ast\ast},c^{\ast\ast}]$. As a result, since
$q^{\ast}<q^{a}$, this deviation lowers $p(t=1\mid a=1)$ and therefore
increases $U_{c}(q^{a},N^{a})$ for any $c$. Moreover, it has no effect on
$U_{c}(q^{\ast},N^{\ast})$ by definition. By revealed preference, the
deviation improves the ex-ante payoff of the deviating types. It follows that
the deviation increases aggregate consumer payoffs, even after taking into
account the equilibrium effects of this deviation due to the data externality.
$\square\vskip 12.0pt plus 4.0pt minus 4.0pt$

It remains to show that $q^{a}>0$ if $c^{\ast}=c^{\ast\ast}$ — i.e., the only
narrative that induces $a=1$ with positive probability is $N^{a}$. We know
from the homogenous case that for every $c<\frac{1}{2}$, $q^{a}>0$ attains
higher utility than $q^{a}=0$ (and recall that the utility from $q^{a}=0$ for
$c>\frac{1}{2}$ cannot be positive). Moreover, since $q^{a}>0$ generates a
higher overall probability of $a=1$, it exerts a positive externality than
$q=0$ on consumers who choose $N^{\emptyset}$ (it has no externality on
consumers who choose $N^{t}$). Therefore, deviating from $q^{a}=0$ to
$q^{a}>0$ is strictly profitable for the media. $\blacksquare$

Proposition 4

The proof proceeds stepwise, taking the characterization in Proposition 3 as a
starting point.

Step 1: $c^{\ast}=c^{\ast\ast}\vskip 3.0pt plus 1.0pt minus 1.0pt$

Assume that $c^{\ast\ast}>c^{\ast}$. The payoffs induced by
$(q^{\ast},N^{\ast})$ and $(q^{a},N^{a})$ at some $c$ are

$\displaystyle U_{c}(q^{\ast},N^{\ast})$ $\displaystyle=$
$\displaystyle\frac{1}{4}-\frac{1+q^{\ast}}{2}c$ $\displaystyle
U_{a}(q^{a},N^{a})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[2-p(t=1\mid a=1)\right]-\frac{1+q^{a}}{2}c$

In the proof of Step 3 of Proposition 3, we showed that $q^{a}>q^{\ast}$.
Since $c\sim U[0,1]$, we can write

$p(t=1\mid
a=1)=\frac{c^{\ast\ast}}{c^{\ast\ast}+c^{\ast}q^{a}+(c^{\ast\ast}-c^{\ast})q^{\ast}}=\frac{c^{\ast\ast}}{c^{\ast\ast}(1+q^{\ast})+c^{\ast}(q^{a}-q^{\ast})}$

At $c^{\ast}$, the indifference between $(q^{\ast},N^{\ast})$ and
$(q^{a},N^{a})$ can be written as follows:

$\frac{1}{2}c^{\ast}(q^{a}-q^{\ast})=\frac{1}{4}\left[1-\frac{c^{\ast\ast}}{c^{\ast\ast}(1+q^{\ast})+c^{\ast}(q^{a}-q^{\ast})}\right]\vskip
6.0pt plus 2.0pt minus 2.0pt$

Observe that if we slightly raise $c^{\ast}$ and lower $q^{a}$ such that
$q^{a}$ is still above $q^{\ast}$ and $c^{\ast}(q^{a}-q^{\ast})$ remains
unchanged, then the indifference condition continues to hold, as long as we
keep $c^{\ast\ast}$ fixed. In this way, $p(t=1\mid a=1)$ remains unchanged.
This modified consumer strategy is an equilibrium and it is strictly
profitable for the media. To see why, note first that $c^{\ast\ast}$ is
unchanged because by construction, $p(a=1)$ and $p(a=1\mid t=1)$ are both
unchanged, hence the payoff from $N^{t}$ or $N^{\emptyset}$ is unchanged.
Since the payoff from $(q^{\ast},N^{\ast})$ is by definition invariant to
$(p(a\mid t))$, the indifference at $c^{\ast\ast}$ continues to hold. Thus,
the set of types who always play $a=0$ and their utility are unaffected. Now
consider the infra-marginal types $c<c^{\ast}$. These types are now better off
thanks to the decrease in $q^{a}$, and since $p(a=1\mid t=1)$ is unchanged.
The types who chose and continue to choose $(q^{\ast},N^{\ast})$ are
unaffected by definition. Therefore, the new equilibrium is an improvement, a
contradiction.

What this step establishes is that we can restrict attention to menus $M$ and
consumer strategies that take either of the two following forms:
$(i)$ $M=\\{\\{q^{a},N^{a}),(q^{t},N^{t})\\}$, all consumer types in
$[0,c^{\ast}]$ choose $(q^{a},N^{a})$ and play $a=s$, and all consumer types
$c>c^{\ast}$ choose $(q^{t},N^{t})$ and play $a=0$; and
$(ii)$ $M=\\{\\{q^{a},N^{a}),(q^{\emptyset},N^{\emptyset})\\}$, all consumer
types in $[0,c^{\ast}]$ choose $(q^{a},N^{a})$ and play $a=s$, and all
consumer types $c>c^{\ast}$ choose $(q^{\emptyset},N^{\emptyset})$ and play
$a=0$. $\square$

Step 2: Completing the characterization when $M$ includes $N^{t}\vskip 3.0pt
plus 1.0pt minus 1.0pt$

Aggregate utility under $M=\\{\\{q^{a},N^{a}),(q^{t},N^{t})\\}$ is

$\int_{0}^{c^{\ast}}U_{c}(q^{a},N^{a})dc+\int_{c^{\ast}}^{1}U_{c}(q^{t},N^{t})dc$

where

$U_{c}(q^{a},N^{a})=\frac{1}{4}\left[2-\frac{1}{1+q^{a}}\right]-\frac{1+q^{a}}{2}c$

and

$\displaystyle U_{c}(q^{t},N^{t})$ $\displaystyle=$ $\displaystyle
p(ty=1)=p(t=1)\cdot p(y=1\mid t=1)\vskip 6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle=$ $\displaystyle\frac{1}{2}\cdot p(a=1\mid
t=1)\cdot\frac{1}{2}(2-1)=\frac{1}{4}c^{\ast}$

Thus, the objective function can be written as

$\displaystyle\int_{0}^{c^{\ast}}\left\\{\frac{1}{4}\left[2-\frac{1}{1+q^{a}}\right]-\frac{1+q^{a}}{2}c\right\\}dc+(1-c^{\ast})\cdot\frac{1}{4}c^{\ast}\vskip
6.0pt plus 2.0pt minus 2.0pt$ $\displaystyle=$ $\displaystyle
c^{\ast}\cdot\frac{1}{4}\left[2-\frac{1}{1+q^{a}}\right]-\frac{1+q^{a}}{2}\cdot\frac{1}{2}(c^{\ast})^{2}+(1-c^{\ast})\cdot\frac{1}{4}c^{\ast}$

The cutoff $c^{\ast}$ satisfies

$\frac{1}{4}\left[2-\frac{1}{1+q^{a}}\right]-\frac{1+q^{a}}{2}c^{\ast}=\frac{1}{4}c^{\ast}$

Plugging this equation into the objective function, we obtain

$\frac{2q^{a}+1}{\left(2q^{a}+3\right)^{2}}$

The optimal value of $q^{a}$ is $\frac{1}{2}$, yielding an aggregate utility
of $\frac{1}{8}$. $\square$

Step 3: Completing the characterization when $M$ includes $N^{\emptyset}\vskip
3.0pt plus 1.0pt minus 1.0pt$

Aggregate utility under
$M=\\{\\{q^{a},N^{a}),(q^{\emptyset},N^{\emptyset})\\}$ is

$\int_{0}^{c^{\ast}}U_{c}(q^{a},N^{a})dc+\int_{c^{\ast}}^{1}U_{c}(q^{\emptyset},N^{\emptyset})dc$

where

$U_{c}(q^{a},N^{a})=\frac{1}{4}\left[2-\frac{1}{1+q^{a}}\right]-\frac{1+q^{a}}{2}c$

and

$\displaystyle U_{c}(q^{\emptyset},N^{\emptyset})$ $\displaystyle=$
$\displaystyle p(t=1)\cdot p(y=1)$ $\displaystyle=$ $\displaystyle
p(t=1)\cdot[p(t=1)\cdot p(y=1\mid t=1)+p(t=0)\cdot p(y=1\mid t=0)]$
$\displaystyle=$ $\displaystyle\frac{1}{2}\cdot[\frac{1}{2}\cdot p(a=1\mid
t=1)\cdot\frac{1}{2}(2-1)+\frac{1}{2}\cdot p(a=1\mid
t=0)\cdot\frac{1}{2}(2-0)]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\cdot[\frac{1}{2}\cdot
c^{\ast}\cdot\frac{1}{2}(2-1)+\frac{1}{2}\cdot
c^{\ast}q^{a}\cdot\frac{1}{2}(2-0)]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\cdot[\frac{1}{4}c^{\ast}+\frac{1}{2}c^{\ast}q^{a}]$
$\displaystyle=$ $\displaystyle\frac{c^{\ast}}{4}[\frac{1}{2}+q^{a}]$

Thus, the objective function can be written as

$\displaystyle\int_{0}^{c^{\ast}}\left\\{\frac{1}{4}\left[2-\frac{1}{1+q^{a}}\right]-\frac{1+q^{a}}{2}c\right\\}dc+(1-c^{\ast})\cdot\frac{c^{\ast}}{4}[\frac{1}{2}+q^{a}]$
$\displaystyle=$ $\displaystyle
c^{\ast}\cdot\frac{1}{4}\left[2-\frac{1}{1+q^{a}}\right]-\frac{1+q^{a}}{2}\cdot\frac{1}{2}(c^{\ast})^{2}+(1-c^{\ast})\cdot\frac{c^{\ast}}{4}[\frac{1}{2}+q^{a}]$

The cutoff $c^{\ast}$ satisfies

$\frac{1}{4}\left[2-\frac{1}{1+q^{a}}\right]-\frac{1+q^{a}}{2}c^{\ast}=\frac{c^{\ast}}{4}[\frac{1}{2}+q^{a}]$

Plugging this equation into the objective function, we obtain

$\frac{3}{4}\left(2q^{a}+1\right)^{2}\frac{2q^{a}+3}{\left(6q^{a}+5\right)^{2}\left(q^{a}+1\right)}$

This expression is monotonically increasing in $q^{a}$, hence the optimal
value of $q^{a}$ is $1$, yielding an aggregate utility of approximately
$0.139$. $\square$

Since the menu characterized by Step 3 yields a higher payoff than the one
characterized by Step 2, the optimal menu includes the denial narrative, and
sets $q^{a}=1$. $\blacksquare$

Proposition 5

First, we establish that without loss of generality, $I_{c}$ is the perfectly
informative signal function for every $c$. The reason is that the maximization
of type $c$’s anticipatory utility takes $p$ as given without taking into
account the effect of the behavior induced by $(I_{c},N_{c})$ on $p_{N}$.
Therefore, $U_{c}$ is effectively the maximum of functions that are linear in
beliefs, hence it is convex in posterior beliefs. It follows that a fully
informative signal maximizes $U_{c}$ (as in the rational-expectations
benchmark). It is the unique maximizer if it induces $a=s$.

Second, we show that all consumers play $a=0$ when $t=0$. This holds under
$N^{t}$ or $N^{\emptyset}$ because under these narratives, $a$ has no causal
effect on $y$. Under $N^{a}$ or $N^{\ast}$, optimal provision of information
implies that when $t=0$ the consumer knows that $ty=0$, and therefore finds
$a=0$ optimal.

An immediate consequence of the previous step is that $p(t=0\mid a=1)=0$. such
that the formulas for $U_{c}$ under $N^{\ast}$ and $N^{a}$ coincide. Thus,
from now on, we will take it for granted that the only narrative that can
induce $a=1$ with positive probability is $N^{\ast}$. Let us denote by
$\sigma$ the fraction of consumers who play $a=1$ when $t=1$.

Third, we will show that $N^{t}$ weakly outperforms $N^{\emptyset}$ for every
consumer type. To see why, let us write down the anticipatory utility under
each of these narratives. The anticipatory utility under $N^{t}$ is

$p(t=1)p(y=1\mid
t=1)=\frac{1}{2}\cdot\sigma\cdot\frac{1}{2}(2-1)=\frac{1}{4}\sigma$

The anticipatory utility under $N^{\emptyset}$ is

$\begin{array}[]{ccc}p(t=1)p(y=1)&=&\frac{1}{2}\cdot\left[\frac{1}{2}\cdot
p(y=1\mid t=1)+\frac{1}{2}\cdot p(y=1\mid t=0)\right]\\\
&=&\frac{1}{4}p(y=1\mid t=1)\\\ &=&\frac{1}{8}\sigma\end{array}$

Note that $p(y=1\mid t=0)=0$ because all consumers play $a=0$ when $t=0$.

Thus, the only narratives we need to consider are $N^{\ast}$ and $N^{t}$.
Moreover, we can assume that any consumer who adopts $N^{\ast}$ will play
$a=1$ when $t=1$, because otherwise he would get zero payoffs, which is below
the payoff he can get from $N^{t}$. A consumer of type $c$ will prefer
$N^{\ast}$ if $\frac{1}{4}-\frac{c}{2}>\frac{1}{4}\sigma$. There is a cutoff
$\bar{c}$ characterized by
$\frac{1}{4}-\frac{\bar{c}}{2}=\frac{1}{4}F(\bar{c})$, such that all
$c<\bar{c}$ choose $N^{\ast}$ and play $a=t$, while all $c>\bar{c}$ choose
$N^{t}$ and always play $a=0$. $\blacksquare$

Proposition 6

Suppose first $N_{nr}\in\\{N^{t},N^{\emptyset}\\}$. Then, type $nr$’s
evaluation of the pair $(q_{nr},N_{nr})$ is independent of $q_{nr}$, and he
always plays $a=0$ in response to this pair. As to type $r$’s evaluation of
the same pair, it is independent of $N_{nr}$ because this type applies
$N^{\ast}$. Thus, type $r$’s evaluation of $(q_{nr},N_{nr})$ only depends on
$q_{nr}$. Moreover, it is decreasing in the degree of media bias $q$. It
follows that if the media collapses the menu
$\\{(q_{r},N_{r}),(q_{nr},N_{nr})\\}$ into a singleton $\\{(0,N_{nr})\\}$, it
will weakly raise consumers’ aggregate anticipatory utility.

Now suppose $N_{nr}=N^{a}$. First, it cannot be the case that type $nr$
responds to $(q_{nr},N_{nr})$ by always playing $a=0$. The reason is that
under this strategy, $U_{nr}(q_{nr},N^{a})=0$. This cannot be optimal, because
the singleton menu $\\{(0,N^{t})\\}$ would outperform it: Since type $r$
responds to this pair by playing $a=t$, $U_{nr}(0,N^{t})=\frac{1}{4}\lambda>0$
— i.e., the singleton menu maximizes $U_{r}$ while inducing $U_{nr}>0$.

We can thus take it for granted that type $nr$ plays $a=s$. If
$(q_{r},N_{r})\neq(q_{nr},N_{nr})$, then in order for type $r$ to favor
$(q_{r},N_{r})$ over $(q_{nr},N_{nr})$, he must find the former pair more
informative — i.e., $q_{r}<q_{nr}$. Therefore, $q_{r}<1$, such that type $r$
plays $a=s$ in response to $(q_{r},N_{r})$.

Then,

$U_{r}(q_{r},N_{r})=\frac{1}{4}-\frac{1+q_{r}}{2}c$

and

$\displaystyle U_{r}(q_{nr},N^{a})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[2-p(t=1\mid a=1)\right]-\frac{1+q_{nr}}{2}c$
$\displaystyle=$ $\displaystyle\frac{1}{4}\left[2-\frac{p(t=1)p(a=1\mid
t=1)}{p(t=1)p(a=1\mid t=1)+p(t=0)p(a=1\mid t=0)}\right]$
$\displaystyle-\frac{1+q_{nr}}{2}c$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[2-\frac{1}{1+\lambda
q_{r}+(1-\lambda)q_{nr}}\right]-\frac{1+q_{nr}}{2}c$

Crucially, $p(t=1\mid a=1)$ is based on the aggregate distribution, which
makes use of both types’ strategies. Denote $\bar{q}=\lambda
q_{r}+(1-\lambda)q_{nr}$. It follows that the aggregate anticipatory utility
is

$\lambda\frac{1}{4}+(1-\lambda)\frac{1}{4}\left[2-\frac{1}{1+\bar{q}}\right]-\frac{1+\bar{q}}{2}c$
(16)

This is exactly the same aggregate utility that would be obtained from a
singleton menu $\\{(\bar{q},N^{a})\\}$, where both types respond to the pair
by playing $a=s$. therefore, there is no loss of generality in restricting
ourselves to singleton menus. $\blacksquare$

Proposition 7

Suppose $\lambda$ is close to $0$ — i.e., the population consists almost
entirely of non-rational consumers. Then, $N^{a}$ is part of an optimal
strategy. This follows from continuity relative to the $\lambda=0$ case. The
derivative of (16) with respect to $\bar{q}$ is

$\frac{1-\lambda}{4(1+\bar{q})^{2}}-\frac{c}{2}$ (17)

The optimal $\bar{q}$ is given by the first-order condition.

Now suppose $\lambda$ close to $1$ — i.e., the population consists almost
entirely of rational consumers. Suppose that $N^{a}$ is part of an optimal
strategy. As $\lambda\rightarrow 1$, (17) becomes negative, hence it is
optimal to set $\bar{q}=0$ — i.e., offering a fully informative signal.
However, when $\bar{q}=0$, $N^{a}$ offers the same utility for non-rational
consumers as $N^{\ast}$, hence it cannot outperform $(0,N^{\ast})$. The only
remaining media strategies we need to check are $(0,N^{t})$ and
$(0,N^{\emptyset})$. Note that

$U_{nr}(0,N^{t})=\frac{1}{4}\lambda>\frac{1}{8}\lambda=U_{nr}(0,N^{\emptyset})$

Obviously, since $\lambda\approx 1$,
$U_{nr}(0,N^{t})>U_{r}(0,N^{t})=\frac{1}{4}-\frac{1}{2}c$. It follows that the
optimal media strategy when $\lambda$ is close to $1$ is $(0,N^{t})$.
$\blacksquare$

Proposition 8

First, observe that for every feasible strategy $(I,N)$, the ex-ante
subjective expectation of $w(t)$ is

$\sum_{s}p(s)\sum_{t^{\prime}}p_{N}(t^{\prime}\mid s)w(t^{\prime})$

Recall that for every feasible narrative $N$, $p_{N}(t^{\prime}\mid s)\equiv
p(t^{\prime}\mid s)$. Therefore, the above expression reduces to

$\sum_{t^{\prime}}p(t^{\prime})w(t^{\prime})=Ew(t)$

regardless of $(I,N)$. Therefore, we can regard $Ew(t)$ as a constant in the
media’s objective function, and focus on the $v$ term. Thus, from now on, we
conveniently set $w(t)=0$ for all $t$ — this without loss of generality.

Consider the narrative $N^{a}$. In this case,

$U_{I,N^{a}}(s,a)=\sum_{t}p(t\mid s)\sum_{y}p(y\mid a)v(a,y)=\sum_{y}p(y\mid
a)v(a,y)$

We can see that $I$ is irrelevant for the consumer’s anticipatory utility from
action $a$. It follows that his ex-ante anticipatory utility can be written as

$\begin{array}[]{ccc}\sum_{a}p(a)\sum_{y}p(y\mid
a)v(a,y)&=&\sum_{a}p(a)\sum_{y}\left(\sum_{t}p(t\mid a)p(y\mid
t,a)\right)v(a,y)\\\ &=&\sum_{t}p(t)\sum_{a}p(a\mid t)\sum_{y}p(y\mid
t,a)v(a,y)\\\ &\leq&\sum_{t}p(t)\max_{a}\sum_{y}p(y\mid t,a)v(a,y)\end{array}$

The final expression is the consumer’s maximal ex-ante anticipatory utility
according to the true narrative $N^{\ast}$. Therefore, $N^{a}$ cannot be part
of a media strategy that outperforms the strategy of providing complete
information and the true narrative.

Now consider the narrative $N^{\emptyset}$. In this case,

$U_{I,N^{\emptyset}}(s,a)=\sum_{t}p(t\mid
s)\sum_{y}p(y)v(a,y)=\sum_{y}p(y)v(a,y)$

Here, too, we can see that $I$ is irrelevant for the consumer’s anticipatory
utility from action $a$. It follows that his ex-ante anticipatory utility can
be written as

$\displaystyle\sum_{a}p(a)\sum_{y}p(y)v(a,y)=\sum_{a}p(a)\sum_{y}\left(\sum_{t}p(t)p(y\mid
t)\right)v(a,y)$ $\displaystyle=$
$\displaystyle\sum_{a}p(a)\sum_{t}p(t)\sum_{y}p_{N^{t}}(y\mid t,a)v(a,y)$

This is equal to the ex-ante anticipatory utility from the mixture over
actions $(p(a))$, when the media conveys the narrative $N^{t}$ and provides no
information. It follows that the maximal anticipatory utility from
$N^{\emptyset}$ can be replicated by the narrative $N^{t}$ (coupled with fully
uninformative signals). $\blacksquare$

Proposition 9

Consider the term $v(t,a)$. As we have observed, $p_{N}(t,s,a)\equiv p(t,s,a)$
for every feasible narrative $N$. Therefore,

$\sum_{s}p(s)\sum_{a}p(a\mid s)E_{N}v(t,a\mid s,a)=E_{N^{\ast}}(v(t,a))$

Now turn to the term $w(y)$. The ex-ante expectation of this term according to
some feasible $(I,N)$ is

$\sum_{y}p_{N}(y)w(y)$

We will now show that $p_{N}(y)\equiv p_{N^{\ast}}(y)$ for every feasible
false narrative. First, observe that

$p_{N}(y)=\sum_{t}p(t)\sum_{s}p(s\mid t)\sum_{a}p(a\mid s)p_{N}(y\mid
t,a)=\sum_{t,a}p(t,a)p_{N}(y\mid t,a)$

Let us now write this expression for each of the three feasible false
narratives. For $N^{a}$,

$\sum_{t,a}p(t,a)p(y\mid a)=\sum_{a}p(a)p(y\mid a)=p(y)$

For $N^{t}$,

$\sum_{t}p(t,a)p(y\mid t)=\sum_{t}p(t)p(y\mid t)=p(y)$

Finally, for $N^{\emptyset}$,

$\sum_{t,a}p(t,a)p(y)=p(y)$

It follows that both terms of $u$ are undistorted by any false narrative.
Therefore, the media cannot outperform the true narrative (coupled with full
information). $\blacksquare$