# The Supply of Motivated Beliefs††thanks: Thank you to Roland Bénabou, Yan
Chen, Ernesto Dal Bo, Christine Exley, Drew Fudenberg, David Laibson, Cesar
Martinelli, Matthew Rabin, Mattie Toma, Leeat Yariv, Sevgi Yuksel, seminar
participants at Exeter, HEC Paris, Imperial, LMU Munich, Michigan, Nottingham,
Princeton, Toronto, UCL, and Warwick, and attendees at the ASSA Annual
Meetings, Belief-Based Utility Workshop, NBER SI Political Economy, SITE:
Psychology and Economics, and others for helpful comments. I am grateful for
funding support from the Pershing Square Fund for the Foundations of Human
Behavior. Preregistration plans for the two experiments are available on
AsPredicted: (Primary experiment: https://aspredicted.org/blind.php?x=VMH_8HW;
Additional experiment: https://aspredicted.org/blind.php?x=GAQ_ZJF.) Both
experiments were approved by the IRB at Princeton University (13114-04 and
13114-05).

Michael Thaler University College London. Email<EMAIL_ADDRESS>
(September 2023)

##  1 Introduction

There has been a proliferation of interest in understanding how people
communicate with others and why much of the news marketplace contains
inaccurate information. “Fake news,” defined by [4] as the intentional
reporting of false information, has been shown to be a contributing factor to
the undermining of trust in public health, historical misconceptions, and the
state of democracy ([46]; [54]; [56]). Given these large societal costs, it is
important to better understand what motivates people to send false information
to others in order to inform efforts to improve news dissemination and
increase trust in credible sources.

False information has been of particular concern in environments where
information comes from unmoderated sources, such as on social media, and when
information is about issues that evoke strong motivated beliefs, such as in
politics ([4]). Meanwhile, there is ample evidence that motivated beliefs
affect people’s demand for information (e.g. [57]; [40]), but much less is
known about their effects on the supply of information.

This paper provides novel experimental evidence to show that motivated
reasoning influences the supply of false information, focusing on political
settings. It highlights the role of two factors that lead senders to be less
truthful: (1) believing that receivers’ motivated beliefs are misaligned with
the truth, and (2) having incentives to be perceived as truthful. Both factors
are prevalent in many political news transmission environments, and can play a
significant role on social media, for two reasons. First, when news senders
have information about their audience, they often can tailor messages to
appeal to their receivers’ politics. If receivers prefer to hold beliefs that
are farther in the direction that aligns with their political party, they will
be motivated to believe that “good news” about their party is true while “bad
news” is false. Second, many social media platforms have users rate other
users, incentivizing news senders to be rated well by others. For instance,
Facebook implemented a policy in 2015 that enabled users to report news as
false, and by 2018 began assigning users a credibility score based on the news
they shared ([26]). These incentives can perversely lead to more
disinformation if users send false messages to appeal to receivers’ motivated
beliefs in order to boost their trustworthiness.

In order to have a controlled environment that is able to cleanly identify the
effects of motivated beliefs and incentives, I turn to the (online) laboratory
and run two large preregistered experiments. The primary experiment recruits
self-reported social media users in the United States. Subjects send and
receive messages about various factual issues that are chosen to evoke
politically-motivated beliefs, as described in Table 1. On each question,
receivers report their prior belief about which of two answers is correct.
Senders learn the true answer to the question and then choose, as a function
of the receiver’s prior, whether to send a message to the receiver that
corresponds to the true answer or to the false answer. Finally, receivers rate
the probability that the message from their sender is truthful. To identify
the role that incentives play, senders are randomly assigned to either be paid
as a function of receivers’ ratings or to have their pay not depend on
receivers’ ratings. To identify the role that motivated beliefs play, senders
are randomly assigned to receivers whose political party is either aligned
with the truthful message, whose party is aligned with the false message, or
whose party is unknown, and can condition on the receivers’ prior beliefs.

Topic | Pro-Democrat Motives | Pro-Republican Motives
---|---|---
Immigrants’ crime rate | Lower than US citizens | Higher than US citizens
Racial discrimination | Severe in labor market | Not severe in labor market
US crime | Got worse under Trump | Got better under Trump
Media bias | Media not mostly Dems | Media mostly Dems
COVID-19 restrictions | Mostly stopped spread | Did not mostly stop spread
Gun reform | Decreased homicides | Did not decrease homicides
Unemployment | Got worse under Trump | Got better under Trump
Wages | Grew slower under Trump | Grew faster under Trump
Undocumented immigrants | Mostly overstaying visas | Mostly illegally entered US
Domestic terrorism | More due to white supremacy | More due to other factors
Poverty rates | Got worse under Trump | Got better under Trump
Illegal immigration | Not historically high | Historically high
Table 1: The list of political topics and hypothesized motives in the
experiments.

The main result from the primary experiment is that senders are more likely to
send false messages when they face incentives to be rated as truthful and when
the receiver’s politics are aligned with the false message. When faced with
receivers with the same prior belief, senders with these incentives are 7.1
percentage points (subject-level clustered s.e. 3.2 pp) more likely to send a
false message when the receiver’s party is aligned with the false message than
when it is aligned with the true message. Senders are also more likely to send
false messages when the receiver’s party is aligned with the false message
than when the receiver’s party is unknown or when the topic is neutral instead
of political. Since senders condition on receivers’ prior beliefs, these
results indicate that senders are tailoring messages to receivers’ politics
separately from their priors.

There are many alternative explanations as to why senders may condition
messages on receivers’ party, such as altruism or innate preferences to be
seen as truthful. These factors are not directionally driving results, as
without incentives, senders no longer condition on receivers’ party.
Relatedly, the second main result is that incentives have a negative effect on
truthfulness overall. Senders are 7.3 pp (s.e. 2.5 pp) more likely to send
false messages when incentivized, an increase of 34 percent. The negative
effect of incentives is driven by cases in which the receiver’s party is
misaligned with the truth.

Results are similar in an additional treatment in which senders cannot
condition on receivers’ priors: incentivized senders condition messages on
receivers’ party even more, but unincentivized senders do not on average, and
incentives lead to significantly more false messages being sent. Results are
also robust to other specifications and cuts of the data.

Next, we turn to receivers’ behavior. Receivers do not incorporate the
negative effects of incentives when rating the truthfulness of senders.
Neither receivers’ ratings nor beliefs are statistically-significantly
affected by senders’ incentives, and the point estimates are tightly estimated
around zero. Survey evidence confirms these patterns: Senders self-report
choosing messages that are more aligned with the receiver’s party, and less
aligned with the truth, when they are incentivized, but receivers’ predictions
of sender behavior are not significantly affected by senders’ incentives.

In an additional experiment, I further identify the mechanism of motivated
reasoning, defined as the distortion of belief updating to favor preferred
states ([45]; [9]; [44]; [68]). My preferred explanation from the primary
experiment is that senders believe that receivers engage in motivated
reasoning, and tailor messages to appeal to this bias. The primary experiment
is able to rule out other preference-based explanations by comparing
incentivized to unincentivized senders, but higher-order belief explanations
are harder to rule out. For instance, if senders believe that receivers expect
senders to distort messages in the direction opposed to receivers’ party,
leading them to trust pro-party messages more (such as in [52]). To identify
the motivated-reasoning mechanism, the additional experiment breaks the
interaction between senders and receivers, isolating the beliefs channel
directly.

I identify motivated reasoning among receivers using a version of the design
from [68]. This design has two main steps. First, each receiver is given a
variety of factual questions with numerical answers. On each question, the
receiver gives a guess that they think is equally likely to be above or below
the correct answer; that is, the median of their belief distribution is
elicited. Second, the receiver is given two binary messages from the computer:
one message is true and and the other is false. Each message tells them
whether the answer was above or below their median. The receiver is not told
whether each source is true or false; instead, they are asked to make
inferences about its veracity from the message content. Since messages relate
the true answer to the receiver’s median, a Bayesian would believe that it is
equally likely for each source to report either message.111That is, the
receiver has stated that they believe the answer is equally likely to be
greater than or less than her median; so, they believe the likelihood that a
true message would report that the answer is greater than their median is 1/2,
and the likelihood that a false message would report that the answer is less
than their median is also equal to 1/2, leading a “greater than” or “less
than” message to be completely uninformative about the veracity of the news
source. On the other hand, a receiver who engages in motivated reasoning will
think the news is more likely to be true if it sends a message that aligns
more with their preferred (political) beliefs. In [68], I argue that this
method is a well-identified and well-powered way to identify motivated
reasoning.222While there are other experimental approaches to identifying
motivated reasoning, such as those in [51] and [27], those approaches have
often found it difficult to cleanly detect motivated reasoning, as
identification of the bias from Bayesian updating and other inference biases
is noisy ([12]; [66]; [67]). Indeed, I find significant motivated reasoning
among receivers here as well.

After receivers play the game above, each sender is given a matched receiver’s
median belief and are asked to select either the “greater than” or the “less
than” message of the computer. The two primary treatment arms are the same as
those in the primary experiment. First, senders are either incentivized to
have the receiver’s rating of the message be implemented for payment, or they
are unincentivized. Second, senders are matched to receivers whose party is
either aligned or misaligned with the truth. The median beliefs provided to
the sender are selected such that at least one Democrat and at least one
Republican has stated that median. This enables random assignment, for a given
prior belief, to a receiver who is a Democrat, is a Republican, or is equally
likely to be of either party.

The main finding from the additional experiment coincides with the main
finding from the primary experiment; senders who are incentivized to be
perceived as truthful are more likely to choose the false message when it is
aligned with the receiver’s party, while unincentivized senders are not.
Incentivized senders are 14.5 percentage points (s.e. 4.5 pp) more likely to
send a false message when the receiver’s party is aligned with the false
message than when it is aligned with the true message. Unincentivized senders
are not directionally affected by this treatment. Incentives again lead to
more false messages chosen. Senders are 8.5 pp (s.e. 2.3 pp) more likely to
send false messages overall when incentivized, a 34 percent increase, which is
driven by the condition in which the receiver’s party is misaligned with the
truth. This result clarifies the form of politically-motivated reasoning that
determines senders’ beliefs and choices of false messages.

The second finding from the additional experiment is that senders demand
information about receivers. A majority of senders are willing to pay a
positive amount to learn the political party of receivers on political
questions, and a majority are not willing to on neutral questions. Senders use
receivers’ party information in order to choose more false news, suggesting
that giving news providers the option to learn about their audience may cause
the audience to receive less truthful pieces of news when motivated reasoning
is at play.

In both experiments, I elicit beliefs about others, finding clear evidence
that senders expect receivers to use their politically-motivated beliefs, in
addition to prior beliefs, when inferring the truthfulness of news. Senders’
beliefs are directionally accurate, but on average they overstate the impact
of party on inference.

Theoretically, the main experimental findings can be explained as equilibria
of a simple sender-receiver game. In this game, the sender prefers to tell the
truth and has incentives for the receiver to believe that she is telling the
truth, and the receiver prefers accurately rating the sender. There is always
a separating Bayesian Nash equilibrium (BNE) in which the sender reports the
truth and the receiver believes that all sent messages are truthful. However,
when the receiver engages in motivated reasoning, directionally distorting his
posteriors in directions he prefers, there may no longer be a separating BNE,
as the sender’s best response is to bias messages in the receiver-preferred
direction.333More technically, I discuss a solution concept in which the
receiver plays a strategy of a BNE in which he has one particular level of
bias, and the sender — who is aware of the receiver’s strategy and beliefs
about his own bias — plays a best response to the receiver under the
assumption that he has another level of bias. When the sender and receiver
have different beliefs about the receiver’s motivated reasoning, and the
receiver is unaware of this difference, incentives can lead to a gap between
the strategy the sender plays and the strategy the receiver plays a best
response to. In particular, the sender’s incentives lead her to send less
truthful messages but do not lead the receiver to rate her as less truthful,
predictions that are consistent with the experimental results.

This paper adds to several strands of literature. It is among the first papers
to unite the growing literature on the demand for motivated beliefs with the
large literature on the supply of misinformation and disinformation. The
definition of motivated reasoning I use was developed in economics by [9] and
was further formalized in a series of subsequent papers by these authors
([10]; [8]; [11]).444The “optimal beliefs” framing discussed in my model is
also similar to the setups of [15] and [51]). I contribute to the
understanding of the relationship between politics and misperceptions has been
studied extensively in applied political settings (e.g. [65]; [3]; [2]). [8],
[47], and [50] theoretically study beliefs about others’ motivated reasoning,
but to my knowledge, my paper is the first to experimentally study such
higher-order beliefs.555My model has overlaps with [47]. He studies a setting
in which a policy-maker exploits the demand for motivated beliefs by voters by
choosing whether to supply information to them and finds that there is often a
truthful equilibrium as well as non-truthful equilibria. In my setting, I find
an even starker setting in which truthful equilibria cannot exist. [42] study
persuasion in a motivated setting, but focus on senders and receivers with
materially-misaligned incentives. The additional experiment further extends
the experimental design of [68] (which also studies politics), following a set
of papers that study motivated reasoning about various topics including gender
differences ([69]), racial discrimination ([30]), and “good news” about the
world ([70]).

These results relate to the experimental cheap talk literature and preferences
for truth-telling. As in [1], I find that people prefer to send truthful
messages but that this preference is malleable. Compared to most cheap-talk
and information-design games in the literature (e.g. [16]; [33]), my game does
have a truth-telling BNE, but still finds considerable lying and
distrust.666This result is quite different from [16], who find excess trust
and truth-telling absent motivated beliefs. A number of other papers
experimentally study the determinants of lying (e.g. [28]; [61]; [39]),
receivers’ behavior in detecting lies in nonpolitical contexts ([62]); and
non-motivated reasons why people may lie to be seen as truthful ([21]; [7];
[64]). I add to this literature by emphasizing behavior of senders and
relating lying to beliefs about others’ motivated reasoning.

This paper also adds to the psychology literature on bias blind spots ([60];
[59]; [58]) and persuasion of news ([23]) — which finds that people see
themselves as less susceptible to biases than others — by considering belief
gaps about motivated reasoning.777In economics, [32] finds similar results in
the context of self-control problems. It also relates to papers that
experimentally study higher-order beliefs about motivated and unmotivated
biases in social settings ([55]; [14]; [35]), and theoretically to literatures
on higher-order belief biases like bounded strategic rationality ([22]),
cognitive hierarchy ([17]), and cursed equilibrium ([29]). [63] find
complementary evidence about what people believe about their own motivated
reasoning. My data provide evidence for bias blind sports about motivated
reasoning, in which people act as if others treat them as if they were less
biased.

Finally, these findings contribute to the understanding of the spread of
disinformation on social media. They show that even small incentives can
significantly change behavior, and that disinformation is especially prevalent
in contexts that evoke motivated beliefs. Related literature analyzes how news
providers distort messages in the direction of their audiences’ current
beliefs ([36]; [37]; [53]); I show that appealing to motivated beliefs is
independently important, and may even provide an alternative explanation for
previous findings. These findings also provide a belief-driven explanation for
pandering, in which senders bias messages towards receivers’ preferences
([49]; [19]; [18]). Here, motivated reasoning impedes truth-telling equilibria
that exist even when receivers have differing priors, a gap that does not
typically occur in individual decision making ([48]).

The rest of the paper proceeds as follows: Section 2 develops the sender-
receiver model. Section 3 presents the design and results of the primary
experiment, and Section 4 presents the additional experiment. Section 5
concludes and proposes directions for future work. Additional results and
study materials are in the appendices.

##  2 Theory

###  2.1 A signaling game

This section introduces a simple sender-receiver game that forms the basis of
the primary experiment. I consider a game with one sender (she) and one
receiver (he). There is a state of the world which is either high
($\theta_{H}$) or low ($\theta_{L}$). The sender observes the true state of
the world and the receiver wants to predict the true state of the world. The
sender has a preference for telling the truth and may receive an additional
benefit for having the receiver think she is telling the truth. The sender may
benefit from being perceived as truthful because the receiver wishes to
consume more news from her or because the receiver will be more likely to
“like” or “share” the message.

After observing the sender’s message, the receiver is asked to rate the
probability that the sender was telling the truth, receiving a payoff that
depends on the accuracy of his rating. These ratings are functionally
equivalent to rating the probability that the state is high or low. The
receiver’s payoff is a reduced-form interpretation of accuracy motives.

More formally, the timing of the game is as follows: First, Nature chooses
whether the state $\theta$ is $\theta_{H}$ (with probability $\pi$) or
$\theta_{L}$ (with probability $1-\pi$). We will later interpret $\pi$ as
reflecting R’s prior belief of the state. Next, S learns the state, and then
chooses whether to send message $x_{H}$ or $x_{L}$ to R. After observing the
message, R takes action $a\in[0,1]$.

The receiver’s expected utility is set so that he is incentivized to report
his true belief: $u_{R}=1-(1-a)^{2}$ if the sender’s message is truthful and
$u_{R}=1-a^{2}$ if the message is false. The sender’s utility includes two
components. She has an intrinsic value to report the truth, receiving $\tau>0$
for truth-telling. She also receives utility that is linear in the receiver’s
rating $a$ of her truthfulness. $\gamma\geq 0$ corresponds to the weight put
on the receiver’s rating.

The expected utility matrix, as a function of $a$, is as follows:

| Nature chooses $\theta_{H}$ | Nature chooses $\theta_{L}$
---|---|---
Sender chooses $x_{H}$ | $\gamma a+\tau,1-(1-a)^{2}$ | $\gamma a,1-a^{2}$
Sender chooses $x_{L}$ | $\gamma a,1-a^{2}$ | $\gamma a+\tau,1-(1-a)^{2}$

The first element in each cell is the sender’s utility, and the second is the
receiver’s utility. Figure 1 describes the game in extensive form.

[scale=2] 18.4mm18.4mm (0)[chance node]<below>N <grow=left>θ_H[a]π[b]
<grow=right>θ_L[a]1-π[b] 12mm20mm (1)(0-1)<180>S <grow=north>x_H[l]
<grow=south>x_L[l] (2)(0-2)<0>S <grow=north>x_H[r] <grow=south>x_L[r] 6mm14mm
[dotted][gray!35] ’[west](a1)(1-1) a[a](γa + τ1-(1-a)2) [west](b1)(1-2)
a[a](γa1-a2) [east](b2)(2-1) a[a](γa1-a2) ’[east](a2)(2-2) a[a](γa +
τ1-(1-a)2) [dashed](a1)(b2)R [dashed](b1)(a2)R

Figure 1: The extensive-form game. S’s payoffs are listed on top; R’s payoffs
are listed on bottom. Dashed lines denote information sets.

We first consider the Bayesian Nash equilibria (BNE) of the game. There is
always a full-information separating BNE in which the sender plays a truthful
strategy — sending $x_{H}$ given $\theta_{H}$ and sending $x_{L}$ given
$\theta_{L}$ — and the receiver chooses $a(x_{H})=1$ and $a(x_{L})=1$. The
sender earns $\gamma+\tau$ and the receiver earns 1, and this BNE is Pareto
optimal.888I focus on BNE, instead of equilibrium refinements, for simplicity.
The truthful BNE is also a perfect Bayesian equilibrium and a sequential
equilibrium. As is common in coordination games, there may be other
equilibria. In this game, the existence of other equilibria depends on
$\tau/\gamma$ being sufficiently small.999Specifically, there is a pooling BNE
in which the sender always sends $x_{H}$ iff $\tau/\gamma\leq\pi$. In such a
BNE, the receiver plays $a(x_{H})=\pi$ and $a(x_{L})=\pi^{\prime}$ for
$\pi^{\prime}<\pi-\tau/\gamma$. Similarly, there is an $x_{L}$-pooling BNE iff
$\tau/\gamma\leq 1-\pi$. The receiver plays $a(x_{H})=1-\pi^{\prime\prime}$
and $a(x_{L})=1-\pi$ for $1-\pi^{\prime\prime}<1-\pi-\tau/\gamma$. There can
also be mixed-strategy equilibria. If $\tau/\gamma\leq 1-\pi$, there is a BNE
in which the sender sends $x_{H}|\theta_{H}$ with probability 1, and sends
$x_{L}|\theta_{L}$ with probability
$\frac{1-\pi-\tau/\gamma}{(1-\pi)(1-\tau/\gamma)}$. The receiver plays
$a(x_{H})=1-\tau/\gamma$ and $a(x_{L})=1$. If $\tau/\gamma\leq\pi$ there is a
BNE in which the sender sends $x_{L}|\theta_{L}$ with probability 1 and sends
$x_{H}|\theta_{H}$ with probability
$\frac{\pi-\tau/\gamma}{\pi(1-\tau/\gamma)}$. The receiver plays
$a(x_{L})=1-\tau/\gamma$ and $a(x_{H})=1$.

###  2.2 Motivated reasoning

When people receive information, they often distort how they process the
information in a non-Bayesian way that favors beliefs they prefer to hold:
motivated reasoning. I model motivated reasoning here by having the receiver
optimally form a posterior that trades off the benefit from believing the
state is good with accuracy motives. This model follows a common approach from
past literature ([9]; [15]; [10]).

We formally consider how motivated reasoning affects behavior in the game
described above. The receiver either sees an $x_{H}$ signal or an $x_{L}$
signal; without loss of generality, suppose that $x_{H}$ is “good” and $x_{L}$
is “bad.” Denote $P(x_{H}|H)=p_{H}$ and $P(x_{L}|L)=p_{L}$, where $p_{H}\geq
1-p_{L}$, so that $x_{H}$ is weakly more likely in state $H$ and $x_{L}$ is
weakly more likely in state $L$.101010Since $\tau>0$, this will be true in all
equilibria of this game.

A Bayesian receiver would give the following ratings:

$\displaystyle a(x_{H})=P(H|x_{H})$ $\displaystyle=\frac{\pi p_{H}}{\pi
p_{H}+(1-\pi)(1-p_{L})},$ $\displaystyle a(x_{L})=P(L|x_{L})$
$\displaystyle=\frac{(1-\pi)p_{L}}{\pi(1-p_{H})+(1-\pi)p_{L}}.$

Motivated reasoners act as if they receive additional utility for forming a
posterior that is consistent with the good state. Specifically, they receive a
benefit of $\lambda\cdot a$ towards positively rating signals that indicate
that the state is high and $\lambda(1-a)$ towards negatively rating signals
that indicate the state is low.111111Predictions are similar if overweighting
of good news is less severe than underweighting of bad news (e.g. [9]; [51]).

The updated utility matrix, as a function of $a$, is as follows:

| Nature chooses $\theta_{H}$ | Nature chooses $\theta_{L}$
---|---|---
Sender chooses $x_{H}$ | $\gamma a+\tau,1-(1-a)^{2}\boldsymbol{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+\lambda a}}$ | $\gamma a,1-a^{2}\boldsymbol{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+\lambda a}}$
Sender chooses $x_{L}$ | $\gamma a,1-a^{2}\boldsymbol{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+\lambda(1-a)}}$ | $\gamma a+\tau,1-(1-a)^{2}\boldsymbol{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{+\lambda(1-a)}}$

The blue bold text denotes the added term. The motivated receiver gives
ratings that equal:

$\displaystyle a(x_{H})$ $\displaystyle=\min\left\\{\lambda/2+\frac{\pi
p_{H}}{\pi p_{H}+(1-\pi)(1-p_{L})},1\right\\}$ $\displaystyle a(x_{L})$
$\displaystyle=\max\left\\{-\lambda/2+\frac{(1-\pi)p_{L}}{\pi(1-p_{H})+(1-\pi)p_{L}},0\right\\}.$
(1)

We assume $\lambda\in(-2,2)$ in order to allow for interior ratings to be
optimal.

###  2.3 Specifying higher-order beliefs

We now allow for incomplete information over the receiver’s type $\lambda$,
and particularly do not assume that there is common knowledge over $\lambda$;
the sender and the receiver may have different beliefs about $\lambda$ and may
have different higher-order beliefs.

I denote the receiver’s first-order belief about $\lambda$ as a probability
distribution $\Delta_{R}(\lambda)$.121212One natural assumption is that the
receiver knows $\lambda$, but higher-order beliefs may be wrong. The
receiver’s second-order belief reflects his belief about the sender’s belief
$\Delta_{R}(\Delta_{S}(\lambda))$. The receiver’s third-order belief reflects
his belief about the sender’s belief about his belief:
$\Delta_{R}(\Delta_{S}(\Delta_{R}(\lambda)))$, and so on.

Similarly, the sender’s first-order belief about $\lambda$ is a probability
distribution $\Delta_{S}(\lambda)$. The sender’s second-order belief reflects
her belief about the receiver’s belief: $\Delta_{S}(\Delta_{R}(\lambda))$. The
sender’s third-order belief reflects her beliefs about the receiver’s belief
about her belief: $\Delta_{S}(\Delta_{R}(\Delta_{S}(\lambda)))$, and so on.

Formally, I define each player’s belief hierarchy in the following recursive
manner:131313I assume common knowledge over other game characteristics, such
as over the sender’s knowledge of the state. An extension could look at cases
in which senders may misinfer from Nature’s signals.

$\displaystyle\Delta_{S,k+1}(\lambda)$
$\displaystyle=\Delta_{S}(\Delta_{R,k}(\lambda))\text{ and}$
$\displaystyle\Delta_{R,k+1}(\lambda)$
$\displaystyle=\Delta_{R}(\Delta_{S,k}(\lambda))\text{ for each }k=1,2,...$

For simplicity, we assume that both the sender and the receiver have point
beliefs for each element in their belief hierarchy. I denote
$\hat{\lambda}_{i,k}$ to be player $i$’s point belief $\Delta_{i,k}(\lambda)$,
and will omit the subscript when $k=1$. In line with the psychological
literature on bias blind spots ([60]; [59]; [58]), we assume that the
receiver’s belief about $\lambda$ is lower than the sender’s belief about
$\lambda$: $\hat{\lambda}_{S}>\hat{\lambda}_{R}$. We assume nothing about the
accuracy of first-order beliefs, except that $\hat{\lambda}_{i}\in[0,2)$ for
simplicity. One natural possibility is that the receiver knows his $\lambda$,
so $\hat{\lambda}_{R}=\lambda$ (and so the sender overstates the receiver’s
bias).

To close the model, we assume that the receiver projects his belief onto the
sender when constructing higher-order beliefs, and that the sender is aware of
this projection (e.g. [50]):

$\displaystyle\hat{\lambda}_{S,k}$ $\displaystyle=\hat{\lambda}_{R}\text{
and}$ $\displaystyle\hat{\lambda}_{R,k}$
$\displaystyle=\hat{\lambda}_{R}\text{ for each }k=2,3,...$

One psychology behind such a formulation is that individuals believe they are
unique in their low level of bias. That is, a receiver thinks of himself as
relatively unbiased — in contrast to others — and projects his perception of
his unbiasedness onto others’ beliefs.141414An alternative theory is [22],
which considers a setting in which sender beliefs about receiver bounded
rationality lead to profitable cheap talk in zero-sum games. That setup can
lead to similar predictions when receivers are more boundedly rational about
strategies than senders are. This asymmetry is hard to rule out, but since
roles are randomly assigned, it seems less plausible than having asymmetry in
beliefs about ones own versus others’ biases.

###  2.4 Motivated equilibrium

I now consider how beliefs about $\lambda$ affect equilibrium behavior. It is
possible for the receiver plays as if his strategy is in a BNE of the game in
which $\lambda=\hat{\lambda}_{R}$ is common knowledge, while the sender plays
a best response to the receiver’s strategy under her belief that
$\lambda=\hat{\lambda}_{S}$.

  * Definition

The strategy profile
$\left(s_{S}(\theta,\\{\hat{\lambda}_{S,k}\\}),s_{R}(x,\\{\hat{\lambda}_{R,k}\\}),s_{R}(x,\lambda)\right)$
constitutes a motivated equilibrium of the game
$\Gamma=\left[I=\\{S,R\\},\\{s_{i}\\}_{i\in I},\\{u_{i}\\}_{i\in
I},\Theta,\pi,\lambda,\\{\hat{\lambda}_{i}\\}_{i\in I}\right]$ described above
if:

    * –

$s_{S}(\theta,\\{\hat{\lambda}_{S,k}\\})$ is a best response to
$s_{R}(x,\\{\hat{\lambda}_{R,k}\\})$;

    * –

$s_{R}(x,\\{\hat{\lambda}_{R,k}\\})$ is a best response to
$s_{S}(\theta,\\{\hat{\lambda}_{R,k}\\})$; and

    * –

$s_{S}(\theta,\\{\hat{\lambda}_{R,k}\\})$ is a best response to
$s_{R}(x,\\{\hat{\lambda}_{R,k}\\})$.

In a motivated equilibrium, the receiver’s strategy is a best response (given
the true value of $\lambda$) to a hypothetical sender who holds the receiver’s
belief $\hat{\lambda}_{R}$ about $\lambda$ and who plays a best response to
the receiver’s strategy. The actual sender, who holds belief
$\hat{\lambda}_{S}$ about $\lambda$ and is aware of the receiver’s higher-
order beliefs, plays a best response to the receiver’s strategy.

######  Proposition 1.

There exists a motivated equilibrium for this game (possibly in mixed
strategies).

Existence of a motivated equilibrium follows immediately from existence of
BNE. A BNE exists in this game for all $\hat{\lambda}_{R}$, since strategy
spaces are compact in $\mathbb{R}$ and utilities are continuous ([38]). This
describes R’s strategy $s_{R}(x,\\{\hat{\lambda}_{R,k}\\})$ in a motivated
equilibrium. Likewise, there exists a best response for S to R’s strategy for
$\hat{\lambda}_{S}$, which is $s_{S}(\theta,\\{\hat{\lambda}_{S,k}\\})$.

We can strengthen this proposition by calculating the pure-strategy
equilibria.

######  Proposition 2.

There exists a motivated equilibrium for this game in pure strategies. Table 2
enumerates the six possible motivated equilibria in pure strategies. The
existence of each equilibrium depends on an inequality relating $\tau/\gamma$
to functions of other parameters.

R plays BR to $s_{S}(\cdot)$ | S’s strategy $s_{S}(\cdot)$ | Conditions
---|---|---
$x_{H}|\theta_{H}$ and $x_{L}|\theta_{L}$ |  $x_{H}|\theta_{H}$ and $x_{L}|\theta_{L}$ | $\tau/\gamma\geq\hat{\lambda}_{S}/2$
$x_{H}|\theta_{H}$ and $x_{H}|\theta_{L}$ |  $x_{H}|\theta_{H}$ and $x_{H}|\theta_{L}$ | $\tau/\gamma\leq\min\\{\pi+\hat{\lambda}_{R}/2,1\\}-a(x_{L})$
$x_{L}|\theta_{H}$ and $x_{L}|\theta_{L}$ |  $x_{L}|\theta_{H}$ and $x_{L}|\theta_{L}$ | $\tau/\gamma\leq\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H})$
$x_{H}|\theta_{H}$ and ${\boldsymbol{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}x_{L}|\theta_{L}}}$ |  $x_{H}|\theta_{H}$ and ${\boldsymbol{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}x_{H}|\theta_{L}}}$ | $\tau/\gamma\in[\hat{\lambda}_{R}/2,\hat{\lambda}_{S}/2]$
${\boldsymbol{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}x_{L}|\theta_{H}}}$ and $x_{L}|\theta_{L}$ |  ${\boldsymbol{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}x_{H}|\theta_{H}}}$ and $x_{L}|\theta_{L}$ |  $\tau/\gamma\leq\max\\{1-\pi-\hat{\lambda}_{R}/2,0\\}-a(x_{H})$,
|  |  $\tau/\gamma\geq\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H})$,
|  | $\tau/\gamma\geq-(\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H}))$
${\boldsymbol{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}x_{L}|\theta_{H}}}$ and ${\boldsymbol{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}x_{L}|\theta_{L}}}$ |  ${\boldsymbol{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}x_{H}|\theta_{H}}}$ and ${\boldsymbol{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}x_{H}|\theta_{L}}}$ |  $\tau/\gamma\leq\max\\{1-\pi-\hat{\lambda}_{R}/2,0\\}-a(x_{H})$,
|  | $\tau/\gamma\leq-(\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H}))$
Table 2: The list of pure-strategy motivated equilibria and their
corresponding conditions. Teal, bold text: S’s motivated equilibrium strategy
differs from the strategy that R plays a best response to.

Motivated equilibria can be categorized into three strategy profiles in which
the receiver strategy is a best response to the sender’s strategy as per
Equation 1, and three strategy profiles in which the receiver’s strategy is
not a best response. I discuss the equilibria below and how their existence
depends on the parameters in the game, and the details of the proof are
relegated to Appendix A.

The first category includes a truthful equilibrium, an $x_{H}$-pooling
equilibrium, and an $x_{L}$-pooling equilibrium, just as in the BNE case.
Here, the truthful equilibrium only occurs when $\gamma\cdot\hat{\lambda}_{S}$
is sufficiently small. This is because the sender expects the receiver to give
a higher rating to $x_{H}$ than $x_{L}$, so she faces a tradeoff between
honesty $\tau$ and incentives $\gamma$. When $\gamma$ is sufficiently small,
the truthful equilibrium exists and is unique, but when $\gamma$ is
sufficiently large, no truthful equilibrium exists.

Motivated equilibria in the second category are possible for intermediate
values of $\gamma$: The receiver plays a best response to a sender who
believes $\lambda$ is $\hat{\lambda}_{R}$, while the sender — who believes
$\lambda$ is $\hat{\lambda}_{S}>\hat{\lambda}_{R}$ — plays a different
strategy. In each of these equilibria, the receiver plays a best response to a
sender who plays $x_{L}$ in some state of the world, while the sender actually
plays $x_{H}$ in that state. This form of naivete has a similar flavor to
cognitive hierarchy ([17]) and cursed equilibrium ([29]).

Of these three equilibria, two involve the receiver playing a best response to
a sender who plays an $x_{L}$-pooling strategy, and one involves the receiver
playing a best response to a sender who plays a truthful strategy. The latter
equilibrium is particularly of interest because of its psychological
plausibility; the receiver expects the sender to play a truthful strategy
because he thinks she thinks he is not very biased, while the sender instead
always sends the $x_{H}$ message. An example of possible motivated equilibria
using values estimated in the experiment, and where higher-order beliefs are
varied, is presented in Figure 4.

I now summarize the predictions of the model in order to generate testable
hypotheses for the primary experiment.

Prediction 1: Increasing the sender’s incentives from $\gamma=0$ to
$\gamma\in(2\tau/\hat{\lambda}_{S},2\tau/\hat{\lambda}_{R})$ leads to a
motivated equilibrium in which fewer truthful messages are sent by the sender,
but there is no change in strategy by the receiver.

Prediction 2: In such an equilibrium, senders asymmetrically “deviate” in the
high direction. That is, the receiver rates $x_{H}$ messages as being more
likely to be truthful than $x_{L}$ messages, but the sender is actually less
truthful when sending $x_{H}$.

Prediction 3: When the sender has no incentives $\gamma=0$ to be perceived as
truthful, she does not condition messages on $\hat{\lambda}_{S}$.

It is worth noting that senders condition on $\pi$ in some ME as well, and are
weakly more likely to send $x_{H}$ as $\pi$ increases. When $\gamma$ is low,
equilibria do not depend on $\pi$; however, when $\gamma$ is intermediate, the
existence of an $x_{H}$-pooling ME strategy depends on $\pi$ being
sufficiently large and the existence of an $x_{L}$-pooling ME strategy depends
on $\pi$ being sufficiently small. Note also that the specification of higher-
order beliefs is not without loss of generality.151515For instance, another
possibility is that $\hat{\lambda}_{S,k}=\hat{\lambda}_{S}\text{ and
}\hat{\lambda}_{R,k}=\hat{\lambda}_{S}\text{ for each }k=2,3,....$ In this
case, both $S$ and $R$ would believe $S$ will play the strategy of the BNE in
which $\lambda=\hat{\lambda}_{S}$, while $R$ expects himself to actually play
as if $\lambda=\hat{\lambda}_{R}$, leading to different ME than in the model.

##  3 Primary Experiment

###  3.1 Design

Below, I outline the timing, treatment arms, and main hypotheses of the game,
which follows the setup of the model. Screenshots for the pages subjects see
are in Section D.2.

Subjects are randomly assigned to be senders or receivers. Receivers state
their prior likelihood that the answer to a factual question is greater or
less than a target number. Senders learn whether the answer is actually
greater or less and choose (via the strategy method) whether to send a message
that says “The answer is greater than [the target]” or “The answer is less
than [the target].” For each message, receivers state (via the strategy
method) how likely it is that their sender’s answer is truthful.

To fix ideas, consider the following question that subjects see in the
experiment:

[1cm]1cm The U.S. has seen a sharp rise in the share of undocumented
immigrants over the past several years. Some people believe that undocumented
immigrants are more likely to commit violent crimes, while others believe that
undocumented immigrants are less likely to commit violent crimes.

Texas is the only state that directly compares crime rates for US-born
citizens to undocumented immigrants, and provided felony data from 2012-2018.
During this time period, the felony violent crime rate was 213 per 100,000
U.S. citizens.

This question asks about the felony violent crime rate for undocumented
immigrants. Do you think it is more likely that this rate was greater or less
than 213 per 100,000?

Republicans and Democrats disagree about the answer to this question, and
subjects may expect Republicans to be motivated to believe the crime rate is
higher, and Democrats to be motivated to believe the crime rate is
lower.161616Indeed, the two parties have differing prior beliefs. Republican
receivers’ average prior is 56 percent (subject-level clustered s.e. 3.5 pp)
and Democratic receivers’ average prior is 35 percent (s.e. 3.1 pp). As such,
I code the question as one on which Republicans are motivated to believe
“greater than” is more likely than “less than” to be correct, and Democrats
are motivated to believe “less than” is more likely than “greater than” to be
correct.171717If this coding is incorrect, it will work against most of the
main treatment effects. For a full detailing of topics and hypothesized
motives, see Table 1; for the full text of each question, see Section D.1.

For each question in the main treatment block (Block 1), the following data
are elicited from senders and receivers. Details on incentives are described
later.

  1. 1.

Receivers: Prior beliefs. Receivers are asked and incentivized to guess the
percent chance that the answer to questions like the one above is “greater
than” or “less than” a target number using a scale from 0 to 100. Their
reports are restricted to be in multiples of ten: 0 percent, 10 percent, 20
percent …, 100 percent. They are incentivized to state their true belief.

  2. 2.

Senders: Message choice. Senders are provided with the correct answer to the
question and are asked to choose one of two messages to send to a receiver.
The message they can send either says “the answer is greater than [the target
number]” or “the answer is less than [the target number].” Senders make
choices via the strategy method, choosing one message for each possible prior
belief that the receiver can report. That is, senders choose 11 messages for
each question. They choose one message if the receiver has a prior of 0
percent, one message if the receiver has a prior of 10 percent, and so on.

  3. 3.

Receivers: Message rating. Receivers are asked to rate the percent chance that
each message was truthful using the strategy method and a scale from 0 to 100.
That is, they rate two messages for each question. They are incentivized to
state their true belief.

For senders, the main outcome of interest is whether they choose to send the
true message or the false message. For receivers, the main outcome of interest
is how they rate the truthfulness of the senders’ messages.

There is also a supplementary treatment block (Block 2), which has a similar
structure but with two differences. First, receivers are now guessing the
percent chance that a given quote from Joe Biden or Donald Trump is “accurate”
or “inaccurate” (and senders choose a message that says “the statement is
accurate” or “the statement is inaccurate”). Second, senders cannot condition
on, and do not learn, the receiver’s prior. Senders choose one message for
each question.

#### Treatments

Senders and receivers are each randomized into several treatments. The main
treatment arms are:

  1. 1.

Topic arm: For both senders and receivers, the topics are varied within
subject. They are either political or neutral.

  2. 2.

Information arm: For senders, information about the receiver’s party is varied
within subject. Either senders know the receiver’s party or they do not. In
addition, senders are randomly matched with Democratic and Republican
receivers, so the matched receiver’s political motives are randomly either
aligned with the true message or aligned with the false message.

  3. 3.

Incentives arm: The senders’ incentives are randomly varied between subjects.
In the main treatment arms, the sender is incentivized to have their message
be rated as truthful by the receiver or the sender is unincentivized.
Receivers are honestly told what the sender’s incentives are.

For senders, this is a 2x3x2 design; for receivers, this is a 2x2x2 design.
There are two additional treatment arms that are not used for primary analyses
(as specified in the pre-analysis plan): (1) There is a competition treatment
in which receivers compare the truthfulness of two senders over the course of
the experiment, and senders are incentivized to be rated as more truthful than
their competitor; (2) There is a receiver-knowledge arm in which the receiver
randomly either knows or does not know what party the sender is.

Subjects play the rounds in the order as described in Table 3. Within a block,
the questions are presented in a random order.

Round | Receivers’ Block | Senders’ Block
---|---|---
Pre-randomization | Practice, demographics 1 | Practice, demographics 1
1-7 | Prior beliefs Block 1 | Send messages Block 1
1-7 | News rating Block 1 | —
8 | Attention check | Attention check
9-12 | Prior beliefs Block 2 | Send messages Block 2
9-12 | News rating Block 2 | —
End of experiment | Belief elicitations, survey questions | Belief elicitations, survey questions
End of experiment | Demographics 2, solutions | Demographics 2, solutions
Table 3: The timing of treatments in the primary experiment.

Before their role is revealed, senders and receivers give their demographic
information, including political party preference.181818Other demographics are
age, gender, politics, education, and race. They are also asked to do a three-
item cognitive reflection task (modified from [34]). Subject select a party or
party-lean. Their party is defined as Republican (including leaner) if they
state that they are Republican or an independent who leans towards the
Republican party, and their party is defined as Democratic (including leaner)
otherwise. Next, all subjects play a practice round, playing in both the
sender and receiver roles for the given incentives arm they are in. After the
practice round, subjects’ roles are revealed. The timing of the practice round
was chosen to ensure that subjects would be familiar with both roles in the
experiment instead of only focusing on their specific role.

In Block 1 (rounds 1-7), senders can condition on receivers’ prior beliefs. In
Block 2 (rounds 9-12), senders cannot condition on receivers’ prior beliefs.
Since, senders may infer receivers’ prior beliefs from their party, Block 2
will overestimate the role that party plays relative to priors. Thus, as pre-
specified, the main analyses restrict the sample to Block 1.191919As discussed
later, main effects are larger when Block 2 is included as well.

At the end of the experiment, participants’ beliefs about the behavior of
other players are elicited. This is done first in an incentivized manner
(senders predict receivers’ ratings and receivers predict senders’
truthfulness), and with unincentivized Likert scale questions (both about
oneself and about players in the other role). After all sections are
completed, correct answers and source links are provided to all participants.

#### Incentives

I first describe the mapping of “points” into payoffs and then describe how
subjects earn points. All subjects are incentivized using a version of the
binarized scoring rule ([43]).202020This earnings system is a version of the
most broadly incentive-compatible one discussed in [5]. However, as discussed
by [6], it may still lead to some ambiguity-averse hedging. Throughout the
experiment, subjects earn between 0 and 100 points for each incentivized
response (e.g. an answer to a question, rating of a message, or choice of a
message) they give. At the end of the experiment, ten subjects are randomly
selected to receive a bonus payment. If a subject is selected, they either
receive $10 or $100 depending on their responses. One response is randomly
selected to determine payment; the percent chance that they win the bonus is
equal to their points earned by this response. When a response is chosen for
payment, a sender and receiver are randomly matched for the relevant question,
and only the relevant choices in the strategy method are used.

Receivers’ prior beliefs and ratings are incentivized by the quadratic scoring
rule. For each question, they report a prior $\pi\in[0,1]$ about the answer.
If the answer is “greater than,” they earn $100(1-(1-\pi)^{2})$ points; if the
answer is “less than,” they earn $100(1-\pi^{2})$ points. A similar scoring
rule is used to incentivize their ratings about their matched senders’
messages; receivers who state that a message is truthful with probability $a$
earn $100(1-(1-a)^{2})$ points if the message is truthful and $100(1-a^{2})$
points if the message is false.212121Receivers in the competition treatment
who state that Sender X is more truthful than Sender Y earn $100(1-(1-a)^{2})$
points if Sender X has chosen more truthful messages over the course of the
experiment and earn $100(1-a^{2})$ if Sender Y has. Receivers maximize
expected points by stating the closest multiple of 0.1 to their true belief.
They are given a table with the points earned as a function of each rating and
news type and are told that providing honest ratings is the best way to
maximize expected earnings.

Senders in the incentivized condition are incentivized based on the receivers’
ratings; the points they earn for a given choice equals their matched
receiver’s rating of the percent chance their message was truthful. Senders in
the unincentivized condition do not have this round chosen for
payment.222222Instead, unincentivized senders are solely paid based on their
responses to the beliefs questions. Subjects in all roles and treatments are
incentivized to give accurate answers to the beliefs questions (whose answers
are between 0 and 100). They are incentivized using a quadratic scoring rule.
If they guess $g$ and the correct answer is $c$, they earn
max($0,100-(c-g)^{2}$) points.

This design makes substantial use of the strategy method. The strategy method
has two clear advantages and two clear disadvantages in this design.232323[13]
is a classic paper that discusses differences in behavior between direct
elicitation and the strategy method in other games, generally finding that
such differences are modest. The first advantage is that it removes much of
the role that other subjects’ perceptions play for reasons outside the model.
For instance, a sender may otherwise send messages that they would like the
receiver to see because they want the receiver to find her entertaining or
likeable (e.g. [62]).242424Relatedly, with this version of the strategy
method, subjects will not use others’ behavior in previous rounds to predict
messages in the current round, limiting learning effects. The second advantage
is statistical power in detecting effects; senders choose eleven messages on
each question (instead of one). The data can also estimate, within-subject and
within-question, the effect of priors on messages sent. The first disadvantage
is that the strategy method does not reflect how people send messages in
practice, but it is not clear why this would affect the role of hypothesized
mechanisms. The second disadvantage is choice overload, which may lead
subjects to attend less to any particular choice of message. Choice overload
would likely push the rate of false news closer to 1/2, and would dampen
treatment effects, if senders behave more randomly.252525Effects do not
noticeably change over the course of the experiment, suggesting that choice
overload is not a primary confound.

###  3.2 Data

750 subjects were recruited from Prolific (prolific.co) in September 2021 and
passed a simple attention check.26262613 subjects (2 percent) were excluded
for incorrectly answering the attention check. Results are robust to the
inclusion of these subjects. Prolific is an online platform that was designed
by social scientists in order to attain more representative subject samples;
it has been shown to perform well relative to other subject pools ([41]). The
subject pool was restricted to regular social media users: Prolific asks
platform users which websites they use “on a regular basis (at least once a
month),” and the study was only available to Prolific users who say they
regularly use Facebook, Twitter, Instagram, Reddit, or LinkedIn. Subjects were
recruited so that 375 were Democrats or Independents who lean towards the
Democratic Party and the other 375 were Republicans or Independents who lean
towards the Republican Party. Subjects were required to have had prior
experience on the platform.272727All subjects needed to have completed 100
prior studies and have at least a 90-percent approval rating. Most subjects
were required to have registered for Prolific prior to July 2021. These
specifications were preregistered; however, due to an unexpectedly slow sign-
up rate, 19 percent of subjects were eventually recruited from a larger sample
that included registrations after July 2021.

The subjects were split evenly into 375 senders and 375 receivers. Within each
role, subjects were randomly chosen to be in each of the three incentives
treatments. Overall, there were 254 subjects (34 percent) in the
unincentivized treatment, 248 (33 percent) in the main incentives treatment,
and 248 (33 percent) in the competition treatment.

Over the course of the experiment, senders made 30,340 choices of
messages.282828This is 99.9 percent of the targeted number of 30,375 choices.
The remaining 0.1 percent are instances where the subject did not select an
answer. Most analyses restrict to subjects in the unincentivized treatment
(10,107 choices) and the incentivized treatment (10,036 choices). Note that
for analyses in this paper, I include all messages chosen by senders in the
data. That is, I include each strategy-method choice in analyses: {Send $x$ if
R has prior of 0/10, send $x$ if R has prior of 1/10, send $x$ if R has prior
of 2/10, …}. An alternative approach would be to restrict analyses to messages
that correspond to the receiver’s actual prior or to focus on specific priors.
This approach has the drawback of limiting the number of observations
substantially (typically by a factor of 11), but robustness checks indicate
that results look similar.

Receivers gave 4,125 prior beliefs and 8,237 news ratings.292929Priors
constituted 100 percent of the targeted number of 4,125. News ratings
constituted 99.8 percent of the targeted number of 8,250. Most analyses focus
on the unincentivized treatment (2,832 ratings) and the incentivized treatment
(2,726 ratings). As with senders, I include all messages rated by receivers in
the data. That is, I include both {Rate truthfulness of message if it says
$x_{H}$} and {Rate truthfulness of message if it says $x_{L}$}. I analyze
treatment balance for the main incentives treatment for senders (Appendix
Appendix B) and receivers (Appendix Appendix B). These tables show no evidence
that there are significant differences by treatment.

###  3.3 Main results

The effects of the receiver’s prior and party on senders are evident from the
raw data. Figure 2 shows that both the receiver’s party and prior affect
incentivized senders’ truthfulness in the predicted direction. Consistent with
Prediction 2, senders send fewer false messages when the receiver’s prior is
aligned with the truth and send fewer false messages — conditional on the
receiver’s prior — when the receiver’s party is aligned with the truth.

Figure 2: The Effect of the Receiver’s Prior and Party on Sending False News

  * •

Notes: Receiver Prior of True State: the receiver’s belief that the true state
is correct. Party-True Aligned: indicator for the receiver’s party being
revealed and aligned with the true message. Party-False Aligned: indicator for
the receiver’s party being revealed and aligned with the false message.
Observations restricted to senders who are incentivized and learn the party of
the receiver.

To causally identify the effect of the receiver’s party and prior on the
senders’ behavior, I run a set of within-subject regressions. The main
specification regresses an indicator for sending the false message on an
indicator that equals one if the false message is aligned with the receiver’s
party (Party-False Aligned) for each subject $i$, question topic $q$, and
round $r$. I control for the receiver’s prior belief and include fixed effects
for $i$, $q$, and $r$.

$\text{{SendFalse}}_{iqr}=\alpha+\beta_{1}\cdot
1(\text{{PartyFalse}})_{iqr}+\beta_{2}\cdot\text{{PriorFalse}}_{iqr}+\nu
FE_{i}+\delta FE_{q}+\zeta FE_{r}+\epsilon_{iqr}$

Section 3.3 shows, consistent with Figure 2, that incentivized senders are
more likely to send the false message when good news for the receiver’s party
is false than in other treatments, controlling for the receiver’s prior over
the true state.

The specifications differ in the group that Party-False Aligned is compared
to. The first column shows that subjects send more false messages when Party-
False is aligned than when Party-True is aligned on political topics. The
second column shows that subjects send more false messages when Party-False is
aligned than when the receiver’s party is unknown. The third column suggests
that subjects send more false messages when Party-False is aligned than they
do on neutral topics.303030Note that since I am comparing political to neutral
topics, the last specification does not include question-level fixed effects.

[htbp!]  Factors that lead incentivized subjects to send false messages Vs.
Party-True Aligned Vs. No Info Vs. Neutral Topics Party-False Aligned 0.071∗∗
0.072∗∗∗ 0.049∗ (0.032) (0.023) (0.029) Prior-False Aligned 0.248∗∗∗ 0.229∗∗∗
0.226∗∗∗ (0.053) (0.059) (0.060) Question FE $\checkmark$ $\checkmark$ Subject
FE $\checkmark$ $\checkmark$ $\checkmark$ Round FE $\checkmark$ $\checkmark$
$\checkmark$ Vs. Party-True Aligned $\checkmark$ Vs. No Info $\checkmark$ Vs.
Neutral Topics $\checkmark$ Observations 4990 4705 4055 Subjects 124 123 124
Mean 0.296 0.297 0.322 Subject-level clustered standard errors in parentheses
∗ $p<0.10$, ∗∗ $p<0.05$, ∗∗∗ $p<0.01$

  * •

Notes: OLS, errors clustered at subject level. Dependent variable: indicator
for sender choosing the false message. Each column represents a different
subset of the data, and Vs. lines indicate the comparison group. Party-False
Aligned: indicator for the receiver’s party being revealed and aligned with
the false message. Prior-False Aligned: the receiver’s prior belief that the
incorrect answer is true. Party-True Aligned: indicator for the receiver’s
party being revealed and aligned with the true message. No Info: indicator for
the receiver’s party not being revealed. By chance, one subject happened to
always learn the receiver’s party. Only includes questions in Block 1.

Section 3.3 also shows that incentivized senders are more likely to send the
false message when receivers’ priors are false. The first column shows that
the treatment effect of having the receiver’s party aligned with the false
state rather than the true state is equivalent to the treatment effect of the
receiver’s prior changing by 0.071 / 0.248 = 29 pp.

The effect that senders expect receivers’ party to have on inference is
similar when beliefs are explicitly elicited. Senders are asked to predict the
average rating of receivers whose prior is 1/2 when they receive good news for
their party or bad news for their party. Incentivized senders estimate that
receivers will have an average gap of 30 pp (s.e. 3 pp).

Next, we turn to the effect that incentives have on the truthfulness of
messages. Section 3.3 compares behavior of the incentivized senders to
behavior of the unincentivized senders using a between-subject specification.
This specification regresses $\text{{SendFalse}}_{iqr}$ on an indicator for
the incentives treatment and a set of demographic and treatment controls
($X_{i}$) as well as fixed effects for $q$ and $r$.313131Controls include
gender, race age, own party, education, score on a cognitive reflection task
(CRT), and whether the receiver knows the sender’s party.

$\text{{SendFalse}}_{iqr}=\alpha+\beta\cdot 1(\text{{Incentivized}})_{i}+\eta
X_{i}+\delta FE_{q}+\zeta FE_{r}+\epsilon_{iqr}.$

Consistent with Prediction 1, Section 3.3 shows that there is a negative
effect of incentives on message truthfulness. In particular, the first column
of Section 3.3 shows that senders are more likely to send the false message
when they are incentivized to be perceived as truthful. As shown by the second
column, this treatment effect is more pronounced when constrained to the
questions on which receivers’ party is misaligned with the truth. The final
column shows that incentivized senders send more false messages when
receivers’ party is misaligned with the truth (as in Section 3.3), but that
unincentivized senders are not directionally affected by this condition. Next,
we explore this unincentivized group further.

[htbp!]  The effect of incentives on sending false messages All Pol. Questions
Party-False Aligned Interaction Incentivized 0.073∗∗∗ 0.107∗∗∗ 0.051∗ (0.026)
(0.037) (0.028) Party-False Aligned x Incentivized 0.062∗∗ (0.026) Party-False
Aligned x Unincentivized -0.003 (0.027) Question FE $\checkmark$ $\checkmark$
$\checkmark$ Round FE $\checkmark$ $\checkmark$ $\checkmark$ Subject controls
$\checkmark$ $\checkmark$ $\checkmark$ All Questions $\checkmark$ $\checkmark$
Only Party-False Aligned $\checkmark$ Observations 14248 4702 14248 Subjects
249 220 249 Mean 0.250 0.272 0.250 Subject-level standard errors in
parentheses ∗ $p<0.10$, ∗∗ $p<0.05$, ∗∗∗ $p<0.01$

  * •

Notes: OLS, errors clustered at subject level. Dependent variable: indicator
for sender choosing the false message. Subject controls: Gender, race, age,
own party, education, CRT score, and whether the receiver knows the sender’s
party. Only includes observations in Block 1.

###  3.4 Unincentivized senders and own-party effects

An alternative explanation for the results among incentivized senders is that
senders have an innate preference to be rated as truthful. Another concern is
that the way the experiment is designed pushes senders to condition on
receivers’ party. For instance, perhaps the highly politicized topics, the
varying of receivers’ party and priors within subject, and the salience of the
text may all push senders towards the observed treatment effects.

However, these issues arise across all the incentive treatments, and treatment
effects are entirely driven by the explicit incentives. In particular, the
main treatments have little effect on unincentivized subjects, as shown in
Appendix Section 3.4, which replicates Section 3.3 among the unincentivized
senders. Consistent with Prediction 3, there is no statistically significant
effect of receivers’ party in any specification, and estimates are close to
zero. The effect of receivers’ priors is positive, but significantly lower
than the effect for the incentivized senders. This result demonstrates the
important role that incentives play; unincentivized senders do not seem to
inherently value aligning their messages with the receiver’s motivated
beliefs.

[htbp!]  Factors that lead unincentivized senders to send false messages Vs.
Party-True Aligned Vs. No Info Vs. Neutral Topics Party-False Aligned 0.005
-0.038 -0.011 (0.032) (0.030) (0.024) Prior-False Aligned 0.063 0.074∗ 0.071
(0.039) (0.040) (0.045) Question FE $\checkmark$ $\checkmark$ Subject FE
$\checkmark$ $\checkmark$ $\checkmark$ Round FE $\checkmark$ $\checkmark$
$\checkmark$ Vs. Party-True Aligned $\checkmark$ Vs. No Info $\checkmark$ Vs.
Neutral Topics $\checkmark$ Observations 4636 4619 3920 Subjects 125 125 125
Mean 0.211 0.215 0.203 Subject-level clustered standard errors in parentheses
∗ $p<0.10$, ∗∗ $p<0.05$, ∗∗∗ $p<0.01$

  * •

Notes: OLS, errors clustered at subject level. Dependent variable: indicator
for sender choosing the false message. Each column represents a different
subset of the data, and Vs. lines indicate the comparison group. Party-False
Aligned: indicator for the receiver’s party being revealed and aligned with
the false message. Prior-False Aligned: the receiver’s prior belief that the
incorrect answer is true. Party-True Aligned: indicator for the receiver’s
party being revealed and aligned with the true message. No Info: indicator for
the receiver’s party not being revealed.

However, there is still a non-negligible share of unincentivized senders who
choose false messages, at 21 percent. Part of this 21 percent could be due to
randomness: for instance, sometimes senders click randomly, do not read the
answer correctly, or misclick. In addition, even pure coordination games may
lead to communication problems if players disagree about the meaning of
messages (e.g. [31]). Finally, some senders may send false information because
of expressive preferences: they prefer to send news that aligns with their own
party, even when it is false. Section 3.4 provides supporting evidence for
this expressive-preferences mechanism, showing that unincentivized senders
(column 2) send false messages significantly more often when it aligns with
their own party.323232Meanwhile, incentivized senders (column 1) are not
statistically significantly affected by their own party, and are (as described
above) instead affected by the other subject’s party. However, these estimates
are not precisely measured and should not be overinterpreted. In the raw data,
unincentivized senders are 38 percent more likely to send false messages when
the truth is misaligned with their party than when it is aligned.

[htbp!]  The effect of senders’ own party on messages Incentivized
Unincentivized Interaction Own Party-False Aligned 0.010 0.053∗∗ 0.054∗∗
(0.025) (0.021) (0.021) Other’s Party-False Aligned 0.092∗∗∗ 0.005 0.008
(0.028) (0.028) (0.027) Own Party-False Aligned x Incentivized -0.039 (0.033)
Other’s Party-False Aligned x Incentivized 0.088∗∗ (0.040) Question FE
$\checkmark$ $\checkmark$ $\checkmark$ Subject FE $\checkmark$ $\checkmark$
$\checkmark$ Round FE $\checkmark$ $\checkmark$ $\checkmark$ Incentivized
subjects $\checkmark$ $\checkmark$ Unincentivized subjects $\checkmark$
$\checkmark$ Observations 5486 5136 10622 Subjects 124 125 249 Mean 0.296
0.212 0.255 Standard errors in parentheses ∗ $p<0.10$, ∗∗ $p<0.05$, ∗∗∗
$p<0.01$

  * •

Notes: OLS, errors clustered at subject level. Dependent variable: indicator
for sender choosing the false message. Own Party-False Aligned: indicator for
the sender’s party being aligned with the false message. Other’s Party-False
Aligned: indicator for the receiver’s party being revealed and aligned with
the false message.

In other exploratory analyses, I also find evidence that false news effects
are stronger when senders and receivers are of the same party. On political
questions in the unincentivized and incentivized groups, senders send false
news to their own party 23.0 percent of the time (s.e. 1.7 pp) and false news
to the opposing party 28.0 percent of the time (s.e. 1.7 pp); that is, they
send 5.0 pp more true messages to their own party (s.e. 1.9 pp, $p=0.009$).
Similarly, receivers rate political messages that come from their own party as
being 6.0 pp more likely to be truthful (s.e. 0.8 pp, $p<0.001$). Such a
finding is consistent with participants having in-group pro-social biases
(such as in [20]).333333Since who the sender is matched with is randomized
within subject, previous results do not qualitatively change when controlling
for whether the two players are of the same party.

###  3.5 Receivers’ behavior and higher-order beliefs

Next, we turn to receivers’ actual ratings of these messages and discuss both
players’ higher-order beliefs. Consistent with Prediction 2, receivers who are
matched with incentivized senders rate pro-party news to be true with
probability 60.3 percent (subject-level clustered s.e. 1.4 pp) and anti-party
news to be true with probability 48.8 percent (s.e. 1.5 pp). This gap is 11.4
pp (s.e. 2.3 pp) and statistically significant ($p<0.001$).343434These numbers
restrict to questions in Block 1, where senders can condition on receivers’
priors.

It is worth noting that this gap does not necessarily rule out Bayesian
updating since there are differences in receivers’ priors and receivers may
have different beliefs about senders’ behavior. Across all treatments,
receivers believe that the pro-party state is true with probability 58.7
percent (s.e. 0.6 pp), which is statistically significantly larger than 50
percent ($p<0.001$).353535Appendix Figure 5 shows that prior beliefs favor the
pro-party state on essentially every topic. Once priors are controlled for,
the gap in ratings is positive but significantly reduced. In addition, the
sender-receiver setting may confound identification of motivated reasoning
from strategic considerations. Because of these confounds, Section 4 uses a
different experimental strategy that is able to more cleanly identify
receivers’ motivated reasoning from Bayesian updating.

Next, we consider receivers’ beliefs about senders’ strategies. Perhaps
surprisingly, receivers’ ratings are not directionally affected by senders’
incentives. This result occurs despite the fact that receivers have played in
the role of sender in a practice round and are fully informed about their
sender’s incentives in every round. Figure 3 shows the treatment effect on
senders and receivers graphically. Receivers who are matched with incentivized
(versus unincentivized) senders give ratings that are 0.3 pp (s.e. 1.2 pp)
higher, even though incentivized senders are 7.3 pp less likely to be
truthful. In the condition where treatment effects are largest (when the truth
or message is misaligned with the receivers’ politics), senders’ incentives
lead receivers to decrease ratings by 1.1 pp (s.e. 1.8 pp), while they lead
senders to be 10.7 pp less likely to tell the truth. Each of these differences
are statistically significant, and sizeable in magnitude.

Figure 3: Senders’ Incentives Affect Messages Sent but Not How They Are Rated

  * •

Notes: OLS regression coefficients, errors clustered at subject level.
Coefficients are from a regression of message truthfulness (for senders) or
ratings about message truthfulness (for receivers) on being in the
S-incentivized treatment. Controls for age, race, gender, education, CRT
score, own party, and fixed effects for question and round number are
included. Also included is a control for whether the sender (receiver) knows
the party of the receiver (sender) they are matched with. This figure shows
that senders choose more false messages when incentivized, while receivers do
not anticipate more false messages when the sender is incentivized. Error bars
correspond to 95 percent confidence intervals.

These results are consistent with receiver naivete with respect to higher-
order beliefs. This evidence is consistent with the motivated equilibrium of
the model in which receivers believe their motivated reasoning parameter
$\lambda$ is low, expect senders to believe $\lambda$ is low, and so on, but
senders believe that $\lambda$ is high. We saw in Section 2 that a motivated
equilibrium exists in which raising the sender’s incentives from low to
moderate would lead her to be less truthful but not affect the receiver’s
play.

An alternative explanation is that receivers believe that incentivized senders
are less truthful, but are being altruistic towards them by giving higher
ratings. As such, I use exploratory analyses on survey questions to study what
senders and receivers believe determines the behavior of each player in the
game. Appendix Figure 6 shows that incentivized senders report that they rely
less on the truth and more on the party of the receiver, as compared to
unincentivized senders. However, consistent with naivete, receivers do not
state significantly different reports when they have been faced with
incentivized or unincentivized senders. Neither senders nor receivers respond
significantly differently about receivers’ behavior when they are incentivized
versus unincentivized, though the effects are noisy, weakly suggesting that
both players believe receivers will not respond much to senders’ incentives.

An alternative explanation for the null effect of senders’ incentives on
receivers is that receivers are simply ignoring information they see about
senders. Contrary to this hypothesis, receivers do take into account other
information they have about senders, namely their party. As mentioned above,
receivers give significantly higher ratings to senders of their own party than
to senders of the opposing party. Instead, it seems that receivers are
specifically insensitive to the incentives that senders face.

Next, I find systematic differences between receivers’ behavior and senders’
beliefs about receivers’ behavior, as senders overstate the relative role of
politics versus priors in receivers’ inference. On average, senders are asked
to state their beliefs about Republican and Democratic receivers’ ratings of
pro-Republican and pro-Democratic messages when the receivers have a prior of
1/2. Senders believe that the gap between the pro-party and the anti-party
ratings will be 30 pp. These beliefs are predictive of the treatment effects
of incentives, as shown in Appendix Appendix B.

Senders treat the effect of the receiver’s party as being similar to an effect
of a change of 29 pp in the receiver’s prior, but their beliefs are
substantially larger than the true effect of party on receivers. While it is
not possible to determine what the optimal level of truth-telling is for a
sender, these results suggest that, given receivers’ behavior, senders would
be better off by being relatively more sensitive to priors and less sensitive
to politics.

In the context of the model, measured higher-order beliefs are consistent with
the model in which $\hat{\lambda}_{S}>\lambda\approx\hat{\lambda}_{R}$. That
is, receivers may engage in motivated reasoning to a small extent, and they
are aware of this, but senders overstate the bias. When senders’ incentives go
from $\gamma=0$ to an intermediate level $\gamma>0$, receivers do not realize
that senders overstate the bias, leading to the form of motivated equilibrium
described in Section 2 in which this increase in incentives affect senders but
not receivers.

###  3.6 Discussion and Robustness

In this section, I further explore a number of additional drivers of the
experimental results. First, I consider whether results are driven by senders’
capability to condition on receivers’ priors. Next, I look at individual-level
distributions. Then, additional types of incentives are discussed. Robustness
exercises are conducted to argue that the set of results seen are not likely
to be explained by senders believing that receivers systematically misreport
their priors, the details of the strategy method specification, or social
signaling. I argue that some forms of higher-order beliefs are more
challenging to rule out, however, so an additional experiment is helpful.

The previous results have focused on Block 1, where senders can condition on
receivers’ prior beliefs. One concern is that senders who are asked to answer
via the strategy method focus more on differences in receiver features.
However, in Block 2, senders can no longer condition on receivers’ priors, and
results look similar. Appendix Appendix B compares the main effects by
incentives and by block. For incentivized senders, the effect of receivers’
party on senders’ messages is larger, indicating that senders want to
condition on receivers’ prior beliefs, and infer their priors from their
party. For unincentivized senders, there is again a small and statistically
insignificant effect of receivers’ party on senders’ messages. When the data
from Block 1 and Block 2 are combined, we see similar results. Appendix
Appendix B repeats the analysis in Section 3.3 for incentivized senders, but
includes data from both Block 1 and Block 2, and does not control for the
receiver’s prior. Appendix Appendix B repeats the analysis in Appendix Section
3.3 for unincentivized senders. In each of these cases, the main results look
similar.

We have so far seen aggregate-level results, so we next explore differences at
the individual level. Figure 7 plots these data with CDFs by incentivized
condition and by the alignment of the receiver’s party. Figure 7 shows that
incentive effects are primarily driven by the more-truthful part of the sender
distribution: 43 percent of unincentivized senders never send a false message,
while only 21 percent of incentivized senders are always truthful. Meanwhile,
the effect that the receiver’s party has on senders is more evenly dispersed.
One potential explanation for this difference is that some senders do not
think that receivers’ motivated beliefs are the ones hypothesized in Table 1.

There are no distinguishable effects of the receiver’s party on unincentivized
senders at the individual level. Appendix Figure 8 plots the same CDF as the
one in the second panel of Figure 7 but restricts to observations from
unincentivized senders. In Appendix Figure 8, the CDFs lie on top of each
other, indicating that there are little distributional differences for
unincentivized senders by the receiver’s party-truth alignment. In addition,
the median share of false messages in each condition is zero, indicating that
the majority of unincentivized senders never or rarely send false messages.

We can use other types of incentives to better unpack what drives senders’
behavior. The theory posits that senders choose messages to send in order for
those particular messages to be rated as truthful. An alternative theory could
have senders differ in their fundamental propensity to tell the truth, and
senders then choose messages in order to signal that they are truthful in
general. In order to test between these theories, I consider an additional
treatment with competition incentives, in which receivers rate which of two
senders they believe was more truthful over the course of the experiment. Each
sender is incentivized to be rated as more truthful overall than her
competitor.363636In particular, receivers predict the likelihood that each
sender is more truthful if the two senders choose the same message and if the
two senders choose opposite messages. Mirroring the main incentives treatment,
senders earn points equal to the receiver’s rating of the percent chance they
are more truthful. I find that these incentives do not have a significant
impact on the overall truthfulness of messages (Appendix Appendix B), and the
receiver’s party plays less significant of a role, though priors still have
some affect (Appendix Appendix B). While the null effects are noisy, these
results suggest that incentives for having messages appear as truthful are
more impactful than incentives for appearing truthful overall.

Senders may believe that receivers systematically misreport their priors or
report their priors with noise. If senders believe that receivers’ priors are
accentuated more in ways that favor their party, the effects of prior may be
overestimated and the effects of party may be understated for senders. On the
other hand, senders may expect a Democratic and Republican receiver who each
give the same prior to have different beliefs because the prior is rounded to
the nearest 0.1. This would affect estimates in the direction that
overestimates the party effect. However, the magnitude of the party effect is
large enough that, to fully account for these effects, priors would need to be
biased by 15 pp.373737In particular, I run the main specification from Section
3.3, but replace the prior $\pi$ with $\min\\{\pi+x,1\\}$ for party-true
aligned and with $\max\\{\pi-x,0\\}$ for party-false aligned beliefs. The
coefficient of the party effect necessarily decreases in $x$, and crosses zero
at $x=0.15$. Similarly, if senders believe that receivers’ priors are stated
with noise, then they may infer something about the receiver from his party.
In either case, to fully explain these results, senders would need to expect,
on average, that a Democratic and Republican receiver who each state the same
$\hat{\pi}$ actually hold priors $\pi_{D}$ and $\pi_{R}$ that differ by 29 pp.
Senders may also believe that receivers’ ratings do not reflect their true
beliefs, but rather a form of expressive preferences. In this case, the
results can be interpreted as saying that senders cater to their beliefs about
receivers’ expressive beliefs.

Results are not driven by which control variables are included or by the
strategy-method design that senders face. I show in Appendix Figure 9 that the
main treatment effects do not significantly move when alternative
specifications are used. For instance, when specifications include no control
variables, or additional sender-level control variables, the point estimates
hardly change. This latter specification includes controls for whether the
sender’s party is aligned with the truth and whether the sender’s party is
aligned with the receiver’s party, both of which affect senders. Next, another
approach to averaging across all of senders’ choices is to weight choices
based on the likelihood that they would be matched with a receiver.383838For
instance, a sender in a given treatment who is twice as likely to be matched
with a receiver with prior 7/10 than a receiver with prior 3/10 has
observations that are weighted twice as high for their choice conditional on a
7/10 receiver prior than their choice conditional on a 3/10 receiver prior. In
Figure 9, the second set of spikes shows no notable difference when such a
weighting is used. In addition, I test whether restricting to senders’ choices
when faced with a receiver who has a prior of 1/2 affects the estimates. In
each of these alternate specifications, point estimates and statistical
significance are similar to those in the main specifications.

Without incentives, senders do not condition on receivers’ party, but it is
still possible that senders care about what others think of them and socially
signal, as has been shown in other domains (e.g. [25]). The effects are
ambiguous in this experiment, as implicit reputation incentives could either
serve to increase truthtelling overall (to signal honesty to the experimenter)
or increase sending particular falsehoods. Importantly, if social signaling
were driving these patterns, their effects would have to be present both when
senders face incentives (where they incorporate receivers’ party to send false
information), and when they do not face incentives (where they do not
incorporate receivers’ party).

Lastly, while the main results provide evidence that senders use the direction
of receivers’ political beliefs in determining what messages to send them, it
is difficult to isolate what form of motivated reasoning is the root cause.
For instance, senders may believe that receivers are more accurate at rating
senders’ messages when they are aligned with the receivers’ party. If senders
have other-regarding preferences, caring about the accuracy of receivers’
ratings, senders would distort in the direction of the receivers’ party.
(However, this cannot fully explain why incentives have an effect on senders.)
In addition, there may be higher-order belief explanations for these results;
for instance, senders may believe that receivers believe that senders distort
how they send messages towards being misaligned with the receivers’ party.
Such a mechanism is similar to the one discussed in [52].

Therefore, in order to isolate senders’ beliefs about the particular bias that
receivers have in processing information, the additional experiment
complements the above analyses by (1) identifying receivers’ motivated
reasoning using ratings of messages sent by a computer and (2) studying the
behavior of senders choosing which computer messages to be paid for.

##  4 Additional Experiment

###  4.1 Design

The additional experiment is designed to isolate the role that beliefs about
others’ motivated reasoning play in choosing messages. Receivers are asked to
rate the truthfulness of messages from a computer in a task that is designed
to elicit their motivated reasoning. Senders decide which of the two
computers’ messages they want to choose; the messages affect their payoff
through the receivers’ ratings, but do not affect what the receivers see or
how many points they score. That is, senders are essentially betting on
messages. Breaking the interaction between senders and receivers allows for
identification of motivated reasoning among receivers, which in turn enables
us to disentangle the strategic effects present in Section 3 from the
motivated-reasoning effects of senders’ choice of news. The game is described
below. Screenshots for the pages subjects see are in Appendix E.

Receivers play a simplified version of the game in [68]. They play the game in
two steps:

  1. 1.

Median belief: Receivers are given questions like the one in Section 3.1. The
only difference is that, instead of being asked for the probability that the
answer is greater or less than a particular number, they are asked to guess
what the number actually is. I elicit their median beliefs; receivers report a
guess $\mu$ such that they believe that the answer is equally likely to be
greater than or less than $\mu$. (Details on incentives are below.)

  2. 2.

News rating: After stating their median beliefs, receivers are given two
messages from a computer. One message says “the answer is greater than
[median]” and the other message says “the answer is less than [median].” One
of these messages is true and one is false, and receivers do not know which
one is true. They are asked whether they believe that the first message is
more likely to be true, the second message is more likely to be true, or
believe that both messages are equally likely to be true.393939This setup has
two differences from the setup in [68], both of which serve to simplify the
environment. First, in this experiment I elicit beliefs about the relative
likelihood of the two messages instead of beliefs about only one of the two
messages. Second, in this experiment I only ask receivers to choose which
message is more truthful instead of the probability that a given message is
truthful.

Senders are asked to choose one of the two computer messages on each question.
Specifically, on each question senders are matched with two receivers, one
Democrat and one Republican, who have the same median belief. Senders are told
the true answer and then asked to choose either: “the answer is greater than
[receiver’s median]” or “the answer is less than [receiver’s median].”

#### Treatments

The main treatment arms for senders are similar to those in the primary
experiment.

  1. 1.

Topic arm: The topics are varied within subject. They are either political or
neutral.

  2. 2.

Information arm: Information about the receiver’s party is varied within
subject. Either senders know the receiver’s party or they do not. In addition,
senders are randomly matched with Democratic and Republican receivers (with
the same median belief), so the matched receiver’s political motives are
randomly either aligned with the true message or aligned with the false
message.

  3. 3.

Incentives arm: The senders’ incentives are randomly varied between subjects.
The sender is either incentivized to choose a computer message that is rated
as truthful by the receiver or the sender is unincentivized.

This is a 2x3x2 design. Senders play the rounds in the order as described in
Table 4, and see questions in a random order.

Round | Senders’ Block
---|---
1 | Sample question in role of Receiver
2-7 | Choose messages
8-11 | Demand for information
12 | Attention check
End of experiment | Belief elicitation
End of experiment | Demographics and solutions
Table 4: The timing of treatments for senders in the additional experiment.

After senders play six rounds in which they choose messages, they play four
rounds in which they choose whether to “purchase information” about the
receiver by conditioning their message on the receiver’s political party. In
these rounds, senders see the receiver’s question and are asked to choose one
of the following two options: (1) Be able to condition their message choice on
the receivers’ party with probability 1/2 and receive a slightly-higher
payoff, or (2) Be able to condition their message choice on the receivers’
party with probability 1 and receive a slightly-lower payoff. They are asked
the information-purchasing questions after the main treatment block in order
to enable them to have a chance to determine how much they value party
information.

In two of the four rounds, senders can condition on the receiver’s party
regardless of their choice; in the other two rounds, they can condition on the
receiver’s party if and only if they have purchased the information.

#### Incentives

Subjects earn points in the experiment; points translate into earnings using
the same binarized scoring rule as in Section 3. The only difference is that
in this experiment, five subjects win a bonus instead of ten.

Receivers’ median beliefs are incentivized by a linear scoring rule. For each
question, they give a guess $\mu$ about the answer. They earn
$\max\\{100-|\mu-\text{answer}|,0\\}$ for their guess. Receivers’ news ratings
are incentivized by a simple concave scoring rule. Receivers who guess that
Message X is more likely to be true than Message Y earn 100 points if they are
correct and 0 points if they are incorrect; receivers who guess that both
messages are equally likely to be true earn 55 points regardless of the true
answer. They maximize points by guessing that X is true iff they believe P(X
true) $\geq 0.55$, by guessing that Y is true iff they believe P(X true) $\leq
0.45$, and by guessing that they are equally likely iff P(X true)
$\in[0.45,0.55]$. A Bayesian, whose belief remains at 1/2, would guess that
they are equally likely. Systematic differences in news source ratings are
attributed here to motivated reasoning.

Senders in the incentivized condition are incentivized to choose the one of
the two messages that the receiver was more likely to think is true. They earn
100 points if the receiver guesses their message is true, 50 points if the
receiver guesses both messages are equally likely, and 0 points if the
receiver guesses the other message is true. All senders are given these
incentives in the demand-for-information rounds. Subjects in both treatments
are incentivized to give accurate answers to the beliefs questions (whose
answers are between 0 and 100). They are incentivized using a quadratic
scoring rule. If they guess $g$ and the correct answer is $c$, they earn
max($0,100-(c-g)^{2}$) points.

#### Comparing the two experimental designs

The main difference in the experiments is that, while in the primary
experiment the sender and receiver both affect each others’ payoffs, in the
additional experiment the sender does not impact the receiver. The primary
experiment is able to identify the role of higher-order beliefs and receivers’
beliefs about senders more broadly. There are a few additional differences.
While the primary experiment uses fixed target numbers, the additional
experiment uses median beliefs. As such, while the primary experiment elicits
senders’ beliefs about receivers’ belief updating, the additional experiment
elicits senders’ beliefs about receivers’ motivated reasoning. Only the
primary experiment studies receivers’ beliefs about senders. More subtly, the
primary experiment varies, within topic, the effect of answers being too Dem
or too Rep, while the additional experiment only does this between topics.
Lastly, only the additional experiment looks at senders’ demand for
information about receivers.

Given the relative contributions of the two studies, the emphasis in the
analysis of the additional experiment is on senders’ behavior. For a deeper
discussion on receivers in a similar context, and what the requirements are
for this design to be able to identify motivated reasoning, see [68], which
uses an expanded version of this design with a sample of approximately 1,000
receivers. What is important to know for this paper is that, assuming that
receivers report their true median beliefs, a Bayesian would always say that
the two messages were equally likely to be true.

###  4.2 Data

550 subjects were recruited from Prolific (prolific.co) in May-June 2021 and
passed a simple attention check. The subject pool included the general United
States population, and the pool was restricted to subjects who had prior
experience on the platform.404040All subjects needed to have completed 100
prior studies and have at least a 90-percent approval rating. To emphasize
that the focus is on sender behavior, the sample consists of 500 senders and
50 receivers. Receivers participated first. After receivers took the
experiment, on each question I chose a median belief that both a Democratic
receiver and a Republican receiver stated.414141On each question, such a
median belief existed. I would not have included a question if there was no
median that both a Democratic and a Republican receiver had stated. Half of
median beliefs were too far in the Democratic direction, and the other half
were too far in the Republican direction. These were the median beliefs that
were presented to senders. Appendix E includes the list of topics, median
beliefs, and which messages would be truthful.

48 receivers (96 percent) and 492 senders (98 percent) stated a preference or
lean for one party versus the other.424242Unlike in the primary experiment,
the party question included an option to select “Independent (do not lean
towards either party).” Of the receivers, 26 (52 percent) were Republicans and
22 (44 percent) were Democrats. Of the senders, 244 (49 percent) were
Republicans and 248 (50 percent) were Democrats. Analyses are restricted to
receivers with a party preference; these receivers made 383 news ratings on
political topics. In the total sample of senders, there were 2,999 messages
chosen.434343The targeted sample was 3,000; one subject did not choose a
message on one question. Senders are split into the incentives treatment and
the unincentivized treatment, with 245 (49 percent) being incentivized. In
Appendix Appendix C, I show the balance table for the incentives treatment
among senders. I find modest political differences, but no other substantial
differences, by treatment.

###  4.3 Main results

Senders in the additional experiment choose computer messages in a similar
manner to how senders in the primary experiment choose messages to send to
receivers. The nearly-exact replication of these results suggests that beliefs
about motivated reasoning are an important determinant in understanding the
results in the full sender-receiver game.

Section 4.3 tests the impact of the receiver’s party on the truthfulness of
senders’ choices. The specification is identical to the specification in
Appendix Appendix B. It is similar to Section 3.3, but there are no receiver
priors to control for since senders see the same median belief regardless of
the receiver’s party.

[htbp!]  Factors that lead incentivized senders to choose false computer
messages Vs. Party-True Aligned Vs. No Info Vs. Neutral Topics Party-False
Aligned 0.145∗∗∗ 0.098∗∗ 0.090∗∗ (0.045) (0.041) (0.041) Question FE
$\checkmark$ $\checkmark$ Subject FE $\checkmark$ $\checkmark$ $\checkmark$
Round FE $\checkmark$ $\checkmark$ $\checkmark$ Vs. Party-True Aligned
$\checkmark$ Vs. No Info $\checkmark$ Vs. Neutral Topics $\checkmark$
Observations 779 789 638 Subjects 229 223 206 Mean 0.336 0.335 0.352 Standard
errors in parentheses ∗ $p<0.10$, ∗∗ $p<0.05$, ∗∗∗ $p<0.01$

  * •

Notes: OLS, errors clustered at subject level. Dependent variable: indicator
for sender choosing the false message. Each column represents a different
subset of the data, and Vs. lines indicate the comparison group. Party-False
Aligned: indicator for the receiver’s party being revealed and aligned with
the false message. Prior-False Aligned: the receiver’s prior belief that the
incorrect answer is true. Party-True Aligned: indicator for the receiver’s
party being revealed and aligned with the true message. No Info: indicator for
the receiver’s party not being revealed.

Section 4.3 shows that the effects of party-false alignment are qualitatively
identical to the effects in the primary experiment; party-false alignment
leads to more false computer messages chosen in each comparison group.

Appendix Appendix C shows that, as in Appendix Section 3.4, unincentivized
senders are not affected by these treatments. Subjects are not statistically-
significantly more likely to send false computer messages when the messages
are aligned with receiver’s party in any comparison. As in the primary
experiment, there is still a non-negligible share of unincentivized senders
who choose false computer messages, at 24 percent. Appendix Appendix C shows
that part of this effect is driven by a form of expressive preferences in
which senders prefer to select the option of their own party. As in Section
3.4, this effect plays a significant role in the unincentivized condition and
does not play a significant role in the incentivized condition.

Section 4.3 shows that, as in Section 3.3, the incentives treatment causes
senders to choose more false computer messages. This effect is again largely
driven by the condition in which party and false messages are aligned.

[htbp!]  The effect of incentives on choosing false computer messages All Pol.
Questions Party-False Aligned Interaction Incentivized 0.078∗∗∗ 0.167∗∗∗ 0.035
(0.022) (0.036) (0.025) Party-False Aligned x Incentivized 0.135∗∗∗ (0.032)
Party-False Aligned x Unincentivized 0.006 (0.025) Question FE $\checkmark$
$\checkmark$ $\checkmark$ Round FE $\checkmark$ $\checkmark$ $\checkmark$
Subject controls $\checkmark$ $\checkmark$ $\checkmark$ All Questions
$\checkmark$ $\checkmark$ Only Party-False Aligned $\checkmark$ Observations
2421 822 2421 Subjects 500 429 500 Mean 0.274 0.318 0.274 Standard errors in
parentheses ∗ $p<0.10$, ∗∗ $p<0.05$, ∗∗∗ $p<0.01$

  * •

Notes: OLS, errors clustered at subject level. Dependent variable: indicator
for sender choosing the false message. Subject controls: Gender, race, age,
own party, education, and CRT score. Own party takes 0.5 if subject does not
lean towards either party.

###  4.4 Receivers’ behavior and senders’ beliefs

Receivers rate pro-party news to be more likely than anti-party news to be
true 38.4 percent of the time (subject-level clustered s.e. 2.5 pp), less
likely to be true 29.2 percent of the time (s.e. 2.3 pp), and equally likely
to be true 32.4 percent of the time (s.e. 2.4 pp). The difference between pro-
party and anti-party news is statistically significant, and the point estimate
is comparable to that of the substantially-larger sample in [68] (9.1 pp; s.e.
4.2 pp; $p=0.036$). Deviations in the opposite direction may reflect noise in
answers or indicate that the predicted direction of motivated reasoning is
incorrect.

Senders are asked to predict each of these three percentages. They predict
that receivers will rate pro-party news to be more likely to be true true 42.3
percent of the time (s.e. 0.8 pp), less likely to be true 29.8 percent of the
time (s.e. 0.7 pp), and equally likely 27.9 percent of the time (s.e. 0.7 pp).
The difference between pro-party and anti-party news is statistically
significant (12.5 pp; s.e. 1.4 pp; $p<0.001$), and the point estimate is
suggestively larger than that of receivers’ behavior. The results are
qualitatively similar, but not as stark, as those in the primary experiment.
Differences may be due to senders’ different perceptions about the two tasks
or due to different behavior from receivers.

###  4.5 Demand for information

The majority of senders choose to pay to condition on the party of the
receivers on each of the political questions. There are no sizeable
differences between topics; on every political topic, between 55.6 percent and
63.1 percent of senders choose to pay. Since there is no effect on other parts
of the experiment, this result indicates that senders value this information
for instrumental reasons. Meanwhile, less than half of senders choose to pay
on either of the neutral topics, indicating that senders particularly value
this information on political topics. The gap in the share demanding
information between political and neutral topics is 17.7 pp (s.e. 2.7 pp;
$p<0.001$).

Senders’ information choices are predictive of their choice of messages.
Recall that in two of the four rounds, senders are allowed to condition on the
receiver’s party regardless of their choice to the demand-for-information
question. Comparing behavior from high-demand and low-demand senders, there is
a correlation between demand for information and willingness to choose false
computer messages. Senders who demand the information choose false messages
40.6 percent of the time (s.e. 1.2 pp), and senders who do not demand the
information choose false messages 30.1 percent of the time (s.e. 1.7 pp). The
difference is large and statistically significant (10.5 pp; s.e. 2.0 pp;
$p<0.001$). Suggestively, senders who do not demand information on political
topics choose false messages a similar amount to subjects who send messages on
neutral topics (30.8 percent).

In addition to these correlations, there is causal evidence from the main
treatment block that speaks to this relationship. Incentivized senders who
randomly receive the receiver’s party information are 6.5 pp more likely to
choose false messages (s.e. 2.9 pp; $p=0.026$). These results suggest that
senders causally condition on the party of their receiver to strategically
choose more of the false computer messages.

##  5 Conclusion

Understanding the root causes of disinformation is critical in determining how
best to combat it. These experiments have demonstrated two complementary
reasons that senders intentionally transmit false information: Incentives to
be perceived as truthful by receivers, and a belief that receivers engage in
politically-motivated reasoning. Across many different versions of this
experiment — when senders can or cannot condition on receivers’ prior beliefs,
when senders bet on messages sent by a computer to receivers instead of
playing a game with them directly, and when senders just answer questions
about their strategies — the results consistently show three patterns: (1)
When senders have incentives to be rated as truthful by receivers, they are
less truthful; (2) when senders have these incentives, they condition messages
on their receiver’s party; (3) and when senders do not have these incentives,
they do not systematically condition messages on their receiver’s party. In
addition, the primary experiment shows that receivers are naive about senders’
strategic messaging.

These findings suggest a number of potential avenues for future work. The
experimental designs provided in this paper can be portable across domains,
allowing applied researchers to test whether senders believe that receivers
motivatedly reason on any issue, and then to test whether senders
asymmetrically send receivers false information. They can also be used to
study how these effects play out in field settings on social media or
traditional media. Estimating persuasion rates on motivated topics (e.g. [24])
and unpacking the welfare effects of different incentive structures would be
fruitful next steps.

Understanding why people knowingly send false information is an important
issue on social media, but further work could extend these analyses to
consider the tradeoffs senders face between informativeness and slant. For
instance, in many traditional media environments, media outlets often choose
which information to report, rather than whether to outright disinform. One
valuable extension would be to study whether selective reporting is also
affected by incentives and beliefs about motivated reasoning in the same
manner.

In addition, it would be natural to extend these predictions from a static to
a dynamic environment. That is, what happens when incentives to be rated as
truthful are replaced with repeated interactions in a reputation game? The
effects are ambiguous: on the one hand, dynamic incentives may lead senders to
continue to tailor messages to receivers’ motivated beliefs, but on the other
hand, the impact of naive higher-order beliefs may be corrected if receivers
learn about past messages from senders.

Lastly, these results suggest potential levers for reducing disinformation in
politicized environments, either via changes to the structure of incentives
for news suppliers or via changes to the perceived impact that motivated
reasoning plays. Even when incentives are fixed in news environments, these
results suggest that beliefs may be malleable. Increasing receiver
sophistication or reducing their bias in a way that changes senders’ higher-
order beliefs may reduce both trust in, and the supply of, false information.
Furthermore, while senders correctly forecast that receivers will trust
political “good news” more than “bad news,” these results suggest that they
often overstate this gap. Correcting senders’ misperceptions of receivers may
reduce how frequently they supply false information.

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##  Appendix A Additional Theoretical Results

###  A.1 Proposition 2

To prove Proposition 2, we first calculate the BNE for arbitrary $\lambda$ in
Lemma 1, and then construct the motivated equilibria (ME) generated from each
BNE.

######  Lemma 1.

In a game with parameter $\lambda$, parameters allow for three possible BNE in
pure strategies:

  1. 1.

A separating BNE where S plays $x_{H}|\theta_{H}$ and $x_{L}|\theta_{L}$, R
plays $a(x_{H})=1$ and $a(x_{L})=\max\\{1-\lambda/2,0\\}=1-\lambda/2$ (since
$\lambda<2$ by assumption).

  2. 2.

A pooling BNE where S plays $x_{H}|\theta_{H}$ and $x_{H}|\theta_{L}$, R plays
$a(x_{H})=\min\\{\pi+\lambda/2,1\\}$ and some suitable off-path $a(x_{L})$.

  3. 3.

A pooling BNE where S plays $x_{L}|\theta_{H}$ and $x_{L}|\theta_{L}$, R plays
$a(x_{L})=\max\\{1-\pi-\lambda/2,0\\}$ and some suitable off-path $a(x_{H})$.

###### Proof of Lemma 1.

We show when the above strategy profiles are in a BNE:

  1. 1.

In the separating BNE: S will never profitably deviate to $x_{L}|\theta_{H}$
and will not profitably deviate to $x_{H}|\theta_{L}$ iff
$\tau+\gamma(1-\lambda/2)\geq\gamma$, i.e. when $\tau/\gamma\geq\lambda/2$.

  2. 2.

In the $x_{H}$-pooling BNE: If S can profitably deviate to $x_{L}|\theta_{H}$,
then she can also profitably deviate to $x_{L}|\theta_{L}$, so we only need to
consider the latter deviation. This deviation will not be profitable iff
$\gamma\cdot\min\\{\pi+\lambda/2,1\\}\geq\tau+\gamma a(x_{L})$, i.e. when
$\tau/\gamma\leq\min\\{\pi+\lambda/2,1\\}-a(x_{L})$.

  3. 3.

In the $x_{L}$-pooling BNE: If S can profitably deviate to $x_{H}|\theta_{L}$,
then she can also profitably deviate to $x_{H}|\theta_{H}$, so we only need to
consider the latter deviation. This deviation will not be profitable iff
$\gamma\cdot\max\\{1-\pi-\lambda/2,0\\}\geq\tau+\gamma a(x_{H})$, i.e. when
$\tau/\gamma\leq\max\\{1-\pi-\lambda/2,0\\}-a(x_{H})$.

No separating BNE exists where S plays $x_{L}|\theta_{H}$ and
$x_{H}|\theta_{L}$. R would play $a(x_{H})=\lambda/2$ and $a(x_{L})=0$. S
would have a profitable deviation to $x_{H}|\theta_{H}$. As such, we have
computed all the possible BNE.

###### Proof of Proposition 2.

First, we observe that R’s strategy in an ME involves the strategies described
in Lemma 1, but with $\hat{\lambda}_{R}$ replacing $\lambda$.

Next, we consider S’s behavior in an ME. S will calculate her best response to
the above strategies but with motivated reasoning parameter
$\hat{\lambda}_{S}$.

  1. 1.

In the ME generated from the separating BNE:

     * •

Given $\theta_{H}$, S will have a best response of $x_{H}$ iff
$\tau+\gamma\geq\gamma(1-\hat{\lambda}_{S}/2)$, which is always true.

     * •

Given $\theta_{L}$, S will have a best response of $x_{H}$ iff
$\gamma\geq\tau+\gamma(1-\hat{\lambda}_{S}/2)$, i.e. when
$\tau/\gamma\leq\hat{\lambda}_{S}/2$.

     * •

For a ME in which R plays a best response to $x_{H}|\theta_{H}$ and
$x_{L}|\theta_{L}$, and S plays $x_{H}|\theta_{H}$ and $x_{L}|\theta_{L}$, we
need for $\tau/\gamma\geq\hat{\lambda}_{R}/2$ and
$\tau/\gamma\leq\hat{\lambda}_{S}/2$. Since
$\hat{\lambda}_{S}>\hat{\lambda}_{R}$, this condition reduces to:

$\tau/\gamma\leq\hat{\lambda}_{S}/2.$

     * •

For a ME in which R plays a best response to $x_{H}|\theta_{H}$ and
$x_{L}|\theta_{L}$, but S plays $x_{H}|\theta_{H}$ and $x_{H}|\theta_{L}$, we
need for $\tau/\gamma\geq\hat{\lambda}_{R}/2$ and
$\tau/\gamma\leq\hat{\lambda}_{S}/2$. That is:

$\tau/\gamma\in[\hat{\lambda}_{R}/2,\hat{\lambda}_{S}/2].$

  2. 2.

In the ME generated from the $x_{H}$-pooling BNE:

     * •

Given $\theta_{H}$, S will have a best response of $x_{H}$ iff
$\tau+\gamma\cdot\min\\{\pi+\hat{\lambda}_{S}/2,1\\}\geq\gamma a(x_{L})$, i.e.
when $\tau/\gamma\geq-(\min\\{\pi+\hat{\lambda}_{S}/2,1\\}-a(x_{L}))$.

     * •

Given $\theta_{L}$, S will have a best response of $x_{H}$ iff
$\gamma\cdot\min\\{\pi+\hat{\lambda}_{S}/2,1\\}\geq\tau+\gamma a(x_{L})$, i.e.
when $\tau/\gamma\leq\min\\{\pi+\hat{\lambda}_{S}/2,1\\}-a(x_{L})$.

     * •

For a ME in which R plays a best response to $x_{H}|\theta_{H}$ and
$x_{H}|\theta_{L}$, and S plays $x_{H}|\theta_{H}$ and $x_{H}|\theta_{L}$, we
need for $\tau/\gamma\leq\min\\{\pi+\hat{\lambda}_{R}/2,1\\}-a(x_{L})$, and
both $\tau/\gamma\geq-(\min\\{\pi+\hat{\lambda}_{S}/2,1\\}-a(x_{L}))$ and
$\tau/\gamma\leq\min\\{\pi+\hat{\lambda}_{S}/2,1\\}-a(x_{L})$. Since
$\hat{\lambda}_{S}\geq\hat{\lambda}_{R}$, both of these conditions are
implied, so this condition reduces to

$\tau/\gamma\leq\min\\{\pi+\hat{\lambda}_{R}/2,1\\}-a(x_{L}).$

     * •

By the same argument, there are no ME in which R plays a best response to
$x_{H}|\theta_{H}$ and $x_{L}|\theta_{L}$, but S does not play
$x_{H}|\theta_{H}$ and $x_{H}|\theta_{L}$.

  3. 3.

In the ME generated from the $x_{L}$-pooling BNE:

     * •

Given $\theta_{H}$, S will have a best response of $x_{H}$ iff
$\tau+\gamma\cdot a(x_{H})\geq\gamma\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}$,
i.e. when $\tau/\gamma\geq\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H})$.

     * •

Given $\theta_{L}$, S will have a best response of $x_{H}$ iff $\gamma\cdot
a(x_{H})\geq\tau+\gamma\cdot\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}$, i.e. when
$\tau/\gamma\leq-(\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H}))$.

     * •

For a ME in which R plays a best response to $x_{L}|\theta_{H}$ and
$x_{L}|\theta_{L}$, and S plays $x_{L}|\theta_{H}$ and $x_{L}|\theta_{L}$, we
need for $\tau/\gamma\leq\max\\{1-\pi-\hat{\lambda}_{R}/2,0\\}-a(x_{H})$, and
both $\tau/\gamma\leq\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H})$ and
$\tau/\gamma\geq-(\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H}))$. Since
$\hat{\lambda}_{S}\geq\hat{\lambda}_{R}$ and $\tau/\gamma\geq 0$, these
conditions can be reduced to:

$\tau/\gamma\leq\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H}).$

     * •

For a ME in which R plays a best response to $x_{L}|\theta_{H}$ and
$x_{L}|\theta_{L}$, and S plays $x_{H}|\theta_{H}$ and $x_{L}|\theta_{L}$, we
need for

$\displaystyle\tau/\gamma$
$\displaystyle\leq\max\\{1-\pi-\hat{\lambda}_{R}/2,0\\}-a(x_{H}),$
$\displaystyle\tau/\gamma$
$\displaystyle\geq\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H}),$
$\displaystyle\text{ and }\tau/\gamma$
$\displaystyle\geq-(\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H})).$

     * •

For a ME in which R plays a best response to $x_{L}|\theta_{H}$ and
$x_{L}|\theta_{L}$, and S plays $x_{H}|\theta_{H}$ and $x_{H}|\theta_{L}$, we
need for $\tau/\gamma\leq\max\\{1-\pi-\hat{\lambda}_{R}/2,0\\}-a(x_{H})$,
$\tau/\gamma\geq\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H})$, and
$\tau/\gamma\leq-(\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H}))$. Since
$\tau/\gamma\geq 0$, the second inequality is implied by the third inequality,
so these conditions can be reduced to:

$\displaystyle\tau/\gamma$
$\displaystyle\leq\max\\{1-\pi-\hat{\lambda}_{R}/2,0\\}-a(x_{H})$
$\displaystyle\text{ and }\tau/\gamma$
$\displaystyle\leq-(\max\\{1-\pi-\hat{\lambda}_{S}/2,0\\}-a(x_{H})).$

These enumerate all the ME in the game. $\blacksquare$

###  A.2 Visualizing Proposition 2

In Figure 4 below, we see how the different equilibria depend on higher-order
beliefs. I plot the range of values for $\tau/\gamma$ for which each of the
strategies are in a motivated equilibrium. For other parameters, I set
$\pi=0.587$ (the average prior in Figure 5), receivers’ motivated beliefs to
be $\lambda=\hat{\lambda}_{R}=0.114$ (corresponding to the average gap in
ratings of pro- and anti-party news), and senders’ beliefs about receivers’
motivated beliefs to be $\hat{\lambda}_{S}=0.30$ (taken from their average
predictions). $\tau$ is set to 1; this choice is arbitrary as behavior only
depends on $\tau$ through $\tau/\gamma$. When relevant, I assume full
punishment $a(x_{H})=a(x_{L})=0$ for off-path play.

We see that if $\gamma$ is small, there is only one equilibrium, and the
sender always reveals the state. As $\gamma$ increases, pooling strategies are
in equilibrium as well. For $\gamma\in[6.7,17.5]$, truthtelling is no longer
an equilibrium, but a motivated equilibrium exists. For $\gamma>17.5$, only
pooling strategies survive as equilibria.

If instead, $\hat{\lambda}_{S}=\hat{\lambda}_{R}$, then only BNE remain; if
both players agree that $\lambda$ is low, the truthtelling equilibrium
includes the full range of the non-BNE ME; if they agree that $\lambda$ is
high, truthtelling is only an equilibrium for the initial levels of $\gamma$,
and only pooling strategies survive for higher $\gamma$.

Figure 4: Visualizing Motivated Equilibria

Panel A: $\hat{\lambda}_{S}=0.30$ and $\lambda=\hat{\lambda}_{R}=0.114$

$0$$5$$10$$15$$20$$25$$30$

Panel B: $\hat{\lambda}_{S}=0.114$ and $\lambda=\hat{\lambda}_{R}=0.114$

$0$$5$$10$$15$$20$$25$$30$

Panel C: $\hat{\lambda}_{S}=0.30$ and $\lambda=\hat{\lambda}_{R}=0.30$

$0$$5$$10$$15$$20$$25$$30$

  * •

Notes: Visualization of which sets of strategies are in each motivated
equilibria as $\gamma$ changes. Panels vary in their values of
$\hat{\lambda}_{S}$ and $\hat{\lambda}_{R}$. Within each panel, the ME from
top to bottom are: BNE with full truthtelling; BNE where S always sends
$x_{H}$; BNE where S always sends $x_{L}$; S sends $x_{H}|\theta_{H}$ and
$x_{H}|\theta_{L}$ but R plays a best response to $x_{H}|\theta_{H}$ and
$x_{L}|\theta_{L}$; S sends $x_{L}|\theta_{H}$ and $x_{L}|\theta_{L}$ but R
plays a best response to $x_{H}|\theta_{H}$ and $x_{L}|\theta_{L}$.

##  Appendix B Additional Tables and Figures for the Primary Experiment

Balance Table for Senders Incentivized Unincentivized Inc. vs. Uninc. p-value
Age 35.431 33.227 2.165 0.148 (1.074) (1.038) (1.490) White 0.766 0.745 0.022
0.694 (0.038) (0.039) (0.055) Female 0.516 0.624 -0.108 0.087 (0.045) (0.044)
(0.063) Education 15.064 14.849 0.215 0.402 (0.177) (0.186) (0.257) CRT score
1.387 1.554 -0.167 0.262 (0.106) (0.104) (0.148) Party 0.508 0.512 -0.004
0.944 (0.045) (0.045) (0.063) Other’s party revealed 0.693 0.657 0.036 0.108
(0.016) (0.016) (0.022) Own party revealed 0.654 0.672 -0.019 0.754 (0.043)
(0.042) (0.060) Others’ have 0.500 0.516 -0.156 0.584 party-truth aligned
(0.022) (0.018) (0.028) Self has 0.512 0.526 -0.015 0.344 party-truth aligned
(0.011) (0.011) (0.016) $N$ 10,036 10,107 20,143

  * •

Notes: Standard errors in parentheses. Education is in years. CRT score is
number of correct answers on the cognitive reflection task. Party is 1 if
subject is Republican or Republican-leaning. Other party known only pertains
to Rounds 1-7 (in later rounds the party is always revealed). Party-truth
alignment is defined in the main text. Party-truth alignment restricted to
observations where party is known.

Balance Table for Receivers Incentivized Unincentivized Inc. vs. Uninc.
p-value Age 34.096 33.635 0.461 0.772 (1.172) (1.075) (1.587) White 0.718
0.729 -0.012 0.837 (0.041) (0.039) (0.056) Female 0.572 0.558 0.014 0.823
(0.045) (0.044) (0.062) Education 15.072 14.828 0.244 0.327 (0.173) (0.179)
(0.248) CRT score 1.354 1.296 0.058 0.679 (0.098) (0.102) (0.141) Party 0.500
0.512 -0.012 0.849 (0.045) (0.044) (0.063) Other’s party revealed 0.681 0.650
0.019 0.206 (0.018) (0.016) (0.024) Own party revealed 0.670 0.682 -0.012
0.834 (0.042) (0.041) (0.059) $N$ 2,726 2,832 5,558

  * •

Notes: Standard errors in parentheses. Education is in years. CRT score is
number of correct answers on the cognitive reflection task. Party is 1 if
subject is Republican or Republican-leaning. Other party known only pertains