#  Information Greenhouse:
Optimal Persuasion for Medical Test-Avoiders

Zhuo Chen Center of Economic Research, Shandong University. Email:
<EMAIL_ADDRESS>
###### Abstract

Patients often delay or reject medical tests due to information avoidance,
which hinders timely reception of necessary treatments. This paper studies the
optimal information policy to persuade an information-avoidant patient to
undergo the test and make the best choice that maximizes his health. The
patient sequentially decides whether to take the test and the optimal
treatment plan. The information provided is about the background knowledge of
the disease, which is complementary with the test result, and disclosure can
take place both before and after the test decision. The optimal information
policy depends on whether the patient is willing to be tested when he is
completely pessimistic. If so, the optimal policy features preemptive warning:
the disclosure only takes place before the test, and the bad news guarantees
the patient to be tested and be treated even without further information. If
not, the optimal policy constructs an information greenhouse: an information
structure that provides high anticipatory utility is committed when the
patient is tested and the test result is bad. I consider extensions to general
information preference and ex ante participation constraint.

JEL classification: C72, D82, D91.

Keywords: Dynamic information design; information incentives; information
avoidance; anticipatory utility; doctor-patient relationship.

“To cure sometimes, to relieve often, to comfort always.”
— On the statue of Dr. Edward Livingston Trudeau

###  1 Introduction

Evidence from medicine, psychology, and economics shows that information
avoidance, the tendency that people deliberately delay or refuse to acquire
(free but) potentially useful information,111Information avoidance is also
called information aversion by other papers. In a decision-theoretic
perspective, it indicates a preference for late resolution of uncertainty. I
select information avoidance as the name because in some other studies
information aversion is used to represent a preference for one-shot resolution
of uncertainty, e.g. Andries and Haddad (2020). is a significant reason
preventing people to participate in necessary medical tests on time (See
Golman et al. (2017), Sweeny et al. (2010) and Hertwig and Engel (2016) for
reviews of empirical evidence). For instance, while the early detection of
breast cancer is widely recognized to have a deterministic impact on a
patient’s five-year survival rate, evidence still suggests that information
avoidance leads many women to delay or resist routine mammography (Caplan,
1995, Meechan et al., 2002, Melnyk and Shepperd, 2012, Zanella and Banerjee,
2016, for example). Undoubtedly, this non-standard information preference
results in significant health and economic losses.

This paper studies how to motivate a medical test-avoider to accept a medical
test through information disclosure, thereby maximizing his survival rate. For
two reasons I focus on the information incentive rather than other material
incentives or behavioral nudges. First, as a form of incentive, communication
per se is widely existing and frequently used. Especially, for patients,
persuasion is the most common way in reality utilized to change their
decisions. Even without deliberate design, the (suboptimal) disclosure rules
in the real word are still motivating the patients’ decisions every day, and
therefore it deserves a primary attention. Second, information avoidance is
commonly attributed to the motivation to manage anticipatory emotions
(Loewenstein, 1987, Caplin and Leahy, 2001), i.e., the emotional experiences
generated by the belief about the future outcomes, such as hope, anxiety and
suspense, and people avoid information because they anticipate that the
information will cause a painful emotional experience. Based on this concern,
the concept of information incentive takes on a more nuanced and profound
meaning: what information can change is not only the patient’s objective
belief, but also the direct emotional feeling attached to the belief. Hence,
information disclosure in this context will result in several new patterns.

This paper provides a stylized model that describes a typical doctor-patient
interaction. The patient observes some preliminary signs of a disease, and
seeks help from a doctor. The doctor then requires the patient to undergo a
medical test, upon which the patient chooses a plan of medical treatment.
However, the patient is reluctant to do so due to information avoidance, and
therefore the doctor, seeking to maximize the patient’s probability of
survival from the disease, needs to provide some additional incentives for the
patient.

I adopt the framework of Bayesian persuasion and information design (Kamenica
and Gentzkow, 2011, Bergemann and Morris, 2019), which helps me to identify
the theoretically optimal strategy for information disclosure. To persuade the
patient to make the desired choices (i.e., accepts the test and choose the
optimal treatment), the doctor provides information about the background
knowledge about the disease, which is independent of, but complementary with,
the test result. That is, the doctor’s information disclosure policy and the
patient’s voluntary information acquisition behavior collectively determine
the patient’s information environment, and the doctor influences the latter by
controlling the former. Real-world examples of such information include the
availability of various treatment options, their success rates, economic
costs, physical discomfort and the potential risks of not undergoing the
treatment. The doctor has two time windows to provide information: the ex ante
disclosure before the test decision, and the interim disclosure after the test
decision but before the treatment decision. Therefore, the optimization
problem is categorized into a two-period dynamic information design program.

The patient’s preference is depicted by the psychological expected utility
model by Caplin and Leahy (2001), which is able to describe information-
avoidance preference while preserving the minimal linearity required by
information design.

####  1.1 Key Tradeoffs and Main Findings

The main contribution of this paper is the characterization of the optimal
information policy. I begin with the restricted case that the ex ante
disclosure is unavailable, and characterizes the optimal interim disclosure.
The result is jointly determined by two key tradeoffs. First, since the
patient is forward-looking, the interim disclosure influence both the
incentives before and after it. Hence, as articulated by Ball (2023), there is
a tradeoff between guiding the future and rewarding the past. At this point,
information disclosure can guide the future by influencing the patient’s
subsequent treatment decision. However, a good signal for guiding the future
may simultaneously be expected to be hurting and frighten the patient away
from accepting the test. Second, when the doctor rewards the past, she also
faces the tradeoff between the instrumental usefulness and the psychological
pain brought by the information. This tradeoff is common in the applied
literature of information avoidance (Kőszegi, 2003, 2006a, 2006b, Barigozzi
and Levaggi, 2010, Schweizer and Szech, 2018).

Taking the tradeoffs into account, the optimal interim disclosure can be
solved by maximizing the doctor’s continuation payoff after the test under the
constraint of voluntary participation. I first recognize that if the patient
refuses the test, it is full disclosure that is optimal for the doctor, which,
termed as punitive revelation, stems from information avoidance. In the
standard model of dynamic information design, when the receiver is detected to
choose against the sender’s recommendation, the sender conceals all the future
information as a punishment, which follows because information is always
valuable. However, in my model, the instrumental value of information
disappears after the patient refuses the test, and his utility is thus
minimized by full disclosure.

If the patient took the test, obviously, the unconstrained optimum survives
when it can ensure voluntary participation, and if not, the participation
constraint must be binding. In this case, I show in Proposition 1 that the
optimal interim disclosure adheres to a simple principle, which is termed as
the “best-good-news” rule, such that the optimal signal must maximize the most
optimistic belief among all the signals capable to ensure voluntary
participation. Thus, the optimal signal can be solved by combining the
participation constraint, the revelation principle and the best-good-news
rule.

As a comparison, I also consider an alternative setting that there is no
information avoidance but taking the test incurs a fixed physical cost, which
is termed as the physical cost setting henceforth.222This setting is closely
similar to the that of Ely and Szydlowski (2020). The result is provided in
Appendix C. Compared to the physical cost setting, the key feature of the
optimal interim disclosure here is that the doctor can provide the patient
with a higher payoff than full disclosure, which comforts the patient when the
test result is bad. Since full disclosure maximizes the instrumental value,
this additional piece of carrot is made by anticipatory utility.

Also, in Proposition 2, I show that the solution of the optimal disclosure can
be extended to the case that the patient’s information preference is
arbitrary. The only difference is that the number of posterior beliefs may be
more than the number of states, which has been observed by a series of papers
on constrained Bayesian persuasion (Le Treust and Tomala, 2019, Boleslavsky
and Kim, 2020, Salamanca, 2021, Doval and Skreta, 2024). In my model, the
additional posterior belief comes from the possibility that the patient is
intrinsic information loving, and doctor thus provides an additional piece of
information to reward the past.

Then I bring the ex ante disclosure back to consider the global optimal
information policy, which can be solved by backward induction based on the
optimal interim disclosure. The following two strategies are the candidates of
the optimal policy, which reflect two different philosophies of persuasion:

  * -

Preemptive warning: The doctor discloses information only before the test, and
if the patient is tested and the result is bad, the bad news guarantees the
patient to accept the treatment even without any further information.

  * -

Committed comfort: The doctor adjust the patient’s belief to a zone that he
feels comfortable to be tested, and commits to provide comfort if the test
result is bad.

Since the test and the treatment are complementary to the patient’s health,
the choice between the two strategies reflects a tradeoff between the
patient’s willingness to be tested and his willingness to treated. On the one
hand, by committed comfort, the doctor can indeed reward voluntary
participation by information incentives, but because of the information, the
patient may become inappropriately optimistic about the prospect when he is
diagnosed to be sick, and then forgo the treatment opportunity. In other
words, committed comfort improves the willingness to be tested at the cost of
the willingness to be treated, On the other hand, preemptive warning is exact
the opposite, which overcomes the the efficiency loss by committed comfort
since the patient, as long as tested, will be treated whenever necessary.
Simultaneously, however, the probability the patient being tested is lowered.

I show in Proposition 3 that the optimal information policy is preemptive
warning if and only if the patient reacts to fear; that is, he is willing to
be tested when he is extremely pessimistic about his health prospect when the
test result is bad and he forgoes the treatment. If not, the doctor is unable
to persuade the patient when he is approximately “standard”. However, when the
patient is sufficiently “psychological”, committed comfort is optimal. In
fact, in the physical cost setting, preemptive warning is also optimal when
the patient reacts to fear, but when he does not, the doctor cannot persuade
the patient at all. Therefore, committed comfort is new in the presence of
anticipatory feelings and information avoidance.

So far, the results rely on the assumption that the reception of the doctor’s
information is mandatory, and finally I extend the model to the case that the
patient can choose not to hear anything from the doctor. Thus, the solution is
constructed upon both an ex ante and an interim participation constraint, and
the the ex ante participation constraint makes policies such like preemptive
warning undesirable. I fully characterize the optimal policy in Proposition 4.
Similar with preemptive warning, I show that by the optimal policy, the doctor
complete information disclosure before the test. However, in this case the
doctor has to solve the tradeoff between guiding the future and rewarding the
past within the ex ante disclosure. Consequently, the optimal signal is
completely analogous to the optimal interim disclosure derived in Proposition
1, which can be characterized by the best-good-news rule.

In summary, there are two scenarios that the doctor benefits from constructing
an information greenhouse, which warms the patient up by informational comfort
when a bad news is received. One is the case that the ex ante participation
constraint is absent and the patient does not react to fear, and the other is
that the ex ante participation constraint is present.

####  1.2 Literature

The concept of anticipatory emotions is introduced to the field of economics
by Akerlof and Dickens (1982) and Loewenstein (1987), and is extended to the
environments with risk and uncertainty by Wu (1999), Caplin and Leahy (2001)
and Epstein (2008). Their works link the of anticipatory emotions, especially
anxiety and hope, with the decision-theoretic models on temporal resolution of
uncertainty (Kreps and Porteus, 1978, for example). Using the decision models,
several papers study the choice of whether to accept an informative test,
where the basic tradeoff is between the instrumental value of and the pain
from information (Caplin and Eliaz, 2003, Kőszegi, 2003, 2006a, Barigozzi and
Levaggi, 2010, Fels, 2015).

Early game-theoretic works concentrate on the impact of anticipatory emotions
and information avoidance in different settings of information communication
other than Bayesian persuasion, including verifiable disclosure (Caplin and
Leahy, 2004) and cheap talk (Kőszegi, 2006a). The most important insight there
is that even if the preferences of the sender and receiver are perfectly
aligned, the “good intention” of the sender to protect the receiver’s
emotional feeling will destroy the credibility of the information she sends,
thereby reducing the receiver’s payoff. The current paper adopts a more
normative perspective by studying the Bayesian persuasion framework, which
helps to better identify the theoretically optimal method of information
communication without having to consider the implications posed by the
requirement of sequential rationality in the aforementioned models.

The study of Bayesian persuasion and information design begins with the path-
breaking work by Kamenica and Gentzkow (2011). Among this literature, there
has been two persuasion papers that explicitly assume that receiver to have
anticipatory feelings and information avoidance. Schweizer and Szech (2018)
study the problem of optimal test design in a doctor-patient interaction. The
optimal information policy maximizes the patient’s utility, and since there is
no conflict of interest, the result can be simply interpreted as the patient’s
optimal choice among information structures. My model shares the spirit that
the sender improve the receiver’s anticipatory feeling by tailoring the
information structure. However, in my model there is indeed a principal-agent
problem and the reception of test result is voluntary. Thus, the tradeoff
between guiding the future and rewarding the past in the current paper is
absent in theirs. Also, the model by Lipnowski and Mathevet (2018) allows the
receiver to have a non-standard information attitude, and there is still no
conflict of interest. They realize that the revelation principle may fail for
such a psychological receiver, but show that it will still hold when he is
information-avoidant.

There are two distinct but deeply connected branches of Bayesian persuasion
papers that identifies the tradeoff between guiding the future and rewarding
the past. One category is Bayesian persuasion under constraints, and the other
is dynamic information design when the receiver is forward-looking. In the
first category, the paper most closely related to the mine is Rosar (2017),
who also explicitly studies voluntary participation of a test. In her model,
after the test it is the doctor that makes a decision, and therefore the
reason that the receiver is reluctant to be tested are extrinsic. Also, the
model is similar with Boleslavsky and Kim (2020), who consider a moral hazard
problem with three players. Between the agent’s action choice and the contract
execution, a sender can transform the agent’s action into messages for the
principal based on the committed rule. In my paper, if we regard the two
selves of the patient making the two choices as two distinct players, there is
no moral hazard problem between them.

In the second category, my paper is especially related to Ely and Szydlowski
(2020), Habibi (2020) and D’Antoni et al. (2024), all of which characterize
the optimal information policy in the two period setting. Their papers
explicitly point out that when the sender has the power to make a credible
commitment about future information policies, information can be used both as
a carrot to reward the past effort and as a guide to the future decisions. In
both my model and theirs, the principal can extract the rent by the interim
disclosure and the ex ante disclosure can be derived by backward induction.
However, there is a subtle difference that their optimal policies, due to the
linearity in preference, satisfies the ex ante participation constraint
automatically, but in my model since information could make the patient worse
off than the no-information benchmark, the optimal policy violates the ex ante
constraint.

The remaining part of this paper is organized as follows. Section 2 provides
the formal description of the model. Section 3 solves the optimal interim
disclosure. Section 4 provides the main characterization results. Section 5
concludes. In the appendices, Appendix A discusses and formally proves the
best-good-news rule. Appendix B provides the proofs. Appendix C solves the
physical cost model for comparison. Appendix D solves the optimal disclosure
when the information is about the test result.

###  2 The Model

This section outlines the formal model.

####  2.1 Economic Environment

Consider a patient (“he”) who suspects he might have a life-threatening
disease and seeks medical advice from a doctor (“she”). The medical
consultation process alters the patient’s initial health status, categorized
as either sick ($\theta=0$) or not sick ($\theta=1$), to a final health
status, which is either healthy or unhealthy. The term “health probability”
will denote the probability that the patient’s final health status is healthy.

The health function. When $\theta=1$, the health probability equals $1$. When
$\theta=0$, following the linear production model by Boleslavsky and Kim
(2020), the health probability is calculated as

$\displaystyle\bar{p}e+\underline{p}(1-e),$

where $e\in\\{0,1\\}$ represents the patient’s treatment action against the
disease, where $e=1$ ($e=0$) is interpreted as (not) receiving a formal
medical intervention, such as taking a medicine or having a surgery. Taking
action $e$ incurs a physical cost $ce$, which can be interpreted as the either
the monetary cost or the physical pain of the treatment. Parameters $\bar{p}$
and $\underline{p}$ ($\bar{p}>\underline{p}$) represent the health
probabilities when the patient is sick, corresponding to $e=1$ and $e=0$,
respectively.

The medical test. Both parties do not observe $\theta$ at first, and the only
way to reveal it is through a fully informative medical test. Let
$\alpha\in(0,1)$ be the common prior that $\theta=1$, and denote by $a=1$ and
$a=0$ respectively as the choices that the patient accepts and does not accept
the medical test. Also, similar to the real-life medical consultation process,
the patient can undergo the treatment only if he is tested and the result is
bad.

Information disclosure. The doctor aims to maximize the patient’s health
probability, and his only instrument is through information disclosure. The
doctor can make a credible commitment about the information policy at the
beginning of the interaction, and the patient fully understands the
credibility. The doctor has two time windows to disclose information: One is
before the patient’s test choice, and the other is either after the test
result is received when $a=1$, or after observing choice $a=0$. The two
instances of information disclosure are referred to as the ex ante and the
interim disclosure, respectively.

In particular, I assume the test result can be publicly observed by both
parties.333In the reality, patients are often unable to observe the test
result due to the lack of specialized knowledge. In this case, the revelation
of the test result can also be an instrument of information disclosure. I
study this setting and characterize the optimal disclosure in Appendix D.
Information the doctor conveys is about parameter $\underline{p}$, the health
probability when $e=0$.444Compared to the other channels of information
disclosure ($\bar{p}$ and $c$), disclosure about the information about
$\underline{p}$ is the most complicated and contains all the theoretical
constructs contained in other channels. I assume parameter $\underline{p}$ is
valued in a binary set $\\{\underline{p}_{H},\underline{p}_{L}\\}$
($\bar{p}>\underline{p}_{H}>\underline{p}_{L}$). The patient begins with an ex
ante belief assigns $\mu_{0}\in(0,1)$ to event
$\underline{p}=\underline{p}_{H}$. Then it is updated to the interim belief
$\mu_{1}$ by the ex ante disclosure, and is subsequently updated to an ex post
belief $\mu_{2}$ in the interim disclosure. Denote the ex ante disclosure as

$\displaystyle\tau_{0}\in\mathcal{R}(\mu_{0})\equiv\left\\{\tau\in\Delta([0,1])\bigg{|}\int\mu\tau(d\mu)=\mu_{0}\right\\}.$

Also, since the doctor can observe the patient’s test result, she can use
different signal in the interim disclosure contingent on the patient’s test
choice. Thus, the interim disclosure is given by pair
$\left<\tau_{1}^{1}(\cdot|\cdot),\tau_{1}^{0}(\cdot|\cdot)\right>$, where the
former is used after the patient is diagnosed to be sick ($\theta=0$) in the
test and the latter is used after observing $a=0$. By Bayesian plausibility,
$\tau^{k}(\cdot|\mu_{1})\in\mathcal{R}(\mu_{1})$ for any
$\mu_{1}\in\text{Supp}(\tau_{0})$ and $k=0,1$.

Thus, an information policy is defined as triple
$\langle\tau_{0},\tau_{1}^{1}(\cdot|\cdot),\tau_{1}^{0}(\cdot|\cdot)\rangle$,
which can be interpreted as either the standard operating procedure (SOP) of
the medical consultation, or the personalized pattern of information
disclosure of the doctor.

The timeline. There are two dates: $1$ and $2$. The game proceeds as follows.
At date $1$, the doctor makes a credible commitment on the information
disclosure rule, and then the ex ante disclosure takes place by implementing
the commitment. Next, the patient chooses whether to accept the medical test
(i.e., choosing $a\in\\{0,1\\}$), and then the doctor carries out the
committed interim disclosure. At date $2$, the agent first decides whether to
accept the treatment (i.e., choosing $e\in\\{0,1\\}$) and pays the physical
cost $ce$. Then the final health status is realized and experienced. The
timeline is depicted in Figure 1.

Date 1Date 2ex antedisclosurethe patientchooses $a$$\theta$ is observedif
$a=1$interimdisclosurerealizedanticipatory utilitythe patientchooses $e$the
outcomereveals$A$ Figure 1: Timing of decisions and outcomes.

Equilibrium concept. The equilibrium concept is the perfect Bayesian
equilibrium, which requires the patient to be sequentially rational at every
decision node given the belief system, and the belief system is generated by
the Bayesian law whenever possible.

####  2.2 The Patient’s Preference

The patient’s preference is modeled as in the psychological expected utility
model by Caplin and Leahy (2001), which is common in the literature. In this
model, the patient’s choices can be understood as a subgame perfect Nash
equilibrium in an intrapersonal game between two subsequent selves.

Formally, in the end of period 1 (point A in Figure 1), the patient’s
subjective health probability is given by

$\displaystyle\eta(a,e*,\mu_{1})=\left\\{\begin{array}[]{cc}\alpha+(1-\alpha)\underline{p}(\mu_{1})&a=0\\\
\bar{p}e^{*}+\underline{p}(\mu_{1})(1-e^{*})&a=1\text{ and }\theta=0\\\
1&a=1\text{ and }\theta=1\end{array}\right.,$

where $e$ is his (rational) expectation of his action at date 2 and
$\underline{p}(\mu)\equiv\mu\underline{p}_{H}+(1-\mu)\underline{p}_{L}$ for
simplicity. Then at date 2, I normalize the physical prize of being eventually
healthy to $1$, and since there is no anticipatory emotions, the patient’s
payoff in date 2 is given by
$u_{2}(a,e^{*},\mu_{1})=\eta(a,e^{*},\mu_{1})-ce$. Denote by
$v_{2}(\alpha,\mu_{1})$ the indirect utility at date 2 that optimizes
$u_{2}(a,\cdot,\mu_{1})$.

The key assumption of the psychological expected utility model is the dynamic
inconsistency arising from the temporal distance between decision making and
uncertainty resolution. Based on this idea, the payoff at the end of date 1 is
given by

$\displaystyle u_{1}(a,\mu_{1})=\phi\circ v_{2}(a,\mu_{1}).$

Function $\phi$ measures the distortion of the patient’s preference by the
anticipatory emotions in period 1, which is continuous, smooth, and strictly
increasing. The patient, at any date-1 decision node of her, maximizes the
expected value of $u_{1}(a,\mu_{1})$ based upon the current belief (as well as
a correct anticipation of her future behavior) whenever possible.

Information attitude is modeled by the curvature of function $\phi$, and the
patient is information-avoidant (resp. loving, neutral) if and only if $\phi$
is concave (resp. convex, linear) in probabilities.555The classical
interpretation of this result is based on the relationship between risk
attitude and time. Suppose the probability of being eventually healthy is
$\beta$. Then if the uncertainty is resolved completely at date 1, the utility
is $\beta\phi(z)$, while if the uncertainty is resolved completely at date 2,
the utility is $\phi(\beta z)$. The former is larger if and only if “immediate
risk” is more desirable than its corresponding “remote risk”. To model
information avoidance, the concavity of $\phi$ is assumed throughout this
paper unless explicitly stated. The concavity of $\phi$ can be interpreted as
a result of diminishing marginal utility of hope. Hope preservation is widely
believed as the as-if psychology of information avoidance (Chew and Ho, 1994,
Ahlbrecht and Weber, 1996, Kocher et al., 2014, Chen, 2022, Masatlioglu et
al., 2023). If we think of $\phi(v)-v$ as the positive emotional feeling of
hope from date-2 payoff $v$, then $\phi$ being concave indicates that the
marginal increase in utility from an additional increase in the hopefulness
(i.e., the belief of the desired outcome) is decreasing.666This pattern is
consistent with the observation that hope is a feeling that is more sensitive
to possibility than probability (Loewenstein and Elster, 1992, Bruininks and
Malle, 2005, Bury et al., 2016, Luo et al., 2022).

###  3 Optimal Interim Disclosure

In this section, I characterize the optimal interim disclosure when the doctor
has no access to the ex ante disclosure.

####  3.1 Preliminaries

If the patient takes the medical test and the result is $\theta=0$, the
patient accepts the treatment if and only if
$\bar{p}-c\geq\underline{p}(\mu)$, which requires

$\mu\leq\mu_{e}\equiv\underline{p}^{-1}\left(\bar{p}-c\right).$

That is, the patient chooses $e=1$ if he is pessimistic about the prospect of
forgoing the treatment. Assume $\mu_{e}\in(0,1)$ to avoid trivial results.

Since if $\theta=1$ is detected, the interaction terminates with a happy
ending for both parties, I only concentrate on the case that $\theta=0$.
Define

$\displaystyle P(\mu)=\left\\{\begin{array}[]{cc}\bar{p}&\mu\leq\mu_{e}\\\
\underline{p}(\mu)&\mu>\mu_{e}\end{array}\right.\text{and
}V(\mu)=\left\\{\begin{array}[]{cc}\alpha+(1-\alpha)\phi(\bar{p}-c)&\mu\leq\mu_{e}\\\
\alpha+(1-\alpha)\phi(\underline{p}(\mu))&\mu>\mu_{e}\end{array}\right.$

as the payoff functions of the doctor and the patient when $\theta=0$,
respectively. For any interim belief $\mu_{1}$ and any pair of signals
$(\tau_{1}^{1},\tau_{1}^{0})\in\mathcal{R}(\mu_{1})\times\mathcal{R}(\mu_{1})$
in the interim disclosure, the patient decides to accept the test if and only
if

$\displaystyle\int_{0}^{1}V(\mu)\tau_{1}^{1}(d\mu)\geq\int_{0}^{1}V_{0}(\mu)\tau_{1}^{0}(d\mu)\equiv\int_{0}^{1}\phi\big{(}\alpha+(1-\alpha)\underline{p}(\mu)\big{)}\tau_{1}^{0}(d\mu).$
(1)

Here, function $V_{0}$ denotes the patient’s payoff if he declines the test.
Then interim disclosure
$\left<\tau_{1}^{1}(\cdot|\cdot),\tau_{1}^{0}(\cdot|\cdot)\right>$ is optimal
if for any $\mu_{1}$, pair
$\left<\tau_{1}^{1}(\cdot|\mu_{1}),\tau_{1}^{0}(\cdot|\mu_{1})\right>$ solves
the following optimization problem:

$\displaystyle\max_{\tau_{1}^{1},\tau_{1}^{0}\in\mathcal{R}(\mu_{1})}:$
$\displaystyle\int_{0}^{1}P(\mu)\tau_{1}^{1}(d\mu)$ (2)
$\displaystyle\text{subject to:
}\int_{0}^{1}V(\mu)\tau_{1}^{1}(d\mu)\geq\int_{0}^{1}V_{0}(\mu)\tau_{1}^{0}(d\mu).$

Say interim belief $\mu_{1}$ motivation available, denoted by
$\mu_{1}\in\mathcal{M}$, if there exists
$(\tau_{1}^{1},\tau_{1}^{0})\in\mathcal{R}(\mu_{1})\times\mathcal{R}(\mu_{1})$
such that inequality (1) holds. Since the health probability cannot be
improved without the test, in the optimal signal the patient chooses to accept
the medical test under whenever $\mu_{1}\in\mathcal{M}$.

######  Lemma 1.

In the interim disclosure, we can focus on signals with at most two
posteriors. That is, for any $\mu_{1}\in[0,1]$ and $k\in\\{0,1\\}$,
$|\text{Supp }\tau^{k}(\cdot|\mu_{1})|\leq 2$. In addition, if $|\text{Supp
}\tau^{k}(\cdot|\mu_{1})|=2$, then $\mu_{e}$ locates between the two
posteriors.

###### Proof.

Since $P$ is piecewise linear and $V$ is piecewise concave, this result can be
derived directly by Doval and Skreta (2024, Proposition 3.2). ∎

This result extends the revelation principle (Kamenica and Gentzkow, 2011,
Proposition 1) to the case with psychological preferences and participation
constraint. It is not trivial because existing works have noticed that if
either the audience is “psychological” (Lipnowski and Mathevet, 2018) or there
are additional constraints (Le Treust and Tomala, 2019, Boleslavsky and Kim,
2020, Salamanca, 2021, Doval and Skreta, 2024), the revelation principle may
not hold and the number of posteriors in the optimal signal may exceed the
number of states. Fortunately, the result can be guaranteed since (i) for each
action $e\in\\{0,1\\}$, the set of beliefs that $e$ is optimal is convex, and
(ii) functions $V$ and $P$ are piecewise-concave.

Since only binary signals deserve consideration, I adopt the notational
convenience that every binary signal $\tau\in\mathcal{R}(\mu_{1})$ is
identified by a pair of posterior beliefs $\\{x,y\\}$ such that
$x\geq\max\\{\mu_{1},\mu_{e}\\}\geq\min\\{\mu_{1},\mu_{e}\\}\geq y$, and the
probabilities that these posteriors realize can be solved by Bayesian
plausibility.

####  3.2 Punitive Revelation

######  Observation 1.

For any $\mu_{1}\in\mathcal{M}$, signal $\tau_{1}^{0}(\cdot|\mu_{1})$ in the
optimal interim disclosure minimizes $E_{\tau}V_{0}(\cdot)$. Since
$V_{0}(\mu)=\phi(\alpha+(1-\alpha)\underline{p}(\mu))$ is concave, the optimal
choice of $\tau_{1}^{0}(\cdot|\mu_{1})$ is $\\{0,1\\}$, i.e., full disclosure.

That is, for any interim belief that is motivation available, it is always
optimal for the doctor to commit to use signal
$\tau_{1}^{0}(\cdot|\mu_{1})=\\{0,1\\}$ to punish the patient if the patient
rejects the medical test. I will refer to the information disclosure strategy
here as punitive revelation. As a comparison, in the models of principal-agent
interaction, the optimal way to punish the agent violating the principal’s
recommendation is concealing all further information (Ely and Szydlowski,
2020, Habibi, 2020, Senkov, 2022, Liu, 2023, D’Antoni et al., 2024, Zhao et
al., 2024, Koh et al., 2024). This strategy works in their models since the
value of information is always positive there.

As a consequence, the expected payoff that the agent chooses $a=0$ and is told
about the true state is given by

$\displaystyle\bar{V}(\mu)=\mu\cdot\phi(\alpha+(1-\alpha)\underline{p}_{H})+(1-\mu)\phi(\alpha+(1-\alpha)\underline{p}_{L}),$

which is a linear function of $\mu$. Since
$\tau_{1}^{1}\in\mathcal{R}(\mu_{1})$, condition (1) can be written as

$\displaystyle\int_{0}^{1}V(\mu)\tau_{1}^{1}(d\mu)\geq\bar{V}(\mu_{1})\Rightarrow\int_{0}^{1}\big{(}V(\mu)-\bar{V}(\mu)\big{)}\tau_{1}^{1}(d\mu)\geq
0.$ (3)

####  3.3 The Tradeoff between Guiding the Future and Rewarding the Past

It remains to specify $\tau^{1}_{1}$. Without constraint (3), the optimal
interim disclosure simply concavifies function $P$. I will refer to this
signal as optimal for guiding the future since it maximizes the doctor’s
continuation payoff.

######  Observation 2.

The optimal signal for guiding the future is $\\{1,\mu_{e}\\}$ when
$\mu_{1}>\mu_{e}$, and is $\\{\mu_{1}\\}$ (no information) otherwise.

Panel (a) of Figure 2 demonstrates the concavification result, where the
supremum of the shaded area identifies the concave closures of the convex hull
of $P$. Therefore, to guide the future, the doctor utilizes a signal with
perfect good news, in order to persuade the patient that the prospect is more
pessimistic than expected. That is, if $\mu_{1}\geq\mu_{e}$, the doctor sends
the bad news with probability $1$ when $\underline{p}=\underline{p}_{L}$, and
randomize between the good and the bad news when
$\underline{p}=\underline{p}_{H}$. As a consequence, the patient update the
belief to $1$ if the good news is received but update the belief down to
$\mu_{e}$ otherwise.

$\mu$$P(\mu)$$\mu_{v}$$A$$\bar{p}$$0$$C$$\underline{p}(\mu_{e})$$\underline{p}_{H}$$D$$\mu_{e}$$1$$B$
(a) Guiding the future.

$\mu$$V(\mu)$$B$$\phi(\underline{p}(1))$$A$$0$$\phi(\underline{p}(\mu_{e}))$$\mu_{v}$$D$$C$$\mu_{e}$$1$
(b) Rewarding the past.

Figure 2: Optimal interim disclosure.

However, the optimal signal for guiding the future may be insufficient to
guarantee $a=1$. In the other extreme, the optimal signal for rewarding the
past maximizes $E_{\tau}V(\cdot)$ and can be solve by concavifying function
$V$.

######  Observation 3.

There exists $\mu_{v}>\mu_{e}$, such that the optimal signal for rewarding the
past is $\\{0,\mu_{v}\\}$ when $\mu_{1}<\mu_{v}$, and is $\\{\mu_{1}\\}$ (no
information) otherwise.

As is illustrated in the panel (b) of Figure 2, to reward the past, the doctor
utilizes a signal with perfect bad news, which splits the patient’s belief
into $0$ and some $\mu_{v}>\mu_{e}$ when $\mu_{1}<\mu_{v}$. To implement this
signal, the doctor always sends the good news when the state is good, and
randomizes between the good and the bad news when the state is bad.
Consequently, the bad news means the probability of staying healthy without
the treatment is truly $\underline{p}_{L}$, but the good news doesn’t imply
that the probability of being healthy without the treatment is
$\underline{p}_{H}$.

Consequently, $\mu_{1}\in\mathcal{M}$ if and only if the expected value of the
optimal signal for rewarding the past exceeds $\bar{V}(\mu_{1})$.

####  3.4 Characterization

The optimal interim disclosure balances the two conflicting motives of guiding
the future and rewarding the past. Identify $\mathcal{F}\subset\mathcal{M}$ as
the set of interim beliefs such that the patient is willing to take the test
if $\tau_{1}^{1}$ is optimal for guiding the future. Also, identify
$\mathcal{D}\subset\mathcal{M}$ as the set of interim beliefs such that the
patient is willing to take the test if $\tau_{1}^{1}$ is full disclosure. The
following definition helps to unravel the basic structure of $\mathcal{M}$,
$\mathcal{D}$, and $\mathcal{F}$.

######  Definition 1.

The patient reacts to fear if he chooses $a=1$ when $\mu_{1}=0$; that is

$c\leq\bar{p}-\phi^{-1}\left(\frac{\phi(\alpha+(1-\alpha)\underline{p}_{L})-\alpha}{1-\alpha}\right),$
(4)

That is, the patient reacts to fear if he is willing to be tested if he is
completely pessimistic about health probability when $\theta=0$
($\underline{p}=\underline{p}_{L}$). Intuitively, this indicates that the fear
generated by the pessimism triggers action instead of avoidance.

######  Lemma 2.

If the patient reacts to fear, $\mathcal{M}=[0,\mu_{\mathcal{M}}]$ for some
$\mu_{\mathcal{M}}<1$. Also, $\mathcal{D}=[0,\mu_{\mathcal{D}}]$ and
$\mathcal{F}=[0,\mu_{\mathcal{F}}]$ for some
$\mu_{\mathcal{F}}\leq\mu_{\mathcal{D}}\leq\mu_{\mathcal{M}}$. If not,
$\mathcal{F}=\mathcal{D}\equiv\emptyset$, and $\mathcal{M}$ is either
$\emptyset$ or some $[\mu^{L}_{\mathcal{M}},\mu^{H}_{\mathcal{M}}]$. Further,
$\mathcal{M}$ is non-empty if and only if

$\displaystyle\sup_{\mu\in[\mu_{v},1]}\big{(}V(\mu)-\bar{V}(\mu)\big{)}\geq
0.$

###### Proof.

See Appendix B.1. ∎

Thus, if either the optimal signal for guiding the future or full disclosure
is committed, the patient is willing to be tested only if the patient reacts
to fear and he is sufficiently pessimistic about $\underline{p}$. Also, Lemma
2 identifies $\mathcal{M}$ as an interval and shows that $\sup\mathcal{M}<1$,
which indicates that no matter how attractive the information reward is, the
patient is never willing to accept the test if he is completely optimistic
about $\underline{p}$.

######  Proposition 1.

Given interim belief $\mu_{1}$, the optimal signal after $a=1$ and $\theta=0$
is the optimal signal for guiding the future if $\mu_{1}\in\mathcal{F}$. If
$\mu_{1}\in\mathcal{D}/\mathcal{F}$, the optimal signal is
$\\{1,l(\mu_{1})\\}$, where

$l(\mu_{1})=1-(1-\mu_{1})\frac{V(1)-V(0)}{V(1)-\bar{V}(\mu_{1})}\leq\mu_{e}.$
(5)

If $\mu_{1}\in\mathcal{M}/\mathcal{D}$, the optimal signal is
$\\{0,h(\mu_{1})\\}$, where $h(\mu_{1})\in(\mu_{v},1)$ and is given implicitly
by

$\left(1-\frac{\mu_{1}}{h(\mu_{1})}\right)V(0)+\frac{\mu_{1}}{h(\mu_{1})}V(h(\mu_{1}))=\bar{V}(\mu_{1}).$
(6)

###### Proof.

See Appendix B.2. ∎

When $\mu_{1}\in\mathcal{F}$, using the unconstrained optimum for guiding the
future will not hinder the patient to accept the test. When
$\mu_{1}\notin\mathcal{F}$, participation constraint (3) binds. The rest part
of the proof applies the aforementioned best-good-news rule, which states that
under certain conditions, which is met in the current optimization program
(2), the posterior belief induced by the good news in the optimal signal is
maximized under the requirement of voluntary participation (3). Thus, for
$\mu_{1}\in\mathcal{M}/\mathcal{F}$, if $\mu_{1}\in\mathcal{D}$, since at
least full disclosure is sufficient to guarantee voluntary participation, the
upper belief must be $1$. Then the lower belief is solved by letting the
participation constraint holds with equality, which yields (5). If
$\mu_{1}\notin\mathcal{D}$, any signal with upper belief equal to $1$ is
unable to guarantee voluntary participation, since fix any $x>\mu_{e}$ as the
upper belief, the lower belief that maximizes the patient’ expected payoff
must be $\\{x,0\\}$. Consequently, the best-good-news rule indicates that the
lower belief is $0$ and the upper belief is derived by letting the
participation constraint holds with equality, which yields (6).

Figure 3 provides an example to illustrate the way the non-conclusive belief
is solved in the case of $\mu_{1}\notin\mathcal{D}$. The shaded area is the
convex hull of function $V$, where its supremum corresponds to the optimal
signal for rewarding the past,777Its infimum, the largest convex function
below $\text{co}(V)$, is exactly the value attained by the optimal signal for
guiding the future and the dashed line is the value of $\bar{V}$. Thus, if
point A denotes the interim belief, $h(\mu_{1})$ can be derived by connecting
point $(0,V(0))$ and $(\mu_{1},\bar{V}(\mu_{1}))$, and extending it until it
touches the concave closure of $\text{co}(V)$, which is point B.

$\mu$$V(0)$$\bar{V}(0)$$B$$A$$0$$\bar{V}(\mu)$$V(\mu)$optimal for guiding the
future$1$$\mu_{\mathcal{F}}$$\mu_{e}$$\mu_{v}$$\mu_{1}$$\mu_{\mathcal{M}}$$h(\mu_{1})$
Figure 3: Constructing the optimal signal when $a=1$ and $\theta_{0}=0$.

Note that in the physical cost setting, the best-good-news rule still applies,
but optimal interim disclosure must be with perfect good news. Therefore, the
possibility of signals with perfect bad news to be optimal is new in the
current model. Such a signal creates a comfort for the patient because it
provides a higher payoff than full disclosure, which maximizes the
instrumental value of information. However, when the bad news is not received,
the patient becomes optimistic about his health prospect when $e=0$.

The following corollary provides simple comparative statics with respective to
$\mu_{1}$, and the proof is omitted since it is straightforward.

######  Corollary 1.

When $\mu_{1}\in\mathcal{D}/\mathcal{F}$, the lower belief $l(\mu_{1})$ is
decreasing with $\mu_{1}$, and when $\mu_{1}\in\mathcal{M}/\mathcal{D}$, the
upper belief $h(\mu_{1})$ is also decreasing with $\mu_{1}$.

The intuition behind Corollary 1 is that the difficulty to motivate $a=1$ is
increasing with $\mu_{1}$, and hence the doctor must be increasingly
compromised to the motive of rewarding the past. Observe that when
$\mu_{1}\in(\mu_{e},\mathcal{F}]$, the optimal signal is optimal for guiding
the future (A and B in Figure 2). When $\mu_{1}$ goes beyond the border of
$\mathcal{F}$, by the best-good-news rule, the doctor compromises to the
motive of rewarding the past by keeping the higher belief unchanged and shifts
the lower belief downward. In this process, the doctor compromise to the
motive to reward the past by increasing the instrumental informativeness in
the signal. The endpoint of this process is when the lower belief decreases to
$0$ (point C in Figure 2), and the signal becomes full disclosure, which
indicates that the interim belief $\mu_{1}$ is on the edge of the set
$\mathcal{D}$. If $\mu_{1}$ continues to increase, it will reach
$\mathcal{M}/\mathcal{D}$, and the doctor continues to surrender to the motive
of rewarding the past by fixing the lower belief at $0$ and simultaneously
reducing the higher belief. Here, the doctor’s positive messages become
increasingly unconvincing, but the patient’s expected payoff of taking the
test continues to increase since the probability of receiving the bad news
keeps decreasing. That is, the doctor turns to reward the past by increasing
the anticipatory feeling of not being treated. Finally, as $\mu_{1}$ reaches
the border of $\mathcal{M}$, the higher belief reaches $\mu_{v}$ (point D in
Figure 2) and the signal shrinks to the optimal signal for rewarding the past.

####  3.5 Unconditional Information Disclosure

In line with Habibi (2020) and D’Antoni et al. (2024), I also consider the
case that the doctor discloses information independent of the patient’s
action. In this case, punitive revelation is unavailable, and therefore the
interim disclosure is a simple signal $\tau\in\mathcal{R}(\mu_{1})$. The
optimal signal is then given by the following optimization problem

$\displaystyle\max_{\tau\in\mathcal{R}(\mu_{1})}:$
$\displaystyle\int_{0}^{1}P(\mu)\tau_{1}^{1}(d\mu)$ (7)
$\displaystyle\text{subject to:
}\int_{0}^{1}\big{(}V(\mu)-V_{0}(\mu)\big{)}\tau(d\mu)\geq 0.$

######  Corollary 2.

Substitute $V(\cdot)$ and $\bar{V}(\cdot)$ in (5) and (6) by
$V(\cdot)-V_{0}(\cdot)$ and $0$, respectively, and the corresponding optimal
signal identified in Proposition 1 solves program (7).

###### Proof.

See Appendix B.3. ∎

Since punitive revelation is unavailable, sets $\mathcal{F}$, $\mathcal{D}$
and $\mathcal{M}$ become smaller. However, the structural similarity between
(2) and (7) indicates that the best-good-news rule also applies to (7), and
therefore the solution can be similarly derived.

####  3.6 General Probability Weighting

The assumption that function $\phi$ is concave allows for a global preference
for late resolution of uncertainty. However, empirical evidence has
demonstrated that the information attitudes in the real world are more
complicated than that. For example, people avoids information often also
exhibit a preference for positively skewed signals and prior dependence (see
Ganguly and Tasoff (2016), Nielsen (2020), Gul et al. (2020) and Masatlioglu
et al. (2023) for recent evidence),888Prior dependence refers to the
information that people prefer early (late) resolution of uncertainty if and
only if the prior belief is optimistic (pessimistic). and this cannot be
captured by a concave function $\phi$.999Instead, to fully reflect these non-
standard patterns of information attitudes, function $\phi$ is better to be
assumed as inverse S-shaped. That is, $\phi$ satisfies: (i)
$\phi^{\prime}(0)>1$ and $\phi^{\prime}(1)>1$, and (ii) there exists a kink
$\tilde{v}\in(0,1)$ such that $\phi$ is concave when $v<\tilde{v}$ and convex
when $v>\tilde{v}$. For example, suppose there are two possible levels date-2
payoff, $v_{H}$ and $v_{L}$, and the prior belief of $v_{H}$ is $\frac{1}{2}$.
Then for signals in $\mathcal{R}(\frac{1}{2})$, $\\{1,\frac{1}{3}\\}$ (a
positively skewed signal) is better than $\\{\frac{2}{3},0\\}$ (a negatively
skewed signal) if $\tilde{v}\in(v_{H}/3+2v_{L}/3,2v_{H}/3+v_{L}/3)$. Also,
comparing full disclosure and non-disclosure, people prefers the former if and
only if $(v_{H}+v_{L})/{2}$, the expected payoff at date 2, is greater than
$\tilde{v}$.

With a mild modification, the method to solve the optimal interim disclosure
above can also be applied for any general payoff function $\phi$ without the
assumption of global concavity. For any interval $I\subset\mathbb{R}$, define
the local convex hull of the graph of $V$ on $I$ as set

$\displaystyle co_{I}(V)=\bigg{\\{}(\mu,v)\bigg{|}\,\mu_{1},\mu_{2}\in
I,\text{ and }\pi\in[0,1],\text{ such that }\pi
V(\mu_{1})+(1-\pi)V(\mu_{2})=v\bigg{\\}}.$

Then the local concavification of $V$ on $I$ is given by

$\displaystyle\text{cav}_{I}(V)(\mu)\equiv\sup\\{v\big{|}\,\mu\in
I,\,(\mu,v)\in co_{I}(V)\\}.$

Define

$\displaystyle\hat{V}(\mu)=\left\\{\begin{array}[]{cc}\text{cav}_{[0,\mu_{e}]}(V)(\mu)&\mu\leq\mu_{e}\\\
\text{cav}_{[\mu_{e},1]}(V)(\mu)&\mu>\mu_{e}\\\ \end{array}\right..$

Then the following Proposition provides the method to solve the optimal
interim disclosure when $V$ is not concave on some subsets of $[0,1]$.

######  Proposition 2.

When the doctor is persuading a patient with payoff function $V$, she can get
exactly the same health probability as when the payoff function is $\hat{V}$.
Moreover, the optimal signal has an at-least-trinary support if and only if
the optimal interim disclosure is $\\{0,\mu\\}$ ($\mu>\mu_{e}$) when the
payoff function is $\hat{V}$ and $\hat{V}(\mu)>V(\mu)$.

###### Proof.

See Appendix B.4. ∎

The proof is a simple application of the Carathéodory theorem. The proposition
states that if $V$ is not concave on $[\mu_{e},1]$, the doctor can solve the
persuasion problem as if the patient’s payoff function is replaced by the
“local concavification” $\hat{V}$, which is piecewise concave. As is shown in
Proposition 1, there exists an optimal signal $\tau^{*}$ consisting of at most
two posterior beliefs, and if the support is binary, the posteriors locate on
the different side of $\mu_{e}$. Then if $\hat{V}(\mu)=V(\mu)$ for any
$\mu\in\text{Supp}(\tau^{*})$, $\tau^{*}$ is also the optimum in the original
program (2). If there exists $\mu\in\text{Supp}(\tau^{*})$ and $\mu>\mu_{e}$
such that $\hat{V}(\mu)>V(\mu)$, we can find two posteriors in $[\mu_{e},1]$
such that splitting $\mu$ to these two posteriors yields the same payoff for
the patient and becomes implementable with respect to $V$. Consequently, the
optimal signal may consist of three posterior beliefs, which is equal to the
summation of the number of states and the number of constraints. This echos
with the earlier finding in constrained information design (Le Treust and
Tomala, 2019, Boleslavsky and Kim, 2020, Salamanca, 2021, Doval and Skreta,
2024).

The existence of the third posterior belief has a straightforward
interpretation in the current context. When $\phi$ is not globally concave,
there exists some interim beliefs at which the patient is intrinsic
information loving; that is, the patient wants more information not for the
instrumental usage, but for some psychological motives, such as anxiety
reduction or curiosity. Since it is costly for the doctor to disclose
information with the constraint that the posteriors must be larger than
$\mu_{e}$, she can use this additional information as a reward to the past
decision accepting the test.

###  4 Characterization of the Optimal Information Policy

The optimal interim disclosure solved in Section 3 helps to characterize the
global optimal policy. This section provides the results. I first solve the
optimal policy when the reception of the doctor’s information is mandatory,
and then turn to the case with the ex ante participation constraint.

####  4.1 Ex ante Disclosure

Suppose first that the patient has no choice but to listen to the doctor,
solving the optimal information policy begins with the following observation.

######  Observation 4.

In the optimal information policy, the ex ante disclosure $\tau_{0}$ can be
solved by backward induction. That is, if triple
$\langle\tau_{0},\tau_{1}^{1},\tau_{1}^{0}\rangle$ is an optimal policy and
pair $\langle\hat{\tau}_{1}^{1},\hat{\tau}_{1}^{0}\rangle$ is an optimal
interim disclosure, then
$\langle\tau_{0},\hat{\tau}_{1}^{1},\hat{\tau}_{1}^{0}\rangle$ is also an
optimal policy.101010In general, backward induction fails in dynamic Bayesian
persuasion when (i) the receiver has multiple choices to make, (ii) the
receiver is forward-looking, and (iii) the sender has a commitment power for
future disclosure rules. This observation holds by the doctor’s objective
function being linear in probabilities. The independence axiom indicates that
changing the strategies at any node in the interim disclosure has no influence
on the behaviors on the other nodes. Thus, if $\mu_{1}\in\text{Supp}(\pi_{0})$
and the signal at $\mu_{1}$, denoted as $\tau$, is not the optimal interim
disclosure, then replace $\tau$ with the optimal interim disclosure is weakly
better than the original strategy.

Define function $P^{*}(\cdot)$: $[0,1]\rightarrow[0,1]$ as the patient’s
health probability when there is no information in the ex ante disclosure and
the interim disclosure is given in Proposition 1. The following proposition
characterizes the ex ante disclosure in the optimal information policy.

######  Proposition 3.

The following is known about the optimal ex ante disclosure:

  * •

If the patient reacts to fear, the optimal ex ante disclosure is
$\\{1,\min\left\\{\mu_{\mathcal{F}},\mu_{e}\right\\}\\}$ if
$\mu_{0}>\min\left\\{\mu_{\mathcal{F}},\mu_{e}\right\\}$ and is
$\\{\mu_{0}\\}$ otherwise.

  * •

If the patient does not react to fear and $\mathcal{M}\neq\emptyset$, there
exists
$\hat{\mu}^{L},\hat{\mu}^{H}\in[\mu_{\mathcal{M}}^{L},\mu_{\mathcal{M}}^{H}]$
($\hat{\mu}^{L}\leq\hat{\mu}^{H}$), such that the optimal ex ante signal is
$\\{1,\hat{\mu}^{H}\\}$ if $\mu_{0}>\hat{\mu}^{H}$ and is is
$\\{0,\hat{\mu}^{L}\\}$ if $\mu_{0}<\hat{\mu}^{L}$. Otherwise, it is
$\\{\mu_{0}\\}$.

###### Proof.

See Appendix B.5. ∎

According to the objective function, the doctor is motivated to avoid two
types of losses. One is the patient refusing the test, while the other is he
forgoing the treatment even though he took the test and the result is bad.
That is, the optimal disclosure must balance the tradeoff between the
willingness to be tested and the willingness to be treated. In this regard,
the optimal interim disclosure maintains the willingness to be tested by
sacrificing some willingness to be treated. However, an alternative
methodology, preemptive warning, which sacrifices the willingness to be tested
to guarantee the willingness to be treated, emerges when the ex ante
disclosure is considered. Preemptive warning requires the doctor to release a
sufficiently clear message at the start of the diagnostic process, and if the
bad news is received, the patient will willingly to accept the treatment with
probability $1$ even without any interim disclosure.

Proposition 3 shows that preemptive warning is optimal if and only if the
patient reacts to fear. The concavification result is provided in panel (a) of
Figure 4, the curve is the doctor’s objective function when there is no ex
ante disclosure but the interim disclosure is optimal. In fact, if we consider
the physical cost setting, it is straightforward that preemptive warning is
still optimal if the patient accepts the test $\mu_{1}=0$. That is, the
underlying philosophy of this strategy is “classical” and information
avoidance only plays a minor role: The patient is willing to take actions
against the health risk if and only if he is pessimistic, and persuasion
maximizes the probability that the patient is pessimistic.

$\mu$$P^{*}(\mu)$$\bar{p}$$0$$\underline{p}_{H}$$1$$\mu_{\mathcal{F}}$$\mu_{\mathcal{D}}$$\mu_{\mathcal{M}}$
(a) (4) holds

$\mu$$P^{*}(\mu)$$\bar{p}$$0$$\underline{p}_{L}$$\underline{p}_{H}$$1$$\mu^{L}_{\mathcal{M}}$$\mu^{H}_{\mathcal{M}}$$\hat{\mu}$
(b) (4) does not hold

Figure 4: Optimal ex ante disclosure.

However, the second part of Proposition 3 shows that once the patient chooses
to bury his head into the sand when $\mu_{1}=0$, preemptive warning becomes
ineffective. In this case, since $\mathcal{D}=\emptyset$, the subsequent
interim signal is with perfect good news, and the patient accepts the test
only when the interim belief is intermediate, at which the patient anticipates
that he will be comforted if the test result is bad. Thus, the only way that
the doctor improves the result is by keeping the patient’s belief within this
area, utilizing the committed comfort as a carrot to reward the patient. The
panel (b) of Figure 4 illustrates the concavification result.

Note that due to the concavity of function $\phi$, inequality (4) holds if and
only if $\alpha$ is sufficiently small. Thus, ceteris paribus, the committed
comfort strategy works only for people how are optimistic about $\theta$.

In the physical cost setting, the doctor is unable to persuade the patient to
take the test, and therefore the committed comfort strategy is new for
psychological patients. Also, when information provision is independent of the
test choice, since punitive revelation is unavailable,
$\mathcal{M}\equiv\emptyset$, and hence due to information avoidance, the
patient will not choose $a=1$ at any interim belief. Consequently, the ex ante
disclosure is also unnecessary.

####  4.2 Ex ante Participation Constraint

The previous discussion imposes no restriction on the ex ante signal, and thus
it cannot guarantee an information-avoidant patient to benefit from the entire
medical consultation process. That is, the ex ante payoff of listening to the
doctor can be smaller than quitting at the first place. This is impossible in
the physical cost setting because people has free disposal of the information.
Particularly, the patient’s interim payoff may be equal to $\bar{V}(\cdot)$,
which, because of punitive revelation, leaves the patient with no rent in the
interim stage, and therefore the ex ante payoff is strictly lower than the
payoff the patient quits at the first moment.

Now I allow the information-avoidant patient to reject the entire medical
consultation process, which is effective to shut down all the channels of
information disclosure by the doctor preemptively. In this case, the ex ante
disclosure will face an additional ex ante participation constraint. That is,
before the ex ante disclosure, the patient now has an additional choice to
refuse the medical consultation and receive the payoff
$V_{0}(\mu_{0})=\phi(\alpha+(1-\alpha)\underline{p}(\mu_{0}))$.

Consequently, given triple $\langle\tau_{0},\tau_{1}^{1},\tau_{1}^{0}\rangle$,
let

$\displaystyle\overline{\mathcal{M}}\equiv\left\\{\mu_{1}\in\text{Supp}(\tau_{0})\bigg{|}\,\int_{0}^{1}V(\mu)\tau_{1}^{1}(d\mu|\mu_{1})\geq\int_{0}^{1}V_{0}(\mu)\tau_{1}^{0}(d\mu|\mu_{1})\right\\}$

be the set of interim beliefs that the patient is willing to be tested. Then
policy $\langle\tau_{0},\tau_{1}^{1},\tau_{1}^{0}\rangle$ must satisfy

$\int_{\overline{\mathcal{M}}}\left(\int_{0}^{1}V_{1}(\nu)\tau_{1}^{1}(d\nu|\mu)\right)\tau_{0}(d\mu)+\int_{\overline{\mathcal{M}}^{C}}\left(\int_{0}^{1}V_{0}(\nu)\tau_{1}^{1}(d\nu|\mu)\right)\tau_{0}(d\mu)\geq
V_{0}(\mu_{0}).$ (8)

The main technical difficulty here is that backward induction no longer holds
with the additional ex ante participation constraint (8). This is because even
in the ex ante disclosure, the doctor also needs to balance the motives of
guiding the future and rewarding the past. Hence, the two instances of
disclosure are both responsible to the ex ante participation constraint.

To solve the problem, the following lemma shows that we can adopt the
theoretical convenience to focus only on simple recommendation policies
without loss of generality, which has been proved by Makris and Renou (2023)
in the setting without the participation constraints.

######  Lemma 3.

Without loss of generality, the optimal ex ante signal is some
$\\{\mu_{H},\mu_{L}\\}$ ($\mu_{H}>\mu_{L}$ and $\mu_{\mathcal{V}}>\mu_{L}$),
where the patient accepts the test if and only if the interim belief is
$\mu_{L}$.

###### Proof.

See Appendix B.6. ∎

Thus, in every instance of information disclosure, the doctor’s information
structure can be reduced to a recommendation of the action taken in the
current choice problem. That is, the doctor only recommends the patient
whether to be tested in the ex ante disclosure, and the recommendation of the
treatment decision is left to the interim disclosure.

I introduce the following notations to solve the problem. First, define
function

$\displaystyle\mathcal{V}(\mu)\equiv\max\left\\{V_{0}(\mu),\alpha+(1-\alpha)\phi(\bar{p}-c)\right\\}$

as the doctor’s payoff function when she commits to disclose no information in
the interim disclosure.111111The patient either takes the test and accepts the
treatment, or refuses to see the doctor in the very beginning. Second, belief
$\mu_{\mathcal{N}}$ is the solution to equation
$\phi(\alpha+(1-\alpha)\underline{p}(\mu_{\mathcal{N}}))=\alpha+(1-\alpha)\phi(\bar{p}-c)$;
that is, $\mu_{\mathcal{N}}$ is the threshold that taking and not taking the
test are indifferent, given there is no information in the interim disclosure.
Third, belief $\mu_{\mathcal{T}}$ is the solution to

$\displaystyle\mu_{\mathcal{T}}\mathcal{V}(1)+(1-\mu_{\mathcal{T}})\mathcal{V}(0)=\mathcal{V}(\mu_{\mathcal{T}}),$

and therefore full disclosure in the ex ante disclosure is sufficient to
motivate voluntary reception of information if and only if
$\mu_{0}\geq\mu_{\mathcal{T}}$. Finally, belief $\mu_{\mathcal{V}}$ is
identified by

$\displaystyle\mu_{\mathcal{V}}=\sup\left\\{\mu\bigg{|}\,\text{cav}\circ\mathcal{V}(\mu)>\mathcal{V}(\mu)\right\\},$

and hence by concavification, if there is no interim disclosure, the patient
can benefit from persuasion if and only if $\mu_{0}<\mu_{\mathcal{V}}$.

I first consider the case that inequality (4) holds, under which it is optimal
to use the preemptive warning strategy without the ex ante participation
constraint.

######  Proposition 4.

If (4) holds, it is still optimal for the doctor to disclose no information in
the interim disclosure. In the ex ante disclosure, the doctor benefit from
persuasion if and only if $\mu_{0}<\mu_{\mathcal{V}}$.

  * •

If $\mu_{0}<\mu_{\mathcal{N}}$, non-disclosure is optimal.

  * •

If $\mu_{0}\in(\mu_{\mathcal{N}},\mu_{\mathcal{T}})$, the optimal ex ante
signal is with perfect good news, such that the lower belief is

$1-(1-\mu_{0})\cdot\frac{\mathcal{V}(1)-\mathcal{V}(0)}{\mathcal{V}(1)-\mathcal{V}(\mu_{0})}.$
(9)

  * •

If $\mu_{0}\in(\mu_{\mathcal{T}},\mu_{\mathcal{V}})$, the optimal ex ante
signal is with perfect bad news, such that the upper belief, denoted by
$\bar{h}(\mu_{0})$, is given implicitly by

$\displaystyle\left(1-\frac{\mu_{0}}{\bar{h}(\mu_{0})}\right)\cdot\mathcal{V}(0)+\frac{\mu_{0}}{\bar{h}(\mu_{0})}\cdot\mathcal{V}\left(\bar{h}(\mu_{0})\right)=\mathcal{V}(\mu_{0}).$

###### Proof.

See Appendix B.7. ∎

Surprisingly, the result is parallel to the optimal interim disclosure derived
in Proposition 1. Indeed, if we substitute functions $V$ and $\bar{V}$
respectively by $\mathcal{V}$ and $V_{0}$, and replace triple
$\langle\mu_{\mathcal{F}},\mu_{\mathcal{D}},\mu_{v}\rangle$ by
$\langle\mu_{\mathcal{N}},\mu_{\mathcal{T}},\mu_{\mathcal{V}}\rangle$, then
the result in Proposition 4 can be regarded as bringing forward the optimal
interim disclosure in Proposition 1 to the ex ante disclosure. That is, the
doctor designs the ex ante signal as if the interim disclosure is absent, and
balances the tradeoff between guiding the future and rewarding the past under
both the ex ante and interim participation constraints.

If the optimal signal is still with perfect good news
($\mu_{0}\in(\mu_{\mathcal{N}},\mu_{\mathcal{T}})$), the policy is still
preemptive warning, and the only difference is that the ex ante signal is more
informative than the case without the ex ante constraint. If the optimal
signal is still with perfect bad news
($\mu_{0}\in(\mu_{\mathcal{T}},\mu_{\mathcal{V}})$), the policy can be
nominated as preemptive comfort since it brings the committed comfort after
the test forward and send the signal before the test.

Proposition 4 can be regarded as the joint consequence of two key properties
of the optimization problem. First, like the case that the ex ante constraint
(8) is absent and inequality (4) holds, the doctor forgoes the opportunity of
the interim disclosure as the solution of the aforementioned tradeoff between
the willingness to be tested and the willingness to be treated. Second, since
the information stick is not available in the ex ante disclosure, the ex ante
participation constraint is more difficult to meet than its interim
counterpart. Thus, any information policy that satisfies the ex ante
constraint must have provided the patient with a sufficiently large payoff,
which guarantees the interim constraint as well.

Finally, when inequality (4) does not hold, the proof of Proposition 4 is
sufficient to show that the doctor cannot persuade the patient to participate
in the medical consultation by any information policy.

###  5 Conclusion

The main contribution of this paper is the characterization of the optimal
information policy to promote a medical test-avoider to undergo a necessary
medical test. The feature of the problem is that the patient has the decision-
making power to acquire some of the information (test result), but is a
passive recipient of the other information. Thus, the sender of the latter
information, namely the doctor, can influence the patient’s information
environment to create the motivation for the patient to take the test and make
the best treatment decision.

The results is summarized in Table 1. The key differences are whether the
patient react to fear and whether the ex ante participation constraint is
present. To make a better comparision, the corresponding results in the
physical cost setting is also listed in the last column.

| Ex ante participation
---
constraint
Reacts to fear | Optimal information policy |  | Counterpart in the
---
physical cost setting
$\times$ | ✓ | preemptive warning | preemptive warning
$\times$ | $\times$ |  | committed comfort
---
or unable to persuade
unable to persuade
✓ | ✓ |  | preemptive warning
---
or preemptive comfort
preemptive warning
✓ | $\times$ | unable to persuade | unable to persuade
Table 1: Summary of the Results

Compared with the physical cost benchmark, this paper emphasizes the question
that whether the information greenhouse policy, which rewards the voluntary
information acquisition by committing to withhold the other information, is
useful to the patient. The positive answer is partially confirmed: It is an
efficient tool either if the patient does not react to fear, or if the patient
faces an ex ante participation constraint, and otherwise it is inferior to
preemptive warning.

Additionally, this paper provides a technical trick, namely the best-good-news
rule, which is particularly useful for constrained information design in two-
stage settings.

It should be noted that in practice, the doctor or other senders of medical
information face more constraints and practical considerations than those
considered in this paper. These additional concerns may render the optimal
policy solved by this paper to be non-optimal. For example, in many cases, the
standard operating procedures or the ethical guidelines of medical
consultation do not allow the doctor to conceal the information she possesses.
However, even so, the model provides a theoretical framework for evaluating
the impact of any information disclosure policy on information-avoidant
patients, and the optimal policy can be regarded as a reference point for
evaluating the use of these policies. By comparing the actual diagnostic and
treatment process with the optimal policy that is theoretically derived, we
can assess to what extent it deters information-avoidant patients and thereby
estimate the welfare loss incurred in this regard.

Finally, there is a need for future theoretical and empirical research to
carefully examine the interaction between information avoidance and other
behavioral biases. A recent paper examines the key predictions in Ely and
Szydlowski (2020) experimentally, and finds that the optimal policy is not as
useful as expected because it ignores the agent’s prosocial preference (Du and
Shahriar, 2024). Such an impact from other behavioral biases may also
contribute to the result of this paper.

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## Appendix

###  Appendix A The Best-Good-News Rule

In this appendix I formally prove the best-good-news rule in the traditional
two-action-two-state Bayesian persuasion problems with a linear constraint.

Consider a game played between a sender and a receiver. The sender designs an
information structure in the test, and then the receiver decides whether to
accept the test. The state space of binary, and the test updates the prior
belief from $\upsilon_{0}\in(0,1)$ to $\upsilon\in[0,1]$, which are the
probabilities assigned to one of the two states. Based on the posterior
belief, the receiver chooses an action $e\in\\{0,1\\}$, and the game ends. The
Bayesian persuasion problem can be written as

$\displaystyle\max_{\tau\in\mathcal{R}(\upsilon_{0})}:$
$\displaystyle\int_{0}^{1}P(\upsilon)\tau(d\upsilon)$ subject to:
$\displaystyle\int_{0}^{1}V(\upsilon)\tau(d\upsilon)\geq\bar{V}(\upsilon_{0})$
$\displaystyle\text{(PC)},$

where $P$ and $V$ are the indirect utility functions of posterior belief
$\upsilon$ for the sender and the receiver, respectively.

Suppose that there exists $\upsilon_{e}\in(0,1)$, such that the ses of beliefs
that $e=1$ and $e=0$ are optimal are $[0,\upsilon_{e}]$ and
$[\upsilon_{e},1]$, respectively. Assume functions $P$ and $V$ are both
concave in both $[0,\upsilon_{e}]$ and $[\upsilon_{e},1]$, and therefore we
can focus on signals in set

$\displaystyle\mathcal{S}(\upsilon_{0})=\bigg{\\{}\\{x,y\\}\bigg{|}1\geq
x\geq\max\\{\upsilon_{0},\upsilon_{e}\\}\geq\min\\{\upsilon_{0},\upsilon_{e}\\}\geq
y\geq 0\bigg{\\}}.$

without loss of generality.

Note that by concavification, the unconstrained optimum is also at-most-
binary. That is, for any $\upsilon_{0}$, the exists
$\\{\hat{x},\hat{y}\\}\in\mathcal{S}(\upsilon_{0})$, such that

$\displaystyle\hat{\tau}=\\{\hat{x},\hat{y}\\}\in\arg\max_{\tau\in\mathcal{R}(\upsilon_{0})}:\int_{0}^{1}P(\upsilon)\tau(d\upsilon).$

I focus on the case that the (PC) constraint is binding in the constrained
optimum, which indicates that signal $\\{\hat{x},\hat{y}\\}$ cannot meet the
requirement of (PC). Thus, the optimization problem can be rewritten as

$\displaystyle\max_{x,y:\,\\{x,y\\}\in\mathcal{S}(\upsilon_{0})}:$
$\displaystyle\,\frac{x-\upsilon_{0}}{x-y}\cdot
P(y)+\frac{\upsilon_{0}-y}{x-y}\cdot P(x)$ subject to:
$\displaystyle\frac{x-\upsilon_{0}}{x-y}\cdot
V(y)+\frac{\upsilon_{0}-y}{x-y}\cdot V(x)=\bar{V}(\upsilon_{0})$
$\displaystyle\text{(PCB)}.$

Here, $(x-\upsilon_{0})/(x-y)$ and $(\upsilon_{0}-y)/(x-y)$ are respectively
the probabilities that posterior $y$ and $x$ realize, which are derived from
Bayesian plausibility.

To formally state the best-good-news rule, I introduce the following
notations. First, for any $x\in[\max\\{\upsilon_{0},\upsilon_{e}\\},1]$,
define

$\displaystyle
D(x)=\left\\{y\in[0,\min\\{\upsilon_{0},\upsilon_{e}\\}]\bigg{|}\,\frac{x-\upsilon_{0}}{x-y}V(y)+\frac{\upsilon_{0}-y}{x-y}V(x)=v\right\\}.$

That is, $D(x)$ is the set of lower beliefs that makes constraint (PCB) binds.
If $D(x)$ is non-empty, define $d(x)$ is the solution to the following
maximization problem:

$\displaystyle d(x)=\sup\bigg{\\{}\arg\max_{y\in
D(x)}:\,\frac{x-\upsilon_{0}}{x-y}\cdot P(y)+\frac{\upsilon_{0}-y}{x-y}\cdot
P(x)\bigg{\\}}.$

Then given $x$, $d(x)$ is the “best lower belief” that makes the (PCB)
constraint binds. Second, for any function $G$, define

$\displaystyle S_{G}(x)=\frac{G(x)-G(d(x))}{x-d(x)},$

as the slope of the segment connecting $(x,G(x))$ and $(d(x),G(d(x)))$. Third,
for any $x\geq\max\\{\upsilon_{e},\upsilon_{0}\\}$, define

$\displaystyle\mathcal{P}(x)=\frac{x-\upsilon_{0}}{x-d(x)}\cdot
P(d(x))+\frac{\upsilon_{0}-d(x)}{x-y}\cdot P(x)$

as the objective function when $y$ is substituted by $d(x)$.

######  Lemma A.1.

$\mathcal{P}^{\prime}(x)\geq 0$ if and only if

$\left(S_{P}(x)-P^{\prime}(x)\right)\leq\frac{S_{V}(x)-V^{\prime}(x)}{S_{V}(x)-V^{\prime}(d(x))}\cdot\big{(}S_{P}(x)-P^{\prime}(d(x))\big{)}.$
(10)

In particular, if (10) holds for all $x$, the optimal signal is
$\\{x^{*},d(x^{*})\\}$, where

$x^{*}=\sup\left\\{x\in[\max\\{\upsilon_{0},\upsilon_{e}\\},1]\bigg{|}\,D(x)\neq\emptyset\right\\}.$
(11)

###### Proof.

Using implicit differentiation,

$d^{\prime}(x)=-\frac{\upsilon_{0}-d(x)}{x-\upsilon_{0}}\cdot\frac{V(x)-V(d(x))-(x-d(x))V^{\prime}(x)}{V(x)-V(d(x))-(x-d(x))V^{\prime}(d(x))}.$
(12)

Also,

$\displaystyle\mathcal{P}^{\prime}(x)=$
$\displaystyle-\left[\frac{\upsilon_{0}-d(x)}{x-d(x)}\cdot\left(\frac{P(x)-P(d(x))}{x-d(x)}-P^{\prime}(x)\right)\right]$
$\displaystyle+d^{\prime}(x)\cdot\left[-\frac{x-\upsilon_{0}}{x-d(x)}\cdot\left(\frac{P(x)-P(d(x))}{x-d(x)}-P^{\prime}(d(x))\right)\right].$

Substituting $d^{\prime}(x)$ by expression (12), $d\mathcal{P}(x,d(x))/dx>0$
if and only if

$\displaystyle S_{P}(x)-P^{\prime}(x)$
$\displaystyle=\frac{P(x)-P(d(x))}{x-d(x)}-P^{\prime}(x)$
$\displaystyle\leq\frac{V(x)-V(d(x))-(x-d(x))V^{\prime}(x)}{V(x)-V(d(x))-(x-d(x))V^{\prime}(d(x))}\cdot\left(\frac{P(x)-P(d(x))}{x-d(x)}-P^{\prime}(d(x))\right)$
$\displaystyle=\frac{S_{V}(x)-V^{\prime}(x)}{S_{V}(x)-V^{\prime}(d(x))}\cdot(S_{P}(x)-P^{\prime}(d(x))),$

which completes the proof. ∎

Lemma A.1 characterizes the condition under which the best-good-news rule is
guaranteed. By this result, if condition (10) is satisfied, the optimal signal
can be derived in the following procedure:

  1. 1.

Check if $\\{\hat{x},\hat{y}\\}$, the unconstrained optimum, satisfies
constraint (PC). If so, it is the optimum in the constrained program.

  2. 2.

If $\\{\hat{x},\hat{y}\\}$ violates constraint (PC), identify the supremum of
upper belief $x$ such that $D(x)$ is non-empty.

  3. 3.

Maximize the objective function by letting $x$ be the maximum value and
selecting $y$ in $D(x)$.

The following example provides a simple example that how to use the best-good-
news rule to solve the optimal interim disclosure.

######  Example 1 (Optimal disclosure about the cost).

Consider a simple Bayesian persuasion problem such that taking the test incurs
a fixed fee. Formally, the receiver is involved in a choice problem
$e\in\\{0,1\\}$, where $e=1$ and $0$ denote action and inaction, respectively.
The benefits of taking the action and inaction are normalized respectively to
$1$ and $0$. Also, the cost of taking the action is uncertain. Let
$c\in\\{c_{H},c_{L}\\}$ ($c_{H}>1>c_{L}$) be the cost and $\upsilon$ is the
belief that $c=c_{H}$. Consequently, the receiver takes the action if and only
if $\upsilon\leq\upsilon_{e}\equiv(1-c_{L})/(c_{H}-c_{L})$.

Suppose $\upsilon_{0}\geq\upsilon_{e}$, which indicates that the receiver does
not intend to take the action at the prior belief. Also, suppose the sender’s
payoff function is given by

$\displaystyle
P(\upsilon)=\left\\{\begin{array}[]{cc}\bar{P}&\upsilon\leq\upsilon_{e}\\\
\underline{P}&\upsilon>\upsilon_{e}\end{array}\right.,$

where $\bar{P}\geq\underline{P}$. In other words, the sender benefit from the
receiver taking the action. In order to change the receiver’s behavior, the
sender commits to a signal $\\{x,y\\}$ in the test, and taking the test incurs
a private cost $\psi>0$ for the receiver. Then the Bayesian persuasion problem
can be expressed as

$\displaystyle\max_{\\{x,y\\}\in\mathcal{S}(\upsilon_{0})}:$
$\displaystyle\,\frac{x-\upsilon_{0}}{x-y}\cdot\bar{P}+\frac{\upsilon_{0}-y}{x-y}\cdot\underline{P}$
$\displaystyle\text{subject to:
}\frac{x-\upsilon_{0}}{x-y}\cdot\big{(}1-\upsilon
c_{H}-(1-\upsilon)c_{L}\big{)}\geq\psi.$

This problem has a wide range of applications in real-world scenarios. For
example, a doctor uses the results of a potentially painful colonoscopy test
to persuade a patient who is reluctant to undergo a colon surgery to change
his decision.

$\upsilon$$P(\upsilon)$$\upsilon_{0}$$\upsilon_{e}$$1$$\bar{P}$$\underline{P}$$x$$d(x)$$0$
(a) Guiding the future.

$\upsilon$$V(\upsilon)$$\upsilon_{0}$$\upsilon_{e}$$1$$1-c_{L}$$0$$\psi$$x$$d(x)$
(b) Rewarding the past.

Figure 5: Checking Whether the Best-Good-News Rule Holds.

Function $P$ and $V$ are depicted on the left and right panel of Figure 5. The
receiver’s payoff function $V$ is convex, which indicates that the result is
not founded upon information avoidance. In the figures, the blue arrows are
the slopes of the original functions, and the red arrows are the slopes of the
lines crossing $(\upsilon_{0},\psi)$ and touches the original functions at
$\upsilon=x$ and $\upsilon=d(x)$. Comparing the slopes, we can observe that

$\displaystyle\frac{S_{V}(x)-V^{\prime}(x)}{S_{V}(x)-V^{\prime}(d(x))}<0.$

whereas $S_{P}(x)-P^{\prime}(x)=S_{P}(x)-P^{\prime}(d(x))=S_{P}(x)<0$. Thus,
the requirement of Lemma A.1 holds automatically.

Now we apply the three-step procedure to solve the problem. First, the
unconstrained optimum is $\\{1,\upsilon_{e}\\}$, which is not sufficient to
motivate the receiver to take the test for all $\upsilon_{0}>\upsilon_{e}$.
Also, for any $\upsilon_{0}\geq\upsilon_{e}$, there always exists
$y\leq\upsilon_{e}$ such that $\\{1,y\\}$ makes constraint (PCB) hold, and
therefore, the upper belief is always $1$. Finally, observe that since $V$ is
piecewise linear, and thus $D(x)$ is a singleton for any $x>\upsilon_{e}$.
Thus, the lower belief can be derived simply by solving equation

$\displaystyle\frac{1-\upsilon_{0}}{1-y}\cdot(1-yc_{H}-(1-y)c_{L})=\psi,$

which implies that the optimal signal is given by

$\displaystyle\left\\{1,\frac{(1-\upsilon_{0})(1-c_{L})-\psi}{(1-\upsilon_{0})(c_{H}-c_{L})-\psi}\right\\}.$

Thus, the lower belief is decreasing with $\psi$, which indicates that the
sender need to provide a more informative signal to reward the past when the
test cost becomes higher. ∎

###  Appendix B Proofs

####  B.1 Proof of Lemma 2

As functions of the interim belief $\mu_{1}$, the expected payoff of the
optimal signal for guiding the future is piecewise linear and convex, and it
equals

$\displaystyle V(1)=\alpha+(1-\alpha)\phi(\underline{p}_{H}).$

at point $\mu=1$. Meanwhile, by the punitive revelation strategy, the payoff
of rejecting the medical test is linear in $\mu$, and at point $\mu=1$, it
equals

$\displaystyle\bar{V}(1)=\phi(\alpha+(1-\alpha)\underline{p}_{H})\geq V(1),$

by the concavity of $\phi$. Therefore, $\mathcal{F}$ is either empty or an
interval $[0,\mu_{\mathcal{F}}]$, which is fully determined by the comparison
at point $\mu=0$. In other words, it is non-empty if and only if

$\displaystyle
V(0)=\alpha+(1-\alpha)\phi(\bar{p}-c)\geq\bar{V}(0)=\phi(\alpha+(1-\alpha)\underline{p}_{L}),$

which implies (4). The some reasoning applies for $\mathcal{D}$.

The expected payoff of the optimal signal for rewarding the past is concave
and linear in $[0,\mu_{v}]$. Again, $\mathcal{M}$ is either empty or an
interval since $\bar{V}$ is linear. If (4) holds, it is
$[0,\mu_{\mathcal{M}}]$ for some $\mu_{\mathcal{M}}<1$ since its value at
$\mu=1$ is still $V(1)<\bar{V}(1)$. If (4) does not hold, by the linearity of
function $\text{cav}\circ V(\mu)$ in interval $[0,\mu_{v}]$,
$\mathcal{M}\subset[0,\mu_{v}]$ can never hold, which completes the proof.

####  B.2 Proof of Proposition 1

By Lemma 1, we can focus on signals with at most two potential posteriors. If
$\mu_{1}\in\mathcal{F}$, the optimality of the optimal signal for guiding the
future holds automatically, and therefore I only consider the case that
$\mu_{1}\in\mathcal{M}/\mathcal{F}$.

I first show that the participation constraint (3) must be binding. Suppose
not, then there exists an optimal signal $\\{x,y\\}$ such that (3) holds with
inequality. For any $\varepsilon>0$ that is sufficiently small, by the
continuity of $V$, signal $\\{x+\varepsilon,y\\}$ also satisfies (3). Note
that the doctor’s payoff of signal $\\{x,y\\}$ after $\theta=0$ is given by

$\displaystyle\frac{x-\mu_{1}}{x-y}\cdot\bar{p}+\frac{\mu_{1}-y}{x-y}\cdot\underline{p}(x),$

which is strictly increasing with $x$. Then signal $\\{x+\varepsilon,y\\}$
dominates $\\{x,y\\}$, which contradicts with the optimality of $\\{x,y\\}$.

Next, I verify the best-good-news rule holds when (3) is binding. In fact, for
any $x>\mu_{e}$ such that $d(x)$ is well defined, $S_{P}(x)<0$ and
$P^{\prime}(\cdot)\geq 0$ almost everywhere. Thus,
$S_{P}(x)-P^{\prime}(d(x))<0$, and the best-good-news rule holds if and only
if

$\displaystyle\frac{S_{P}(x)-P^{\prime}(x)}{S_{P}(x)-P^{\prime}(d(x))}\geq\frac{S_{V}(x)-V^{\prime}(x)}{S_{V}(x)-V^{\prime}(d(x))}.$

This inequality holds by observing that

$\displaystyle\frac{S_{P}(x)-P^{\prime}(x)}{S_{P}(x)-P^{\prime}(d(x))}=\frac{\frac{\underline{p}(x)-\bar{p}}{x-d(x)}-(\underline{p}_{H}-\underline{p}_{L})}{\frac{\underline{p}(x)-\bar{p}}{x-d(x)}}=1+\frac{(\underline{p}_{H}-\underline{p}_{L})(x-d(x))}{\bar{p}-\underline{p}(x)}>1,$

and

$\displaystyle\frac{S_{V}(x)-V^{\prime}(x)}{S_{V}(x)-V^{\prime}(d(x))}=\frac{\frac{V(x)-V(0)}{x-d(x)}-V^{\prime}(x)}{\frac{V(x)-V(0)}{x-d(x)}}=1-V^{\prime}(x)\frac{x-d(x)}{V(x)-V(0)}<1,$

by the monotonicity of $V$.

For any $\mu_{1}\in\mathcal{D}$, when the upper belief is $1$, since
$\\{0,1\\}$ is optimal for rewarding the past among this subcategory and
$\\{0,1\\}$ is sufficient to motivate $a=1$, there exists a lower belief
$y\leq\mu_{e}$ such that $\\{1,y\\}$ makes the participation constraint (3)
holds with equality. Based on the best-good-news rule, given that the upper
belief attains its maximum and the participation constraint binds, $\\{1,y\\}$
is optimal. Thus, $V(y)=V(0)$ and therefore $y$ can be derived by equation

$\displaystyle\frac{1-\mu_{1}}{1-y}\cdot V(0)+\frac{\mu_{1}-y}{1-y}\cdot
V(1)=\bar{V}(\mu_{1}),$

which gives the required expression.

For $\mu_{1}\in\mathcal{M}/\mathcal{D}$, no information structure with the
upper belief being $1$ is sufficient to meet the participation constraint.
Note that for any $x>\mu_{e}$, the expected payoff of the patient, which is

$\displaystyle\frac{x-\mu_{1}}{x-y}\cdot\underbrace{V(y)}_{=V(0)}+\frac{\mu_{1}-y}{x-y}\cdot
V(x),$

is obviously a decreasing function of $y$ since $V(x)>V(0)$. Therefore, to
maximize $x$, $y$ must be minimized, and consequently the optimal signal is
given by equation

$\displaystyle\frac{1-\mu_{1}}{x}\cdot V(0)+\frac{\mu_{1}}{x}\cdot
V(x)=\bar{V}(\mu_{1}),$

which gives the required expression.

####  B.3 Proof of Corollary 2

It suffices to show that the best-good-news rule applies for (7) when the
participation constraint holds with equality. When $\mu\leq\mu_{e}$,

$\displaystyle
V(\mu)-V_{0}(\mu)=\alpha+(1-\alpha)\phi(\bar{p}-c)-\phi\big{(}\alpha+(1-\alpha)\underline{p}(\mu)\big{)}.$

The first derivative is obviously negative. When $\mu>\mu_{e}$,

$\displaystyle
V(\mu)-V_{0}(\mu)=\alpha+(1-\alpha)\phi(\underline{p}(\mu))-\phi\big{(}\alpha+(1-\alpha)\underline{p}(\mu)\big{)}.$

The first derivative is given by

$\displaystyle(1-\alpha)(p_{H}-p_{L})\big{(}\phi^{\prime}(\underline{p}(\mu))-\phi^{\prime}\big{(}\alpha+(1-\alpha)\underline{p}(\mu)\big{)},$

which is positive since
$\underline{p}(\mu)<\alpha+(1-\alpha)\underline{p}(\mu)$ and $\phi$ is
concave. Observe that for any $x>\mu_{e}$,
$V^{\prime}(x)-V_{0}^{\prime}(x)>0>V^{\prime}(d(x))-V_{0}^{\prime}(d(x))$, and
thus

$\displaystyle\frac{S_{V-V_{0}}(x)-V^{\prime}(x)+V_{0}^{\prime}(x)}{S_{V-V_{0}}(x)-V^{\prime}(d(x))+V_{0}^{\prime}(d(x))}<1.$

Also,

$\displaystyle\frac{S_{P}(x)-P^{\prime}(x)}{S_{P}(x)-P^{\prime}(d(x))}>1,$

and hence the by the same reasoning in Proposition 1, the best-good-news rule
applies, and all the results can be derived in a similar manner.

####  B.4 Proof of Proposition 2

If $a=0$, it is still optimal for the doctor to minimize the patient’s payoff.
Let

$\displaystyle\bar{V}(\mu)=-\text{cav}\circ\left(-\phi\big{(}\alpha+(1-\alpha)\underline{p}(\mu)\big{)}\right),$

and then by concavification $\bar{V}(\cdot)$ is the largest convex function
that is lower than function
$\phi\big{(}\alpha+(1-\alpha)\underline{p}(\mu)\big{)}$. In other words,
$\bar{V}(\cdot)$ is the lowest payoff that can be achieved when the patient
rejects the test.

Consider the following optimization problem:

$\displaystyle\max_{\tau\in\mathcal{R}(\mu_{1})}:$
$\displaystyle\int_{0}^{1}P(\mu)\tau_{1}^{1}(d\mu)$ (13)
$\displaystyle\text{subject to:
}\int_{0}^{1}\hat{V}(\mu)\tau(d\mu)\geq\bar{V}(\mu_{1}).$

Here, if $\tau$ satisfies the participation constraint, since
$\hat{V}(\mu)\geq V(\mu)$ for all $\mu\in[0,1]$, it must satisfy the
participation constraint of program (2). Thus, the solution of (13) is weakly
larger than the solution of (2).

Then it is sufficient to show that the solution of (13) can be achieved by a
signal that is feasible in persuading the patient with payoff function $V$.
Let $\mu_{H}$ and $\mu_{L}$ be the upper and lower beliefs in the signal of
problem (13) ($\mu_{H}>\mu_{e}\geq\mu_{L}$), and
$\pi=(\mu_{H}-\mu_{1})/(\mu_{H}-\mu_{L})$ is the probability of posterior
$\mu_{L}$. If $V(\mu)=\hat{V}(\mu)$, $\\{\mu_{H},\mu_{L}\\}$ is also an
optimum of program (2). If $V(\mu)<\hat{V}(\mu)$, by the definition of local
concavification and the Carathéodory theorem, there exists two posteriors
$\mu_{H}^{1}$ and $\mu_{H}^{2}$, both larger than $\mu_{e}$, and a probability
$\rho$, such that $\rho\mu_{H}^{2}+(1-\rho)\mu_{H}^{1}=\mu_{H}$ and $\rho
V(\mu_{H}^{2})+(1-\rho)V(\mu_{H}^{1})=\hat{V}(\mu_{H})$. That is, lottery of
posterior belief
$(\mu_{L},\pi;\mu_{H}^{2},(1-\pi)\rho;\mu_{H}^{1},(1-\pi)(1-\rho))$ is
feasible for program (2). By the piecewise linearity of $P$, it generates the
same health probability as in (13), which completes the proof.

####  B.5 Proof of Proposition 3

Suppose inequality (4) holds. If
$\mu_{0}\leq\min\\{\mu_{e},\mu_{\mathcal{F}}\\}$, non-disclosure is optimal
for guiding the future and sufficient to motivate $a=1$. Thus, non-disclosure
is optimal and guarantees $e=1$ with probability $1$. Consequently, there is
no need to send any messages in the ex ante disclosure as well.

Suppose $\mu_{0}>\min\\{\mu_{e},\mu_{\mathcal{F}}\\}$. By the proposed policy,
the health probability is given by

$P^{*}(\mu_{0})=\frac{1-\mu_{0}}{1-\min\\{\mu_{e},\mu_{\mathcal{F}}\\}}\cdot\bar{p}+\frac{\mu_{0}-\min\\{\mu_{e},\mu_{\mathcal{F}}\\}}{1-\min\\{\mu_{e},\mu_{\mathcal{F}}\\}}\cdot\underline{p}_{H}.$

By concavification, it suffices to show that for any
$\mu_{0}>\min\\{\mu_{e},\mu_{\mathcal{F}}\\}$, the consequent health
probability by the optimal interim disclosure derived in Proposition 1,
denoted by $\bar{P}(\mu_{0})$, is no larger than $P^{*}(\mu_{0})$. Here, since
the optimal interim disclosure changes continuously with interim belief
$\mu_{1}$, $\bar{P}(\cdot)$ is continuous.

First, if $\mu\in[\mu_{e},\mu_{\mathcal{F}}]$, the optimal interim disclosure
is $\\{1,\mu_{e}\\}=\\{1,\min\\{\mu_{e},\mu_{\mathcal{F}}\\}\\}$, which is
also optimal for guiding the future, and the health probability is exactly
$P^{*}(\mu)$.

Second, if $\mu\in\mathcal{M}/\mathcal{D}$, the optimal interim signal is with
perfect bad news. For any signal with perfect bad news
$\\{0,x\\}\in\mathcal{R}(\mu)$, the corresponding health probability is given
by

$\displaystyle\bar{P}(\mu)=\left(1-\frac{\mu}{x}\right)\cdot\bar{p}+\frac{\mu}{x}\cdot\underline{p}(x).$

The first-order derivative with respect to $x$ is
$\mu(\bar{p}-\underline{p}(\mu)+(\underline{p}_{H}-\underline{p}_{L})x)/x^{2}>0$,
and therefore

$\displaystyle\bar{P}(\mu)$
$\displaystyle<\left(1-\mu\right)\cdot\bar{p}+\mu\underline{p}_{H}$
$\displaystyle<\frac{1-\mu}{1-\min\\{\mu_{e},\mu_{\mathcal{F}}\\}}\cdot\bar{p}+\frac{\mu-\min\\{\mu_{e},\mu_{\mathcal{F}}\\}}{1-\min\\{\mu_{e},\mu_{\mathcal{F}}\\}}\cdot\underline{p}_{H}=P^{*}(\mu).$

Third, if
$\mu\in\mathcal{D}/\mathcal{F}=[\mu_{\mathcal{F}},\mu_{\mathcal{D}}]$, the
interim signal is with perfect good news, and thus the health probability is

$\displaystyle\frac{1-\mu}{1-l(\mu)}\cdot\bar{p}+\frac{\mu-l(\mu)}{1-l(\mu)}\cdot\underline{p}_{H},$

where $l(\cdot)$ is given by (5). By simple algebra, the health probability
can be written as

$\displaystyle\bar{P}(\mu)=\frac{V(1)-\bar{V}(\mu)}{V(1)-V(0)}\cdot\bar{p}+\frac{\bar{V}(\mu)-V(0)}{V(1)-V(0)}\cdot\underline{p}_{H}.$

Observe that $\bar{P}(\mu)$ is linear and continuous in $\mu$. Thus, since
$\mathcal{D}/\mathcal{F}$ is an interval, $\bar{P}(\mu)<P^{*}(\mu)$ for
$\mu\in\mathcal{D}/\mathcal{F}$ because
$\bar{P}(\mu_{\mathcal{F}})=P^{*}(\mu_{\mathcal{F}})$ but
$\bar{P}(\mu_{\mathcal{D}})<P^{*}(\mu_{\mathcal{D}})$.

Last, if $\mu\notin\mathcal{M}$, the result holds directly since
$\bar{P}(\cdot)$ is increasing and $P^{*}(1)=\bar{P}(1)$, which completes the
proof when (4) holds.

Suppose now that (4) does not hold. Then for any
$\mu\in[\mu_{\mathcal{M}}^{L},\mu_{\mathcal{M}}^{H}]$,

$\displaystyle\bar{P}(\mu)=\left(1-\frac{\mu}{h(\mu)}\right)\cdot\bar{p}+\frac{\mu}{h(\mu)}\cdot\underline{p}(h(\mu)),$

where $h(\mu)$ is derived by equation (6). By implicit differentiation,
equation (6) indicates

$\displaystyle(h^{\prime}(\mu)-1)\cdot V(0)+V(h(\mu))+\mu
V^{\prime}(h(\mu))h^{\prime}(\mu)=\bar{V}(\mu)h(\mu)+h^{\prime}(\mu)\bar{V}(\mu),$

which yields

$h^{\prime}(\mu)=\frac{h(\mu)}{\mu}\cdot\frac{V(h(\mu))-V(0)-\bar{V}_{N}^{\prime}(\mu)\cdot
h(\mu)}{V(h(\mu))-V(0)-V^{\prime}(h(\mu))\cdot
h(\mu)}\equiv\frac{h(\mu)}{\mu}\cdot\frac{D_{2}(\mu)}{D_{1}(\mu)},$ (14)

where

$\displaystyle\begin{array}[]{ll}D_{1}(\mu)=V(h(\mu))-V(0)-V^{\prime}(h(\mu))\cdot
h(\mu),\text{ and}\\\ D_{2}(\mu)=V(h(\mu))-V(0)-\bar{V}^{\prime}(\mu)\cdot
h(\mu).\end{array}$

By the mean-value theorem and the concavity of $V$, $D_{1}(\mu)>0$.

The second-order derivative is given by

$\displaystyle\frac{\mu
h^{\prime}(\mu)^{2}\left(h(\mu)^{2}\left(-V^{\prime\prime}(h(\mu))\right)+2h(\mu)V^{\prime}(h(\mu))-2V(h(\mu))+2V(0)\right)}{h(\mu)}$
$\displaystyle=\left(h(\mu)V^{\prime}(h(\mu))-V(h(\mu))+V(0)\right)\left(\mu
h^{\prime\prime}(\mu)+2h^{\prime}(\mu)\right).$

Substituting $h^{\prime}(\mu)$ with expression (14), we have

$h^{\prime\prime}(\mu)=\frac{D_{2}(\mu)h(\mu)\left(2D_{1}(\mu)(D_{2}(\mu)-D_{1}(\mu))+D_{2}(\mu)h(\mu)^{2}V^{\prime\prime}(h(\mu))\right)}{\mu^{2}D_{1}(\mu)^{3}}.$
(15)

We have

$\displaystyle\bar{P}^{\prime\prime}(\mu)=$
$\displaystyle(1-\alpha)\bigg{[}\frac{2h(\mu)h^{\prime}(\mu)-2\mu
h^{\prime}(\mu)^{2}+\mu h(\mu)h^{\prime\prime}(\mu)}{h(\mu)^{3}}\cdot\bar{p}$
$\displaystyle+\frac{\left(h(\mu)\underline{p}^{\prime}(h(\mu))-\underline{p}(h(\mu))\right)\left(\mu
h(\mu)h^{\prime\prime}(\mu)+2h^{\prime}(\mu)\left(h(\mu)-\mu
h^{\prime}(\mu)\right)\right)}{h(\mu)^{3}}\bigg{]}.$

Substituting $h^{\prime}(\mu)$ and $h^{\prime\prime}(\mu)$ by (14) and (15),
and we have

$\displaystyle\bar{P}^{\prime\prime}(\mu)=(1-\alpha)\bigg{[}\frac{D_{2}(\mu)^{2}h(\mu)V^{\prime\prime}(h(\mu))}{\mu
D_{1}(\mu)^{3}}\cdot\underbrace{\left(h(\mu)\underline{p}^{\prime}(h(\mu))+\bar{p}-\underline{p}(h(\mu))\right)}_{>0}\bigg{]}.$

Since $D_{1}(\mu)>0$, $\bar{P}^{\prime\prime}(\mu)<0$.

When $\mu\notin[\mu_{\mathcal{M}}^{L},\mu_{\mathcal{M}}^{H}]$, the patient
rejects the test and therefore $\bar{P}(\mu)=\underline{p}(\mu)$, which is
increasing and linear. Also, $\bar{P}(\cdot)$ is continuous at
$\mu_{\mathcal{M}}^{H}$ but is not at $\mu_{\mathcal{M}}^{L}$. By
concavification, the optimal signal must have the proposed structure.

####  B.6 Proof of Lemma 3

Suppose there exists an optimal policy such that the ex ante signal has $N\geq
2$ different posteriors, i.e.,
$\text{Supp}(\tau_{0})=\\{\mu^{1},\mu^{2},\dots,\mu^{N}\\}$, where the
probability of posterior $\mu^{i}$ is $\pi_{i}$ and the subsequent interim
policy of interim belief $\mu^{i}$ is $\tau_{i}\in\mathcal{R}(\mu^{i})$ for
every $i=1,2,\dots,N$. Among the posteriors, suppose also that $\mu^{i}$ is
sufficient to motivate $a=1$ if and only if $i\leq n$ ($n<N$), and then the
policy satisfies the ex ante participation constraint if and only if

$\displaystyle\sum_{i=1}^{n}\pi_{i}\int_{0}^{1}V(\mu)\tau_{i}(d\mu)+\sum_{i={n+1}}^{N}\pi_{i}V_{0}(\mu^{i})\geq
V_{0}\left(\sum_{i=1}^{n}\pi_{i}\mu^{i}\right).$

The corresponding health probability is given by

$\displaystyle\sum_{i=1}^{n}\pi_{i}\int_{0}^{1}P(\mu)\tau_{i}(d\mu)+\sum_{i={n+1}}^{N}\pi_{i}\big{(}\alpha+(1-\alpha)\underline{p}(\mu^{i})\big{)}.$

Define another information policy with ex ante signal $\\{\mu_{H},\mu_{L}\\}$,
where

$\displaystyle\mu_{L}=\frac{\sum_{i=1}^{n}\pi_{i}\mu^{i}}{\sum_{i=1}^{n}\pi_{i}},\text{
and }\mu_{H}=\frac{\sum_{i=n+1}^{N}\pi_{i}\mu^{i}}{\sum_{i=n+1}^{N}\pi_{i}},$

and the corresponding interim policies are respectively

$\displaystyle\tau_{L}=\sum_{i=1}^{n}\pi_{i}\tau_{i},\text{ and
}\tau_{H}=\sum_{i=n+1}^{N}\pi_{i}\tau_{i}.$

Then by the linearity of the reduction operator, the new policy is well-
defined, and it generates exactly the same health probability since $P$ is
piecewise linear. Meanwhile, since $V_{0}$ is piecewise concave, the new
policy yields a higher ex ante payoff for the patient since it conveys less
information. Thus, the new signal also meets the participation constraint, and
consequently $\\{\mu_{H},\mu_{L}\\}$ is also optimal.

####  B.7 Proof of Proposition 4

I complete the proof in six steps.

Step 1: There exists an informative information policy that meets the ex ante
participation constraint if and only if $\mu_{0}<\mu_{\mathcal{V}}$.

For the if part, consider the information policy such that
$\tau_{0}\in\\{0,\mu_{\mathcal{V}}\\}$ and no information is disclosed in the
interim disclosure. This policy is informative when
$\mu_{0}<\mu_{\mathcal{V}}$ and it satisfies the ex ante participation
constraint.

For the only if part, it suffices to show that there is no informative policy
can yield a strictly higher payoff for the patient than non-disclosure when
$\mu_{0}\geq\mu_{\mathcal{V}}$. The patient’s highest payoff by an information
policy can be solved by backward induction. That is, for any interim belief
$\mu_{1}$ such that the patient is willing to accept the test there, the
interim disclosure is optimal for rewarding the past. If we let
$V^{*}(\mu_{1})$ denote that patient’s payoff function under the signal,

$\displaystyle
V^{*}(\mu_{1})=\left\\{\begin{array}[]{cc}\left(1-\frac{\mu_{1}}{\mu_{v}}\right)\cdot
V(0)+\frac{\mu_{1}}{\mu_{v}}\cdot V(\mu_{v})&\mu_{1}<\mu_{v},\\\
V(\mu)&\mu_{1}\geq\mu_{v}.\end{array}\right.$

Then if there is no ex ante disclosure, the highest payoff that can be
attained by any information policy at belief $\mu_{0}$ is
$V^{**}(\mu_{0})=\max\\{V^{*}(\mu_{0}),V_{0}(\mu_{0})\\}$. Thus, the payoff of
any information policy is no higher than $\text{cav}\circ V^{**}(\mu_{0})$.
Also, note that $V^{*}(\mu_{v})=V(\mu_{v})<V_{0}(\mu_{v})$, and the statement
holds by observing that the slope of the line connecting
$(0,\alpha+(1-\alpha)\phi(\bar{p}-c))$ and $(\mu_{v},V(\mu_{v}))$ is smaller
than that of the line connecting $(0,\alpha+(1-\alpha)\phi(\bar{p}-c))$ and
$(\mu_{v},V_{0}(\mu_{v}))$, and the latter is further smaller than the slope
of the line connecting $(0,\alpha+(1-\alpha)\phi(\bar{p}-c))$ and
$(\mu_{\mathcal{V}},V_{0}(\mu_{\mathcal{V}}))$.

Step 2: The optimal disclosure when $\mu_{0}<\mu_{\mathcal{N}}$ is non
disclosure.

The unconstrained optimum without the ex ante participation constraint is
still optimal when it satisfies the ex ante participation constraint (8). When
inequality (4) holds, the ex ante payoff of this policy is

$\displaystyle\left\\{\begin{array}[]{cc}V(0)&\mu_{0}<\min\\{\mu_{e},\mu_{\mathcal{F}}\\}\\\
\frac{(\mu-\mu_{e})V(1)+(1-\mu)V(0)}{1-\mu_{e}}&\mu_{e}\leq\mu\leq\mu_{\mathcal{F}}\\\
\bar{V}(\mu)&\mu\geq\mu_{\mathcal{F}}\end{array}\right..$

By the concavity of $V_{0}$ and the fact that $V_{0}(1)=\bar{V}(1)$, there
exists a unique intersection $\mu_{\mathcal{N}}^{\prime}$ such that the
unconstrained optimum is sufficient for the ex ante participation if and only
if $\mu_{0}<\mu_{\mathcal{N}}^{\prime}$. To show that
$\mu_{\mathcal{N}}^{\prime}=\mu_{\mathcal{N}}$, it suffices to show that
$\mu_{\mathcal{N}}<\min\\{\mu_{\mathcal{F}},\mu_{e}\\}$. If
$\mu_{e}\geq\mu_{\mathcal{F}}$, $\mu_{\mathcal{N}}<\mu_{\mathcal{F}}$ since

$\displaystyle\bar{V}(\mu_{\mathcal{F}})=\alpha+(1-\alpha)\phi(\bar{p}-c)=V_{0}(\mu_{\mathcal{N}})>\bar{V}(\mu_{\mathcal{N}}).$

If $\mu_{e}<\mu_{\mathcal{F}}$, $\mu_{\mathcal{N}}<\mu_{e}$ since

$\displaystyle
V(\mu_{e})=\alpha+(1-\alpha)\phi(\bar{p}-c)=V_{0}(\mu_{\mathcal{N}})>V(\mu_{\mathcal{N}}).$

In conclusion, when $\mu_{0}<\mu_{\mathcal{N}}$, the unconstrained optimum is
non-disclosure and it meets the ex ante participation constraint.

Step 3: If $\mu_{0}\in(\mu_{\mathcal{N}},\mu_{\mathcal{V}})$, the interim
signal in the optimal policy is also at-most-binary, and it is either with
perfect good news or with perfect bad news.

When $\mu_{0}\in(\mu_{\mathcal{N}},\mu_{\mathcal{V}})$, since the
unconstrained optimum is no longer able to motivate ex ante participation, the
ex ante participation constraint (8) must be binding. Also, by Step 3, if at
interim belief $\mu_{H}$ the doctor recommends the patient not to take the
test, then using non-disclosure in the interim disclosure is optimal for both
guiding the future and rewarding the past. Thus, the optimization problem can
be expressed as follows

$\displaystyle\max_{\begin{subarray}{c}\mu_{H},\mu_{L}\in[0,1]\\\
\tau\in\mathcal{R}(\mu_{L})\end{subarray}}:\,\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot$
$\displaystyle\int_{0}^{1}P(\mu)\tau(d\mu)+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\underline{p}(\mu_{H})$
subject to: $\displaystyle\int_{0}^{1}V(\mu)\tau(d\mu)\geq\bar{V}(\mu_{L})$
$\displaystyle V(\mu_{H})\leq V_{0}(\mu_{H})$
$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\int_{0}^{1}V(\mu)\tau(d\mu)+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot
V_{0}(\mu_{H})=V_{0}(\mu_{0}).$

Since $V(\mu_{H})<V_{0}(\mu_{H})$ and

$\displaystyle\int_{0}^{1}V(\mu)\tau(d\mu)=\frac{\mu_{H}-\mu_{L}}{\mu_{H}-\mu_{0}}\cdot\left(V_{0}(\mu_{0})-\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot
V_{0}(\mu_{H})\right)\geq V_{0}(\mu_{L})\geq\bar{V}(\mu_{L}),$

the first two constraints must be not binding in the optimum. Thus, if
$\langle\mu_{H},\mu_{L},\tau\rangle$ is the optimal policy, then $\tau$, the
interim signal, must be a solution to problem

$\displaystyle\max_{\tau\in\mathcal{R}(\mu_{L})}:$
$\displaystyle\,\int_{0}^{1}P(\mu)\tau(d\mu)$ $\displaystyle\text{subject to:
}\int_{0}^{1}V(\mu)\tau(d\mu)=\frac{\mu_{H}-\mu_{L}}{\mu_{H}-\mu_{0}}\cdot
V(\mu_{0})-\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{0}}\cdot V(\mu_{H}).$

Similar with Proposition 1, the optimal signal is at-most-binary. Also, it can
be solved by the best-good-news rule, and thus it is either with perfect good
news or with perfect bad news.

Step 4: If there is no interim disclosure
($\tau(\mu_{1})\equiv\\{\mu_{1}\\}$), then the optimal ex ante disclosure can
be derived by the best-good-news rule.

If there is no interim disclosure, the optimization problem becomes

$\displaystyle\max_{\begin{subarray}{c}\mu_{H},\mu_{L}\in[0,1]\\\
\tau\in\mathcal{R}(\mu_{L})\end{subarray}}:\,\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot$
$\displaystyle
P(\mu)+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\underline{p}(\mu_{H})$
subject to: $\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot
V(\mu_{L})+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot
V_{0}(\mu_{H})=V_{0}(\mu_{0}).$

By the concavity of $V_{0}$, we must have $V(\mu_{L})>V_{0}(\mu_{L})$, which
indicates $\mu_{L}<\mu_{\mathcal{N}}$. That is,
$V(\mu_{L})=\mathcal{V}(\mu_{L})$ and $V(\mu_{H})=\mathcal{V}(\mu_{H})$. By
the same reasoning in Proposition 1, the best-good-news rule holds.

Step 5: The optimal disclosure when
$\mu_{0}\in(\mu_{\mathcal{N}},\mu_{\mathcal{T}})$.

When $\mu_{0}\in(\mu_{\mathcal{N}},\mu_{\mathcal{T}})$, by the proposed
information policy, the doctor commits to disclose no information after the
test. In the ex ante signal, the upper belief is $1$ and the lower belief,
denoted as $\mu_{L}^{*}$, is given by (9). Then the ex ante signal satisfies
the ex ante participation constraint with equality.

The optimality of this signal is proved by contradiction. Suppose there exists
another policy $\langle\mu_{H},\mu_{L},\tau\rangle$ that strictly dominates
the proposed policy. Then, the health probability must exceeds

$\frac{1-\mu_{0}}{1-\mu_{L}^{*}}\cdot\bar{p}+\frac{1-\mu_{0}}{1-\mu_{L}^{*}}\cdot\underline{p}_{H}.$
(16)

Then I show that it cannot satisfy the ex ante participation constraint.

First, by the best-good-news rule, there is no ex ante signal in
$\mathcal{R}(\mu_{0})$ that can dominate the proposed policy without the
interim disclosure.

Second, suppose $\mu_{L}<\mu_{L}^{*}$, then the health probability is given by

$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot$
$\displaystyle\int_{0}^{1}P(\mu)\tau(d\mu)+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\underline{p}(\mu_{H})$
$\displaystyle\leq\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\bar{p}+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\underline{p}(\mu_{H})\leq\frac{1-\mu_{0}}{1-\mu^{*}_{L}}\cdot\bar{p}+\frac{\mu_{0}-\mu_{L}^{*}}{1-\mu^{*}_{L}}\cdot\underline{p}_{H}.$

That is, even though the policy is implementable, the health probability of
such a triple $\langle\mu_{H},\mu_{L},\tau\rangle$ is still unable to exceed
the proposed policy.

Third, if $\mu_{L}$ is larger than $\mu_{L}^{*}$ and $\tau$ is informative,
then by Step 4, $\tau$ is with perfect news. If it is with perfect bad news,
i.e., $\tau=\\{0,\mu_{h}\\}$, then the health probability is

$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\int_{0}^{1}P(\mu)\tau(d\mu)+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\underline{p}(\mu_{H})$
$\displaystyle=$
$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\left(\frac{\mu_{h}-\mu_{L}}{\mu_{h}}\cdot\bar{p}+\frac{\mu_{L}}{\mu_{h}}\cdot\underline{p}(\mu_{h})\right)+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\underline{p}(\mu_{H})$
$\displaystyle<$
$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\left((1-\mu_{L})\bar{p}+\mu_{L}\underline{p}_{H}\right)+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\underline{p}(\mu_{H})$
$\displaystyle<$
$\displaystyle\frac{1-\mu_{0}}{1-\mu_{L}^{*}}\cdot\bar{p}+\frac{1-\mu_{0}}{1-\mu_{L}^{*}}\cdot\underline{p}_{H}.$

Thus, if the policy that dominates the proposed policy exists, then $\mu_{L}$
must be larger than $\mu_{L}^{*}$ and $\tau$ must be with perfect good news. I
denote $\mu_{l}$ as the lower belief, and by this signal the health
probability is given by

$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\bigg{(}\frac{1-\mu_{L}}{1-\mu_{l}}\cdot\bar{p}+\frac{\mu_{L}-\mu_{l}}{1-\mu_{l}}\cdot\underline{p}_{H}\bigg{)}+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\underline{p}(\mu_{H}).$

If this expression exceeds (16), then we must have

$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\frac{1-\mu_{L}}{1-\mu_{l}}\geq\frac{1-\mu_{0}}{1-\mu_{L}^{*}},$

which requires

$\mu_{l}\geq
1-\frac{(\mu_{H}-\mu_{0})(1-\mu_{L})(1-\mu_{L}^{*})}{(\mu_{H}-\mu_{L})(1-\mu_{0})}.$
(17)

However, if (17) holds, the patient’s ex ante payoff is given by

$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\bigg{(}\frac{1-\mu_{L}}{1-\mu_{l}}\cdot
V(0)+\frac{\mu_{L}-\mu_{l}}{1-\mu_{l}}\cdot
V(1)\bigg{)}+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\mathcal{V}(\mu_{H}),$

which is decreasing with respect to $\mu_{l}$. Thus, substituting $\mu_{l}$ by
expression (17), the ex ante payoff is no larger than

$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\bigg{[}\frac{(\mu_{H}-\mu_{L})(1-\mu_{0})}{(1-\mu_{L}^{*})(\mu_{H}-\mu_{0})}\cdot
V(0)+\left(1-\frac{(\mu_{H}-\mu_{L})(1-\mu_{0})}{(1-\mu_{L}^{*})(\mu_{H}-\mu_{0})}\right)\cdot
V(1)\bigg{]}$
$\displaystyle+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\mathcal{V}(\mu_{H})$
$\displaystyle=$ $\displaystyle\frac{1-\mu_{0}}{1-\mu_{L}^{*}}\cdot
V(0)+\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\left(1-\frac{(\mu_{H}-\mu_{L})(1-\mu_{0})}{(1-\mu_{L}^{*})(\mu_{H}-\mu_{0})}\right)\cdot
V(1)+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\mathcal{V}(\mu_{H}).$

Since $V(1)<\mathcal{V}(1)$ and $\mathcal{V}(\mu_{H})\leq V(1)$, this
expression is smaller than

$\displaystyle\frac{1-\mu_{0}}{1-\mu_{L}^{*}}\cdot
V(0)+\frac{\mu_{0}-\mu_{L}^{*}}{1-\mu_{L}^{*}}\cdot\mathcal{V}(1)$
$\displaystyle=\frac{1-\mu_{0}}{1-\mu_{L}^{*}}\cdot\mathcal{V}(0)+\frac{\mu_{0}-\mu_{L}^{*}}{1-\mu_{L}^{*}}\cdot\mathcal{V}(1)$
$\displaystyle=\frac{1-\mu_{0}}{1-\mu_{L}^{*}}\cdot\mathcal{V}(\mu_{L}^{*})+\frac{\mu_{0}-\mu_{L}^{*}}{1-\mu_{L}^{*}}\cdot\mathcal{V}(1)<\mathcal{V}(\mu_{0}),$

which yields a contradiction.

Step :6 The optimal disclosure when
$\mu_{0}\in(\mu_{\mathcal{T}},\mu_{\mathcal{V}})$.

When $\mu_{0}\in(\mu_{\mathcal{T}},\mu_{\mathcal{V}})$, the proposed policy
discloses no information in the interim disclosure and the ex ante signal is
$\\{0,\mu_{H}^{*}\\}$ for some $\mu_{H}^{*}>\mu_{\mathcal{V}}$. Thus, the
patient’s ex ante payoff is

$\displaystyle\left(1-\frac{\mu_{0}}{\mu^{*}_{H}}\right)\cdot
V(0)+\frac{\mu_{0}}{\mu_{H}^{*}}\cdot\mathcal{V}(\mu^{*}_{H})=\mathcal{V}(\mu_{0}),$

and the corresponding health probability is

$\displaystyle\left(1-\frac{\mu_{0}}{\mu^{*}_{H}}\right)\cdot\bar{p}+\frac{\mu_{0}}{\mu_{H}^{*}}\cdot\underline{p}(\mu_{H}^{*}).$

I prove by contradiction that there exists no policy that satisfies the
constraints and yields a higher health probability. Suppose not, and then
denote the optimal signal by triple $\langle\mu_{H},\mu_{L},\tau\rangle$.

First, if $\tau$ is non-disclosure, by the best-good-news rule, it cannot be a
better policy.

Second, if $\tau$ is informative, by Step 4, it is either with perfect goods
news or with perfect bad news. Suppose first it is with perfect bad news. Then
the interim signal is $\\{0,\mu_{h}\\}$ for some $\mu_{h}>\mu_{e}$. Therefore,
the health probability is

$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\left(\frac{\mu_{L}}{\mu_{h}}\cdot\underline{p}(\mu_{h})+\left(1-\frac{\mu_{L}}{\mu_{h}}\right)\cdot\bar{p}\right)+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\underline{p}(\mu_{H})$
$\displaystyle=$
$\displaystyle\underbrace{\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\left(1-\frac{\mu_{L}}{\mu_{h}}\right)}_{\equiv\pi}\cdot\bar{p}+\left[\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\frac{\mu_{L}}{\mu_{h}}\cdot\mu_{h}+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\mu_{H}\right]\cdot\underline{p}_{H}$
$\displaystyle+\left[\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\frac{\mu_{L}}{\mu_{h}}\cdot(1-\mu_{h})+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot(1-\mu_{H})\right]\cdot\underline{p}_{L}$
$\displaystyle=$
$\displaystyle\pi\cdot\bar{p}+(1-\pi)p\left(\frac{\mu_{0}}{1-\pi}\right).$

Thus, this is also the health probability if the doctor discloses no
information in the interim disclosure and sending ex ante signal
$\\{0,\mu_{0}/(1-\pi)\\}$. Compare the two policies with an identical health
policy, the ex ante payoff of the patient under the latter policy is

$\displaystyle\pi\cdot\mathcal{V}(0)+(1-\pi)\cdot\mathcal{V}\left(\frac{\mu_{0}}{1-\pi}\right)$
$\displaystyle\geq$
$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\left(\frac{\mu_{L}}{\mu_{h}}\cdot\mathcal{V}(\mu_{h})+\left(1-\frac{\mu_{L}}{\mu_{h}}\right)\cdot
V(0)\right)+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\mathcal{V}(\mu_{H})$
$\displaystyle\geq$
$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\left(\frac{\mu_{L}}{\mu_{h}}\cdot
V(\mu_{h})+\left(1-\frac{\mu_{L}}{\mu_{h}}\right)\cdot
V(0)\right)+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\mathcal{V}(\mu_{H}),$

which is exactly the ex ante payoff of the former policy. Since
$\\{0,\mu_{h}\\}$ cannot be better than $\\{0,\mu_{H}^{*}\\}$ if it is
implementable, $\langle\mu_{H},\mu_{L},\tau\rangle$ cannot, either.

Finally, if $\tau$ is informative and $\tau$ is with perfect good news, then
the maximization problem becomes:

$\displaystyle\max_{\mu_{H},\mu_{L},\mu_{l}}:\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot$
$\displaystyle\left(\frac{1-\mu_{L}}{1-\mu_{l}}\cdot\bar{p}+\frac{\mu_{L}-\mu_{l}}{1-\mu_{l}}\cdot\underline{p}_{H}\right)+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\underline{p}(\mu_{H})$
subject to:
$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}\cdot\left(\frac{1-\mu_{L}}{1-\mu_{l}}\cdot
V(0)+\frac{\mu_{L}-\mu_{l}}{1-\mu_{l}}\cdot V(1)\right)$
$\displaystyle+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\mathcal{V}(\mu_{H})=\mathcal{V}(\mu_{0}),$

where $\mu_{l}$ is the lower belief of the interim signal. Solving the
constraint directly, we get

$\displaystyle\mu_{l}=1-\frac{(1-\mu_{L})(\mu_{H}-\mu_{0})(V(1)-V(0))}{\mathcal{V}(\mu_{H})(\mu_{0}-\mu_{L})+(\mu_{H}-\mu_{0})V(1)-(\mu_{H}-\mu_{L})\mathcal{V}(\mu_{0})}.$

Substituting it to the objective function, it becomes

$\displaystyle\frac{\mu_{H}-\mu_{0}}{\mu_{H}-\mu_{L}}$
$\displaystyle\cdot\frac{V(1)\bar{p}-V(0)\underline{p}_{H}}{V(1)-V(0)}$
$\displaystyle+\frac{\mu_{0}-\mu_{L}}{\mu_{H}-\mu_{L}}\cdot\left(\frac{(\bar{p}-\underline{p}_{H})\mathcal{V}(\mu_{H})}{V(1)-V(0)}+\underline{p}(\mu_{H})\right)-\frac{\bar{p}-\underline{p}_{H}}{V(1)-V(0)}\cdot\mathcal{V}(\mu_{0}).$

The first derivative with respect to $\mu_{L}$ is given by

$\displaystyle\frac{\mu_{H}-\mu_{0}}{(\mu_{H}-\mu_{L})^{2}(V(1)-V(0)}\cdot\big{(}(\bar{p}-\underline{p}(\mu_{H}))\cdot
V(1)-(\bar{p}-\underline{p}_{H})\mathcal{V}(\mu_{H})-(\underline{p}_{H}-\underline{p}(\mu_{H}))V(0)\big{)},$

which is positive if and only if

$\displaystyle\mathcal{V}(\mu_{H})<\frac{\bar{p}-\underline{p}(\mu_{H})}{\bar{p}-\underline{p}_{H}}\cdot
V(1)-\frac{\underline{p}_{H}-\underline{p}(\mu_{H})}{\bar{p}-\underline{p}_{H}}\cdot
V(0).$

Since the right-hand side of the inequality is decreasing with $\mu_{H}$, it
holds when $\mu_{H}$ is sufficiently small. Note that the criterion is
irrelevant with $\mu_{L}$, in the optimal policy $\mu_{L}$ is either $0$ or
the highest $\mu_{L}$ that makes the ex ante participation constraint binds.
In the latter case, the interim signal must be full disclosure ($\mu_{l}=0$)
because the first derivative of $\mu_{l}$ with respect to $\mu_{L}$ is
proportional to

$\displaystyle(\mu_{H}-\mu_{0})V(1)+(1-\mu_{H})\mathcal{V}(\mu_{0})-(1-\mu_{0})\mathcal{V}(\mu_{H})$
$\displaystyle<(\mu_{H}-\mu_{0})\mathcal{V}(1)+(1-\mu_{H})\mathcal{V}(\mu_{0})-(1-\mu_{0})\mathcal{V}(\mu_{H})<0.$

However, full disclosure is also with perfect bad news, and therefore by the
former discussion that $\tau$ is with perfect good news, it cannot be optimal.
Therefore, $\mu_{L}=0$, and consequently the proposed policy is optimal by the
fact that the ex ante participation constraint is binding, which completes the
proof.

###  Appendix C The Optimal Policy in Ely and Szydlowski (2020)

To have a better understanding about the optimal policy, let us consider an
alternative setting such that there is no information avoidance ($\phi(v)=v$)
but taking the test incurs a fixed fee $\psi$. Then the patient’s payoff
functions when $a=1$ and $a=0$ become

$\displaystyle\tilde{V}(\mu)=\left\\{\begin{array}[]{cc}\alpha+(1-\alpha)(\bar{p}-c)&\mu\leq\mu_{e}\\\
\alpha+(1-\alpha)\underline{p}(\mu)&\mu>\mu_{e}\end{array}\right.\text{ and
}\tilde{V}_{0}(\mu)=\alpha+(1-\alpha)\underline{p}(\mu),$

respectively, and the optimization problem (2) shrinks to:

$\displaystyle\max_{\tau_{1}^{1},\tau_{1}^{0}\in\mathcal{R}(\mu_{1})}:\,$
$\displaystyle\int_{0}^{1}P(\mu)\tau_{1}^{1}(d\mu)$
$\displaystyle\text{subject to:
}\int_{0}^{1}\tilde{V}(\mu)\tau_{1}^{1}(d\mu)-\psi\geq\int_{0}^{1}\tilde{V}_{0}(\mu)\tau_{1}^{0}(d\mu).$

Still, the optimal policy can be derived by backward induction.

In fact, this setting is closely similar to the two-period model of Ely and
Szydlowski (2020), with the only exception that the principal (doctor) always
prefers the agent (patient) to hold a pessimistic belief and therefore there
is no need to “move the goalpost”.

######  Proposition 5.

In the ES model, identify

$\displaystyle\tilde{\mu}_{\mathcal{M}}=\frac{\bar{p}-c-\psi-\underline{p}_{L}}{\bar{p}-c-\underline{p}_{L}}\text{
and
}\tilde{\mu}_{\mathcal{F}}=\frac{\bar{p}-c-\psi-\underline{p}_{L}}{\underline{p}_{H}-\underline{p}_{L}},$

such that the following statements hold.

  * •

The doctor benefits from the interim disclosure if and only if
$\mu_{1}\leq\tilde{\mu}_{\mathcal{M}}$.

  * •

The optimal interim disclosure is $\\{\mu_{1}\\}$ (non-disclosure) if
$\mu_{1}\leq\tilde{\mu}_{\mathcal{F}}$, and is $\\{1,l(\mu_{1})$ if
$\mu_{1}\in(\tilde{\mu}_{\mathcal{F}},\tilde{\mu}_{\mathcal{M}}]$, where
$l(\cdot)$ is given by (5).

  * •

If $\bar{p}-c\geq\underline{p}_{L}$, the optimal ex ante disclosure is
$\\{\tilde{\mu}_{\mathcal{F}}\\}$ if $\mu_{0}\leq\tilde{\mu}_{\mathcal{F}}$,
and is $\\{1,\tilde{\mu}_{\mathcal{F}}\\}$ otherwise. If
$\bar{p}-c<\underline{p}_{L}$, the patient will never take the test.

###### Proof.

The reasoning in Lemma 1 also applies here, and therefore I only focus on
binary signals. Then it is straightforward that $\tilde{V}$ is convex, and the
optimal signal for rewarding the past is full disclosure. Also, the optimal
signal for guiding the future is still $\\{\mu_{e},1\\}$ when
$\mu_{0}>\mu_{e}$ and non-disclosure otherwise. If the optimal signal for
guiding the future is sufficient to generate $a=1$, it is optimal. If
$\bar{p}-c<\underline{p}_{L}$, then it is impossible to satisfy this
condition, and so is any $E_{\tau}\tilde{V}$ ($\tau\in\mathcal{R}(\mu_{1})$)
by concavification. If $\bar{p}-c\geq\underline{p}_{L}$, the optimal signal
for guiding the future satisfies the participation constraint if and only if
$\mu_{1}<\tilde{\mu}_{\mathcal{F}}<\mu_{e}$.

By Doval and Skreta (2024, Theorem 3.2 and 3.3), there exists a Lagrangian
multiplier $\lambda\geq 0$ such that the optimal interim disclosure
concavifies $P(\cdot)+\lambda\tilde{V}(\cdot)$. Since both $P$ and $\tilde{V}$
are with perfect good news, the optimal interim signal must also be with
perfect good news when $\lambda>0$. Let $y$ be the lower belief, and then if
$\lambda>0$, the participation constraint is binding, which indicates

$\displaystyle\frac{1-\mu_{1}}{1-y}\tilde{V}(y)+\frac{\mu_{1}-y}{1-\mu_{1}}\tilde{V}(1)=\tilde{V}_{0}(\mu_{1})+\psi\Rightarrow
y=\frac{(1-\mu_{1})(\bar{p}-c-\underline{p}_{L})-\psi}{(1-\mu_{1})(\underline{p}_{H}-\underline{p}_{L})-\psi}.$

Also, the interim disclosure is sufficed to generate $a=1$ if and only if full
disclosure is able to do this, which indicates

$\displaystyle(1-\mu_{1}))\tilde{V}(0)+\mu_{1}\tilde{V}(1)\Leftrightarrow\mu_{1}<\tilde{\mu}_{\mathcal{M}}.$

Now consider the ex ante disclosure. By the optimal interim disclosure,
$P^{*}(\mu_{1})$ is $\bar{p}$ when $\mu_{1}\leq\tilde{\mu}_{\mathcal{F}}$, and
is $\underline{p}(\mu)$ when $\mu_{1}>\tilde{\mu}_{\mathcal{M}}$. When
$\mu_{1}\in[\tilde{\mu}_{\mathcal{F}},\tilde{\mu}_{\mathcal{M}}]$,

$\displaystyle
P^{*}(\mu)=\frac{1-\mu_{1}}{1-y}\cdot\bar{p}+\frac{\mu_{1}-y}{1-y}\cdot\underline{p}_{H}=\frac{(\bar{p}-\underline{p}_{H})(\underline{p}_{H}\mu_{1}+\underline{p}_{L}(1-\mu_{1})+\psi)-\underline{p}_{H}c}{\bar{p}-c-\underline{p}_{H}}.$

Note that in this interval $P^{*}$ is decreasing
($\bar{p}-c<\underline{p}_{H}$) and linear in $\mu_{1}$, and
$P^{*}(\tilde{\mu}_{\mathcal{F}})=\bar{p}$. Thus, the optimal ex ante
disclosure is $\\{\mu_{0}\\}$ if $\mu_{0}\leq\tilde{\mu}_{\mathcal{F}}$ and is
$\\{\tilde{\mu}_{\mathcal{F}},1\\}$ otherwise. ∎

###  Appendix D Optimal Test

In this appendix, I consider an alternative setting in which the doctor
directly sends the test result instead of other complementary information to
maximize the health probability. This setting is highly consistent with the
former setting and can constitute a direct comparison with the result in
Schweizer and Szech (2018).

Assume both parties do not observe $\theta$, the patient’s initial health
status, at first. Let $\alpha_{0}\in(0,1)$ be the common prior that
$\theta=1$. The test result ($\theta=0$ or $1$) can only be privately observed
by the doctor. Simultaneously, all the other aforementioned parameters,
$\bar{p}$, $\underline{p}$, and $c$, are common knowledge to both parties.
Since no message can be sent before the test, there is only the interim
disclosure. That is, the doctor’s choice is an information disclosure rule
that maps each test result to a probability distribution over a set of
messages. Thus, with a slight abuse of notation, the information disclosure
rule is mathematically equivalent to a lottery of beliefs, which is a
probability measure $\tau$ defined on $\Delta([0,1])$, and the Bayesian
plausibility constraint requires

$\displaystyle\tau\in\mathcal{R}(\alpha_{0})\equiv\left\\{\tau\in\Delta([0,1])\bigg{|}\int\alpha\tau(d\alpha)=\alpha_{0}\right\\}.$

Assume $\bar{p}-\underline{p}\geq c$. Then if the patient accepts the medical
test and the posterior belief is $\alpha$, the patient then accepts the
treatment if and only if

$\alpha+(1-\alpha)\bar{p}-c\geq\alpha+(1-\alpha)\underline{p}\Rightarrow\alpha\leq\alpha_{e}\equiv
1-\frac{c}{\bar{p}-\underline{p}}.$ (18)

That is, he is pessimistic about her initial health status. Consequently,
given test $\tau$, the patient accepts the test if and only if

$\displaystyle\int_{0}^{\alpha_{e}}\phi\big{(}\alpha+(1-\alpha)\bar{p}-c\big{)}\tau(d\alpha)+\int_{\alpha_{e}}^{1}\phi\big{(}\alpha+(1-\alpha)\underline{p}\big{)}\tau(d\alpha)\geq\phi\big{(}\alpha_{0}+(1-\alpha_{0})\underline{p}\big{)}.$
(19)

Again, by Doval and Skreta (2024, Proposition 3.2), we can only consider
signals with an at-most-binary support. Thus, I still use $\\{x,y\\}$
($x\geq\alpha_{0}\geq y$) to denote the binary signal where $x$ and $y$ are
the upper and lower beliefs, respectively.

Identify

$\displaystyle
P(\alpha)=\left\\{\begin{array}[]{cc}\bar{p}&\alpha\leq\alpha_{e}\\\
\underline{p}&\alpha>\alpha_{e}\end{array}\right.\text{ and
}V(\alpha)=\left\\{\begin{array}[]{cc}\phi\big{(}\alpha+(1-\alpha)\bar{p}-c\big{)}&\alpha\leq\alpha_{e}\\\
\phi\big{(}\alpha+(1-\alpha)\underline{p}\big{)}&\alpha>\alpha_{e}\end{array}\right.$

as the payoff functions of the doctor and the patient when the patient’s
belief is $\alpha$, respectively.

######  Observation 5.

In the absence of constraint (19), the doctor benefits from persuasion if and
only if $\alpha_{0}\geq\alpha_{e}$. If so, signal $\\{\alpha,1\\}$ is optimal
for any $\alpha\leq\alpha_{e}$.

That is, without the participation constraint, the doctor only needs to
(partially) reveal the test result when the patient is “too optimistic” about
her initial health status.

$\alpha$$V(\alpha)$$\alpha_{v}$$\alpha_{e}$$1$$1$$\phi(\bar{p}-c)$

$\alpha$$V(\alpha)$$1$$1$$\alpha_{e}$$\alpha_{v}$$l(\alpha_{v})$$\phi(\bar{p}-c)$

Figure 6: Optimal signal for rewarding the past.

The optimal signal for rewarding the past is derived by concavifying function
$V$, a continuous and increasing function with two concave pieces.

######  Observation 6.

There exists $\alpha_{v}\geq\alpha_{e}$ and a signal
$\tau\in\mathcal{R}(\alpha_{0})$, such that $E_{\tau}V(\cdot)>V(\alpha_{0})$
if and only if $\alpha_{v}\geq\alpha_{0}\geq l(\alpha_{v})$, where

$\displaystyle
l(\alpha)\equiv\max_{\alpha^{\prime}\in[0,\alpha_{e}]}:\frac{V(\alpha)-V(\alpha^{\prime})}{\alpha-\alpha^{\prime}}.$

Here, for any $\alpha>\alpha_{e}$, $l(\alpha)$ is either $0$, or the tangent
point of the line that is tangent to the left piece of $V$ and originates from
point $(\alpha,V(\alpha))$. A signal with perfect good news, which is optimal
for guiding the future, is not necessarily optimal for rewarding the past, and
Figure 6 provides two examples. In the left panel, the optimal signal for
rewarding the past is with perfect bad news, and the doctor can improve the
patient’s utility of taking the test if and only if $\alpha_{0}<\alpha_{v}$.
In the right panel, the doctor can improve the patient’s utility of taking the
test if and only if $\alpha_{0}\in(l(\alpha_{v}),\alpha_{v})$, and in the
optimal signal, there is no perfect news.

The optimal test is characterized by the following proposition.

######  Proposition 6.

The doctor benefits from persuasion if and only if
$\alpha_{e}<\alpha_{0}<\alpha_{v}$, and the optimal signal there is
$\\{\alpha^{*},l(\alpha^{*})\\}$, where

$\displaystyle\alpha^{*}=$
$\displaystyle\sup\left\\{\alpha\in(\alpha_{e},1]\bigg{|}\frac{\alpha-\alpha_{0}}{\alpha-l(\alpha)}\cdot
V(l(\alpha))+\frac{\alpha_{0}-l(\alpha)}{\alpha-l(\alpha)}\cdot
V(\alpha)\geq\phi\big{(}\alpha_{0}+(1-\alpha_{0})\underline{p}\big{)}\right\\}.$

Proposition 6 first states that an informative signal benefits the doctor only
when the prior belief is intermediate. When the patient is “too optimistic”
($\alpha_{0}\geq\alpha_{v}$), constraint (19) can be satisfied only by the