Double Directional Persuasion

Fan Wu, Jie Zheng∗

First Version September 23, 2018; Updated June 2nd, 2019

Abstract

We study a double directional persuasion game between two players: a doctor and

a patient. The health condition of the patient, either severely ill or mildly ill, is known

to the doctor but not to the patient, while the type of valuation (or willingness to

pay) for the treatment, either high or low, is the patient’s private information. The

doctor’s payoff is merely the price she charges, while the patient’s payoff depends on

his health condition, his type, and the price charged by the doctor. This sequential

game consists of two stages: the ex ante design stage and the ex post implementation

stage. We consider all reasonable orders of play and fully characterize the equilib-

rium in each situation whenever an equilibrium exists. Our results imply that when

design is sequential, an equilibrium always exists and being the first mover at the de-

sign stage is weakly dominated by being the second mover, regardless who is the first

mover. In contrast, when design is simultaneous, there may not exist an equilibrium

under some parameter environment, and the equilibrium when the doctor sends signal

first is the equilibrium when the patient sends signal first and also coincides with the

equilibrium when design is sequential and the doctor designs (and sends signal) first.

Furthermore, comparing the situations where the patient sends signal first, we show

that the equilibrium under sequential design payoff dominates the one under simul-

taneous design, whenever an equilibrium exist. Our work applies to other scenarios

when sellers have professional knowledge regarding customers’ needs. The finding is

consistent with many real world observations and provides a better understanding of

the situations where persuasion can be sequential and double directional.

Keywords. Bayesian Persuasion, Sequential Persuasion, Two-sided Asymmetric Infor-

mation, Information Design

JEL Codes. C72, C73, D82, D83

∗Fan Wu: PBC School of Finance, Tsinghua University(email: wuf.17@pbcsf.tsinghua.edu.cn); Jie Zheng: School

of Economics and Management, Tsinghua University (email: jie.academic@gmail.com). For helpful comments and

insights, we thank Zhuoqiong Chen, Hong Feng, Matthew Gentzkow, Xiangting Hu, Xiaoxi Li, Xiang Sun, Jean Tirole,

Ruixin Wang, Mu Zhang, Mofei Zhao, and seminar participants at Harbin Institute of Technology (Shenzhen) and

Wuhan University. This research is funded by National Natural Science Foundation of China (Projects No.61661136002

and No.71873074), and Tsinghua University Initiative Scientific Research Grant (Project No.20151080397). All errors

are our own.

1

1 Introduction

Suppose two types of patients (H or L) with different willingness to pay under the

same health condition go to see the doctor (she). The doctor provides treatment and

charges a price. For the doctor, she always wishes to extract the full surplus from

the patient, which is the first best outcome for her. On the other hand, the patient

(he) of type H might try to convince the doctor that he is type-L in order to pay a

lower price. However, he cannot indulge his desire to lie. In other words, there is a

constraint that he cannot do so to pass the threshold when the doctor is indifferent

between two valuations (as the extremal market in Bergemann, Brooks and Morris

(2015)). The optimal way to persuade is Bayesian persuasion.

Now consider a different scenario. Suppose only one type of patients but with

different health conditions (S or M) go to see the doctor. After diagnosis, the doctor

would disclose his health condition and provide treatment accordingly. In addition,

the treatment for condition S is more advanced and thus more expensive. As a result,

the doctor always has the incentive to convince the patient that he is under condition

S which requires superior treatment. Therefore, she might benefit from Bayesian

persuasion due to her information superiority, i.e., professional knowledge.

What if we combine these two forces at play under one setup? That is to say, the

patient might be under condition S or M which the doctor knows, being type H or L

which is his private information. On the one hand, the H type patient might resort to

persuasion to mimic type L patient. On the other hand, the doctor might persuade the

patient regarding his health condition. As a whole, these two forces might interact

with each other. In this paper, we fully characterize the equilibrium for such a

double directional persuasion game whenever an equilibrium exists, by considering

different orders of play. Depending on whether the design stage is sequential move

or simultaneous move and which player sends the signal first at the implementation

stage, there are in total 4 sensible orders of play: (a) doctor being the first mover both

at the design stage and the implementation stage; (b) patient being the first mover

both at the design stage and the implementation stage; (c) simultaneous design with

doctor being the first mover at the implementation stage; (d) simultaneous design

with patient being the first mover at the implementation stage.

The first set of our results relates to the existence of equilibrium. For the two

setups with sequential design an equilibrium always exists, while for the two setups

with simultaneous design an equilibrium may not always exist and the space of pa-

rameter values under which equilibrium exists for the setup where the doctor sends

2

the signal first is a subset of that for the setup where the patient sends the signal

first.

The second set of our results relates to the welfare comparison for players. Our

Theorem 1 shows that being the first mover at the design stage (and the implementa-

tion stage) is weakly dominated by being the second mover, regardless of the identity

of the first mover. In contrast, Theorem 2 states that the equilibrium with simultane-

ous design and the doctor being the first signal sender is the same as the equilibrium

with simultaneous design and the patient being the first signal sender whenever the

equilibrium exists. The comparison of equilibria between the setup of simultaneous

design with the doctor being the first signal sender and the setup of sequential design

with the doctor being the first signal designer (and signal sender) yields a similar

result (Theorem 3). Finally, if we compare the setups of patient being the first sig-

nal sender, the equilibrium under sequential design payoff dominates the one under

simultaneous design, whenever an equilibrium exists (Theorem 4).

Our work belongs to the rising literature on Bayesian persuasion, following the

seminal work by Kamanica and Gentzkow (Kamenica and Gentzkow (2011)). Since in

the game we consider, more than one player sends the signal and more than one player

receives the signal, our work is related to both the studies on Bayesian persuasion

with multiple senders (Li and Norman (2017), Gentzkow and Kamenica (2017a),

Gentzkow and Kamenica (2017b)) and those on Bayesian persuasion with multiple

receivers (Alonso and Camara (2016a), Alonso and Camara (2016b), Bardhi and Guo

(2018), Chan et al. (2019), Wang (2015), Marie and Ludovic (2016)). However, the

key distinction between our work and those mentioned above is that in our setup the

receivers of some message are indeed the senders of a different message which means

the information transmission is double directional. Since both players have private

information in our setup, our work is also related to studies on Bayesian persuasion

of a privately informed receiver (for example, Kolotilin et al. (2017)), while we differ

from that literature by providing the privately informed receiver an opportunity to

persuade the sender.

The outline of this paper is as follows. In Section 2 we setup the model. In Sections

3 and 4, we analyze the setup of sequential design and consider the case where the

doctor moves first and the case where the patient moves first, respectively. Sections

5 and 6 consider the setup of simultaneous design where the doctor sends the signal

first and that where the patient sends the signal first, respectively. In Section 7, we

conduct welfare comparisons between different setups and deliver the main results.

Section 8 concludes.

3

2 The model

We consider a double directional sequential persuasion problem with double-sided

information asymmetry, between a doctor and a patient. The health condition of

the patient, either severely ill (j = S) or mildly ill (j = M), is known to the doctor

but not to the patient, while the type of valuation (or willingness to pay) for the

treatment, either high (i = H) or low (i = L), is the patient’s private information.

This sequential game consists of two stages: the ex ante signal design stage and the

ex post singal implementation stage. The players become privately informed between

these two stages, that is, after players design their persuasion strategies and before

they implement their persuasion strategies, nature draws patient’s type i ∈ {H,L}(high or low) and his health condition j ∈ {S,M}. Patient becomes aware of his

type while his condition is revealed to the doctor (she). Both players are assumed to

commit to their designed strategy when implementing their strategy at the second

stage. The information structure, the order of play, and the commitment assumption

are all common knowledge.

In the information design stage, we consider both sequential and simultaneous

commitment. For the information realization stage, the patient sends signal i while

the doctor sends signal j. In addition, we endogenize the timing of their commit-

ments (if sequential) and their reports. After reporting, the doctor would charge a

corresponding price P (i, j) for the treatment of his disease depending on the reported

type and condition. The information reported from both sides is unverifiable.

The value vij of the treatment to the patient is given by

vij H L

S xa a

M xb b

where a ≥ b is dictated by the definition of j while the patient’s type implies x ≥ 1.

We further assume that xb ≥ a to guarantee that the willingness to pay for type H

is always higher than type L, i.e., there is no price inversion under any parameter

settings represented by the following matrix which is the distribution of the patient’s

type and health condition:

Nij H L

S qn n

M pm m

where Nij indicates the relative amount of type i patients under condition j. There is

4

an alternative interpretation that this matrix provides the probability of the patient

being type i under condition j once you normalize them.

Preferences: The patient’s utility ui (where i ∈ {H,L}) relies on his type, his

condition reported j, and the price P (i, j) charged by the doctor

ui(i, j) =∑j

π(j|j) · vij − P (i, j)

with the outside option ui = 0 if he rejects the treatment. The doctor’s utility Ud

depends on the price P (i, j) of the treatment given the patient’s acceptance of the

treatment. Here we normalize the cost of this treatment to zero (or we could interpret

that P denotes the profit of the treatment after taking the cost into consideration).

Ud =∑i,j,i,j

Nij (i, j) · 1ui (i,j)≥0 · P (i, j)

where Nij (i, j) denote the number of type i, j patients under signal realization i, j.

The first best outcome for the doctor is

Ud0 =

∑i,j

Nijvij

which extracts the full surplus from the patient. We use Ud0 as a benchmark to

characterize the doctor’s relative welfare loss

U ≜ Ud − Ud0 .

Persuasion: The doctor’s persuasion signal is denoted by j. An arbitrary signal

j would take the form of

π(j = S|j = S) = t π(j = S|j = M) = r

π(j = M |j = S) = 1− t π(j = M |j = M) = 1− r

where 0 ≤ t ≤ 1, 0 ≤ r ≤ 1.

Similarly, an arbitrary signal i of the patient would be

π(i = H|i = H) = T π(i = H|i = L) = R

π(i = L|i = H) = 1− T π(i = L|i = L) = 1−R

where 0 ≤ T ≤ 1, 0 ≤ R ≤ 1.

5

Figure 1: Sequential commitment-Doctor moves first

0 1 2 3

Doctor

commits

Patient

commits

Doctor sends signal,

then patient sends

signal

Doctor charges

price P, then

patient chooses

to accept or not

We shall define two persuasion thresholds πm and πs which would prove to be

useful later.

πm =1

(x− 1)p

πs =1

(x− 1)q.

Solution concept: We focus on subgame perfect equilibria. To simplify the anal-

ysis without loss of generality, we restrict our attention to the straightforward signal

under persuasion, which indicates the same meaning as Kamenica and Gentzkow

(2011).

3 Sequential Commitment-Doctor Moves First

In this section, the doctor takes the lead and the patient acts subsequently (both in

information design stage and information realization stage). In the information de-

sign stage, she is free to commit to any straightforward signal to send to the patient

regarding his condition in the first period. Notice that her commitment is indepen-

dent on the patient’s type, since she is unaware of that information at this point.

Then, with the doctor’s commitment in mind, the patient will choose his commit-

ment concerning his type. His strategy hinges upon both his type and the doctor’s

incoming signal j.

In the information generalization stage, the doctor sends her signal first. In the

second period, the patient sends his signal. In the third period, the doctor arrives

at the optimal P price to charge given patient’s report and her persuasion strategy.

Equilibrium is achieved when the doctor chooses the commitment to maximize her

utility given the anticipated commitment of the patient while the patient chooses the

optimal commitment strategy given the doctor’s commitment.

6

We characterize the equilibrium by backward induction. First, we address the

case where only one condition exists to see how the patient would act accordingly.

Second, we relax this assumption to analyze both player’s actions.

First, suppose only one condition exists, which implies that the doctor does not

need to diagnose. The patient has the incentive to convince her that his type is L

to reveal a lower willingness to pay. Thus, for type L patient, he would report so

with probability 1. However, type H patient might mimic type L where his uniquely

optimal strategy is a binary signal under which the doctor is indifferent between

charging vH and vL if she receives signal L. Therefore, we immediately obtain the

following lemma.

Lemma 1. If the market contains two types (H or L) of customers whose relative

amount N and willingness to pay v are represented by

H L

N NH NL

v vH vL,

the optimal persuasion strategy for the customer would be

π(i = L|i = H) = min[1

NH

NL(vHvL

− 1), 1] (1)

π(i = L|i = L) = 1 (2)

and the seller’s best response is

P =

vH , if i = H,

vL, if i = L.

The economic intuition is simple. Type L patient has no incentive to misreport

his type. However, type H patient could mimic type L patient to lower his payment.

But he could not indulge himself for doing so. Otherwise, the doctor may charge vH if

the market with i = L is mostly filled with type H patient. The threshold is reached

when the doctor is exactly indifferent between charging vH and vL. On the other

hand, the doctor’s best response to his action is to charge the price corresponding to

his report.

By Lemma 1, if j = S is truthfully revealed to the patient, his optimal signal i

for persuasion is

π(i = L|i = L) = 1

7

π(i = L|i = H) =

1(x−1)q

= πs, if πs ≤ 1,

1, o.w.

and the doctor’s best response is

P =

vHS, if i = H,

vLS, if i = L.

On the other hand, if j = M is truthfully revealed to the patient, his optimal signal

for persuasion is

π(i = L|i = L) = 1

π(i = L|i = H) =

1(x−1)p

= πm, if πm ≤ 1,

1, o.w.

and the doctor’s best response is

P =

vHM , if i = H,

vLM , if i = L.

From here we can see that (1) and (2) fully characterize patient’s best response,

which means that the doctor would lead the patient to react accordingly. What’s left

is the doctor’s persuasion strategy.

From Lemma 1, it is clear that if the patient’s type is H, he always attempts to

persuade the doctor. However, there is a natural constraint that

π(i = L|i = H) ≤ 1.

Suppose the doctor reveals the truth. We call the case where πs>1 as full persuasion

under condition S, since this constraint is binding and π(L|H) = 1. Otherwise,

π(L|H) = πs ≤ 1, which we refer to as partial persuasion. The same goes for

the patient under condition M where πm>1 is called full persuasion while partial

persuasion indicates π(L|H) = πm ≤ 1. As a result, we will address the issue case by

case: double partial persuasion, double full persuasion, single full persuasion.

Now we allow the doctor to persuade as well. Under an arbitrary persuasion

strategy by the doctor

π(j = S|j = S) = t π(j = S|j = M) = r,

8

the patient would be divided into two markets (or groups) j ∈ {S,M} where the first

one j = S is described by:

H L

N tqn+ rpm tn+ rm

v x tqna+rpmbtqn+rpm

tna+rmbtn+rm

(3)

Define

NH(t, r) ≜ tqn+ rpm

NL(t, r) ≜ tn+ rm

vH(t, r) ≜ xtqna+ rpmb

tqn+ rpm

vL(t, r) ≜tna+ rmb

tn+ rm

∆v(t, r) ≜ vH(t, r)− vL(t, r).

v relies on the persuasion because the patient would take the doctor’s persuasion

strategy into account and adjust his willingness to pay. It turns out that either one

of these two sub-market admits the following property.

Define

π(t, r) ≜ 1NH(t,r)NL(t,r)

(vH(t,r)vL(t,r)

− 1).

Lemma 2.

min[πs, πm] ≤ π(t, r) ≤ max[πs, πm]

always holds for any j.

That is to say, the persuasion threshold for either one is bounded between two

original thresholds.

3.1 Double Partial Persuasion

In this case, we have

πm ≤ 1

πs ≤ 1.

If the doctor reveals j truthfully,

U = −qnπs(xa− a)− pmπm(xb− b)

= −qn1

(x− 1)q(xa− a)− pm

1

(x− 1)p(xb− b)

= −na−mb.

9

However, it turns out that this result not only holds if the doctor is telling the truth,

but for all possible scenarios as well.

Lemma 3. If πs ≤ 1 and πm ≤ 1, the doctor’s welfare loss is

U = −na−mb

regardless of her own persuasion strategy.

This lemma directly leads us to the following proposition.

Proposition 1. Given doctor moves first, if πm ≤ 1 and πs ≤ 1,

1. the doctor is indifferent with any of her persuasion strategy.

2. the patient’s persuasion follows (1) and (2) under j = S and j = M , respec-

tively.

This proposition is easy to interpret. Since it is partial persuasion (interior so-

lution) under both conditions, it remains so for any reported condition j. Recall

that if it is interior persuasion, the doctor will suffer a loss which is equivalent to

the surplus of the L type patient. Combining both reported conditions together, her

welfare loss is equal to the result from lemma 3, which is a constant independent on

her persuasion strategy.

3.2 Double Full Persuasion

In this case, we have

πm>1

πs>1.

By Lemma 2,

π(t, r) ≥ min[πs, πm]>1,

π(1− t, 1− r) ≥ min[πs, πm]>1.

Thus, for both group of patients, they will employ full persuasion again

π(i = L|i = H) = 1

which leads to the following proposition.

10

Proposition 2. If πm>1 and πs>1, the doctor will always befully uninformative (t = r = 1), if p ≥ q,

fully informative (t = 1, r = 0), o.w.

and the patient’s best response is π(L|H) = 1.

From our previous case in double partial persuasion, we see that the persuasion

from the doctor’s side is ineffective to improve her utility. However, it is a different

story under double full persuasion case. Here, the patient will employ full persuasion

regardless of the doctor’s persuasion. If the doctor indeed bunch patients under

two conditions together, these patients would harbor the same willingness to pay.

Therefore, we could intuitively interpret this proposition. If p = q, the doctor’s

utility would remain the same, since now the willingness to pay for type H would be

x times of type L, which is the same as vHS

vLSand vHM

vLM. Nevertheless, if p>q, the ratio

of vH(1,1)vL(1,1)

would be smaller, hence a smaller price gap between type H and L, which

implies a lower welfare loss for the doctor (recall that her welfare loss derives from

and only from the price gap under double full persuasion).

3.3 Single Full Persuasion

Single full persuasion implies

min[πm, πs] ≤ 1<max[πm, πs].

Since it is either p<q or q<p, we will address them case by case.

Case 1: p<q

In this case,

πs ≤ 1<πm.

Proposition 3. If πs ≤ 1<πm, the doctor will always tell the truth and the patient’s

best response is

π(L|H) =

1, if j = M,

πs, o.w.

The intuition comes from two observations. A. It is not optimal to bunch patients

under two conditions into an interior solution π(t, r) ≤ 1. Otherwise, it would waste

the corner solution condition πm>1 where the patient is unable to tap into the full

potential of his persuasion. B. It is not a good choice to bunch patients together into

11

a corner solution, either. To see this, we could view such a bunching process as two

steps. 1. Bunching to the point where π(t, r) just hits the corner solution π(t, r) = 1.

2. Bunching further to pass the corner point. It is obvious that the first step will

take a toll on her (see point A). The doctor will suffer a loss in second step as well

(see Proposition 2). Therefore, she is better off revealing the truth when p<q.

Case 2: q<p

In this case,

πm ≤ 1<πs.

Define π1 and π0 as

(a− b)(p− q) = (m+ n

n)2b[p(x− 1)− 1

π1

] (4)

(a− b)(p− q) = b[p(x− 1)− 1

π0

] (5)

By their definitions, π1<π0 is self-obvious. If you plug π1 = πm into the RHS of (4),

LHS>0 = RHS

which implies πm<π1. If you plug π0 = πs into the RHS of (5),

LHS = (a− b)(p− q) ≤ b(x− 1)(p− q) = RHS

due to a ≤ xb. Therefore, π0 ≤ πs. Thus,

πm<π1<π0 ≤ πs. (6)

Proposition 4. If πm ≤ 1<πs, the doctor’s optimal strategy is

t = 1

r =

0, if π0 ≤ 1,

1, if π1 ≥ 1,

nm{√

(a−b)(p−q)b[p(x−1)−1]

− 1}, o.w.

and the patient’s best response is

π(L|H) =

1, if j = S,

πm, o.w.

12

Figure 2: Sequential commitment-Patient moves first

0 1 2 3

Patient

commits

Doctor

commits

Patient sends signal,

then doctor sends

signal

Patient charges

price P, then

doctor chooses

to accept or not

To understand this proposition, it is crucial to recall two observations. First, if

r is too large, it might render an interior solution π(L|H)<1, which is not desirable

because it allows the patient to exploit the full potential of his persuasion. Second, in

Proposition 2, it is optimal to bunch different types altogether since q<p. Therefore,

the doctor’s final utility hinges upon the trade-off between those two forces. In

addition, we have a natural constraint that 0 ≤ r ≤ 1, which leads to our proposition

above.

4 Sequential Commitment-Patient Moves First

For now we consider the case where the patient makes the first move. For the infor-

mation design stage, the patient commits first and the doctor commits afterwards.

For the information generalization stage, the patient sends signal first and the doctor

sends second. Then, the doctor is going to charge the patient for the treatment.

Equilibrium is achieved when the patient chooses the optimal commitment strategy

given the doctor’s following commitment strategy while the doctor chooses to commit

to maximize her utility given the patient’s commitment.

If the patient always reveals his type truthfully to the doctor, the doctor would

be able to obtain the first best outcome which is the full surplus from the market.

In addition, every type H patient has the incentive to convince the doctor that he is

type L in order to show a lower willingness to pay. However, he cannot indulge his

desire to lie to the doctor without any kind of restraint because the doctor is aware

of this fact and might charge vH as she sees fit. Besides, the patient is unaware of

his condition when he commits first. Thus, his commitment strategy is independent

on his condition. In this setup, it is the game between the doctor and the H type

patient. L type patient has no incentive to convince the doctor that his type is H.

Nor does he have any bargaining power in group i = L because he either needs to

13

offer his full willingness to pay or leave when vH is charged. It is the same even if

doctor would misreport. In this sense, L type patient is at the mercy of the doctor

and type H patient and always obtains zero surplus.

π(i = L|i = L) = 1

Therefore, there is only one variable in the strategy of type H patient we need to pin

down, π(i = L|i = H) which we will refer to as π(L|H). Under π(L|H), the doctor

is going to face two markets where the first one i = H is represented by

Nij H L

S [1− π(L|H)]qn 0

M [1− π(L|H)]pm 0

while the second one i = L is mixing

Nij H L

S π(L|H)qn n

M π(L|H)pm m.

Now we introduce the following lemma to pinpoint the sub-range which the opti-

mal π(L|H) belongs to.

Lemma 4. If the patient moves first,

1. The patient’s utility is monotonically increasing with π(L|H) over the sub-range

π(L|H) ∈ [0,min[πm, πs]] given any persuasion strategy of the doctor.

2. The patient’s persuasion strategy π(L|H) ∈ (0,min[πm, πs]] would strictly dom-

inate persuasion design where π(L|H)>max[πm, πs], if max[πm, πs]<1.

This lemma is easy to interpret. In the sub-range π(L|H) ∈ [0,min[πm, πs]], since

it has not reached the threshold yet, he could improve his persuasion power through

increasing π(L|H). However, in the sub-range π(L|H)>max[πm, πs], type H patient

would end up with zero surplus, since the doctor is going to charge vH regardless of

her persuasion design. Thus, π(L|H)>max[πm, πs] is always dominated by a small

value of π(L|H).

By the second part of this lemma, π(L|H)>max[πm, πs] is ruled out because it

is inferior to the choice 0<π(L|H) ≤ min[πm, πs]. Besides, π(L|H) = min[πm, πs]

dominates all the other solutions π(L|H) ∈ [0,min[πm, πs]]. Thus, optimal π(L|H)

satisfies

min(πm, πs) ≤ π(L|H) ≤ max(πm, πs) (7)

14

if max(πm, πs) ≤ 1.

To mirror our analysis in the previous section, we will continue to address this

issue case by case with the same method to categorize them. That is, if doctor reveals

his condition truthfully first, the patient under S and M would resort to: double full

persuasion, double partial persuasion, single full persuasion (notice this is only to

facilitate categorizing different parameter settings).

Before coming to our discussion, we need to point out that this section is closely

related to the previous one by the following lemma, which implies that some conclu-

sions in Section 3 could greatly simplify our upcoming analysis.

Lemma 5. Surplus loss equivalence: If the market contains two types (H or L) of

customers whose relative amount N and willingness to pay v are represented by

H L

N NH NL

v vH vL,

the producer’s surplus loss would be equivalent in two setups: 1. Directly charge a

single optimal price. 2. Let the patient persuade first and then decide the optimal

prices to charge separately.

The doctor is going to face market i = L

Nij H L

S π(L|H)qn n

M π(L|H)pm m

after the patient’s persuasion. Suppose there are two scenarios. In the first one,

the doctor employs persuasion and then charge prices directly. For another one,

the doctor employs persuasion and then let the patient persuade. Afterwards, she

charges prices based on the patient’s signal. By Lemma 5, the doctor’s payoff would

be identical in those two scenarios. That is to say, we could borrow the results from

Section 3 if we replace the p and q with π(L|H)p and π(L|H)q, respectively, to analyze

the doctor’s best response.

4.1 Double Partial Persuasion

In this case,

πm ≤ 1,

15

πs ≤ 1.

Case 1: p ≤ q (πs ≤ πm ≤ 1)

Proposition 5. If the patient moves first and πs ≤ πm ≤ 1, the patient’s optimal

action is

π(L|H) =

πs, if πs

πm>1− na

mb,

πm, o.w.

and the doctor will always reveal the truth (t = 1, r = 0).

By Lemma 4, the patient’s persuasion should satisfy π(L|H) ≤ max(πs, πm), which

would fall into our previous discussions in Proposition 1 and Proposition 3. Thus,

the doctor will always reveal the truth. The patient needs to consider the relative

probability of his health condition. Intuitively speaking, if condition S is highly

possible, he should bet on that result to choose π(L|H) = πs and vice versa.

Case 2: p>q (πm<πs ≤ 1)

Proposition 6. If the patient moves first and πm<πs ≤ 1, there exists a cutoff π∗

that the patient’s optimal

π(L|H) =

π1, if π1 ≥ π∗,

πs, o.w.

while the doctor’s best response is t = 1,

r =

1, if π(L|H) = π1,

0, if π(L|H) = πs.

In this case, the patient’s payoff is convex in sub-range [π1, πs], which leads to

corner solution. If the probability for him being under condition S is sufficiently

high, he should choose πs and abandon his persuasion benefit under condition M . If

q is relatively small compared to p, he should pick π1, which partially gives up his

persuasion potential under condition S.

4.2 Double Full Persuasion

In this case,

πm>1

πs>1.

16

Previously in Section 3.2 where doctor moves first, patient will always employ full

persuasion, i.e.,

π(L|H) = 1.

Notice that in that setup, doctor has already taken the patient’s action into account.

Therefore, if the patient does so beforehand, her action would remain the same as

Section 3.2.

As for the patient, by Lemma 4 his utility is monotonically increasing with π(L|H)

over the sub-range

π(L|H) ∈ [0,min(πm, πs)].

Since

min(πm, πs)>1,

he would choose π(L|H) = 1.

To sum up, the equilibrium is exactly the same as Section 3.2, which is character-

ized by Proposition 2.

4.3 Single Full Persuasion

Single full persuasion implies

min[πm, πs] ≤ 1<max[πm, πs].

Case 1: p<q (πs ≤ 1<πm)

Proposition 7. If patient moves first and πs ≤ 1<πm, patient’s optimal action is

π(L|H) =

πs, if 1−πs

πm≤ na

mb,

1, o.w.

and the doctor will always reveal the truth.

The intuition for this proposition is almost the same as Proposition 5. If the

probability of condition S is sufficiently large, the patient should pick π(L|H) = πs.

Otherwise, π(L|H) = 1.

Case 2: p>q (πm ≤ 1<πs)

Proposition 8. Given patient moving first and πm ≤ 1<πs, the optimal strategy of

the players is as follows:

1. If π1 ≥ 1, the patient would choose π(L|H) = 1 and the doctor chooses t = r =

1.

17

Figure 3: Simultaneous commitment-Doctor sends signal first

0 1 2 3

Simultaneous

commitment

Doctor sends

signal

Patient sends

signal

Patient charges

price P, then

doctor chooses

to accept or not

2. If π1<1<π0, there exists a cutoff π∗∗ that the patient would choose

π(L|H) =

π1, if π1 ≥ π∗∗,

1, o.w.

The doctor’s best response is t = 1,

r =

1, if π(L|H) = π1,

nm{√

(a−b)(p−q)b[p(x−1)−1]

− 1}, if π(L|H) = 1.

3. If π0 ≤ 1, the patient would choose

π(L|H) =

π1, if π1 ≥ π∗

πs,

1, o.w.

The doctor’s best response is t = 1,

r =

1, if π(L|H) = π1,

0, if π(L|H) = 1.

π1 and π0 are defined in (4) and (5).

The intuition of this proposition derives from Section 4.2 and Proposition 6. Since

the patient always choose corner solution to maximize his utility, the doctor only needs

to respond accordingly.

5 Simultaneous Commitment-Doctor sends signal

First

5.1 Double Partial Persuasion

Proposition 9. Under simultaneous commitment with doctor sending her signal first,

given πm ≤ 1 and πs ≤ 1,

18

1. if p = q, there is no equilibrium.

2. if p = q, equilibria exist with only this form such that the patient chooses

π(L|H) = πs(= πm) while the doctor is indifferent with her persuasion strategy.

With simultaneous commitment, the existence of equilibrium is more delicate.

That is to say, equilibrium may not exist under some parameter settings. Proposition

9 is easy to interpret. The case for p = q is self-obvious and we shall illustrate the

case for p = q (πs = πm). Given any commitment of the doctor, the patient is always

able to achieve his first-best outcome as in Lemma 3, which is the worse case scenario

for the doctor. However, now the doctor would always deviate to improve her welfare,

i.e., to switch two groups when t = r or to fully separate the patient (reveal the truth)

when t = r. That is to say, both players desire to obtain the second-mover advantage,

which makes it impossible to reach an equilibrium under simultaneous setup.

5.2 Double Full Persuasion

By Proposition 2, although the doctor would lead the patient’s commitment, she is

already fully aware of the patient’s action, i.e., π(L|H) = 1. Thus, under simultaneous

commitment setup, this is the unique equilibrium.

Proposition 10. Under simultaneous commitment with doctor sending her signal

first, given πm>1 and πs>1, the doctor will always the doctor will always befully uninformative (t = r = 1), if p ≥ q,

fully informative (t = 1, r = 0), o.w.

and the patient’s best response is π(L|H) = 1.

Proof of Proposition 10. The proof follows directly from previous discussions.

5.3 Single Full Persuasion

Proposition 11. Under simultaneous commitment with doctor sending her signal

first, if πs ≤ 1<πm, there is no equilibrium.

The economic intuition is similar to Proposition 9. No equilibrium exists due to the

fact that both players desires to play wait and see to gain the second-mover advantage.

To be more specific, given any doctor’s commitment strategy, the best response of

the patient would cause the doctor to deviate from her original commitment.

19

Proposition 12. Under simultaneous commitment with doctor sending her signal

first, given πm ≤ 1<πs,

1. If π1 ≥ 1, equilibrium exists: the doctor would pool the patient into one group

(t = r = 1) while the patient chooses π(L|H) = 1.

2. Otherwise, no equilibrium.

We could compare this proposition with Proposition 4 to facilitate our under-

standing. If π1 ≥ 1, equilibrium remains exactly the same as sequential commitment.

(Actually, it is essentially the same as the case in Proposition 10.) Otherwise, equi-

librium does not exist due to the second-mover advantage for both players.

πm

πs

1

10

π1 = 1

Figure 4: Simultaneous Commitment Doctor First

Before concluding this section, we combines all possible parameter settings into

figure 4. The shaded area identifies the parameter values where the equilibrium does

not exist. The curve represents π1 = 1.

6 Simultaneous Commitment-Patient sends signal

First

6.1 Double Partial Persuasion

Case 1: p ≤ q (πs ≤ πm ≤ 1)

Recall Proposition 5. Given the patient’s persuasion

min(πm, πs) ≤ π(L|H) ≤ max(πm, πs)

20

Figure 5: Simultaneous commitment-Patient sends signal first

0 1 2 3

Simultaneous

commitment

Patient sends

signal

Doctor sends

signal

Patient charges

price P, then

doctor chooses

to accept or not

by Lemma 4, fully separating the patient would become the dominant strategy for

the doctor by the proof of Proposition 5. Meanwhile, the patient’s choice is the

best response given the doctor’s strategy. Thus, under simultaneous commitment

with patient sending his signal first, the unique equilibrium remains the same as

Proposition 5.

Proposition 13. Under simultaneous commitment with patient sending his signal

first, given πs ≤ πm ≤ 1, the patient’s optimal action is

π(L|H) =

πs, if πs

πm>1− na

mb,

πm, o.w.

and the doctor will always reveal the truth (t = 1, r = 0).

Proof of Proposition 13. The proof follows directly from previous discussions.

Case 2: p>q (πm<πs ≤ 1)

Proposition 14. Under simultaneous commitment with patient sending his signal

first, given πm<πs ≤ 1, the unique equilibrium exists if and only if

πm

πs

≤ 1− mb

na

Under this equilibrium, the patient’s optimal action is π(L|H) = πs and the doctor

will reveal the truth (t = 1, r = 0).

The economic intuition is as follows. Since the patient sends signal first, his

commitment is independent on the doctor’s signal j. That is to say, he commits a

single π(L|H) for both j = S and j = M . To maximize his utility, he either chooses

π(t, r) or π(1 − t, 1 − r). If he chooses the smaller one, the doctor would deviate

to pooling all patients together in the commitment stage. If he chooses the larger

21

one ( = πs), the doctor would deviate to fully separate all patients (reveal the truth).

Therefore, the only possible equilibrium exists when the doctor reveals the truth and

the patient chooses π(L|H) = πs. The condition for the existence of such equilibrium

dictates that the patient would prefer πs over πm.

Notice in this equilibrium, type L patient under condition M is crowded out of

the market, since price vHM is charged. Thus, social welfare is not optimal and there

is some welfare losses due to transaction failure.

6.2 Double Full Persuasion

In this case, the equilibrium is exactly the same as Proposition 10.

Under simultaneous commitment with doctor sending her signal first, given πm>1

and πs>1, the doctor will always befully uninformative (t = r = 1), if p ≥ q,

fully informative (t = 1, r = 0), o.w.

and the patient’s best response is π(L|H) = 1.

6.3 Single Full Persuasion

Case 1: πs ≤ 1<πm

Recall Proposition 7. Once more, fully separating the patient would become

the dominant strategy for the doctor by the proof of Proposition 7. Meanwhile,

the patient’s choice is the best response given the doctor’s strategy. Thus, under

simultaneous commitment with patient sending his signal first, the unique equilibrium

remains the same as Proposition 7.

Proposition 15. Under simultaneous commitment with patient sending his signal

first, given πs ≤ 1<πm, the patient’s optimal action is

π(L|H) =

πs, if 1−πs

πm≤ na

mb,

1, o.w.

and the doctor will always reveal the truth (t = 1, r = 0).

Proof of Proposition 15. The proof follows directly from previous discussions.

Case 2: πm ≤ 1<πs

22

Proposition 16. Under simultaneous commitment with patient sending his signal

first, given πm ≤ 1<πs,

1. If π1 ≥ 1, the unique equilibrium is such that the patient would choose π(L|H) =

1 and the doctor chooses t = r = 1.

2. If π1<1<π0, there exists a cutoff Π∗ that the unique equilibrium exists if and

only if

πm ≤ Π∗.

Under this equilibrium, the patient would choose

π(L|H) = 1.

The doctor’s best response is t = 1,

r =n

m{

√(a− b)(p− q)

b[p(x− 1)− 1]− 1}.

3. If π0 ≤ 1, the unique equilibrium exists if and only if

1− πm

πs

≥ mb

na.

Under this equilibrium, the patient would choose

π(L|H) = 1.

The doctor’s best response is

t = 1, r = 0.

The readers could turn back to Proposition 8 to gain more insight into this propo-

sition. Equilibrium exists only if π(L|H) = 1. Recall that the patient either chooses

π(t, r) or π(1 − t, 1 − r) in equilibrium. If π(L|H) = 1, the doctor would deviate to

reveal the truth (bunch all the patients together) when he chooses the larger (smaller)

one of π(t, r) and π(1− t, 1− r). The condition in the Proposition 16 is to guarantee

that the patient would not deviate from commitment π(L|H) = 1.

In figure 6, the shaded area identifies the parameter values where the equilibrium

does not exist. Two curves represent π1 = 1 and π0 = 1, respectively.

7 Comparison and Result

7.1 Comparison between Sequential Games

In this section, we are ready to draw a comparison between two setups in Section 3

and Section 4. That is, how would the patient and the doctor choose between those

23

πm

πs

1

10

1− mbna

π1 = 1

π0 = 1

Figure 6: Simultaneous Commitment Patient First

two setups? Would they choose to move first to gain some first-mover advantages?

Or they prefer to wait and observe to obtain second-mover advantage? With the

following theorem, we shall illustrate that if the doctor plays wait and see, her utility

is weakly better than the case where she takes the first move. What’s more, the same

goes for the patient. In other words, for both players, the strategy to wait always

dominates moving first.

Theorem 1. Both parties weakly prefer to play wait and see, i.e., choosing to let the

other one move first always weakly dominates taking the first move.

The readers should be familiar with both players’ utilities by now and the intuition

is simple. If there is only one possible type of patient (H or L), the doctor’s persuasion

would serve no purposes because the patient would take her action into consideration

and adjust his willingness to pay accordingly. Therefore, she would end up with the

same surplus which is her first best outcome. By contrast, the patient’s persuasion

is indeed effective, since the type H patient could disguise themselves as type L and

thereby lower the payment. Thus, in this double directional game, if the doctor is

strictly better off with her persuasion, this improvement must derive from type H

patient. To be more specific, her persuasion could mitigate her surplus loss caused

by the patient’s persuasion. In other words, her persuasion is able to keep type H

patient’s persuasion in check.

If the patient moves first, without knowing which group j he will end up being, he

needs to take both possibilities into account. However, the maximum π(L|H) allowed

by the doctor varies with j. By the proof of Theorem 1, the patient’s utility function

is convex over π(L|H) ∈ [π1, π0]. Thus, he will choose corner solution as a result.

24

If it is the smaller corner solution, compared to the situation where patient moves

second, the doctor is better off while type H patient is worse off, since his persuasion

power does not reach its full potential, relatively speaking. Type L patient remains

the same. If it is the larger corner solution, compared to the situation where patient

moves second, type H patient is worse off because some of them end up paying H

price while crowding out corresponding type L patient. The doctor remains the same.

7.2 Comparison between Simultaneous Games

Theorem 2. Under simultaneous commitment, if equilibrium exists under both regimes

(doctor sending first versus patient first), then equilibrium for two regimes remains

the same.

Comparing two setups under simultaneous commitment, it is clear that the space

of parameter values under which equilibrium exists when doctor sends signal first is a

subset of that when patient sends signal first. (In this subset, the equilibrium under

two regimes are identical.) That is to say, it is more difficult to reach an equilibrium

when doctor sends signal first. A closer look at Proposition 9 and Proposition 13

would shed light on this issue. Recall that when doctor sends signal first, the patient’s

commitment is more intricate. His commitment is contingent on the signal j he will

receive. In other words, he is at the liberty to commit to different π(L|H) for j = S

and j = M , respectively, which provides him with more commitment power. However,

is this a blessing? The answer is negative. From the doctor’s perspective, she could

use it to her advantage to improve her welfare. In Proposition 9, she could deviate

to switch j = S and j = M to obtain a positive gain relative to full loss. However,

when the patient sends signal first (forgoes such conditional commitment), the doctor

finds it more difficult to deviate to profit. Binding the hands the patient nullifies the

doctor’s incentive to deviate, which renders it easier to reach an equilibrium.

7.3 Comparison between Simultaneous and Sequential Com-

mitment

Theorem 3. If equilibrium exists under both regimes (simultaneous commitment and

sequential commitment) where doctor moves first, the equilibrium in those two regimes

are the same.

Once again, the space of parameter values under which equilibrium exists under

simultaneous commitment is a subset of that under sequential commitment. (In this

25

subset, the equilibrium under two regimes are identical.) This result is easy to in-

terpret. For sequential commitment, equilibrium always exists that the doctor would

lead the patient to respond (commit). That is to say, regardless of the doctor’s strat-

egy, the patient’s best response always exists. So does the equilibrium. Nevertheless,

under simultaneous commitment, given the patient’s best response, we need to check

whether the doctor would deviate or not, which kicks in one more necessary condition

for the equilibrium to hold and shrinks the parameter space befitting the equilibrium.

Theorem 4. If equilibrium exists under both regimes (simultaneous commitment and

sequential commitment) where the patient moves first, the sequential-commitment

setup Pareto dominates the simultaneous-commitment one.

If equilibrium exists, we could focus on the case πm<πs, since otherwise equilibria

remain the same under two regimes. For the sequential-commitment regime, the pa-

tient leads the doctor to respond and chooses between two corner solutions of π(L|H).

If he prefers the larger one, the equilibrium is identical to that of simultaneous-

commitment setup. On the other hand, if he prefers the smaller corner solution, both

players are better off. For the patient, this is due to his larger choice set. For the

doctor, her utility is monotonically decreasing in π(L|H) (see the proof of Theorem

1). As a result, sequential-commitment regime Pareto dominates the simultaneous-

commitment one. How does the social welfare improve in this game? The answer

lies within the participation of type L patient. If he faces an over-charged price, he

would forgo the transaction and leave, which could only be caused by the persuasion

strategy of type H patient. However, this is not ideal in terms of social welfare, since

the positive gain of trade vLS or vLM is lost. Taking the smaller corner solution of

π(L|H) means to bring the type L patient into the game and let them participate in

the transaction. Their participation provides cover for the type H patient in a sense

that type H patient could more easily disguise themselves as type L and thus lower

their payment. However, for type H patient, in this generous strategy, there is one

drawback that sometimes the persuasion does not reach its full potential, since it is

the smaller corner solution by design. Therefore, whether both players are strictly

better off hinges upon the tension between those two forces.

8 Conclusion

In this paper, we study the double directional Bayesian persuasion in a setting of

two-sided asymmetric information where both players are rational Bayesian updaters

26

and can commit to their persuasion strategies designed before the private information

is received. As far as we know, this is the first work on sequential persuasion with

different directions, and the first work incorporating the order of play into the design

stage (and the implementation stage) of the persuasion strategies.

Depending on whether the design stage is sequential move or simultaneous move

and which player sends the signal first at the implementation stage, we consider in

total four reasonable orders of play and fully characterize the equilibrium in each

setup whenever an equilibrium exists. For the two setups with sequential design we

show the existence of equilibrium, while for the two setups with simultaneous design

an equilibrium may not always exist and the space of parameter values under which

equilibrium exists for the setup where the doctor sends the signal first is a subset of

that for the setup where the patient sends the signal first.

By comparing the equilibria between different setups, we show that being the

first mover at the design stage (and the implementation stage) is weakly dominated

by being the second mover, regardless who is the first mover. By contrast, under

simultaneous design, the equilibrium for the doctor being the first signal sender is the

same as the equilibrium with the patient being the first signal sender whenever the

equilibrium exists. The comparison of equilibria between the setup of simultaneous

design with the doctor being the first signal sender and the setup of sequential design

with the doctor being the first signal designer (and signal sender) yields a similar

result. Finally, if we compare the setups of patient being the first signal sender,

the equilibrium under sequential design payoff dominates the one under simultaneous

design, whenever an equilibrium exists.

Our analysis for the doctor-patient interaction could easily extend to other sce-

narios where the seller has professional knowledge regarding the customer’s needs.

On the one hand, the customer desires to signal a lower willingness to pay in order

to lower the payment. On the other hand, the seller wishes to sell the customer a

more advanced (fancier) product in order to earn a higher profit. Both phenomena

coincide with real life experiences that customers like to haggle and might end up

buying something flashier (more expensive) than previously expected. Within both

processes, Bayesian persuasion could serve as a useful tool.

Notice that we assume commitment for both players in our study, which is a very

strong assumption. Alternative setups to analyze such a double directional persuasion

game may either have both players cheap talk or let one player cheap talk while the

other employ a committed strategy. As another direction for future study, we may

consider how our result is related to the bargaining literature where two privately

27

informed players interact via sequential bargaining.

28

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29

A Omitted Proofs in Section 3

Proof of Lemma 1. We only need to prove i is the patient’s best response, since the

rest is obvious. For the producer to be indifferent between charging vH and vL if

i = L.

π(i = H |i = L)NHvH = [π(i = H |i = L)NH +NL]vL

π(i = H |i = L) =1

NH

NL(vHvL

− 1),

if 1NHNL

(vHvL

−1)≤ 1. Otherwise,

π(i = H |i = L) = 1.

Proof of Lemma 2. It suffices to prove

min[(x− 1)q, (x− 1)p] ≤ 1

π(t, r)≤ max[(x− 1)q, (x− 1)p]. (8)

1

π(t, r)=tqn+ rpm

tn+ rm(x

tqna+ rpmb

tqn+ rpm

tn+ rm

tna+ rmb− 1)

=xtqna+ rpmb

tna+ rmb− tqn+ rpm

tn+ rm

=xp− p+ (q − p)tn[xa

tna+ rmb− 1

tn+ rm]

=xq − q + (p− q)rm[xb

tna+ rmb− 1

tn+ rm]

To prove (8), it suffices to prove

xa

tna+ rmb− 1

tn+ rm≥ 0 (9)

xb

tna+ rmb− 1

tn+ rm≥ 0. (10)

Notice that (9) is already guaranteed by a ≥ b and x ≥ 1 while (10) is equivalent to

x ≥ tna+ rmb

tnb+ rmb

which is assured by xb ≥ a.

The proof for the other group of patient under j = M is almost the same if you

replace t and r with 1− t and 1− r, respectively.

30

Proof of Lemma 3. For j = S, by (1)

π(i = L|i = H) = min[π(t, r), 1].

From Lemma 2

π(t, r) ≤ max[1

(x− 1)q,

1

(x− 1)p] ≤ 1

Therefore, the doctor’s welfare loss for the group of patients under j = S would be

Uj=S = −NH(t, r)π(i = L|i = H)∆v(t, r)

= −NH(t, r)1

NH(t,r)NL(t,r)

(vH(t,r)vL(t,r)

− 1)(vH(t, r)− vL(t, r))

= −tna− rmb.

The same logic goes for market 2 where j = M ,

Uj=M = −(1− t)na− (1− r)mb.

Thus, we obtain the total welfare loss

U = Uj=S + Uj=M = −na−mb.

Lemma 6. If 0 ≤ r ≤ 1 and 0 ≤ t ≤ 1,

1

m+ n=

rt

rm+ tn+

(1− r)(1− t)

(1− r)m+ (1− t)n+

mn(t− r)2

(m+ n)(rm+ tn)[(1− r)m+ (1− t)n].

Proof of Proposition 2. If the doctor reveals the truth, her welfare loss is

U = −qn(x− 1)a− pm(x− 1)b.

If she chooses t = r = 1, by Lemma 2,

π(1, 1) ≥ min[πs, πm]>1

U ′ = −(qn+ pm)∆v(1, 1).

The difference between these two strategies is denoted by ∆U

∆U = U ′ − U

= qn[(x− 1)a−∆v(1, 1)] + pm[(x− 1)b−∆v(1, 1)]

= qn[b− a

m+ nm− x

b− a

qn+ pmpm] + pm[

a− b

m+ nn− x

a− b

qn+ pmqn]

=mn

m+ n(a− b)(p− q)

31

Under persuasion strategy t, r by the doctor, the difference between doctor’s persua-

sion and her revealing the truth would be

∆U = (a− b)(p− q)[rmtn

rm+ tn+

(1− r)m(1− t)n

(1− r)m+ (1− t)n].

If p<q, ∆U<0. Thus, her best strategy would be revealing the truth.

If p ≥ q, ∆U ≥ 0. Besides, by Lemma 6

(a− b)(p− q)[rmtn

rm+ tn+

(1− r)m(1− t)n

(1− r)m+ (1− t)n] ≤ (a− b)(p− q)

mn

m+ n.

Therefore, her optimal choice is to group them together by always reporting only one

condition (t = r = 1).

Lemma 7. In the case of single full persuasion, if doctor persuades the patients under

under j = S and j = M to form a group where

π(t, r) ≤ 1,

from this group she will sustain a strict loss compared to revealing the truth.

Proof of Lemma 7. We shall only prove the case where p<q because the proof for the

scenario p>q is the same.

Suppose p<q. Since patient will employ single full persuasion if doctor reveals the

truth, we have1

(x− 1)q≤ 1<

1

(x− 1)p.

Suppose π(t, r) ≤ 1. Under persuasion,

U ′ = −NH(t, r)π(t, r)∆v(t, r)

= −NL(t, r)vL(t, r)

= −tna− rmb

Without doctor’s persuasion, her welfare loss from this group of patients is

U =− rpm(x− 1)b− tqnπs(x− 1)a

>− rmp1

(x− 1)p(x− 1)b− tqn

1

(x− 1)q(x− 1)a

=− tna− rmb

=U ′

where the inequality holds since 1(x−1)p

>1.

If this group is j = M , the proof is quite similar where you only need to replace

r and t with 1− r and 1− t.

32

Proof of Proposition 3. Under persuasion strategy t, r by the doctor, if

π(t, r) ≤ 1,

doctor would sustain a strict loss from this group by Lemma 7. If it is the case for

both groups j = S and j = M , the doctor’s persuasion would take a toll on her.

Otherwise, there is at least one group where

π(t, r)>1

or π(1− t, 1− r)>1. Suppose it is group j = S.

U ′ = −(prm+ qtn)∆v(t, r).

For 1(x−1)q

≤ 1< 1(x−1)p

, if doctor reveals condition truthfully, her welfare loss from this

group would be

U = −prm(x− 1)b− tna.

The difference between persuading them together and telling the truth is

∆U =U ′ − U

=prm[(x− 1)b−∆v(t, r)] + tn[a− q∆v(t, r)]

=prm[a− b

rm+ tntn− x

a− b

qtn+ prmqtn] + tnq[

a

q− (x− 1)a+ rm

b− a

rm+ tn− x

b− a

qtn+ prmprm]

=− tna[(x− 1)q − 1] +rmtn

rm+ tn(a− b)(p− q)

<0

where the inequality holds since p<q and (x− 1)q ≥ 1.

If this group is j = M , the proof is quite similar where you only need to replace

r and t with 1− r and 1− t.

Lemma 8. If πm ≤ 1<πs and t = 1, the doctor’s optimal strategy is

r =

0, if π0 ≤ 1,

1, if π1 ≥ 1,

nm{√

(a−b)(p−q)b[p(x−1)−1]

− 1}, o.w.

Proof of Lemma 8. If t = 1, there is only one variable r left for the doctor to maximize

her utility. If doctor reveals the truth,

U = −qn(x− 1)a−mb (11)

33

If she resorts to persuasion π(j = S|j = M) = r, for group j = S we have

H L

Ni qn+ rpm n+ rm

vi x qna+rpmbqn+rpm

na+rmbn+rm

By Lemma 7,

r<r∗ (12)

where r∗ is defined by

π(1, r∗) = 1. (13)

Otherwise, the doctor would be better off refraining from persuasion. Thus, her utility

would be

U ′ = −(qn+ rpm)∆v(1, r)− (1− r)mb (14)

The difference between persuasion and telling the truth is

∆U = U ′ − U

= qn[(x− 1)a−∆v(1, r)] + rm[b− p∆v(1, r)]

= qn[b− a

rm+ nmr − x

b− a

qn+ pmrpmr] + rmp[

b

p− (x− 1)b− xqn

a− b

qn+ pmr+ n

a− b

rm+ n]

= −rmb[p(x− 1)− 1] +mrn

rm+ n(a− b)(p− q)

∆U = m{−rb[p(x− 1)− 1] +rn

rm+ n(a− b)(p− q)} (15)

∂∆U

∂r= m{−b[p(x− 1)− 1] + (a− b)(p− q)

n2

(n+ rm)2} (16)

∂2∆U

∂r2<0

Therefore, there is a unique maximizer in the sub-range r ∈ [0, 1].

If (a− b)(p− q) ≤ b[p(x− 1)− 1],

∂∆U

∂r≤ 0.

Thus, ∆u reaches its maximum when r = 0.

If b[p(x− 1)− 1]<(a− b)(p− q)<(m+nn

)2b[p(x− 1)− 1], ∆u reaches its maximum

when

b[p(x− 1)− 1] = (a− b)(p− q)n2

(n+ rm)2.

34

If (a− b)(p− q) ≥ (m+nn

)2b[p(x− 1)− 1],

∂∆U

∂r≥ 0, 0 ≤ r ≤ 1

Thus, ∆U reaches its maximum when r = 1.

Notice that for the optimal r, by (15)

m{−rb[p(x− 1)− 1] +rn

rm+ n(a− b)(p− q)} ≥ 0.

Now we come back to illustrate that the optimal r already satisfies the constraint

r<r∗. Suppose r ≥ r∗, we arrive at

π(1, r) ≤ 1.

By Lemma 2. Thus,

U ′ = −(qn+ rpm)π(1, r)(vH − vL)− (1− r)mb ≥ −(qn+ rpm)(vH − vL)− (1− r)mb

Therefore, by (15)

∆U = U ′ − U ≥ m{−rb[p(x− 1)− 1] +rn

rm+ n(a− b)(p− q)}

However, by Lemma 7,

∆U<0

m{−rb[p(x− 1)− 1] +rn

rm+ n(a− b)(p− q)} ≤ ∆U<0,

a contradiction.

Lemma 9. If 1(x−1)p

≤ 1< 1(x−1)q

, the doctor’s persuasion strategy

π(j = S|j = S) = 1

r =

0, if (a− b)(p− q) ≤ b[p(x− 1)− 1],

1, if (a− b)(p− q) ≥ (m+nn

)2b[p(x− 1)− 1],

nm{√

(a−b)(p−q)b[p(x−1)−1]

− 1}, o.w.

will always dominate the strategy to pool all patient together

t = r

Proof of Lemma 9. Doctor’s strategy to bunch the patient together is same as r = 1

in Lemma 8. However, the optimal r in Lemma 8 maximize her utility over the

sub-range r ∈ [0, 1] which contains the case r = 1.

35

Proof of Proposition 4. We are about to prove that the doctor’s strategy in the propo-

sition dominates an arbitrary persuasion strategy

π(j = S|j = S) = t π(j = S|j = M) = r (17)

π(j = M |j = S) = 1− t π(j = M |j = M) = 1− r. (18)

If the doctor is able to divide the patient into two markets where the first one is

represented by:

Nij H L

S qtn tn

M prm rm

while second one isNij H L

S q(1− t)n (1− t)n

M p(1− r)m (1− r)m.

Furthermore, if she is able to independently employ persuasion in those two markets.

By Lemma 9, the strategy to dictate t1,r1 in market 1 and t2, r2 in market 2 where

t1 = 1

r1 =

0, if (a− b)(p− q) ≤ b[p(x− 1)− 1],

1, if (a− b)(p− q) ≥ ( rm+tntn

)2b[p(x− 1)− 1],

tnrm

{√

(a−b)(p−q)b[p(x−1)−1]

− 1}, o.w.

t2 = 1

r2 =

0, if (a− b)(p− q) ≤ b[p(x− 1)− 1],

1, if (a− b)(p− q) ≥ ( (1−r)m+(1−t)n(1−t)n

)2b[p(x− 1)− 1],

(1−t)n(1−r)m

{√

(a−b)(p−q)b[p(x−1)−1]

− 1}, o.w.

would dominate the original persuasion strategy indicated by (17) and (18). There-

fore, it suffices to prove that the persuasion strategy

t = 1

r =

0, if (a− b)(p− q) ≤ b[p(x− 1)− 1],

1, if (a− b)(p− q) ≥ (m+nn

)2b[p(x− 1)− 1],

nm{√

(a−b)(p−q)b[p(x−1)−1]

− 1}, o.w.

36

in the whole market weakly dominates persuading the patient by t1,r1 in market 1

and t2, r2 in market 2, respectively. Without loss of generality, we assume t ≥ r

(Otherwise, you could switch t and r).

r1 ≥ r ≥ r2

Case 1: (a− b)(p− q) ≤ b[p(x− 1)− 1]

In this case,

r1 = r2 = r = 0.

Thus, the doctor will always reveal the truth in market 1, 2 and the whole market.

Whether the segmentation of the whole market exists or not does not matter and we

are done here.

Case 2: b[p(x− 1)− 1]<(a− b)(p− q)<(m+nn

)2b[p(x− 1)− 1]

In this case,

0<r<1.

Suppose r and t are such that r1<1. Let ∆U1, ∆U2 and ∆U denote the benefit

of doctor’s persuasion in market 1, market 2, and the whole market, respectively. By

(15),

∆U = −rmb[p(x− 1)− 1] +rmn

rm+ n(a− b)(p− q)

Define C1 as

C1 = {

√(a− b)(p− q)

b[p(x− 1)− 1]− 1}.

r =n

mC1

Define C2 as

C2 = −C1b[p(x− 1)− 1] +C1

C1 + 1(a− b)(p− q)

∆U =− C1nb[p(x− 1)− 1] +C1n

2

C1n+ n(a− b)(p− q)

=− C1nb[p(x− 1)− 1] +C1n

C1 + 1(a− b)(p− q)

=n{−C1b[p(x− 1)− 1] +C1

C1 + 1(a− b)(p− q)}

=C2n

To derive ∆U1 from ∆U , we only need to replace n, m, r with tn, rm, r1 in (15).

∆U1 = C2tn

37

∆U2 = C2(1− t)n

∆U = ∆U1 +∆U2

Thus, we would be done here.

Otherwise, r1 = 1. Define r′1 as

r′1 =t

rr.

r′1 ≥ 1

By (16), ∆U1 reaches its unique maximum at r1 = r′1,

∆U1

∣∣∣∣r1=r′1

≥ ∆U1

∣∣∣∣r1=1

= ∆U1

Since

∆U = C2n = ∆U1

∣∣∣∣r1=r′1

+∆U2,

∆U ≥ ∆U1 +∆U2

which concludes the discussion for this case.

Case 3: (a− b)(p− q) ≥ (m+nn

)2b[p(x− 1)− 1]

In this case,

r1 = r = 1.

By (15),

∆U = m{−rb[p(x− 1)− 1] +rn

rm+ n(a− b)(p− q)}

= −b[p(x− 1)− 1]m+nm

m+ n(a− b)(p− q)

Define C3 = b[p(x− 1)− 1], C4 = (a− b)(p− q) where

C4>0.

∆U = −C3m+ C4nm

m+ n

∆U1 = −C3rm+ C4rmtn

rm+ tn

Suppose r and t are such that r2 has reached corner solution r2 = 1.

∆U2 = −C3(1− r)m+ C4(1− r)m(1− t)n

(1− r)m+ (1− t)n

38

By Lemma 6,

∆U1+∆U2 = −C3m+C4[rmtn

rm+ tn+

(1− r)m(1− t)n

(1− r)m+ (1− t)n] ≤ −C3m+C4

nm

m+ n= ∆U

Otherwise, r2<1.

∆U2 = C2(1− t)n

∂U2

∂r= 0

Since r1 = 1, by (15)

∆U1 = −rmb[p(x− 1)− 1] +rmtn

rm+ tn(a− b)(p− q)

∂∆U1

∂r= −mb[p(x− 1)− 1] + (

tn

rm+ tn)2m(a− b)(p− q)

By the definition of r1 = 1,

(a− b)(p− q) ≥ (rm+ tn

tn)2b[p(x− 1)− 1]

(tn

tn+ rm)2(a− b)(p− q) ≥ b[p(x− 1)− 1]

Thus,∂∆U1

∂r≥ 0

∂∆U1 +∆U2

∂r≥ 0

As a result, the doctor would be better off increasing r for a given t. Since

r2 =(1− t)n

(1− r)mC1,

this process would increase r2 to the point where r2 = 1, which has been addressed

earlier.

B Omitted Proofs in Section 4

Lemma 10. If the market contains two types (H or L) of customers whose relative

amount N and willingness to pay v are represented by

H L

N NH NL

v vH vL,

39

and they all claim to be type L, the optimal price to charge would be

P =

vL, if 1

NHNL

(vHvL

−1)≥ 1,

vH , o.w.

Proof of Lemma 10. Charging vL is optimal if and only if

vL(NH +NL) ≥ NHvH

NLvL ≥ NH(vH − vL)

1 ≥ NH

NL

(vHvL

− 1)

1NH

NL(vHvL

− 1)≥ 1

Proof of Lemma 4. 1. Suppose π(L|H) ∈ [0,min[πm, πs]].

In market i = H (market 1),

Nij H L

S [1− π(L|H)]qn 0

M [1− π(L|H)]pm 0

where the doctor would easily extract the full surplus from him. For instance, the

doctor could reveal the truth and charge vHS and vHM accordingly. Or she could

persuade them together by always reporting j = S (or M) and charge vH = x qna+pmbqn+pm

.

That is to say, the patient would end up with zero utility if he reports H.

In market i = L (market 2) where

Nij H L

S π(L|H)qn n

M π(L|H)pm m.

Under an arbitrary persuasion strategy of the doctor, this market would be divided

into two groups j ∈ {S,M}. For j = S,

H L

Ni (tqn+ rpm)π(L|H) tn+ rm

vi x tqna+rpmbtqn+rpm

tna+rmbtn+rm

40

Thus,

NH(t, r) = (tqn+ rpm)π(L|H)

By the proof of Lemma 2,

min[πs, πm]

π(L|H)≤ π(t, r) ≤ max[πs, πm]

π(L|H)

(Notice here the setup is slightly different where NH is π(L|H) times what it used to

be in Lemma 2.) Since π(L|H) ≤ min[πm, πs],

π(t, r) ≥ min[πs, πm]

π(L|H)≥ 1.

By Lemma 10, the doctor would always charge vL. Due to

min(xa, xb) ≥ xb ≥ a ≥ tna+ rmb

tn+ rm= vL,

the type H patient’s willingness to pay is always above the price charged. The proof

for another group j = M is almost the same where you need to replace r and t with

1 − r and 1 − t. Thus, he will always obtain a positive surplus in market i = L. As

a result, end up being in market i = L (market 2) is always preferred over market

i = H (market 1), which means the patient’s utility is monotonically increasing in

π(L|H).

2. Suppose π(L|H)>max(πm, πs). By the proof of Lemma 2,

min[πs, πm]

π(L|H)≤ π(t, r) ≤ max[πs, πm]

π(L|H)<1.

By Lemma 10, the doctor would charge vH in market 2 (i = L) leaving zero surplus

for type H patient. He would thereby be wiser to choose 0<π(L|H) ≤ min(πm, πs)

so that she would charge vL in market i = L where he would fare better.

Proof of Lemma 5. Case 1: 1NHNL

(vHvL

−1)≥ 1

If the producer directly charge a single optimal price, she would charge vL by

Lemma 10. Thus, the surplus loss would be

U = −NH(vH − vL).

If we let the customer to persuade first,

π(L|H) = min[1

NH

NL(vHvL

− 1), 1] = 1

41

which indicates the same surplus loss for the producer because she would still charge

vL and we are done here.

Case 2: 1NHNL

(vHvL

−1)<1

If the producer directly charge a single optimal price, she would charge vH by

Lemma 10. Thus, the surplus loss would be

U = −NLvL.

If we let the customer to persuade first,

π(L|H) = min[1

NH

NL(vHvL

− 1), 1] =

1NH

NL(vHvL

− 1).

The doctor would charge vL and her surplus loss is

U = −NH1

NH

NL(vHvL

− 1)(vH − vL) = −NLvL.

Proof of Proposition 5. The doctor is going to face market i = L

Nij H L

S π(L|H)qn n

M π(L|H)pm m

to decide whether to persuade or not. We need to point out that this structure bears

quite the resemblance with what we encounter in Section 3, except that we need to

replace q and p in Section 3 with π(L|H)p and π(L|H)q. Furthermore, under any

persuasion strategy adopted by the doctor, Lemma 5 holds for any j (S or M). That

is to say, we could rely on propositions in Section 3 to analyze her strategy if we take

the nuance, π(L|H), into account.

Since the optimal π(L|H) satisfy (7)

πs ≤ π(L|H) ≤ πm

due to πs ≤ πm ≤ 1, for group j = S we have

1NHS

NLS(vHS

vLS− 1)

=1

(x− 1)π(L|H)q=

πs

π(L|H)≤ 1

while for group j = M we have

1NHM

NLM(vHM

vLM− 1)

=1

(x− 1)π(L|H)p=

πm

π(L|H)≥ 1.

42

πs

π(L|H)≤ 1 ≤ πm

π(L|H)

We shall first prove that the doctor will always tell the truth and then analyze

the patient’s choice.

Case 1: πm

π(L|H)= 1

In this case,πs

π(L|H)≤ πm

π(L|H)= 1.

By Lemma 5, we could rely on Proposition 1 which indicates that the doctor is

indifferent with any of her persuasion strategy. Thus, revealing the truth is obviously

one of her best response.

Case 2: πm

π(L|H)>1

In this case,πs

π(L|H)≤ 1<

πm

π(L|H).

By Lemma 5 and Proposition 3, the doctor strictly prefer to reveal the truth.

Given telling the truth as the doctor’s best response, if

πs<π(L|H) ≤ πm,

by Lemma 10 the doctor is going to charge vHS for patients under j = S in market

i = L because

1NHS

NLS(vHS

vLS− 1)

=1

(x− 1)π(L|H)q=

πs

π(L|H)<1.

But the doctor will charge vLM for patients under j = M in market i = L because

1NHM

NLM(vHM

vLM− 1)

=1

(x− 1)π(L|H)p=

πm

π(L|H)≥ 1.

If type H patient is in market i = H (see proof of Lemma 4) or stays in market i = L

under condition j = S, he will end up with zero surplus. However, he will fare better

if he end up being in market i = L under condition j = M . Therefore, the patient’s

utility is monotonically increasing with π(L|H) over the sub-range (πs, πm]. Thus,

π(L|H) = πm

dominates all the other choice of π(L|H) where π(L|H) ∈ (πs, πm]. Consequently,

only two choices left: π(L|H) = πm or π(L|H) = πs.

If the patient chooses π(L|H) = πs,

pm(πm − πs)(x− 1)b ≤ na

43

mb(1− πs

πm

) ≤ na

πs

πm

≥ 1− na

mb

Therefore, the patient’s optimal action would be

π(L|H) =

πs, if πs

πm≥ 1− na

mb,

πm, o.w.

Proof of Proposition 6. By (7), optimal π(L|H) satisfies

πm ≤ π(L|H) ≤ πs.

For group j = S we have

1NHS

NLS(vHS

vLS− 1)

=1

(x− 1)π(L|H)q=

πs

π(L|H)≥ 1

while for group j = M we have

1NHM

NLM(vHM

vLM− 1)

=1

(x− 1)π(L|H)p=

πm

π(L|H)≤ 1.

πm

π(L|H)≤ 1 ≤ πs

π(L|H)

As we mentioned earlier in Proposition 5, we could use the results in Section 3 to an-

alyze the doctor’s response once we replace their p and q with π(L|H)p and π(L|H)q,

which is assured by Lemma 5.

If πs

π(L|H)= 1,

πm

π(L|H)≤ πs

π(L|H)= 1

where we could rely on Proposition 1 which indicates that revealing the truth is

obviously one of her best response.

If πs

π(L|H)>1,

πm

π(L|H)≤ 1<

πs

π(L|H)

implying that we could use the result of Proposition 4 which illustrates the doctor’s

best response as

t = 1

44

r =

0, if (a− b)(p− q) ≤ b[p(x− 1)− 1

π(L|H)], (19)

1, if (a− b)(p− q) ≥ (m+ n

n)2b[p(x− 1)− 1

π(L|H)], (20)

n

m{√

(a− b)(p− q)

b[p(x− 1)− 1π(L|H)

]− 1}, o.w. (21)

which concludes the discussion for the doctor’s best response. Next, we analyze the

patient’s action.

First, we point out that the patient’s utility is monotonically increasing with

π(L|H) over the sub-range [πm, π1]. Suppose π(L|H) ∈ [πm, π1]. By the definition of

π1 in (4),

(a− b)(p− q) ≥ (m+ n

n)2b[p(x− 1)− 1

π(L|H)]

which indicates the doctor’s response is

r = 1

by (20). Therefore, market i = L becomes one group

H L

Ni π(L|H)(pm+ qn) n+m

vi xpmb+qnapm+qn

mb+nam+n

.

By the definition of r∗ in (12) and (13), we have

π(t, r)>1.

By Lemma 10, the doctor is about to charge vL. Since

xb ≥ a>mb+ na

m+ n= vL,

the patient always prefer market i = L over market i = H where he end up with zero

surplus (see proof of Lemma 4). Thus, his utility is monotonically increasing with

π(L|H) over the sub-range [πm, π1].

Second, we shall prove that the patient’s utility is monotonically increasing with

π(L|H) over the sub-range [π0, πs]. Suppose π(L|H) ∈ [π0, πs]. By the definition of

π0 in (5),

(a− b)(p− q) ≤ b[p(x− 1)− 1

π(L|H)]

which indicates the doctor’s response is

r = 0

45

by (19). Therefore, she will always reveal the truth and the market i = L is thereby

separate between two conditions. For group j = S, we have

1NHS

NLS(vHS

vLS− 1)

=1

(x− 1)π(L|H)q=

πs

π(L|H)≥ 1

where she charges vLS by Lemma 10. For group j = M , we have

1NHM

NLM(vHM

vLM− 1)

=1

(x− 1)π(L|H)p=

πm

π(L|H)<1,

since πm<π0 ≤ π(L|H) which is guaranteed by (6). Thus, she charges vHM . If the

patient stays in market i = H (see proof of Lemma 4), he will return home with zero

surplus. By contrast, if he is in market i = L, he might obtain positive surplus, since

he could be under condition S. Thus, his utility is monotonically increasing over the

sub-range [π0, πs].

Third, we shall prove that the patient’s utility is convex over sub-range π(L|H) ∈[π1, π0]. Suppose π(L|H) ∈ [π1, π0]. By (21), we have

(a− b)(p− q)(n

n+ rm)2 = b[p(x− 1)− 1

π(L|H)] (22)

(n+ rm)2[p(x− 1)− 1

π(L|H)]− a− b

b(p− q)n2 = 0

Define G(r, π(L|H)) as

G = (n+ rm)2[p(x− 1)− 1

π(L|H)]− a− b

b(p− q)n2

∂G

∂r= 2m(n+ rm)[p(x− 1)− 1

π(L|H)]

∂G

∂π(L|H)= [

n+ rm

π(L|H)]2

By implicit function theorem,

dr

dπ(L|H)= −

∂G∂π(L|H)

∂G∂r

= − 1

[π(L|H)p(x− 1)− 1]

n+ rm

2mπ(L|H)<0 (23)

where the inequality is due to π(L|H) ≥ π1>πm. Now we characterize the patient’s

gain uH from persuasion. Recall that only by staying in market 2 under j = S could

46

he obtain a positive gain. Define β = pq.

uH =π(L|H)qn+ pmr

qn+ pm(x

qna+ rpmb

qn+ rpm− na+ rmb

n+ rm)

=π(L|H)

qn+ pm[xqna+ xrpmb− na+ rmb

n+ rm(qn+ pmr)]

=π(L|H)

qn+ pm[xqna+ xrpmb− qnb− rpmb− n

a− b

n+ rm(qn+ pmr)]

=qπ(L|H)

qn+ pm[xna+ xrβmb− nb− rβmb− n

a− b

n+ rm(n+ βmr)]

=π(L|H)

n+mβ[xna− nb+ (x− 1)rβmb− βn(a− b) + n2 a− b

n+ rm(β − 1)]

Define Z(r) as

Z(r) = xna− nb+ (x− 1)rβmb− βn(a− b) + n2 a− b

n+ rm(β − 1). (24)

Thus,

uH =π(L|H)

n+mβZ(r). (25)

By (22),dZ(r)

dr= (x− 1)βmb−mn2 a− b

(n+ rm)2(β − 1) =

bm

π(L|H)q.

dZ(r)

dπ(L|H)=dZ(r)

dr

dr

dπ(L|H)

=bm

π(L|H)q

dr

dπ(L|H)

=− bm

q

1

[π(L|H)p(x− 1)− 1]

n+ rm

2mπ2(L|H)

=− b(n+ rm)

2π2(L|H)q

1

[π(L|H)p(x− 1)− 1]

=− b(n+ rm)

2π2(L|H)q

b(n+ rm)2

π(L|H)(a− b)(p− q)n2

=− b2

2n2q(a− b)(p− q)

(n+ rm)3

π3(L|H)

where the third equality is due to (23) while the fifth one derives from (22).

duH

dπ(L|H)=

1

n+ βm{Z(r) + π(L|H)

dZ(r)

dπ(L|H)}

=1

n+ βm{Z(r)− b2

2n2q(a− b)(p− q)

(n+ rm)3

π2(L|H)}

47

d2uH

dπ2(L|H)=

1

n+ βm{ dZ(r)

dπ(L|H)− b2

2n2q(a− b)(p− q)

d (n+rm)3

π2(L|H)

dπ(L|H)}

=1

n+ βm{− b2

2n2q(a− b)(p− q)

(n+ rm)3

π3(L|H)− b2

2n2q(a− b)(p− q)

d (n+rm)3

π2(L|H)

dπ(L|H)}

=1

n+ βm

b2

2n2q(a− b)(p− q){−(n+ rm)3

π3(L|H)−

d (n+rm)3

π2(L|H)

dπ(L|H)}

=1

n+ βm

b2

2n2q(a− b)(p− q){−(n+ rm)3

π3(L|H)+ 2

(n+ rm)3

π3(L|H)− 1

π2(L|H)

d(n+ rm)3

dπ(L|H)}

=1

n+ βm

b2

2n2q(a− b)(p− q){(n+ rm)3

π3(L|H)− 1

π2(L|H)

d(n+ rm)3

dπ(L|H)}

>0

where the second equality utilize the result from dZ(r)dπ(L|H)

while the inequality is guar-

anteed by (23). That is to say, patient’s utility is convex over sub-range π(L|H) ∈[π1, π0], which means his optimal choice over this sub-range is either π(L|H) = π1 or

π(L|H) = π0.

To sum up, the first, second and the third part indicates π(L|H) = π1, π(L|H) =

π1 orπ0, π(L|H) = πs as the optimal solution over the sub-range [πm, π1], [π1, π0],

[π0, πs], respectively. The best choice for him over the entire range [πm, πs] would be

either π(L|H) = π1 or π(L|H) = πs. If π(L|H) = π1,

uH =q

qn+ pmπ1Z(1)

by (25). If π(L|H) = πs,

uH =qn

qn+ pmπs(x− 1)a =

n

qn+ pma.

Thus, the optimal strategy for the patient would be

π(L|H) =

π1, if π1 ≥ naqZ(1)

≜ π∗,

πs, o.w.

where Z(r) is defined by (24). The best response of the doctor is t = 1,

r =

1, if π(L|H) = π1,

0, if π(L|H) = πs.

48

Proof of Proposition 7. Here, we have πs ≤ 1<πm. Besides,since the optimal π(L|H)

satisfy (7),

πs ≤ π(L|H) ≤ 1<πm.

πs

π(L|H)≤ 1<

πm

π(L|H).

As we discussed in the proof of Proposition 5, the doctor strictly prefer to reveal the

truth by Lemma 5 and Proposition 3. Since the patient’s utility is monotonically

increasing with π(L|H) over the sub-range (πs, 1]. Thus,

π(L|H) = 1

dominates all the other choice of π(L|H) where π(L|H) ∈ (πs, 1]. Consequently, only

two choices left: π(L|H) = πs or π(L|H) = 1.

If the patient chooses π(L|H) = πs,

pm(1− πs)(x− 1)b ≤ na

1− πs

πm

≤ na

mb

Therefore, the patient’s optimal action would be

π(L|H) =

πs, if 1−πs

πm≤ na

mb,

1, o.w.

Proof of Proposition 8. In the proof of Proposition 6, the patient’s utility is mono-

tonically increasing over the sub-range [πs, π1] and [π0, 1]. In addition, his utility is

convex over the sub-range [π1, π0]. We can directly borrow these results to simplify

our analysis here. The only difference is that in this case πs ≤ 1<πm which indicates

that the relative order of π1, π0 and 1 is undetermined and we shall address it case

by case.

Case 1: π1 ≥ 1

As in the proof of Proposition 6, the patient’s utility is increasing over π(L|H) ∈[πs, π1], which results in π(L|H) = 1 as the patient’s optimal choice. By the definition

of π1 in (4), π1 ≥ 1 indicates (a− b)(p− q) ≥ (m+nn

)2b[p(x− 1)− 1]. Since π(L|H) ≤1 ≤ π1, by (20) we have

t = r = 1

as the doctor’s best response.

49

Case 2: π1<1<π0

By the definition of π1 and π0 in (4) and (5), π1<1<π0 indicates b[p(x− 1)− 1] ≤(a− b)(p− q)<(m+n

n)2b[p(x− 1)− 1]. As in the proof of Proposition 6, the patient’s

utility is convex over the sub-range π(L|H) ∈ [π1, π0]. Thus, the patient would choose

either π(L|H) = π1 or π(L|H) = 1 as the optimal choice.

By (25), if π(L|H) = π1,

uH =q

qn+ pmπ1Z(1).

If π(L|H) = 1,

uH =q

qn+ pmZ(

n

m{

√(a− b)(p− q)

b[p(x− 1)− 1]− 1}).

Thus, the optimal strategy for the patient would be

π(L|H) =

π1, if π1 ≥Z( n

m{√

(a−b)(p−q)b[p(x−1)−1]

−1})Z|r=1

≜ π∗∗,

1, o.w.

where Z(r) is defined by (24).

By (20) and (21), the doctor’s best response is t = 1,

r =

1, if π(L|H) = π1,

nm{√

(a−b)(p−q)b[p(x−1)−1]

− 1}, if π(L|H) = 1.

Case 3: π0 ≤ 1

By the definition of π0 in (5), π0 ≤ 1 indicates (a − b)(p − q)<b[p(x − 1) − 1].

As in the proof of Proposition 6, the patient would choose either π(L|H) = π1 or

π(L|H) = 1 as the optimal choice.

By (25), if π(L|H) = π1,

uH =q

qn+ pmπ1Z(1).

If π(L|H) = 1,

uH =qn

qn+ pm(x− 1)a.

Thus, the optimal strategy for the patient would be

π(L|H) =

π1, if π1 ≥ π∗

πs,

1, o.w.

50

By (19) and (20), the doctor’s best response is t = 1,

r =

1, if π(L|H) = π1,

0, if π(L|H) = 1.

C Omitted Proofs in Section 7.1

Lemma 11. Given patient moves (π(L|H)) first, if p>q and π1<1, the doctor’s gain

from her persuasion strategy is monotonically decreasing with π(L|H) over the sub-

range [π1,min(π0, 1)].

Proof of Lemma 11. Suppose π(L|H) ∈ [π1,min(π0, 1)]. The doctor’s gain from her

persuasion is delivered by (15). If the patient’s first move π(L|H) is taken into account

in market i = L (we need to replace p and q with π(L|H)p and π(L|H)q),

∆U = −rmb[π(L|H)p(x− 1)− 1] +rmn

rm+ n(a− b)(p− q)π(L|H).

Since r(π(L|H)) is the unique maximizer given π(L|H), by envelope theorem

d∆U

dπ(L|H)=

∂∆U

∂π(L|H)

∣∣∣∣r=r(π(L|H))

=− rmbp(x− 1) +rmn

rm+ n(a− b)(p− q)

=rm[−bp(x− 1) +n

rm+ n(a− b)(p− q)]

By (22),

n

n+ rm=

√b[p(x− 1)− 1

π(L|H)]

(a− b)(p− q)

d∆U

dπ(L|H)= rm[−bp(x− 1) +

√b[p(x− 1)− 1

π(L|H)](a− b)(p− q)] (26)

b[p(x− 1)− 1

π(L|H)](a− b)(p− q)<bp(x− 1)(a− b)(p− q)

≤bp(x− 1)(xb− b)(p− q)

<b2p2(x− 1)2

where the first inequality derives from π(L|H)>πm and the second one is due to

xb ≥ a. Thus, √b[p(x− 1)− 1

π(L|H)](a− b)(p− q)<bp(x− 1)

51

By (26),d∆U

dπ(L|H)<0.

Lemma 12. Given doctor moves first, if p>q and π1<1, the doctor’s gain from her

persuasion strategy is equivalent to the type H patient’s loss incurred by the doctor’s

persuasion.

Proof of Lemma 12. Suppose doctor moves first. By Lemma 1, the optimal persua-

sion strategy for the customer would be

π(L|H) = min[π(t, r), 1].

We need to point out that the type H patient’s gain by his persuasion strategy derives

from the doctor. Recall that if doctor moves first, the L type patient will always accept

the deal with zero surplus left. That is to say, their situation will remain the same

with or without type H patient’s persuasion. However, type H patient will always

accept the deal as well. If he gains some surplus by his persuasion, this gain must

derives from the doctor. By the proof of Lemma 3, the doctor’s loss is

U = −NH(t, r)π(L|H)∆v(t, r)

which is exactly the type H patient’s gain ∆uH by his persuasion

∆uH = NH(t, r)π(L|H)∆v(t, r).

Thus, if the doctor would somehow benefit from her persuasion, her gain must comes

from the type H patient. To be more specific, she gains some surplus by reducing

her loss which is caused by the patient’s action. By the proof of Lemma 8, if doctor

reveals the truth,

U = −qn(x− 1)a−mb.

If she optimally persuades,

U ′ = −(qn+ rpm)∆v(1, r)− (1− r)mb

Therefore, her gain from persuasion U ′ − U is the type H patient’s loss

U ′ − U = −(∆uH ′ −∆uH)>0.

52

Lemma 13. Suppose p>q, π1<1. If we switch the setup from the doctor first to

patient first,

1. the type H patient is strictly worse off.

2. the doctor is at least as good as before.

Proof of Lemma 13. In this proof, we shall construct a new model which acts as a

bridge between two setups. Since this lemma is critical for the proof of the next

theorem, we will elaborate this one here in detail.

The Model

There are two types of patients H or L on the market.

N i = i = H i = H, i = L i = i = L

j = S [1− π(L|H)]qn π(L|H)qn n

j = M [1− π(L|H)]pm π(L|H)pm m.

In the first period, the patient reports his type through i by Bayesian persuasion. In

the second period, the doctor persuades i = L patient. (There is no need to persuade

the i = H patient, since she could easily extract the full surplus by the proof of

Lemma 4). In the third period, type i = H, i = L patient could choose to turn

back to i = i = H (to eat their words) with some positive probability to guarantee

that vL(t, r) (vL(1− t, 1− r)) will be charged instead of vH(t, r) (vH(1− t, 1− r)) in

the corresponding sub-market. In the fourth period, the doctor chooses an optimal

price to charge in sub-markets i = L, j = S and i = L, j = M , which concludes our

fictitious model.

Our first setup where the doctor moves first is equivalent to this fictitious model

where π(L|H) = 1. Suppose π(L|H) = 1. The doctor is going to face

Nij H L

S qn n

M pm m

and persuades them. Then what happens in the third period in our fictitious model

is the patient’s persuasion at work.

On the other hand, our second setup is the same as our fictitious model if we

restrain type i = H, i = L patient from his action in the third period. That is, he has

no right to turn back on his words and the doctor is going to charge immediately.

Now we analyze our fictitious model. Let uHk and Uk denote the type H patient

and the doctor’s utility in the following step k. Notice they are functions of π(L|H).

We focus our attention on the sub-range π(L|H) ∈ [π1, 1].

53

In step 1, suppose we restrict the doctor from persuasion in the second period. If

so, uH1 is strictly increasing with π(L|H) over the sub-range [0,min(πs, 1)] and remains

a constant over [min(πs, 1), 1]. The logic is straightforward. If π(L|H) ∈ [0, πm], he

strictly benefits from π(L|H) regardless of his condition. If π(L|H) ∈ [πm,min(πs, 1)],

he would be strictly better off under condition S. Otherwise, π(L|H) ∈ [min(πs, 1), 1],

increasing π(L|H) is futile, since either π(L|H) = 1 ≤ πs reaches its corner solution or

πs<1 implies that all the increment over πs needs to be offset in the third period due

to π(L|H) ≥ πs>πm. On the other hand, U1 is strictly decreasing over [0,min(πs, 1)]

and remains a constant over [min(πs, 1), 1].

In step 2, we relax the restriction and allow the doctor to persuade in the second

period. Notice that this would make a difference only over [π1,min(π0, 1)] where the

doctor does persuade by (19), (20), (21) and the definition of π1 and π0 in (4), (5).

By Lemma 11, the doctor’s gain from her persuasion ∆U is strictly decreasing over

the sub-range [π1,min(π0, 1)] and becomes zero over [min(π0, 1), 1]. Thus,

U2 = U1 +∆U

is strictly decreasing over [π1,min(πs, 1)] and remains a constant over [min(πs, 1), 1]

due to πm<π1<π0 ≤ πs. On the other hand, the type H patient’s loss incurred by

her persuasion is the same as ∆U by Lemma 12, which means

uH2 = uH

1 −∆U

is strictly increasing over [π1,min(πs, 1)] and remains a constant over [min(πs, 1), 1].

In step 3, we restrain the patient from turning back to i = i = H (eating his

words) in period 3 to mirror our second setup. If so, this would have no effects on

the doctor’s utility over the sub-range [π1, 1] by Lemma 5. Thus,

U3 = U2.

By contrast, this would have a significant impact on the patient side. By (21), the

doctor would persuade the patient with π(S|M) = r over π(L|H) ∈ [π1,min(π0, 1)].

For those who left in j = M , she will charge vHM due to π(L|H)>πm by Lemma 10.

Therefore, it would incur welfare loss (1− r)mb for the type H patient by Lemma 1.

For π(L|H) ∈ [π0, 1] if π0<1, his welfare loss is mb with r = 0.

uH3 = uH

2 − (1− r)mb

.

54

Since it must be one of π1<1 ≤ πs, πs<1, we shall compare both players’ welfare

under two setups case by case.

Case 1: π1<1 ≤ πs

In this case, the type H patient will choose either π1 or 1 when he moves first.

If he chooses π1 under the second setup, then his welfare is strictly getting worse

than the first setup (doctor first), since

uH3 (π1) = uH

2 (π1)<uH2 (1)

while type L patient remains the same. In addition, the doctor is strictly better than

when she moves first

U3(π1) = U2(π1)>U2(1)

If he chooses 1 under the second setup, then his welfare is strictly getting worse

than the first setup (doctor first), since

uH3 (1) = uH

2 (1)− (1− r)mb<uH2 (1)

while type L patient under j = M gives up the transaction. In addition, the doctor’s

welfare remains the same

U3(1) = U2(1)

Case 2: πs<1

In this case, the type H patient will choose either π1 or πs when he moves first.

If he chooses π1 under the second setup, then his welfare is strictly getting worse

than the first setup (doctor first), since

uH3 (π1) = uH

2 (π1)<uH2 (1)

while type L patient remains the same. In addition, the doctor is strictly better than

when she moves first

U3(π1) = U2(π1)>U2(1)

If he chooses πs under the second setup, then his welfare is strictly getting worse

than the first setup (doctor first), since

uH3 (πs) = uH

2 (πs)−mb = uH2 (1)−mb<uH

2 (1)

while type LM patient gives up the transaction due to r = 0. In addition, the doctor’s

welfare remains the same

U3(πs) = U2(πs) = U2(1)

55

Proof of Theorem 1. We shall address this issue case by case, including double full

persuasion, double partial persuasion, single full persuasion. In the Section 4.2, we

have seen that there is no difference between those two setups under double full

persuasion (πm>1, πs>1). For the following part, we shall address another two cases.

We shall go through this proof by switching from the first setup where doctor moves

first to the second setup to see how the welfare of both parties change accordingly.

Case 1: Double Partial Persuasion p ≤ q

In this case, πs ≤ πm ≤ 1. First, suppose doctor moves first. By Lemma 3, her

utility is

U = −na−mb.

By proposition 5, if we switch to the second setup, the doctor will always prefer to

reveal the truth and the patient shall choose either π(L|H) = πs or π(L|H) = πm.

If he chooses π(L|H) = πs (see the proof of Proposition 5 for the condition under

which he does so), the doctor would charge vLS and vLM for those under condition S

and M , respectively, by Lemma 10. Thus, her utility is higher

U = −na− pmπs(x− 1) ≥ −na−mb,

since πs ≤ πm ≤ 1. Meanwhile, type H patient’s utility is lower and type L patient

remains the same.

If he chooses π(L|H) = πm (see the proof of Proposition 5 for the condition under

which he does so), the doctor would charge vHS and vLM for those under condition

S and M , respectively, by Lemma 10. Thus, her utility is the same as before

U = −na−mb,

since πs ≤ πm ≤ 1. Meanwhile, type LS patient gives up the transaction and type LM

patient remains the same. The type H patient is strictly worse than before because

the doctor’s income does not change and some type L patients quit the transaction.

Case 2: Double Partial Persuasion p>q

In this case, πm<πs ≤ 1. Since π1<πs ≤ 1, by Lemma 13, the type H patient is

strictly worse off and the doctor is weakly better than setup 1.

Case 3: Single Full Persuasion p ≤ q

In this case, πs ≤ 1<πm. This case is quite similar to the case 1. First, suppose

doctor moves first. By proof of Lemma 1, her utility is

U = −na− pm(x− 1)b.

56

By proposition 7, if we switch to the second setup, the doctor will always prefer

to reveal the truth and the patient shall choose either π(L|H) = πs or π(L|H) = 1.

If he chooses π(L|H) = πs (see the proof of Proposition 7 for the condition under

which he does so), the doctor would charge vLS and vLM for those under condition S

and M , respectively, by Lemma 10. Thus, her utility is higher

U = −na− pmπs(x− 1)b ≥ −na− pm(x− 1)b,

since πs ≤ 1<πm. Meanwhile, type H patient’s utility is lower and type L patient

remains the same.

If he chooses π(L|H) = 1 (see the proof of Proposition 7 for the condition under

which he does so), the doctor would charge vHS and vLM for those under condition

S and M , respectively, by Lemma 10. Thus, her utility is the same as before

U = −na− pm(x− 1)b,

since πs ≤ 1<πm. Meanwhile, type LS patient gives up the transaction and type LM

patient remains the same. The type H patient is strictly worse than before because

the doctor’s income does not change and some type L patients quit the transaction.

Case 4: Single Full Persuasion p>q

In this case, πm ≤ 1<πs.

If π1 ≥ 1 which indicates (a− b)(p− q) ≥ (m+nn

)2b[p(x− 1)− 1], those two setups

are exactly the same. By Proposition 4, if doctor moves first, r = 1 and π(L|H) = 1.

By the proof of Proposition 6, if patient moves first, his utility is strictly increasing

with π(L|H) over the sub-range [πm, π1], which implies π(L|H) = 1, r = 1.

Otherwise, π1<1. By Lemma 13, the type H patient is strictly worse off and the

doctor is weakly better than before.

D Omitted Proofs in Section 5

Let π(t, r) denote 1NH (t,r)

NL(t,r)(vH (t,r)

vL(t,r)−1)

in those two groups created by the doctor’s signal.

π(t, r) =1

x tqna+rpmbtqn+rpm

tn+rmtna+rmb

− 1· tn+ rm

tqn+ rpm(27)

π(t, r) only relies on tr. Therefore, we take the limit and define

π(0, 0) = π(1, 1).

57

The intuition is simple. For strategy t = r = 1, it is exactly the same as t = r = 0.3

that pooling into one group is the same as pooling into one and further divide to two

homogenous groups.

Proof of Proposition 9. Since the patient sends his signal after the doctor’s signal,

his best response for commitment is just π(t, r) and π(1 − t, 1 − r) for those two

realizations of j, respectively. If p = q, it degenerates to a trivial case and the result

is obvious. Now we focus on p = q.

First, suppose π(t, r) = π(1−t, 1−r). Without loss of generality, let π(t, r)>π(1−t, 1 − r). Given the patient’s best response, the doctor would sustain full loss by

Lemma 3 and always has incentive to switch t and r with 1− t and 1− r, i.e., swap

those two groups. After swapping, patient only commit π(1− t, 1− r) in group t, r.

Since

π(1− t, 1− r)<π(t, r),

it would generate a strict positive gain for the doctor relative to full loss

∆U = (tqn+ rpm)[π(t, r)− π(1− t, 1− r)]∆v(t, r).

In another group, doctor’s surplus remains the same. Thus, she would deviate to

profit.

Second, suppose π(t, r) = π(1 − t, 1 − r). This case is identical to the situation

where the doctor would pool all the patient together. Given the patient’s best re-

sponse, the doctor would deviate to fully separating strategy. After that, patient only

commit π(t, r) in group S and group M . Since

π(t, r)<max(πs, πm),

the logic is the same and she would deviate to fully separate the patient to profit.

Proof of Proposition 11. First, suppose π(t, r) = π(1 − t, 1 − r). Without loss of

generality, let π(t, r)>π(1− t, 1− r). If π(1− t, 1− r)<1, the doctor will always have

incentive to swap two groups, since she will gain a strict positive surplus

∆U = (tqn+ rpm)[min[1, π(t, r)]− π(1− t, 1− r)]∆v(t, r).

If π(1− t, 1− r) ≥ 1, the patient would choose π(L|H) = 1. Given π(L|H) = 1, the

doctor would choose to reveal the truth (fully separate patients). To see this, notice

in Proposition 3, patient’s strategy is

π(L|H) =

1, if j = M,

πs, o.w.

58

However, for the doctor, this is identical to

π(L|H) =

1, if j = M,

1, o.w.

Therefore, by Proposition 3, the doctor would choose to reveal the truth. Conse-

quently, t = 0 and r = 1 would render π(1− t, 1− r)<1, a contradiction.

Second, suppose π(t, r) = π(1 − t, 1 − r). If π(1, 1)<1, π(L|H) = π(t, r) =

π(1, 1)<1. Given π(L|H)<1, the doctor would sustain full loss and would deviate to

fully separating the patient. By doing so, she could gain

∆U = pm(πm − π(1, 1))∆v(0, 1)

relative to full loss. If π(1, 1) ≥ 1, π(L|H) = 1. Given π(L|H) = 1, the doctor would

choose to reveal the truth, a deviation from π(t, r) = π(1− t, 1− r).

Proof of Proposition 12. First, suppose π(t, r) = π(1 − t, 1 − r). Without loss of

generality, let π(t, r)>π(1− t, 1− r). If π(1− t, 1− r)<1, by the proof of Proposition

11, the doctor would deviate to swap two groups. If π(1 − t, 1 − r) ≥ 1, the patient

would choose π(L|H) = 1. Given π(L|H) = 1, the doctor would choose

r =

0, if (a− b)(p− q) ≤ b[p(x− 1)− 1],

1, if (a− b)(p− q) ≥ (m+nn

)2b[p(x− 1)− 1],

nm{√

(a−b)(p−q)b[p(x−1)−1]

− 1}, o.w.

by Proposition 4. However, all these cases would violate either π(t, r) = π(1−t, 1−r)

or π(1− t, 1− r) ≥ 1. Thus, no equilibrium.

Second, suppose π(t, r) = π(1 − t, 1 − r). If π(t, r) = π(1, 1)<1, π(L|H) =

π(t, r)<1. Given π(L|H)<1, the doctor would deviate to fully separating the patient

by the proof of Proposition 11. If π(1, 1) ≥ 1, π(L|H) = 1. Given π(L|H) = 1,

equilibrium exists if and only if the doctor chooses r = 1. Therefore, by Proposition

4, the unique equilibrium only exists under condition (a− b)(p− q) ≥ (m+nn

)2b[p(x−1)− 1].

Proof of Proposition 14. First, suppose π(t, r) = π(1 − t, 1 − r). Under t = r, the

patient would choose π(L|H) = π(t, r) = π(1, 1). Given π(L|H) = π(1, 1), the doctor

would deviate to fully separating strategy to obtain a strictly positive gain

∆U = qn[πs − π(1, 1)]∆v(1, 0)

59

relative to full loss. Thus, she will deviate to profit.

Second, suppose π(t, r) = π(1−t, 1−r). Without loss of generality, let π(t, r)>π(1−t, 1− r) (i.e., t>r). Facing this, the patient would choose either π(L|H) = π(t, r) or

π(L|H) = π(1− t, 1− r). We shall address this issue case by case.

Case 1: π(L|H) = π(1− t, 1− r)

In this case, the doctor would deviate to pooling the patient into one group if and

only if

(tqn+rpm)[π(t, r)−π(1−t, 1−r)]∆v(t, r)<(qn+pm)[π(1, 1)−π(1−t, 1−r)]∆v(1, 1)

(tna+ rmb)[1− π(1− t, 1− r)

π(t, r)]<(na+mb)[1− π(1− t, 1− r)

π(1, 1)]

na+mb

π(1, 1)<tna+ rmb

π(t, r)+

(1− t)na+ (1− r)mb

π(1− t, 1− r)

By (27),

(na+mb)[xqna+ pmb

na+mb− qn+ pm

n+m]<(tna+ rmb)[x

tqna+ rpmb

tna+ rmb− tqn+ rpm

tn+ rm]

+(t′na+ r′mb)[xt′qna+ r′pmb

t′na+ r′mb− t′qn+ r′pm

t′n+ r′m]

where we use t′ and r′ to denote 1− t and 1− r.

(na+mb)qn+ pm

n+m>(tna+ rmb)

tqn+ rpm

tn+ rm+ (t′na+ r′mb)

t′qn+ r′pm

t′n+ r′m

By Lemma 6,

(na+mb)(qn+ pm)mn(t− r)2

n+m>(t′n+ r′m)[qnna(t− r)t+ pmmb(r − t)r]

+(tn+ rm)[qnna(t′ − r′)t′ + pmmb(r′ − t′)r′]

(na+mb)(qn+ pm)mn(t− r)

n+m>(t′n+ r′m)[qnnat− pmmbr]

+(tn+ rm)[−qnnat′ + pmmbr′]

(na+mb)(qn+ pm)mn(t− r)

n+m>naqmn(tr′ − t′r) +mbpmn(tr′ − rt′)

(na+mb)(qn+ pm)

n+m>naq +mbp

mnap+mnbq>mnaq +mnbp

ap− aq>bp− bq

a>b

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which is satisfied trivially. Therefore, the doctor would deviate to pooling the patient

together. No equilibrium under this case.

Case 2: π(L|H) = π(t, r)

If π(t, r)<πs, the doctor would deviate to fully separating the patient, since she

would obtain a strictly positive gain

∆U = qn[πs − π(t, r)]∆v(1, 0).

With π(t, r) = πs (i.e., t = 1, r = 0), the doctor would have no incentive to deviate.

We need to check if π(L|H) = πs is indeed the patient’s best response given t = 1, r =

0.

The patient would choose π(L|H) = πs if and only if

qn[πs − πm](x− 1)a ≥ mb

na(1− πm

πs

) ≥ mb

πm

πs

≤ 1− mb

na

Proof of Proposition 16. First, suppose π(t, r) = π(1 − t, 1 − r). Under t = r, the

patient would choose π(L|H) = min[1, π(t, r)]. If π(t, r)<1, given π(L|H) = π(t, r),

the doctor would sustain full loss and have incentive to deviate to fully separating

strategy (t = 1, r = 0) to obtain a strictly positive gain

∆U = qn[πs − π(t, r)]∆v(1, 0)

relative to full loss. Thus, she will deviate to profit. If π(t, r) ≥ 1, given π(L|H) = 1,

the doctor’s best response is characterized in Proposition 8. Thus, only if π1 ≥ 1 that

equilibrium would exist in this case. Otherwise, the doctor would deviate to r<1, a

deviation for t = r. Under π1 ≥ 1, unique equilibrium exists such that π(L|H) = 1

and t = r = 1.

Second, suppose π(t, r) = π(1−t, 1−r). Without loss of generality, let π(t, r)>π(1−t, 1 − r) (i.e., t>r). Facing this, the patient would choose π(L|H) = π(t, r) or

π(L|H) = π(1− t, 1− r) or π(L|H) = 1. We shall address this issue case by case.

Case 1: π(L|H) = π(1− t, 1− r)<1

If π(t, r) ≤ 1, the doctor would deviate to fully pooling strategy by the proof of

Proposition 14.

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Besides, if π(t, r)>1 or π(1, 1)>1, the same logic still holds and the corner solution

does not bite our previous proof in Proposition 14 whatsoever. The reason is that we

only consider the doctor’s surplus relative to full loss. Thus, the doctor would deviate

to fully pooling strategy (t = r) no matter what. No equilibrium as a result.

Case 2: π(L|H) = π(t, r)<1

Facing π(L|H) = π(t, r)<1, the doctor would sustain full loss and have incentive

to deviate to fully separating strategy t = 1, r = 0 to gain a strictly positive gain

∆U = qn[πs − π(t, r)]∆v(1, 0).

No equilibrium as a result.

Case 3: π(L|H) = 1

Facing π(L|H) = 1, the doctor’s best response is characterized in Proposition 8.

The equilibrium exists if and only if π(L|H) = 1 is indeed the patient’s best response.

If π1<1<π0, the patient would choose π(L|H) = 1 instead of π(L|H) = πm if and

only if1− πm

n+mβ· Z(r) ≥ pm(1− r)

pm+ qnπm(x− 1)b

by (11).

(1− πm)Z(r) ≥mb(1− r)

q

πm ≤ 1− mb(1− r)

qZ(r)≜ Π∗

If π0 ≤ 1, the patient would choose π(L|H) = 1 instead of π(L|H) = πm if and

only if

qn(1− πm)(x− 1)a ≥ mb

1− πm

πs

≥ mb

na

E Omitted Proofs in Section 7.2

Proof of Theorem 2. From Section 5, equilibrium exists only under three cases. Case

1. πs = πm ≤ 1. Case 2. πs>1, πm>1. Case 3. πs>1, πm ≤ 1, π1 ≥ 1.

By Proposition 9 and Proposition 13, if p = q, for two setups equilibrium re-

mains the same where π(L|H) = πs(= πm) and the doctor is indifferent between her

persuasion strategy.

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By Proposition 10 (Proposition 12) and Section 6.2 (Proposition 16), it is clear

that for two setups equilibrium (if exists) remains the same where t = r = 1 and

π(L|H) = 1.

F Omitted Proofs in Section 7.3

Proof of Theorem 3. The proof follows directly from the comparison between Propo-

sition 2 and Proposition 10; Proposition 4 and Proposition 12; Proposition 1 and

Proposition 9.

Proof of Theorem 4. By the comparison between Proposition 5 and Proposition 13,

Proposition 7 and Proposition 15, Section 4.2 and Section 6.2, the equilibrium remains

the same for those two setups. Only two cases left: 1. πm<πs ≤ 1. 2. πm ≤1<πs, π1<1.

Case 1: πm<πs ≤ 1

By the proof of Proposition 6, the patient would choose between two corner so-

lutions π(L|H) = π1 or π(L|H) = πs in sequential-commitment game. However, by

Proposition 14, equilibrium only exists for π(L|H) = πs in simultaneous-commitment

game.

If the patient chooses π(L|H) = π1 in sequential-commitment game, his utility

would be at least as good as choosing π(L|H) = πs which is the action he takes

in simultaneous-commitment game. Otherwise, his utility is the same for those two

setups. Therefore, the patient weakly prefers sequential game where his choice set is

larger.

If the patient chooses π(L|H) = π1 instead of π(L|H) = πs in sequential-commitment

game, the doctor’s surplus will increase as well due to the monotonicity of U3 (see the

Proof of Lemma 13). That is to say, the doctor also weakly prefers sequential setup.

Case 2: πm ≤ 1<πs, π1<1

The logic is the same as Case 1 that the patient would choose between two corner

solutions of π(L|H) under sequential-commitment setup. If he chooses the smaller

one, both players’ payoff would be higher in sequential-commitment game than in the

simultaneous-commitment one.

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