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Journal of Economic Behavior & Organization 103 (2014) 1–16

Contents lists available at ScienceDirect

Journal of Economic Behavior & Organization

j ourna l h om epa ge: w ww.elsev ier .com/ locate / jebo

xpert advice for amateurs�

rnest K. Lai ∗

epartment of Economics, Lehigh University, United States

r t i c l e i n f o

rticle history:eceived 18 January 2013eceived in revised form 17 March 2014ccepted 20 March 2014vailable online 13 April 2014

EL classification:728283

eywords:mateurheap talkxpertnformed decision maker

a b s t r a c t

A biased, perfectly informed expert advises a partially and privately informed decisionmaker using cheap-talk message. The decision maker can tell whether the state is “high”or “low” relative to a private threshold that divides the unit-interval state space into twosubintervals. The decision maker’s response to the expert’s advice becomes less sensitiveunder the former’s own information. In response, the expert provides advice that is con-sidered more biased, relative to the case when decision maker is uninformed. For sometypes of decision maker, this negative, strategic effect of their own information outweighsits direct, positive effect—being informed makes them worse off. Examples show, however,that evaluated before the realization of her type, the opportunity to access information isalways beneficial to the decision maker when the expert has moderate bias.

© 2014 Elsevier B.V. All rights reserved.

“A little Learning is a dang’rous Thing;Drink deep, or taste not the Pierian spring.”—Alexander Pope

. Introduction

With a great deal of information only a few clicks away, the boundary between experts and novices blurs. Once therivilege of experts, specialized knowledge is now widely available on encyclopedic websites and through search engines.

n the medical arena, for example, websites such as www.webmd.com have rendered patients much more informed andophisticated than their counterparts a decade ago. We can perhaps jump a step ahead by saying that novices no longer existoday, and amateurs—those who know but do not know enough to dispense with the help from experts—have emerged toll the void.

� This paper is adapted from the first chapter of my Ph.D. dissertation at the University of Pittsburgh. I am indebted to Andreas Blume for his guidance,upport, and encouragement. I also owe Oliver Board, Pierre Liang and Lise Vesterlund for their advice and support. I am grateful to Sourav Bhattacharya, Yinghen, Kim-Sau Chung, Vincent Crawford, James Dearden, Wooyoung Lim, Deirdre McCloskey, Tymofiy Mylovanov, Jack Ochs, Marla Ripoll, Joel Sobel andhe anonymous referees for their valuable comments and suggestions. Thanks are also due to seminar participants at Indiana University-Purdue Universityndianapolis, Lehigh University, University of Pittsburgh, the Third World Congress of the Game Theory Society and the Fall 2008 Midwest Economic Theoryonference for the helpful comments and discussions. I gratefully acknowledge financial support from the Arts & Sciences Summer Research Fellowship athe University of Pittsburgh.∗ Correspondence to: 621 Taylor Street, Bethlehem, PA 18015, United States. Tel.: +1 610 758 5726.

E-mail address: kwl409@lehigh.edu

http://dx.doi.org/10.1016/j.jebo.2014.03.023167-2681/© 2014 Elsevier B.V. All rights reserved.

2 E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16

Is evolving from novices to amateurs beneficial to patients and other decision makers? Advocates for consumer educationwould answer affirmatively; the underlying proposition of consumer education is that more information may help consumersdefend against fraud and deception and in general leads to better decision.1 Yet it is a well-known result in informationeconomics that more information is not necessarily better. In, for instance, the classic lemons model of Akerlof (1970),information, when asymmetrically distributed, can eliminate trades that are otherwise mutually beneficial. In this paper,I examine the effects of decision makers’ information in strategic information transmission (Crawford and Sobel, 1982), asetting that captures the interactions between experts and decision makers. Within the confine of the specified environment,two questions are explored: (1) how a biased expert responds to a decision maker who is (partially) informed; and (2) whetherand under what circumstances becoming amateur may benefit or hurt the decision maker.

I start with Crawford and Sobel’s (1982) model (the “CS model”). An expert (he), after privately observing the state ofthe world distributed uniformly on [0,1], sends a message (advice) to a decision maker (she). “Talk is cheap”—the messageitself has no payoff consequence. After receiving a message, the decision maker takes an action that affects the payoff ofboth. Interests are misaligned: while the decision maker wants to take an action that matches the state, the expert’s mostpreferred action is higher than the state by a fixed bias parameter.

The novelty of my model—which I call the amateur model—lies in the decision maker being an “amateur” who is partiallyinformed. The decision maker does not directly observe the state but can tell whether it is “high” or “low”: she is informedabout in which interval of a binary partition of [0,1] lies the true state. Her definition of “high” and “low”—the cutoff inthe partition or the threshold—is a private information constituting her type. The realization of the threshold, uniformlydistributed on [0,1], privately determines the amateur’s partitional information structure.

The way amateurs access and use information, which I attempt to capture with the model, may be illustrated with patientuse of online information.2 Fox (2006) reports that eight out of ten Internet users in the United States, accounting for some113 million adults, have searched for health information on the Internet. These users may have access to only limited sourcesof information. Even when they have at their disposal the same information available to the professionals (e.g., by using theGoogle Scholar), as amateurs they typically lack the ability to interpret the information and sort out the relevant from theirrelevant. As one doctor puts it, “There’s so much information (as well as misinformation) in medicine—and, yes, a lot of itcan be Googled—that one major responsibility of an expert is to know what to ignore.”3 Given these extrinsic and intrinsicconstraints, all a patient can get out of the websites may amount only to a rough idea as to whether her condition callsfor serious attention (“high”) or not (“low”). And with different sources of information and individual interpretations, it isplausible that even for the same underlying condition different individuals may arrive at different conclusions.

Fox (2006) reports that only one-third of the respondents mentioned their online findings during doctor’s visits; a doctorfacing a “Googler-patient” is likely to offer advice in the midst of some private information on the patient’s part. Suppose thedoctor reports a diagnosis biased toward inducing more intense and expensive treatments than are necessary.4 A patientwho believes that her condition is serious may consider the biased diagnosis a confirmation of her findings and proceedwith an expensive treatment. Otherwise, she may request for other options or even seek a second opinion.5 Even for thesame piece of advice, once decision makers have their own information, it is inevitable that different responses will ensue.

To illustrate how different interpretations of advice arise under the decision maker’s partitional information structure,consider, in the context of doctor-patient interaction, two types of patient, Wimp and Stoic. The diagnosis, observed only bythe doctor, is represented by a point in [0,1]. The patients do not observe the exact diagnosis. But from what she learns fromthe Internet, Wimp is able tell that her condition is “not serious” if the underlying diagnosis lies in [0, 1

3 ) and “serious” if itlies in

[13 , 1]. Stoic, interpreting information differently, considers [0, 2

3 ) as “not serious” and[

23 , 1]

as “serious.” Suppose the

true, exact diagnosis is 12 for both of them. Then, even for the same diagnosis, Wimp will consider her condition as “serious”

while Stoic will deem otherwise.Suppose the doctor provides a “vague” advice that the true diagnosis lies in

[14 ,

712

]; he is not telling the exact truth but

is not deceiving either because[

14 ,

712

]contains 1

2 . In light of her knowledge that the diagnosis lies in[

13 , 1], Wimp will

1 Ben Bernanke was once quoted on the Fed Education website: “In today’s complex financial markets, financial education is central to helping consumersmake better decisions for themselves and their families.” As one of its missions, the Bureau of Consumer Protection “empowers consumers with freeinformation to help them exercise their rights and spot and avoid fraud and deception”; they believe “education is the first line of defense against fraudand deception; it can help you make well-informed decisions before you spend your money.”

2 Cheap talk models have been applied to study the interactions between doctors and patients. For example, Koszegi (2006) uses one to explore theemotional aspect of doctors’ advice, extending on Caplin and Leahy’s (2004) emotional model of certifiable information. Applications can also be found inother areas, e.g., political science (Gilligan and Krehbiel, 1989; Krishna and Morgan, 2001b) and finance (Benabou and Laroque, 1992; Morgan and Stocken,2003). The questions that motivate this paper arise in these areas as well. For instance, with widely available online financial information, investors mayno longer rely exclusively on the information provided by investment advisors.

3 This controversial Time magazine article, “When the Patient Is a Googler,” is written by an orthopedist Haig (2007) who reports his unpleasant experiencewith a “Googler-patient” whom he describe as “brainsucker.” The doctor eventually decided not to treat the patient.

4 The supplier-induced demand hypothesis in health economics (Evans, 1974) posits that doctors recommend more health care purchases than patientswould buy if they had the same information. This coincides with the bias of the expert in the model.

5 Thirty percent of the respondents in Fox (2006) indicate that online information led them to ask further questions to their doctors or seek a secondopinion. In an article on salon.com, “Is There a Doctor in the Mouse?,” the pediatrician-author Parikh (2008) mentions that some parents refused to vaccinatetheir children after being exposed to stories on autism websites about the dangers of vaccinating children.

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E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16 3

nterpret the advice to mean that the exact diagnosis lies in[

13 ,

712

]. Since Stoic’s information—that the diagnosis lies in

0, 23 )—is even vaguer than the advice, she will take the advice as it is. To Wimp, the advice is a complementary advice—it

dds on but does not supersede her information. The same advice is, however, a substituting advice to Stoic—it substituteshat she knows. Should the doctor provide a completely vague advice that the diagnosis can be anywhere in [0, 1], bothimp and Stoic will ignore it as they both know better; it is a redundant advice to them.6 Finally, suppose the doctor deceivesith advice

[78 , 1]

which does not contain 12 . While the advice will be considered by Wimp as substituting, Stoic can detect

hat it is a false advice because it contradicts what she knows. This out-of-equilibrium event represents our model analog ofdefense against fraud and deception.”

I analyze the partitonal equilibria of the amateur model, the same kind of equilibria found in the CS model. The expertartitions [0, 1] into a finite number of intervals (steps) and transmits information by revealing which step contains the truetate. Comparing across the two models, the expert in the amateur model provides less informative advice: for a fixed bias,he maximum number of step a partitional equilibrium can have is weakly lower in the amateur model; when the numbersf steps are equal, the partition in the amateur model is less even. An upwardly biased expert has a tendency to exaggerateo as to induce a higher action. Complete separation is infeasible in equilibrium, because then no matter how small thepward bias is the expert will at least find the next higher inducible action more preferable. The key behind the workingf partitional equilibria is to make the next higher actions too high to be preferable, when only finite actions situated farnough away from each other are induced, each by a step. The required distances between the actions in turn determineow the steps are partitioned.

In deciding what action to take, the amateur combines her information with the advice, which attenuates the expert’snfluence on her; the actions induced on the amateur are, on average, not as high as those induced on the “novice” in the CS

odel. The next higher actions that were too high to be desirable will now be within the ranges to be favorable when theecision maker is informed. Equilibrium restores by having the steps adjust downward, resulting in less even partitions orlimination of steps in the lower end. The downward adjustment means that some states will be relocated to a higher step,hich may be interpreted as the expert exaggerating more in the amateur model. In line with the objectives of consumer

ducation, in the model the information in the hands of the amateur makes her less susceptible to the exaggeration of aiased expert. Yet the analysis suggests that that is not the end of the story: it is precisely the lower susceptibility that leadso more exaggeration, which in turn lowers the usefulness of the expert’s advice. Some doctors are reported to become lesselpful if the patient is a “Googler.”7 While there are certainly many different factors driving the doctors’ responses, theesult provides a rational, strategic account for why some doctors become less helpful in terms of offering less useful advice.

With the expert providing less informative advice in response, the evolution from novices to amateurs does not benefitvery (type of) decision maker; in the circumstances that it does, the decision maker’s information has to be sufficientlyifferent from the one that would be provided by the expert. In, for example, partitional equilibria with two steps, themateur’s information structure confers a positive strategic value only if it has a relatively high threshold. A high thresholdeans that the amateur’s information by itself is not so useful. However, since the two-step partition provided by the expert

as a low cutoff under his upward bias, combined with her information structure with a threshold situated far enough awayrom the expert’s cutoff the amateur is making decision under an evenly spaced three-step partition, enough to compensateor the expert’s negative response. Before the realization of the threshold, the amateur is endowed with a “lottery” overnformation structures. Despite the negative strategic value of some information structures, examples show that the accesso the lottery is always beneficial when the expert’s bias is moderate.

The paper proceeds as follows. Section 2 describes the model and provides some preliminary analysis. Section 3 analyzeshe equilibria, articulating the sense in which the expert provides less informative advice to the amateur. Section 4 addresseshe strategic value of the amateur’s information structures.

Seidmann (1990) and Watson (1996) are the first to incorporate informed receiver into cheap-talk games, followed bylszewski (2004). To the best of my knowledge, the subject matter has since been largely dormant until more recently. With

ome alternations to otherwise standard cheap-talk games, these early papers show that a decision maker’s informationan improve communication or even lead to full elicitation of information. This paper shows, quite differently, that, in theanonical cheap-talk framework of Crawford and Sobel (1982), a decision maker’s information not only does not elicit fullevelation but can indeed worsen communication. A few recent papers also obtain similar results. To better compare this

aper with these related papers, I postpone the literature review to Section 5, intertwined with discussion and examples.ection 6 concludes. Proofs and calculations are relegated to the appendix.8

6 This “babbling advice” will also be ignored by an uninformed decision maker. While I use it as an example of redundant advice, there is non-babblingdvice in the amateur model that can nevertheless be redundant.7 In Fox and Fallows (2003), for example, a patient responds to the survey: “...they [doctors] became irritated with me for having knowledge of the

ondition beyond what they chose to share with me...they became defensive and short with me when I would question...what I had found out on my ownn the Internet.” Despite the incentives to protect their professional images, negative responses are also noted from the doctors’ side (Ahmad et al., 2006):So, if your patient [is] having a $15 visit, you’re not going to sit for 15 minutes going through all this [patient’s findings], you’re going to get them out ofhe office.”

8 An online appendix, which is available on the author’s website, contains analysis of how off-equilibrium beliefs relate to the existence of equilibria inhe amateur model.

4 E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16

2. The model

2.1. Expert and amateur

There are two players, an expert (e) and a decision maker (d). The expert is perfectly informed about the state of theworld, �, which is commonly known to be uniformly distributed on � = [0, 1].9 After privately observing �, the expert sendsa cheap-talk message (advice) m ∈ M to the decision maker, who then takes an action a ∈ R.10

Payoffs adopt the following form:

Ue(a, �, b) = −(a − (� + b))2,

Ud(a, �) = −(a − �)2,

where b > 0 is a commonly known parameter measuring the misaligned interests between the parties. Together with theuniformly distributed state, the quadratic payoff functions constitute the uniform-quadratic model introduced by Crawfordand Sobel (1982). A specification adopted by a majority of subsequent work especially in applications, the model offerstractability as well as a benchmark to compare the results in this paper with those in the literature.

The ideal action of the expert when he observes � is ae(�, b) = � + b. If the decision maker could observe � perfectly, herideal action would be ad(�) = �. With ae(�, b) − ad(�) = b > 0, the misaligned interests between the parties manifest specificallyas: for a given state, the expert prefers an action that is b higher than what the decision maker would prefer were she ableto observe the state. Hereafter, b will be referred to as the expert’s bias.

Take the doctor–patient relationship as an example. The state of the world may refer to a diagnosis of a certain disease,with a higher � representing a more serious diagnosis. The doctor privately observes the true diagnosis and delivers hisreported diagnosis, m, to the patient. The reported diagnosis induces the patient to take certain action, a, which can beinterpreted as a treatment or procedure choice. A higher action corresponds to a more intense treatment. Consistent withthe supplier-induced demand hypothesis in health economics (Evans, 1974), the biased doctor therefore prefers treatmentsthat are more intense—and thus in general more expensive—than are ideal from the patient’s perspective. Alternatively, wecan interpret a as the patient’s lifestyle choice such as hours of exercises per week. With a stricter standard on what count ashealthy habits, the doctor prefers a more disciplined lifestyle than is deemed optimal by the patient. In this case, b capturesthe doctor’s “paternalistic bias.”

The decision maker does not observe � but has partial information about it. She privately observes a threshold, t, commonlyknown to be uniformly distributed on T = [0, 1], independently of �. A threshold realized in the interior of T divides � = [0, 1]into two non-degenerate intervals, one low tl = [0, t) and one high th = [t, 1]. The decision maker receives a private signal, s,indicating to her in which interval the true � lies:

s(�|t) ={l, if � < t,

h, if � ≥ t.

I call the decision maker an amateur; she knows less than an expert but almost always more than an uninformed novice.11

The set of thresholds T coupled with the set of signals {l, h} defines the type space of the amateur, T × {l, h}, with genericelement ts; th is collectively referred to as the high-interval types and tl the low-interval types.12

The game begins with nature drawing t, privately observed by the decision maker. The realization of � and the simul-taneous generation of s then follow, where the former is privately observed by the expert and the latter by the decisionmaker.13 The expert then sends a message. The game ends with the decision maker’s taking an action and the distributionof payoffs.

2.2. Equilibrium solution

The solution concept employed is perfect Bayesian equilibrium, where, succinctly stated, strategies are optimal given

beliefs and beliefs are derived from Bayes’ rule whenever possible. A behavior strategy of the expert, � : � → �M, specifiesthe distribution of message he sends for each � ∈ �. Denote ��(m) = {� ∈ � : �(m|�) > 0}, the set of � for which the expertsends message m with positive probability under �. Note that when there are messages in M that are not used under �, ��(·)is an empty set for those messages.

9 Throughout the paper, I shall interchangeably refer to � as the “state” or the “type of the expert.”10 There is no restriction on M except that it should be sufficiently large. Any infinite set will more than suffice.11 An amateur knows nothing more than a novice in the measure-zero event that t = 0 or t = 1. Accordingly, a novice is a special type of amateur with t = 0

or 1.12 The choice of only “high” and “low” is guided by parsimony and tractability. The analysis will become exponentially complicated if only “medium” is

added.13 The property of the signal is a common knowledge. As such, that the signal realization is private to the decision maker hinges on the threshold being a

private information; if the expert could observe t, then upon observing � he would know which signal, l or h, the decision maker has received.

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E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16 5

A pure strategy of the decision maker, � : M × T × {l, h} → R, specifies for each combination of received message andnterval type an action she takes.14 Her belief function, � : M × T × {l, h} → ��, specifies for each combination of received

essage and interval type a distribution over �.The decision maker updates beliefs using two sources of information, one credible and one that may not be. And this leads

o a difference between unused and out-of-equilibrium messages in the amateur model. In the CS model, unused messagesre synonyms for out-of-equilibrium messages, and vice versa. Since in cheap-talk models messages do not directly affectayoffs, for every equilibrium (in which there may be unused messages) there exists another equilibrium with the sameutcome in which all messages are used. One can therefore restrict attention to equilibria with no unused message, and anypecifications of off-equilibrium beliefs would have no effect on the equilibrium outcomes.15 Crawford and Sobel (1982)nlist all the messages by assuming that the expert randomizes uniformly over supports of distinct sets that exhaust theessage space.In the amateur model, even when there is no unused message, out-of-equilibrium messages can still arise when the

ecision maker receives a false advice. Before establishing this observation formally, the following lemma facilitates thexposition:

emma 1. A type-ts amateur is in an information set unreached in equilibrium if and only if she receives message m ∈ M suchhat ��(m)∩ ts = ∅, where � is an equilibrium strategy.

The next lemma establishes that the amateur’s information set can be empty—that there are out-of-equilibriumessages—even when all messages are used:

emma 2. Suppose there is an informative equilibrium in the amateur model with no unused message. There exists a deviationf the expert from this equilibrium that leaves some interval types of amateur in information sets unreached in equilibrium.

As illustrated in Section 1, there are four types of advice: substituting advice, complementary advice, redundant advice,nd false advice. Given that cheap-talk messages have no intrinsic meaning, advice also has no intrinsic type; the type ofn advice is determined with respect to how it is interpreted, i.e., who (ts) is receiving it. The first three types, includinghe useless redundant advice, result from the application of Bayes’ rule. The out-of-equilibrium event described in Lemma

pertains to the false advice, which is an out-of-equilibrium but nevertheless used message. The intuition behind maye better illuminated with a discretized version of the model. Suppose there are three possible states of the world: {bad,ediocre, good}. The amateur’s partition of the state space is {{bad}, {mediocre, good}}; she can tell whether the state is,

iterally, “bad” or “not bad” but cannot distinguish between “mediocre” and “good.” Suppose in an equilibrium the expertends m1 when he observes either “bad” or “mediocre” and m2 when he observes “good.” Consider a deviation in which thexpert sends m2—a message used in equilibrium—when the true state is “bad.” Since the amateur knows that it is a “bad,”he expert’s message contradicts what she knows. Bayes’ rule cannot be applied, and m2 is out-of-equilibrium even thought is used. In general, ��(m) ∩ ts is the information set an interval type ts finds herself in after receiving m, and Bayes’ rulean be applied only when it is non-empty.16

Finally, the following lemma states that, same as the CS model, it is without loss of generality to consider that all messagesre used in the amateur model:

emma 3. In the amateur model, for every equilibrium (�, �, �) there exists another equilibrium (� ′, �′, �′) with no unusedessage of which the equilibrium outcome is equivalent to that of (�, �, �).

Guided by Lemmas 1–3, the decision maker’s beliefs after receiving the expert’s message can be specified as follows:pon receiving m′, a type-ts decision maker updates his beliefs according to

�(�|m′, ts) =

⎧⎪⎪⎨⎪⎪⎩

�(m′|�)�(�|ts)∫ 1

0

�(m′|�′)�(�′|ts)d�′, if ��(m′) ∩ ts /= ∅,

(�), if ��(m′) ∩ ts = ∅ and ��(m′) /= ∅,here

�(�|th) ={

0, for � ∈ [0, t),1

1 − t, for � ∈ [t, 1],

and �(�|tl) =

⎧⎨⎩

1t, for � ∈ [0, t),

0, for � ∈ [t, 1].

nd (�) is any distribution supported on [0, 1].The concept of perfect Bayesian equilibrium places no restriction on the decision maker’s beliefs in the event that a false

dvice or an unused message is received. Any distribution supported on � can be a candidate for off-equilibrium beliefs.

14 That Ud11( · ) < 0 guarantees that only pure strategy will be adopted by the decision maker.15 See, for example, the discussion in Farrell (1993).16 An empty information set will never arise in the CS model when all messages are used in equilibrium.

6 E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16

In the equilibrium definition that follows, however, the off-equilibrium beliefs of a type-ts decision maker are restricted tobe supported on ts. It is thus ruled out at the level of definition that a type-ts decision maker would contradict herself (byputting positive density on � \ ts) after being deceived or surprised.17

Definition 1 (Perfect Bayesian equilibrium). A perfect Bayesian equilibrium of the amateur model is a pair of strategies (�,�) and a set of beliefs � such that

1. the expert maximizes his expected payoff given the decision maker’s strategy: for all � ∈ �, if m ∈ supp[�(· |�)], then

m ∈ argmaxm′∈M

Ve(m′, �, b) =∫ �

0

Ue(�(m′, th), �, b)dt +∫ 1

�

Ue(�(m′, tl), �, b)dt,

2. the decision maker maximizes her expected payoff given her beliefs: for all m ∈ M and all ts ∈ T × {l, h},

�(m, ts) = argmaxa′

∫ 1

0

Ud(a′, �)�(�|m, ts)d�, and

3. the decision maker updates her beliefs using Bayes’ rule whenever possible, taking into account the expert’s strategy andher interval type: for all m ∈ M and all ts ∈ T × {l, h},

�(�|m, ts) =

⎧⎪⎪⎨⎪⎪⎩

�(m|�)�(�|ts)∫ 1

0

�(m|�′)�(�′|ts)d�′, if ��(m) ∩ ts /= ∅,

(�|ts), if ��(m) ∩ ts = ∅,

where (�|ts) is any distribution supported on ts.

3. Less informative advice for the amateur

3.1. A simple illustration

To demonstrate the basic intuition of why the expert provides less informative advice to the amateur, which originatesfrom the fact that the amateur’s action becomes less sensitive to the expert’s message when she is informed, I start with asimple example where the threshold, t, is fixed and common knowledge.

Example 1. (a) A two-step equilibrium in the CS model: the expert has bias b = 18 , who reveals whether � lies in

[0, 1

4

]or

in ( 14 , 1]. The two actions induced on the novice are a1 = 1

8 and a2 = 58 . (b) The decision maker’s threshold in the amateur

model is commonly known at t = 34 .

In the CS model, for any � realized in[0, 1

4

], the expert reveals

[0, 1

4

]instead of ( 1

4 , 1], because for these �’s the expert’s

ideal action, � + 18 , lies closer to a1 than to a2; the expert correctly but vaguely reveals that the state lies in

[0, 1

4

]because

a2—the consequence of revealing that � ∈ ( 14 , 1]—lies too far away to be desirable.

Now with the decision maker having t = 34 , if the expert with � ∈

[0, 1

4

]deviates to reveal � ∈ ( 1

4 , 1], combining with her

own information the decision maker will take a′2 = 1

2 instead of a2 = 58 . In other words, the amateur’s action becomes less

sensitive to the expert’s message. This magnifies the incentives to exaggerate, where some types of expert with � ∈[0, 1

4

]now have an incentive to reveal that � ∈ ( 1

4 , 1]. Equilibrium restores by having the steps adjust downward to[0, 1

8

]and

( 18 , 1].

3.2. Less informative advice for the amateur

This section analyzes the complete model in which t ∈ [0, 1] is the decision maker’s private information. I analyze theeffects of the decision maker’s private information within the class of partitional equilibria, the same kind of equilibria

17 Apart from being a natural restriction in the current context, this also shares the spirit of equilibria with “support restrictions,” which requires that thesupport of beliefs at an information set be a subset of that at the preceding information sets (see, e.g., Madrigal et al., 1987).

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E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16 7

ound in CS.18 In a partitional equilibrium with N steps (N-step equilibrium), the expert partitions the state space [0, 1] into finite number of N intervals {Ii}Ni=1 = {(�i−1, �i]}Ni=1, �0 = 0 and �N = 1, and reveals in which Ii the true � lies.19 When N ≥ 2, thequilibrium is informative. The end points of the intervals, �i, i = 1, . . ., N − 1, are called boundary types, who, in equilibrium,ust be indifferent between sending the messages designated by the two adjacent intervals.20 In the CS model, to sustain a

wo-step equilibrium with one boundary type �cs1 , the expert’s bias b has to be less than 14 . Two-step equilibria are considered

he least informative among informative equilibria. The requirement for such equilibria to exist is more restrictive in themateur model:

roposition 1. In the amateur model, a two-step equilibrium exists if and only if b < 16 .

While the expert provides informative advice to the novice when b ∈ [ 16 ,

14 ), he babbles to the amateur. The indifference

ondition dictates that �1 = 12 (1 − 6b) in the amateur model and �cs1 = 1

2 (1 − 4b) in the CS model. Even when b ∈(

112 ,

16

)so

hat the most informative partitional equilibria in both models are of two steps, the boundary type in the amateur modelies on the left of that in the CS model, and their distance is exactly equal to the expert’s bias.

The rationale of the expert’s response to the amateur in the complete model is similar to that in Example 1. Consider againhat b = 1

8 and the expert provides two different advice, whether the situation is serious or not. In the amateur model, thexpert advises that it is “not serious” if � ∈

[0, 1

8

], while “serious” is reserved for � ∈ ( 1

8 , 1]. But suppose in the amateur model

he expert uses the CS cutoff, 14 , for issuing the two advice. By saying that it is “serious,” the expert induces a distribution

f actions, �(“serious′′, t) = 1/4+t2 , t ∈ ( 1

4 , 1], which, on average, is lower than the 58 induced on the novice. Even though the

not serious” advice induces a distribution of actions �(“notserious′′, t) = t+1/42 , t ∈

[0, 1

4

], that on average is higher than the

18 induced on the novice, such upward distortion is more than countered by the downward distortion given that the secondtep ( 1

4 , 1] is longer than the first step[0, 1

4

]. When dealing with an amateur with private t, the expert will find that the two

ifferent advice both induces actions that on average are closer to his ideal, but the “serious” advice induces actions that areloser among the two. Accordingly, as in Example 1 the consequence of exaggeration becomes attractive to some types ofxpert with � ∈

[0, 1

4

], and in equilibrium the cutoff moves leftward to 1

8 .21

Let N(b) (Ncs(b)) be the maximum number of steps an equilibrium in the amateur model (CS model) can have when thexpert’s bias is b. The discussion above suggests that the results should not be limited to two-step equilibria:

roposition 2. For all b > 0, N(b) ≤ Ncs(b). Furthermore, for a given b > 0, in the respective partitional equilibria of the amateurodel and the CS model in which N = Ncs ≥ 2, the boundary types in the two models are such that �i < �cs

i, i = 1, . . ., N − 1.

In the equilibria with the maximum number of steps allowed by a given b, the expert in the amateur model partitionshe state space into (weakly) lower numbers of steps. When the numbers of steps are equal, the steps are less even in themateur model.22 The question of how much information the expert provides, i.e., the informativeness of the equilibrium,s addressed in Crawford and Sobel (1982) by evaluating the decision maker’s (ex-ante) expected payoff achieved under theartitional information structure provided by the expert. To use the same criterion in the amateur model and to comparehe informativeness of equilibria across the two models, it is imperative that we disentangle the payoff contribution of themateur’s exogenous information structure from that of the expert’s endogenous information structure. Toward this end,

use the novice’s expected payoffs as the yardstick for evaluating the endogenous information structures in the amateurodel. The contribution of the amateur’s information structure will come back into the picture in the next section.With a slight abuse of notations, I use I(N(b)) and I(N (b)) to denote the partitional information structures with N(b) and

cs

cs(b) steps. The following proposition compares the expected payoffs the novice receives in the CS model with what sheould hypothetically receive in the amateur model if the expert was treating her as an amateur:

18 Given that in the amateur model off-equilibrium beliefs arise which are absent in the CS model, it is virtually impossible to characterize all equilibriander all possible off-equilibrium beliefs. Refer to the online appendix for the relation between off-equilibrium beliefs and the existence of partitionalquilibrium. See Chen (2009) for an example of non-monotone equilibrium (non-partitional equilibrium) in a CS model with informed decision makerith a different information structure. Even though a non-partitional equilibrium is conceivable in the amateur model, a general analysis is unlikely under

he immense possibilities of off-equilibrium beliefs. The value of focusing on partitional equilibrium is that it allows us to obtain systematic comparisonsetween the CS and the amateur models.19 In terms of message uses, it is without loss of generality to consider that the expert partitions the message space M into N distinct and exhaustive sets,

i , i = 1, . . ., N, and, if � is realized in Ii , he randomizes uniformly over messages in Mi . Insofar as equilibrium outcomes are concerned, we can without lossf generality assume that all messages are used (Lemma 3). The correspondence between the index of the message sets and that of the intervals is alsoonessential.20 This indifferent condition, while being necessary and sufficient for the existence of partitional equilibria in the CS model, is no longer sufficient in themateur model; off-equilibrium beliefs have to be considered. The online appendix contains the specification of off-equilibrium beliefs that is adopted forhe analysis in this section.21 Since the expert’s payoff is a concave function of distance from ideal action, risk aversion also plays a role. With the second step longer than the firsttep, the variance of the distribution of actions �(“ serious′′ , t) is larger than that of �(“ notserious′′ , t). This makes the consequence of exaggerating not asttractive as it would be if the expert was not risk averse. However, the result suggests that this risk aversion effect is dominated by the effect of actionistortions due to the amateur’s information, resulting in a “net benefit” of exaggerating.22 Unlike the CS uniform-quadratic model, in which the indifference condition reduces to a solvable second-order linear difference equation, the corre-ponding difference equation in the amateur model is non-linear; the lack of closed-form solution makes beyond reach an explicit characterization of theelationship between b and N(b) in the amateur model.

8 E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16

Novice’s expected payoff under ( ( ) )

Amateur’s expected payoff under ( ) and exogenous informa�on structure with thres hold

Indirect effect Direct effect

Indirect effect + Direct effect: Strategic value of the a mateu r’s

Novice’s expected payoff und er ( )

exogen ous informa tion structure

Fig. 1. The effects of the amateur’s information structures.

Proposition 3. For b ∈(

0, 14

), the novice’s expected payoff is strictly lower under partitional information structure I(N(b)) than

under I(Ncs(b)).

Proposition 3 provides a basis for concluding that, for a given b > 0, the most informative equilibrium in the amateurmodel is less informative than that in the CS model.23 This also implies that if the game is augmented with a stage for thedecision maker to communicate to the expert whether she is one of novice or amateur, no truthful communication will result.A novice will never want the expert to mistaken her as an amateur, because then the expert will provide less informativeadvice that leads to lower expected payoff.

4. The strategic value of the amateur’s information structures

With the wide variety of resources on the Internet, the information obtained by one Googler-patient may be very differentfrom that by another. If doctors respond to informed patients indiscriminately by providing less informative advice, whethera patient does overall benefit from visiting particular medical websites may depend on what information she obtains. In themodel, this amounts to evaluating the strategic value of the amateur’s information structures.

Fig. 1 dissects the two effects involved, the direct effect due to more information at the decision maker’s disposal andthe indirect effect resulting from the expert’s reaction. The effects are decomposed conceptually by evaluating one changeat a time. Using the novice’s expected payoff and switching the endogenous information structure from I(Ncs(b)) to I(N(b)),Proposition 3 establishes that the indirect effect is negative. On the other hand, fixing the endogenous information structureat I(N(b)) and evaluating the amateur’s payoff instead of the novice’s, it is straightforward that the direct effect cannot benegative: keeping the expert’s behavior unchanged, having additional information can never hurt.

Combining the two effects returns the strategic value of the amateur’s information structure. The intuition of the analysisis indeed simple. Given that the expert provides less informative advice, the amateur’s own information has a positive valuein the game only when it is useful enough to compensate for the loss. What may be less obvious is that the usefulness of herinformation has to be assessed with respect to the information provided by the expert. If what the amateur knows by herselfwill also be told by the expert, her information will be completely useless—being informed will indeed make her worse off.If the amateur is to benefit, she must secure certain knowledge that the expert will not tell, which is equivalent to sayingthat relative to what she knows the expert has to be providing some complementary advice.

The rest of the section makes precise the above intuition. The amateur’s information structure is defined by the threshold;her expected payoffs used in the analysis are evaluated before � but after t is realized.24 Such expected payoff in the N(b)-stepequilibrium is compared to the novice’s expected payoff in the Ncs(b)-step equilibrium in the CS model. When the formeris higher (lower) than the latter, the strategic value of the amateur’s information structure is positive (negative), and theamateur is better (worse) off compared to the novice.

The following proposition, which follows immediately from Proposition 3, formally states that there exists a set ofthresholds associated with which the amateur’s information structure is not useful enough to benefit her in the game:

Proposition 4. For b ∈(

0, 14

), there exists an open subset T(b) ⊂ T such that the strategic value of the amateur’s information

structures defined by t ∈ T(b) is strictly negative.

Had this been a single-agent decision problem, a not that useful information structure of the decision maker would havebeen characterized only by thresholds that are close to 0 or 1, which is also the case in the amateur model when b ∈ [ 1

6 ,14 ).

However, for b ∈(

0, 16

), “not-so-useful information” gains additional meaning. Even when t lies deep in the interior of [0, 1], if

it is close to one of the boundary types, the associated information structure will not be useful in the game. In the extreme case

when t coincides with one of the boundary types, the information structure will be completely useless. A negative strategicvalue results so long as the exogenous information structure is not useful enough to compensate for the less informativeendogenous information structure. With no close-form solution to the difference equation that defines the relation between

23 The result is in line with Theorems 3 and 4 of Crawford and Sobel (1982), who show under a different comparative statics that equilibria with a highernumber of or more even steps are more informative.

24 See the next section for examples where payoffs are evaluated before t is realized.

E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16 9

Strategic Value of the

Amateur 's InformationStructure

Amateur 'sExpectedPayoff

Novice 'sExpected Payoff

θ1

H

K1 K20.2 0.4 0.6 0.8 1.0

t

–0.04

–0.03

–0.02

–0.01

0.01

bpi

EpowT

ah

sa

[

tiBgac

5

5

stwe

Et

ie

Fig. 2. Strategic value of the amateur’s information structure when b = 111 .

and N(b), it is generally impossible to characterize the explicit values of t that would result in information structures withositive or negative strategic value. By focusing on two-step equilibria with b ∈

(1

12 ,14

), however, the following example

llustrates Proposition 4 with specific values of t.

xample 2. Strategic value of the amateur’s information structures for b ∈(

112 ,

14

). Recall that when b ∈ [ 1

6 ,14 ), the expert

rovides “two-step advice” to the novice but no information to the amateur. Even in the game, the amateur is making decisionn her own. Her expected payoff is the highest if her information structure has a threshold at 1

2 and decreases monotonicallyhen t moves away from 1

2 . In the CS model, the boundary type defining the endogenous information structure is �cs1 = 12 − 2b.

he amateur is thus better off compared to the novice if and only if t is closer to 12 than is �cs1 , i.e., when t ∈

[12 − 2b, 1

2 + 2b]. In

single-agent decision problem, even t ∈(

0, 12 − 2b

)∪(

12 + 2b, 1

)represents valuable information structures. In the game,

owever, the indirect effect renders them a strictly negative strategic value.

When b ∈(

112 ,

16

), the most informative partitional equilibria in both models are of two steps. The amateur’s information

tructure, which effectively adds a step to the endogenous two-step partition, has positive strategic value if and only if thedditional step can more than make up for the less even endogenous partition. This happens when the amateur’s threshold t ∈t′, t

′], where t′ = 3+12b−36b2−

√1+24b+56b2−96b3+1296b4

4(1+6b) and t′ = 3+12b−36b2+

√1+24b+56b2−96b3+1296b4

4(1+6b) . Fig. 2 further illustrates

he case when b = 111 , where the boundary type in the amateur model is �1 = 5

22 . Given that the novice’s expected payoff isndependent of t, the strategic value of the amateur’s information structure is a vertical displacement of her expected payoff.oth reach maximums when t is farthest away from 0, �1 and 1 at t = 5

44 and t = 2744 , where the latter corresponds to the

lobal maximum. Note only does “not-so-useful information” gain new meaning in the game, but the “optimal information”lso becomes different: without the expert the optimal information structure for the amateur has t = 1

2 ; in the game ithanges to t = 27

44 . The strategic value is positive only when, with approximations, t ∈ [0.304, 0.923].

. Discussion and related literature

.1. The decision maker’s incentive to communicate her threshold

Section 3 ends with a discussion about whether the decision maker has an incentive to communicate to the expert ifhe is a novice or an amateur. Another natural question is whether the decision maker has an incentive to communicatehe exact value of t to the expert. A full analysis is beyond the scope of this paper. In the following, I provide an examplehich illustrates using a two-step equilibrium that the decision maker truthfully communicating her t cannot constitute an

quilibrium.

xample 3. I adopt the parameter values in Example 1, where b = 1 , t = 3 , and the boundary type �t= 3

4 = 1 . The boundary

8 4 1 8ype is obtained from the condition �t1 = 1

2 (t − 12 ). Denote t to be the t revealed by the decision maker. In this example,

f the decision maker truthfully communicates her t, her expected payoff will be EUd(t = t = 34 , b = 1

8 ) = −0.02181. How-ver, the decision maker has an incentive to reveal a higher t, because then the expert’s boundary type increases, which

10 E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16

results in a more even partition of [0, 1], increasing her expected payoff. For example, if she reveals t = 45 , �

t= 45

1 = 320 , and

EUd(t = 3

4 , t = 45 , b = 1

8

)= −0.01958 > −0.02181.

5.2. A behavioral interpretation

The amateur’s information structure may also be given a behavioral interpretation. Consider again the example of doctorsand patients, where the state refers to the severity of certain disease. The severity increases in �, and a partially informedpatient considers � ∈ [0, t) as “not serious” and � ∈ [t, 1] as “serious.” A patient with a high t may then be interpreted as acarefree, optimistic type: her condition has to be very severe before she considers it “serious.” Similarly, a patient with alow t is like a hypochondriac who would consider her condition “serious” even when it is not so severe.25 Since the steps ina partitional equilibrium are increasingly longer, the amateur’s information structure has the highest strategic value whent is relatively high, at the middle of the longest step. This suggests that when consulting a doctor with a tendency to presenta pessimistic picture, a relatively optimistic patient is most likely to benefit from self-acquired information. The behavioralinterpretation is reversed if a higher � represents a good news, as in a CEO reporting profit forecasts to investors, where theprofitability is increasing in �. Suppose the two forecast categories that an informed investor can distinguish are “strong”and “average.” An investor with a high t is more conservative: � has to be really high in order to be considered “strong.” Andit is a relatively conservative investor who benefits the most from acquiring information when the CEO is inclined to painta rosy picture.

5.3. Related literature

Unlike most of the recent papers, earlier papers on cheap talk with informed receiver show that the receiver’s informationcan facilitate communication. Seidmann (1990) obtains fully revealing equilibrium under a discrete state space and state-independent ideal action. Also obtaining full revelation in a discrete setting, Watson’s (1996) results exploit correlationbetween the sender’s and the receiver’s information. Olszewski (2004) obtains full revelation with a different driving force,where the sender has reputational concern for honesty that the receiver evaluates with her information.

A paper that is related to the present paper in terms of the rationale of the expert’s response is Morgan and Stocken(2008). They consider a multiple-sender environment, where a policymaker’s (decision maker’s) decision depends on thepoll.26 As the poll size becomes larger, the policymaker reacts less and less to each constituent’s (expert’s) message, which inturn diminishes each constituent’s incentive to report truthfully. While the decision maker being “less sensitive to message”underlies the expert’s response in both papers, in the present paper the analysis shows that a similar result can be obtainedin a single-sender environment where the lower sensitivity originates from the decision maker’s own partial information.The contexts of and the questions asked by the two papers are also different.

Similar to the present paper, Chen (2009, 2010) introduce informed receiver into the CS model; the receiver has access toone of two information structures with priors that satisfy the monotone likelihood ratio. The major finding in Chen (2009)is that the receiver cannot credibly reveal to the sender which information structure she has. Chen (2010) also provides abehavioral interpretation of the information structures, considering the two different priors as representing the receiver’sdegree of optimism. The paper shows that more information can be transmitted under the more optimistic prior and exploresthe applications thereof.

Three papers that are most related to the present paper in terms of the questions are Chen (2012), Ishida and Shimizu(2012b), and Moreno de Barreda (2012). In contrast to earlier papers, we all show in a single-sender but otherwise differentenvironments that the receiver’s information can impede communication. Chen (2012) considers a model with binary statewhere the receiver has access to a public signal. Ishida and Shimizu (2012b) study a model with more general discrete stateand signal that go beyond binary. The present paper contributes to understanding of informed receiver in a CS setting. Morenode Barreda (2012) also extends the CS model, where the receiver privately observes a noisy signal affiliated with the state. Thedifferent information structure allows her to show that as the precision of the signal increases there is less communicationin equilibrium. In terms of welfare, she provides examples with fixed b showing that being informed decreases the decisionmaker’s ex-ante welfare. The different information structure considered in the present paper allows me to conduct ananalysis of interim welfare (Proposition 4), which is not considered in Moreno de Barreda (2012). An example of ex-antewelfare similar to that in Moreno de Barreda (2012) but with contrasting result will be provided in Example 4 below.

Chen (2012) shows that the receiver’s expected payoff is not increasing in the informativeness of her signal. The finding

is complementary to the analysis in the last section. The strategic value of the amateur’s information structure is, however,assessed after a component of her private information is drawn, whereas in Chen (2012) the value of the receiver’s informa-tion is assessed before any move of nature. A general evaluation with the same timing presents technical difficulty in models

25 White and Horvitz (2009) provide the first systematic study of “cyberchondria.” In line with the idea that psychology may play a role in the interpretationof information, they find that the conclusions a user of medical websites draws could depend on her predisposition to escalate or seek more reasonablemedical explanations.

26 Other papers that study models with multiple senders include Austen-Smith (1993), Krishna and Morgan (2001a), Battaglini (2002), Ambrus andTakahashi (2008), Morgan and Stocken (2008), Hagenbach and Koessler (2010), and Galeotti et al. (2013).

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b

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E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16 11

ith continuous state space when equilibria are partitional and no closed-form solution exists for the boundary types dueo the distributions of actions induced on the privately informed receiver. For the amateur model, examples focusing on aestricted set of bias levels may still allow us to gain some insight amid the tractability issue:

xample 4. Strategic value of the amateur’s information opportunity for b ∈(

112 ,

14

)and b = 1

20 . Before t is realized, the

mateur is endowed with an information opportunity, a “lottery” over information structures. For b ∈(

112 ,

14

), the novice’s

xpected payoff is EUdcs(b) = − 148 − b2. For b ∈ [ 1

6 ,14 ), the amateur relies only on her information and has an expected payoff

f EUd(b) = − 124 before t is realized. After two-step advice, such expected payoff becomes EUd(b) = − 1+72b2−432b4

64 when

∈(

112 ,

16

). In both cases, EUd(b) > EUdcs(b); the amateur’s information opportunity has a strictly positive strategic value

hen b ∈(

112 ,

14

). For b = 1

11 , Fig. 2 provides a graphical illustration where the area associated with information structuresith positive value (H) is larger than that with negative value (K1 + K2).

Consider next an example of three-step equilibria with b = 120 , which is also illustrated in Crawford and Sobel (1982). The

oundary types in the CS and the amateur models are, respectively, {�cs1 , �cs2 } = {0.133, 0.467} and {�1, �2} = {0.09, 0.404}.

he novice’s expected payoff is EUdcs(

120

)= −0.01593. In terms of information structures, EUd

(t, b = 1

20

)> −0.01593 only

hen approximately t ∈ (0.458, 0.946). Before t is realized, EUd(b = 1

20

)= −0.01463 > −0.01593; the strategic value of the

nformation opportunity is again strictly positive.As an analogy, a Googler-patient who obtained certain knowledge from specific websites may be worse off under the

octor’s unfavorable response to informed patients. However, the above examples suggest that, at least when the doctors moderately biased, patients are on average better off with the Internet, which provides them a channel for gatheringnformation.

In another paper that introduces informed receiver into the CS model, Ishida and Shimizu (2012a) find a receiver’snformation structure under which communication can be improved. Other than that on informed receiver, the paper fitsnto another strand of literature that studies how introducing additional components into the CS environment can improveommunication. Krishna and Morgan (2004) introduce additional round of communication. Blume et al. (2007) introduceoise into the communication channel. Ottaviani and Squintani (2006), Kartik et al. (2007) and Kartik (2009) introduceredulous receiver and/or cost of lying. Goltsman et al. (2009) and Ivanov (2010) introduce mediator. Broadly speaking, thedditional components in most of these papers distort the receiver’s actions in ways that bring into closer alignment theffective interests of the players, allowing communication to be improved. In the present paper, an opposite result obtainsecause the amateur’s information distorts actions in a different direction exacerbating the interest misalignment.

. Concluding remarks

This paper explores how information of a decision maker who interacts with a biased expert may affect the decisionutcomes. The question is perhaps asked in an apt time, when recent developments in the Internet have created a groupf amateurs who have access to information once available only to experts. Simply put, the analysis suggests that thisevelopment may not benefit every decision maker involved. When a decision maker also relies on a biased expert inaking decision, becoming an amateur could indeed backfire because the expert may respond strategically by withholdingore information. In the case that the decision maker’s own information is not that useful on top of what the expert offers

n advice, the negative strategic effect can dominate the benefit of being informed.While motivated by practical questions, this paper contributes toward a bigger theoretical picture by providing an instance

f information structure under which the receiver’s information can be detrimental to cheap-talk communication. Given theariety of receiver’s information structures considered by different authors with potentially different impacts on communi-ation, I believe that an important question is under what general properties of information structures communication cane improved or harmed. This awaits further explorations.

ppendix. Proofs and calculations

.1. Proofs

roof (Proof of Lemma 1). For all m sent on the equilibrium paths, by definition ��(m) contains the realized �. Furthermore, ifhe amateur is of type ts, the realized � ∈ ts. Thus, for any m that ts receives in information sets reached in equilibrium, ��(m)nd ts contain at least one common element. Conversely, if ��(m) and ts contain some common elements, in equilibrium ts

eceives m with positive probability. The lemma follows by the contrapositive of the above. �

roof (Proof of Lemma 2). When there is an informative equilibrium with no unused message, for all m ∈ M, ��(m) is a stricton-empty subset of �. Suppose the expert deviates from the equilibrium by sending m′ for �′′ /∈ ��(m′); ��(m′) being atrict subset of � ensures that there exists such a �′′. We can find a ts such that �′′ ∈ ts and ts ⊂ � \ ��(m′): if �′′ > sup ��(m′),hoose a th with t ∈ (sup ��(m′), �′′), and if �′′ < inf ��(m′) choose a tl with t ∈ (�′′, inf ��(m′)); ��(m′) being non-empty

12 E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16

ensures that sup ��(m′) and inf ��(m′) are well-defined in �. When ts receives m′ in the deviation, because ��(m′)∩ ts = ∅,she will, by Lemma 1, be in an information set unreached in equilibrium even though m′ is a used message.

�

Proof (Proof of Lemma 3). Consider an equilibrium (�, �, �) in which there may be unused messages. Denote the set of suchunused messages by M0. Consider a message m ∈ M \ M0 that induces the set of actions A(m) in this equilibrium. Constructanother equilibrium (� ′, �′, �′) as follows: prescribe � ′ to the expert in which he is asked to randomize over M0 ∪ {m} for all� ∈ ��(m) such that every m′ ∈ M0 ∪ {m} is used with positive probability. Upon receiving any m′ ∈ M0 ∪ {m}, every intervaltype that the expert could face updates beliefs �′ with the same conclusion as when m is received. Given that �′ best respondsto �′, A(m′) ≡ A(m). It is thus a best response to �′ that the expert adopts the prescribed � ′. We then have an equilibrium(� ′, �′, �′) with the same outcome as (�, �, �), but every m ∈ M is used with positive probability.

�

Proof (Proof of Proposition 1). Consider N = 2, and the expert sends m ∈ M1 for � ∈ I1 and m ∈ M2 for � ∈ I2. The induced actionsare

�(m, th) =

⎧⎪⎨⎪⎩t + �1

2, for t ∈ I1 and m ∈ M1,

�1 + 12

, for t ∈ I1 and m ∈ M2,

and �(m, tl) =

⎧⎪⎨⎪⎩�1

2, for t ∈ I2 and m ∈ M1,

�1 + t

2, for t ∈ I2 and m ∈ M2.

In order for �1 to be indifferent between sending messages in M1 and M2, we require:∫ �1

0

([t + �1

2− (�1 + b)

]2

−[�1 + 1

2− (�1 + b)

]2)dt +

∫ 1

�1

([�1

2− (�1 + b)

]2

−[�1 + t

2− (�1 + b)

]2)dt = 0.

(A.1)

Solving (A.1) for �1 gives �1 = 12 (1 − 6b); N = 2 requires �1 > 0, satisfied if and only if b < 1

6 .I verify next that it is a best response for all � ∈ I1 to send messages in M1 and all � ∈ I2 to send messages in M2. Note

first that the induced actions under redundant advice are: �(m, th) = t+12 for t ∈ I2 and m ∈ M2, and �(m, tl) = t

2 for t ∈ I1 andm ∈ M1. Assuming that the off-equilibrium beliefs after false advice coincide with the amateur’s pre-communication beliefs,the corresponding actions are: �(m, th) = t+1

2 for t ∈ I2 and m ∈ M1, and �(m, tl) = t2 for t ∈ I1 and m ∈ M2.

When � ∈ [0, �1] sends m ∈ M1 and when he deviates by sending m ∈ M2, his expected payoffs are, respectively,∫ �

0

−[t + �1

2− (� + b)

]2

dt +∫ �1

�

−[t

2− (� + b)

]2dt +

∫ 1

�1

−[�1

2− (� + b)

]2

dt, and (A.2)

∫ �

0

−[�1 + 1

2− (� + b)

]2

dt +∫ �1

�

−[t

2− (� + b)

]2dt +

∫ 1

�1

−[�1 + t

2− (� + b)

]2

dt. (A.3)

Subtracting (A.2) from (A.3) we have, after imposing the equilibrium condition �1 = 12 (1 − 6b),

D1 = �(27� − 10�2 − 3) + b(9 + 96� − 30�2) + 36b2(4 − 3�) − 324b3 − 424

.

I show that D1 ≤ 0 for all (�, b) ∈[0, 1

2 (1 − 6b)]

×(

0, 16

). Note first that D1 is convex in � for (�, b) ∈

[0, 1

2 (1 − 6b)]

×(

0, 16

).

Thus, if the value of D1 at the boundaries of[0, 1

2 (1 − 6b)]

is non-positive for all b ∈(

0, 16

), then D1 ≤ 0 for all (�, b) ∈[

0, 12 (1 − 6b)

]×(

0, 16

). By the indifference condition, D1 = 0 when � = 1

2 (1 − 6b). When � = 0, D1 = (1+6b)(33b−54b2−4)24 , and

this can easily be verified to be negative for all b ∈(

0, 16

).

When � ∈ (�1, 1] sends m ∈ M1 and m ∈ M2, his expected payoffs are, respectively,∫ �1

0

−[t + �1

2− (� + b)

]2

dt +∫ �

�1

−[t + 1

2− (� + b)

]2dt +

∫ 1

�

−[�1

2− (� + b)

]2

dt, and (A.4)

∫ �1

0

−[�1 + 1

2− (� + b)

]2

dt +∫ �

�1

−[t + 1

2− (� + b)

]2dt +

∫ 1

�

−[�1 + t

2− (� + b)

]2

dt. (A.5)

Subtracting (A.4) from (A.5) we have, after imposing the equilibrium condition �1 = 12 (1 − 6b),

2 2

D2 = (1 − �)[10� + 7� + 30b(1 + �) − 5] + (1 − 6b)[9� + 3b(5 + 6�) + 54b − 5]

24.

Similar to the above, it can be verified that D2 ≥ 0 for all (�, b) ∈[

12 (1 − 6b), 1

]×(

0, 16

).

�

Pi

w

u

P

P

w

re

ppt

�T

S

N

U

fVN{

s

E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16 13

roof (Proof of Proposition 2). For clarity, I shall use Ue(·) to denote the expert’s quadratic payoff. In the CS model, thendifference condition reduces to: for i = 1, . . ., N − 1

Vcs(�csi−1, �csi , �csi+1, b) = Ue(�csi

+ �csi+1

2, �csi , b

)− Ue

(�csi−1 + �cs

i

2, �csi , b

)=∫ 1

0

[Ue(�csi

+ �csi+1

2, �csi , b

)

−Ue(�csi−1 + �cs

i

2, �csi , b

)]dt = 0, (A.6)

ith �cs0 = 0 and �csN = 1.The proof below parallels the proofs of Lemmas 4–6 in Crawford and Sobel (1982). The following two properties of the

niform-quadratic model are used. The first corresponds to the monotonicity condition in their Condition (M).

roperty P1. Suppose {�csi

}Ni=0

and {�cs′i

}Ni=0

are two solutions to (A.6), and �cs0 = �cs′0 and �cs1 > �cs′1 . Then, �csi> �cs′

ifor all i ≥ 2.

roperty P2. Suppose {�csi

}Ni=0

is a solution to (A.6). Then, (�csi+1 − �cs

i) > (�cs

i− �cs

i−1), i = 1, . . ., N − 1.

In the amateur model, the indifference condition reduces to: for i = 1, . . ., N − 1

V(�i−1, �i, �i+1, b) =∫ �i−1

0

[Ue(�i + �i+1

2, �i, b

)− Ue

(�i−1 + �i

2, �i, b

)]dt +

∫ �i

�i−1

[Ue(�i + �i+1

2, �i, b

)−

Ue(t + �i

2, �i, b

)]dt +

∫ �i+1

�i

[Ue(�i + t

2, �i, b

)− Ue

(�i−1 + �i

2, �i, b

)]dt +

∫ 1

�i+1

[Ue(�i + �i+1

2, �i, b

)− Ue

(�i−1 + �i

2, �i, b

)]dt = 0, (A.7)

ith �0 = 0 and �N = 1.Note that the indifference condition (A.7) is necessary but in general not sufficient for the determination of equilib-

ium. We proceed here without considering this issue. Refer to the online appendix for the relationship between thexistence of equilibria and off-equilibrium beliefs. Let �cs(K) = {�cs

i(K)}K

i=0with �cs

i−1(K) < �csi

(K), i = 1, . . ., K, be a partial

artition with K + 1 elements that satisfy (A.6). Similarly, let �(K) = {�i(K)}Ki=0 with �i−1(K) < �i(K), i = 1, . . ., K, be a partialartition with K + 1 elements that satisfy (A.7). I suppress “K” as the argument for the solution when it is clear what K equalso.

I first prove the second part of the proposition by proving: for K ∈ N, if �cs0 (K) = �0(K) and �csK (K) = �K (K), then �csi

(K) >

i(K) for i = 1, . . ., K − 1. The proof proceeds by induction on K. The statement is true vacuously for K = 1. So, consider K = 2.hen, (A.6) becomes

Vcs(�cs0 , �cs1 , �cs2 , b) = Ue(�cs1 + �cs2

2, �cs1 , b

)− Ue

(�cs0 + �cs1

2, �cs1 , b

)=∫ 1

0

[Ue(�cs1 + �cs2

2, �cs1 , b

)

−Ue(�cs0 + �cs1

2, �cs1 , b

)]dt = 0. (A.8)

ubstituting {�cs0 , �cs1 , �cs2 } into V(�i−1, �i, �i+1, b) in (A.7), we have, after using (A.8),

V(�cs0 , �cs1 , �cs2 , b) =∫ �cs

1

�cs0

[Ue(�cs1 + �cs2

2, �cs1 , b

)− Ue

(t + �cs1

2, �cs1 , b

)]dt +

∫ �cs2

�cs1

[Ue(�cs1 + t

2, �cs1 , b

)

−Ue(�cs0 + �cs1

2, �cs1 , b

)]dt. (A.9)

ote thatt+�cs

12 >

�cs0

+�cs1

2 for t ∈ (�cs0 , �cs1 ). Thus, given Ue11( · ) < 0, (A.8) implies that, for t ∈ (�cs0 , �cs1 ), Ue(�cs

1+�cs

22 , �cs1 , b

)−

e(t+�cs

12 , �cs1 , b

)< 0. Similarly, that

�cs1

+t2 <

�cs1

+�cs1

2 for t ∈ (�cs1 , �cs2 ) implies that Ue(�cs

1+t

2 , �cs1 , b)

− Ue(�cs

0+�cs

12 , �cs1 , b

)> 0

or t ∈ (�cs1 , �cs2 ). Property P2 implies that on balance V(�cs0 , �cs1 , �cs2 , b) > 0. It follows that there exists � > �cs2 such that(�cs0 , �cs1 , �, b) = 0. Since V(�0, �1, �2, b) = 0, under the hypothesis that �cs0 = �0 and �cs2 = �2, we have V(�cs0 , �1, �cs2 , b) = 0.

cs cs cs

ote that, given Property P2 of the solution to (A.6), {�0 , �1 , �2 }, Property P1 also holds for the two solutions to (A.7),�cs0 , �cs1 , �} and {�cs0 , �1, �cs2 }. Thus, that � > �cs2 implies that �cs1 > �1.

As the induction hypothesis, suppose the statement is true for K − 1. Consider partial partitions �cs(K) = {�csi

(K)}Ki=0

that

atisfies (A.6) and �(K) = {�i(K)}Ki=0 that satisfies (A.7). Note first that {�csi

(K)}K−1i=0

is a partial partition with K elements

14 E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16

that satisfies (A.6). Suppose that �cs0 (K) = �0(K) = 0 and �csK (K) = �K (K). If {�′i}K−1i=0

is a partial partition with K elementsthat satisfies (A.7) and �′

0 = �cs0 (K) = 0 and �′K−1 = �csK−1(K), then by the induction hypothesis �cs

i(K) > �′

i, i = 1, . . ., K − 2.

Since V(�csK−2(K), �csK−1(K), �csK (K), b) > 0, we have that V(�′K−2, �′

K−1, �csK (K), b) > 0. Thus, there exists � > �csK (K) such thatV(�′

K−2, �′K−1, �, b) = 0. Given that V(�K−2(K), �K−1(K), �K (K), b) = V(�K−2(K), �K−1(K), �csK (K), b) = 0, that � > �csK (K) implies

that �′i> �i(K), i = 1, . . ., K − 1. Thus, that �cs

i(K) > �′

i, i = 1, . . ., K − 2 and �csK−1(K) = �′

K−1 imply that the statement is true for K.Since the statement holds for any two partial partitions with K + 1 elements so long as �cs0 (K) = �0(K) and �csK (K) = �K (K),

it holds for �cs0 (K) = �0(K) = 0 and �csK (K) = �K (K) = 1. The second part of the proposition then holds when we set K = N = Ncs.For the first part, note first that the second part, coupled with Property P1 of the solutions to (A.7) which follows fromProperties P1 and P2 of the solutions to (A.6), implies that if �cs0 (K) = �0(K) and �cs1 (K) = �1(K), then �cs

i(K) < �i(K), i = 2, . . .,

K. Suppose �(N′) is an equilibrium partition in the amateur model with N′ + 1 elements. Further suppose �cs(N′) is a partitionsatisfying (A.6) with �cs0 (N′) = �0(N′) and �cs1 (N′) = �1(N′). Then, �cs

i(N′) < �i(N′), i = 2, . . ., N′. Thus, if �cs(N′) is an equilibrium

partition, it has at least N′ + 1 elements. Hence, Ncs(b) ≥ N(b).�

Proof (Proof of Proposition 3). The notational convention in the proof of Proposition 2 is followed. The proof parallelsthat of Theorem 3 in Crawford and Sobel (1982). Let �(x) = {�i(x)}Ki=0 be a partial partition that satisfies (A.7) for i = 2,. . ., N, with �0(x) = �0, �K−1(x) = x, and �K(x) = �K. Let � be the type such that �1(�) = �0 and � be the type such that �(�)satisfies (A.7) for i = 1, . . ., N. Note that under the partition �(x) the novice’s (ex-ante) expected payoff on [�0, �K] is

EUdcs(x) =∑K

i=1

∫ �i(x)�i−1(x)

Ud(�i−1(x)+�i(x)

2 , �)d�. Since �i−1(x)+�i(x)

2 is a result of optimization and �0(x) and �K(x) are fixed,

applying the envelope theorem we have

dEUdcs(x)dx

=K−1∑i=1

[Ud(�i−1(x) + �i(x)

2, �i(x)

)− Ud

(�i(x) + �i+1(x)

2, �i(x)

)]d�i(x)dx

.

Given that Properties P1 and P2 of the solutions to (A.6) in the proof of Proposition 2 imply that Property P1 also holdsfor the solutions to (A.7), we have d�i(x)

dx > 0. Since �i+1(x)+�i(x)2 >

�i−1(x)+�i(x)2 , �i+1(x)+�i(x)

2 ≥ t+�i(x)2 for t ∈ (�i−1(x), �i(x)], and

�i(x)+t2 ≥ �i−1(x)+�i(x)

2 for t ∈ (�i(x), �i+1(x)], the satisfaction of (A.7) and that Ue(·)13 > 0 imply that Ud(�i−1(x)+�i(x)

2 , �i(x))

−

Ud(�i(x)+�i+1(x)

2 , �i(x))> 0 for i = 2, . . ., K − 1. Also, Property P1 of the solutions to (A.7) and that Ue(·)13 > 0 imply that

Ud(�0(x)+�1(x)

2 , �1(x))

− Ud(�1(x)+�2(x)

2 , �1(x))> 0 for x ∈ [�, �]. Hence EUdcs(x) is strictly increasing in x ∈ [�, �].

I first prove the statement that for Ncs = N = K ≥ 2 under a given b, the novice’s expected payoff is strictly higher underpartition �cs(K) than under partition �(K). The proof proceeds by induction on K. Given that in two-step equilibria, �1 =12 (1 − 6b) and �cs1 = 1

2 (1 − 4b), the argument in the last paragraph implies that the statement is true for K = 2 when we set �0 = 0

and �K = 1. As the induction hypothesis, suppose the statement is true for K − 1. Consider two partitions �cs(K) = {�csi

(K)}Ki=0

and �(K) = {�i(K)}Ki=0, that satisfy, respectively, (A.6) and (A.7), where �cs0 = �0 = 0 and �csK = �K = 1. We can find a partition

�cs(K) = {�csi

(K)}Ki=0

such that �cs1 = �1 and (A.6) is satisfied for {�csi

(K)}Ki=2

. It follows from the argument in the last paragraph

that the novice’s expected payoff is strictly higher under �cs(K) than under �cs(K). For �cs(K) and �(K), it follows from theproof of Proposition 2 that �cs

i> �i, i = 2, . . ., K − 1. That the statement is true for K = 2 and the induction hypothesis imply

that the novice’s expected payoff is strictly higher under �cs(K) than under �(K). The statement then follows. Since it is truefor any K ≥ 2, it is true for Ncs(b) = N(b) ≥ 2.

I show next that if Ncs(b) > N(b) ≥ 2, the novice’s expected payoff is strictly higher under partition �cs(Ncs(b)) than under�(N(b)). (The case for Ncs(b) > N(b) = 1 is obvious, and the proof is omitted.) Construct a new partition �(N(b)) by extractingthe elements {�cs1 , . . ., �cs

Ncs(b)−N(b)} ∈ �cs(Ncs(b)) and imposing them into �(N(b)). Note that then N(b) = Ncs(b). Since �(N(b))

is finer than �(N(b)) (i.e., �(N(b)) ⊂ �(N(b))), the novice’s expected payoff is strictly higher under �(N(b)) than under �(N(b)).Now, for �cs(Ncs(b)) and �(N(b)), Property P1 and the construction above guarantee that �cs

i≥�i, i = 1, . . ., Ncs(b) − 1. By an

induction argument similar to the above, the novice’s expected payoff is strictly higher under �cs(Ncs(b)) than under �(N(b)).The result then follows.

Note that for b ∈(

0, 14

), Ncs(b) ≥ 2 and N(b) ≥ 1. It thus follows from the above that for b ∈

(0, 1

4

)the novice’s expected

payoff is strictly lower under information structure I(N(b)) associated with partition �(N(b)) than under I(Ncs(b)) associatedwith �cs(Ncs(b)).

�

Proof (Proof of Proposition 4). It follows from Proposition 3 that if t ∈ {�i}N(b)i=0 , the amateur’s information structure has a

strictly negative strategic value. Since the amateur’s payoff is continuous in t, there exists other t′s in the neighborhoods ofthe elements in {�i}N(b)

i=0 such that the associated information structures also have a strictly negative strategic value.�

A

P

w

t

�

i

A

t

I

w

t

h

P

F

a

E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16 15

.2. Calculations

roof (Calculation for Example 2). The novice’s expected payoff is

EUdcs(b) = − 1

12Ncs(b)2− b2(Ncs(b)2 − 1)

3, (A.10)

here Ncs(b) =⌈− 1

2 + 12

(1 + 2

b

)1/2⌉

(Crawford and Sobel, 1982). For b ∈(

112 ,

14

), Ncs(b) = 2, and thus EUdcs(b) = − 1

48 − b2.

The amateur’s expected payoff if t ∈ (�1, 1] is

EUd(t, b) =∫ �1

0

−(�1

2− �

)2

d� +∫ t

�1

−(�1 + t

2− �

)2

d� +∫ 1

t

−(t + 1

2− �)2d�. (A.11)

For b ∈ [ 16 ,

14 ), �1 = 0, and (A.11) becomes EUd(t, b) = − 1

12 (3t2 − 3t + 1). Thus, the strategic value of the amateur’s informa-ion structure is strictly positive if and only if − 1

12 (3t2 − 3t + 1) −(− 1

48 − b2)> 0 or t ∈

(12 − 2b, 1

2 + 2b)

. For b ∈(

112 ,

16

),

1 > 0. Using �1 = 12 (1 − 6b), (A.11) becomes EUd(t, b) = − 1

48 − b2 + A(t, b), where A(t, b) = t(3−2t)+12bt(1−t)+4b2(4−9t)−116 . The

nformation structure has a strictly positive strategic value if and only if A(t, b) > 0. Note that since for all b ∈(

112 ,

16

),(

12 (1 − 6b), b

)= A(1, b) = − 5b2

4 < 0, ∂2A(t,b)∂t2

< 0, and maxt′∈[

12 (1−6b),1

]A(t′, b) = 1+18b−52b2−216b3

128 > 0, solving A(t, b) = 0 gives

hat A(t, b) > 0 if and only if t ∈ (t′, t′), where

t′ = 3 + 12b − 36b2 −√

1 + 24b + 56b2 − 96b3 + 1296b4

4(1 + 6b)>

12

(1 − 6b), and

t′ = 3 + 12b − 36b2 +

√1 + 24b + 56b2 − 96b3 + 1296b4

4(1 + 6b)< 1.

f t ∈ [0, �1], the amateur’s expected payoff is

EUd(t, b) =∫ t

0

−(t

2− �)2d� +

∫ �1

t

−(t + �1

2− �

)2

d� +∫ 1

�1

−(�1 + 1

2− �

)2

d� = − 148

− b2 + B(t, b), (A.12)

here B(t, b) = t(1−2t)−12bt(1−t)−4b2(5−9t)16 . Note that B(0, b) = B

(12 (1 − 6b), b

)= − 5b2

4 < 0 and ∂2B(t,b)∂t2

< 0 for b ∈(

112 ,

16

). Fur-

hermore, for b ∈(

112 ,

16

), maxt′∈[

0, 12 (1−6b)

]B(t′, b) = 1−18b−52b2−216b3

128 < 0. Thus, information structures with t ∈[0, 1

2 (1 − 6b)]

ave strictly negative strategic value.�

roof (Calculation for Example 3). For b ∈ [ 16 ,

14 ), EUd(b) = − 1

24 is obtained by integrating (A.11) with respect to t with �1 = 0.

or b ∈(

112 ,

16

), EUd(b) = − 1+72b2−432b4

64 is obtained by integrating (A.11) and (A.12) with respect to t in the respective ranges

nd summing them up while setting �1 = 12 (1 − 6b).

When b = 120 , the indifference condition in the amateur model reduces to

∫ �1

0

([�1 + �2

2−(�1 + 1

20

)]2

−[t + �1

2−(�1 + 1

20

)]2)dt +

∫ �2

�1

([�1 + t

2−(�1 + 1

20

)]2

−[�1

2−(�1 + 1

20

)]2)dt +

∫ 1

�2

([�1 + �2

2−(�1 + 1

20

)]2

−[�1

2−(�1 + 1

20

)]2)dt = 0; (A.13)

∫ �1

0

([�2 + 1

2−(�2 + 1

20

)]2

−[�1 + �2

2−(�2 + 1

20

)]2)dt +

∫ �2

�1

([�2 + 1

2−(�2 + 1

20

)]2

−[t + �2

2−(�2 + 1

20

)]2)dt +

∫ 1

�2

([�2 + t

2−(�2 + 1

20

)]2

−[�1 + �2

2−(�2 + 1

20

)]2)dt = 0. (A.14)

16 E.K. Lai / Journal of Economic Behavior & Organization 103 (2014) 1–16

While in the amateur model the indifference condition is not sufficient for the existence of partitional equilibria, usingthe steps similar to the proof of Proposition 6 (online appendix), it can be verified that the relevant solution to (A.13) and(A.14), �1 = 0.090131 and �2 = 0.403562, constitutes the boundary types of a three-step equilibrium.

�

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