# Why do experts give simple advice?

Benjamin Davies  Department of Economics, Stanford University;
<EMAIL_ADDRESS>I thank Yunus Aybas, Doug Bernheim, Steve Callander,
Matt Jackson, Spencer Pantoja, and Anirudh Sankar for helpful discussions.

(Draft version: September 2022)

###### Abstract

An expert tells an advisee whether to take an action that may be good or bad.
He may provide a condition under which to take the action. This condition
predicts whether the action is good if and only if the expert is competent.
Providing the condition exposes the expert to reputational risk by allowing
the advisee to learn about his competence. He trades off the accuracy benefit
and reputational risk induced by providing the condition. He prefers not to
provide it—i.e., to give “simple advice”—when his payoff is sufficiently
concave in the posterior belief about his competence.

JEL codes: D82, D83

Keywords: advice, career concerns, experts, information disclosure,
reputation, science communication

##  1 Introduction

This paper discusses how reputational concerns impact the type of advice
experts provide. I frame advice as a “rule” or “model” that maps conditions to
actions. It makes a prediction of the _ex post_ “correct” action based on _ex
ante_ unobserved conditions. For example, it tells a patient whether to pursue
medical treatment if they develop specific symptoms. I assume that

  1. (i)

the _ex post_ correct action cannot be predicted perfectly, and

  2. (ii)

the advisee does not know what conditions are relevant if the expert does not
include them in their advice.

These assumptions reflect the situation faced by public health officials and
policy-makers early in the COVID-19 pandemic. Assumption (i) reflects how the
mapping between medical conditions (e.g., being immunocompromised) and optimal
policies (e.g., getting vaccinated) was uncertain because the relevant science
was unsettled. It implies that advice can be “wrong” (i.e., recommend the _ex
post_ incorrect action) even if it includes all relevant conditions.
Assumption (ii) reflects how the novelty of the relevant science prevented
advisees from comparing experts’ advice with public knowledge. It implies that
_if_ advice is wrong then experts cannot be punished for excluding relevant
conditions. Intuitively, advisees cannot observe bad outcomes and attribute
them to causes experts “should have known about” because advisees do not know
the set of possible causes.111 This became less true as the pandemic
progressed and people started sharing (scientific and anecdotal) information
about vaccination outcomes.

Under assumptions (i) and (ii), experts make strategic choices about the type
of advice they provide. Advice may be unconditional (e.g., “do $x$”) or have
conditions (e.g., “do $x$ if $y$ happens and $z$ does not”). Including
conditions tailors the advice and makes experts appear more confident.
However, this confidence may hurt experts’ reputations if their advice turns
out to be wrong. Consequently, experts may exclude relevant conditions—that
is, “simplify” their advice—to hide their confidence and preserve their
reputation. I analyze this tension between providing better advice and
managing reputational risk.

Section 2 embeds this tension in a simple model. An expert (he) tells an
advisee (she) whether to take an action that may be good or bad. Both parties
want the advisee to take the action if and only if it is good. The expert may
provide a binary condition under which to take the action. This condition
predicts whether the action is good if the expert is competent but not if he
is incompetent. Neither the advisee nor the expert know if he is competent. He
chooses between a “simple rule,” which says to take the action, and a “complex
rule,” which says to take the action under the provided condition. If he
chooses the complex rule then the advisee learns about the expert’s competence
by whether the rule predicts the correct action. If he chooses the simple rule
then she learns nothing about the expert’s competence. The expert trades off
the complex rule’s superior predictive power with its ability to reveal
information about his competence.

Section 3 analyzes the expert’s trade-off. Choosing the complex rule over the
simple rule induces a mean-preserving spread in the posterior belief about the
expert’s competence. This spread makes him better off if his payoff is convex
in the posterior and worse off if it is concave in the posterior. In the
convex case, the expert always chooses the complex rule because it is more
accurate and induces an expected reputational benefit. In the concave case, he
chooses the complex rule if and only if its accuracy benefit exceeds its
reputational cost. Increasing the condition’s predictive power makes the
complex rule more accurate and more diagnostic of the expert’s competence. If
his payoff is concave in the posterior then this increase in predictive power
makes the expert more willing to choose the complex rule if and only if his
reputational concerns are sufficiently weak.

Section 4 considers a setting in which the expert earns a wage if and only if
the posterior belief about his competence exceeds a threshold. If this
threshold equals the prior belief, then the expert chooses the complex rule if
and only if the wage is sufficiently low. If the advisee wants to solicit the
complex rule but cannot change the wage, then she can (i) lower the threshold
and guarantee earning the wage, or (ii) raise the threshold and guarantee
_not_ earning the wage. Both (i) and (ii) remove the reputational risk that
prevents the expert from choosing the complex rule.

Section 5 considers some extensions of my analysis and Section 6 compares it
to related literature. Section 7 concludes. Appendix A contains proofs of my
mathematical claims.

##  2 Model

#### Environment

There are two players: an expert (he) and an advisee (she).

The expert tells the advisee whether to take an action. The action is either
good ($A=1$) or bad ($A=0$). The expert and advisee know $\Pr(A=1)=0.5$.

The expert may provide a condition under which to take the action. This
condition predicts $A$ if the expert is competent ($\theta=1$) but not if he
is incompetent ($\theta=0$). The condition $X_{\theta}\in\\{0,1\\}$ has
prevalence $\Pr(X_{\theta}=1)=x\in(0,1)$ and covariance
$\operatorname{Cov}(A,X_{\theta})=\theta\sigma$ with $\sigma\in(0,0.25]$.

The expert wants the advisee to take the action if and only if it is good. He
offers one of two rules $r\in\\{S,C\\}$:

  1. 1.

“take the action” (the “simple” rule $r=S$);

  2. 2.

“take the action if and only if $X_{\theta}=1$” (the “complex” rule $r=C$).

Thus, he provides the condition if and only if he chooses the complex rule.
The advisee can verify the condition’s prevalence and covariance if and only
if the expert provides it. If he provides it then she observes its value
between receiving and implementing his advice.

Let $Y\in\\{0,1\\}$ indicate whether the advisee takes the “correct” action.
Then

$\displaystyle\Pr(Y=1\,|\,r,\theta)$
$\displaystyle=\begin{cases}\Pr(A=1)&\text{if}\ r=S\\\
\Pr(A=X_{\theta})&\text{if}\ r=C\end{cases}$
$\displaystyle=\begin{cases}0.5&\text{if}\ r=S\\\ 0.5+2\theta\sigma&\text{if}\
r=C\end{cases}$ (1)

for each $\theta\in\\{0,1\\}$.

#### Beliefs

Neither the advisee nor the expert know whether he is competent. They have
common prior $\Pr(\theta=1)=\pi_{0}\in(0,1)$. Hence

$\displaystyle\Pr(Y=1\,|\,r)$
$\displaystyle=\Pr(Y=1\,|\,r,\theta=1)\Pr(\theta=1)+\Pr(Y=1\,|\,r,\theta=0)\Pr(\theta=0)$
$\displaystyle=\begin{cases}0.5&\text{if}\ r=S\\\
0.5+2\sigma\pi_{0}&\text{if}\ r=C.\end{cases}$ (2)

The expert’s ignorance of $\theta$ prevents him from trying to signal it.
Therefore, his chosen rule $r$ is independent of $\theta$.222 This remains
true (in equilibrium) if the expert knows $\theta$—see Section 5.3.

The players use (1), (2), and Bayes’ rule to form a posterior belief

$\displaystyle\Pr(\theta=1\,|\,r,Y)$
$\displaystyle=\frac{\Pr(Y\,|\,r,\theta=1)\Pr(\theta=1)}{\Pr(Y\,|\,r)}$
$\displaystyle=\begin{cases}\pi_{0}&\text{if}\ r=S\\\
\frac{(1+4\sigma)\pi_{0}}{1+4\sigma\pi_{0}}&\text{if}\ r=C\ \text{and}\ Y=1\\\
\frac{(1-4\sigma)\pi_{0}}{1-4\sigma\pi_{0}}&\text{if}\ r=C\ \text{and}\
Y=0\end{cases}$ (3)

about $\theta$ after observing the chosen rule $r$ and the outcome $Y$.

For convenience, I let

$\pi_{1}(Y)\equiv\Pr(\theta=1\,|\,r=C,Y)$

denote the random posterior belief about $\theta$ when the expert chooses the
complex rule. This belief has expected value

$\displaystyle\operatorname{E}[\pi_{1}(Y)]$
$\displaystyle=\pi_{1}(1)\Pr(Y=1\,|\,r=C)+\pi_{1}(0)\Pr(Y=0\,|\,r=C)$
$\displaystyle=\Pr(Y=1\,|\,r=C,\theta=1)\pi_{0}+\Pr(Y=0\,|\,r=C,\theta=1)\pi_{0}$
$\displaystyle=\pi_{0}.$ (4)

Thus, choosing the complex rule leads to unchanged beliefs _on average_ ,
while choosing the simple rule leads to unchanged beliefs independently of
whether the advisee takes the correct action.

#### Payoffs

The expert’s payoff has two components:

  1. 1.

A unit “accuracy payoff” from the advisee taking the correct action;

  2. 2.

A “reputation payoff” $\psi(\Pr(\theta=1\,|\,r,Y))$ that depends on the
posterior belief (3) about $\theta$.

I assume $\psi:[0,1]\to\mathbb{R}$ is non-decreasing but has
$\psi(0)<\psi(1)$. This function captures the expert’s extrinsic incentive to
be viewed as competent. This incentive may reflect the continuation value of
employment in a dynamic setting with replaceable experts (see Section 4).
Alternatively, it may reflect public officials (e.g., Anthony Fauci) wanting
to foster trust in government institutions (e.g., the Centres for Disease
Control).

The expert’s total payoff equals the sum of his accuracy and reputation
payoffs. He chooses the rule $r\in\\{S,C\\}$ that maximizes his expected
payoff

$\Phi(r,\sigma,\pi_{0})\equiv\begin{cases}0.5+\psi(\pi_{0})&\text{if}\ r=S\\\
0.5+2\sigma\pi_{0}+\Psi(\sigma,\pi_{0})&\text{if}\ r=C,\end{cases}$

where

$\displaystyle\Psi(\sigma,\pi_{0})$
$\displaystyle\equiv\operatorname{E}[\psi(\pi_{1}(Y))]$
$\displaystyle=(0.5+2\sigma\pi_{0})\psi\left(\frac{(1+4\sigma)\pi_{0}}{1+4\sigma\pi_{0}}\right)+(0.5-2\sigma\pi_{0})\psi\left(\frac{(1-4\sigma)\pi_{0}}{1-4\sigma\pi_{0}}\right)$

is his expected reputation payoff from choosing the complex rule. He breaks
ties by choosing the complex rule.333 This is an arbitrary assumption. It is
not consequential for any of my results.

The advisee’s payoff equals $Y$. Therefore, she always follows the expert’s
advice.

#### Timing

The model timing is as follows:

  1. 1.

The expert chooses the simple or complex rule.

  2. 2.

If the expert chooses the simple rule then the advisee takes the action. If he
chooses the complex rule then she observes $X_{\theta}$ and takes the action
if and only if $X_{\theta}=1$.

  3. 3.

The expert and advisee observe $Y$, and form a posterior belief (3) about
$\theta$.

  4. 4.

Payoffs are realized.

###  2.1 Discussion of modeling assumptions

#### Behavioral limitations

I assume the advisee is boundedly rational. Her behavioral limitation is not
knowing the set of conditions that could predict whether the action is good.
This knowledge is what makes the expert an “expert.” However, if the expert
provides a condition then I assume the advisee can verify its potential
predictive power. This prevents the expert from mis-reporting the predictive
power of the condition he provides. Thus, assuming verifiability allows me to
focus on the expert’s decision to provide the condition and abstract from his
decision about how much uncertainty to communicate.

Intuitively, the expert in my model generates hypotheses about the
relationship between conditions and actions. Choosing the complex rule
corresponds to stating a hypothesis and referencing the scientific literature
on which it is based. The advisee reads that literature and verifies that the
hypothesis is “reasonable.” However, she cannot tell if the hypothesis is
true. That skill is unique to competent experts.

The form of bounded rationality I assume is intentionally unrealistic. For
example, one reason we outsource vaccine guidance to public health agencies,
rather than have the public engage with the relevant virological science
themselves, is because that science is difficult to understand. This
difficulty may lead the public to endure cognitive costs or make mistakes that
agencies want to prevent. Thus, one explanation for why experts provide
“simple” advice is that they internalize the advisee’s cognitive burden and
avoid sharing “too much detail.” However, this explanation is unsatisfying
because one can rationalize any amount of detail suppression with a suitable
assumption about the advisee’s cognitive costs. In contrast, my reputation-
based explanation does not appeal to cognitive costs. It is meant to be
idealized: to show that experts may simplify their advice even if the advisee
has _zero_ cognitive costs.

Section 5.2 considers an extension to my model in which the advisee makes
mistakes when implementing the complex rule.

#### Expert does not know $\theta$

I assume the expert does not know if he is competent. This allows me to
abstract from competence signaling motives because the expert has no
information about his competence to signal. It makes the expert’s choice of
model uninformative about his competence, and isolates the tension between
predictive power and reputational risk. Section 5.3 considers the implications
of letting the expert know $\theta$.

#### Expert has aligned preferences

I assume the expert is “unbiased” in that he wants the advisee to take the
action if and only if she thinks it is good. This assumption is appropriate in
the science communication setting that motivates my analysis: experts are
intrinsically “benevolent” but face extrinsic incentives to preserve their
reputation. It allows me to abstract from persuasion motives (as studied,
e.g., by Kamenica and Gentzkow, , 2011) and lie detection (see, e.g.,
Balbuzanov, , 2019).

##  3 Analysis

###  3.1 Expert’s choice

Proposition 1 characterizes the expert’s choice between the simple and complex
rules. For convenience, I let

$\displaystyle\Delta\Phi(\sigma,\pi_{0})$
$\displaystyle\equiv\Phi(C,\sigma,\pi_{0})-\Phi(S,\sigma,\pi_{0})$
$\displaystyle=2\sigma\pi_{0}+\Psi(\sigma,\pi_{0})-\psi(\pi_{0})$

denote the gain in the expert’s expected payoff from choosing the complex rule
over the simple rule. He chooses the complex rule if and only if this gain is
non-negative:

######  Proposition 1.

The expert chooses the complex rule if and only if

$2\sigma\pi_{0}\geq\psi(\pi_{0})-\Psi(\sigma,\pi_{0}).$ (5)

The left-hand side of (5) equals the gain in the expert’s expected accuracy
payoff from choosing the complex rule over the simple rule. The right-hand
side equals the loss in his expected reputation payoff from choosing the
complex rule over the simple rule.

If $\psi$ is convex then the expert’s reputation gain from advising the
correct action exceeds his reputation loss from advising the incorrect action.
As a result, he chooses the complex rule because it is more accurate and
offers an expected reputation _benefit_ :

######  Corollary 1.

If $\psi$ is linear or convex, then the expert chooses the complex rule.

In contrast, if $\psi$ is concave then the reputation gain from advising the
correct action is smaller than the reputation loss from advising the incorrect
action. As a result, choosing the complex rule incurs an expected reputation
_cost_. Choosing the simple rule avoids this cost because it prevents the
advisee from learning about the expert’s competence. He chooses the simple
rule if this incentive to hide his competence dominates his incentive to
advise the correct action.

Suppose the expert’s reputation payoff comprises a wage $w\geq 0$ earned with
some probability $R(\pi)$ that depends on the posterior belief $\pi$ about his
competence. For example, the advisee may decide whether to re-hire the expert
based on his perceived competence relative to other experts (see Section 4).
Corollary 2 states that if $R$ is concave in $\pi$ (e.g., since competition
among more competent experts is more aggressive) then the expert chooses the
complex rule if and only if $w$ is sufficiently low.

######  Corollary 2.

Let $w\geq 0$ and suppose $\psi(\pi)=w\,R(\pi)$ for all beliefs $\pi\in[0,1]$,
where $R:[0,1]\to\mathbb{R}$ is non-decreasing and concave. Then the expert
chooses the complex rule if and only if

$w\leq\frac{2\sigma\pi_{0}}{R(\pi_{0})-\operatorname{E}[R(\pi_{1}(Y))]}.$

Figure 1: Values of $\Delta\Phi(\sigma,\pi_{0})$ when $\psi(\pi)=w\sqrt{\pi}$,
$\sigma\in\\{0.1,0.2\\}$, $\pi_{0}\in\\{0.25,0.5,0.75\\}$, and $w\in[0,10]$

For example, suppose $R(\pi)\equiv\sqrt{\pi}$. Figure 1 plots the
corresponding value of $\Delta\Phi(\sigma,\pi_{0})$ for combinations of
$\sigma\in\\{0.1,0.2\\}$, $\pi_{0}\in\\{0.25,0.5,0.75\\}$, and $w\in[0,10]$.
This value falls when $w$ rises because the expected reputational cost of
choosing the complex rule rises. If $\sigma=0.2$ and $\pi_{0}=0.25$ (i.e.,
competence is valuable but rare) then the complex rule is good at diagnosing
the expert’s probable incompetence. Consequently, the expert chooses the
complex rule if and only if $w<3.07$; that is, his reputational concerns are
sufficiently _weak_. This threshold value of $w$ rises when $\sigma$ falls to
0.1, which makes competence less valuable but also makes the complex rule less
diagnostic. The threshold also rises when $\pi_{0}$ rises, in part because
$\psi$ becomes less concave near $\pi_{0}$. This fall in local concavity
lowers the reputational loss from advising the incorrect action relative to
the gain from advising the correct action.

In this example, the expert chooses the complex rule only if the maximum
reputation payoff $w$ is many times larger than his accuracy payoff. This
difference in payoff magnitudes is plausible if $w$ represents the
continuation value of employment. This value includes the present value of the
accuracy payoffs accrued from future advisory events. If there are many such
events and the expert is patient then $w$ can be large. For example, if he
offers advice at each date $t\in\\{0,1,2,\ldots,10\\}$ and has inter-temporal
discount factor $\delta=0.95$ then the date 0 value of his date $t\geq 1$
accuracy payoffs is $\delta(1-\delta^{10})/(1-\delta)\approx 7.62$ times
larger than the value of his date 0 accuracy payoff.

###  3.2 Changes in parameter values

I now analyze how the posterior belief $\pi_{1}(Y)$ and gain in expected
payoff $\Delta\Phi(\sigma,\pi_{0})$ from choosing the complex rule depend on
the parameters $\sigma$ and $\pi_{0}$.

######  Lemma 1.

$\pi_{1}(1)$ is increasing in $\sigma$ and $\pi_{0}$, whereas $\pi_{1}(0)$ is
decreasing in $\sigma$ and non-decreasing in $\pi_{0}$.

Figure 2: Posterior belief $\pi_{1}(Y)$ for $Y\in\\{0,1\\}$ and
$\sigma\in\\{0.1,0.2\\}$

Figure 2 shows how $\pi_{1}(1)$ and $\pi_{1}(0)$ grow with $\pi_{0}$ when
$\sigma\in\\{0.1,0.2\\}$. As $\sigma$ rises, the complex rule becomes less
likely to lead to the incorrect action if the expert is competent. Then
observing $Y=0$ becomes more diagnostic of his _in_ competence if he chooses
the complex rule. Whether this dissuades him from choosing that rule depends
on the shape of $\psi$:

######  Proposition 2.

If $\psi$ is linear or convex, then $\Delta\Phi(\sigma,\pi_{0})$ is increasing
in $\sigma$.

Increasing the covariance $\sigma$ makes the complex rule (i) more likely to
lead to the correct action and (ii) more diagnostic of the expert’s
competence. These two forces on $\Delta\Phi(\sigma,\pi_{0})$ act in the same
direction when $\psi$ is convex. In that case, the expert gains more from
being seen as more competent than he loses from being seen as less competent.
Consequently, making the complex rule better at diagnosing his competence
makes him more willing to choose that rule.

In contrast, forces (i) and (ii) act in _opposite_ directions when $\psi$ is
concave. In that case, the expert gains _less_ from being seen as more
competent than he loses from being seen as less competent. Consequently,
making the complex rule better at diagnosing his competence makes him more
willing to choose that rule if and only if his reputation concerns are
sufficiently weak. Figure 1 demonstrates this condition: if
$\psi(\pi)=w\sqrt{\pi}$ then $\Delta\Phi(0.2,0.5)>\Delta\Phi(0.1,0.5)$ when
$w$ is small but $\Delta\Phi(0.2,0.5)<\Delta\Phi(0.1,0.5)$ when $w$ is large.

Increasing $\pi_{0}$ raises the probability that the complex rule has superior
predictive power. It also raises the expert’s reputation payoff at each
posterior belief about his competence. However, this rise in reputation
payoffs applies to the complex rule _and_ the simple model. Consequently, the
overall effect on the expert’s willingness to choose the complex rule depends
on how changing $\pi_{0}$ changes his reputation payoffs under each rule.
Proposition 3 characterizes this overall effect.

######  Proposition 3.

Suppose $\psi$ is differentiable at $\pi_{0}$, $\pi_{1}(1)$, and $\pi_{1}(0)$.
Then $\Delta\Phi(\sigma,\pi_{0})$ is increasing in $\pi_{0}$ if and only if

$2\sigma+2\sigma\left[\psi(\pi_{1}(1))-\psi(\pi_{1}(0))\right]+\operatorname{E}\left[\psi^{\prime}(\pi_{1}(Y))\frac{\partial\pi_{1}(Y)}{\partial\pi_{0}}\right]>\psi^{\prime}(\pi_{0}),$
(6)

where $\psi^{\prime}$ denotes the derivative of $\psi$.

The left-hand side of (6) decomposes the effect of increasing $\pi_{0}$ on the
expert’s expected payoff from choosing the complex rule. The first term equals
the increase in his expected accuracy payoff. The second term equals the
increase in his expected reputational payoff from being more likely to be
competent. The third term equals the increase in his expected reputation
payoff induced by the increase in the posteriors $\pi_{1}(1)$ and
$\pi_{1}(0)$. The right-hand side of (6) equals the increase in the reputation
payoff from choosing the simple rule. Increasing $\pi_{0}$ increases
$\Delta\Phi(\sigma,\pi_{0})$ if and only if the sum of the three terms on the
left-hand side of (6) exceeds the term on the right-hand side.

Proposition 4 analyzes the class of reputation payoff functions
$\psi(\pi)=w\,R(\pi)$ considered in Corollary 2, replacing the concavity
restriction with a differentiability restriction.

######  Proposition 4.

Let $w\geq 0$ and suppose $\psi(\pi)=w\,R(\pi)$ for all beliefs $\pi\in[0,1]$,
where $R:[0,1]\to\mathbb{R}$ is non-decreasing and differentiable. Then either

  1. 1.

$\Delta\Phi(\sigma,\pi_{0})$ is increasing in $\pi_{0}$, or

  2. 2.

There exists $w^{*}>0$ such that $\Delta\Phi(\sigma,\pi_{0})$ is increasing in
$\pi_{0}$ if and only if $w<w^{*}$.

Intuitively, increasing $\pi_{0}$ raises the payoff from choosing the simple
rule by a _certain_ amount but the payoff from choosing the complex rule by an
_uncertain_ amount. If $R$ is “not too concave” or the stakes are sufficiently
low (i.e., $w$ is small) then the uncertain amount is more attractive.

##  4 Wages and replacement thresholds

This section considers a setting in which the expert earns wage $w>0$ if and
only if the posterior belief about his competence exceeds a threshold
$\pi^{*}\in(0,1)$. This setting arises, for example, when the advisee faces an
action in each of two periods, and she hires the expert to provide advice in
the second period if and only if she believes he is more likely to be
competent than another expert drawn from the market. Then the expert’s
reputation payoff is given by

$\psi(\pi)=\begin{cases}w&\text{if}\ \pi\geq\pi^{*}\\\
0&\text{otherwise}\end{cases}$ (7)

for all beliefs $\pi\in[0,1]$. Lemma 2 defines the expert’s expected payoffs
from choosing the simple and complex rules in this setting.

######  Lemma 2.

Suppose the reputation payoff function $\psi$ is defined as in (7). Then the
expert’s expected payoff from choosing the complex rule equals

$\Phi(C,\sigma,\pi_{0})=\begin{cases}0.5+2\sigma\pi_{0}&\text{if}\
\pi_{0}<\underline{\pi}\\\ 0.5+2\sigma\pi_{0}+w(0.5+2\sigma\pi_{0})&\text{if}\
\underline{\pi}\leq\pi_{0}<\overline{\pi}\\\ 0.5+2\sigma\pi_{0}+w&\text{if}\
\overline{\pi}\leq\pi_{0},\end{cases}$ (8)

where

$\underline{\pi}\equiv\frac{\pi^{*}}{1+4\sigma(1-\pi^{*})}$ (9)

and

$\overline{\pi}\equiv\frac{\pi^{*}}{1-4\sigma(1-\pi^{*})}.$ (10)

His expected payoff from choosing the simple rule equals

$\Phi(S,\sigma,\pi_{0})=\begin{cases}0.5+w&\text{if}\ \pi_{0}\geq\pi^{*}\\\
0.5&\text{otherwise}.\end{cases}$ (11)

Suppose the expert chooses the simple rule. Then the posterior belief about
his competence equals the prior belief because no information about his
competence is revealed. Consequently, he receives the wage $w$ if and only if
his prior exceeds the threshold $\pi^{*}$.

Now suppose the expert chooses the complex rule. If $\pi_{0}<\underline{\pi}$
then the expert is guaranteed a reputation payoff of zero because the
posterior belief $\pi_{1}(1)$ induced by the advisee taking the correct action
is less than $\pi^{*}$. Conversely, if $\pi_{0}\geq\underline{\pi}$ then the
expert is guaranteed a reputation payoff of $w$ because the posterior belief
$\pi_{1}(0)$ induced by the advisee taking the _incorrect_ action is greater
than $\pi^{*}$. If $\pi_{0}\in[\underline{\pi},\overline{\pi})$ then the
expert receives a positive reputation payoff if and only if the advisee takes
the correct action.

Figure 3: Graphs of $\Phi(C,\sigma,\pi_{0})$ and $\Phi(S,\sigma,\pi_{0})$ with
$\psi$ defined as in (7) and with $\sigma\in\\{0,1\\}$, $w\in\\{0.5,1\\}$, and
$\pi^{*}=0.5$

Figure 3 plots the expected payoffs (8) and (11) against $\pi_{0}$ when
$\sigma\in\\{0.1,0.2\\}$, $w\in\\{0.5,1\\}$, and $\pi^{*}=0.5$. It
demonstrates three cases:

  1. 1.

$\Phi(C,\sigma,\pi_{0})$ exceeds $\Phi(S,\sigma,\pi_{0})$ for all $\pi_{0}$
(as when, e.g., $(\sigma,w)=(0.2,0.5)$).

  2. 2.

$\Phi(C,\sigma,\pi_{0})$ and $\Phi(S,\sigma,\pi_{0})$ “cross” at the
replacement threshold $\pi_{0}=\pi^{*}$ and the discontinuity
$\pi_{0}=\overline{\pi}$ (as when, e.g., $(\sigma,w)=(0.1,0.5)$);

  3. 3.

$\Phi(C,\sigma,\pi_{0})$ and $\Phi(S,\sigma,\pi_{0})$ intersect at some
$\pi_{0}\in[\underline{\pi},\overline{\pi})$ (as when, e.g.,
$(\sigma,w)=(0.2,1)$).

Proposition 5 states that these are the _only_ three possible cases.444
Corollary 2 does not apply in this setting because $\psi$ is not concave.

######  Proposition 5.

Suppose the reputation payoff function $\psi$ is defined as in (7). Define

$\pi^{\dagger}\equiv\frac{w}{4\sigma(1+w)}.$ (12)

If $\pi^{\dagger}\leq\pi^{*}$ then the expert chooses the complex rule. If
$\pi^{\dagger}>\pi^{*}$ then he chooses the _simple_ rule if and only if
$\pi^{*}\leq\pi_{0}<\min\\{\pi^{\dagger},\overline{\pi}\\}$, where
$\overline{\pi}$ is defined as in Lemma 2.

If the wage $w$ is sufficiently low or the threshold $\pi^{*}$ is sufficiently
high, then the expert receives a greater expected payoff from choosing the
complex rule regardless of the prior $\pi_{0}$. If $w$ is high and the prior
lies between $\pi^{*}$ and $\min\\{\pi^{\dagger},\overline{\pi}\\}$, then the
reputational risk induced by choosing the complex rule dissuades the expert
from making that choice.

If $\pi_{0}$ is the proportion of employable experts who are competent and the
advisee replaces the expert with a random competitor, then she maximizes the
probability of employing a competent expert by setting $\pi^{*}=\pi_{0}$. In
that case, if she wants the expert to choose the complex model then she must
limit his reputational risk by offering a low wage:

######  Corollary 3.

Suppose the reputation payoff function $\psi$ is defined as in (7). If
$\pi^{*}=\pi_{0}$ then the expert chooses the complex rule if and only if
$w\leq 4\sigma\pi_{0}/(1-4\sigma\pi_{0})$.

Alternatively, the wage may be fixed (due to, e.g., legal or institutional
constraints). Then the advisee’s only way to solicit the complex rule is to
vary the replacement threshold $\pi^{*}$. She has two choices: set it low
enough to guarantee that the expert receives the wage, or set it _high_ enough
to guarantee that expert _does not_ receive the wage. Both choices remove the
expert’s reputational risk. The first choice removes the risk by fully
insuring it. The second choice removes the risk by making explicit that the
advisee seeks the expert’s advice as a “one-off,” effectively imposing a
single-period term limit on the expert’s employment.

##  5 Extensions

This section considers several standalone extensions to my analysis in
Sections 2 and 3.

###  5.1 $\Pr(A=1)\not=0.5$

Sections 2 and 3 assume $\Pr(A=1)=0.5$. This equates the simple and complex
rules’ expected accuracy payoffs when the expert is incompetent. Consequently,
it isolates the tension between the complex rule’s superior predictive power
and its ability to reveal the expert’s competence. If $\Pr(A=1)=a\in(0,1)$
then

$\Pr(A=X_{\theta})=ax+(1-a)(1-x)+2\theta\sigma$

and the condition (5) under which the expert chooses the complex rule becomes

$2\sigma\pi_{0}-(2a-1)(1-x)\geq\psi(\pi_{0})-\Psi(\sigma,\pi_{0}).$ (13)

The left-hand side of (5) exceeds the left-hand side of (13) if and only if
$a>0.5$. In that case, conditioning on $X_{0}$ lowers the expert’s accuracy
payoff because it lowers the probability that the advisee takes the action
when it is good. Conditioning on $X_{1}$ raises the expert’s accuracy payoff
if and only if the covariance $\sigma$ of $A$ and $X_{1}$ is sufficiently
high. This covariance depends on $a$ since it depends on the variance
$\operatorname{Var}(A)=a(1-a)$. Therefore, increasing $a$ changes both terms
on the left-hand side of (13).555 If $A$ and $X_{1}$ have correlation $\rho$
then $\sigma=\rho\sqrt{a(1-a)x(1-x)}$. Then the left-hand side of (13) is
increasing in $a$ if and only if
$\rho\pi_{0}(1-2a)>2(1-x)\sqrt{\frac{a(1-a)}{x(1-x)}}.$ If $a\geq 0.5$ then
this condition never holds. If $a<0.5$ then it holds if and only if
$\rho\pi_{0}$ is sufficiently large.  Increasing $a$ also changes the right-
hand side by changing the distribution of posterior beliefs $\pi_{1}(Y)$ and,
in turn, the expected reputation payoff $\Psi(\sigma,\pi_{0})$. The overall
effect of increasing $a$ on $\Delta\Phi(\sigma,\pi_{0})$ is ambiguous; it
depends on the parameters $x$, $\sigma$ and $\pi_{0}$, and the shape of
$\psi$. I do not elaborate on this dependency because it is not central to my
analysis.

###  5.2 Measurement errors

One reason real-world experts may prefer the simple rule is that the advisee
may make mistakes when implementing the complex rule. For example, suppose the
expert provides condition $X_{\theta}$ but the advisee observes
$\hat{X}_{\theta}\in\\{0,1\\}$ with
$\Pr(\hat{X}_{\theta}\not=X_{\theta})=\epsilon\in(0,0.5]$. Proposition 6
states that such measurement errors decrease the complex rule’s accuracy.666
If $\Pr(A=1)\not=0.5$ then whether measurement errors decrease the complex
rule’s accuracy depends on the joint distribution of $A$ and $X_{\theta}$, and
on the relative rates of false positives and negatives.

######  Proposition 6.

Let $\epsilon\in[0,0.5]$. Suppose $\hat{X}_{\theta}\in\\{0,1\\}$ is
distributed such that $\Pr(\hat{X}_{\theta}\not=X_{\theta})=\epsilon$ for each
$\theta\in\\{0,1\\}$. Then

$\Pr(A=\hat{X}_{\theta})=0.5+2\theta(1-2\epsilon)\sigma.$ (14)

Measurement errors decrease the covariance between the observed condition and
the correct action. By Proposition 2, this decrease makes the expert less
willing to choose the complex rule if the reputation payoff function $\psi$ is
linear or convex. However, if $\psi$ is concave then measurement errors
increase the expected reputation payoff from choosing the complex rule because
they make that rule less diagnostic of the expert’s competence. Thus,
introducing measurement errors may lead the expert to switch to the complex
rule if the reputation benefit exceeds the accuracy cost.

In practice, measurement errors may arise exogenously from, for example,
imprecise medical tests. They may also arise endogenously from the advisee
optimally allocating their scarce attention and mental resources.777 This
possibility relates to Oprea, ’s (2020) claim that complex rules are
cognitively costly to implement.  For example, suppose the advisee receives
benefit $b>0$ from taking the correct action and zero from taking the
incorrect action. Suppose further that she endures cost $c(0.5-\epsilon)^{2}$
from observing $\hat{X}_{\theta}$ with error $\epsilon\in[0,0.5]$, where
$c\in(0,2b\sigma\pi_{0})$.888 This upper bound on $c$ ensures
$\epsilon^{*}<0.5$.  If the expert chooses the complex rule then the advisee
chooses the error $\epsilon^{*}$ that maximizes

$b\left(0.5+2\sigma(1-2\epsilon)\pi_{0}\right)-c(0.5-\epsilon)^{2},$

which has unique maximizer

$\epsilon^{*}\equiv 0.5-\frac{2b\sigma\pi_{0}}{c}.$

Substituting $\epsilon=\epsilon^{*}$ into (14) then gives

$\Pr(A=\hat{X}_{\theta})=0.5+\frac{4b\theta\sigma^{2}\pi_{0}}{c}.$

Thus, in this example, the accuracy of the complex rule is increasing in the
advisee’s benefit-cost ratio $b/c$. Now suppose the expert knows $b$ and $c$,
and anticipates the advisee’s choice of error $\epsilon^{*}$ when choosing
between rules. Then it is possible the advisee is willing to “pay” for low
errors because her benefit-cost ratio is high, but this payment leads the
expert to avoid the complex rule because it becomes too diagnostic of his
competence. If the advisee wants to learn the expert’s competence (e.g.,
because she will face a higher stakes choice in the future) then she may
prefer to under-invest in measuring the condition correctly. This would
encourage the expert to choose the complex rule and reveal information about
his competence.

###  5.3 Expert knows $\theta$

Sections 2 and 3 assume that the expert does not know whether he is competent.
This assumption allows me to isolate his trade-off between providing accurate
advice and managing reputational risk. If the expert knows his competence type
then he also has a signaling motive. I analyze that motive below.

Suppose the expert knows his competence type $\theta$ but the advisee does
not. Then his choice of rule may reveal information about $\theta$. Lemma 3
describes how the advisee incorporates that information into her posterior
belief about $\theta$.

######  Lemma 3.

Suppose the expert knows $\theta$ but the advisee does not. Let
$\hat{q}_{\theta}\equiv\Pr(r=C\,|\,\theta)$ be the probability with which the
advisee believes an expert with type $\theta$ chooses the complex rule. Then
her posterior belief about $\theta$ after observing the chosen rule
$r\in\\{S,C\\}$ and its outcome $Y\in\\{0,1\\}$ is given by

$\Pr(\theta=1\,|\,r,Y)=\begin{cases}\pi_{S}&\text{if}\ r=S\\\
\frac{(1+4\sigma)\pi_{C}}{1+4\sigma\pi_{C}}&\text{if}\ r=C\ \text{and}\ Y=1\\\
\frac{(1-4\sigma)\pi_{C}}{1-4\sigma\pi_{C}}&\text{if}\ r=C\ \text{and}\
Y=0,\end{cases}$ (15)

where

$\pi_{r}=\begin{cases}\frac{(1-\hat{q}_{1})\pi_{0}}{(1-\hat{q}_{1})\pi_{0}+(1-\hat{q}_{0})(1-\pi_{0})}&\text{if}\
r=S\\\
\frac{\hat{q}_{1}\pi_{0}}{\hat{q}_{1}\pi_{0}+\hat{q}_{0}(1-\pi_{0})}&\text{if}\
r=C\end{cases}$ (16)

is her “intermediate” belief about $\theta$ after observing $r$ but before
observing $Y$.

Equations (3) and (15) coincide when $p_{0}=p_{1}$; that is, when the advisee
believes that competent and incompetent experts choose between rules with the
same probabilities. Then the choice of rule is uninformative and so
$\pi_{r}=\pi_{0}$ for each $r\in\\{S,C\\}$.

I now study the perfect Bayesian equilibrium (henceforth “equilibrium”) of the
game in which the expert chooses a rule and the advisee forms a posterior
belief about $\theta$. In this game, the expert’s strategy comprises (i) a
probability $p_{\theta}$ with which he chooses the complex rule given his type
$\theta$ and (ii) a conjecture $\hat{p}_{1-\theta}$ about the corresponding
probability for the other type. The advisee’s strategy comprises a conjecture
$\hat{q}_{\theta}$ about the probability for each type. She uses this
conjecture to form her posterior belief (15). In equilibrium, all conjectures
are correct: $p_{\theta}=\hat{p}_{\theta}=\hat{q}_{\theta}$. As a shorthand, I
describe an equilibrium by the pair $(p_{0},p_{1})$ of probabilities that
define it, and refer to $p_{\theta}$ as the $\theta$-type expert’s “strategy.”
This strategy is “pure” if $p_{\theta}\in\\{0,1\\}$ and “mixed” if
$p_{\theta}\in(0,1)$. It is his best-response to the strategy played by the
other type, given the advisee’s subsequent inference about $\theta$ and its
reputational consequences.

Lemma 4 defines the expected payoff that experts of each type maximize to
determine their best-response.

######  Lemma 4.

Suppose the expert knows $\theta$ but the advisee does not. Let $p_{\theta}$
be the equilibrium probability with which an expert with type $\theta$ chooses
the complex rule. Then such an expert has expected payoff

$\Omega_{\theta}(\sigma,\pi_{0},p_{0},p_{1})\equiv
p_{\theta}\Phi_{\theta}(C,\sigma,\pi_{0},p_{0},p_{1})+(1-p_{\theta})\Phi_{\theta}(S,\sigma,\pi_{0},p_{0},p_{1}),$
(17)

where

$\Phi_{\theta}(C,\sigma,\pi_{0},p_{0},p_{1})\equiv
0.5+2\sigma\theta+(0.5+2\sigma\theta)\psi\left(\frac{(1+4\sigma)\pi_{C}}{1+4\sigma\pi_{C}}\right)+(0.5-2\sigma\theta)\psi\left(\frac{(1-4\sigma)\pi_{C}}{1-4\sigma\pi_{C}}\right)$
(18)

is his expected payoff from choosing the complex rule with certainty,

$\Phi_{\theta}(S,\sigma,\pi_{0},p_{0},p_{1})\equiv
0.5+\psi\left(\pi_{S}\right)$ (19)

is his expected payoff from choosing the simple rule with certainty, and
$\pi_{C}$ and $\pi_{S}$ are defined as in Lemma 3.

I first show that there are no pure separating equilibria. This is because
incompetent experts never play strategies that allow the advisee to diagnose
their incompetence with certainty.

######  Proposition 7.

Suppose the expert knows $\theta$ but the advisee does not. If a pure strategy
equilibrium exists, then it is pooling.

The game may have two, one, or no pure pooling equilibria. It may also have
mixed strategy equilibria. The number and nature of equilibria depend on the
parameters $\sigma$ and $\pi_{0}$, and the shape of the reputation payoff
function $\psi$.

To gain traction, I focus on the “replaceable expert” setting discussed in
Section 4. I assume the replacement threshold equals $\pi^{*}=\pi_{0}$. Lemma
5 defines the expert’s expected payoffs in this setting.

######  Lemma 5.

Suppose the expert knows $\theta$ but the advisee does not. Let the
probabilities $p_{\theta}$ and functions
$\Phi_{\theta}(r,\sigma,\pi_{0},p_{0},p_{1})$ be defined as in Lemma 4, and
let the reputation payoff function $\psi$ be defined as in (7) with
$\pi^{*}=\pi_{0}$. Then

$\Phi_{\theta}(C,\sigma,\pi_{0},p_{0},p_{1})=\begin{cases}0.5+2\sigma\theta&\text{if}\
p_{1}(1+4\sigma)<p_{0}\\\ 0.5+2\sigma\theta+(0.5+2\sigma\theta)w&\text{if}\
p_{1}(1-4\sigma)<p_{0}\leq p_{1}(1+4\sigma)\\\ 0.5+2\sigma\theta+w&\text{if}\
p_{0}\leq p_{1}(1-4\sigma)\end{cases}$ (20)

and

$\Phi_{\theta}(S,\sigma,\pi_{0},p_{0},p_{1})=\begin{cases}0.5+w&\text{if}\
p_{0}\geq p_{1}\\\ 0.5&\text{otherwise}.\end{cases}$ (21)

Proposition 8 determines competent and incompetent experts’ best responses by
maximizing their expected payoff $\Omega_{\theta}(\sigma,\pi_{0},p_{0},p_{1})$
under the conditions specified in Lemma 5.

######  Proposition 8.

Under the conditions specified in Lemma 5, we have

$\operatorname*{arg\,max}_{p}\Omega_{0}(\sigma,\pi_{0},p,p_{1})=\\{p_{1}\\}$
(22)

and

$\displaystyle\operatorname*{arg\,max}_{p}\Omega_{1}(\sigma,\pi_{0},p_{0},p)$
$\displaystyle=\begin{cases}\\{0\\}&\text{if}\ 1-p_{0}<4\sigma<1\ \text{and}\
\max\left\\{2\sigma,4\sigma/(1-4\sigma)\right\\}<w\\\
\\{0\\}\cup\left[\frac{p_{0}}{1+4\sigma},p_{0}\right]\cup\\{1\\}&\text{if}\
1-p_{0}<4\sigma<1\ \text{and}\ 2\sigma<w=4\sigma/(1-4\sigma)\\\
\\{1\\}&\text{otherwise}.\end{cases}$ (23)

Incompetent experts best-respond by mimicking competent experts. Thus, in any
equilibrium, the advisee learns nothing from the choice of rule because
competent and incompetent experts play the same strategies. Yet she may learn
from the _outcome_ of the complex rule—i.e., whether it leads to the correct
action—if the expert chooses it. Competent experts have a dominant strategy to
choose the complex rule if (i) $w\leq 2\sigma$ or (ii) $4\sigma<1$ and
$w<4\sigma/(1-4\sigma)$—i.e., their reputation concerns are sufficiently weak.
Otherwise, they want to differentiate themselves from incompetent experts.

Figure 4: Experts’ best-response curves when $\sigma\in\\{0.1,0.2\\}$ and
$w\in\\{0.5,1,5\\}$

Figure 4 shows the best response curves for experts of each type
$\theta\in\\{0,1\\}$ when $\sigma\in\\{0.1,0.2\\}$ and $w\in\\{0.5,1,5\\}$.
These curves do not depend on the prior $\pi_{0}$. Their intersections
correspond to equilibria. If the wage $w$ is small and the covariance $\sigma$
is large, then the unique equilibrium involves pure pooling on the complex
rule. Increasing $w$ or decreasing $\sigma$ destroys this equilibrium. Indeed,
Corollary 4 states that if the wage $w$ is large relative to the covariance
$\sigma$ then the game has no equilibria:

######  Corollary 4.

Under the conditions specified in Lemma 5,

  1. (i)

If $4\sigma<1$ and $\max\left\\{2\sigma,4\sigma/(1-4\sigma)\right\\}<w$, then
there are no equilibria;

  2. (ii)

If $4\sigma<1$ and $2\sigma<w=4\sigma/(1-4\sigma)$, then the pair
$(p_{0},p_{1})$ defines an equilibrium if and only if
$p_{0}=p_{1}\in(1-4\sigma,1]$.

Otherwise, the pair $(p_{0},p_{1})=(1,1)$ defines the unique equilibrium.

Consider the knife-edge case in which the competent expert’s condition is
imperfectly predictive (i.e., $4\sigma<1$) and
$2\sigma<w=4\sigma/(1-4\sigma)$. If he can mimic an incompetent expert who
chooses the complex rule with sufficiently high probability, then he is
indifferent between doing so and choosing the complex rule for sure.
Otherwise, and outside this knife-edge case, the game either has no
equilibrium or a unique equilibrium with pure pooling on the complex rule.
This pooling equilibrium exists only when the expert’s reputational concerns
(indexed by $w$) are small relative to his accuracy concerns (indexed by
$\sigma$). Thus, letting the expert know his type preserves the main insight
from Sections 2–4: experts give complex advice only if they face low downside
reputational risk.999 This insight echoes an insight from the innovation
literature that failure tolerance tends to improve performance (see, e.g.,
Azoulay et al., , 2011; Manso, , 2011; Tian and Wang, , 2014).

##  6 Related literature

My analysis is closely related to the literature on reputational cheap talk.
That literature explores how the incentive to appear informed distorts the
equilibria of cheap talk games in which a sender shares his signal of an _ex
post_ observable state. Ottaviani and Sørensen, 2006a ; Ottaviani and
Sørensen, 2006b show that, in equilibrium, the sender does not report his
signal truthfully if it depends on whether he is informed. Instead, he
manipulates his report to induce the most favorable belief about his
informedness. For example, he may bias his report toward the receiver’s prior
belief about the state to hedge against the report disagreeing with the
state.101010 Klein and Mylovanov,  (2017) argue that this behavior is limited
to models with short time horizons. If the expert has a long horizon, then he
may forgo “impressing” the receiver in the short run if he becomes privately
pessimistic about his ability to impress in the long run.  This hedging
behavior corresponds to choosing the simple rule in my model. Indeed, one
could interpret my model as an example of reputational cheap talk in which the
expert reports information about the predictive power of the condition he
identifies. However, the expert in my model has reputational concerns
associated with the correctness of the action taken as a consequence of his
report, rather than the correctness of the report itself.111111 This
distinction between actions and information is the focus of Andrews and
Shapiro,  (2021), who study statistical decision theoretic models of science
communication.  The innovation of my paper is to show how reputational
concerns influence the type of advice experts provide: do they make “blanket
recommendations,” or do they provide conditions under which to take different
actions?121212 Levit and Tsoy,  (2020) argue that experts make blanket
recommendations to conceal their bias.

Backus and Little,  (2020) offer another model similar to mine: one of an
unbiased expert who provides advice about an action but has reputational
concerns. In their model, the correct action may be impossible to identify and
only competent experts know if identification is possible. This makes
competent experts averse to admitting ignorance because doing so makes them
look incompetent. Whereas in my model, the correct action _is_ possible to
identify but only competent experts can identify it. This makes competent
experts _pretend to be ignorant_ (by choosing the simple rule) if the
reputational cost of appearing incompetent is sufficiently large. Backus and
Little,  and my conclusions differ because our definitions of “competence”
differ.

My definition of “expertise” is similar to that used by Callander et al.,
(2021). They note that experts tend to provide “referential” advice that
combines suggestions for what to do with examples of inferior alternatives.
The authors posit that expertise consists in knowing the relative merits of
different suggestions. Similarly, I posit that expertise consists in knowing
conditions under which different suggestions are preferable. Callander et al.,
and my analyses differ in what motivates the expert. They focus on the
expert’s desire to retain an informational advantage over their advisee. In
contrast, I allow the expert to forfeit this advantage because it makes his
advice more likely to induce the correct the action and may yield reputational
benefits.

Callander et al.,  (2021) build on the literature on disclosing verifiable
information. The strand of that literature closest to my analysis is due to
Dye,  (1985). It considers the possibility that the sender is uninformed. Then
the receiver cannot distinguish between the sender withholding information and
having none (see, e.g., Dziuda, , 2011). Dye,  studies a model in which the
receiver knows the sender has information but does not know its value. This is
similar to my model, in which the advisee knows the expert is aware of
predictive conditions but does not know their predictive power. Dye,  and my
models differ in that the expert I study has no notion of whether his
information is “good” or “bad” for the advisee. So it is not that competent
experts “can partially conceal bad news by pooling with those who are unable
to disclose” (Grubb, , 2011, p. 226). Instead, competent and incompetent
experts avoid disclosure because they want to avoid its reputational
consequences.

My assumption that the expert does not know his competence type is typical in
the literature on career concerns. That literature explores how agents’ desire
to appear competent drives a wedge between their risk preferences and
principals’ (Holmström and Ricart i Costa, , 1986; Holmström, , 1999).131313
Dewatripont et al.,  (1999) generalize Holmström, ’s (1999) model through the
lens of information design.  This wedge can lead to inefficient information
seeking (Levy, , 2004; Milbourn et al., , 2001; Scharfstein and Stein, , 1990)
and risk-taking (Aghion and Jackson, , 2016; Chevalier and Ellison, , 1997;
Prendergast and Stole, , 1996). For example, Aghion and Jackson,  (2016)
analyze replacement schemes in a dynamic setting similar to that described in
Section 4.141414 Tsuyuhaha,  (2012) also analyzes this setting in a two-period
model. He focuses on replacing biased experts, whereas I focus on replacing
incompetent experts.  They study a “leader” who takes a materially safe or
risky action, where the cost of the risky action being bad exceeds the benefit
of it being good. This concavity in material payoffs drives the leader toward
the safe action, which is less beneficial on average but reveals no
information about his type. In contrast, the expert in my model chooses
between _reputationally_ safe and risky advice, and I show how his choice
depends on the concavity in his reputational payoffs.151515 Rappoport,  (2022)
also distinguishes between material and reputational payoffs.

Many insights from the career concerns literature apply to my model. For
example, Gibbons and Murphy,  (1992) explain that compensation contracts
should be strongest for workers close to retirement because they have the
weakest career concerns. The advisee in my model may prefer an older expert
with lower expected competence $\pi_{0}$ to a younger expert with higher
expected competence. This is because the older expert has a lower continuation
value of employment, making him more willing to risk losing that value by
choosing the complex rule. I analyze the relationship between expected
competence, continuation values, and rule choices in Propositions 4 and 5.

More generally, the reputation payoff function $\psi$ acts as a reduced-form
expression for the expert’s compensation scheme. Its shape determines the
expert’s choice. For example, if $\psi$ is close to constant then the expert
prefers to choose the complex rule and risk being “wrong” (see Corollary 1).
This echoes another insight from the career concerns literature: weakening
incentives, such as by offering tenure (Aghion and Jackson, , 2016) or
severance pay (Bonatti and Hörner, , 2017), can lead to more efficient
behavior.

##  7 Conclusion

This paper discusses how reputational concerns impact the type of advice that
experts provide. I show that experts may “simply” their advice by excluding
relevant conditions if their payoff is sufficiently concave in posterior
beliefs about their competence. This simplification makes advice worse by
pooling states in which actions should and should not be taken. Advisees can
prevent experts from giving simple advice by limiting their reputational risk:
by reducing the payoff from being perceived as competent, or by making payoffs
independence of perceived competence.

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##  Appendix A Proofs

###  A.1 Results in Section 3

###### Proof of Proposition 1.

The result follows from the definition of $\Delta\Phi(\sigma,\pi_{0})$. ∎

###### Proof of Corollary 1.

If $\psi$ is linear or convex then
$\Psi(\sigma,\pi_{0})=\operatorname{E}[\psi(\pi_{1}(Y))]\leq\psi(\operatorname{E}[\pi_{1}(Y)])=\psi(\pi_{0})$
by Jensen’s inequality. Then (5) holds since the left-hand side is positive
and the right-hand side is non-positive. ∎

###### Proof of Corollary 2.

The result follows from (5) and the definition of $\Psi(\sigma,\pi_{0})$. ∎

###### Proof of Lemma 1.

Now

$\frac{\partial\pi_{1}(1)}{\partial\sigma}=\frac{4\pi_{0}(1-\pi_{0})}{(1+4\sigma\pi_{0})^{2}}$

and

$\frac{\partial\pi_{1}(1)}{\partial\pi_{0}}=\frac{1+4\sigma}{(1+4\sigma\pi_{0})^{2}}$

are positive. In contrast,

$\frac{\partial\pi_{1}(0)}{\partial\sigma}=-\frac{4\pi_{0}(1-\pi_{0})}{(1-4\sigma\pi_{0})^{2}}$

is negative, while

$\frac{\partial\pi_{1}(0)}{\partial\pi_{0}}=\frac{1-4\sigma}{(1-4\sigma\pi_{0})^{2}}$

equals zero if $\sigma=0.25$ and is positive otherwise. ∎

###### Proof of Proposition 2.

Now $2\sigma\pi_{0}$ is increasing in $\sigma$, while $\psi(\pi_{0})$ is
constant in $\sigma$. Moreover, by (4) and Lemma 1, increasing $\sigma$
induces a mean-preserving spread in the distribution of $\pi_{1}(Y)$. This
spread weakly increases $\Psi(\sigma,\pi_{0})$ when $\psi$ is linear or convex
(Rothschild and Stiglitz, , 1970, Theorem 2). ∎

###### Proof of Proposition 3.

We have

$\displaystyle\frac{\partial\Psi(\sigma,\pi_{0})}{\partial\pi_{0}}$
$\displaystyle=\frac{\partial}{\partial\pi_{0}}\left[(0.5+2\sigma\pi_{0})\psi(\pi_{1}(1))+(0.5-2\sigma\pi_{0})\psi(\pi_{1}(0))\right]$
$\displaystyle=2\sigma\psi(\pi_{1}(1))+(0.5+2\sigma\pi_{0})\psi^{\prime}(\pi_{1}(1))\frac{\partial\pi_{1}(1)}{\partial\pi_{0}}$
$\displaystyle\quad-2\sigma\psi(\pi_{1}(0))+(0.5-2\sigma\pi_{0})\psi^{\prime}(\pi_{1}(0))\frac{\partial\pi_{1}(0)}{\partial\pi_{0}}$
$\displaystyle=2\sigma\left[\psi(\pi_{1}(1))-\psi(\pi_{1}(0))\right]+\operatorname{E}\left[\psi^{\prime}(\pi_{1}(Y))\frac{\partial\pi_{1}(Y)}{\partial\pi_{0}}\right]$

and therefore

$\displaystyle\frac{\partial\Delta\Phi(\sigma,\pi_{0})}{\partial\pi_{0}}$
$\displaystyle=\frac{\partial}{\partial\pi_{0}}\left[2\sigma\pi_{0}+\Psi(\sigma,\pi_{0})-\psi(\pi_{0})\right]$
$\displaystyle=2\sigma+2\sigma\left[\psi(\pi_{1}(1))-\psi(\pi_{1}(0))\right]+\operatorname{E}\left[\psi^{\prime}(\pi_{1}(Y))\frac{\partial\pi_{1}(Y)}{\partial\pi_{0}}\right]-\psi^{\prime}(\pi_{0}).$
(24)

The result follows. ∎

###### Proof of Proposition 4.

Substituting $\psi(\pi)=w\,R(\pi)$ into (24) gives

$\frac{\partial\Delta\Phi(\sigma,\pi_{0})}{\partial\pi_{0}}=2\sigma+w\left(2\sigma\left[R(\pi_{1}(1))-R(\pi_{1}(0))\right]+\operatorname{E}\left[R^{\prime}(\pi_{1}(Y))\frac{\partial\pi_{1}(Y)}{\partial\pi_{0}}\right]-R^{\prime}(\pi_{0})\right).$
(25)

If (6) does not hold then the bracketed term in (25) must be negative because
$\sigma>0$. In that case, defining

$w^{*}\equiv\frac{2\sigma}{R^{\prime}(\pi_{0})-\operatorname{E}\left[R^{\prime}(\pi_{1}(Y))\frac{\partial\pi_{1}(Y)}{\partial\pi_{0}}\right]-2\sigma\left[R(\pi_{1}(1))-R(\pi_{1}(0))\right]}$

yields the result. ∎

###  A.2 Results in Section 4

###### Proof of Lemma 2.

Suppose the expert chooses the complex rule. If

$\pi^{*}>\pi_{1}(1)=\frac{(1+4\sigma)\pi_{0}}{1+4\sigma\pi_{0}}$ (26)

then he always receives a reputation payoff of zero. Substituting
$\pi_{0}=\underline{\pi}$ into (26) and rearranging for $\underline{\pi}$
yields (9). Conversely, if

$\pi^{*}\leq\pi_{1}(0)=\frac{(1-4\sigma)\pi_{0}}{1-4\sigma\pi_{0}}$ (27)

then he always receives a reputation payoff of $w$. Substituting
$\pi_{0}=\overline{\pi}$ into (27) and rearranging for $\overline{\pi}$ yields
(10). Finally, if $\pi_{1}(0)\leq\pi^{*}<\pi_{1}(1)$ then the expert receives
a reputation payoff of $w$ if and only if $Y=1$, which occurs with probability
$(0.5+2\sigma\pi_{0})$. Equation (8) follows from the definition of
$\Phi(C,\sigma,\pi_{0})$.

Now suppose the expert chooses the simple rule. Then he receives a reputation
payoff of $w$ if and only if $\pi_{0}\geq\pi^{*}$. Equation (11) follows from
the definition of $\Phi(S,\sigma,\pi_{0})$. ∎

###### Proof of Proposition 5.

Fix $\sigma$ and consider the thresholds $\underline{\pi}$ and
$\overline{\pi}$ defined in Lemma 2. Now $\underline{\pi}<\pi^{*}$ and
$\overline{\pi}>\pi^{*}$, so if $\pi_{0}<\pi^{*}$ then

$\displaystyle\Phi(C,\sigma,\pi_{0})-\Phi(S,\sigma,\pi_{0})$
$\displaystyle>\Phi(C,\sigma,\pi^{*})-\Phi(S,\sigma,\pi_{0})$
$\displaystyle=(0.5+2\sigma\pi^{*})(1+w)-0.5$ $\displaystyle>0$

and if $\pi_{0}\geq\overline{\pi}$ then
$\Phi(C,\sigma,\pi)-\Phi(S,\sigma,\pi)=2\sigma\pi_{0}>0$. Therefore, if the
expert chooses the simple rule then $\pi^{*}\leq\pi_{0}<\overline{\pi}$. For
such values of $\pi_{0}$ we have
$\Phi(C,\sigma,\pi_{0})=(0.5+2\sigma\pi_{0})(1+w)$ and
$\Phi(S,\sigma,\pi_{0})=0.5+w$. These functions of $\pi_{0}$ coincide when
$\pi_{0}=\pi^{\dagger}$ with $\pi^{\dagger}$ defined as in (12). If
$\pi^{\dagger}\leq\pi^{*}$ then
$\Phi(C,\sigma,\pi_{0})\geq\Phi(S,\sigma,\pi_{0})$ for all $\pi_{0}\in(0,1)$.
On the other hand, if $\pi^{\dagger}>\pi^{*}$ then
$\Phi(C,\sigma,\pi_{0})<\Phi(S,\sigma,\pi_{0})$ if and only if
$\pi^{*}\leq\pi_{0}<\min\\{\pi^{\dagger},\overline{\pi}\\}$. The result
follows. ∎

###### Proof of Corollary 3.

Let $\pi^{\dagger}$ be defined as in (12). If $\pi^{*}=\pi_{0}$ then
$\overline{\pi}>\pi_{0}$ and so the expert chooses the complex rule if and
only if $\pi^{\dagger}\leq\pi^{*}=\pi_{0}$. This occurs if and only if $w\leq
4\sigma\pi_{0}/(1-4\sigma\pi_{0})$. ∎

###  A.3 Results in Section 5

###### Proof of Proposition 6.

By the Law of Total Probability, we have

$\displaystyle\Pr(A=\hat{X}_{\theta})$
$\displaystyle=\Pr(A=1,\hat{X}_{\theta}=1)+\Pr(A=0,\hat{X}_{\theta}=0)$
$\displaystyle=\Pr(A=1,\hat{X}_{\theta}=1\,|\,X_{\theta}=1)\Pr(X_{\theta}=1)+\Pr(A=1,\hat{X}_{\theta}=1\,|\,X_{\theta}=0)\Pr(X_{\theta}=0)$
$\displaystyle\quad+\Pr(A=0,\hat{X}_{\theta}=0\,|\,X_{\theta}=1)\Pr(X_{\theta}=1)+\Pr(A=0,\hat{X}_{\theta}=0\,|\,X_{\theta}=0)\Pr(X_{\theta}=0).$

Now $A$ and $\hat{X}_{\theta}$ are conditionally independent given
$X_{\theta}$, so

$\Pr(A,\hat{X}_{\theta}\,|\,X_{\theta})=\Pr(A\,|\,X_{\theta})\Pr(\hat{X}_{\theta}\,|\,X_{\theta})$

and therefore

$\Pr(A,\hat{X}_{\theta}\,|\,X_{\theta})\Pr(X_{\theta})=\Pr(A,X_{\theta})\Pr(\hat{X}_{\theta}\,|\,X_{\theta})$

by the definition of $\Pr(A\,|\,X_{\theta})$. Thus

$\displaystyle\Pr(A=\hat{X}_{\theta})$
$\displaystyle=\Pr(A=1,X_{\theta}=1)\Pr(\hat{X}_{\theta}=1\,|\,X_{\theta}=1)+\Pr(A=1,X_{\theta}=0)\Pr(\hat{X}_{\theta}=1\,|\,X_{\theta}=0)$
$\displaystyle\quad+\Pr(A=0,X_{\theta}=1)\Pr(\hat{X}_{\theta}=0\,|\,X_{\theta}=1)+\Pr(A=0,X_{\theta}=0)\Pr(\hat{X}_{\theta}=0\,|\,X_{\theta}=0).$

Equation (14) follows from substituting in $\Pr(A,X_{\theta})$,
$\Pr(\hat{X}_{\theta}\,|\,X_{\theta})$, and $a=0.5$. ∎

###### Proof of Lemma 3.

The advisee first observes the chosen rule $r$ and forms an intermediate
belief

$\displaystyle\pi_{r}$ $\displaystyle\equiv\Pr(\theta=1\,|\,r)$
$\displaystyle=\frac{\Pr(r\,|\,\theta=1)\Pr(\theta=1)}{\Pr(r)}$
$\displaystyle=\begin{cases}\frac{(1-p_{1})\pi_{0}}{(1-p_{1})\pi_{0}+(1-p_{0})(1-\pi_{0})}&\text{if}\
r=S\\\ \frac{p_{1}\pi_{0}}{p_{1}\pi_{0}+p_{0}(1-\pi_{0})}&\text{if}\
r=C.\end{cases}$

She then observes the outcome $Y$ and uses her updated belief $\pi_{r}$,
rather than her prior $\pi_{0}$, to form her posterior as in (3). ∎

###### Proof of Lemma 4.

Equations (18) and (19) follow from Lemma 3 and the definition of the expected
payoff $\Phi_{\theta}(r,\sigma,\pi_{0},p_{0},p_{1})$. Equation (17) then
follows from the Law of Total Expectation. ∎

###### Proof of Proposition 7.

If $(p_{0},p_{1})=(0,1)$ then $\pi_{C}=1$ and $\pi_{S}=0$, and so

$\displaystyle\Phi_{0}(C,\sigma,\pi_{0},p_{0},p_{1})$
$\displaystyle=0.5+\psi(1)$ $\displaystyle>0.5+\psi(0)$
$\displaystyle=\Phi_{0}(S,\sigma,\pi_{0},p_{0},p_{1})$

since $\psi(0)<\psi(1)$ by assumption. Thus, an incompetent expert can
profitably deviate by choosing the complex rule with non-zero probability.
Similarly, if $(p_{0},p_{1})=(1,0)$ then an incompetent expert can profitably
deviate by choosing the simple rule with non-zero probability. ∎

###### Proof of Lemma 5.

Replacing $\pi_{0}$ with $\pi_{r}$ in the proof of Lemma 2 gives

$\Phi_{\theta}(C,\sigma,\pi_{0},p_{0},p_{1})=\begin{cases}0.5+2\sigma\theta&\text{if}\
\pi_{C}<\underline{\pi}\\\ 0.5+2\sigma\theta+(0.5+2\sigma\theta)w&\text{if}\
\underline{\pi}\leq\pi_{C}<\overline{\pi}\\\ 0.5+2\sigma\theta+w&\text{if}\
\overline{\pi}\leq\pi_{C}\end{cases}$ (28)

and

$\Phi_{\theta}(S,\sigma,\pi_{0},p_{0},p_{1})=\begin{cases}0.5+w&\text{if}\
\pi_{S}\geq\pi^{*}\\\ 0.5&\text{otherwise}.\end{cases}$

If $\pi^{*}=\pi_{0}$ then, by the definitions of $\pi_{C}$, $\underline{\pi}$,
and $\overline{\pi}$, we have (i) $\pi_{C}<\underline{\pi}$ if and only if
$p_{1}(1+4\sigma)<p_{0}$ and (ii) $\pi_{C}<\overline{\pi}$ if and only if
$p_{1}(1-4\sigma)<p_{0}$. Substituting (i) and (ii) into (28) yields (20).
Finally, by the definition of $\pi_{S}$, we have $\pi_{S}\geq\pi_{0}$ if and
only if $p_{0}\geq p_{1}$. Equation (21) follows. ∎

###### Proof of Proposition 8.

Consider an incompetent expert. Suppose he chooses the complex rule with
probability $p$. Then, by (20) and (21), he receives expected payoff

$\displaystyle\Omega_{0}(\sigma,\pi_{0},p,p_{1})$
$\displaystyle=p\Phi_{0}(C,\sigma,\pi_{0},p,p_{1})+(1-p)\Omega_{0}(S,\sigma,\pi_{0},p,p_{1})$
$\displaystyle=\begin{cases}0.5+pw&\text{if}\ p\leq p_{1}(1-4\sigma)\\\
0.5+0.5pw&\text{if}\ p_{1}(1-4\sigma)<p<p_{1}\\\ 0.5+(1-0.5p)w&\text{if}\
p_{1}\leq p\leq p_{1}(1+4\sigma)\\\ 0.5+(1-p)w&\text{if}\
p_{1}(1+4\sigma)<p,\end{cases}$

which is uniquely maximized by $p=p_{1}$. Equation (22) follows.

Now consider a competent expert. Suppose he chooses the complex rule with
probability $p$. For convenience, let

$\displaystyle\Omega(p)$
$\displaystyle\equiv\Omega_{1}(\sigma,\pi_{0},p_{0},p)$
$\displaystyle=p\Phi_{1}(C,\sigma,\pi_{0},p_{0},p)+(1-p)\Phi_{1}(S,\sigma,\pi_{0},p_{0},p)$

denote his objective given the (exogenous to him) parameters $\sigma$,
$\pi_{0}$, $p_{0}$, and $w$. There are several cases to consider:

  1. 1.

Suppose $4\sigma=1$. Then

$\Phi_{1}(C,\sigma,\pi_{0},p_{0},p)=\begin{cases}1&\text{if}\
p<\frac{p_{0}}{2}\\\ 1+w&\text{if}\ \frac{p_{0}}{2}\leq p\end{cases}$

by (20) and hence

$\displaystyle\Omega(p)$
$\displaystyle=\begin{cases}0.5+(0.5-w)p+w&p<\frac{p_{0}}{2}\\\
0.5+0.5p+w&\frac{p_{0}}{2}\leq p\leq p_{0}\\\ 0.5+0.5p+wp&p_{0}<p\end{cases}$

by (21). But if $p<p_{0}$ then $\Omega(p)<\Omega(p_{0})$, and if $p_{0}<1$
then $\Omega(p_{0})<\Omega(1)$. It follows that
$\operatorname*{arg\,max}_{p}\Omega(p)=\\{1\\}$ in this case.

  2. 2.

Suppose $4\sigma<1$. Then $1-4\sigma>0$, and so

$\displaystyle\Omega(p)$
$\displaystyle=\begin{cases}0.5+w+p(2\sigma-w)&p<\frac{p_{0}}{1+4\sigma}\\\
0.5+w+p(2\sigma(1+w)-0.5w)&\frac{p_{0}}{1+4\sigma}\leq p\leq p_{0}\\\
0.5+p(2\sigma(1+w)+0.5w)&p_{0}<p<\frac{p_{0}}{1-4\sigma}\\\
0.5+p(2\sigma+w)&\frac{p_{0}}{1-4\sigma}\leq p\end{cases}$

by (20) and (21). Now $\Omega(p)$ is increasing in $p$ when $p>p_{0}$. Also,
it is increasing, constant, or decreasing in $p$ when $p<p_{0}/(1+4\sigma)$
and when $p_{0}/(1+4\sigma)\leq p\leq p_{0}$. It follows that

$\operatorname*{arg\,max}_{p}\Omega(p)\subseteq\\{0\\}\cup\left[\frac{p_{0}}{1+4\sigma},p_{0}\right]\cup\\{1\\},$

and it suffices to compare the values

$\displaystyle\Omega(0)$ $\displaystyle=0.5+w,$
$\displaystyle\Omega\left(\frac{p_{0}}{1+4\sigma}\right)$
$\displaystyle=0.5+w+\frac{p_{0}(2\sigma(1+w)-0.5w)}{1+4\sigma},$
$\displaystyle\Omega(p_{0})$ $\displaystyle=0.5+w+p_{0}(2\sigma(1+w)-0.5w),$
and $\displaystyle\Omega(1)$
$\displaystyle=\begin{cases}0.5+2\sigma+w&\text{if}\ p_{0}\leq 1-4\sigma\\\
0.5+2\sigma(1+w)+0.5w&\text{if}\ 1-4\sigma<p_{0}.\end{cases}$

Again, there are two cases:

    1. (a)

Suppose $p_{0}\leq 1-4\sigma$. Then

$\Omega(1)>\max\left\\{\Omega(0),\Omega\left(\frac{p_{0}}{1+4\sigma}\right),\Omega(p_{0})\right\\}$

since $2\sigma\in(0,0.5)$ and so $2\sigma>\max\\{0,2\sigma(1+w)-0.5w\\}$. Thus
$\operatorname*{arg\,max}_{p}\Omega(p)=\\{1\\}$ in this case.

    2. (b)

Suppose $p_{0}>1-4\sigma$. Then $\Omega(1)-\Omega(0)=2\sigma(1+w)-0.5w$
exceeds zero if and only if $2\sigma(1+w)>0.5w$. If this condition holds then

$\Omega(1)-\Omega(p)=\left(2\sigma(1+w)-0.5w\right)(1-p)$

also exceeds zero for $p\in[p_{0}/(1+4\sigma),p_{0}]$. The condition always
holds when $2\sigma\geq w$. Hence, there are three cases:

      1. i.

If $2\sigma\geq w$ or $2\sigma(1+w)>0.5w$ then
$\operatorname*{arg\,max}_{p}\Omega(p)=\\{1\\}$.

      2. ii.

If $2\sigma<w$ and $2\sigma(1+w)<0.5w$, then

$\Omega(0)>\max\left\\{\Omega\left(\frac{p_{0}}{1+4\sigma}\right),\Omega(p_{0}),\Omega(1)\right\\}$

and so $\operatorname*{arg\,max}_{p}\Omega(p)=\\{0\\}$.

      3. iii.

If $2\sigma<w$ and $2\sigma(1+w)=0.5$, then $\Omega(p)=0.5+w$ for all
$p\in\operatorname*{arg\,max}_{p}\Omega(p)=\\{0\\}\cup\left[\frac{p_{0}}{1+4\sigma},p_{0}\right]\cup\\{1\\}$.

Combining these cases and subcases yields

$\displaystyle\operatorname*{arg\,max}_{p}\Omega(p)$
$\displaystyle=\begin{cases}\\{0\\}&\text{if}\ 4\sigma<1\ \text{and}\
p_{0}>1-4\sigma\ \text{and}\ 2\sigma<w\ \text{and}\ 2\sigma(1+w)<0.5w\\\
\\{0\\}\cup[\frac{p_{0}}{1+4\sigma},p_{0}]\cup\\{1\\}&\text{if}\ 4\sigma<1\
\text{and}\ p_{0}>1-4\sigma\ \text{and}\ 2\sigma<w\ \text{and}\
2\sigma(1+w)=0.5w\\\ \\{1\\}&\text{otherwise}.\end{cases}$

Equation (23) follows from rewriting $2\sigma(1+w)\leq 0.5w$ as $w\leq
4\sigma/(1-4\sigma)$. ∎

###### Proof of Corollary 4.

Cases (i) and (ii) correspond to the first two pieces of (23). I consider
these two cases separately:

  1. (i)

Suppose $4\sigma<1$ and
$\max\left\\{2\sigma,\frac{4\sigma}{1-4\sigma}\right\\}<w$. Assume towards a
contradiction that $(p_{0},p_{1})$ defines an equilibrium. Then $p_{0}=p_{1}$
by Proposition 8. Moreover, by the same proposition, if $p_{0}\leq 1-4\sigma$
then $p_{1}=1>p_{0}$, whereas if $p_{0}>1-4\sigma$ then $p_{1}=0<p_{0}$. Thus
$p_{1}\not=p_{0}$; a contradiction. It follows that no equilibrium exists in
this case.

  2. (ii)

Suppose $4\sigma<1$ and $2\sigma<w=\frac{4\sigma}{1-\sigma}$. There are two
subcases to consider:

    1. (a)

Suppose $p_{0}\leq 1-4\sigma$. Assume towards a contradiction that
$(p_{0},p_{1})$ defines an equilibrium. Then $p_{0}=p_{1}$ and $p_{1}=1$ by
Proposition 8. Hence $p_{1}>p_{0}$; a contradiction.

    2. (b)

Suppose $p_{0}\in(1-4\sigma,1]$. Then, by Proposition 8, the pair
$(p_{0},p_{0})$ defines an equilibrium in which both competent and incompetent
experts play best responses. Moreover, no other equilibria exist because every
other pair involves at least one player not best-responding.

It follows that $\\{(p,p):1-4\sigma<p\leq 1\\}$ defines the set of equilibria
in this case.

Outside of these two cases, competent experts have a dominant strategy to play
$p_{1}=1$, to which incompetent experts best-respond by playing $p_{0}=1$.
Then $(p_{0},p_{1})=(1,1)$ defines the unique equilibrium. ∎