# Bayesian Elicitation

Mark Whitmeyer Department of Economics, University of Texas at Austin
Email<EMAIL_ADDRESS>Thanks to V. Bhaskar, Gleb Domnenko, William
Fuchs, Rosemary Hopcroft, Vasudha Jain, Meg Meyer, Vasiliki Skreta, Max
Stinchcombe, Yiman Sun, Tymon Tatur, Alex Teytelboym, Thomas Wiseman, and
Joseph Whitmeyer. Thanks also to various seminar and conference audiences.
This paper is a shorter version of the job market paper of the same name.

###### Abstract

How can a receiver design an information structure in order to elicit
information from a sender? We study how a decision-maker can acquire more
information from an agent by reducing her own ability to observe what the
agent transmits. Intuitively, when the two parties’ preferences are not
perfectly aligned, this garbling relaxes the sender’s concern that the
receiver will use her information to the sender’s disadvantage. We
characterize the optimal information structure for the receiver. The main
result is that under broad conditions, the receiver can do just as well as if
she could commit to a rule mapping the sender’s message to actions:
information design is just as good as full commitment. Similarly, we show that
these conditions guarantee that ex ante information acquisition always
benefits the receiver, even though this learning might actually lower the
receiver’s expected payoff in the absence of garbling. We illustrate these
effects in a range of economically relevant examples.

Keywords: Costly Signaling, Cheap Talk, Information Design, Bayesian
Persuasion
JEL Classifications: C72; D82; D83

###### Contents

  1. 1 Introduction
  2. 2 The Model
  3. 3 Opacity = Commitment
  4. 4 The Structure of Optimal Transparency and the Rewards to Ex Ante Learning
  5. 5 Examples
  6. 6 Discussion
  7. A Section 3 Proofs
  8. B Section 4 Proofs
  9. C Section 5 (Examples) Derivations

##  1 Introduction

> In that open field
> If you do not come too close, if you do not come too close,
> On a summer midnight, you can hear the music…
>
> T.S. Eliot
> East Coker

There is a decision-maker faced with choosing an action in an uncertain world.
She does not have direct access to information about the state of the world,
but there is a second person who does. The second person (or sender) observes
the state (equivalently, his type) before sending a message111We term the
sender’s action a message in order to distinguish it from the receiver’s
action. In some settings, like for instance cheap talk games, the moniker is
fitting. In others, like for instance the Spence (1978) [31] model of
signaling through education attainment, where the sender chooses his level of
education, labeling his action a message is less appropriate., which the
decision-maker (or receiver) observes before taking an action. This scenario
is called a communication game (or a signaling game), and the class of such
games includes cheap talk games, like that studied in Crawford and Sobel
(1982) [10]; and costly signaling games, like that studied in Spence (1978)
[31].

In nearly all of the literature, and for much of this paper, the message
chosen by the sender does not enter into the receiver’s utility function;
hence, the sender’s message has only instrumental value to the receiver.
Consequently, the receiver’s only concern is information transmission:
she222Throughout, to simplify the language, we arbitrarily impose that the
receiver is female and the sender is male. values only the information content
of the message, and so naturally her welfare increases as the sender’s
messages become more informative. However, there are a number of potential
frictions that could impede this information transmission. First, the sender’s
and the receiver’s preferences over the action taken may be imperfectly
aligned. Second, the messages may be costly to the sender, which costs affect
the messages that are chosen in equilibrium. As a result, less than full
information may be transmitted at equilibrium, and the frictions may even be
so severe that no information is transmitted.

Now, suppose that the receiver is not forced to observe the sender’s message
directly, but may commit ex ante to observe a noisy signal of the message
instead. That is, suppose the receiver may choose the degree of transparency,
or information structure, in the game. Can less than full transparency help
the receiver? Moreover, if less than full transparency is beneficial, then
what does the (receiver-)optimal degree of transparency look like, and how can
the receiver solve the problem of designing the optimal information structure?

If this were a decision problem in which the information were exogenous, i.e.
if there were no sender and instead the message sent to the receiver about the
state were from nature, then the answer to the first question would clearly be
no. Namely, if the message were exogenous, then the receiver would always (at
least weakly) prefer to observe the message itself rather than some noisy
signal. Conversely, here, the message is not exogenous, but is instead an
equilibrium choice of the sender. Crucially, the sender is aware of the degree
of transparency, which thus affects the message, or distribution over
messages, that he sends at equilibrium.

There is an important trade-off present in the receiver’s choice of
information structure. By choosing a more informative signal of the message,
the receiver obtains more information for any fixed strategy vector of the
sender types. On the other hand, the vector of strategies chosen by the sender
types, and hence the information, is an endogenous choice of the sender that
he makes cognizant of the information structure. Thus, less transparency may
beget a more informative vector of strategies at equilibrium. The optimal
degree of transparency arises as a consequence of these trade-offs, and it may
strictly benefit the receiver to choose a less informative signal of the
sender’s message. In short, less than full transparency can strictly benefit
the receiver.

The following example illustrates this observation. Sender and receiver are
employees of a firm: the receiver is part of upper management, the Chief
Operations Officer (COO), say; and the sender is a (local) branch manager. The
COO is contemplating whether to close the branch (action $C$) or not (action
$O$). The branch is either viable (state $\theta_{G}$) or not (state
$\theta_{B}$) and the COO would like to close the branch if the state is
$\theta_{B}$ and keep the branch open otherwise. Explicitly, the COO’s payoffs
are

$u_{R}(O,\theta_{G})=u_{R}(C,\theta_{B})=1,\quad\text{and}\quad
u_{R}(O,\theta_{B})=u_{R}(C,\theta_{G})=0$

The COO is unable to observe directly the viability of the branch. Instead,
the branch manager is “on the ground" and observes the state. The branch
manager must make an investment decision, whether to buy new equipment, say;
and all else equal, the branch manager would prefer to invest (message $I$) if
and only if the state is $\theta_{G}$. On the other hand the branch manager
would prefer the branch be kept open no matter the state; and, crucially, the
payoffs are such that the branch manager would rather choose the incorrect
investment decision for the state and have the branch be kept open than choose
the correct investment level and have the branch be closed. The state-
dependent utilities for the branch manager for each message, action
combination are

| $(I,O)$ | $(I,C)$ | $(N,C)$ | $(N,O)$
---|---|---|---|---
$\theta_{G}$ | $3$ | $1$ | $0$ | $2$
$\theta_{B}$ | $2$ | $0$ | $1$ | $3$

Both COO and branch manager share the common prior
$\mu_{0}\coloneqq\Pr\left(\Theta=\theta_{G}\right)=2/3$: the branch is more
likely to be viable than not.

The full transparency scenario is depicted in Figure 1. There, no equilibria
exist in which any information is transmitted. The only equilibria are
pooling–those in which the branch manager chooses the same message no matter
the state–and so the COO’s posterior is the same as her prior. The logic
behind this is simple: there can be no equilibrium in which both $I$ and $N$
are sent in such a way that the COO strictly prefers to take a different
action after each message. In such a circumstance, the local manager who knows
the state is $\theta_{B}$ always prefers to deviate to the message that is
followed by the branch being kept open ($O$). The COO’s payoff is $2/3$.

Figure 1: Whether to Close the Branch, Full Transparency

Suppose we introduce a neutral third party, a middle manager, say, who
oversees the branch manager. The COO no longer observes the message of the
branch manager; instead, the middle manager witnesses the message before
communicating to the COO. Because the middle manager is neutral, we may model
him as a signal, a stochastic map $\pi:M\to\Delta(X)$ where $X$ is some
(finite) set of signal realizations. For any message $m$ sent by the branch
manager, the middle manager sends signal realization $x$ to the COO with
probability $\pi(x|m)$.

One possible signal is one that is completely uninformative and involves just
one signal realization, the statement (everything is ) $fine$–no matter what
message the sender chooses, the middle manager says that “everything is fine."
Formally, the set of signal realizations is $X=\left\\{fine\right\\}$, and

$\pi(fine|I)=\pi(fine|N)=1$

We term this signal no transparency, and this scenario is depicted in Figure
2. As in the full transparency case, no information is transmitted at
equilibrium, but now the branch managers separate and send different messages.
In particular, the branch manager in the low state does not invest, since he
knows that the receiver will not be able to identify him from this choice due
to the garbling by the signal. As in the full transparency case, the COO’s
payoff is $2/3$.

Figure 2: Whether to Close the Branch, No Transparency

No transparency is too extreme, and the COO can increase her payoff by
choosing a more moderately uninformative signal. Indeed, the optimal signal is
one in which the middle manager sends one of two signal realizations;
$X=\left\\{fine,bad\right\\}$. If the branch manager invests then the middle
manager always sends fine, and if he doesn’t invest then the middle manager
sends fine half of the time and bad the other half of the time. Formally, the
optimal signal is

$\begin{split}\pi\left(bad|N\right)&=\frac{1}{2},\qquad\pi\left(bad|I\right)=0\\\
\pi\left(fine|N\right)&=\frac{1}{2},\qquad\pi\left(fine|I\right)=1\\\
\end{split}$

The optimal signal is depicted in Figure 3. After $fine$ the COO keeps the
branch open (using Bayes’ law she believes that the branch is viable with
probability $4/5$), and after $bad$ she closes the branch (she is certain that
the branch is nonviable). The signal begets a separating equilibrium–one in
which local manager chooses not to invest if the branch is not viable and
chooses to invest if it is. This gain in informativeness outweighs the
garbling by the signal, and the COO obtains a strictly higher equilibrium
payoff than with full transparency ($5/6$ versus $2/3$).

The signal is pinned down by the incentives of the branch manager in state
$\theta_{B}$. The COO, in designing the information structure, gives local
management just enough enough incentive to be willing to separate. She is able
to minimize the amount of time she chooses the “wrong” action in relation to
the state, and obtains more information due to the branch manager’s
willingness to separate.

Figure 3: Whether to Close the Branch, Optimal Transparency

The COO benefits by not having direct oversight of the branch manager and
instead acquiring information through an intermediary. As stated above, the
third party could be internal, part of the “chain of command" within the
company, or it could be an external auditor. If the COO could directly observe
the branch manager’s decision, she would not be able to credibly commit to not
exploiting the information provided by a branch manager. Thus, with full
transparency, she would obtain no information. She would benefit if instead
she had a go-between who observed the branch manager’s decision before
providing her with a recommendation.

Throughout this paper, we explore the problem of solving for the receiver-
optimal333We focus throughout on the receiver-optimal Perfect Bayesian
Equilibrium. Henceforth, by equilibrium we refer to the PBE that is best for
the receiver. degree of transparency in communication games. We restrict
attention to the environment in which the sender and receiver share a common
prior, and allow for arbitrary state-dependent preferences for the sender and
receiver over both the sender’s message and the receiver’s action. For most of
this paper we focus on what we term simple signaling games, which are those in
which the receiver has no intrinsic preferences over the message chosen by the
sender. That is, they are games in which the message chosen by the sender is
not an argument of the receiver’s utility function–e.g. in the example with
the firm above, the COO is indifferent about whether the local manager chooses
to make an investment outside of the information that the choice contains.

Solving for the receiver-optimal information structure is difficult in
general. The receiver solves an optimization problem with two sets of
constraints: given the signal and vector of strategies by the sender types,
the receiver’s action must be sequentially rational; and the vector of
strategies by the sender types must itself be an equilibrium. This is a
maximization problem with incentive compatibility constraints for the
different sender types and obedience constraints for the receiver.

There is a useful upper bound for the solution to this problem: the commitment
solution for the receiver, which corresponds to the scenario in which the
receiver can commit to a distribution over actions conditioned on the sender’s
choice of message. This is just the optimal transparency problem with the
obedience (sequential rationality) constraints relaxed. Hence, if the
commitment strategy that maximizes the receiver’s payoff also satisfies the
receiver’s obedience constraints, then it must correspond to the optimal
information structure. The commitment problem is much simpler to solve, and we
establish that it reduces to a simple linear program.

In the main result of this paper, Theorem 3.3, we establish that in any simple
signaling game with two actions, “Opacity equals Commitment”. Namely, for any
number of states and messages, provided the receiver has at most two actions,
the optimal commitment solution satisfies the receiver’s obedience
constraints. Thus, to solve the information design problem we need only search
for a solution to the much simpler commitment problem. Furthermore, such
results need not be limited merely to simple signaling games. We also
establish sufficient conditions that guarantee that in non-simple signaling
games with two receiver actions the receiver’s optimal transparency solution
is equivalent to the commitment solution. Moreover, these conditions are quite
natural and hold for a variety of settings, including a paradigmatic game in
biology, the Sir Philip Sidney game444Optimal transparency in this game is the
subject of Whitmeyer (2019) [34]., and a political setting, which we explore
in Section 5.2.

These ideas have implications and applications in numerous additional
scenarios, due to the pervasiveness of situations in which signaling plays a
role. For instance, the advertising choices made by firms famously serve as
signals (see the seminal papers Nelson 1974, Milgrom and Roberts 1986) [27,
24], as do the warranties that firms offer (Gal-Or 1989) [14]. Here, our
results suggest that consumers may benefit from recommender or ratings systems
or “expert" advice, which serve as intermediaries and aid consumers by
providing coarse advice. Another venue is venture-capital, where the equity-
retainment decisions by entrepreneurs signal their private information (Leland
and Pyle 1977) [23]. Mandatory full disclosure or total transparency about the
entrepreneur’s stake; thus, may be to the investor’s detriment.

Headhunting and staffing services are widespread, and can aid hiring firms by
their role as an information intermediary. Namely, the hiring firm may benefit
by delegating the job of acquiring information about the prospective
candidate’s education to the headhunter. The Beer-Quiche game of Cho and Kreps
(1987) [9] can be viewed as an allegory for an incumbent firm signaling to a
potential entrant through its pricing decisions. Accordingly, there is a role
for third party consultants who can condense and even obfuscate to the
aspiring entrant. Finally, policy decisions by governments convey private
information about their competence or aims. Hence, this work suggests that a
populace may benefit from a free press that occasionally hides the incumbent’s
actions. Sunlight may not be the best disinfectant–sometimes it is better to
leave a few things in the dark.

In the companion paper to this one, Whitmeyer (2019) [33], we explore the
value of information in two player communication games with full transparency.
While it is clear that in any setting, information acquisition may benefit the
receiver, our goal in that paper is to uncover in which settings ex ante
learning is always beneficial. To put another way, suppose that the receiver
may, prior to the communication game, acquire public information in the form
of the realization of some Blackwell experiment (Blackwell 1951 [4]). It is
obvious that some experiments benefit the receiver, e.g. a fully informative
experiment always does; but is it true that any experiment benefits the
receiver?

Surprisingly, even in simple games with just two actions ex ante information
acquisition may hurt the receiver if she cannot choose the information
structure in the ensuing game. However, as we find in this paper, in two
action simple games, if the receiver may choose the optimal information
structure, then information acquisition is always beneficial. Thus, in the
class of binary action simple communication games, the ability to choose the
optimal degree of transparency ensures that the value of information is always
positive, even though with full transparency it may not be.

All-in-all, we discover that information design is remarkably powerful. In
two-action simple communication games, it allows the receiver to obtain a
payoff as high as that which she could achieve with commitment. Moreover, it
guarantees that ex ante information can only help her, even though without
information design it may harm.

Section 2 presents the formal model, Section 3 explores the connection between
the receiver-optimal information structure and commitment, and Section 4
explores the possible benefits and drawbacks to ex ante learning. Section 5
illustrates applications of these ideas in finance/accounting, political
economy, and the academic job market in economics; and Section 6 concludes.

###  1.1 Related Work

The past decade has seen a rapid proliferation of works that explore the
underlying ideas from Kamenica and Gentzkow (2011) [20] in different ways. In
that paper, as well as in many of the works that followed, the underlying
information is exogenous, and the persuader’s (or persuaders’) problem is how
to transmit this information in a particular way in order to induce one or
many receivers to take actions favorable (or at least as favorable as
possible) to the persuader(s). In some sense; then, this paper can be viewed
as the inverse of the Bayesian Persuasion problem. Instead of a sender aiming
to persuade a receiver; here, a receiver seeks to elicit information from a
sender–hence, “Bayesian Elicitation".

More recently, commencing with Boleslavsky and Kim (2017) [8], the literature
has explored the situation in which the information is endogenous (in fact, in
Boleslavsky and Kim, the state itself is endogenous). That is, there is now
some information generation process, which is itself affected by the
information structure or signal chosen by the persuader. In Boleslavsky and
Kim, this is manifested in the form of a moral hazard problem for the
agent–the signal must not only convince the receiver, but provide incentives
for the agent as well.

In this paper, the principal is, herself, the receiver, and so the signal is
in part designed to persuade her. However, the sender is also conscious of the
signal, and his choice of action is shaped by the signal. Accordingly, the
optimal signal is chosen not just with persuasion in mind but incentive
provision as well. This dual objective is also present in Asriyan, Fuchs, and
Green (2017) [2]. In their model, $N+1$ sellers have an indivisible asset
whose value is private information, and have two periods in which they may
trade it. In the portion of the paper relevant to this one, they ask how a
planner “should disclose trade behavior to maximize social welfare". As these
authors note, persuasion is not the only objective, since the information
policy “affects the information content of trading, and hence affects trading
itself". Other papers that involve similar trade-offs include Le Treust and
Tomala (2017) [22] and Giorgadis and Szentes (2017) [16]. To help deal with
some technical issues related to this idea of a “constrained information
design" problem, several authors have written notes, see Doval and Skreta
(2018) [11] and Zhong (2018) [35].

Ball (2019) [3] investigates the design of a scoring rule in order to elicit
information from a sender who can distort multiple features (what the receiver
can observe) about himself. Ball’s paper echoes the trade-off that we consider
here, that, “the intermediary must consider how the scoring rule motivates the
sender to distort her features.” Analogously, he finds that coarser
information can benefit the receiver due to the endogeneity of the
information, which is produced by the sender.

Another paper that bears mention is Salamanca (2017) [30]. There the author
relaxes the commitment assumption endemic to the Bayesian persuasion
literature and introduces a mediator, who facilitates communication.
Communication is cheap talk, though as in this paper there is the interplay
between the persuasion motive and the obedience/incentive compatibility
requirement for the information designer.

The paper closest to this one is Rick (2013) [28], who forwards the idea that
mis-communication–what we in this paper refer to as limited transparency–can
be useful in communication games, which category encompasses costly signaling
games (e.g. Spence (1978) [31]), cheap talk (e.g. Crawford and Sobel (1982)
[10]), and games with verifiable messages (e.g. Grossman (1981) [18]). He
assigns Pareto weights to the different sender types and receiver, and asks
how mis-communication can be helpful (welfare improving). It can improve
equilibrium information transmission without raising communication costs
and/or it can reduce communication costs without changing the quality of
information transmission.

Rick focuses on communication games in which the receiver does not have
preferences over the message sent by the sender (what we term simple signaling
games in this paper) and the sender’s payoff is additively separable in his
benefit from the sender’s action and his cost of sending a message. As Rick
shows, benefits from mis-communication, if there are any, must come from at
least one of two sources: 1. mis-communication can remove some profitable
deviations to unused messages and 2. mis-communication can expand the set of
strategy profiles that implements a given distribution of posteriors in
equilibrium.

In contrast to Rick’s paper, we explore the receiver’s problem, which allows
for different results. In particular, if the receiver has two actions we find
an equivalence between the commitment problem and the information design
problem (for simple signaling games). In short, the focus of this work is not
that the receiver may benefit from less than full transparency but instead how
she should choose the degree of transparency optimally.

There are other papers in the literature that explore the benefits of noise
and look at optimal information structures in cheap talk settings. Myerson
(1991) [26] famously describes a cheap talk game in which messages are sent
via carrier pigeon. Remarkably, a somewhat wayward pigeon, one who
occasionally becomes lost, improves communication between the two players. In
a well known paper, Forges (1990) [13] considers mediation in a job-market
example, in which the signals about the prospective candidate’s type are cheap
talk. As in the other papers in this literature, this introduction of a
mediator enlarges the size of the set of equilibrium payoffs.

Quite a few other papers–Goltsman, Hörner, Pavlov, and Squintani (2009) [17],
Ganguly and Ray (2011) [15], Ivanov (2009) [19], and Blume, Board, and
Kawamura (2007) [5]–that look at mediation in the context of cheap talk.
However, due to the difficulty of the problem, these papers focus on the
uniform-quadratic setting from Crawford and Sobel (1982) [10], and Blume et
al. restricts mediation further to a specific form of noise: after the sender
chooses a message $m$ there is an error with some probability $\epsilon$,
after which the receiver observes message $m^{\prime}$ drawn from the uniform
distribution on $[0,1]$, independent of the chosen message $m$. With
probability $1-\epsilon$, there is no error and the receiver observes the
chosen message $m$. They show (in conjunction with Goltsman et al.) that in
the uniform quadratic setting this corresponds to the receiver-optimal
information structure.

Because the messages are cheap talk, these papers, as does Salamanca, view the
mediation problem as a centralized, mechanism design problem. They use the
idea of a Communication Equilibrium, as formulated in Myerson (1986) [25] and
Forges (1986) [12]. Each sender reports his type to a mediator, who then sends
a (possibly random) recommendation to the receiver. In this paper, here,
provided the game is cheap talk and the set of messages is sufficiently large,
the optimal signal $\pi$ is equivalent to the centralized mediator-driven
problem analyzed in these cheap talk papers (we can think of the separating
equilibrium as the reporting of the sender’s type). However, if the message
set is not sufficiently large, then this equivalence is lost. Moreover, if the
messages are costly (i.e. the game is not cheap talk), then the problem we
explore is different to the centralized problems of the literature.

There is experimental evidence that senders respond to different information
structures. Blume, Lai, and Lim (2019) [6] examine a particular class of
information structures, randomized response,555Originally introduced by Warner
(1965) [32], Blume et al. (2019) [6] use a simple version of the technique:
the sender is asked one of two yes or no questions before an intermediary
reveals the answer to the receiver. However, the intermediary may only
partially reveal the question to which the answer corresponded, thereby adding
noise to the response. and find that randomized response can induce senders to
be significantly more truthful. Note that they also find that the information
loss due to the intermediary’s garbling may outweigh the gain in information
due to the increased frequency of truthfulness. However, in a subsequent
paper, Blume, Lai, and Lim (2019) [7] explore mediated cheap talk
experimentally and find that mediation pushes sender types toward separation,
which increases the receiver’s payoff. Crucially, both papers find that the
degree of transparency does affect the sender’s behavior.

Finally, the notion that noise can improve communication has been mentioned in
the biology literature as well: Lachmann and Bergstrom (1998) [21] examine a
specific signaling game, the “Sir Philip Sidney" game and illustrate through
an example that the receiver may benefit by having some degree of perceptual
error. That is, they consider the game with an exogenous garbling of the
sender’s signal and argue that such an environment may be beneficial for the
receiver. In addition, in concurrent work to this one, Whitmeyer (2019) [34]
looks at strategic inattention in the Sir Philip Sidney game, a paradigmatic
signaling game in biology. There, attention is restricted to specific
information structures, those corresponding to inattention on the part of the
receiver; and we find that the receiver always weakly (and often strictly)
benefits from being somewhat inattentive.

##  2 The Model

The setup is a version of the standard communication game. There are two
players: a sender, $S$; and a receiver, $R$. The sender has private
information, his type (or the state) $\theta\in\Theta$. He observes his type
and chooses a message, $m$, from the set of messages $M$. The receiver
observes $m$, but not $\theta$, updates her belief about the sender’s type and
message using Bayes’ law, then chooses a mixture over actions. We assume that
sets $M$ and $\Theta$ are finite.

$S$ and $R$ share a common prior over the state of the world,
$\mu_{0}\in\Delta(\Theta)$, where $\mu_{0}(\theta)=\Pr(\Theta=\theta)$. Each
player, $S$ and $R$, has state-dependent preferences over the message sent,
and the action taken, which are represented by the continuous utility
functions $u_{i}$, $i\in\left\\{S,R\right\\}$: $u_{i}:M\times
A\times\Theta\to\Re$.

The timing of the game is as follows. First, $S$ observes his private type (or
the state) $\theta\in\Theta$, and chooses a message $m\in M$ to send to $R$.
$R$ observes $m$, updates her belief, and chooses action $a\in A$.

We extend the utility functions for the players to behavioral strategies. A
behavioral strategy for $S$, $\sigma(\cdot|\theta)$ is a probability
distribution over $M$. Similarly, a behavioral strategy for $R$,
$\rho(\cdot|m)$ is a probability distribution over $A$. $\sigma(m|\theta)$ is
the probability that a type $\theta$ sender sends message $m$.

We focus on receiver-optimal Perfect Bayesian Equilibrium, which we define in
the standard manner. As noted in the introduction, by equilibrium or PBE, we
refer to those particular equilibria. In addition, throughout this paper, we
focus primarily on signaling games that fall into the following class:

######  Definition 2.1.

A communication game is Simple if the receiver has preferences over the action
taken, $a$, and the state (or sender’s type), $\theta$, but not over the
message chosen by the sender, $m$. Equivalently, a game is simple provided the
receiver’s preferences are represented by the continuous utility function
$u_{R}:A\times\Theta\to\Re$.

###  2.1 Information Manipulation

The principal purpose of this paper is to explore receiver-optimal information
structures, alternatively termed the optimal degree of transparency, in the
context of signaling games. Suppose that $R$ can commit to an information
structure in the following sense. There is a compact set of signal
realizations $X$, where $|X|\geq|\Delta(\Theta)|$, i.e. the receiver is
unconstrained by the size of this set. A signal $\pi$ is a mapping
$\pi:M\to\Delta\left(X\right)$, where
$\pi\left(x|m\right)\coloneqq\Pr\left(X=x|M=m\right)$. Instead of observing
$m$, $R$ instead observes $x$, before choosing a behavioral strategy
$\rho:X\to\Delta(A)$ so as to maximize her expected utility.

The receiver’s value function is

$V=\sum_{i=1}^{n}\mu_{0}(\theta_{i})\sum_{j=1}^{t}\sigma_{i}(m_{j})\sum_{e=1}^{u}\pi\left(x_{e}|m_{j}\right)\sum_{l=1}^{k}\rho\left(a_{l}|x_{e}\right)u_{R}\left(m_{j},a_{l},\theta_{i}\right)$

and she solves

$\sup_{\pi,\sigma,\rho}\left\\{V\right\\}$

such that

$\rho(a_{l}|x_{e})\in\operatorname*{arg\,max}_{\rho(a_{l}|x_{e})}\left\\{\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)\sum_{j=1}^{t}\sigma_{i}(m_{j})\pi\left(x_{e}|m_{j}\right)\sum_{l=1}^{k}\rho\left(a_{l}|x_{e}\right)u_{R}\left(m_{j},a_{l},\theta_{i}\right)\right\\}$

for all $x_{e}\in X$; and

$\sigma_{i}\in\operatorname*{arg\,max}_{\sigma_{i}}\left\\{\sum_{j=1}^{t}\sigma_{i}(m_{j})\sum_{e=1}^{u}\pi\left(x_{e}|m_{j}\right)\sum_{l=1}^{k}\rho\left(a_{l}|x_{e}\right)u_{S}\left(m_{j},a_{l},\theta_{i}\right)\right\\}$

for all $\theta_{i}\in\Theta$.

As we will shortly see, if the game is simple and the receiver only has two
actions, the maximization problem can be reduced to a finite collection of
linear programming problems. Moreover, we also introduce the following
definition and a subsequent “Revelation Principle"-like result, which greatly
simplifies the receiver’s problem

######  Definition 2.2.

A signal, $\pi$, is Direct if $X=A$. That is, a direct signal recommends
actions.

Then, the following proposition establishes that it is without loss of
generality to restrict attention to direct signals, which recommend actions to
the receiver.

######  Proposition 2.3.

For any equilibrium triple, $(\pi,\sigma,\rho)$, that yields a payoff of $v$
to the receiver, there is another equilibrium triple,
$(\pi^{\prime},\sigma^{\prime},\rho^{\prime})$, that yields the same payoff,
$v$, to the receiver; where $\pi^{\prime}$ is a direct signal,
$\sigma=\sigma^{\prime}$, and $\rho^{\prime}(a|a^{\prime})=1$ for
$a^{\prime}=a$ and $\rho(a|a^{\prime})=0$ for $a^{\prime}\neq a$.

###### Proof.

Consider any equilibrium triple $(\pi,\sigma,\rho)$. Now introduce for each
$x_{e}$ and action $a_{l}$ in the support of $\rho(\cdot|x_{e})$ two new
mappings:

  1. 1.

$\hat{\pi}:M\to\Delta\left(\hat{A}^{\prime}\right)$, where
$\hat{\pi}(a_{l}^{e}|m_{j})\coloneqq\pi(x_{e}|m_{j})\rho(a_{l}|x_{e})$; and

  2. 2.

$\rho^{\prime}$, where $\rho^{\prime}(a_{l}|a_{l}^{e})=1$ and
$\rho^{\prime}(a_{l}|a_{m}^{e})=0$.

That is, $a^{e}_{l}$ is the instruction to play $a_{l}$ that induces the same
belief as $x_{e}$. Clearly set $\hat{A}$ may be larger than $A$–it may have
multiple “duplicate" recommendations.

By construction, for each $a_{l}^{e}$, action $a_{l}$ is a best response.
Moreover, it is easy to see that both the obedience and IC constraints are
satisfied and that the expected value for the receiver is the same. Finally,
introduce garbling $g:\hat{A}\to\Delta\left(A\right)$, where
$g(a_{i}|a_{l}^{e})=1$ for $i=l$ and $g(a_{i}|a_{l}^{e})=0$ for $i\neq l$.
Define $\pi^{\prime}\coloneqq g\circ\hat{\pi}$. It is easy to see that the IC
constraints remain satisfied (since each message will lead to the same
distribution of actions chosen by the receiver). Moreover, the obedience
constraints must be satisfied as well since $\pi^{\prime}$ is less Blackwell
informative than $\hat{\pi}$: if it were optimal for the receiver to choose an
action other than $a_{l}$ after observing recommendation $a_{l}$ (recall that
it is optimal for the receiver to choose $a_{l}$ after $a_{l}^{e}$ for any
$e$) then the receiver would have a higher payoff under the less informative
distribution, which contradicts Blackwell’s Theorem (Blackwell 1951 [4]).
Hence $(\pi^{\prime},\sigma^{\prime},\rho^{\prime})$ is also an equilibrium,
and the receiver’s expected payoff remains the same. ∎

In the remainder of the paper, we restrict attention to direct signals, $\pi$,
and thus the receiver’s problem can be reduced to

$\sup_{\pi,\sigma}\left\\{\sum_{i=1}^{n}\mu_{0}(\theta_{i})\sum_{j=1}^{t}\sigma_{i}(m_{j})\sum_{l=1}^{k}\pi\left(a_{l}|m_{j}\right)u_{R}\left(m_{j},a_{l},\theta_{i}\right)\right\\}$

such that

$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)\sum_{j=1}^{t}\sigma_{i}(m_{j})\pi\left(a_{l}|m_{j}\right)u_{R}\left(m_{j},a_{l},\theta_{i}\right)\geq\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)\sum_{j=1}^{t}\sigma_{i}(m_{j})\pi\left(a_{l}|m_{j}\right)u_{R}\left(m_{j},a_{l^{\prime}},\theta_{i}\right)$

for all $a_{l}$, $a_{l^{\prime}}$, $l,l^{\prime}=1,\dots,n$; and

$\sigma_{i}\in\operatorname*{arg\,max}_{\sigma_{i}}\left\\{\sum_{j=1}^{t}\sigma_{i}(m_{j})\sum_{l=1}^{k}\pi\left(a_{l}|m_{j}\right)u_{S}\left(m_{j},a_{l},\theta_{i}\right)\right\\}$

for all $\theta_{i}$, $i=1,\dots,n$.

Importantly, as the following result states, it is not without loss of
generality to restrict the sender types to pure strategies.

######  Proposition 2.4.

For three or more (receiver) actions, there is not always a solution to the
receiver’s optimal transparency problem in which each type of sender chooses a
pure strategy.

###### Proof.

Proof is via counterexample. We revisit the game from Lemma 4.3 in Whitmeyer
(2019) [33]. There are three types of sender; $\theta_{L}$, $\theta_{M}$, and
$\theta_{H}$. A belief as a triple $(\mu_{L},\mu_{M},\mu_{H})$, and the prior
is $(1/4,1/4,1/2)$.

This game is cheap talk with transparent motives: each sender type gets
utility $1$ if the receiver chooses $l$ or $s$, and $0$ if the receiver
chooses $x$. The receiver’s preferences are given as follows:

Action | $\theta_{L}$ | $\theta_{M}$ | $\theta_{H}$
---|---|---|---
$l$ | $0$ | $1$ | $2$
$s$ | $13/24$ | $13/24$ | $1$
$x$ | $1$ | $0$ | $1$

Suppose that there are just two messages, $g$ and $b$. As we ascertain in
Whitmeyer (2019) [33], the receiver optimal equilibrium with full transparency
is one in which $\theta_{H}$ and $\theta_{L}$ choose different messages and
$\theta_{M}$ mixes between those messages ($g$ and $b$). The receiver’s payoff
is $67/52$. On the other hand, if we look for the receiver optimal equilibrium
in which each type chooses a pure strategy under any information structure, it
is easy (though tedious) to verify that the maximum payoff the receiver can
obtain is the pooling payoff, $5/4<67/52$.

∎

While this counterexample used three states, it is possible to construct a
counterexample with just two states and three actions. As the following result
illustrates, a two state counterexample would require that the game not be
cheap talk, however. That is, in any cheap talk game it is without loss of
generality to restrict the sender types to pure strategies provided there are
sufficiently many messages–in fact there is always an optimal transparency
solution in which they separate. This is extremely useful since it ensures
that the receiver need merely solve a linear program.

######  Proposition 2.5.

In a cheap talk game, let the number of messages, $t$, be weakly greater than
the number of states, $n$. Then, if the receiver can achieve some payoff, $v$,
under some information structure and equilibrium, ($\pi$, $\sigma$), then he
can achieve that payoff under some information structure and equilibrium
$\hat{\pi}$, $\hat{\sigma}$ where $\hat{\sigma}$ corresponds to a fully
separating equilibrium.

###### Proof.

Consider some arbitrary type $\theta_{i}$. He must have

$W\coloneqq\sum_{l=1}^{k}\pi\left(a_{l}|m_{a}\right)u_{S}\left(a_{l},\theta_{i}\right)=\sum_{l=1}^{k}\pi\left(a_{l}|m_{b}\right)u_{S}\left(a_{l},\theta_{i}\right)$

for all $m_{a},m_{b}$ in the support of his mixed strategy; and

$W=\sum_{l=1}^{k}\pi\left(a_{l}|m_{a}\right)u_{S}\left(a_{l},\theta_{i}\right)\geq\sum_{l=1}^{k}\pi\left(a_{l}|m_{c}\right)u_{S}\left(a_{l},\theta_{i}\right)$

for all $m_{a}$ in the support of his mixed strategy, $\sigma_{i}$, and
$m_{c}$ not in the support of his mixed strategy. Next, define
$\hat{\sigma}_{i}(m_{j})=1$ if $j=i$ and $0$ otherwise (each type is now
separating), and call this vector of strategies $\hat{\sigma}$. Moreover,
define $\hat{\pi}$ by

$\hat{\pi}(a_{l}|m_{i})\coloneqq\sum_{j=1}^{t}\sigma_{i}(m_{j})\pi(a_{l}|m_{j})$

It is easy to verify via direct substitution that the receiver’s payoff is the
same and that the obedience constraints are satisfied. It remains to verify
that the IC constraints for the sender types are satisfied. Observe that type
$\theta_{i}$, in choosing any $m_{d}$ obtains

$\begin{split}\sum_{l=1}^{k}\hat{\pi}\left(a_{l}|m_{d}\right)u_{S}\left(a_{l},\theta_{i}\right)&=\sum_{l=1}^{k}\sum_{j=1}^{t}\sigma_{d}(m_{j})\pi(a_{l}|m_{j})u_{S}\left(a_{l},\theta_{i}\right)\\\
&=\sum_{j=1}^{t}\sigma_{d}(m_{j})\sum_{l=1}^{k}\pi(a_{l}|m_{j})u_{S}\left(a_{l},\theta_{i}\right)\end{split}$

If $d=i$ then this expression is

$\sum_{j=1}^{t}\sigma_{i}(m_{j})\sum_{l=1}^{k}\pi(a_{l}|m_{j})u_{S}\left(a_{l},\theta_{i}\right)=W$

by the fact that $\sigma_{i}$ was his previous equilibrium strategy and hence
is nonzero only for messages which yield an expected payoff of $W$ (see
equation 2.1). If $d=r\neq i$ then this expression is

$\sum_{j=1}^{t}\sigma_{r}(m_{j})\sum_{l=1}^{k}\pi(a_{l}|m_{j})u_{S}\left(a_{l},\theta_{i}\right)\leq
W$

since the highest expected payoff any message can yield is $W$ and since
$\sigma_{r}$ may place nonzero probability on messages that yield the receiver
an expected payoff that is less than $W$ (see inequality 2.1). ∎

We end the section with the following result, which pertains to belief-based
equilibrium refinements.

######  Lemma 2.6.

Suppose $(\sigma,\rho)$ is an equilibrium in the game with full transparency
that is eliminated by some belief-based refinement. Then, there exists a
signal $\pi$ such that $(\sigma,\pi)$ is an equilibrium with $\pi=\rho$ that
is robust to any belief-based refinement provided $\rho$ has full support over
$A$; i.e. provided every action is played with positive probability.

###### Proof.

Under full transparency suppose that there is some equilibrium under which
message $m^{\prime}$ is not played with positive probability on the
equilibrium path. Thus, there must be some behavioral strategy $\rho(a|\cdot)$
sustained by some belief $\mu$ such that no type can deviate to $m^{\prime}$
profitably. Suppose that there there is a belief-based equilibrium refinement
such that there exists no $\mu$ that sustains such a $\rho$.

Introduce signal $\pi$ and suppose that every action $a^{\prime}$ needed to
head off a deviation is played with positive probability on the equilibrium
path. Then, simply choose $\pi$ such that following a deviation to an off path
$m^{\prime}$, the signal $\pi(a|m)$ is such that the resulting distributions
over actions is just the $\rho(a|m^{\prime})$ needed to sustain the
equilibrium.

If $\rho$ has full support over $A$, the supposition in the previous paragraph
obviously holds (we’ve actually proved a slightly stronger result). ∎

##  3 Opacity = Commitment

In this section, we establish one of the two main results of the paper, that
in simple communication games in which the receiver has two actions,
information design is as good as commitment.

We begin with the following proposition, which states that any payoff that the
receiver can obtain through information design, she can achieve with the
ability to commit to mixtures over actions as a function of the sender’s
message.

######  Proposition 3.1.

If a receiver can achieve a payoff at equilibrium under a particular
information structure, then the receiver can achieve that payoff at
equilibrium in a game in which ex ante she commits to a distribution of
actions conditioned on the sender’s message $m$.

###### Proof.

Take any garbling, $\pi$, of the message $m$, and the optimal response by the
receiver to this garbling. Every message, $m$, will lead to a distribution of
actions by the receiver. The same payoff can be achieved by the receiver
committing to the same distribution of actions conditioned on each message,
$m$. ∎

Another way to deduce this result is to note that the receiver’s choice of
commitment strategy is a choice of mapping $\pi:M\to\Delta\left(A\right)$, and
in her commitment problem she chooses $\pi$ and $\sigma$ to maximize

$\sum_{i=1}^{n}\mu_{0}(\theta_{i})\sum_{j=1}^{t}\sigma_{i}(m_{j})\sum_{l=1}^{k}\pi\left(a_{l}|m_{j}\right)u_{R}\left(m_{j},a_{l},\theta_{i}\right)$

such that

$\sigma_{i}\in\operatorname*{arg\,max}_{\sigma_{i}}\left\\{\sum_{j=1}^{t}\sigma_{i}(m_{j})\sum_{l=1}^{k}\pi\left(a_{l}|m_{j}\right)u_{S}\left(m_{j},a_{l},\theta_{i}\right)\right\\}$

for all $\theta_{i}$, $i=1,\dots,n$. That is, her commitment problem is her
information design problem with the obedience constraints relaxed.

As the next result illustrates, the commitment problem is much easier to solve
than the information design problem, since it reduces to a finite collection
of linear programs.

######  Lemma 3.2.

There is a receiver-optimal equilibrium with commitment in which each type of
sender chooses a pure strategy.

###### Proof.

As noted above, in the receiver’s commitment problem, she maximizes 3 such
that constraints 3 are satisfied for all types $\theta_{i}$. Suppose that at
the optimum there is a type mixing, say $\theta_{1}$. He must be indifferent
over each message in the support of his mixed strategy: suppose that two such
messages are $m_{1}$ and $m_{2}$. Then, since the receiver’s objective is
linear in $\sigma_{i}(m_{j})$ we have

$\begin{split}\frac{\partial{V}}{\partial{\sigma_{1}(m_{1})}}&=\mu_{0}(\theta_{1})\sum_{l=1}^{k}\pi\left(a_{l}|m_{1}\right)u_{R}\left(m_{1},a_{l},\theta_{1}\right)\eqqcolon\xi\\\
\frac{\partial{V}}{\partial{\sigma_{1}(m_{2})}}&=\mu_{0}(\theta_{1})\sum_{l=1}^{k}\pi\left(a_{l}|m_{2}\right)u_{R}\left(m_{2},a_{l},\theta_{1}\right)\eqqcolon\psi\end{split}$

where $\xi$ and $\psi$ are constants. It is easy to see that we must have
$\xi=\psi$ since by construction $V$ is being maximized. Hence, we may set
either $\sigma_{1}(m_{1})$ or $\sigma_{1}(m_{2})$ to $0$. We may do the same
with each other message in support of the mixed strategy until finally we
arrive at one message chosen with probability $1$ and the rest with
probability $0$. The receiver’s payoff is unchanged and the sender is still
unwilling to deviate. We may do likewise with each other type $\theta_{i\neq
1}$, until each type is choosing a pure strategy.

∎

One consequence of this result is that there is a commitment solution for the
receiver in which at most $t=|M|$ messages are used. Moreover, for a fixed
vector of pure strategies, the receiver’s commitment problem is a linear
program. Thus, the receiver’s commitment problem is merely a finite collection
of linear programming problems. Next, we use this lemma to establish one of
our main results.

######  Theorem 3.3 (Opacity Equals Commitment).

Let $|A|=2$. Then, in any simple signaling game, the optimal transparency
solution coincides with the commitment solution.

###### Proof.

If the same action is optimal in every state then the proof is trivial. Hence
suppose that each action is strictly optimal in at least one state. Let the
number of actions, $k=2$, and for simplicity write $v_{i}\coloneqq
u_{R}\left(a_{1},\theta_{i}\right)$, $w_{i}\coloneqq
u_{R}\left(a_{2},\theta_{i}\right)$,
$\pi_{j}\coloneqq\pi\left(a_{1}|m_{j}\right)$, and
$1-\pi_{j}\coloneqq\pi\left(a_{2}|m_{j}\right)$.

Next, observe that the payoff upon observing nothing for the receiver (i.e.
should she simply choose her optimal action based on the prior) is

$\hat{V}=\max\left\\{\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i},\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}\right\\}$

From Lemma 3.2, in searching for the optimal commitment solution it is without
loss to restrict our search to those in which each type of sender chooses a
pure strategy. Hence, in the obedience constraint, we may reduce
$\sum_{j=1}^{t}\sigma_{i}(m_{j})\pi_{j}$ to just $\pi_{i}$ and
$\sum_{j=1}^{t}\sigma_{i}(m_{j})\left(1-\pi_{j}\right)=1-\pi_{i}$. Thus, in
the optimal commitment solution, the receiver’s payoff may be rewritten as

$\sum_{i=1}^{n}\mu_{0}(\theta_{i})\left[\pi_{i}v_{i}+\left(1-\pi_{i}\right)w_{i}\right]$

and the obedience constraints that must be satisfied are

$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}\pi_{i}\geq\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}\pi_{i}$
$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}\left(1-\pi_{i}\right)\geq\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}\left(1-\pi_{i}\right)$

Suppose first that the receiver chooses each action with positive probability
on path (and so Bayes’ law in the obedience constraints is defined). Moreover,
without loss of generality, let
$\hat{V}=\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}$, i.e.
$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}\geq\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}$:
should she observe nothing, the receiver weakly prefers action $a_{1}$.
Consequently, the first obedience constraint must be slack and we must also
have

$\sum_{i=1}^{n}\mu_{0}(\theta_{i})\left[\pi_{i}v_{i}+\left(1-\pi_{i}\right)w_{i}\right]\geq\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}$

or

$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}\left(1-\pi_{i}\right)\geq\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}\left(1-\pi_{i}\right)$

and thus the obedience constraints are satisfied.

Second, suppose that the receiver does not choose each action with positive
probability on path, and without loss of generality let
$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}\geq\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}$.
Then, it is clear that the receiver must be told to play action $1$ and so
Bayes’ law used to derive the second obedience constraint must not be defined.
Since there is at least one state in which the second action is optimal, we
can always stipulate an off path belief such that constraint 3 is satisfied.
Finally, by construction the first obedience constraint is satisfied. ∎

This result allows us to solve for the optimal degree of transparency in any
$2$ action simple game with ease. We need simply solve the receiver’s
commitment problem, which consists of at most two signal realizations.
Moreover, thanks to Lemma 3.2, this problem can itself be reduced–it suffices
to merely consider the cases in which the different types of sender choose
pure strategies.

###  3.1 The Optimal Signal in the Intra-firm Example

Here we solve for the optimal information structure given in the introductory
example, so as to illustrate the usefulness of the above results. Armed with
Theorem 3.3, our problem is simple: we need only solve the receiver’s
commitment problem. Since the receiver has just two actions, we can describe
her commitment strategy with just two variables, $p$, and $q$, where

$p\coloneqq\pi(C|I),\quad\text{and}\quad q\coloneqq\pi(C|N)$

Namely, the COO commits to close the branch with probability $p$ following
message $I$, and with probability $q$ following message $N$.

Next, from Lemma 3.2 we need only search for the optimum for the cases in
which the sender in each state chooses a pure strategy. First, suppose that
the sender in the good state chooses $I$ and that the sender in the bad state
chooses $N$.

The receiver’s objective function is

$\max_{p,q}\left\\{\mu_{0}\left[1-p\right]+(1-\mu_{0})q\right\\}$

and the two incentive compatibility constraints for the sender (in each state)
are:

$p+3(1-p)\geq 2(1-q)$ $q+3(1-q)\geq 2(1-p)$

It is clear that only the second constraint binds and so the optimal $p$ and
$q$ are $p=1/2$ and $q=1$ for $\mu_{0}\leq 1/2$ and $p=0$ and $q=1/2$ for
$\mu_{0}\geq 1/2$. These yield payoffs to the receiver of $1-\mu_{0}/2$ and
$(1+\mu_{0})/2$, respectively.

This commitment equilibrium is obviously better for the receiver than a
commitment equilibrium in which the sender chooses the same message regardless
of the state (such an equilibrium begets the pooling payoff). Likewise, it is
easy to verify that there is no commitment equilibrium in which the senders
separate in the opposite manner. Thus, this commitment equilibrium is optimal
and by Theorem 3.3, opacity equals commitment, and so the receiver’s
commitment strategy corresponds to the optimal information structure.

Explicitly, the optimal information signal consists of two signal
realizations, $X=\left\\{c,o\right\\}$ (recommendations “close" and “keep
open," respectively), and mapping $\pi:M\to\Delta(X)$, where

$\begin{split}\pi\left(c|N\right)&=1,\qquad\pi\left(c|I\right)=\frac{1}{2}\\\
\pi\left(o|N\right)&=0,\qquad\pi\left(o|I\right)=\frac{1}{2}\\\ \end{split}$

for $\mu_{0}\leq 1/2$, and

$\begin{split}\pi\left(c|N\right)&=\frac{1}{2},\qquad\pi\left(c|I\right)=0\\\
\pi\left(o|N\right)&=\frac{1}{2},\qquad\pi\left(o|I\right)=1\\\ \end{split}$

for $\mu_{0}\geq 1/2$.

###  3.2 Robustness of Opacity = Commitment

Naturally, there are many games that fall under the umbrella of communication
games that nevertheless cannot be termed simple. That is, while information
transmission is a key component of such games, the receiver has preferences
over the messages themselves as well. Here we provide a generalization of
Theorem 3.3.

There are $n$ states and $t$ messages, indexed by $i$ and $j$, respectively.
The receiver has two actions, $a_{1}$ and $a_{2}$, and for convenience we
write the receiver’s utilities from the two actions and $t$ messages as
$v_{j}^{i}\coloneqq u(a_{1},\theta_{i},m_{j})$ and $w_{j}^{i}\coloneqq
u(a_{2},\theta_{i},m_{j})$. We introduce the following condition.

######  Condition 3.4.

For any state $\theta_{i}$, $v_{1}^{i}\geq\max_{j}\left\\{v_{j}^{i}\right\\}$
and $w_{1}^{i}\geq\max_{j}\left\\{w_{j}^{i}\right\\}$.

This condition has a straightforward meaning. Simply, the receiver has a
“favorite" message, which, all else equal, she prefers that the sender choose.
Such a condition is natural in many contexts. One such instance is in the Sir
Philip Sidney game, the topic of Whitmeyer (2019) [34]. There, the relatedness
parameter, $k$, ensures that the game is not simple. However, Condition 3.4 is
satisfied: regardless of the state the receiver would prefer that the sender
remain silent.

Indeed, in many settings, a stronger condition holds; that $v_{1}^{i}\geq
v_{2}^{i}\geq\cdots\geq v_{n}^{i}$ and $w_{1}^{i}\geq w_{2}^{i}\geq\cdots\geq
w_{n}^{i}$. This corresponds to a scenario in which, all else equal, the
receiver has the same preferences over the message chosen by the sender. The
political example included later on in this paper satisfies this strong
condition (and thus also Condition 3.4). There, irrespective of whether the
incumbent is good or bad, the receiver prefers that he choose the lowest
amount of policy frictions.

Yet another instance of a game where this strong condition is natural is a
version of the Spence (1978) [31] education model in which education enhances
productivity. There, we might suppose that regardless of the state, the
employer would prefer that the worker have more education. Of course, all
simple signaling games satisfy the strong condition.

######  Proposition 3.5 (Opacity Equals Commitment II).

Let Condition 3.4 hold. Then, if absent strategic concerns, each sender type
prefers message $m_{1}$, the optimal transparency solution coincides with the
commitment solution.

###### Proof.

The proof is analogous to that for Theorem 3.3 and so is left to Appendix A.1.
∎

Although Proposition 3.5 allows us to extend Theorem 3.3 beyond simple
signaling games, it has a number of conditions as prerequisites. These
assumptions are not innocuous, and without them, the result may fail. It is
easy to see that in general for non-simple games, the receiver may benefit if
she had the ability to commit to an incredible threat. Indeed, consider the
following modification of the Spence (1978) [31] signaling model. The receiver
is a firm with a binary decision: whether to hire an applicant. The applicant
is the sender, who can choose his level of education, but education is costly
and all else equal less education is better for each type.

Suppose that the game is non-simple: education is productivity-enhancing; and,
regardless of the applicant’s type, a more educated worker is better for the
firm. Finally, suppose that no matter the type or the level of education, the
firm would prefer to hire the worker. Thus, with the ability to commit to
actions, the firm might want to commit to only hire workers with a
sufficiently high level of education, yet such a protocol would not be
sequentially rational: no signal would persuade the firm to follow through on
such a threat. Hence, commitment is strictly better than opacity.

In addition, Theorem 3.3 is robust. Namely, if we perturb the receiver’s
preferences so that she has slight preferences over the message chosen by the
sender, the opacity equals commitment equivalence remains, provided in the
unperturbed game, the receiver can obtain a payoff that is strictly higher
than the pooling payoff (the payoff should there be no strategic interaction
in which the receiver simply took the action optimal under the prior). To wit,

######  Proposition 3.6.

Consider any two action, simple, signaling game in which each action is
strictly optimal in at least one state, and in which the receiver can obtain a
payoff that is strictly higher than the pooling payoff. Then for any non-
simple perturbation of this game in which the receiver’s payoffs are
sufficiently close to her payoffs in the unperturbed game, opacity is as good
as commitment.

###### Proof.

We prove the following formal result. Consider any two action, simple,
signaling game with payoffs denoted by $v_{i}\coloneqq
u_{R}(a_{1},\theta_{i})$ and $w_{i}\coloneqq u_{R}(a_{2},\theta_{i})$; and let
there exist some $\theta_{i}$ such that $v_{i}>w_{i}$ and some $\theta_{k}$
such that $w_{k}>v_{k}$. Moreover, let

$V>\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i},\quad\text{and}\quad
V>\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}$

In any non-simple perturbation of this game, with
$u_{R}(a_{1},m_{j},\theta_{i})=v_{i}+\omega(a_{1},m_{j},\theta_{i})$ and
$u_{R}(a_{2},m_{j},\theta_{i})=w_{i}+\omega(a_{2},m_{j},\theta_{i})$, there
exists a $\kappa>0$ such that if $|\omega(a_{l},m_{j},\theta_{i})|\leq\kappa$
for all $a_{l}\in A$, for all $m_{j}\in M$ and for all $\theta_{i}\in\Theta$,
the solution to the receiver’s commitment problem satisfies her obedience
constraints. Viz, opacity equals commitment.

The proof may be found in Appendix A.2. ∎

$\theta_{W}$$(4,1/2)$$h$$(0,1)$$f$$(4,0)$$nf$$B$$(6,0)$$nf$$(1,1)$$f$$(6,1/2)$$h$$Q$$1-\mu_{0}$$\theta_{S}$$(1,1/2)$$h$$(1,0)$$f$$(6,1)$$nf$$B$$(4,1)$$nf$$(0,0)$$f$$(0,1/2)$$h$$Q$$\mu_{0}$$R$$R$
Figure 4: A $3$ Action Game

Alas, if the receiver has three or more actions then Theorem 3.3 may fail to
hold. Consider the modified game depicted in Figure 4. This is a version of
the Beer-Quiche game of Cho and Kreps (1987) [9] with the addition of action
$h$ for the receiver. As this game illustrates, our previous result (Theorem
3.3) cannot be extended to the case in which the receiver has three actions.

Recall the Beer-Quiche game: there are two types, wimp ($\theta_{W}$) and
strong ($\theta_{S}$). In the original formulation, there are just two actions
for the receiver: it is a basic match-the-state game in which the receiver
obtains a payoff of $1$ from fighting the wimp or not fighting the strong
type, and $0$ otherwise. Now, the receiver gets a payoff of $1/2$ from
choosing $h$ no matter the state. As for the sender, in the original setting,
wimps prefer Quiche to Beer and vice-versa for the strong types. In addition,
both types prefer not being fought to being fought, and crucially, they would
rather have their least favorite meal and not be fought than have their
favorite meal and be fought. These preferences continue to hold in the
modification; and there, $h$ yields the same payoff to the wimp as $nf$, and
$h$ yields the same payoff to the strong type as $f$.

Throughout this example, we assume that $\mu_{0}>1/2$. For completeness, we
leave detailed analysis to Appendix A.3. There, we derive that in the
commitment solution the senders separate: type $\theta_{W}$ chooses $Q$ and
type $\theta_{S}$ chooses $B$. After $B$, the receiver chooses $nf$ with
probability $1$; and after $Q$, the receiver chooses $f$ with probability
$2/5$ and $h$ with probability $3/5$.

However, in the transparency problem, it is immediately clear that given the
corresponding direct signal realization, “play $h$”, the obedience constraint
for the receiver will be satisfied only if her belief given the signal is that
the sender is the wimp with probability $1/2$. Instead, she believes that the
sender is the wimp with probability $1$, and so the optimal commitment
strategy is not sequentially rational.

##  4 The Structure of Optimal Transparency and the Rewards to Ex Ante
Learning

We begin this section by revisiting the introductory example involving intra-
firm communication. Suppose that prior to the communication game the COO has a
chance to acquire public information. She may for instance commission an
independent report, or even visit the local branch herself. Is it true that,
given this opportunity, the COO would always choose to take it? That is, would
any ex ante information acquisition benefit the COO? Observe that an
equivalent question is whether the COO’s payoff as a function of the (prior)
belief in the signaling game is convex in the prior.

The companion paper to this one, Whitmeyer (2019) [33] looks at a general
version of this question when the receiver is restricted to full transparency.
As we discover there, for two state, two action simple communication games
like the intra-firm example, the answer is yes. But what if the COO may choose
the optimal degree of transparency in the ensuing communication game?

We denote the receiver’s payoff with optimal transparency as a function of the
prior $V=V(\mu)$,666Note that until now, we have referred to the prior belief
as $\mu_{0}$. Henceforth, $\mu_{0}$ will denote the prior belief before ex
ante learning, and we denote the prior belief in the communication game by
$\mu$ (which is the posterior belief resulting from the initial fact finding).
and denote the receiver’s payoff with full transparency as $V^{T}=V^{T}(\mu)$.

It is straightforward to verify that $V$ and $V^{T}$ are

$V(\mu)=\begin{cases}1-\frac{\mu}{2}&\mu\leq\frac{1}{2}\\\
\frac{1+\mu}{2}&\mu\geq\frac{1}{2}\end{cases},\quad\text{and}\quad
V^{T}(\mu)=\begin{cases}1-\mu&\mu\leq\frac{1}{2}\\\
\mu&\mu\geq\frac{1}{2}\end{cases}$

These two functions are depicted in Figure 5, and are clearly convex. Hence,
regardless of whether the COO can choose the optimal signal, ex ante learning
is always (at least weakly) beneficial.

Figure 5: Optimal Transparency and Full Transparency Payoffs, Intra-Firm
Example

However, is this true more generally with optimal transparency? As we will see
shortly, in simple two action games, then ex ante information acquisition is
always good, provided the receiver can choose the information structure in the
ensuing game. If she cannot, then Theorem 4.1. in Whitmeyer (2019) [33] states
that information up front may actually hurt the receiver. That is, there is
information that she would refuse, even if it was free.

Formally, we model ex ante information acquisition or fact-finding as follows.
Fix a simple communication game, and suppose that prior to participating the
game, the receiver may acquire information, which is public. That is,
initially the receiver and the sender share some prior, $\mu_{0}$, and there
is some finite (or at least compact) set of signal realizations $Y$ and a
signal or Blackwell experiment, mapping $\zeta:\Theta\to\Delta(Y)$ whose
realization is public. This experiment leads to a distribution over
posteriors, where the posterior following signal realization $y$ is $\mu_{y}$.
Following the realization of the experiment, the sender and receiver then take
part in the signaling game, where the common prior for the game is $\mu_{y}$.
Call $\zeta$ the Initial Experiment.

######  Example 4.1.

Next, we revisit the example from Lemma 4.2 in Whitmeyer (2019) [33] to show
how information design ensures a positive value of information to the receiver
in a game where information may have a negative value with full transparency.
There are four states,
$\Theta=\left\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\right\\}$, and a
belief is a quadruple $(\mu_{1},\mu_{2},\mu_{3},\mu_{4})$, where
$\mu_{i}\coloneqq\Pr\left(\Theta=\theta_{i}\right)$ for all $i=1,2,3,4$ and
$\mu_{1}+\mu_{2}+\mu_{3}+\mu_{4}=1$.

The belief can be fully described with just three variables; hence, depicting
the receiver’s payoff as a function of the belief requires four dimensions.
The current medium of this paper renders this impossible, so instead we
restrict attention to a family of experiments that involve learning on just
one dimension. That is, we fix $\mu_{1}=1/3$ and $\mu_{3}=1/8$, and consider
only the receiver’s payoff as a function of her (prior) belief about states
$\theta_{2}$ and $\theta_{4}$. Learning is on just one dimension, and so
(abusing notation) we rewrite the receiver’s belief $\mu_{2}$ as $\mu$ and
$\mu_{4}$ as $13/24-\mu$, where $\mu\in[0,13/24]$.

In states $\theta_{1}$ and $\theta_{2}$, action $a_{2}$ is the correct action
for the receiver; and in states $\theta_{3}$ and $\theta_{4}$, action $a_{1}$
is correct:

Action | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$ | $\theta_{4}$
---|---|---|---|---
$a_{1}$ | $0$ | $0$ | $1$ | $2$
$a_{2}$ | $1$ | $1$ | $0$ | $0$

Likewise, the sender’s state (type)-dependent payoffs from message, action
pairs are given as follows:

type | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$ | $\theta_{4}$
---|---|---|---|---
message |  $m_{1}$ $m_{2}$ $m_{3}$ |  $m_{1}$ $m_{2}$ $m_{3}$ |  $m_{1}$ $m_{2}$ $m_{3}$ |  $m_{1}$ $m_{2}$ $m_{3}$
$a_{1}$ |  $1$ $0$ $0$ |  $0$ $1$ $0$ |  $0$ $1$ $3$ |  $-1$ $2$ $-2$
$a_{2}$ |  $1$ $0$ $0$ |  $0$ $1$ $0$ |  $2$ $4$ $0$ |  $5/4$ $0$ $-1$

Note that types $\theta_{1}$ and $\theta_{2}$ have messages that are strictly
dominant ($m_{1}$ and $m_{2}$, respectively), and that $\theta_{4}$ has a
message that is strictly dominated ($m_{3}$).

Figure 6: Optimal Transparency and Full Transparency Payoffs, Non-convex
Counterexample

In Figure 6 we depict the receiver’s equilibrium payoff as a function of $\mu$
with both full transparency ($V^{T}$) and optimal transparency ($V$).
Explicitly, those functions are

$V(\mu)=\begin{cases}\frac{37}{24}-2\mu&\mu\leq\frac{13}{36}\\\
\frac{59}{72}&\frac{13}{36}\leq\mu\leq\frac{35}{72}\\\
\frac{1}{3}+\mu&\frac{35}{72}\leq\mu\leq\frac{13}{24}\end{cases},\quad\text{and}\quad
V^{T}(\mu)=\begin{cases}\frac{37}{24}-2\mu&\mu\leq\frac{13}{36}\\\
\frac{1}{3}+\mu&\frac{13}{36}<\mu\leq\frac{13}{24}\end{cases}$

where $V^{T}$ was derived in Lemma 4.2 of Whitmeyer (2019) [33]. By Theorem
3.3, to obtain $V$ we need only solve the commitment problem. Thus, its
derivation reduces to a simple (though tedious) linear program, which is
omitted.

As Figure 6 illustrates, the issue engendered by ex ante learning–that the
resulting belief may beget a strictly worse equilibrium in the communication
game–is ameliorated if the receiver can choose the optimal information
structure. Thus, in the example, the receiver’s payoff with optimal
transparency is convex. There, information design ensures that ex ante
information acquisition is always helpful.

This result is general, and holds for all simple games with two actions.
First, we establish that commitment guarantees a positive value of information
in any communication game.

######  Lemma 4.2.

In the commitment problem, the receiver’s payoff under the optimal commitment
strategy is convex in the prior $\mu$.

###### Proof.

The proof is left to Appendix B.1. ∎

Second, Theorem 3.3 and Lemma 4.2 combine to yield

######  Theorem 4.3.

In simple games, if the receiver has two actions, then the receiver’s payoff
with optimal transparency is convex in the prior $\mu$.

This result is striking: in simple binary action games, optimal transparency
guarantees that the value of information is always positive, even though with
full transparency the value of information may be negative. Unfortunately, as
the next result illustrates, with three or more states and actions ex ante
information acquisition is not always beneficial for the receiver, even with
optimal transparency.

######  Proposition 4.4.

If there are at least three states, three actions and two messages then the
receiver’s payoff is not generally convex in the prior.

###### Proof.

Proof is via counterexample. We revisit the game from Proposition 2.4 from
this paper, which itself was taken from Lemma 4.3 in Whitmeyer (2019) [33].

We endow the sender types with access to three messages ($g$, $m$, and $b$),
and remind ourselves that this game is cheap talk with transparent motives,
where the sender gets $1$ from $l$ or $s$ and $0$ from $x$, no matter his type
or message choice. The prior is $\mu_{0}$ and learning consists of a binary
initial experiment that results in the beliefs $\mu_{1}$ and $\mu_{2}$, each
with probability $1/2$. Explicitly,

$\begin{split}\mu_{0}&\coloneqq\left(\frac{1}{4},\frac{1}{4},\frac{1}{2}\right),\qquad\mu_{1}\coloneqq\left(\frac{1}{12},\frac{1}{4},\frac{2}{3}\right),\quad\text{and}\quad\mu_{2}\coloneqq\left(\frac{5}{12},\frac{1}{4},\frac{1}{3}\right)\end{split}$

By Proposition 2.5, since there are three messages it is without loss to
restrict attention to equilibria in which the sender types separate in the
optimal transparency solution. We solve the resulting linear program and
obtain that the receiver’s maximal payoffs are $67/52$ and $83/52$ at beliefs
$\mu_{0}$ and $\mu_{1}$, respectively; and $127/132$ at belief $\mu_{2}$.
Comparing the receiver’s payoff at $\mu_{0}$ to her expected payoff resulting
from the initial experiment $\zeta$–

$\frac{67}{52}>\frac{2195}{1716}=\frac{1}{2}\cdot\frac{83}{52}+\frac{1}{2}\cdot\frac{127}{132}$

–we see that this information strictly lowers the receiver’s payoff.

Figure 7:  Proposition 4.4 Optimal Posteriors

If the receiver has commitment power, then her commitment solution is one in
which $\theta_{L}$ separates and chooses $m$, and $\theta_{M}$ and
$\theta_{H}$ pool on $g$. The receiver commits to choosing $s$ after $m$, $l$
after $g$, and $x$ after $b$, and her payoff is $133/96$ at $\mu_{0}$, which
equals her expected payoff with learning,

$\frac{1}{2}\times\frac{469}{288}+\frac{1}{2}\times\frac{329}{288}$

Thus, as we learned from Lemma 4.2, commitment ensures that information cannot
hurt the receiver.

It’s clear that this commitment solution is not sequentially rational: after
being instructed to choose $s$, the receiver strictly prefers to choose $x$
since she knows the sender is $\theta_{L}$. Hence, we have stumbled upon
another illustration that when the number of actions is three or greater,
opacity $\neq$ commitment.

Finally, Figure 7 depicts the receiver’s posteriors resulting from the
receiver-optimal equilibria at each belief, both when the receiver can choose
the optimal degree of transparency (in black) and when the receiver has
commitment power (in green). Each $x$ denotes a posterior distribution.

∎

##  5 Examples

Here, we illustrate the earlier results through three examples. In the first,
we explore the decision of an investor choosing whether to invest in a firm
with uncertain potential. In the second, we analyze a regime change problem,
where a populace must choose whether to replace a possibly bad incumbent. In
the third, we explore the problem of a hiring school at the American Economic
Association academic job market. All three examples suggest different
interpretations of the signal $\pi$. In the investment example, $\pi$ could be
disclosure rules and regulations, or due diligence insisted upon by the
investor. Conversely, in the political example, $\pi$ could be the reporting
actions of the free press. In the job market example, $\pi$ could be an email
filter.

###  5.1 Benchmarks and Disclosure Rules

The receiver is a prospective investor who is choosing whether to purchase a
firm, whose CEO is the sender. The CEO has private information, the viability
of the firm: with probability $\gamma$ the firm’s future cash flow will be
high, and with probability $1-\gamma$ the firm’s future cash flow will be low.
A high cash flow delivers the receiver an income stream of $1$ in perpetuity,
and a low cash flow delivers the receiver a perpetual stream of $0$.

Hence, the sender’s type is the probability $\gamma$ that the project is
viable. Suppose that there are just three possible types,
$\Gamma\coloneqq\left\\{3/10,3/5,9/10\right\\}$, and that both sender and
receiver share a common prior that each type is realized with probability
$1/3$. The receiver discounts the future by $\delta=9/10$.

The posted price for the project is $7$, and if the receiver buys the company
with viability $\gamma$ her expected payoff is

$\frac{\gamma}{1-\delta}-7$

Hence the receiver will buy only if

$\frac{\mathbb{E}[\gamma]}{1-\delta}-7\geq 0$

or

$\mathbb{E}[\gamma]\geq\frac{7}{10}$

The CEO receives a lump sum payoff of $3/5$ in the event that the company is
bought, and $0$ otherwise.

There is a signaling component to this interaction: prior to the purchasing
decision of the receiver, the CEO must choose how much of a budget of $1$ to
allocate between a risky venture and a safe venture. The safe venture repays
the amount invested of $s$ with probability $1$, and the risky venture repays
double the amount invested $r$ with probability $\gamma$ and repays $0$ with
probability $1-\gamma$. Thus, the short term profitability of the risky
venture is perfectly correlated with the likelihood that the firm will have a
high future cash flow.

With regard to the short-run risky venture, the CEO is risk averse and his
utility over his terminal wealth is $u(w)=\sqrt{w}$. Overall, given an
investment of $r$ in the risky venture and $1-r$ in the safe venture, the
CEO’s utility from the scenario is

$u_{S}=\gamma\sqrt{1-r+2r}+(1-\gamma)\sqrt{1-r}+\frac{3}{5}$

if the receiver buys the firm and

$u_{S}=\gamma\sqrt{1-r+2r}+(1-\gamma)\sqrt{1-r}$

if the receiver does not buy the firm.

Here is a synopsis of the scenario’s timing: first, the state $\gamma$ is
drawn uniformly from $\Gamma$ and revealed to the sender; second, the sender
chooses a short term investment level $r$, which is observed by the receiver;
third, the receiver decides whether to purchase the firm, and then payoffs are
realized.

Absent strategic concerns, the CEO’s short run investment strategy is simple.
If $\gamma=3/10$, he chooses $r=0$; if $\gamma=3/5$, he chooses $r=5/13$; and
if $\gamma=9/10$, he chooses $r=40/41$. However, such separation does not
beget an equilibrium in the game with full transparency. Both types $3/10$ and
$3/5$ would prefer to deviate and mimic type $9/10$.

Indeed, it is easy to see that there can be no equilibrium in this game with
full transparency in which there are messages777Recall that we refer to the
sender’s action as a message–here the investment level, $r$, is his message.
chosen after which the receiver strictly prefers different actions. For that
to be the case, at least one of type $3/10$ or $3/5$ must have some support of
his mixed strategy on a message that is followed by the decision to not buy.
However, then that type would deviate profitably to the message that is
followed by the decision to buy. Consequently, the best payoff the receiver
can obtain is the pooling payoff. Since $\mathbb{E}[\gamma]=3/5<7/10$ the
receiver’s payoff is $0$.

Now consider the commitment problem in which the receiver can choose a mapping
$\pi:M\to\Delta\left\\{buy,not\right\\}$. We obtain that the optimal
equilibrium is one in which each type of sender separates: type $3/10$ chooses
$0$, type $3/5$ chooses $40/41$, and type $9/10$ chooses $1$. One signal that
engenders such an equilibrium is

$\begin{split}\pi\left(buy\bigg{|}\frac{40}{41}\right)&=\frac{5}{3}-\frac{17}{3\sqrt{41}}\approx.78,\quad\pi(buy|1)=\frac{5}{3}-\sqrt{2}+\frac{4}{\sqrt{41}}\approx.88,\\\
&\text{and}\quad\pi(buy|r)=0,\quad\forall r\neq\frac{40}{41},1\end{split}$

which yields a payoff of

$.03\approx\frac{5-6\sqrt{2}+\sqrt{41}}{90}>0$

For a detailed derivation of the solution see Appendix C.1. From Theorem 3.3,
opacity equals commitment, and hence this is the optimal information
structure.

Note that there are several curious properties of the optimal signal and
equilibrium. First, both types $3/5$ and $9/10$ are forced to over-invest in
the signaling stage, in order to negate a profitable deviation from the lowest
type (which is the type that the prospective buyer really does not want to
purchase). Second, the information is coarsened by the signal: now the signal
occasionally provides both false positives and false negatives, so as to
optimally incentive the low types.

We can interpret $\pi$ literally as a combination of a benchmark and a
disclosure rule (or an accounting rule). The investor can design a protocol so
that if the required investment level(s) are not met, she will observe a
signal realization instructing her to refrain from buying. Even if the
benchmark (the required investment levels) are met, then the accounting rule
is such that she is still occasionally instructed to not buy.

One way this could be implemented, for instance, is via an online form that
initially asks the CEO (or his team of accountants) explicitly whether the
firm has either a) invested $40/41$ or $1$, or b) anything else (where
truthfulness could be legally enforced). Then, provided an answer of a) to the
first question the form would ask a follow-up question $10$ percent of the
time asking the sender to state whether he invested $40/41$ or $1$. There, if
$40/41$ was answered, the firm would be flagged as $bad$ and if $1$ was
answered, the firm would be rated $excellent$.

The remaining $90$ percent of the time, no additional question would be asked
and conditional on no additional question being asked, the firm would be
assessed as $excellent$ $85$ percent of the time, and $bad$ otherwise. If b)
were answered to the initial question then the firm would be flagged as $bad$.
Finally, the prospective investor would invest following an $excellent$
rating, but not following $bad$.

###  5.2 A Free Press that Censors

> Publicity is justly commended as a remedy for social and industrial
> diseases. Sunlight is said to be the best of disinfectants.
>
> Justice Louis Brandeis

Next, we apply the ideas promulgated earlier to a basic regime change game in
the spirit of Angeletos, Hellwig and Pavan (2006) [1]. There is an incumbent
in power whose type is a (real-valued) binary random variable $X$, which takes
values in the set $\left\\{x_{g},x_{b}\right\\}$, corresponding to “good" and
“bad", respectively. We impose that $x_{g}>1>x_{b}$. There is single
representative receiver–a median voter.888Angeletos, Hellwig and Pavan (2006)
[1], as do many other papers that look at games of regime change, model the
scenario as a global game. Of course, arguably the most compelling aspect of a
global game is the coordination problem faced by the populace. However, here
this issue is moot: the mapping $\pi$ eliminates possible coordination
problems; and moreover, we are searching for the optimal equilibrium for the
receiver. The incumbent’s type is private information and the receiver’s prior
belief about the incumbent’s type, $\mu_{0}\coloneqq\Pr(X=x_{b})$, is correct
and is common knowledge.

The incumbent chooses $r\in[\underaccent{\bar}{r},\bar{r}]\subset(0,1)$ which
corresponds to the amount of (policy) frictions that impede regime change.
That is, the higher $r$ the more costly it is for the receiver to attack the
regime. Note that $r$ is directly payoff relevant for the receiver: the game
is not simple.

There is news in this model: there is a binary signal
$s\in\left\\{G,B\right\\}$, where $b\coloneqq\Pr(B|x_{b})$, and
$g\coloneqq\Pr(B|x_{g})$. We suppose that the situation is one of “perfect bad
news": $1>b>g=0$. That is, bad news will never arise for a good incumbent.
Formally, the receiver’s set of messages is
$M\coloneqq[\underaccent{\bar}{r},\bar{r}]\times\left\\{G,B\right\\}$.999Note
that in the proof of Theorem 3.3 the realized message was a deterministic
outcome of the receiver’s choice. Here, the realized message is a random
outcome of the receiver’s choice. Nevertheless, it is simple to verify that
the result continues to hold–this more general formulation was not included in
the paper since it adds cumbersome notation and is analogous to the simpler,
more concise, version that this paper contains instead.

The game proceeds as follows: first, the incumbent chooses $r$; second, signal
$s$ is generated and the receiver observes both $s$ and $r$ before choosing
action $a\in\left\\{0,1\right\\}$, corresponding to “not attack" or “attack"
the regime, respectively; and third, the incumbent observes the choice of the
receiver and chooses a regime outcome, $d\in\left\\{0,1\right\\}$. The
incumbent’s utility function is

$U_{I}=(1-d)(x-a)-r$

If the incumbent is bad, an attack will be successful and the incumbent will
choose $d=0$. Conversely, if an incumbent is good, an attack will never be
successful, though it will be costly. The receiver’s payoffs are
straightforward: if an attack is successful, the receiver obtains payoff $1-r$
and if an attack is unsuccessful, the receiver obtains payoff $-r$. If the
receiver does not attack, she gets $0$.

We impose that $x_{b}>\bar{r}-\underaccent{\bar}{r}$ and that $b$ satisfies

$b<\frac{x_{b}-\left(\bar{r}-\underaccent{\bar}{r}\right)}{x_{b}}$

i.e. that $\bar{r}-\underaccent{\bar}{r}<x_{b}(1-b)$, which two conditions
ensure that there are no separating equilibria. Indeed, suppose there is such
an equilibrium, in which type $x_{b}$ chooses some $r$ and type $x_{b}$
chooses some $r^{\prime}\neq r$. Type $x_{b}$’s incentive constraint reduces
to $r^{\prime}-r\geq x_{b}$, which is never satisfied under our imposition. We
also impose that

$\mu_{0}(1-b)(1-\underaccent{\bar}{r})+(1-\mu_{0})(-\underaccent{\bar}{r})<0$

which ensures that there is a pooling equilibrium in which both types of
incumbent pool on $\underaccent{\bar}{r}$. Indeed, in such an equilibrium,
following news $B$, the receiver will attack, and following news $G$, the
receiver will not attack. It is easy to see; moreover, that this is the
equilibrium that maximizes the receiver’s payoff, and she obtains
$V=b(1-\underaccent{\bar}{r})$.

On the other hand, the receiver’s commitment solution, and hence the optimal
degree of transparency (via Theorem 3.3) is given by the direct signal
$\pi:M\to\Delta\left\\{0,1\right\\}$, where

$\begin{split}\pi\left(1|(\bar{r},G)\right)&=0,\quad\pi\left(1|(\underaccent{\bar}{r},G)\right)=\frac{\bar{r}-\underaccent{\bar}{r}}{x_{b}(1-b)},\\\
&\text{and}\quad\pi\left(1|(m,s)\right)=1,\quad\forall(r,s)\neq(\bar{r},G),(\underaccent{\bar}{r},G)\end{split}$

In the receiver-optimal equilibrium under this degree of transparency the
senders separate: sender $x_{b}$ chooses $\underaccent{\bar}{r}$ and sender
$x_{g}$ chooses $\bar{r}$. The full derivation is contained in Appendix C.2.

In this setting, one natural interpretation of the signal $\pi$ is as the
actions or reporting of the “free press". That is, the specific $\pi$
described above can be viewed in general as particular equilibrium play by a
neutral mediator, which in the political setting naturally corresponds to the
neutral press. Such a press observes the level of frictions chosen by the
sender ($r$) and a news event ($G$ or $B$) and then reports on those two
things to the populace.

Hence, in this scenario, the populace is better off if the press occasionally
obfuscates. With all due respect to Justice Brandeis, sunlight may not be the
best disinfectant (at least in general). Occasionally, even though the
incumbent has done something that only a bad incumbent would do (chosen
$\underaccent{\bar}{r}$), the free press should give him a pass and not report
it to the populace.

###  5.3 Filtering Emails

The American Economic Association (AEA) has a mechanism for the academic job
market in which each job market candidate can select up to two schools to
designate as recipients of a “signal" (henceforth a “wave", in order to keep
things clear) expressing the candidate’s interest in being interviewed at the
Allied Social Science Associations (ASSA) meetings in January. We explore a
simplified version of this problem.

In order to avoid the problem of waving to multiple receivers, a scenario
beyond the scope of this paper, we suppose that there is just a single school,
and one candidate with a single wave at his access. The candidate has a type,
random variable $\Theta$, which takes values in the unit interval, $[0,1]$.
Both school and the candidate share a common prior over the candidate’s type,
which has full support on $[0,1]$ and is distributed according to absolutely
continuous cdf $F(\theta)\coloneqq\Pr\left(\Theta\leq\theta\right)$.

The school’s decision is binary: interview the candidate ($i$) or not ($n$).
The school obtains a payoff of $\theta$ if they interview a candidate with
type $\Theta=\theta$, and the school has an outside option $d\in(0,1)$. The
candidate has value $x(\theta)$ of being interviewed by the school where
$x:\Theta\to[0,1]$ is increasing and twice continuously differentiable on
$[0,1]$. Moreover, $x(0)=0$ and $x(1)=1$.

The candidate has a simple choice as well: wave to the school ($w$) or not
($r$). There is a cost $k\in(0,1)$ to the candidate of waving to the
school–think of this as an opportunity cost: if the candidate waves to this
school, he101010Remember, in this paper, sender types are male and the
receiver is female. cannot wave to another.

We search for the commitment solution. By Lemma 3.2 we may restrict the
candidate types to pure strategies. Moreover, we can fully describe the
commitment strategy, $\pi$, with variables $p$ and $q$ where

$\begin{split}p&\coloneqq\pi(i|w),\quad\text{and}\quad
q\coloneqq\pi(i|r)\end{split}$

Let’s look for an equilibrium in which positive measures of types choose each
message (remember, the sender’s message is a choice of wave or not). All types
$\theta$ who choose $w$ have the following IC constraint:

$px(\theta)\geq qx(\theta)+k$

Hence, we must have $p>q$. Rearranging this, we have

$\left(p-q\right)x(\theta)\geq k$

Likewise, all types $\theta^{\prime}$ who choose $r$ have the following IC
constraint

$\left(p-q\right)x(\theta^{\prime})\leq k$

######  Claim 5.1.

If there is an equilibrium in which some type $\hat{\theta}$ chooses $w$ and
some type $\theta^{\dagger}$ chooses $r$, then all $\theta\geq\hat{\theta}$
must choose $w$ and all $\theta\leq\theta^{\dagger}$ must choose $r$.

###### Proof.

Let type $\hat{\theta}$ choose $w$, and let $\theta$ be any type strictly
greater than $\hat{\theta}$. Then, we have

$\left(p-q\right)x(\theta)>\left(p-q\right)x(\hat{\theta})\geq k$

since $x(\cdot)$ is increasing and $p-q>0$. Hence, $\theta$ must choose $w$.
The analogous procedure suffices for types that are choosing $r$. ∎

Consequently, if there is an equilibrium in which positive measures of types
choose different messages then there must be some threshold value
$\hat{\theta}$; where all $\theta<\hat{\theta}$ choose $r$, all
$\theta>\hat{\theta}$ choose $w$, and $\hat{\theta}$ himself is indifferent.
Note that because $k>0$, there must always be some positive measure of types
in equilibrium who strictly prefer and hence choose $r$, no matter the signal.
The receiver’s payoff is

$V=\int_{0}^{\hat{\theta}}\left(q\theta+(1-q)d\right)dF(\theta)+\int_{\hat{\theta}}^{1}\left(p\theta+(1-p)d\right)dF(\theta)$

and the IC constraints are

$\left(p-q\right)x(\theta)\leq k$

for all $\theta<\hat{\theta}$, and

$\left(p-q\right)x(\theta)\geq k$

for all $\theta>\hat{\theta}$. The receiver’s payoff can be rewritten as

$V=q\int_{0}^{\hat{\theta}}\left(\theta-d\right)dF(\theta)+p\int_{\hat{\theta}}^{1}\left(\theta-d\right)dF(\theta)+d$

Because the IC constraint of type $\hat{\theta}$ must bind,

$p=\frac{k}{x\left(\hat{\theta}\right)}+q$

Likewise, since the objective is linear in $p$ and $q$, the optimal signal,
$\pi$, is described by either

$p=1,\quad\text{and}\quad q=1-\frac{k}{x\left(\hat{\theta}\right)}$

or

$q=0,\quad\text{and}\quad p=\frac{k}{x\left(\hat{\theta}\right)}$

Consequently, the optimal commitment solution (and hence the optimal
information structure) is either i. given by a signal $\pi$, with $p$ and $q$
described as in the two pairs of equations 5.3 and 5.3, and the corresponding
$\hat{\theta}$ that maximizes $V$; or ii. given by an equilibrium in which the
receiver obtains the pooling payoff (which she can always obtain trivially).

Now, let’s examine the example with the following parameters and functional
forms. Let $F$ be the uniform distribution, $d$ be $2/3$, $k$ be $1/3$ and
$x(\theta)=\theta$. Then, the value function reduces to

$V=q\frac{\hat{\theta}^{2}}{2}-\frac{2}{3}q\hat{\theta}+p\frac{1-\hat{\theta}^{2}}{2}-\frac{2}{3}p\left(1-\hat{\theta}\right)+\frac{2}{3}$

Substituting in $p=1$ and $q=1-1/(3\hat{\theta})$, we have

$V=\left(1-\frac{1}{3\hat{\theta}}\right)\frac{\hat{\theta}^{2}}{2}-\frac{2}{3}\left(1-\frac{1}{3\hat{\theta}}\right)\hat{\theta}+\frac{1-\hat{\theta}^{2}}{2}-\frac{2}{3}\left(1-\hat{\theta}\right)+\frac{2}{3}$

which reduces to

$V=-\dfrac{3\hat{\theta}-13}{18}$

Hence $\hat{\theta}=1/3$, $p=1$ and $q=0$. This yields the receiver a payoff
of $2/3$.

On the other hand, if $q=0$ and $p=1/(3\hat{\theta})$, we have

$V=\left(\frac{1}{3\hat{\theta}}\right)\frac{1-\hat{\theta}^{2}}{2}-\frac{2}{3}\left(\frac{1}{3\hat{\theta}}\right)\left(1-\hat{\theta}\right)+\frac{2}{3}$

which reduces to

$V=-\dfrac{\hat{\theta}}{6}-\dfrac{1}{18\hat{\theta}}+\dfrac{8}{9}$

This expression is maximized at $\hat{\theta}=1/\sqrt{3}$. Hence
$p=1/\sqrt{3}$, $q=0$, and the receiver obtains a payoff of
$\left(8-\sqrt{3}\right)/9>2/3$.

Because of the digital format of the AEA mechanism, the optimal signal $\pi$
can be implemented quite naturally: via an email filter. Hence, with this
formulation of the problem and in this parameter universe, schools would do
better by using a filter that only reveals the candidate’s wave $1/\sqrt{3}$
of the time. As a result, upon receiving a wave from the candidate, the school
will grant him an interview. If no such wave arrives, then the school will
not.

##  6 Discussion

The main feature of the optimal information structure, or degree of
transparency, in simple signaling games with two actions is that the receiver-
optimal equilibrium under the optimal degree of transparency enables the
receiver to achieve the same payoff as in the case wherein she had the power
of committing ex ante to a (mixed) strategy conditioned on the signal choice
of the sender. This is both remarkable, that for a broad class of games
information design is as good as commitment; and useful, since the receiver’s
commitment problem is merely a linear program.

We also find that in simple games with two actions, information design
guarantees that value of information is always positive–the receiver always
benefits from ex ante information, no matter the form it takes. This contrasts
sharply with the findings in Whitmeyer (2019) [33], where we discover that
with full transparency information may hurt the receiver, even in binary
action simple games.

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##  Appendix A Section 3 Proofs

###  A.1 Proposition 3.5 Proof

There are $n$ states, $t$ messages and $2$ actions for the receiver. From
Lemma 3.2, in searching for the optimal commitment solution it is without loss
to restrict our search to those in which each type of sender chooses a pure
strategy. Hence, we may write the receiver’s objective function as

$V=\sum_{i=1}^{n}\mu_{0}(\theta_{i})\left(\pi_{i}v_{j(i)}^{i}+\left(1-\pi_{i}\right)w_{j(i)}^{i}\right)$

where $j(i)$ is the message chosen by sender $i$, and $\pi_{i}$ is the
probability the receiver plays $a_{1}$ after message $j(i)$. The obedience
constraints are

$\sum_{i}^{n}\mu(\theta_{i})\pi_{i}v_{j(i)}^{i}\geq\sum_{i}^{n}\mu(\theta_{i})\pi_{i}w_{j(i)}^{i}$
$\sum_{i}^{n}\mu(\theta_{i})(1-\pi_{i})w_{j(i)}^{i}\geq\sum_{i}^{n}\mu(\theta_{i})(1-\pi_{i})v_{j(i)}^{i}$

provided both actions are recommended with positive probability.

Since, absent strategic concerns, each sender type prefers $m_{1}$, if the
receiver chooses no transparency then each sender type will choose $m_{1}$.
Moreover without loss of generality suppose that in such an equilibrium the
receiver’s optimal action is $a_{1}$; hence,
$\sum_{i=1}^{n}\mu(\theta_{i})v_{1}^{i}\geq\sum_{i=1}^{n}\mu(\theta_{i})w_{1}^{i}$.

There are three cases to consider; 1a, 1b, and 2. Case 1: Each action is
optimal for some state, message combination; and Case 1a: In the commitment
solution, each action is recommended with positive probability. Then, the
receiver must be able to do (at least weakly) better than the no transparency
equilibrium in her commitment solution:

$\begin{split}\sum_{i=1}^{n}\mu_{0}(\theta_{i})\left(\pi_{i}v_{j(i)}^{i}+\left(1-\pi_{i}\right)w_{j(i)}^{i}\right)&\geq\sum_{i=1}^{n}\mu(\theta_{i})w_{1}^{i}\\\
\end{split}$

Or,

$\begin{split}\sum_{i}^{n}\mu(\theta_{i})\pi_{i}v_{j(i)}^{i}&\geq\sum_{i}^{n}\mu(\theta_{i})\pi_{i}w_{j(i)}^{i}+\sum_{i=1}^{n}\mu(\theta_{i})\left(w_{1}^{i}-w_{j(i)}^{i}\right)\\\
&\geq\sum_{i}^{n}\mu(\theta_{i})\pi_{i}w_{j(i)}^{i}\end{split}$

where the last line follows from Condition 3.4. We conclude that the first
obedience constraint is always satisfied under the commitment solution.
Likewise,

$\begin{split}\sum_{i=1}^{n}\mu_{0}(\theta_{i})\left(\pi_{i}v_{j(i)}^{i}+\left(1-\pi_{i}\right)w_{j(i)}^{i}\right)&\geq\sum_{i=1}^{n}\mu(\theta_{i})v_{1}^{i}\\\
\end{split}$

Or,

$\begin{split}\sum_{i}^{n}\mu(\theta_{i})(1-\pi_{i})w_{j(i)}^{i}&\geq\sum_{i=1}^{n}\mu(\theta_{i})\left(v_{1}^{i}-\pi_{i}v_{j(i)}^{i}\right)\\\
&\geq\sum_{i}^{n}\mu(\theta_{i})(1-\pi_{i})v_{j(i)}^{i}\end{split}$

where the last line follows from Condition 3.4. Thus, the second obedience
constraint is also always satisfied under the commitment solution.

Case 1b: Each action is not recommended with positive probability in the
commitment solution. Following the recommendation of the on-path action, the
proof is identical to that of case 1a. Following the recommendation of the
off-path action, the receiver’s belief is undefined. However, since there is
some message/state combination in which the off-path action is optimal, there
is always a belief that we can stipulate such that the directive to play the
off-path action would be obeyed.

Case 2: For all state and message combinations only one action is optimal, say
action $a_{1}$. Then, since there is an equilibrium under no transparency in
which the senders pool on the receiver’s favorite message, $m_{1}$, this must
be the optimal commitment solution as well (the best message and the best
action). Moreover, this must satisfy obedience as well since the directive to
choose $a_{2}$ will never be sent.

The result is proved: the commitment solution coincides with the optimal
transparency solution.

###  A.2 Proposition 3.6 Proof

###### Proof.

Consider any two action, simple, signaling game with payoffs denoted by
$v_{i}\coloneqq u_{R}(a_{1},\theta_{i})$ and $w_{i}\coloneqq
u_{R}(a_{2},\theta_{i})$; and let there exist some $\theta_{i}$ such that
$v_{i}>w_{i}$ and some $\theta_{k}$ such that $w_{k}>v_{k}$. Moreover, let

$V(\mu_{0})>\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}$

and

$V(\mu_{0})>\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}$

By construction, from Inequalities A.2 and A.2 the obedience constraints must
be slack (simply rearrange them as in the proof of Theorem 3.3), i.e.

$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}\pi_{i}>\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}\pi_{i}$
$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}\left(1-\pi_{i}\right)>\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}\left(1-\pi_{i}\right)$

Or,

$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}\pi_{i}=\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}\pi_{i}+\delta_{1}$
$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}\left(1-\pi_{i}\right)=\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}\left(1-\pi_{i}\right)+\delta_{2}$

for constants $\delta_{1}>0$ and $\delta_{2}>0$.

Then, consider any non-simple perturbation of the game with
$u_{R}(a_{1},m_{j},\theta_{i})=v_{i}+\omega(a_{1},m_{j},\theta_{i})$ and
$u_{R}(a_{2},m_{j},\theta_{i})=w_{i}+\omega(a_{2},m_{j},\theta_{i})$. As in
the proof of Proposition 3.5, denote by $j(i)$ the message chosen by sender
$i$. Hence, we may rewrite $\omega(a_{1},m_{j},\theta_{i})$ as
$\epsilon_{j(i)}$ and $\omega(a_{2},m_{j},\theta_{i})$ as $\eta_{j(i)}$.
Accordingly,

$u_{R}\left(a_{1},m_{j},\theta_{i}\right)=u_{R}\left(a_{1},\theta_{i}\right)+\epsilon_{j(i)}=v_{i}+\epsilon_{j(i)}$

and

$u_{R}\left(a_{2},m_{j},\theta_{i}\right)=u_{R}\left(a_{2},\theta_{i}\right)+\eta_{j(i)}=w_{i}+\eta_{j(i)}$

Hence, in the perturbed game, the obedience constraints, A.2 and A.2, become

$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)\left(v_{i}+\epsilon_{j(i)}\right)\pi_{i}\geq\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)\left(w_{i}+\eta_{j(i)}\right)\pi_{i}$
$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)\left(w_{i}+\eta_{j(i)}\right)\left(1-\pi_{i}\right)\geq\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)\left(v_{i}+\epsilon_{j(i)}\right)\left(1-\pi_{i}\right)$

Or,

$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}\pi_{i}\geq\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}\pi_{i}+\gamma$
$\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)w_{i}\left(1-\pi_{i}\right)\geq\sum_{i=1}^{n}\mu_{0}\left(\theta_{i}\right)v_{i}\left(1-\pi_{i}\right)+\sigma$

where

$\gamma=\sum_{i=1}^{n}\mu_{0}(\theta_{i})\pi_{i}\left(\eta_{j(i)}-\epsilon_{j(i)}\right),\quad\text{and}\quad\sigma=\sum_{i=1}^{n}\mu_{0}(\theta_{i})\left(1-\pi_{i}\right)\left(\epsilon_{j(i)}-\eta_{j(i)}\right)$

Then, since $\delta_{1}$ and $\delta_{2}$ are positive constants, there must
be some $\tau>0$ such that if $|\omega(a_{l},m_{j},\theta_{i})|\leq\tau$ for
all $a_{l}\in A$, for all $m_{j}\in M$ and for all $\theta_{i}\in\Theta$ then
$\gamma\leq\delta_{1}$ and $\sigma\leq\delta_{2}$; and hence the obedience
constraints are satisfied.

Of course, it remains to verify optimality in the perturbed game. However,
recall that in the commitment problem, for each vector of pure strategies of
the sender types, the receiver solves a linear program. The constraint set is
a convex polytope and hence the optimum must lie at a vertex of this object.
There are finitely many vertices, and finitely many messages, hence finitely
many possible vectors of pure strategies. Thus, there are only finitely many
combinations of these things that could constitute an optimum. Without loss of
generality, suppose that in the simple game, the optimum is unique. That is,
the receiver’s payoff, $v_{r*}$, at the optimum is strictly better than her
payoff from any other vertex, vector of messages combination. Equivalently,

$\begin{split}v_{r^{*}}\coloneqq
V(\mu_{0})&=\sum_{i=1}^{n}\mu_{0}(\theta_{i})\left[\pi_{i}v_{i}+\left(1-\pi_{i}\right)w_{i}\right]>v_{r}\\\
\end{split}$

for all $r\in R$ where $R$ is the (finite) set of all vertex, pure-strategy
message vector combinations except for the optimal combination ($r^{*}$), and
$v_{r}$ is the receiver’s payoff for vertex, pure-strategy message vector
combination $r$. Inequality A.2 can be rewritten as

$\begin{split}v_{r^{*}}=v_{r}+\iota_{r}\\\ \end{split}$

where for each $r$, $\iota_{r}>0$ is a constant. Now, consider the perturbed
game, where for $r^{*}$ we have

$\begin{split}\hat{v}_{r^{*}}&=\sum_{i=1}^{n}\mu_{0}(\theta_{i})\left[\pi_{i}\left(v_{i}+\epsilon_{j(i)}\right)+\left(1-\pi_{i}\right)\left(w_{i}+\eta_{j(i)}\right)\right]\\\
&=\sum_{i=1}^{n}\mu_{0}(\theta_{i})\left[\pi_{i}v_{i}+\left(1-\pi_{i}\right)w_{i}\right]+\beta\\\
&=v_{r^{*}}+\beta\end{split}$

where $\hat{v}_{r*}$ is the receiver’s payoff in the perturbed game for the
(previously optimal) $r^{*}$ and

$\beta=\sum_{i=1}^{n}\mu_{0}(\theta_{i})\left[\pi_{i}\epsilon_{j(i})+\left(1-\pi_{i}\right)\eta_{j(i)}\right]$

Likewise, for any other $r$, we have in the perturbed game
$\hat{v}_{r}=v_{r}+\alpha_{r}$. Hence, if $v_{r^{*}}+\beta\geq
v_{r}+\alpha_{r}$ for all $r$, then $r^{*}$ is optimal.

Then, since $\iota_{r}$ is a positive constant for any $r$, there must be some
$\rho>0$ such that if $|\omega(a_{l},m_{j},\theta_{i})|\leq\rho$ for all
$a_{l}\in A$, for all $m_{j}\in M$ and for all $\theta_{i}\in\Theta$, then
$\alpha_{r}-\beta\leq\iota_{r}$ for all $r$; and hence $r^{*}$ is optimal.
Finally, define $\kappa\coloneqq\min\left\\{\tau,\rho\right\\}$, and the
result is shown.

We have made the implicit assumption that each recommendation will be sent
with positive probability on path in the simple game. However, if that is not
the case, then upon the instruction to choose the off-path action, we may
assign the receiver’s belief so that the action is sequentially rational since
if the perturbations are sufficiently small then there are beliefs such that
actions $a_{1}$ and $a_{2}$ are sequentially rational.

The result is proved: the commitment solution coincides with the optimal
transparency solution provided the game is almost simple.

∎

###  A.3 Opacity $\neq$ Commitment for Three or More Actions, in Detail

Here, we proceed through the modified Beer Quiche example of Section 3.2 in
detail. In the commitment problem, the receiver may commit to a behavioral
strategy $\pi:M\to\Delta(A)$. We may write $\pi$ in the form of four
probabilities, $p$, $r$, $q$, and $s$, with

$p\coloneqq\pi\left(f|B\right),\quad r\coloneqq\pi\left(nf|B\right),\quad
q\coloneqq\pi\left(f|Q\right),\quad\text{and}\quad
s\coloneqq\pi\left(nf|Q\right)$

From Lemma 3.2 it is without loss of generality to restrict our search for the
receiver-optimal commitment equilibria to ones in which each sender type
chooses a pure strategy. Moreover, it is clear that equilibria in which the
senders pool will beget a payoff to the receiver of $\mu_{0}$. Let’s consider
the separating equilibrium in which the strong guy chooses beer and the wimp,
quiche: $\sigma_{S}=1$ and $\sigma_{W}=0$. Hence, the receiver’s problem can
be rewritten as

$\max_{p,r,q,s}\left\\{\frac{1}{2}\left[1+\mu_{0}\left(r-p\right)+\left(1-\mu_{0}\right)\left(q-s\right)\right]\right\\}$

such that

$5r+1\geq 4s$ $4p+2\geq 5q$

We also have the (possibly slack) constraints that $p+r\leq 1$ and $q+s\leq
1$. This is straightforward to solve and for $\mu_{0}>1/2$, $p=0$, $r=1$,
$q=2/5$, and $s=0$. The corresponding payoff is

$V=\frac{1}{2}\left(1+\mu_{0}+\left(1-\mu_{0}\right)\frac{2}{5}\right)=\frac{1}{2}\left(\frac{7}{5}+\frac{3}{5}\mu_{0}\right)$

It is easy to verify that the other separating equilibrium is not as fruitful
for the receiver. Of course, the question remains as to whether the receiver
could achieve the same payoff under some equilibrium in which the senders mix.
We need this because we are providing a counter-example to opacity $=$
commitment for three actions–otherwise we would leave open the possibility
that there is a commitment solution in which some type(s) mix that can be
achieved through the optimal degree of transparency. However, it is simple to
verify that there is no optimal equilibrium under commitment in which at least
one sender type chooses a non-degenerate mixed strategy. That is, the
equilibrium above, in which the strong type chooses beer, the wimp chooses
quiche, and the receiver commits to $p=s=0$, $r=1$ and $q=2/5$, is uniquely
optimal.

##  Appendix B Section 4 Proofs

###  B.1 Proposition 4.2 Proof

###### Proof.

Recall that from Lemma 3.2 it is without loss of generality in the receiver’s
commitment problem to restrict the sender to pure strategies. Hence, the
receiver solves

$\max_{\pi,s}\left\\{\sum_{i=1}^{n}\mu_{0}(\theta_{i})\sum_{l=1}^{k}\pi\left(a_{l}|m_{i}\right)u_{R}\left(m_{i},a_{l},\theta_{i}\right)\right\\}$

such that

$\sum_{l=1}^{k}\pi\left(a_{l}|m_{i}\right)u_{S}\left(m_{i},a_{l},\theta_{i}\right)\geq\sum_{l=1}^{k}\pi\left(a_{l}|m_{i}^{\prime}\right)u_{S}\left(m_{i}^{\prime},a_{l},\theta_{i}\right)$

for all $\theta_{i}$, $m_{i}^{\prime}$, where $s$ is a vector of pure
strategies chosen by the sender types and message $m_{i}$ is the message
chosen by type $\theta_{i}$ (of course it is possible that $m_{i}=m_{k}$ for
$k\neq i$ if types $\theta_{i}$ and $\theta_{k}$ choose the same message). For
a fixed vector of pure strategies this is a linear program.

Naturally, the payoffs of the game may be such that for some vector of pure
strategies, there exists no signal that is incentive compatible. Note also
that because this is the commitment problem, incentive compatibility is
independent of the prior $\mu_{0}$. Next define set $F$ as the set of all
pairs of signals and strategy vectors that are incentive compatible, with
element $f\coloneqq(\pi,s)$. For any $f\in F$, we have

$V_{f}(\mu_{0})=\sum_{i=1}^{n}\mu_{0}(\theta_{i})\sum_{l=1}^{k}\pi\left(a_{l}|m_{i}\right)u_{R}\left(m_{i},a_{l},\theta_{i}\right)$

and observe that

$\begin{split}V_{f}\left(\lambda\mu_{0}^{\prime}+(1-\lambda)\mu_{0}^{\prime\prime}\right)&=\sum_{i=1}^{n}\left(\lambda\mu_{0}^{\prime}(\theta_{i})+(1-\lambda)\mu_{0}^{\prime\prime}(\theta_{i})\right)\sum_{l=1}^{k}\pi\left(a_{l}|m_{i}\right)u_{R}\left(m_{i},a_{l},\theta_{i}\right)\\\
&=\lambda\sum_{i=1}^{n}\mu_{0}^{\prime}(\theta_{i})\sum_{l=1}^{k}\pi\left(a_{l}|m_{i}\right)u_{R}\left(m_{i},a_{l},\theta_{i}\right)\\\
&+(1-\lambda)\sum_{i=1}^{n}\mu_{0}^{\prime\prime}(\theta_{i})\sum_{l=1}^{k}\pi\left(a_{l}|m_{i}\right)u_{R}\left(m_{i},a_{l},\theta_{i}\right)\\\
&=\lambda
V_{f}\left(\mu_{0}^{\prime}\right)+(1-\lambda)V_{f}\left(\mu_{0}^{\prime\prime}\right)\end{split}$

for any $\lambda\in[0,1]$. Therefore, $V_{f}$ is convex in $\mu_{0}$.

Furthermore, it is easy to see that $V^{*}(\mu_{0})\coloneqq
max_{\pi,s}V(\mu_{0})=\max_{f}V_{f}(\mu_{0})$. Since $V_{f}$ is convex,
$\operatorname*{epi}V_{f}$ is also convex. Then,
$\operatorname*{epi}V^{*}=\operatorname*{epi}\max_{f}V_{f}=\cap_{f\in
F}\operatorname*{epi}V_{f}$. Since the intersection of convex sets is also
convex, $\operatorname*{epi}V^{*}$ is also convex, hence $V^{*}(\mu_{0})$ is
convex in $\mu_{0}$. Note also that $V^{*}(\mu_{0})$ is continuous in
$\mu_{0}$ and since it is a “proper convex function" (Rockafellar (1970) [29])
it must be differentiable almost everywhere (Theorem 25.5, Rockafellar [29]).

Finally, by Jensen’s inequality, the receiver always (at least weakly) prefers
to obtain information ex ante. ∎

##  Appendix C Section 5 (Examples) Derivations

###  C.1 Investor Example (Section 5.1) Derivation

We first look for the optimal commitment strategy should the senders separate
fully. We make the ansatz that type $9/10$’s IC constraint doesn’t bind, in
which case we want to have him choose $r$ to be as high as possible so that
the other types have the least incentive to imitate him: so $9/10$ chooses
$r=1$. Likewise, it is also clear that we want to have type $3/10$ choose his
myopic optimum so that he has the least incentive to deviate: so $3/10$
chooses $r=0$. Finally, we have $3/5$ choose $r\in(0,1)$.

We summarize $\pi$ in terms of $3$ variables, $k$, $l$, and $m$, where

$k\coloneqq\pi(buy|0),\quad l\coloneqq\pi(buy|r),\quad\text{and}\quad
m\coloneqq\pi(buy|1)$

and the CEO maximizes

$V=\frac{1}{3}k\left(\frac{3}{10}-\frac{7}{10}\right)+\frac{1}{3}l\left(\frac{3}{5}-\frac{7}{10}\right)+\frac{1}{3}m\left(\frac{9}{10}-\frac{7}{10}\right)$

such that

$1+\frac{3}{5}k\geq\frac{3}{10}\sqrt{1+r}+\frac{7}{10}\sqrt{1-r}+\frac{3}{5}l$
$1+\frac{3}{5}k\geq\frac{3}{10}\sqrt{2}+\frac{3}{5}m$

and

$\frac{3}{5}\sqrt{1+r}+\frac{2}{5}\sqrt{1-r}+\frac{3}{5}l\geq\frac{3}{5}\sqrt{2}+\frac{3}{5}m$

where we assume (for now) that the other constraints are slack. It is
straightforward to solve this and obtain

$l=\frac{5}{3}-\frac{17}{3\sqrt{41}}\approx.78,\quad
m=\frac{5}{3}-\sqrt{2}+\frac{4}{\sqrt{41}}\approx.88,\quad
k=0,\quad\text{and}\quad r=\frac{40}{41}\approx.98$

Likewise, it is simple to verify that the other IC constraints hold. Moreover,
we can stipulate that following any off-path message the receiver is never
recommended to buy the firm, and so the sender never wishes to deviate to an
off path message. Thus, this constitutes an equilibrium.

The payoff from this separating equilibrium is strictly higher than the
pooling payoff, so it remains to check that there are no better equilibria in
which two types of sender pool. It is easy to verify that none exist that
yield the receiver a payoff that is as high as she can get from full
separation (although there is an equilibrium in which $3/5$ and $9/10$ pool
that is almost as good). Hence, we conclude that the $\pi$ described above, in
conjunction with a fully separating equilibrium, is optimal.

###  C.2 Regime Change Example (Section 5.2) Derivation

This scenario satisfies the conditions for Proposition 3.5 to hold. Thus,
opacity $=$ commitment. Hence, let us solve the commitment problem. Suppose
the incumbent types separate, $x_{b}$ chooses $r$ and $x_{g}$ chooses
$r^{\prime}>r$. The receiver commits to the following: she chooses $1$ with
probability $p$ following $(r,B)$, with probability $q$ following $(r,G)$,
with probability $x$ following $(r^{\prime},B)$, and with probability $y$
following $(r^{\prime},G)$.

The receiver’s value function is

$V=\mu_{0}\left\\{bp(1-r)+(1-b)q(1-r)\right\\}+(1-\mu_{0})y(-r^{\prime})$

and the incentive compatibility constraints are

$\begin{split}b\left[p(-r)+(1-p)(x_{b}-r)\right]&+(1-b)\left[q(-r)+(1-q)(x_{b}-r)\right]\\\
&\geq
b\left[x(-r^{\prime})+(1-x)(x_{b}-r^{\prime})\right]+(1-b)\left[y(-r^{\prime})+(1-y)(x_{b}-r^{\prime})\right]\end{split}$

and

$\begin{split}\left[y(x_{g}-1-r^{\prime})+(1-y)(x_{g}-r^{\prime})\right]&\geq\left[q(x_{g}-1-r)+(1-q)(x_{g}-r)\right]\end{split}$

Suppose that the second constraint is slack. The first constraint may be
rewritten as

$x_{b}(1-b)(y-q)+x_{b}b(x-p)+r^{\prime}-r\geq 0$

It is clear that we can set $x=1$, and that this constraint must bind. Thus,

$q=y+\frac{b}{1-b}(1-p)+\frac{r^{\prime}-r}{x_{b}(1-b)}$

We substitute in for $q$ into the value function and obtain

$V=\mu_{0}\left\\{(1-b)y(1-r)+b(1-r)+\frac{(r^{\prime}-r)}{x_{b}}(1-r)\right\\}+(1-\mu_{0})y(-r^{\prime})$

The derivative of $V$ with respect to $y$ is

$\mu_{0}(1-b)(1-r)+(1-\mu_{0})(-r^{\prime})<\mu_{0}(1-b)(1-\underaccent{\bar}{r})+(1-\mu_{0})(-\underaccent{\bar}{r})<0$

where the second inequality follows from Inequality 5.2. Likewise, the
derivative with respect to $r$ is

$-\mu_{0}\left((1-b)y+b+\frac{1-r+r^{\prime}-r}{x_{b}}\right)<0$

and with respect to $r^{\prime}$ is

$\mu_{0}\frac{1-r}{x_{b}}-(1-\mu_{0})y$

which is positive for $y$ sufficiently small. Thus, it is clear that $V$ is
maximized at $p=1$, $y=0$, $r=\underaccent{\bar}{r}$, $r^{\prime}=\bar{r}$,
and

$q=\frac{\bar{r}-\underaccent{\bar}{r}}{x_{b}(1-b)}$

Moreover, recall $x=1$. The incentive compatibility constraint for $x_{g}$
reduces to

$\left(\bar{r}-\underaccent{\bar}{r}\right)\left(1-x_{b}(1-b)\right)\geq 0$

which obviously holds. Deviations to off-path $r$ can be neutralized by
imposing that the receiver is recommended to overthrow should any $r$ other
than $\underaccent{\bar}{r}$ or $\bar{r}$ be chosen.

It is easy to verify that this equilibrium results in a strictly higher payoff
than any pooling equilibrium, or any separating equilibrium in which
$r^{\prime}<r$. Hence, this degree of transparency and this equilibrium are
optimal for the receiver.