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Communication Breakdown: consultation or
delegation from experts with uncertain bias∗
By Anthony Rush†
, Vladimir Smirnov and Andrew Wait‡
May 11, 2009
Abstract
When communicating with an uninformed decision maker, the motives behind
an expert’s message are often unclear. To explore this situation and investigate
its impact on organizational design we extend the cheap-talk model of Crawford
and Sobel (1982) to allow for uncertainty over the expert’s bias. We find that,
in contrast to Dessein (2002), it is possible that the decision maker prefers
communication to delegation; that is, it can be optimal for a decision maker to
retain control and to solicit advice from the expert.
Key words: delegation, communication, uncertainty, bias, cheap talk.
JEL classifications: D23, D83, L23.
∗We would like to thank Oleksii Birulin, Wouter Dessein, Kunal Sengupta and participants at the
27th Australasia Economic Theory Workshop 2009. This paper represents the views of the authors
and does not necessarily represent the views of the Reserve Bank of Australia or the University of
Sydney. The authors are responsible for any errors.
†Reserve Bank of Australia, Sydney NSW 2000, arush@rba.gov.au
‡Economics, University of Sydney, NSW 2006 Australia, v.smirnov@econ.usyd.edu.au and
a.wait@econ.usyd.edu.au
1
1 Introduction
Having the relevant specific knowledge is critical for effective making (Jensen and
Meckling (1995)). However, those responsible for this task often lack key information.
A decision maker might be uninformed because of the highly specialized knowledge
needed to choose between potential projects. In such situations, the uninformed deci-
sion maker may seek the advice of an informed expert about what actions she should
take – for example, the senior manager of a plant might ask a shop-floor manager
advice on a new project or a CEO of a software firm could solicit a recommendation
of its research department regarding the development of a new product.
In these situations, more often than not, the project selected affects the welfare of
both the principal and the agent. For instance, a middle manager’s division is likely
to be affected by a CEO’s decision; similarly, shareholders often rely on the advice
of financial analysts, who in turn may have a vested interest in the prevailing stock
price. Where the preferences of the agent and the principal do not coincide, the agent
will have an incentive to behave strategically by distorting or obfuscating the truth
in any message they communicate. This paper analyzes the strategic communication
process between an expert advisor and an uninformed principal and its impact on an
organization’s decision making and communication protocols.
Crawford and Sobel (1982) (hereafter CS) analyzed strategic communication in
a cheap-talk game between a perfectly-informed sender (or agent) and a receiver
(principal). The sender observes the state, which takes the value of a random variable,
before sending a message to the receiver. Upon observing the message, the receiver
takes an action. The state and the receiver’s action determine the payoffs for both
players. The preference divergence, or the sender’s bias, is captured by a constant
parameter b, which is known by both agents.
When the preferences of the two agents differ, CS found that regardless of the
size of the bias, there cannot be an equilibrium in which the sender reveals the true
state. Instead, the equilibrium involves a partition of all of the states of nature and
2
the agent’s message only identifies a partition element that includes the true state
of nature. As the sender’s noisy signal is credible, the receiver in turn chooses the
action that maximizes his expected utility given his (correct) probabilistic belief of
the distribution of the state. CS show that more informative equilibria are preferred
by the receiver, as communication is noisy, and that the receiver’s welfare is lower
than if he were perfectly informed.
As an alternative that avoids this strategic communication, the decision maker in
an organization could delegate his decision-making rights to the expert. Extending
Crawford and Sobel (1982), Dessein (2002) found that with standard assumptions
over the decision maker’s best actions, delegation is optimal ‘whenever informative
communication is possible’ (p.822). In general, the loss of information in communi-
cation leads to a greater reduction in welfare for the receiver than the loss of control
under delegation, especially when biases are small (Dessein, 2002, p812).
In reality, however, the choice between communication and delegation is often
much more complex. The experience of Nestl´ehighlights this point. Nestl´einitially
decided to delegate more decisions; this strategy, however, was not successful and
the company decided to revert to centralized decision making, relying on the transfer
of the requisite information to these key decision makers (The Economist, 5 August
2004). Similarly, the marketing divisions of General Motors and Dell Computers have
made conflicting choices:
General Motors announced this summer that it will merge its 5 marketing
divisions into one [communication], ... Meanwhile, Dell Computer actively
decentralizes [delegates] its marketing by assigning fewer market segments
to divisions as they grow. Dell has 12 marketing divisions now, compared
with 4 in 1994. (Donath, 1998, p.9)
One of the key assumptions of CS and Dessein (2002) is that the preference di-
vergence between the two parties is common knowledge; that is, while the decision
maker is uninformed about the appropriate project to implement for the realized state
of the world, he has perfect knowledge about the expert’s bias. In the language of
3
Dessein (2002), there is a systematic and predictable difference between the prefer-
ences of the principal and the agent. In many contexts, however, the decision maker
will be unsure of the underlying objectives of the expert – that is, the bias of the
agent is not systematic and predictable. Again, examples abound: a political leader
may be uncertain about the political leaning of an advisor; the CEO might be unsure
whether the manager is an empire builder or whether he is effort averse; and only
some financial analysts wish to short sell a stock.
We model the situation when: the principal is uninformed; the principal is uncer-
tain regarding the bias of the expert; and the action taken affects the utility of both
the principal and the agent. To this end we extend CS to allow for the receiver to
be uncertain about the sender’s bias; specifically, we allow the sender’s bias, which is
private information, to take on two possible values. We also assume, partly to reflect
the potential coarseness and imprecise nature of communication in real organizations,
that only two possible messages can be sent by the sender (either Low or High).
By allowing for uncertainty with respect to the sender’s bias, we increase the
range of biases for which communicative equilibria are possible. In order to further
investigate these equilibria, we define two different types of expert/sender: (1) an
informative sender, who in equilibrium is willing to send both types of messages de-
pending on the realized state of the world; and (2) an uninformative sender, who only
sends one message across all states of the world. Further to that, even in situations
when the sender will always be informative, a sender with a small – or moderate –
bias compared to the principal will behave differently to an informative sender with
a larger – or extreme – bias, affecting the relative performance of communication to
delegation.1
We find that if the two biases have the same sign, the result of Dessein (2002) that
delegation is always preferred to communication holds. This is not always the case,
however. For example, if the seller’s potential biases have opposite signs, it is possi-
ble that the principal prefers communication over delegation. The intuition for this
1These two terms, moderate and extreme, are defined precisely in Section 3.
4
result is that uncertainty over the expert’s preferences can mute the strategic effects
of communication, encouraging a biased expert to send more informative messages.
The reduction in the loss of information can be sufficient to allow communication to
dominate delegation. The model also shows that if both types of experts are extreme,
it is better to communicate. Moreover, in the case in which one of the types of ex-
pert is moderate and one is extreme, there is a threshold probability (for the type
being extreme) above which the principal will prefer to communicate (and maintain
centralized decision making). Intuitively, by retaining control rights under communi-
cation the decision maker can maintain incentives for information transmission from
unbiased experts, whilst insuring against (very biased) experts that might want to
implement a project not in his interest.
Several other papers have also addressed the communication-or-delegation ques-
tion. Ivanov (2008) showed that if the receiver can ‘optimally’ restrict the sender’s
information, communication always dominates delegation. Krahmer (2006) compared
communication and delegation when utility is transferable between the sender and
receiver and contracts are only partially incomplete. We take a different but com-
plementary approach. Rather than allowing the receiver to have some control on
communication, we focus on the effect of introducing greater uncertainty in terms of
the sender’s bias.
In another related study, Blume et al. (2007) examined information transmission
when, with positive probability, the message sent is misinterpreted by the receiver
and, as a result, is uninformative. Interestingly, they find that adding some noise to
the sender’s signal can almost always improve the welfare of all parties as the noise
creates incentives for the sender to reveal more information, and the value of this
additional information outweighs that utility loss from misinterpretation.
Other papers have focussed on the bias of the sender. For example, Hughes and
Sankar (2006) found that the partition equilibrium in the CS model is robust to
uncertainty with respect to the sender’s preferences. Specifically, they showed the
existence of a partitional equilibrium when the manager (sender) is biased, which
arises in their model due to differences in the manager’s aversion to litigation and
5
the investor’s (or receiver’s) exposure given their level of insurance and that this
partitional equilibrium is robust to investor uncertainty with respect to the man-
ager/sender’s bias (the sender’s bias can take one of two values). Our focus here is
not on the existence of partitional equilibrium, but rather the relative advantages of
communication over delegation. Dimitrakas and Sarafidis (2005) generalized the CS
model to allow for uncertainty with respect to a sender’s bias and they found that
all equilibria are partitional equilibria. Moreover, In a similar way, Dimitrakas and
Sarafidis (2005) found that a partitional equilibrium with a very limited of partitions
(for example 2 partitions) comes very close to the limit equilibrium, suggesting little
information is lost when the sender is restricted to a small number of messages, as
in the model here. These results allows us to simplify our analysis by focussing on a
2 message model, which is akin to a 2-partition equilibrium. Using a similar set-up
to our model, Li and Madarasz (2008) examined whether, prior to communication,
disclosure by the expert of their bias should be mandatory. They find, as shown by
CS, that all communicative equilibria are partitional once the expert’s bias is known,
meaning that communication inherently involves a loss of information. Moreover,
communication without mandatory reporting (so that there is uncertainty regarding
the sender’s bias) can make communication itself more informative, improving the
welfare of both the expert and the decision maker. While the design of their model
is similar to ours, Li and Madarasz (2008) do not address the impact of uncertainty
over the expert’s preferences and organizational structure.
Morgan and Stocken (2003) analyzed communication when an investor/receiver
is uncertain as to whether the stock analyst (sender) is unbiased (has a bias equal to
zero) or has a positive bias, in that she wishes to inflate the value of the stock. In
equilibrium, uncertainty about the sender’s incentives means that it is not possible
to credibly reveal good news about the valuation of the stock, even if the sender
is unbiased. They also show that institutional restrictions on the communication
process, in that the analyst can only send a broad messages that rank the stocks (for
example, sell, hold or buy ) arise endogenously in equilibrium.
In a paper closely related to ours, Ottaviani (2000) constructed a model where the
6
sender’s bias is symmetric. The sender’s bias can take a positive or negative value
with equal probability (bor −b) — capturing situations where preference divergences
are random rather than systematic. He found that the receiver’s welfare is always
higher under communication than it is under delegation. In an important departure
from Ottaviani (2000) we take a more general approach in that we do not require
that the biases are symmetric.
2 The Model
Consider, in turn, the model under: (a) communication, when the principal opts to
retain her decision-making rights and consults an informed agent or expert; and (b)
delegation, when the principal relinquishes her decision-making rights regarding the
project selection to the agent.
Communication
Here, the underlying problem is that an uninformed decision maker must choose a
project. Given she is uninformed, the decision maker can elicit a costless message
from an informed expert. However, this expert is biased in that his preferences over
these projects differ to the preferences of the decision maker.
Consequently, in the model there are two players, a sender and a receiver. The
sender observes the state sthat is a random variable drawn from the interval S= [0,1].
The distribution of the state is assumed to be uniform. The sender also observes her
bias, b, that may take two values, b1or b2where −b1≤b2≤b1. There is a probability
of p(1 −p) that the sender’s bias is b1(b2). The prior distributions of the sender’s
bias and the state are common knowledge to both sender and receiver. In this paper,
we will refer to the sender’s bias as her type. Note that as there are two types of
uncertainty in the model (over the state and the sender’s bias) this labeling is made
for the purpose of clarity.2
2It is also worth mentioning that the majority of the literature uses the term ‘type’ to describe
the sender’s observation of the state.
7
The timing of the game is as follows. The sender, of type biwhere i= 1,2, observes
s, and sends a message, m∈M, to the receiver. After receiving m, the receiver
chooses an action athat determines the utilities of both players. For simplicity, the
sender’s utility from action ais:
US=−|a−(s+b)|(1)
while the receiver’s utility from action ais:
UR=−|a−s|.(2)
Note that here, for ease of exposition, we depart from many of the applications of the
CS mode in that we assume linear utilities; for example, the leading example of CS
(p.1440) uses a quadratic utility. This departure is not critical as it is the difference
between a party’s preferred action and the implemented action that matters.
The following standard assumptions are made: sending a message is costless for
the sender; the message cannot be verified by the receiver; and the receiver also
cannot commit to a decision rule ex ante. We also make one additional assumption.
Upon observing the state, the sender can send one of only two messages - Low or
High, specifically M={Low,High}. That is, while the state of nature and the
number of actions that can be taken is infinite, the number of reports that can be
sent is restricted. There are several reasons as to why the message space might be
restricted. As noted above, Morgan and Stocken (2003) found that a sender may
wish to implement a restriction on the possible messages that can be sent to simple
ranking (Low or High for example). This result arises endogenously in their model
and, furthermore, does not necessarily result in any loss if information. Moreover,
even when the message space is unrestricted in the original CS model, there is an
equilibrium in which only two distinct messages are sent in equilibrium — a two-
partition model — that is equivalent to the equilibrium in our model when b1=b2.3
3We utilize this equivalence to compare the results from this model to the two-partition CS model
8
Another reasons for our binary message space could be that the communication
technology is coarse. It is often not possible to precisely describe exactly the type of
project required, particularly as we are assuming a costless communication process.
Dwyer (1999, cited in Joiner et al., 2002) argued that if the sender must resort to
technical language to be more precise, this might actually reduce the clarity of the
message. Others have also made a similar assumption: for example Tak´ats (2007)
assumes that the sender cannot report all her information. This coarseness could
arise from the sender’s inability to precisely articulate or from the receiver’s lack of
expertise of comprehension, leading the receiver to understand the rough jist of the
message, while being cloudy on the detail. However, it might be that case that the
sender can communicate in broad terms about the project she recommends, be it
‘small’, ‘large’, ‘locate in region X’, etc.4
The solution concept we shall employ is the Bayesian Nash Equilibrium. The deci-
sion maker’s beliefs, P(·|Low) and P(·|H igh), be formed using Bayes’ rule for possible
messages Low and High. The decision maker’s actions, XLand XH, maximize his
expected utility given his beliefs P(·|Low) and P(·|High). The expert’s messages,
Low or High, maximize her utility for all sgiven the decision maker’s strategy.
Our model satisfies the properties of Dimitrakas and Sarafidis (2005), who found
that all equilibria of the model are partitional equilibria. Consequently, we do not
fully characterize the equilibria of our model. Rather, as our purpose in this paper is
to generate a counter example to the delegation result of Dessein (2002), we focus on
the simple 2-message equilibrium that is equivalent to a two-partition equilibrium.
This means that we are comparing the optimality of delegation to communication
in the least informative (and least advantageous) communicative equilibrium, sug-
gesting any dominance of communication would only be enhanced with more refined
communication.
(hereafter the benchmark model).
4One could argue that the uninformed receiver could expend effort to understand the message,
but this would be equivalent to investing effort into becoming informed, as in Aghion and Tirole
(1997), possibly eliminating the need to communicate with an expert. Here we follow CS and Dessein
(2002) in assuming the information structure as exogenous.
9
Delegation
The model of delegation is as follows. Before the state of nature is revealed, the
receiver delegates control of the action to the sender. Upon observing the state of
nature, the sender chooses an action that determines the utility of both players.
Utility functions for sender and receiver are unchanged under delegation. As the
sender’s signal is perfect, it is assumed she can choose any action with complete
precision under delegation. Thus, the sender will choose her optimal action, s+b,
yielding utilities for of US= 0 and UR=−bfor the sender and receiver respectfully.
Model in context
It is worth placing our model in the context of the previous literature on strategic
communication and the optimality of delegation. CS showed that if there is certainty
over sender’s bias – which we termed the benchmark case where b1=b2) – when
the state is distributed uniformly communication is informative for |b|<1
4. Follow-
ing from this, Dessein (2002) showed that when preferences are linear-quadratic the
receiver optimally delegates to the sender whenever communication is informative.
In the case where b1=−b2, and p= 0.5, Ottaviani (2000) showed that communi-
cation is informative when |b|<1
2(hereafter, the symmetric case) and that when
preferences are linear-quadratic, the receiver always prefers communication over del-
egation. While using a different utility function, Morgan and Stocken (2003) also
allow for some uncertainty with respect to the sender’s bias. In their model b1= 0
and b2=b. Here we relax the assumptions of Dessein (2002), Ottaviani (2000) and
Morgan and Stocken (2003) by modelling communication for arbitrary b1,b2and p.
That is, we provide a complete characterization of the parameter values for which
informative communication is possible when the receiver is uncertain of the sender’s
bias.
10
3 Solving the model
Upon receiving the report Low (High), the receiver chooses the action XL(XH) to
maximise his utility. Without loss of generality, XL< XH. A sender of type bireports
Low if US
i(XL)≥US
i(XH) and reports High if US
i(XH)> US
i(XL), where i= 1,2. As
noted there are many equilibria in this game. We focus on equilibria with the feature
that if a sender with a bias b0sends Low in equilibrium, every sender with a bias
b < b0sends Low in equilibrium; and if sender with bias b0sends High in equilibrium,
every type of sender with b>b0will also send message High in equilibrium. This
allows us to focus on the sender of bias bthat is indifferent between sending either
message.
The state where Type biis indifferent between reporting Low and High is Ti. Type
i’s indifference implies that at state Ti
(Ti+bi)−XL=XH−(Ti+bi)
where the payoff to type bisending message Low or sending message High are dis-
played on the left-hand side and the right-hand side, respectively. Solving for Ti:
Ti=1
2(XL+XH)−bi(3)
Equation (3) implies that Type i reports Low for all states less than Ti, and High for
all states greater than Ti. Given that b1> b2,T1< T2. Intuitively, Type 1’s higher
bias implies that she prefers the higher action at a lower state of nature.
The following definitions are introduced to describe the reporting behavior of the
two types:
Definition 1. Informative - A sender type is informative if she reports, depending
on the state of the world, both Low and High in equilibrium. That is, her indifference
point lies between 0 and 1.
Definition 2. Uninformative - A sender type is uninformative if in equilibrium she
11
only sends one message across all states - that is, her reports are uninformative.
Two further definitions are introduced to describe the reporting behavior of in-
formative types. These will be useful in organizing and explaining the results.
Definition 3. Moderate - An informative type is classed as moderate if her indif-
ference point lies between XLand XH. Intuitively, if a sender’s bias is small than her
indifference point is close to the midpoint of XLand XH.
Definition 4. Extreme - An informative type is extreme if she reports both Low
and High in equilibrium, but reports High when s=XL, or Low when s=XH. As
a type’s bias increases, her indifference point moves further from the midpoint of XL
and XH.
Obviously, communication is only possible when at least one type is informative.
For T1< T2, at least one of the following conditions must hold:
1. Both types are informative, i.e. 0 < T1< T2<1;
2. Type 1 is uninformative and Type 2 is informative, i.e. T1<0< T2<1;
3. Both types are uninformative, that is no effective communication is feasible.
We investigate equilibria in each of the following conditions in turn.
3.1 Both types informative
There are 3 potential subcases where both types are informative.
3.1.1 Subcase 1 — b1and b2moderate
If b1and b2are close to zero, then T1and T2are close to the midpoint of XLand XH.
Both players are moderate when 0 < XL< T1< T2< XH<1. The sender’s problem
is shown in Figure 1. As preferences are linear and because the receiver takes T1and
T2as given, he solves the following problem upon observing a Low signal:
min
XL
p
2£X2
L+ (T1−XL)2¤+1−p
2£X2
L+ (T2−XL)2¤.(4)
12
0
XLT1T2XH
1
¡¡¡¡¡¡¡¡
@
@
@
@
@
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@
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@
@
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@
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@
q
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Figure 1: Both sender types (b1and b2) are moderate
That is, if the receiver observes a Low signal, with probability pthe state is between
0 and T1and with probability 1 −pthe state is between 0 and T2. The receiver then
chooses XLto minimise his welfare loss, given the distribution of states in which Low
may be sent.
The receiver’s problem upon receiving High is:
min
XH
p
2£(XH−T1)2+ (1 −XH)2¤+1−p
2£(XH−T2)2+ (1 −XH)2¤.(5)
If the receiver is sent a High signal, with probability pthe state is between T1and
1 and with probability 1 −pthe state is between T2and 1. Similarly, the receiver
chooses XHto minimise his welfare loss over the distribution of states in which High
is reported.
The receiver’s optimal actions as a function of each type’s indifference points is
obtained by finding the FOCs for (4) and (5) respectively:
XL=p
2T1+1−p
2T2,(6)
XH=1
2+p
2T1+1−p
2T2=1
2+XL.(7)
The optimal actions are obtained by the simultaneous solution of the first order
conditions to the receiver’s optimization problems, equations (6) and (7), and each
13
type’s indifference point, equation (3). We derive the following results.
XL=1
4−pb1−(1 −p)b2,and XH=3
4−pb1−(1 −p)b2,(8)
T1= (1 −p)(b1−b2) + 1
2−2b1,and T2= (2 −p)(b1−b2) + 1
2−2b1.(9)
The restrictions on the equilibrium in the subcase when both types are moderate
are 0 < XL< T1< T2< XH<1. From 0 < XLand XH<1 it follows that
−1
4< pb1+ (1 −p)b2<1
4.(10)
From XL< T1< T2< XHit follows that
−1
4< b2< b1<1
4.(11)
It is easy to see that restrictions (10) follow from restrictions (11). The following
result summarizes the above discussion.
Result 1. An equilibrium where both types are informative and moderate can be
supported iff |bi|<1
4for i= 1,2.
Remark 1. Benchmark Case. CS have shown that with linear quadratic preferences,
when there is certainty over the sender’s bias, informative communication is only
possible if the sender’s bias is less than 1
4. As way of comparison, in our model in the
benchmark case when b1=b2=b, from equation (8)
XL=1
4−b, XH=3
4−b, T1=T2=1
2−2b
and communication can only be supported if b < 1
4.
3.1.2 Subcase 2 — b1extreme, b2moderate
As Type 1’s bias increases above 1
4,T1decreases below XLand b1becomes an extreme
sender. The restrictions for this case are 0 < T1< XL< T2< XH<1.
14
0
T1XLT2XH
1
¡¡¡¡¡¡¡¡¡¡¡
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@
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Figure 2: b1extreme, b2moderate
The receiver solves the following minimization problems for Low (see Figure 2):
min
XL
p
2£X2
L−(XL−T1)2¤+1−p
2£X2
L+ (T2−XL)2¤,(12)
and High:
min
XH
p
2£(XH−T1)2+ (1 −XH)2¤+1−p
2£(XH−T2)2+ (1 −XH)2¤.(13)
The receiver’s optimal actions as a function of each type’s indifference points is
obtained by finding the FOCs for (12) and (13) respectively:
XL=1
2T2−p
2(1 −p)T1,(14)
XH=1
2+p
2T1+1−p
2T2.(15)
The optimal actions are obtained by the simultaneous solution of the first order
conditions to the receiver’s optimization problems, equations (14) and (15), and each
15
type’s indifference point, equation (3).
XL= (1−p)(b1−b2) + (1 −4b1)(1 −2p)
2(2 −p), XH=1
2+(1−p)(b1−b2)+ (1 −4b1)(1 −p)
2(2 −p),
(16)
T1= (1−p)(b1−b2)+ (1 −4b1)(1 −p)
2−p, T2= (2−p)(b1−b2)+ (1 −4b1)(1 −p)
2−p.(17)
The restrictions on the equilibrium in this subcase subcase where type one is
extreme and type 2 is moderate are 0 < T1< XL< T2< XH<1. T1>0 is satisfied
if
b1<1
2 + p−2−p
2 + pb2.
T1< XLif
b1>1
4.
T2< XHif
b1<1
2p+2−p
pb2.
The following result provides a summary of when this communicative equilibrium
is possible.
Result 2. When both types of sender are informative with b1extreme and b2
is moderate a communicative equilibrium if −1
4< b2<1
4and 1
4< b1<
min ³1
2+p−b22−p
2+p,1
2p+b22−p
p´.
Although Type 1’s bias is greater than 1
4, communication is still possible from
both types as the receiver is maximizing expected utility over the reporting strategies
of all sender types. In contrast to the original CS model, uncertainty over the sender’s
bias allows sender types with larger biases to still be informative.
3.1.3 Subcase 3 — b1extreme, b2extreme
As the first type’s bias increases and the second type’s bias decreases, communication
is still possible with T1< XLand XH< T2. That is, if the biases of both types spread
apart, communication can still be sustained when both types’ biases are greater in
magnitude than 1
4. Type 1 reports Low in extremely low states, and Type 2 reports
16
High in extremely high states. The restrictions for this case are 0 < T1< XL< XH<
T2<1. The receiver solves the following minimization problems (see Figure 3):
0
T1XLXHT2
1
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Figure 3: b1and b2extreme
min
XL
p
2£X2
L−(XL−T1)2¤+1−p
2£X2
L+ (T2−XL)2¤,(18)
and High:
min
XH
p
2£(XH−T1)2+ (1 −XH)2¤+1−p
2£(1 −XH)2−(T2−XH)2¤.(19)
Using the same steps as in the previous subcases one can derive:
XL= 2p(1 −p)(b1−b2) + (1 −4pb1)(1 −2p)
2,(20a)
XH=1
2p−2(1 −p)2(b1−b2) + (1 −4pb1)(2p−1)(1 −p)
2p,(20b)
T1= (1−p)(1+b2(1−2p)−b1(1+2p)), T2= (1−p)(1+b2(1−2p)−b1(1+2p))+b1−b2.
(21)
The restrictions on the equilibrium in the subcase where both types are extreme
17
are 0 < T1< XL< XH< T2<1. T1< XLif
b1>1
2(1 + p)+1−p
1 + pb2.
T1>0 is satisfied if
b1<1
2p+ 1 −2p−1
2p+ 1b2.
T2<1 if
b2>−1
3−2p+2p−1
3−2pb1.
T2> XHif
b2<−1
2(2 −p)+p
2−pb1.
The following result provides a summary of when this communicative equilibrium
is possible.
Result 3. If both types are extreme, a communicative equilibrium exists if 1
2(1+p)+
1−p
1+pb2< b1<1
2p+1 −2p−1
2p+1 b2and −1
3−2p+2p−1
3−2pb1< b2<−1
2(2−p)+p
2−pb1.
3.2 One type informative and one type uninformative
In the following subcases, there is one type whose bias is so large that she sends only
one message in equilibrium - i.e. she is uninformative. There are two such subcases.
3.2.1 Subcase 4 — b1uninformative, b2moderate
Type 1 is the uninformative type. As b1> b2, this implies only Type 2 sends the
message Low. In this subcase, Type 2 is a moderate sender (see Figure 4).
The receiver’s minimization problems are:
min
XL
p
2[0] + 1−p
2£X2
L+ (T2−XL)2¤(22)
min
XH
p
2£X2
H+ (1 −XH)2¤+1−p
2£(XH−T2)2+ (1 −XH)2¤(23)
18
0
XLT2XH
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Figure 4: Type 1 is uninformative, Type 2 is moderate
Following the same steps as in previous cases to solve for XLand XH:
XL=1
2(2 + p)−2
(2 + p)b2, XH=3
2(2 + p)−2(1 −p)
(2 + p)b2(24)
T1<0, T2=1−4b2
2 + p.(25)
Notice that as she only sends one message, the uninformative type’s bias is irrelevant
to the choice of XLand XH. However, the probability of her type is relevant: as p
increases, XHconverges to 1
2.
In terms of restrictions on equilibrium in this subcase, type 1 is uninformative if
T1<0, which results in the following constraint:
b1>1
2 + p−(2 −p)b2
2 + p.
Type 2 is moderate if XL< T2< XH.T2< XHplaces the lower bound on b2:
b2>−1
4 (1 + p).
19
T2> XLplaces the upper bound on b2:
b2<1
4.
This analysis is summarized by the following result.
Result 4. If Type 1 is uninformative and Type 2 is informative and moderate, a
communicative equilibrium is supportable if b1>1
2+p−(2−p)b2
2+pand −1
4(1+p)< b2<1
4.
3.2.2 Subcase 5 — b1uninformative, b2extreme
Now consider the case when Type 1 is uninformative and Type 2 is an extreme sender,
as shown in Figure 5.
0
XLXHT2
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Figure 5: Type 1 is uninformative, Type 2 is extreme
The receiver’s minimization problems are:
min
XL
p
2[0] + 1−p
2£X2
L+ (T2−XL)2¤; (26)
and
min
XH
p
2£X2
H+ (1 −XH)2¤+1−p
2£(1 −XH)2−(T2−XH)2¤.(27)
20
In this subcase, the receiver’s optimal equilibrium actions are:
XL=1−4pb2
2(2p+ 1), XH=1
2p−1−p
p
1−4pb2
2(2p+ 1) (28)
where
T1<0, T2=1−4pb2
2p+ 1 .(29)
In terms of restrictions on the equilibrium, Type 1 is uninformative if T1<0, which
results in the following constraint:
b1>1
2p+ 1 −2p−1
2p+ 1b2.
Type 2 is an extreme sender if XH< T2<1. T2> XHis satisfied if:
b2<−1
4(1 + p).
Type two is informative if T2<1:
b2>−1
2.
This analysis is summarized by the following result.
Result 5. If Type 1 is uninformative and Type 2 is informative and extreme, a
communicative equilibrium exists if b1>1
2p+1 −2p−1
2p+1 b2and −1
2> b2>−1
4(1+p).
3.3 No Communication
If the biases of both types are too large the only equilibrium is no communication.
As the distribution of the state is uniform, the receiver’s utility maximizing action is
1
2, regardless of the message sent. This occurs in three distinct subcases.
21
3.3.1 Subcase 6
Type 1 sends the message High, and Type 2 the message Low. This implies T1<0
and T2>1, and XL=XH=1
2. This is the unique equilibrium iff b1≥1
2and
b2≤ −1
2.
3.3.2 Subcase 7
Both types send the message High.T1, T2<0 and XH=1
2.XLis unrestricted. This
is the unique equilibrium iff b2>1
4.
All equilibria of the model with two sender types have been dealt with. In the
next section, a summary of the set of equilibria under uncertainty is provided.
3.4 Characterization of the Message Space
Using the three previous subsections, Figure 6 presents a complete graphical charac-
terization of the message space for b1> b2and p= 0.5.5While the exact charac-
terization depends on the probability of each type, the layout remains qualitatively
unchanged for different values of p.
The red region corresponds to a communicative equilibrium where both types are
informative; the aqua and green areas correspond to the cases where Type 1 and Type
2 are informative, respectively. Communication is not possible if b1and b2are in the
yellow regions.
From section 3.3, the values b1and b2under which communication is impossible
never change. On the other hand, as discussed in section 3.1, the critical level of bias
at which either Type 1 or 2 becomes uninformative depends on the probability of her
type, and the bias of the other type.
The following result summarizes the effect that uncertainty over the sender’s bias
has on informative communication.
5The characterization for b1< b2is merely the mirror image of the Figure 6 around the line
b1=b2.
22
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
b2
b1
5 4 7
6
3 2
1
Figure 6: The set of communicative equilibria for p= 0.5
Result 6. Under uncertainty, there is no longer a unique upper bound on the bias
that can support informative communication. In contrast to Crawford and Sobel
(1982), it is now possible for informative signalling if a sender’s bias is |bi|>1
4.
The borderlines between no communication and uninformative cases provide the
upper bound for the set of biases that support informative signalling from one sender
type, and the borders between informative and uninformative cases provides the upper
bound for the set of biases that support informative communication from all sender
types.
An Example
The previous subsections have provided a complete characterization of communication
when there is uncertainty about the sender’s bias. An example is now provided to
more clearly show that communication can be supported for a wider range of biases
under uncertainty.
23
Example 1. Suppose b1= 0.4, b2=−0.1 and p= 0.5. The communicative equi-
librium corresponds to subcase 3.1.2. From the Appendix, the receiver’s equilibrium
actions are:
XL= 0.25 nd XH= 0.65.
Substituting the receiver’s actions into equation (3), Type 1 reports Low in states
s∈(0,0.05) and High for states s∈(0.05,1). Type 2 reports Low for s∈(0,0.55)
and High over s∈(0.55,1).
4 Communication versus delegation
While the focus of this section is to compare communication and delegation, does the
receiver always prefers communication to no communication?
Proposition 1. The receiver’s expected utility under communication is always greater
than without communication.
Proof. The proof is a revealed-preference argument. If the receiver believes that the
signal is uninformative with probability 1, then his utility maximizing actions are
XL=XH=1
2. This set of actions will always yield the receiver an expected utility
of −1
4.
Under informative communication, the receiver’s utility maximizing actions are
XL<1
2and XH>1
2. As the receiver is still able to choose the set XL=XH=1
2and
obtain on average a utility of −1
4, not doing so must yield a strictly higher expected
utility.
Turning to the receiver’s choice of communication or delegation, for each commu-
nicative subcase of section 3 the receiver’s expected utility, UC, will be computed and
compared to the utility under delegation. Note that given the assumptions over the
distribution of biases and the sender’s action under delegation, the receiver’s expected
utility under delegation, UD, is
−p|b1| − (1 −p)|b2|.(30)
24
4.1 Subcase 1 — b1and b2moderate
When both types of senders are moderate, UCis equal to negative sum of functions
(4) and (5):
UC=−p
2£X2
L+ (T1−XL)2+ (XH−T1)2+ (1 −XH)2¤
−1−p
2£X2
L+ (T2−XL)2+ (XH−T2)2+ (1 −XH)2¤.
(31)
Substitute values of XL,XHfrom (8) and values of T1and T2from (9) to derive
UC=−1
8−pb2
1−(1 −p)b2
2−(pb1+ (1 −p)b2)2.(32)
Communication is preferred by the receiver if UC> UD. In this subcase, commu-
nication is optimal when:
1
8< p|b1|+ (1 −p)|b2| − pb2
1−(1 −p)b2
2−(pb1+ (1 −p)b2)2.(33)
Analysing the above inequality, if b1and b2are biased in the same direction,
then delegation always dominates communication. To see that, note that for b16=b2
the following inequality holds pb2
1+ (1 −p)b2
2>(pb1+ (1 −p)b2)2. Labeling ¯
b=
pb1+ (1 −p)b2, the RHS of inequality (33) is less than |¯
b| − 2¯
b2. On the other hand,
|¯
b| − 2¯
b2≤1
8.
If b1and b2are biased in opposite directions, the same logics does not quite
work and it is possible that communications dominates delegation. For example,
for p= 0.5, b1= 0.2, b2=−0.2 inequality (33) is satisfied. This information is
summarised in the following result.
Result 7. When both types have biases in the same direction, delegation dominates
communication. When both types have biases in the opposite directions, communica-
tion may dominate delegation.
Sender types with opposing biases allow the receiver to take a higher Low action
relative to communication only with Type 1, and a lower High action relative to
communication only with Type 2. As both types’ biases are relatively small, this
25
shifts both types’ indifference points towards 1
2, which reduces welfare loss under
communication. If b1>0 and b2<0 then this effect can be strong enough for
communication to be optimal.
Remark 2. Benchmark Case. Dessein (2002) have shown that when there is cer-
tainty over the sender’s bias, delegation dominates communication whenever infor-
mative communication is possible. If b1=b2then both types have biases in the same
direction, which means the above result applies, i.e. delegation dominates communi-
cation.
4.2 Subcase 2 — b1extreme, b2moderate
In this case UCis equal to minus sum of functions (12) and (13):
UC=−p
2£X2
L−(XL−T1)2+ (XH−T1)2+ (1 −XH)2¤
−1−p
2£X2
L+ (T2−XL)2+ (XH−T2)2+ (1 −XH)2¤.
(34)
Substitute values of XL,XHfrom (16) and values of T1and T2from (17) to derive
UC=−1
4+p(1 −p)(1 −4b1)
(2 −p)2+(1−p)µpb1+ (2 −p)b2+1
2¶µb1−b2+(1 −4b1)(2 −3p)
2(2 −p)2¶.
(35)
Communication is preferred to delegation if:
1
4−p|b1| − (1 −p)|b2|<p(1−p)(1−4b1)
(2−p)2+
(1 −p)¡pb1+ (2 −p)b2+1
2¢³b1−b2+(1−4b1)(2−3p)
2(2−p)2´.
(36)
Result 8. For any b1and b2where Type 1 is extreme and Type 2 moderate, there
exists a p∗where communication is optimal for all p∗< p < 1.
Proof. When pis increased, area 2 could either change or stay the same. If
area stays the same then we use the following proof. Type 1 is extreme only when
b1>1
4, implying that for sufficiently high pthe payoff from delegation is lower than
−1
4. However, from Proposition 1, the receiver’s payoff from communication must
be greater than −1
4. Consequently, given that inequality (36) is continuous, if pis
sufficiently high then communication is prefered to delegation.
26
If the area changes, it will change to either area 5 or area 6. Specifically, both
boundaries b1=1
2+p−b22−p
2+pand b1=1
2p−b22−p
pbecome weaker when pincreases.
The first boundary gives b0
1=4b2−1
(2+p)2<0, the second boundary gives b0
1=−1−4b2
2p2<0.
In area 6 communication dominates delegation, see result (4.5). In area 5, a similar
result holds, see result (4.4). This observation ends the proof. ¤
4.3 Subcase 3 — Both types extreme
For all b1and b2where both types are extreme, p|b1|+(1−p)|b2|>1
4. From Proposition
1, communication is always optimal in this case.
Result 9. Communication is always optimal whenever both types are extreme.
4.4 Subcase 4 — b1uninformative, b2moderate
In this case UCis equal to minus sum of functions (22) and (23):
UC=−p
2£X2
H+ (1 −XH)2¤−1−p
2£X2
L+ (T2−XL)2+ (XH−T2)2+ (1 −XH)2¤.
(37)
Substitute values of XL,XHfrom (24) and value of T2from (25) to derive
UC=−2p+ 1
4(2 + p)−4(1 −p)b2
2
2 + p.(38)
The receiver’s expected utility from communication is greater than her expected
utility from delegation if:
−2p+ 1
4(2 + p)−4(1 −p)b2
2
2 + p>−p|b1| − (1 −p)|b2|
Result 10. For any b1and b2where Type 1 is uninformative and Type 2 moderate,
there exists a p∗where communication is optimal for all p∗< p < 1.
Proof. Similarly to Subcase 4.2, Type 1 is uninformative if b1>1
4, implying that
for sufficiently high pthe payoff from delegation is lower than −1
4.¤
27
4.5 Subcase 5 — b1uninformative, b2extreme
For all b1and b2where b1is uninformative and b2is extreme, p|b1|+ (1 −p)|b2|>1
4.
From Proposition 1, communication is always optimal in this case.
Result 11. Communication is always optimal whenever one type is uninformative
and the other extreme.
4.6 Summary
The previous subsections show that communication can be optimal in every com-
municative subcase. Moreover, when there is uncertainty about the sender’s bias,
communication can be optimal even when both types are informative, and when
both types biases are less than 1
4.Figure 7 presents values of b1and b2under which
communication dominates delegation, for p= 0.5.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
b2
b1
3
4
5
6 7
2
1
Figure 7: Regions where communication dominates delegation
The shaded regions represent the regions where the receiver’s expected utility
is greater under communication. Comparing Figure 7 to Figure 6, it is clear that
28
communication is optimal for a significant proportion of the values of b1and b2that
support communication.
The following section provides an in-depth discussion of the results of the paper.
5 Discussion
This paper has introduced uncertainty about the sender’s bias to the Crawford and
Sobel (1982) model. There are two key findings.
The first result is that communication is possible with a larger range of biases.
Even when there is a single sender who may be of multiple types, there is no longer a
unique upper bound on the level of bias required for communicative signalling. Indeed,
sender types can be informative when their bias is greater than 1
4. The second result
is that the receiver’s welfare under communication can now be greater than under
delegation.
The reason for these results is how uncertainty about the sender’s bias affects the
incentives for strategic behaviour.
First, uncertainty over the bias can mute the strategic effects of communication.
As the receiver chooses actions based on the reporting strategies of all types, sender
types with opposing biases allow the receiver to take more moderate actions. In
effect, the opposing biases allow the receiver to take a higher Low action and a lower
High action. In response, this shifts both types’ indifference points towards 1
2, which
expands the set of biases that supports communication and secondly improves welfare.
The second effect of communication under uncertainty is that it allows for infor-
mative signalling from informative types, while minimising the receiver’s welfare loss
from uninformative types. By retaining control over the action, an acutely biased
type is prevented from taking actions that are not in the receiver’s interest. At the
same time, the receiver can still maintain incentives for relatively unbiased sender
types to be informative.
When the proportion of uninformative types is high, the receiver’s loss of utility
from communication will be small compared to his loss of utility from relinquishing
29
decision rights. Communication can be optimal when the receiver believes that a
sender’s bias might be high.
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