Non-Fully Strategic Information Transmission
Marco Ottaviani
London Business School
Economics Subject Areay
Francesco Squintani
University College London
Department of Economicsz
First Preliminary Version: September 2002
This Version: July 2005
Abstract
Building on Crawford and Sobel’s (1982) general communication model, this paper introduces
the possibility that players are non-strategic. The sender may be honest and have a preference for
speaking truthfully. The receiver may be naive and erroneously believe that the sender is truthful.
In contrast to the predictions of the fully-strategic model, we show that there is an equilibrium where
communication is in‡ated, information is not lost because of strategic reasons, and the equilibrium
outcome is biased. Our …ndings are relevant to understanding communication by …nancial analysts.
JEL Classi…cation: C72 (Noncooperative Games), D82 (Asymmetric and Private Information), D83 (Search, Learning, Information and Knowledge).
We thank two referees, the Associate Editor, Sandeep Baliga, Marco Battaglini, Thomas de Garidel-Thoron, Harrison
Hong, John Morgan, Wojciech Olszewski, Werner Ploberger, Daniel Seidmann, Peter Norman Sørensen, and especially
Joel Sobel for very useful conversations.
y
Regent’s Park, London NW1 4SA, UK.
z
Gower Street, London WC1E 6BT, UK.
1
1
Introduction
Con‡icts of interests plague communication across economic and political spheres. For example,
adverts try to convince the public to purchase goods, politicians try to impose their partisan agenda
on the Parliament, and academic advisers help their students to get good jobs. In these contexts,
language in‡ation and deception are often observed, which is di¢ cult to reconcile with fully strategic
equilibrium behavior. In this paper, we show how language in‡ation and deception arise naturally in
the presence of heterogeneous strategic sophistication.
For the sake of concreteness, we introduce our analysis in the context of …nancial forecasts. Fed by
accounts in the press, a growing body of evidence has shown that …nancial analysts’recommendations
tend to be overoptimistic. This bias is imputed mostly to the presence of con‡icts of interest between
analysts and investors.1 Regulators have shown their concern about protecting investors through
the institution of so-called “Chinese walls,” aimed at separating the research from the investment
banking divisions of …nancial …rms. Empirical evidence also points to systematic di¤erences in the
investors’responses to the same recommendation, that can be attributed to heterogeneous strategic
sophistication.2
It is di¢ cult to reconcile these facts with the theoretical predictions of Crawford and Sobel’s (1982)
benchmark model of strategic information transmission. In that model, a privately informed sender
(the analyst) recommends an action to a receiver (the investor) with misaligned preferences. Because
the receiver is aware of the con‡ict of interest, she is able to undo any bias in the sender’s message,
and she takes equilibrium actions that are unbiased conditional on the information transmitted. The
only e¤ect of the sender’s biased incentives is a reduction in the amount of information transmitted in
equilibrium. In the context of …nancial advice, this implies that no investor should be deceived, and
that …nancial …rms would try to credibly commit to minimizing the bias in their …nancial analysts’
incentives. For example, …rms may choose to sever any link between the research and the investment
banking divisions, spontaneously adopting the “Chinese walls” that in fact have been imposed by
regulators.
This paper introduces the possibility that players have heterogeneous strategic sophistication into
1
These con‡icts are possibly due to the analysts’ incentives to generate more investment banking business, increase
the brokerage commissions for the trading arms of their …nancial …rms, or obtain more accurate information from the
management of the …rms they cover. On these three motives see respectively Michaely and Womack (1999), Konrad and
Greising (1989), and Lim (2001).
2
See Malmeinder and Shanthikumar (2003). Heterogenenous strategic sophistication in responses to the same message
has also been found in controlled laboratory experiments by Dickhaut, McCabe and Mukherjee (1995) and Cai and Wang
(2003).
1
Crawford and Sobel’s (1982) general communication model. We …rst introduce the possibility that
the receiver may be naively unaware of the sender’s incentives and erroneously believe that the sender
always reports truthfully.3 Following Kartik (2005), we then extend our results to the case where the
sender is actually honest and su¤ers strictly increasing and convex costs for lying, whereas the receiver
is fully strategic.4 Finally, we consider the possibility that a sophisticated sender communicates to a
large pool of receivers, partly naive and partly sophisticated, and cares about the average response
to her recommendation. We assume that the state space is unbounded, and show that there is an
equilibrium such that the communication function is fully invertible.5 This is unlike in the fullystrategic communication model by Crawford and Sobel (1982), where all equilibria are partitional.
This equilibrium is calculated by hypothesizing that for any state of the world, the sender reports
a message that reveals the state, according to an in‡ated invertible message function, and by checking
that she has no incentive to deviate. Consider the case where the sender is fully strategic and the
receiver may be naive with positive probability. If sophisticated, the receiver correctly de-biases the
sender’s message, and determines the actual state of the world. Hence, the sender has an incentive
to add more bias to her already in‡ated report. As long as the equilibrium message is already above
the sender’s bliss point, however, this further bias will damage the sender in the case that the receiver
is naive and blindly follows the sender’s recommendation. The equilibrium function equalizes the
marginal bene…t induced by the strategic type of receiver with the marginal cost induced by the
naive type. In sum, naive receivers turn the fully-strategic cheap talk game into a game with costly
signalling and a continuum of types. Because the action of the naive receiver follows the sender’s
recommendation, the sender’s payo¤ is directly in‡uenced by her message regardless of the equilibrium
strategies, as is the case in games with costly signalling and unlike in fully-strategic cheap talk games.6
3
Unlike in fully-strategic communication games, in our framework messages have a natural meaning. The message
space of the model by Crawford and Sobel (1982) may be in principle completely unrelated with the state space. It is
only in equilibrium, that the players assign a meaning to the messages used by the sender. By introducing naive receivers
in the analysis, we implicitly assume that messages have an exogenous natural meaning in the game. We suppose that
the state space and the message space coincide, and that a naive receiver believes that a message m means that the state
x coincides with m; regardless of the equilibrium communication strategy. The concept of natural meaning has been
introduced in a di¤erent context by Farrell (1993).
4
Formally, our results in this paper subsume our previous results on naive receivers (Ottaviani and Squintani, 2002)
and Kartik’s Proposition B.1, which covers the unbounded state space case. We discuss the relationship between his
work and ours in detail later in the paper.
5
In a previous extended version of this paper, we show how to extend our results to the cases where the the state space
is bounded and where the sender’s strategic sophistication is heterogeneous. We discuss these results in the conclusion.
6
The equilibrium construction is analogous when the sender su¤ers direct costs for lying. The receiver correctly debiases the sender’s message, hence the sender has an incentive to add more bias to her messages. But the sender’s costs
for lying increase as the lies increase. The equilibrium function equalizes the marginal bene…t induced by manipulating
the receiver with the marginal cost of lying. Evidently, lying costs immediately turn the fully-strategic cheap talk game
into a game with costly signalling.
2
Our model provides a simple explanation for why analysts are given biased incentives. Suppose
that the sender (i.e. analyst) is employed by a principal (investment bank) who would like the receiver
(investor) to take biased actions. As is customary, the sender’s payo¤ may be represented by a “hillshaped”loss function, because analysts trade o¤ investors’performance with the principals’interests.
Suppose that the principal’s utility function is strictly increasing (and concave) in the receiver’s action.
In the fully-strategic model, the equilibrium action of the receiver is unbiased regardless of the bias
level. Biased incentives only make the …nal outcome noisier and reduce the principal’s expected utility.
It is optimal for the principal to credibly commit to minimize the sender’s bias. In our non-fully
strategic model instead, the average receiver’s action is biased. Hence the principal …nds it optimal to
give the sender biased instructions.
Our work is most closely related to the following papers. Kartik (2004) studies the general bounded
state space Crawford and Sobel (1982) model when the sender may send both costly reports and cheaptalk messages. He fully characterizes equilibria subject to a forward-induction re…nement. When costs
become small, he shows that only the most informative, Pareto dominant, Crawford and Sobel (1982)
equilibrium survives.7 For the unbounded state space case without cheap talk messages, he shows
that a fully-separating equilibrium exists when the cost of lying in the report is convex. While not
globally single crossing, the sender’s utility is the convex combination of two concave single-crossing
loss functions: the lying costs and the utility component that depends on the receiver’s action. Hence
a separating equilibrium can be found extending the …rst-order approach by Mailath (1987). Our
results in this paper subsume this result, together with our previous, logically unrelated, result that
a fully-separating equilibrium exists when the sender is sophisticated, but the receiver is naive with
positive probability.
Chen (2004) studies a bounded-support quadratic-loss preferences communication model with small
fractions of honest senders who are always truthful and naive receivers who always blindly believe the
sender. As we noted, this turns the cheap talk game in a continuous types costly signalling game:
Following Manelli (1996), she proves the existence and uniqueness of monotonic equilibrium in the
cheap-talk extension of the game, where the players also communicate with an additional message
about which they are fully strategic. She shows that as the fraction of naive players become small,
the equilibrium converges to the most informative Crawford and Sobel (1982) equilibrium.8 Blanes
7
Blume Kim and Sobel (1993) develop evolutionary based re…nements for …nite-state signalling games, where the
sender can communicate at a cost smaller than the smallest payo¤ di¤erence. For games of partial common interest
(such as a …nite-state version of the Crawford and Sobel (1982) model), they …nd that the babbling equilibrium cannot
be evolutionary stable.
8
In a previous extended version of this paper, we study a bounded-support quadratic loss preferences model with small
3
(2003) studies communication from a possibly truthful sender to a fully strategic receiver. He assumes
a normally distributed prior. Thus, as the message becomes large, the receiver’s belief that the
sender is truthful drops very fast. This prevents full revelation: unlike in our case, the equilibrium
communication function is invertible only below a certain threshold.9
More distantly related, Morgan and Stocken (2003) study a bounded-state space communication
model where players are fully strategic, but there are two types of senders: upward biased and unbiased.
They …nd a semi-responsive equilibrium that is fully-revealing for low states and partitional for high
states. Intuitively, the biased sender has no interest in mimicking the unbiased one, when the latter
reveals that the state is low. She prefers to pool with biased and unbiased senders signaling that
the state is high. But unlike in our model, there is no fully-revealing equilibrium in the unbounded
state space version of their model. For any state x; there is a smaller state x0 such that the biased
sender would like to pretend that the state is x when, in fact, it is x0 :10 Olszewski (2004) studies a
communication model in which players are fully-strategic, but the sender is motivated by both the
receiver’s choice and the receiver’s belief about her own honesty. When the latter component dominates
the former, he shows that truthful communication is the unique equilibrium if the receiver observes
a signal informative about the sender’s signal. Instead, we show that a fully-revealing equilibrium
exists, even if the sender’s utility is not directly in‡uenced by the receiver’s belief, as long as the
sender believes that the receiver believes that she is truthful.
2
2.1
Analysis
Fully Strategic Communication
Our fully-strategic communication model extends Crawford and Sobel (1982) to the case of unbounded
state space. After being privately informed of the state of the world x 2 R
X; with full support
cumulative distribution function F; with density f 2 C 2 ; a sender (S) sends a message m 2 R
to a receiver (R): Upon receiving the message, R takes a payo¤-relevant action y 2 R
M
Y: The von
fractions of naive receivers. We consider the “opposite polar case” of holding the fraction of naive receivers constant,
while taking the size of state space to in…nity. We show that the resulting equilibrium mimics our unbounded state space
fully revealing equilibrium, on all the state space but a set whose relative measure vanishes.
9
While our paper focuses on the e¤ect of naivete on communication of information, Crawford (2003) introduces naive
players in the study of communication of actions planned (intentions) in a subsequent asymmetric matching-pennies
game.
10
Sobel (1985) studies a dynamic communication model where the players’ preferences may be either identical or
opposed. Credibility is built in equilibrium by consistently reporting truthfully, but leads to the opportunity for strategic
misrepresentation. Benabou and Laroque (1992) study a dynamic model where the sender may be honest and truthfully
reveal her signal with positive probability. Consequently, dishonest senders are able to manipulate receivers through
misleading announcements.
4
Neuman-Morgenstern utilities of the players are U S (y; x; b) 2 C 2 ; and U R (y; x) 2 C 2 ; where the bias
i < 0
b 2 R is common knowledge among the players. For each i = S; R; player i’s utility satis…es U11
i > 0; and has a unique peak at y i such that U R (y R ; x) = 0
as well as the single-crossing condition U12
1
and U1S (y S ; x; b) = 0. We shall refer to the sender’s and receiver’s optimal action as a function of the
state x and the bias b as y S (x; b) and y R (x) :
A message strategy is a family ( ( jx))x2R ; where for each x; ( jx) is a c.d.f. on the message space.
Given the message strategy, the beliefs are a family ( ( jm))m2R where for each m;
( jm) is a c.d.f.
on the state space. The equilibrium concept is perfect Bayesian equilibrium. In equilibrium, beliefs
are derived by means of Bayes’ rule whenever possible. An action strategy is a function s : m 7! y
R < 0; the receiver does not ever play a mixed strategy). When ( jx) is degenerate for all x;
(since U11
we represent ( ( jx))x2R by means of a function
: x 7! m:
A key result of the fully-strategic model is Lemma 1 of Crawford and Sobel (1982). It states that
if the players’ bliss points di¤er, i.e. y S (x; b) 6= y R (x) for all x; then there must be an " > 0 such
that ju
vj
"; for any pair of actions u; v played in equilibrium. The equilibrium outcome is a
step function, and hence any equilibrium is partitional. When the state space is unbounded, the same
result requires a slightly stronger condition, and is stated as follows. If there is a uniform
that y S (x; b)
y R (x)
for all x; then there is " > 0 such that ju
vj
> 0 such
"; for any pair of actions
u; v played in equilibrium.11 We shall henceforth assume the sender is biased upwards: there is an
> 0 such that for any x; y S (x; b)
2.2
y R (x)
S > 0:
, and U13
Non-Fully Strategic Communication
We modify the sender’s utility with respect to the fully-strategic case, so as to follow this speci…cation:
(1
The parameter
) U S (s (m) ; x; b) + G (g(m); x; b) ;
2 (0; 1) denotes the intensity of the departure from the fully-strategic model. The
function s (m) is the equilibrium strategy of a sophisticated receiver. The functions g (m) and G (m)
are exogenous to the equilibrium analysis. We make the following assumptions.
Condition 1 The function G 2 C 2 . For any x; there is y …nite such that G1 (y; x; b) = 0: Furthermore,
G12 > 0; G13
0; and G11 < 0:
Condition 2 The function g 2 C 2 and g 0 > 0:
11
The proof of this result is omitted as it is an immediate extension of Lemma 1 in Crawford and Sobel (1989).
5
Condition 1 requires that G is a single-peaked strictly concave loss function that satis…es a single
crossing condition. Note that this means that G belongs to the same family of functions as U S (m; x; b) :
The assumptions that g 0 > 0; G11 < 0 and G12 > 0 imply that there is a unique y = y (x; b) such
that G1 (y; x; b) = 0 and that y1 (x; b) > 0: Condition 2 requires that the sender’s utility component
G depends on m through g monotonically. The functions g and G allow for the following di¤erent
interpretations.
1. (Naive Receivers) Suppose that the receiver is strategic with probability 1
probability
and naive with
:12 A fully naive receiver blindly believes in the sender’s message m and matches
her action y with y R (m): A partially naive receiver does not necessarily blindly believe the
sender. Upon receiving any message m; she cautiously formulates an estimate
true state such that
(m) < m; where
(m) of the
is a strictly increasing, continuously di¤erentiable,
and unbounded function.13 In equilibrium, the strategic receiver knows the sender’s message
strategy , her strategy s is a best response to : The sender does not know if her opponent is
naive or strategic, hence her equilibrium utility for choosing message m is:
) U S (s (m) ; x; b) + U S y R ( (m)); x; b :
(1
This formulation is covered by our construction, by letting G (y; x; b) = U S (y; x; b) and g (m) =
(y R (m)); because g 0 =
0
0
S = G
S
S
y R > 0; U12
12 > 0; U13 = G13 > 0; U11 = G11 < 0; and
U S (y; x; b) = G1 (y; x; b) = 0 for y = y S (x; b) : If the receiver is fully naive,
(m) = m for all m:
2. (Honest Sender) Suppose that the sender is partially honest, and su¤ers a cost for lying. Following Kartik (2005), we let the lying costs be kC (m; x) ; where C is a strictly concave loss
function, and the parameter k denotes the intensity of the lying cost. Speaking is costless only
when one speaks the truth: C (x; x) = 0: Letting k = = (1
) G (g (m) ; x; b) = C (m; x) ; with
g (m) = m for all m; and G1 (x; x; b) = 0; we recover Kartik’s utility speci…cation
U S (s (m) ; x; b) + kC (m; x; b) ;
because G11 = C11 < 0 and G12 = C12 > 0:
12
It is not necessary for our results that the receiver is truly naive. It is enough that the sender believes (maybe
wrongly) that the receiver could be naive.
13
Monotonicity is a natural assumption: a naive receiver believes the state is higher, the higher the message she
receives. Smoothness is only a technical assumption. If were to be bounded, there would be a state of the world x
such that a naive receiver would never believe that the state x is larger than x regardless of the message received. This
is equivalent to assuming a bounded state space, as considered in a previous version of this paper.
6
The main …nding of this section is Proposition 1. For any
> 0; there exists an equilibrium in
which the sender fully reveals the state to strategic receivers. This occurs independently of the prior
signal distribution.
Informally the result is explained by hypothesizing that for any state of the world x; in equilibrium
the sender reports the message m that reveals the state x according to an invertible message function
; and by checking that she has no incentive to deviate. To …x the intuition suppose that G (y; x; b) =
U S (y; x; b) and g (m) = y R (m): the sender is fully strategic and the receiver may be fully naive with
probability : Since the sophisticated receiver correctly de-biases the sender’s message, and determines
the state x =
1 (m),
the sender has an incentive to add more bias to the report
(x). But if she does
so, the naive receiver will believe this and end up damaging the sender, as long as y R ( (x)) is already
above the sender’s bliss point y S (x; b): For any
the …nal outcome, it is possible to …nd
> 0; since the sender’s utility is strictly concave in
(x) large enough so that the rate at which the sender’s utility
drops because of the naive receiver’s response is fast enough to make up for the gain achieved through
the response of the sophisticated sender.14 This marginal condition de…nes an ordinary di¤erential
equation in
: It is left to show that the ordinary di¤erential equation has a solution
that is in
S < 0 (concavity) and that U S > 0 (singlefact invertible. This follows from the assumptions that U11
12
crossing property), and concludes existence of fully-revealing equilibrium.15
Proposition 1 also shows that an increment in the bias makes the …nal outcome more biased; hence
it is optimal for a biased third party to assign biased incentives to the sender. Furthermore, equilibrium
communication becomes more in‡ated as the fraction of naive receivers (the intensity cost for lying)
decreases. Analogously, the sender is forced to add more bias to her communication strategy
the naive receiver’s estimate
when
(m) becomes more conservative with respect to the received message
m:
Proposition 1 Suppose that
communication strategy
> 0: Under Conditions (1) and (2), there exist equilibria where the
is a di¤ erentiable strictly-increasing function, and hence s
for any x: In any such equilibrium,
pointwise; if G1 (g (x) ; x; b)
0; then
(x) > x and
decreases pointwise in
(x) = y R (x)
and when g increases
increases pointwise in b:
14
While apparently this requires that U1S be unbounded below when
is small. Direct inspection of the proof of
Proposition 1 shows that this assumption is unnecessary. If U1S is bounded below, then the derivative of the equilibrium
communication strategy 0 increases as decreases.
15
In fact, the proof of Proposition 1 shows that there is a continuum of fully separating equilibria. This is because, in
our unbounded state space model, the equilibrium strategy is not pinned down by any initial condition.
7
To substantiate our general result, we present a simple explicit example where the sender is fully
strategic and the receiver may be fully naive with probability :
Example 1 In this example, the receiver and sender have quadratic utilities with bliss points respectively x and x + b; formally U R (y; x) =
x)2 and U S (y; x; b) =
(y
in equilibrium, the sender adopts a di¤erentiable invertible function
(x + b))2 : Suppose that
(y
as her communication strategy.
1 (m)
When a message m is sent, the strategic receiver correctly infers the state
and the naive
receiver plays the action m regardless of strategic considerations. Hence the sender will not deviate
from the strategy
only if for any x;
(x) 2 arg max
m
(1
)
(m)
x
1
(m)
(x + b)
2
(m
(x + b))2 :
The …rst order condition, is
2 (1
1 (m)
By substituting
)
1
with x and m with
2 (1
) (x
x
1
b
(m)
0
2 (m
x
b) = 0:
(x) we obtain the di¤erential equation
b)
2 ( (x)
x
b)
0
(x)
= 0:
Among the possible solutions, this ODE has a linear strictly-increasing solution
(x) = x +
b
:
This strategy may be interpreted as revealing the actual state of the world and in‡ating the communication by an amount b= : The factor by which communication is in‡ated is inversely proportional
to the fraction of naive receivers in the population.
We conclude this section by looking at the case for small
(i.e. the receiver is naive with small
probability, or the sender’s lying costs are small). While this is not likely to be the case for our
motivating scenario of …nancial advice, this question is of some theoretical interest. We show in the
Appendix that for
U1 is bounded
0 (x)
! 0; message in‡ation explodes: when U1 is unbounded
(x) ! +1; and when
! +1: In order to communicate the state of the world, in equilibrium the sender
will use increasingly in‡ated messages. Under a simple condition satis…ed for example when G is a
power loss function of the kind G (y; x; b) =
jy
(x
to negative in…nity.
8
b)jq with q > 0, the sender’s utility diverges
2.3
Communication to a pool of receivers
In the previous analysis we maintained the basic structure of the model in Crawford and Sobel (1982),
and supposed that the sender makes a recommendation to a single receiver: her utility depends on
this single receiver’s action. In the context of stock recommendations, the analyst often announces her
forecast to the entire market and cares about the aggregate response of the whole market following the
announcement. It is natural to study a variation of the communication model where the sender makes
a recommendation to a pool of receivers of heterogeneous sophistication, and cares about the average
aggregate response of the receivers. We assume that the sender is fully strategic, and that fraction
of receivers is naive whereas a fraction (1
) is sophisticated. Each receiver’s action depends
only on her strategic sophistication, and on the message of the sender. Hence, the sender’s payo¤ in
equilibrium is:
U S ((1
) s (m) + g (m) ; x; b) ;
when sending message m: We assume that the naive receivers’strategy g satis…es condition (2).
We introduce the analysis in the context of the quadratic-loss example.
Example 2 Suppose that the sender adopts a di¤erentiable invertible function
as her communica-
tion strategy. When a message m is sent, each strategic receiver correctly infers the state x =
1 (m)
and each naive receiver plays the action m: If the sender cares about the average response of the
receivers, for any x; she chooses
(x) 2 arg max
m
(1
)
1
(m) + m
(x + b)
2
:
Di¤erentiating the sender’s value, we obtain
2( (
x)
b)((1
The linear strictly-increasing solution of this ODE,
)
1
0
+ ) = 0:
(x) = x + b= ; coincides with the linear solution
of Example 1.
Our main results extend to this variation of the model under a restriction on the players’utilities:
we require that the sender’s bliss point is at least as sensitive to a change in the state as the receiver’s
bliss point.
Proposition 2 Suppose that y1S (x; b)
0
y R (x): Under condition (2), for any
unique equilibrium where the communication strategy
9
> 0, there is a
is a di¤ erentiable strictly-increasing function,
and hence s
(x) = y R (x) for any x: In this equilibrium,
decreases pointwise in
3
(x) > b;
increases pointwise in b; and
and when g increases pointwise.
Conclusion
We have formulated a communication model in which players are possibly non strategic. Our model
allows for the sender to be honest (so that lying is costly), or for the receiver to be naive (and so
either blindly implement the sender’s recommendation or believe that the sender is truthful). We
have shown that in these instances equilibrium communication may be in‡ated and results in ex-ante
biased outcomes. These results may serve to build a simple disequilibrium theory of persuasion, where
the receiver, while strategic and rational, is partially persuaded by the sender and hence unable to
correctly invert the in‡ated communication language.
In a previous, extended version of this paper, we studied the following issues. We …rst extended
the analysis to the case of naive receivers where the state space is bounded, restricting attention for
simplicity to quadratic loss utilities. We found that full revelation on the entire state space is generally
impossible in equilibrium. In order to penalize a sender’s deviation from an equilibrium invertible
message function, it is necessary that the equilibrium message is more biased than the sender’s bliss
point. Typically, this would require the sender to send messages that lie outside the state space, when
the state of the world is close enough to the upper bound of the state space.16 This complication
has no e¤ect on the equilibrium construction when the state of the world is small. We constructed
an equilibrium fully revealing in a low range of the state space and partitional in the top range. We
showed that the relative size of the fully revealing range increases in the fraction of naive receivers
and in the size of the state space, whereas it decreases in the bias level. We argued that this results
are broadly consistent with the experimental evidence by Dickhaut, McCabe, and Mukherjee (1995)
(DMM) and Cai and Wang (2003) in Crawford and Sobel’s games with …nite number of states.17
We then considered existence of partitional equilibria for the case of naive receivers, both when the
16
When the state space is bounded, our construction embeds the naive receivers with some degree of strategic sophistication. Say that the state space is [0; U ] : In principle, the sender may send a message m larger than U: We implicitly
assume that the receiver maintains her prior belief that the state x belongs to [0; U ] : If a naive receiver were to believe
that the state x coincided with the message m even when m is larger than U; then a fully revealing equilibrium would
exist even though the state space is bounded.
17
The addition of non-strategic behavior can explain the otherwise puzzling …nding that communication is more
informative than is predicted by the fully-rational model (see, DMM table 3 and CW Section 4). While DMM do not
report the messages used by their subjects, CW’s data display overwhelming language in‡ation (e.g., the highest message
(9) is sent 57% of the time when the bias is large), consistently with our results. As in our equilibrium, the average action
is monotonically increasing in the message and messages bunched at the top of the message set are less informative than
low messages (see CW Table 9).
10
state space is bounded and when it is unbounded. We found that, while not all partitional equilibria
survive the introduction of naive receivers, all partitional equilibrium outcomes are robust when the
fraction of naive receivers is small. Informally, we constructed an equilibrium by requiring that, when
the state belongs to an interval (ai
1 ; ai )
of the partition, the sender sends the message mi that
corresponds to the strategic receiver’s optimal choice (ai
1 ; ai )
given that she only knows that the
state is in the interval. For any other message m in the interval, o¤ the equilibrium path, we assign
beliefs to the sophisticated receiver leading to an action s (m) such that the expected action of the
receiver (which could be naive or sophisticated) equals to (ai
1 ; ai ):
The sender’s deviation is thus
deterred because her utility U S (y; x; b) is strictly concave in y:
Finally, we extended our analysis to the case where the sender’s strategic sophistication is heterogeneous, and private information. We assumed that with some probability the sender may be fully
honest and always report the truth, otherwise the sender is fully strategic and bears no lying costs. For
simplicity, we restricted attention to the unbounded state space case, and assumed the receiver fully
strategic. Unlike the case where sender is not fully strategic, but her sophistication is common knowledge, here existence of a fully-revealing equilibrium depends on the prior state distribution. Under
a simple regularity condition on the players’utilities, we showed that this model has an equilibrium
where the strategic sender plays an invertible communication strategy, provided that the tails of the
prior state density f do not drop too fast. When this condition is violated, the receiver will put little
probability on the sender being honest when receiving a large message. Because she acts as if facing
the strategic sender, the logic of Crawford and Sobel (1982) prevents full revelation in equilibrium.
Appendix: Proofs
Proof of Proposition 1. Given the strategic receiver’s equilibrium choice s; for any x 2 R; in
equilibrium, the sender must choose
(x) 2 arg max (1
m2R
) U S (s (m) ; x; b) + G (g(m); x; b) :
Suppose that s is di¤erentiable, the …rst order condition for the sender’s program is
(1
) U1S (s (m) ; x; b) s0 (m) + G1 (g (m) ; x; b) g 0 (m) = 0:
Suppose that s0 > 0; and that s ( (x)) = y R (x) for any x: This implies that s (m) = y R (
any m: Hence we can rewrite condition (1) as
(1
0
) U1S y R (x) ; x; b y R (x) + G1 (g( ); x; b) g 0 ( )
This expression is an ordinary di¤erential equation in
0 (x) > 0 and
is well de…ned on the whole range R.
11
0
= 0:
: We want to …nd a solution
(1)
1 (m))
for
(2)
such that
0
Because, for any x; 0 < (1
) U1S y R (x) ; x; b y R (x) < +1 and for any m; 0 < g 0 (m) < +1;
by condition (2), the ODE (2) admits a local solution for any pair of states and messages (x0 ; m0 )
such that
G1 (g(m0 ); x0 ; b) 6= 0:
For any x; the solution
(x) is strictly increasing (decreasing) whenever
G1 (g( (x)); x; b) < (>)0;
and furthermore
0
(x) /
[G1 (g( (x)); x; b)]
1
:
For any x; the function G1 (g(m); x; b) is continuous and strictly decreasing in m; because g 0 (m) ; by
condition (2), and G11 < 0; by condition (1). For any x; let the m (x; b) be the message m such that
G1 (g(m); x; b) = 0; note that such message is unique because g 0 > 0 and that m1 (x; b) > 0 because
g 0 > 0; G11 < 0 and G12 > 0: We pick (x0 ; m0 ) such that G1 (g(m0 ); x0 ; b) < 0; and hence m0 >
m (x0 ; b) ; because G1 (g(m0 ); x0 ; b) 6= 0; there is a local solution of the ODE (2) such that (x0 ) =
m0 ; because G1 (g(m0 ); x0 ; b) < 0; 0 (x0 ) > 0: Because m (x; b) is strictly increasing in x; 0 (x) /
[G1 (g( (x)); x; b)] 1 ; and for any x; G1 (g(m(x; b)); x; b) = 0; limm!m(x;b)+ G1 (g(m); x; b) = 0 and
limm!m(x;b) G1 (g(m); x; b) = 0+ ; it follows that cannot ever cross the boundary m (x; b) : Hence the
solution is extended on the range ( 1; x0 ) and 0 (x) > 0 for all x 2 ( 1; x0 ) : The extension of on
the range (x0 ; +1) only requires that does not reach a vertical asymptote (limx0 !x (x0 ) = +1) for
some x > x0 : This is ruled out because limx0 !x 0 (x0 ) ! +1 only if limx0 !x (x0 ) = m (x; b) < +1:
Let us identify the sender’s strategy with ; and let the strategic receiver adopt the strategy
1 as hypothesized. To show that for any x; the sender’s equilibrium message is indeed (x)
s = yR
we proceed as in Mailath (1987). Suppose by contradiction that
(x) 2
= arg max (1
m2R
) U S (s (m) ; x; b) + G (g(m); x; b) ;
because (1
) U S (s (m) ; x; b) + G (g(m); x; b) is C 2 in m; it must be that the …rst-order condition
(1) holds at the maximum m : Letting x be the state such that (x ) = m ; this means that
U1S y R (x ) ; x ; b > 0; that G1 (g(m ); x ; b) < 0 and that
(1
0
) U1S y R (x ) ; x ; b y R (x )
=
G1 (g(m ); x ; b) g 0 (m )
0
(x ) =
(1
0
) U1S y R (x ) ; x; b y R (x )
G1 (g(m ); x; b) g 0 (m )
But this equality cannot hold because
!
U1S y R (x ) ; x; b
@
S
/ U12
y R (x ) ; x; b G1 (g(m ); x; b) G12 (g(m ); x; b) U1S y R (x ) ; x; b < 0
@x
G1 (g(m ); x; b)
for U1S y R (x ) ; x; b < 0 and G1 (g(m ); x; b) > 0; and because there cannot be any x such that
U1S y R (x ) ; x; b > 0 and G1 (g(m ); x; b) < 0; as U12 > 0 and G12 > 0:
To prove the comparative statics and characterization results, …rst note that in equilibrium
0
U1 y R (x) ; x; b > 0 and G1 (g( (x)); x; b) < 0: Because y R > 0; G11 < 0 and g 0 > 0 direct inspection of the ODE (2) then implies that decreases pointwise in and if g increases pointwise;
12
S > 0 and G
that decreases pointwise in b because U13
13
G1 (g(x); x; b) 0:
0; and …nally that
(x) > x whenever
Proposition A.1 For any > 0; let
be the set of sender’s equilibrium strategies of game
such
R
0
that s
(x) = y (x) for any x: If G1 is bounded below, then
(x) ! 1; for any x: Suppose that
G1 is unbounded below, then
(x) ! 1; and the equilibrium sender’s utility
(1
) U S y R (x); x; b + G1 (g(
(x)); x; b)
diverges to negative in…nity whenever
G1 (y; x; b)
= 0;
y!+1 G (y; x; b)
(3)
lim
as is the case for G (y; x; b) =
jy
b)jq ; and any q > 0:
(x
Proof. We have concluded in the proof of Proposition 1 that GS1 (g( (x)); x; b) < 0: As ! 0;
equation (2) is thus satis…ed either if 0 ! +1 or if G1 (g( (x)); x; b) ! 1: If G1 is bounded below,
then 0 ! +1: If G1 is unbounded below, since G12 > 0; it follows that (x) ! +1: Equation (2) requires that lim !0 G1 (g( (x)); x; b) < 0: Condition (3) then requires that lim !0 G (g( (x)); x; b) =
1; which concludes the result. Condition (3) is satis…ed by power loss utility functions, because
lim
y!1
G1 (y; x; b)
= lim
y!1 y
G (y; x; b)
q
(x
b)
= 0:
Proof of Proposition 2. Given the strategic receiver’s equilibrium choice s; for any x 2 R; in
equilibrium, the sender must choose
(x) 2 arg max U S ((1
) s (m) + g (m) ; x; b) :
m2R
Suppose that s is di¤erentiable, the …rst order condition for the sender’s program is
U1S ((1
) s (m) + g (m) ; x; b) (1
) s0 (m) + g 0 (m) = 0:
Suppose that s0 > 0; and that s ( (x)) = y R (x) for any x: This implies that s (m) = y R (
any m: Hence we can rewrite condition (4) as
U1S (1
) y R (x) + g ( ) ; x; b
(1
0
) y R (x) + g 0 ( )
0
0
Because y R > 0; g 0 > 0 and we want to select a strictly increasing solution
always possible. The only possibly admissible solution is
U1S (1
) y R (x) + g ( ) ; x; b = 0;
i.e. (1
) y R (x) + g ( ) = y S (x; b) :
Hence
(x) = g
1
y S (x; b)
13
y R (x)
+ y R (x) ;
= 0:
(4)
1 (m))
for
(5)
; the second term is
(6)
this solution is admissible: because g is onto,
any x;
0
is well de…ned on the whole range R; furthermore, for
1
(x) =
g0
y S (x;b) y R (x)
+ y R (x)
y1S (x;b) y R 0 (x)
+ y R 0 (x)
> 0;
0
because y R (x)
y1S (x; b) and g 0 (m) > 0 by assumption (2). The result is thus proved by letting
= ; and s (m) = y R ( 1 (m)) for any m: Then the strict second order condition
S
0 > U11
((1
+U1S ((1
) s (m) + g (m) ; x; b) (1
) s (m) + g (m) ; x; b) (1
) s0 (m) + g 0 (m)
2
) s00 (m) + g 00 (m)
S < 0: This concludes that (x) maximizes
is locally satis…ed because of condition (6) and because U11
U S ((1
) s (m) + g (m) ; x; b) ; because this function is C 2 in m and, for any x; the message m =
(x) is the unique message satisfying the …rst-order condition (4).
The results that (x) > b; increases pointwise in b; and decreases pointwise in
increases pointwise follow from direct inspection of Condition (6).
and when g
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