GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 255

Can More Information Facilitate Communication?

Junichiro Ishida and Takashi Shimizu

April 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Can More Information Facilitate Communication? ∗Junichiro Ishida†and Takashi Shimizu‡
April 1, 2012

Abstract
In this paper we analyze a cheap talk model with a partially informed receiver. In clear
contrast to the previous literature, we find that there is a case where the receiver’s priorknowledge enhances the amount of information conveyed via cheap talk. The point of departure is our explicit focus on the “dual role” of the sender’s message in this context: when thereceiver has imperfect private information of her own, the sender’s message provides information about the true state as well as about the reliability of the receiver’s private information.
This feature gives rise to the asymmetric response of the receiver’s action, where the receiverreacts less to the truthful message and more to the misrepresented one, which is essential in
disciplining the sender to be more truthful.
Keywords: Cheap talk, Informed receiver.
JEL Classification Number: D23; D82.

∗

This research is financially supported by Grant-in-Aid for Scientific Research from JSPS and MEXT, Kansai
University’s Overseas Research Program for the year of 2009 and the Global COE (GCOE) program of Osaka
University. Of course, any remaining errors are our own.
†
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki-shi, Osaka 567-0047
JAPAN (E-mail: jishida@iser.osaka-u.ac.jp)‡
Faculty of Economics, Kansai University, 3-3-35 Yamate-cho, Suita-shi, Osaka 564-8680 JAPAN (E-mail:tshimizu@kansai-u.ac.jp)

11Introduction

This paper analyzes a cheap talk model with a partially informed receiver. More precisely,
we extend the canonical model of Crawford and Sobel [3] (hereafter, CS) to a setting where
the receiver is also endowed with some private information of her own, on top of the sender’smessage which can be observed subsequently. Within this framework, we ask how the receiver’s
prior knowledge affects the strategic nature of communication and, in particular, the amount of
information conveyed by the sender via cheap talk messages.
Recently, several papers have explored this problem, i.e., how the nature of communication
alters when the receiver becomes more informed in a broad sense (Chen [2], Lai [7], Moreno deBarreda [4]).1 While they differ in their ways to add the receiver’s private information to themodel, all of these studies by and large show that the more informed the receiver is, the lessinformation she can extract from the sender. A similar conclusion is also obtained in models withmultiple senders (Austen-Smith [1], Morgan and Stocken [8], Galeotti et al. [5]).2 For instance,
Morgan and Stocken [8] examine information aggregation in polls and show that truth telling is
impossible when the size of a poll is sufficiently large.
The main logic behind this result is fairly simple, if we carefully dissect why any information
can be conveyed via cheap talk messages. To see this, consider the standard setup of CS where
the state of nature is denoted by t ∈ [0, 1]. The receiver’s bliss point is t whereas the sender’sis t + b, b > 0, meaning that the sender always prefers a larger action than the receiver, andhence has an incentive to exaggerate his message. The key insight of CS is that even in thiscase, the receiver can still extract some information from the sender by dividing the state space
into intervals. These intervals endogenously create the cost of exaggeration because if the senderexaggerates and sends a message in the next interval, the resultant action could move further
to the right (towards one) and away from his bliss point. In other words, what makes thisstrategy work is the sensitivity of the receiver’s action to the sender’s (misrepresented) message.
Since the receiver naturally becomes less sensitive to the sender’s message when she has private
information of her own, the sender’s incentive to exaggerate his information is magnified and, asa consequence, the quality of communication deteriorates.
1

In a model with the discrete state space, Ishida and Shimizu [6] also show that the receiver’s prior information
becomes an impediment to efficient communication.
2
Since the model with an informed receiver does not specify the source of her private information, it inherently
has a close connection with the model with multiple senders.

2So, does this mean that the impact of the receiver’s prior knowledge on the quality of communication is invariably negative? While this conclusion, and the reasoning behind it, appear
fairly robust, it is still hard to believe that this single mechanism is all there is to this problem:
intuition certainly suggests that there are also situations where it is more difficult to lie to a better informed receiver, so that the receiver’s prior knowledgable facilitates, rather than impedes,
communication. If this is the case, i.e., there is a route through which more information facilitates communication, it means that the existing literature overlooks some critical link between
the receiver’s information on one hand and the quality of communication on the other. The aimof this paper is to find this “missing piece” in this class of problems, if any, which hopefully gives
us a clearer insight on the role of information in strategic communication.
The point of departure from the existing literature is our explicit focus on the “dual role” ofthe sender’s message: when the receiver is partially informed, the sender’s message can provide
information not only about the true state but also about the reliability of the receiver’s private
information. The latter aspect, which has been largely neglected in the literature, is the driving
force of our model. To see how this works, note that the receiver knows that her private information is less reliable than the sender’s.3 If the sender’s message is consonant with what she
privately knows, she thinks that her private information is more likely to be correct and, consequently, places more weight on it. As the receiver relies more on her own information, her action
necessarily becomes less sensitive to the sender’s message. If the message is not consonant withher private information, on the other hand, she loses her confidence in her private informationand her action becomes more sensitive to the message. Since the sender’s message is more likely
to be consonant with the receiver’s private information when the sender truthfully reveals his
information, the receiver reacts less to the truthful message and more to the misrepresented one.
As we will clarify in more detail later, this asymmetric response of the receiver is essential in
disciplining the sender to be more truthful and hence facilitating communication between them.
We think that adding private information to the receiver’s side is a natural extension of theexisting literature and poses an intriguing question in itself, as it is not a priori clear whether
the quality of communication improves or deteriorates when the receiver has more precise information. The extension also yields more practical implications when how much information tocollect on her own is the receiver’s endogenous choice. If the quality of communication diminishes as the receiver becomes more informed, then information acquisition and communication
3

As in CS and most cheap talk models, we assume that the sender knows the true state with precision.

3

are substitutes, and the incentive to collect information certainly diminishes when the sender’sinformation is expected to be reliable. In contrast, if the quality of communication enhances
as the receiver becomes more informed, information acquisition and communication are complements, and the incentive to collect information intensifies. Our analysis shows that informationacquisition and communication can be complements, rather than substitutes as the previous
literature indicates, depending on the information structure on the receiver’s side.
The paper is organized as follows. Section 2 outlines the model which is an extension of CS.
Section 3 characterizes equilibria of the model and discusses their implications. Finally, section
4 offers some concluding remarks. All the proofs are relegated to Appendix.

2

The Model

We consider an extended version of CS’s uniform-quadratic model. There are two players, thesender (male) and the receiver (female), and the model goes as follows.
1. Nature randomly draws the state of nature t ∈ [0, 1] from the uniform distribution. The
state is the sender’s private information.
2. The sender sends a message m ∈ [0, 1] to the receiver.
3. The receiver observes a private signal r ∈ [0, 1] which is drawn according to the following
probability:
P (r ∈ A|t) = qI(t ∈ A) + (1 − q)λ(A),where I is the indicator function and λ is Lebesgue measure. In other words, the signal
reflects the true state, i.e., r = t, with probability q while it is randomly drawn from theuniform distribution on [0, 1] with probability 1 − q.
4. Upon observing m and r, the receiver chooses an action a ∈ [0, 1].
The payoff for the receiver is
U R (t, a) = −(t − a)2 ,whereas that for the sender is
U S (t, a) = −(t + b − a)2 .
4We call b the bias and assume b ∈ (0, 0.5).
The only difference from the original CS model is that we allow for the possibility thatthe receiver observes a possibly informative signal of the state of nature. The signal is either
perfectly informative (with probability q) or noisy (with the remaining probability), but thereceiver cannot tell whether any given signal is informative or noisy. What is critical in thisspecification is the way noise is introduced into the receiver’s signal: in the current setting,there is a positive probability that the observed signal is a complete noise containing no useful
information.4 This feature is essential in giving rise to the dual role of the sender’s message as
we detail below. We interpret q as the accuracy of the signal where the model is equivalent to
the original CS model when q = 0.

3

Analysis

3.1

The equilibrium concept

Throughout the analysis, we focus on the class of monotone partition equilibria, which is a subset
of perfect Bayesian equilibria, defined as below.5
Definition 1 Let µt denote the type t sender’s strategy (i.e., a probability distribution over themessage space [0, 1]). A monotone partition strategy (MPS) is the sender’s strategy where there
exists a partition of [0, 1], {Ti }i∈I (I is some index set) such that
• Ti is a non-empty interval for any i ∈ I,
• µt = µt0 for any i ∈ I and any t, t0 ∈ Ti , and• Supp µt ∩ Supp µt0 = ∅ for any distinctive i, i0 ∈ I and any t ∈ Ti , t0 ∈ Ti0 .
A monotone partition equilibrium (MPE) is a perfect Bayesian equilibrium where the sender’sstrategy is an MPS.
4

The assumption that the signal contains no information with some probability is made only for analytical
simplicity and not essential. Our results holds in a qualitative sense as long as there is some positive probability
that the signal is sufficiently weakly correlated with the true state.
5
The need for this focus arises from a special feature of cheap talk models with an informed receiver. Whenthe receiver is endowed with some information of her own, the sender typically induces lotteries over actions, notactions themselves. This feature produces some complicated equilibria that never exist in the original CS model:
for example, Chen [2] shows that there exist a non-monotone equilibrium in a version of cheap talk models withan informed receiver. We rule out this possibility given the question we set out to solve, although it is certainly
intriguing as a theoretical possibility. Note that CS shows that in the case of q = 0, there exists only monotone
partition equilibria.

5

We can show that any MPE consists of finite intervals.
Proposition 1 For any MPE, I is a finite set.
Based on this proposition, we denote any MPE partition by {Tn }n=1,...,N where N is somenatural number and sup Tn = inf Tn+1 for n = 1, . . . , N − 1. We call Tn the nth interval.
Furthermore, we identify an MPS with a partition {Tn }n=1,...,N as a vector t = (t0 , . . . , tN )defined as(0tn =sup Tnfor n = 0,
for n = 1, . . . , N.

d

We call such tn a threshold. Note that 0 = t0 < t1 < · · · < tN = 1. Let τn = tn − tn−1 denote thelength of the nth interval which, as usual, is taken as a measure of the quality of communication:
the shorter each interval is, the more information is conveyed via cheap talk in equilibrium.

3.2

The receiver’s problem

In any MPS, the receiver has two sources of information: her own signal r and the sender’smessage m which indicates in which interval the true state is lying. Given this, the receiver’s
equilibrium strategy is pure and we denote it by α(m, r). Furthermore, the actions induced onany equilibrium path are determined as follows:((1−q)τn tn +tn−1q2
q+(1−q)τn r + q+(1−q)τnα(m, r) = tn +tn−12

if ∃n ∀t ∈ Tn µt (m) > 0 and r ∈ Tn ,if ∃n ∀t ∈ Tn µt (m) > 0 and r ∈/ Tn .

Note that the receiver uses her own information only when the sender’s message falls into thesame interval. On the other hand, the receiver sees her private information as a noise anddisregards it altogether if the message does not agree with her private signal, given that thesender plays the equilibrium (truth-telling) strategy. This is the critical feature of the current
model: when the message is “close” to the receiver’s signal, she places more confidence in her
signal; when it is “further away”, she relies less on it and more on the sender’s message. As
we will see later, this asymmetric response is what disciplines the sender to be more truthful.
The current model provides a setup which captures this dual role of the sender’s message in a
relatively tractable manner.
Remark 1 This argument implies that a subtle difference in the information structure could
result in a large qualitative change in equilibrium outcomes. To elaborate more on the difference
6from the existing literature, consider an alternative information structure in which the receiver
can observe whether the state is low [0, 0.5] or high (0.5, 1], as assumed in Lai [7].6 Now supposethat there exists an equilibrium with two intervals, {[0, t1 ], (t1 , 1]}, and moreover that t ≤ t1 < 0.5
so that the receiver observes a “low” signal. In this environment, if the sender deviates andclaims that the state is in (t1 , 1], the receiver thinks that the state must lie in (t1 , 0.5] andchooses a = 2t14+1 .7 Since the uninformed receiver would think that the state must lie in (t1 , 1]
and choose a = t12+1 , the presence of the receiver’s prior information makes her less sensitive tothe misrepresented message, which magnifies the incentive to exaggerate.

3.3

The equilibrium conditions

Given the receiver’s strategy, we can now identify the conditions for an MPE with a partition t(an MPE with t for short) by checking the sender’s incentives. In particular, what we need tosee is that given some partition t and the true state t, the sender has no incentive to deviate bysending a “nearby” message. To this end, define ∆(t; tn−1 , tn , tn+1 ) as follows:
Z 1
Z 1
2
[t + b − α(mn+1 , r)] P (dr | t) +[t + b − α(mn , r)]2 P (dr | t),∆(t; tn−1 , tn , tn+1 ) = −00where mn is any message sent (with positive probability) when the true state lies in the nthinterval. Then, a necessary condition for the equilibrium with N intervals is that(≤0
for t ∈ (tn−1 , tn )∆(t; tn−1 , tn , tn+1 )≥0
for t ∈ (tn , tn+1 )holds for any n = 1, . . . , N − 1. Moreover, a sufficient condition is that(≤0
for t < tn∆(t; tn−1 , tn , tn+1 )≥0
for t > tn
holds for any n = 1, . . . , N − 1.
In the original CS model, the length of each partition must satisfy certain conditions inequilibrium: in the linear-quadratic specification, each interval must be exactly 4b longer thanthe last. The following result establishes the conditions along this line in our extended setup.
6

Strictly speaking, in Lai [7], the threshold, which is exogenously set at 0.5 in this example, is also only private
known and drawn from the uniform distribution.
7
In contrast, if the sender reveals truthfully, the receiver ignores her own signal and chooses a = t21 . We do
not focus much on this side because there is no qualitative difference in the way the receiver updates her belief.
Even though the receiver ignores her own signal, this is to some extent a figment of the simplified information
structure as assumed here. For instance, if the receiver’s signal space is partitioned into three intervals, very low[0, ε), low [ε, 0.5) and high [0.5, 1], and the true state lies in [ε, 0.5), she uses both pieces of her information and1chooses a = ε+ton the equilibrium path.
2

7

Proposition 2 Define
G+ (τn , τn+1 ; q)d

= (τn+1 + τn )(τn+1 − τn − 4b)3
q 2 (1 − q)τn+1

2q 2 (1 − q)τn3
q 2 τn (τn + 4b),++
q + (1 − q)τn
3 (q + (1 − q)τn+1 )2 3 (q + (1 − q)τn )2
G− (τn , τn+1 ; q)+d

= (τn+1 + τn )(τn+1 − τn − 4b)−3
2q 2 (1 − q)τn+1

q 2 (1 − q)τn3
q 2 τn+1 (τn+1 − 4b).
−−q + (1 − q)τn+1
3 (q + (1 − q)τn+1 )2 3 (q + (1 − q)τn )2(i) A necessary and sufficient condition for an MPE with MPS {Tn }n=1,...,N is G+ (τn , τn+1 ; q) ≥0 and G− (τn , τn+1 ; q) ≤ 0 for n = 1, . . . , N − 1.
(ii) For any τn ∈ (0, 1) and q ∈ [0, 1), there exists τ (τn , q) and τ (τn , q) such that G+ (τn , τn+1 ; q) ≥0 and G− (τn , τn+1 ; q) ≤ 0 if and only if τ (τn , q) ≤ τn+1 ≤ τ (τn , q).
(iii) For any τn ∈ (0, 1) and q ∈ [0, 1), G+ (τn , τ (τn , q); q) = 0 and G− (τn , τ (τn , q); q) = 0.
(iv) For any τn ∈ (0, 1) and q ∈ [0, 1), 2b < τ (τn , q) ≤ τn + 4b ≤ τ (τn , q) and
q = 0 ⇔ τ (τn , q) = τn + 4b = τ (τn , q),
q > 0 ⇔ τ (τn , q) < τn + 4b < τ (τn , q).
The proposition shows that the equilibrium conditions are less stringent when q > 0 in thatthe length of each partition only needs to be in some range. With more breathing room, we
can construct “more informative” equilibria. To see the intuition behind this result, consider an
equilibrium with two intervals, {[0, t1 ], (t1 , 1]}. When q = 0, it is straightforward to computeα(m0 , r) = t1 /2 and α(m1 , r) = (t1 + 1)/2 regardless of r. At t = t1 , the sender must be
indifferent between the two messages, i.e., his bliss point must be at the midpoint of α(m0 , r)and α(m1 , r). Let b0 denote the bias which satisfies!
1 − 2t1
t
t
+
1
1
1
1
⇔ b0 =+.
t1 + b0 =
2 2
2
4

8

It is well known that the second interval must be exactly 4b longer than the first in this linearquadratic specification.
We now let q increase above zero and see how that changes the sender’s incentives. There
are two effects at work, which we call the information effect and the risk effect for expositional
clarity, depending on whether the receiver observes a correct signal or not. These effects areabsent when the receiver is uninformed (q = 0), and mark a key departure from the original CS
model.
First, suppose that the true state is t = t1 , and also that the receiver happens to observe thetrue state, i.e., r = t1 (though she does not know it for sure). If the sender reveals truthfully,i.e., m = m0 , the receiver’s private signal is consonant with the sender’s message, and thereceiver combines the two pieces of evidence to determine her action: the resultant action ishence necessarily gravitated towards the true state t1 away from what the message indicates,i.e., t1 /2. If the sender chose to deviate and send m = m1 , on the other hand, the receiver’s
reaction would totally be different, now that the sender’s message is dissonant with the receiver’s
signal. Under the presumption that the sender plays the equilibrium strategy, the receiver must
think that her signal is a noise and places zero weight in Bayesian updating: the resultant action
hence stays at (t1 +1)/2 regardless of q. We refer to this as the information effect of the receiver’s
prior knowledge, which works to discipline the sender to be more truthful.
Second, suppose that the receiver observes r 6= t, in which case she (mistakenly) uses her
private signal with some positive probability. Given that τ1 > τ0 , r ∈ τ1 is the more dominantcase. In this case, if the sender follows the equilibrium strategy, the receiver now ignores her signal
and deterministically chooses an action regardless of r; if he deviates, the receiver combines thewrong signal with the message and hence produces a stochastic action. The stochastic natureof the latter case introduces irrelevant noise into the receiver’s action, thereby reducing the
expected payoff. We refer to this as the risk effect which also works to discipline the sender tobe more truthful.8
With these two effects and the consequent asymmetric response, t = t1 is no longer on theborder. It is now strictly better for the sender with t = t1 to send m = m0 , meaning that thethreshold can be “pushed further to the right.” Even with the same number of intervals, we
can construct a more informative equilibrium by having more equally divided intervals. We will
8

Obviously, applying the same logic, the risk effect works in a way to make the deviation more attractive when
the receiver observes a wrong signal but that happens to fall into the right interval. In a typical MPE, however,this is less likely to occur and its effect is usually dominated.

9formalize the above discussion in the next subsection.

3.4

Can more information facilitate communication?

We are now ready to address our main question of how a change in q affects the nature ofcommunication. We first establish that an increase in q per se enhances the payoffs of bothplayers. To this end, define V i (t; q), i = S, R, as player i’s ex ante expected payoff. Then, we
can obtain the following result.
Proposition 3∂V i> 0 for i = S, R.
∂qThis result is somewhat straightforward as it simply states that the players benefit fromhaving access to more information. What is more interesting is whether an increase in q canafford a more efficient way of communication, or a more efficient configuration of the equilibrium
partition. To this end, it is convenient to associate each MPE explicitly with the information
accuracy q and denote it by an MPE-q. Given this, we define the following notions.
Definition 2 We say that:(i) A partition t is more efficient than t0 at q if V i (t, q) > V i (t0 , q) for i = S, R.
(ii) More information facilitates communication at (q, q 0 ), q > q 0 , if for any MPE-q 0 with t0 ,there exists an MPE-q with a partition t which is more efficient than t0 at q 0 .
Given this, we can obtain the following result which is derived directly from Proposition 2.
Proposition 4 For any MPE-0 with t and any q > 0, t also constitutes an MPE-q. Furthermore,
more information facilitates communication at (q, 0) for any q > 0.
Remark 2 While we show the above result by comparing among the MPEs with the same
number of intervals, we can also easily construct an example where an increase in q results in
an increase in the maximum number of intervals. For instance, suppose that b = 0.25, in whichcase the unique equilibrium is babbling when q = 0. This conclusion does not hold, however,when q > 0. Since G+ (0, 1) > 0 and G− (0, 1) < 0, for a sufficiently small t1 > 0, (0, t1 , 1) alsosatisfies G+ (τn , τn+1 ) > 0 and G− (τn , τn+1 ) < 0 due to the continuity of G+ and G− . In otherwords, there always exists an MPE-q, q > 0, with two intervals when b = 0.25.

10Proposition 4 establishes that more information facilitates communication at (q, 0) for any
q > 0, but does not show that more information monotonically facilitates communication. The
next question is thus whether this condition holds for any arbitrary pair (q, q 0 ) for q > q 0 > 0. To
do so, we need to identify the most efficient MPE-q (an MPE with the most efficient partition).
If τn < τn+1 holds for n = 1, . . . N − 1, the most efficient equilibrium requires τn = τ (τn , q)
for n = 1, . . . , N − 1 (Lemma 6 in the Appendix). Unfortunately, in our setting, τn < τn+1 isviolated in some MSE. See the following example:
Example 1 When b < 81 and q is sufficiently large, there exists an MPE with two intervals
where τ1 ≥ τ2 . This is verified from the fact that G+ (0.5, 0.5; q) > 0 holds for a sufficiently largeq.
However, it is verified that a partition of this kind never constitutes an MPE whenever q
is sufficiently small (Lemma 7 in the Appendix). Moreover, it is also verified that τ is strictly
decreasing in q whenever q is sufficiently small (Lemma 8 in the Appendix). From these facts, weconclude that more information facilitates communication whenever we focus on the situation
where the accuracy is sufficiently low.
Proposition 5 There exists a q > 0 such that for any q ∈ (0, q] and q 0 < q, more information
facilitates communication at (q, q 0 ).
As a final note, we would like to present some counterexamples to show that the effect of thereceiver’s prior knowledge does not monotonically improve the quality of communication. Figure
1 focuses on MPEs which admit two intervals, and depicts the length of the first interval of themost efficient MPE for different values of b. Since these lengths are below 0.5, they also directly
represent the efficiency level. The figure clearly shows that for b = 0.225 and b = 0.25, more
information results in a less efficient partition when q is sufficiently close to one. The reason forthis is that as q increases, the receiver’s signal becomes more accurate and more likely to fall
into the right interval. An increase in q thus makes the risk effect less relevant, which works as
the first-order effect. Since the risk effect is more salient when the bias is large, this first-ordereffect eventually dominates and lowers the quality of communication as q approaches one.
It is important to note, however, that our focus is generally on the case where q is relatively
small because there is little point in communicating when the receiver already knows the true
state with sufficient precision. We can thus argue that in a class of situations where communication is relevant and beneficial, the effect of the receiver’s prior knowledge on the quality11

of communication is largely positive, if the underlying information structure has a feature thatgives rise to the dual role of the sender’s message.

4

Conclusion

In this paper we analyze a cheap talk model with a partially informed receiver. In clear contrast
to the previous literature, we find that there is a case where the receiver’s prior knowledge
enhances the amount of information conveyed via cheap talk messages. This contrasting result
is mainly due to the structure of the receiver’s private information we assume. While we do notintend to insist that the information structure of the current form is a necessary feature in thisclass of problems, it provides an insight that the information structure matters for the impactof the receiver’s information on the quality of communication. In future, it seems worthwhile
to explore more on this point, as to when and under what conditions the receiver’s information
facilitates communication.

Appendix
Proof of Proposition 1:
The proof closely follows that of Proposition 1 of Moreno de Barreda [4] and directly stems
from the following lemma.
Lemma 1 The width of an interval Ti in Difinition 1 is longer than or equal to 2b unless
inf Ti = 0.
Proof:
Suppose to the contrary that there exists an interval Ti such that0 < inf Ti ≤ sup Ti < inf Ti + 2b.
We divide the situation into the following two cases: {inf Ti } ∈/ Ti or {inf Ti } ∈ Ti .
We first consider the case of {inf Ti } ∈/ Ti . In this case the point inf Ti belongs to anotherinterval, say Tj , andinf Tj ≤ sup Tj = inf Ti < sup Ti

12holds. Pick any two distinctive messages mi and mj such thatµt (mi ) > 0 iff t ∈ Ti ,µt (mj ) > 0 iff t ∈ Tj .
Then,
sup Ti > α(mi , r) > inf Ti ≥ α(mj , r)holds for any r ∈ [0, 1]. This implies that|inf Ti + b − α(mi , r)| < b ≤ |inf Ti + b − α(mj , r)|holds for any r ∈ [0, 1]. However, this eliminates the sender’s incentive to send a message mj at
t = inf Ti . This is a contradiction.
Next, we consider the case of {inf Ti } ∈ Ti . In this case there exists an interval Tj such that
inf Tj < sup Tj = inf Ti ≤ sup Tj .
Pick any two distinctive messages mi and mj such thatµt (mi ) > 0 iff t ∈ Ti ,µt (mj ) > 0 iff t ∈ Tj .
Then,
sup Ti ≥ α(mi , r) ≥ inf Ti > α(mj , r)holds for any r ∈ [0, 1]. This implies that|inf Ti + b − α(mi , r)| < b < |inf Ti + b − α(mj , r)|holds for any r ∈ [0, 1]. Then, there exists a sufficiently small  > 0 such that {inf Ti − } ∈ Tj
and|inf Ti −  + b − α(mi , r)| < |inf Ti −  + b − α(mj , r)|holds. However, this eliminates the sender’s incentive to send a message mj at t = inf Ti − .
This is a contradiction.
13

Proof of Proposition 2:
For t ∈ Tn ,∆(t; tn−1 , tn , tn+1 ) =
tn+1 + tn 2
− t+b−2Zq(1 − q)tn+1 + tn−
r−q + (1 − q)τn+1 r∈Tn+1
2




tn+1 + tnqqr − 2t − 2b + 2 −dr×
q + (1 − q)τn+1
q + (1 − q)τn+1
2

tn + tn−1 2+ t+b−
2


 


2
q
tn + tn−1
tn + tn−1
q
+
t−
t−
− 2−
− 2b
q + (1 − q)τn2
q + (1 − q)τn2
Z
q(1 − q)tn + tn−1+r−
q + (1 − q)τn r∈Tn
2




tn + tn−1qqr − 2t − 2b + 2 −dr.
×
q + (1 − q)τn
q + (1 − q)τn
2
∂∆
2q 2
= τn+1 + τn −∂t
q + (1 − q)τn



q
2−
q + (1 − q)τntn + tn−1
t−
2





+b .

Then, if limt↑tn ∂∆∂t ≥ 0, or equivalently
2q 2 bqq2≤ τn+1 + τn 1 −2−
q + (1 − q)τn
q + (1 − q)τn
q + (1 − q)τn(1)

holds, then ∂∆∂t ≥ 0 for t ∈ Tn . Therefore, ∆(t; tn−1 , tn , tn+1 ) ≤ 0 for t ∈ Tn if and only iflimt↑tn ∆(t; tn−1 , tn , tn+1 ) ≤ 0. This is written as
G+ (τn , τn+1 ) ≥ 0.

(2)

Clearly, (2) is also a necessary condition. Here, we obtain the following results by direct calculation:

14

Lemma 2 For any τn ∈ [0, 1] and τn+1 ,∂ 2 G+ (τn , τn+1 )
> 0,
2
∂τn+1
∂G+ (τn , 0)
< 0,
∂τn+1
∂G+ (τn , 2b)
≥ 0,
∂τn+1
G+ (τn , 0) < 0,
G+ (τn , 2b) < 0,
G+ (τn , τn + 4b) ≥ 0,”=” holds iff q = 0.

This lemma implies that for any τn ∈ [0, 1] and τn+1 ≥ 0, (2) holds if and only if τn+1 ≥ τ (τn , q)
where τ (τn , q) is defined as the unique solution τ̃ ∈ [0, ∞) of G+ (τn , τ̃ ) = 0. Moreover, τ (τn , q) ∈(2b, τn + 4b] and τ (τn , q) = τn + 4b if and only if q = 0.
Since τn+1 ≥ 2b implies that (1) holds for any τn ≥ 0, (1) is redundant for any τn ∈ [0, 1] andτn+1 ≥ τ (τn , q).
For t ∈ Tn+1 ,∆(t; tn−1 , tn , tn+1 ) =
tn+1 + tn 2
− t+b−
2


 


2
q
tn+1 + tn
tn+1 + tn
q
−
t−
t−
− 2−
− 2b
q + (1 − q)τn+1
2
q + (1 − q)τn+12
Z
q(1 − q)tn+1 + tn−
r−q + (1 − q)τn+1 r∈Tn+1
2




tn+1 + tnqqr − 2t − 2b + 2 −×dr
q + (1 − q)τn+1
q + (1 − q)τn+1
2

tn + tn−1 2+ t+b−2Z
tn + tn−1q(1 − q)r−+
q + (1 − q)τn r∈Tn
2




q
tn + tn−1
q×r − 2t − 2b + 2 −dr.
q + (1 − q)τn
q + (1 − q)τn
2
∂∆
2q 2
= τn+1 + τn +∂t
q + (1 − q)τn+1



2−

q
q + (1 − q)τn+1
15



t−

tn+1 + τn2+ b > 0.

Then, ∆(t; tn−1 , tn , tn+1 ) ≥ 0 for t ∈ Tn+1 if and only if limt↓tn ∆(t; tn−1 , tn , tn+1 ) ≥ 0. This is
written as
G− (τn , τn+1 ) ≤ 0.

(3)

Clearly, (3) is also a necessary condition. Here, we obtain the following results by direct calculations:Lemma 3 For any τn ∈ [0, 1] and τn+1 ≥ 0,9∂ 3 G− (τn , τn+1 )
≥ 0,
3
∂τn+1
∂ 2 G− (τn , 2b)
> 0,
2
∂τn+1
∂G− (τn , 2b)
≤ 0,
∂τn+1
∂G− (τn , τn+1 )> 0,limτn+1 →∞∂τn+1
G− (τn , 2b) < 0,G− (τn , τn + 4b) ≤ 0,”=” holds iff q = 0.

This lemma implies that for any τn ∈ [0, 1] and τn+1 ≥ 2b, (3) holds if and only if τn+1 ≤τ (τn , q) where τ (τn , q) is defined as the unique solution τ̃ ∈ [2b, ∞) of G− (τn , τ̃ ) = 0. Moreover,τ (τn , q) ≥ τn + 4b and the equality holds if and only if q = 0.
9The second inequality holds under the assumption that b < 0.5.

16For t ∈ T /(Tn ∪ Tn+1 ),∆(t; tn−1 , tn , tn+1 ) =
tn+1 + tn 2
− t+b−2Z
tn+1 + tn
q(1 − q)r−−
q + (1 − q)τn+1 r∈Tn+1
2




q
tn+1 + tn
q×r − 2t − 2b + 2 −dr
q + (1 − q)τn+1
q + (1 − q)τn+1
2

tn + tn−1 2+ t+b−2Zq(1 − q)tn + tn−1+r−
q + (1 − q)τn r∈Tn
2




tn + tn−1qqr − 2t − 2b + 2 −dr.
×
q + (1 − q)τn
q + (1 − q)τn2
Note that∂∆= τn+1 + τn > 0.
∂tThen, ∆(t; tn−1 , tn , tn+1 ) ≤ 0 for t ∈ Tn0 and n0 < n if and only if limt↑tn0 ∆(t; tn−1 , tn , tn+1 ) ≤ 0.
The latter holds if and only if limt↑tn−1 ∆(t; tn−1 , tn , tn+1 ) ≤ 0 since limt↑tn0 ∆(t; tn−1 , tn , tn+1 ) =limt↓tn0 ∆(t; tn−1 , tn , tn+1 ) for n0 < n − 1. It is verified that (3) assures this.
Similarly, ∆(t; tn−1 , tn , tn+1 ) ≥ 0 for t ∈ Tn0 and n0 > n+1 if and only if limt↓tn0 ∆(t; tn−1 , tn , tn+1 ) ≥0. The latter holds if and only if limt↓tn+1 ∆(t; tn−1 , tn , tn+1 ) ≥ 0 since limt↓tn0 ∆(t; tn−1 , tn , tn+1 ) =limt↑tn0 ∆(t; tn−1 , tn , tn+1 ) for n0 > n + 1. It is verified that (2) assures this.

Proof of Proposition 3:
This proposition is obtained from the following lemma:
S

Lemma 4 V (t; q) = −NX(1 − q)(q + τn )τ 3n

12 (q + (1 − q)τn )
n=1

2

R

− b and V (t; q) = −This lemma is in turn obtained by direct calculation.

NX(1 − q)(q + τn )τ 3n

12 (q + (1 − q)τn )n=1

Proof of Proposition 4:
This proposition is obtained from Proposition 2 and the following lemma:
17Lemma 5 Given t = (t0 , . . . tN ), t0 = (t00 , . . . t0N ), and n̂ such that• tn̂ > t0n̂ ,• tn = t0n for n 6= n̂, and• τn̂ < τn̂+1 .
Then, t is more efficient than t0 under any q.
Proof:We denotedM (τ, q) = −(q + τ )τ 3.
q + (1 − q)τ

Then, it suffices to show that0M (τn , q) + M (τn+1 , q) − M (τn0 , q) − M (τn+1, q) > 0.

It is verified that


∂ ∂M
< 0,
∂τ
∂q


∂ 2 ∂M< 0.
∂τ 2 ∂q
These imply that∂ 
0M (τn , q) + M (τn+1 , q) − M (τn0 , q) − M (τn+1, q) > 0.
∂q

Combining this with the fact that0M (τn , 0) + M (τn+1 , 0) − M (τn0 , 0) − M (τn+1, 0) > 0,

we obtain the lemma.

Proof of Proposition 5:
The proof is based on the following series of lemmas.
Lemma 6 Given b and q > 0. If t is the most efficient MPE and τn < τn+1 for n = 1, . . . , N − 1,then τn+1 = τ (τn , q) for n = 1, . . . , N − 1.
18

Proof:
Suppose the contrary. Define n̂ such thatdn̂ = min {n|τn+1 > τ (τn , q)} .
Then, n̂ = 1 orτn̂ = τ (τn̂−1 , q) < τ (τn̂−1 , q).
Thus, we can find another MPE t0 where t0n̂ is slightly higher than tn̂ , t0n = tn for n 6= n̂, andthe presupposition of Lemma 5 holds at n = n̂. Then the lemma is obtained from Lemma 5.
Lemma 7 There exists q̂ > 0 such that τ (τ, q) > τ for any τ ∈ (0, 1) and q ∈ [0, q̂].
Proof:
By Proposition 2, τ (τ, 0) = τ + 4b > τ for any τ ∈ (0, 1). Then, there must exist q̂ > 0 such thatτ̂ (τ, q) > τ for any τ ∈ (0, 1) and q ∈ [0, q̂].
Lemma 8 There exists q̌ > 0 such that∂τ< 0 for any τ ∈ (0, 1) and q ∈ (0, q̌].
∂q

Proof:++∂G∂G
Since ∂τ> 0 by Lemma 2.
> 0 for τn+1 > 2b, ∂τ|
n+1
n+1 τn+1 =τ (τn )On the other hand,
1 ∂G+
=
q ∂q
 3
−q 2 + (1 − q)(2 − q)τn+1 τn+1
3

3 (q + (1 − q)τn+1 )+
2 −q 2 + (1 − q)(2 − q)τn τn33

3 (q + (1 − q)τn )+(q + (2 − q)τn ) τn (τn + 4b).
(q + (1 − q)τn )2Then, Lemma 2 implies
2
1 ∂G+
4
|q=0,τn+1 =τ (τn ) = τ (τn ) + τn + 2(τn + 4b)
q ∂q
3
3
2
> 2b + 8b
3
28
=
b.
3
+

Therefore, there exists q̌ such that ∂G∂q |τn+1 =τ (τn ) > 0 for any q ∈ (0, q̌].
d

We now return to the proof of Proposition 5. Letting q = min{q̂, q̌}, the proposition thenfollows from Lemmas 5, 6, 7, and 8.

19

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20

Figure 1:21