GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.62

Cheap Talk with an Informed Receiver

Junichiro Ishida and Takashi Shimizu

June 2009; revised June 2015

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

Cheap Talk with an Informed Receiver
Junichiro Ishida∗ and Takashi Shimizu†
June 11, 2015

Abstract
This paper considers a model of strategic information transmission with an imperfectly informed receiver and provides a simple logic by which the receiver’s prior knowledge becomes an impediment to efficient communication. We show that the extent ofcommunication is severely limited as the receiver becomes more informed. Moreover, in a
simple example with two signals, we show that no information can be conveyed via cheap
talk for an arbitrarily small degree of preference incongruence. This result draws sharp
contrast to the case with an uninformed receiver which always yields a fully separating
equilibrium as long as the preferences are sufficiently congruent.
Keywords: Cheap talk, Informed receiver, Information gap, Preference congruence.
JEL Classification Number: D23; D82.

∗

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)†
Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho Nada-ku, Kobe, 657-8501 JAPAN(E-mail: shimizu@econ.kobe-u.ac.jp)

1

Introduction

Since the seminal work of Crawford and Sobel [5], a substantial amount of attention has been
paid to strategic aspects of communication between players with conflicting interests. Despite
many notable developments that have been made over the years, however, most existing works
focus on the case where the sender who knows the state of nature with precision sends a
message to the receiver who knows nothing about it. The latter assumption that the receiver
has no information of her own seems particularly restrictive as the receiver often has accessto alternative sources of information.
In this paper, we consider a simple model of strategic information transmission to investigate how the receiver’s prior information affects the extent of communication. The situation
we consider is as follows. The sender observes a signal which imperfectly reflects the true
state and then sends a costless message to the receiver. Upon receiving the message, thereceiver then “double-checks” it by using her own source of information and chooses someaction. Within this environment, we show that the receiver’s ability to gain some information on her own, aside from the conflict of interests, can be a major impediment to effective
communication. Moreover, in a two-signal example, we show that no information can beconveyed via cheap talk for an arbitrarily small preference incongruence when the receiver’s
source of information becomes as reliable as the sender’s. This result contrasts sharply withthe case with an uninformed receiver where there is always an informative equilibrium aslong as the degree of preference incongruence is sufficiently small.
There are now a handful of works which examine the nature of strategic information transmission when the receiver, broadly defined, has alternative sources of information. Amongthem, the paper is more closely related to Morgan and Stocken [15], Galeotti et al. [7], Lai[13] and Moreno de Barreda [6]. The first two consider models with multiple imperfectly informed senders,1 while the latter two consider cases with an imperfectly informed receiver.2
While these works do not necessarily concentrate on the questions we pose here, they theoretically have one common feature: the incentive for truthful communication diminishes as
1
Also see Austen-Smith [1], Gilligan and Krehbiel [8], Krishna and Morgan [12], Battaglini [2], and Li [14]for works with multiple senders. Kawamura [10] considers a setting in which senders have an incentive to“exaggerate” their preferences and show that the use of binary messages is a robust mode of communication.
Kawamura [11] also uses a similar setting to analyze how the sample size affects the quality of communication.
2
Some early attempts to model an informed receiver are provided by Seidmann [16] and Watson [17]. See
Chen [3, 4] for more recent works on models with an imperfectly informed receiver.

1the receiver accumulates more information.3 An important feature of these studies is thatthe action space is continuous. In such an environment, when the receiver becomes more informed, her resultant action necessarily becomes less sensitive to the sender’s message, whichyields an effect equivalent to an increase in the preference bias. This implies that the original
insight of Crawford and Sobel [5] directly applies to each of these cases: in fact, when thesignal space is discrete as in Morgan and Stocken [15], truthful communication is always
possible as long as the preferences are sufficiently congruent.4

2

Model

We consider a simple model of cheap talk to illustrate how the receiver’s (imperfect) priorknowledge becomes an impediment to effective communication. There are two players, Player
0 (male) and Player 1 (female), to whom we interchangeably refer as the sender and thereceiver, respectively. The game proceeds as follows:
1. Nature randomly draws the state t ∈ {0, 1}. Denote P (t = 1) by π.
2. Each player n, n = 0, 1, observes a private signal sn ∈ Sn = {1, . . . , Kn }. Given therealized state, s0 and s1 are independently distributed.
3. Upon observing s0 , Player 0 costlessly sends a message m ∈ M = S0 to Player 1.
4. Upon observing s1 and m, Player 1 chooses an action a ∈ {0, 1}.5
5. Player n receives a payoff un (t, a) given byun (t, a) = I(a = t) + bn I(a = n),where I is the indicator function, and bn is Player n’s private benefit of implementing
3

In contrast, Ishida and Shimizu [9] consider a setting in which the receiver observes a signal which carries
no information with some positive probability. It is shown that there is a case where the receiver’s priorknowledge enhances the amount of information conveyed via cheap talk.
4
In a variant of our model where the action space is continuous while the signal space is discrete, we canshow that truthful communication is always possible if the preference bias is sufficiently small.
5
An essential feature of our model is that the action space is discrete. For practical purposes, there areseveral interpretations of this structure. One is that the underlying situation is such that optimal actions
diverge to extreme points (the case of corner solutions), i.e., when an action is to be taken, it is always
optimal to go all the way to the end point. Another possible interpretation is that there are some technicalrestrictions on the set of feasible actions, so that the decision maker can only take some pre-specified points
in the action space.

2

his or her preferred action.6 Let βn := (1 − bn )/2, where we assume bn ∈ (0, 1) or,
equivalently, βn ∈ (0, 0.5).
Let ℓn denote the likelihood ratio whereℓn (i) =P (sn = i|t = 1).
P (sn = i|t = 0)

Without loss of generality, we order the signals by the likelihood ratio:ℓn (1) ≥ · · · ≥ ℓn (Kn ) n = 0, 1.

(1)Each player’s signal structure is thoroughly characterized by a pair of conditional density
functions or, equivalently, a vector of the likelihood ratios derived from them. In what
follows, therefore, we identify player n’s signal structure with Ln = (ℓn (1), ..., ℓn (Kn )).
Definition 1 A signal structure Ln is said to be informative if the corresponding likelihood
ratios satisfy ℓn (1) > 1 > ℓn (Kn ) and uninformative otherwise.7 We in particular denote
Ln = U when Player n’s signal structure is uninformative.
Throughout the analysis, we place three restrictions on the sender’s signal structure.
First, for ease of exposition, we restrict our attention to a class of signal structures which
satisfy (1) with strict inequalities.8 Second, we also assume that none of the sender’s signals
gives a posterior belief that assigns equal probabilities over states, i.e.,∄i ∈ S0 s.t. P (t = 1|s0 = i) = 0.5, or equivalently ℓ0 (i) =

1−π
.
π

Third, we assume that it is not possible that the sender has a posterior belief that one stateis always more likely irrespective of his signal, i.e.,
P (t = 1|s0 = 1) > 0.5 > P (t = 1|s0 = K0 ), or equivalently ℓ0 (1) >61−π> ℓ0 (K0 ).
π

Note that in the current setup, the preference bias is deterministic in that it is attached to a particularaction irrespective of the realized state. While this may appear a departure from the standard setup, this israther a direct consequence of the fact that the action space is discrete. To be more precise, what ultimately
matters in a discrete setup like this is the threshold belief at which a player is indifferent between the two
actions. Since all of our subsequent results are characterized by the relationship between the threshold beliefs
and the signal structures, two models are strategically equivalent if they lead to the same threshold beliefs,
given a pair of preference bias parameters. Given this, we can always construct a strategically equivalentmodel even if the state space is continuous and the biases are defined in probabilistic terms.
7
It is easily verified that a signal structure is uninformative if and only if ℓn (1) = · · · = ℓn (Kn ) = 1.
8
If there are any two signals with the same likelihood ratio, we can in principle merge them into one signal.

3

These assumptions guarantee the existence of i∗ ∈ S0 \{K0 } such that(
> 0.5
if i ≤ i∗
P (t = 1|s0 = i)
< 0.5
if i > i∗ ,or equivalentlyℓ0 (i)

(

> 1−π
π
< 1−π
π

if i ≤ i∗
if i > i∗ .

0
Let F ⊂ RK
+ denote the set of signal structures that satisfy these assumptions.

As for the receiver’s side, note that there would be no point in soliciting information fromthe sender if her signals were highly informative relative to the sender’s.9 We thus restrict
our attention to a set of signal structures that are relatively weak in the following sense
Definition 2 The receiver’s signal structure is said to be relatively weak ifℓ0 (i∗ )ℓ0 (i∗ + 1)
>
ℓ
(1)
and
ℓ
(K
)
>
1
1
1
ℓ0 (i∗ + 1)
ℓ0 (i∗ )(2)

1
hold. Let W(L0 ) ⊂ RK
+ denote the set of signal structures (with K1 signals) that satisfy (2)for a given L0 .
Note that U ∈ W(L0 ), i.e., the uninformative signal structure is relatively weak.

3

Main result

We denote the sender’s strategy by M : S0 → M, and the receiver’s by A : M × S1 → {0, 1}.
In order to make our contention most succinctly, throughout the analysis, we restrict ourattention to when full separation can be achieved in equilibrium. We in particular focus
on what we call a fully informative equilibrium, which is a fully separating equilibrium inwhich communication matters, i.e., an equilibrium in which the sender reveals his private
information truthfully and the receiver’s choice of action depends on the sender’s message
with positive probability.10
Definition 3 An equilibrium is said to be fully informative if
9That is, if the receiver’s private information is highly accurate by itself, her choice of action depends only
on her own information, irrespective of the sender’s message, and communication plays no role.
10
The latter requirement is necessary to exclude truth-telling equilibria in which the receiver’s action is
independent of the sender’s message, and communication hence plays no role.

4• it is truth-telling, i.e., M (i) = i for any i ∈ S0 , and• it is influential, i.e., there exist i, i′ ∈ S0 and j ∈ S1 such that A(i, j) 6= A(i′ , j).
We start our analysis with the receiver’s problem over the choice of action. Define Pij :=P (t = 1|s0 = i, s1 = j), which coincides with the receiver’s posterior belief under the sender’struth-telling strategy. Then, the receiver’s best response is expressed as follows:11(
1
if Pij > β1A(i, j) =0
if Pij < β1 .

(3)

We now shift our attention to the sender’s problem. Define
S1 (i, i′ ) = {j|A(i, j) = 1, A(i′ , j) = 0}.
It is verified that if the receiver’s strategy A satisfies (3), then for any i, i′ ∈ S1 , one of thefollowing holds:(i) S1 (i, i′ ) = S1 (i′ , i) = ∅.
(ii) S1 (i, i′ ) 6= ∅ and S1 (i′ , i) = ∅.
(iii) S1 (i, i′ ) = ∅ and S1 (i′ , i) 6= ∅.
Among them, we can show that the sender’s incentive matters only in case (ii) when s0 = i.
More precisely, one can show that

holds if and only if
E [u0 (t, A(i, j))|s0 = i] ≥ E u0 (t, A(i′ , j))|s0 = iPj∈S1 (i,i′ ) Pij P (s1 = j|s0 = i)P

j∈S1 (i,i′ ) P (s1 = j|s0 = i)

≥ 1 − β0 .

(4)These results straightforwardly lead to the conditions that must be satisfied in any truthtelling equilibrium.
Proposition 1 There exists a truth-telling equilibrium if and only if• (3) holds for any i ∈ S0 and j ∈ S1 , and
11

The details of the derivation of the equilibrium conditions are relegated to Appendix A.

5• (4) holds for any i, i′ ∈ S0 satisfying S1 (i, i′ ) 6= ∅.
Moreover, it is influential if and only if there exist i, i′ ∈ S0 satisfying S1 (i, i′ ) 6= ∅.
One implication of the proposition is that there may arise a serious conflict of interests
between the sender and the receiver when the resulting posterior belief Pij falls into the range(β1 , 1−β0 ). More precisely, there exists no truth-telling equilibrium only if there exists i ∈ S0
and j ∈ S1 satisfying β1 < Pij < 1 − β0 . Given this fact, we can now make the following
statement which is the main result of this paper.
Theorem 1 Fix L0 ∈ F. Then, for all L1 ∈ W(L0 ), there always exists (b0 , b1 ) such that
there exists a fully informative equilibrium if and only if L1 = U , i.e., the receiver’s signal
structure is uninformative.
Proof: We set (b0 , b1 ) such that1−ππℓ0 (i∗ ) + 1 − π
πℓ0 (i∗ + 1)β1 =.
πℓ0 (i∗ + 1) + 1 − π
β0 =Here, (b0 , b1 ) is chosen so that the sender is indifferent between the two messages after
observing i∗ , and the receiver is indifferent between the two actions if the sender’s signal
is i∗ + 1, given that the receiver’s signal structure is uninformative, i.e., L1 = U . Then, from
Proposition 1, there exists a fully informative equilibrium if L1 = U .
On the contrary, suppose that L1 is informative. Then, there exist j, j ∈ S1 such that
> 1ℓ1 (j) < 1

=1

if j ≤ j
if j ≥ jotherwise.

It follows from (2) that {j, . . . , K1 } ⊆ S1 (i∗ , i∗ + 1) ⊆ {j + 1, . . . , K1 } andPi∗ j ≤ 1 − β0∀j ∈ S1 (i∗ , i∗ + 1), in particular,Pi∗ j < 1 − β0∀j ≥ j.

It then turns out that (4) is violated, and therefore, by Proposition 1, there exists no fully
informative equilibrium.

6

Theorem 1 states that we can always find a pair (b0 , b1 ) such that truthful communicationis possible if and only if the receiver has no information of her own, meaning that better
information (on the receiver’s side) can lead to worse information transmission.12

4

A two-signal example

To illustrate the intuition behind our result, consider a simple example in which there are
only two signals: S0 = S1 = {h, l}. Suppose that P (sn = h | t = 1) = P (sn = l | t = 0) = rn
where rn ∈ [0.5, 1] measures the accuracy of Player n’s prior information. For this example,
we assume that π = 0.5, b0 = b1 = b and r0 > 1 − β := (1 + b)/2, but do not impose theassumption of relatively weak signals.
The receiver is totally uninformed if r1 = 0.5, in which case there always exists a fully
informative equilibrium as long as the preference bias is sufficiently small.13 In contrast, thesituation changes rather starkly once we allow for the possibility that the receiver’s own signal
is also partially informative. Now suppose that the information gap (r0 − r1 ) is positive but
sufficiently small such that
r1 >

βr0
.
(1 − r0 ) + β(2r0 − 1)(5)Note that under this condition, we have Phh > Plh > β and 1 − β > Phl > β, meaning thatthe second condition in Proposition 1 is violated, and therefore, there is no fully informativeequilibrium.14 A striking fact is that as r1 → r0 , (5) collapses to 1 > 2β, which is satisfied
by any positive b.15
An important implication of this result is that the receiver may be better off by remaining
uninformed about the current state than by obtaining imperfect information. To see this,
suppose that the receiver has a chance to acquire her own information at the beginning. If
the receiver chooses to do so, then the accuracy of her signal is given by r1 = r ∈ (0.5, r0 ); if
12

It should be noted that the theorem itself does not constitute a direct proof that better information leads
to worse information transmission, because it is a non-generic result for this purpose. However, the argument
can easily be extended to show that it is not a non-generic phenomenon in our environment (see Theorem 2
in Appendix B).
13
To verify this, note that when r1 = 0.5, the receiver’s best response must be A(h, j) = 1 and A(l, j)
for j = h, l. Given this, one can readily verify that there is no incentive for the sender to make a false
recommendation if r0 > 1 − β.
14One can also show that there is no partially informative equilibrium in which the sender adopts a mixed
strategy, meaning that no information can be conveyed via cheap-talk communication under this condition.
15
To be more precise, one can show that there exists no fully informative equilibrium for any b > 0 if r1 ≥ r0 .

7

not, the signal contains no information and r1 = 0.5. Ifr>

βr0≥ 0.5,(1 − r0 ) + β(2r0 − 1)

the receiver faces a cumbersome tradeoff: she can receive an informative message from thesender only if she foregoes the chance to acquire her own information. When r0 > r as we
assume throughout, the receiver can raise her payoff by committing to remain uninformed.
To understand this result, one must look at the sender’s cost and benefit of misrepresenting information when he observes the signal indicating that his preferred action is unlikely
to be optimal, i.e., s0 = h. First, the benefit is relatively clear, as the sender may sway thereceiver towards his preferred action by making a false recommendation. This information
manipulation comes at a cost, however, because it necessarily entails inefficient use of information. Given the receiver’s strategy, a potential loss arises when the receiver observes s1 = l,in which case the sender’s message becomes pivotal. That is, the cost of misrepresenting information is determined by how much the sender loses by recommending his preferred action
when s0 = h and s1 = l, which is precisely captured by Phl . As the receiver’s information
becomes as accurate as the sender’s, Phl converges towards one half. At this point, therefore,
it is a fair bet, one way or the other, as far as the probability of choosing the right action isconcerned. The sender can only gain from lying by the margin of the preference bias, and noinformation can be conveyed at all even when the preference bias is arbitrarily small.
The key to this result is the fact that the receiver can now overrule the sender’s recommendation based on her own information. The problem is that this overruling does not occur
randomly: the receiver overrules the sender’s recommendation when she believes, judging
from her own information, that the recommendation is more likely to be wrong. The receiverthus effectively functions as a gatekeeper to sort out bad information, but this capability, or
the lack of commitment not to use her own information, diminishes the sender’s incentive toreport truthfully. Since the sender knows that he would be corrected whenever he is way off
the mark, the salience of the private benefit is magnified, which renders communication lessinformative.

5

Conclusion

This paper explores the extent to which cheap-talk communication can credibly convey
meaningful information when the receiver is partially informed. As it turns out, the receiver’s
8

prior knowledge matters and makes a non-trivial difference in the quality of informationthat can be extracted from the sender. As a general rule, communication becomes lessefficient as the receiver becomes more informed.

This result yields a critical implica-

tion: in order to facilitate communication, it may be advisable for the receiver to refrain
herself from acquiring her own information, even if it can be done with a relatively small cost.

Acknowledgements: This research is financially supported by JSPS Grant-in-Aid for Scientific Research (C) (No.26380252 and 15K03352), 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.

9

References[1] David Austen-Smith. Interested experts and policy advice: Multiple referrals under openrule. Games and Economic Behavior, 5(1):3–43, 1993.
[2] Marco Battaglini. Multiple referrals and multidimensional cheap talk. Econometrica,
70(4):1379–1401, 2002.
[3] Ying Chen. Communication with two-sided asymmetric information. mimeo, 2009.
[4] Ying Chen. Value of public information in sender-receiver games. Economics Letters,
114(3):343–345, 2012.
[5] Vincent P. Crawford and Joel Sobel. Strategic information transmission. Econometrica,
50(6):1431–1451, 1982.
[6] Ines Moreno de Barreda. Cheap talk with two-sided private information. mimeo, 2010.
[7] Andrea Galeotti, Christian Ghiglino, and Francesco Squintani. Strategic informationtransmission in networks. Journal of Economic Theory, 148(5):1751–1769, 2013.
[8] Thomas Gilligan and Keith Krehbiel. Asymmetric information and legislative rules witha heterogeneous committee. American Journal of Political Science, 33(2):459–490, 1989.
[9] Junichiro Ishida and Takashi Shimizu. Can more information facilitate communication?
ISER Discussion Paper No. 839, 2012.
[10] Kohei Kawamura. A model of public consultation: Why is binary communication so
common? Economic Journal, 121(553):819–842, 2011.
[11] Kohei Kawamura. Eliciting information from a large population. Journal of Public
Economics, 103:44–54, 2013.
[12] Vijay Krishna and John Morgan. A model of expertise. Quarterly Journal of Economics,
116(2):747–775, 2001.
[13] Ernest K. Lai. Expert advice for amateurs. Journal of Economic Behavior and Organization, 103:1–16, 2014.

10

[14] Ming Li. Advice from multiple experts: A comparison of simultaneous, sequential, andhierarchical communication. B.E. Journal of Theoretical Economics, 10(1):Article 18,
2010.
[15] John Morgan and Phillip C. Stocken. Informal aggregation in polls. American Economic
Review, 98(3):864–896, 2008.
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11

Appendix A: Proof of Proposition 1
The Equilibrium Conditions for the Receiver’s Strategy: Given the sender’s truthtelling strategy, we haveE [u1 (t, 1)|s0 = i, s1 = j] − E [u1 (t, 0)|s0 = i, s1 = j]
= P (t = 1|s0 = i, s1 = j)(1 + b1 ) − P (t = 0|s0 = i, s1 = j)(1 − b1 )= 2 [P (t = 1|s0 = i, s1 = j) − β1 ] .
This implies that the receiver’s best response is given by (3).
The Equilibrium Conditions for the Sender’s Strategy:
We first show that S1 (i, i′ ) 6= ∅ ⇒ S1 (i′ , i) = ∅ necessarily holds as long as the receiver’s strategy A satisfies (3). Suppose to the contrary that there exists (i, i′ , j, j ′ ) suchthat A(i, j) = 1, A(i′ , j) = 0, A(i, j ′ ) = 0, and A(i′ , j ′ ) = 1. If i < i′ , thenβ1 ≥ Pij ′ > Pi′ j ′ ≥ β1 .
This is a contradiction. Similarly, if i > i′ , thenβ1 ≥ Pi′ j > Pij ≥ β1 .
This is also a contradiction, Therefore, it is verified S1 (i, i′ ) 6= ∅ ⇒ S1 (i′ , i) = ∅. It thenfollows that for any i, i′ ∈ S0 , one of the following holds in truth-telling equilibrium:(i) S1 (i, i′ ) = S1 (i′ , i) = ∅.
(ii) S1 (i, i′ ) 6= ∅ and S1 (i′ , i) = ∅.
(iii) S1 (i, i′ ) = ∅ and S1 (i′ , i) 6= ∅.
Furthermore, it is clear that S1 (i, i′ ) = S1 (i′ , i) = ∅ implies E [u0 (t, A(i, j))|s0 = i] =12

E [u0 (t, A(i′ , j))|s0 = i]. Now suppose S1 (i, i′ ) 6= ∅ and S1 (i′ , i) = ∅. Then we have
E [u0 (t, A(i, j))|s0 = i] − E u0 (t, A(i′ , j))|s0 = i
X=[P (t = 1, s1 = j|s0 = i)(1 − b0 ) − P (t = 0, s1 = j|s0 = i)(1 + b0 )]j∈S1 (i,i′ )



= 2



= 2



= 2


= 2



= 2



= 2

X

j∈S1 (i,i′ )X

j∈S1 (i,i′ )X

j∈S1 (i,i′ )X

j∈S1 (i,i′ )X

j∈S1 (i,i′ )X

j∈S1 (i,i′ )"Pj∈S1 (i,i′ ) P (t = 0, s1 = j|s0 = i)"Pj∈S1 (i,i′ ) P (t = 0, s0 = i, s1 = j)



P

"

P

"

P

P (s1 = j|s0 = i) β0 −P (s1 = j|s0 = i) β0 −P (s1 = j|s0 = i) β0 −P (s1 = j|s0 = i) β0 −P (s1 = j|s0 = i) β0 −
 "PP (s1 = j|s0 = i)

P

j∈S1 (i,i′ ) P (s1 = j|s0 = i)P

j∈S1 (i,i′ ) P (s0 = i, s1 = j)j∈S1 (i,i′ )

P

#
#


P (t=0,s0 =i,s1 =j)
P (s0 =i,s1 =j) P (s0 = i, s1 = j) j∈S1 (i,i′ ) P (s0 = i, s1 = j)j∈S1 (i,i′ ) (1 − Pij )P (s0 = i, s1 = j)P

j∈S1 (i,i′ ) P (s0 = i, s1 = j)P

j∈S1 (i,i′ ) P (s1 = j|s0 = i)j∈S1 (i,i′ )(1 − Pij )P (s1 = j|s0 = i)j∈S1 (i,i′ ) Pij P (s1 = j|s0 = i)P

j∈S1 (i,i′ ) P (s1 = j|s0 = i)

13

#

#
#

− (1 − β0 ) .

Similarly, suppose S1 (i, i′ ) = ∅ and S1 (i′ , i) 6= ∅. Then we have
E [u0 (t, A(i, j))|s0 = i] − E u0 (t, A(i′ , j))|s0 = i
X=[P (t = 0, s1 = j|s0 = i)(1 + b0 ) − P (t = 1, s1 = j|s0 = i)(1 − b0 )]j∈S1 (i′ ,i)



= 2



= 2


= 2


= 2


= 2



= 2

X

j∈S1 (i′ ,i)X

j∈S1 (i′ ,i)X

j∈S1 (i′ ,i)X

j∈S1 (i′ ,i)X

j∈S1 (i′ ,i)X

j∈S1 (i′ ,i)

> 0.

 "P

P (s1 = j|s0 = i)

 "P

P (s1 = j|s0 = i)

j∈S1 (i′ ,i) P (t = 0, s1 = j|s0 = i)

 "P

P (s1 = j|s0 = i)

 "P

P (s1 = j|s0 = i)

"j∈S1 (i′ ,i) P (s1 = j|s0 = i)P

j∈S1 (i′ ,i) P (s0 = i, s1 = j)j∈S1 (i′ ,i) P (t = 0, s0 = i, s1 = j) PP (s1 = j|s0 = i) 

P

j∈S1 (i′ ,i)

P

− β0

#

− β0

#

P (t=0,s0 =i,s1 =j)
P (s0 =i,s1 =j) P (s0 = i, s1 = j)j∈S1 (i′ ,i) P (s0 = i, s1 = j)j∈S1 (i′ ,i) (1 − Pij )P (s0 = i, s1 = j)P

j∈S1 (i′ ,i) P (s0 = i, s1 = j)P

j∈S1 (i′ ,i) P (s1 = j|s0 = i)j∈S1 (i′ ,i) (1 − Pij )P (s1 = j|s0 = i)

P

− β0

− β0

j∈S1 (i′ ,i) P (s1 = j|s0 = i)− β0 

#j∈S1 (i′ ,i) Pij P (s1 = j|s0 = i)

P (s1 = j|s0 = i) (1 − β0 ) − P

#



#

The last inequality follows from the fact that j ∈ S1 (i′ , i) implies Pij ≤ β1 < 1 − β0 .

Appendix B
Theorem 2 Fix L0 ∈ F and L̃1 ∈ W(L0 ) \ {U }. Then, there exist non-degenerate intervalsof b0 and b1 such that• there exists a fully informative equilibrium under L1 = U , while• there exists no fully informative equilibrium under L1 = L̃1 .
Proof: Similarly in the proof of Theorem 1, we can show that if
1−ππℓ0 (i∗ ) + 1 − π
πℓ0 (i∗ + 1).
β1 ≥πℓ0 (i∗ + 1) + 1 − π
β0 ≥

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then there exists a fully informative equilibrium under L1 = U .
Also similarly in the proof of Theorem 1, we can show that if
Pi∗ K1 < 1 − β0 < Pi∗ 1
Pi∗ +1.K1 < β1 < Pi∗ +1.1
Pi∗ +1.1 < 1 − β0
Pi∗ K1 > β1 ,or equivalently
1−π

>β >

1−π

0
πℓ0 (i∗ )ℓ1 (K1 ) + 1 − π
πℓ0 (i∗ )ℓ1 (1) + 1 − π
πℓ0 (i∗ + 1)ℓ1 (K1 )
πℓ0 (i∗ + 1)ℓ1 (1)< β1 <∗
πℓ0 (i + 1)ℓ1 (K1 ) + 1 − π
πℓ0 (i∗ + 1)ℓ1 (1) + 1 − π

1−ππℓ0 (i∗ + 1)ℓ1 (1) + 1 − π
πℓ0 (i∗ )ℓ1 (K1 )β1 <πℓ0 (i∗ )ℓ1 (K1 ) + 1 − π
β0 <hold, then there exist no fully informative equilibrium under L1 = L̃1 .
Combining these results, it is verified that



1−π
1−π
1
1−π
, min
,
,
and
β0 ∈
πℓ0 (i∗ ) + 1 − π
πℓ0 (i∗ )ℓ1 (K1 ) + 1 − π πℓ0 (i∗ + 1)ℓ1 (1) + 1 − π 2
πℓ0 (i∗ + 1)
πℓ0 (i∗ + 1)ℓ1 (1)πℓ0 (i∗ )ℓ1 (K1 )
1
β1 ∈
,
min
,
,
,
∗
∗
∗
πℓ0 (i + 1) + 1 − π
πℓ0 (i + 1)ℓ1 (1) + 1 − π πℓ0 (i )ℓ1 (K1 ) + 1 − π 2or equivalentlyπℓ0 (i∗ ) − (1 − π)πℓ0 (i∗ )ℓ1 (K1 ) − (1 − π) πℓ0 (i∗ + 1)ℓ1 (1) − (1 − π),,0 ,b0 ∈ maxand
πℓ0 (i∗ )ℓ1 (K1 ) + (1 − π) πℓ0 (i∗ + 1)ℓ1 (1) + (1 − π)πℓ0 (i∗ ) + (1 − π)
1 − π − πℓ0 (i∗ + 1)ℓ1 (1) 1 − π − πℓ0 (i∗ )ℓ1 (K1 )1 − π − πℓ0 (i∗ + 1)
b1 ∈ max,,0,,
1 − π + πℓ0 (i∗ + 1)ℓ1 (1) 1 − π + πℓ0 (i∗ )ℓ1 (K1 )1 − π + πℓ0 (i∗ + 1)
are the intervals stated in the theorem. It can also be shown that these intervals are nondegenerate.

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