Strategic Behavior in Acquiring and Revealing Costly Private
Information
January 8, 2015
Abstract
When an agent has the opportunity to access public information, whether or not to acquire
costly private information is a strategic decision. In this case, to study the informativeness of the
revealed information, it is essential to consider the processes of both information acquisition and
revelation together. Using a model in which two agents sequentially forecast the true state of a
forthcoming period, this paper studies an agent’s strategic behavior in acquiring and revealing
costly private information in the presence of the payo¤ externality due to their competition. If
the penalty for an incorrect forecast is greater than the reward for being right, the forecast cannot
be informative because there is no investment in private information. Otherwise, costly private
information is acquired as long as the information quality is moderate. Hence, the payo¤ structure
under which the reward is greater than the penalty is a necessary condition for the investment
in costly private information. Moreover, the acquired information is always revealed truthfully
without herding or anti-herding. That is, the introduction of endogenous costly information raises
the issue of "free-riding" rather than the issue of "herding or anti-herding".
Keyword: Costly information acquisition, Asymmetric reward and penalty, Truthfulness of revealed information
JEL classi…cation: D81, D82, D83
0
1
Introduction
Professional forecasters are paid for providing information regarding future market situations. The
information those agents provide is assumed to be di¢ cult to obtain, either because access to it
is relatively limited or specialized knowledge is required for interpretation. Therefore, the market
usually relies on their forecasts while remaining unaware of the detailed processes involved, providing
an opportunity for agents to act strategically for their own bene…t. For this reason, whether or not
the forecasts agents provide are informative has been a key concern.
The information agents provide is costly to acquire even for themselves. If agents must choose to
sacri…ce a large amount of money, time or e¤ort to acquire meaningful information, while they have
costless access to existing public information, then it would be rational for them to avoid the costly
acquisition of private information and use public information instead. That is, costly information
may induce agents to act strategically, especially during the information acquisition process. This
strategic behavior can be relatively common as it is somewhat di¢ cult to detect. It appears, then,
that studying the informativeness of forecasts requires us to look at both information acquisition and
revelation at the same time.
Thus, this paper studies the topic of the informativeness of an agent’s forecast when her informative private signal is costly to acquire. We discuss this issue considering the case in which a payo¤
externality is present due to the agents’competition.1 We consider a simple sequential model in which
two players compete each other in forecasting the true state of the forthcoming period. The leader
always observes the informative private signal. On the other hand, it is costly for the follower to acquire her informative private signal. Before announcing her own forecast, the follower can observe the
leader’s forecast. As the follower can use it without acquiring her own costly signal, the investment in
a costly private signal is her endogenous decision. After both players sequentially issue forecasts, the
true state is revealed. Then according to the accuracy of their forecasts, each player earns a reward
or a penalty. We assume that both players observe the signals with the same precision, in order to
explore the payo¤ e¤ects without appealing to a di¤erence in the ability of players. If both players
are of the same ability, no one’s (potential) information should be ignored. Addressing the question
of under what condition the follower invests in costly information to obtain an informative forecast
is well worth considering, especially when players have equal precision.
The main results of our model can be summarized as follows. As the leader’s signal is private
information, she can strategically decide whether to follow her signal or not in announcing forecast.
As to this leader’s strategic behavior, we restrict our attention only to the separating equilibrium.
1
There is no doubt that forecasting agents compete with each other and are evaluated frequently. These evaluations
consider not only the individual agent’s own performance, but also her performance relative to her competitors’ performances. Aggarwal and Samwick (1999), Antle and Smith (1986), Gibbons and Murphy (1990), Janakiraman, and
Lambert and Larcker (1992) provide empirical …ndings which support that the relative evaluation is widely used in the
market. In particular, according to Mikhail, Walther and Willis (1999), the analysts who are inferior relative to their
peers are more likely to be …red. However, the absolute accuracy of the forecast has little e¤ect on the probability of
layo¤. Given that the relative evaluation is used, if an agent is successful but her peers are also successful, her reward
may not be greater than if she alone were successful. On the other hand, if an agent fails but her competitors also fail,
the blame can be shared, resulting in a less negative evaluation than if she alone were unsuccessful. It would be natural
then to raise the question of how the payo¤ structures a¤ect the informativeness of the agents’forecasts.
1
This restriction can be justi…ed by the argument that we are mainly interested in the case where
there is credible public information the follower can free ride on. In our setting, this denotes the case
where the follower can infer the leader’s private signal perfectly. We show that there exists an unique
separating equilibrium where the leader always reveals her signal truthfully. The follower, therefore,
can infer the leader’s private signal and it can be used as truthful public information in this separating
equilibrium. Thus, the follower makes a strategic decision on whether to make use of the leader’s
announced forecast or to acquire a costly private signal. She should also consider whether or not to
be truthful in revealing the private signal if she intends to acquire it.
The follower’s equilibrium strategy varies according to the payo¤ structure and the information
quality. If the penalty is greater than the reward, the follower always imitates the leader’s forecast
without acquiring a costly private signal. Hence, her forecast is not informative at all. Even if the
reward is greater than the penalty and the information cost is relatively low, the acquisition of a
private signal is not guaranteed. A su¢ ciently high or low information quality strongly signals the
accuracy or inaccuracy of public information. Thus, the follower makes use of public information
by imitating or deviating from it, without costly investment in a private signal. If the information
quality is not extreme, on the other hand, the follower acquires the costly private signal. Moreover,
if she does so, she always reveals it truthfully without herding or anti-herding: As the signal is costly
to acquire, the follower compares the expected gain from observing the costly private signal and the
gain from making use of existing public information. The follower acquires the costly signal only if
the former is greater than the latter. Hence, if a costly signal is acquired, there is no incentive for her
to ignore it. If the follower had any incentive to ignore the acquired private signal, she would have
avoided paying a cost to observe it.
To sum up, although the public information can be used for free, there exists an equilibrium in
which a costly private signal is acquired and revealed truthfully without herding or anti-herding. The
acquisition of a costly signal is not monotone with respect to the information quality: it happens as
long as the public information is moderately accurate. The payo¤ structure under which the reward
is greater than the penalty is a necessary condition for the existence of this equilibrium.
We also consider the benchmark case, in which the private signal is given exogenously. The
following points are notable …ndings derived from the comparison of two cases of exogenous- and
endogenous information. First, if information is given exogenously, the follower exhibits herding
when the penalty is larger than the reward and anti-herding in the opposite case. On the other hand,
if information is costly to acquire, there is no herding and anti-herding for the acquired information.
Given that not much information is free to acquire in the real world, this proposes that if we are
interested in the truthfulness of provided information, we have to pay more attention to the issue of
whether or not an agent invests in costly information, rather than the issue of the ignorance of the
acquired information through herding and anti-herding. Second, if information is given exogenously,
the symmetric reward and penalty is an optimal payo¤ structure for inducing the follower’s truthful
revelation of a private signal. Our results show that this does not work if information is costly
to acquire. The asymmetric payo¤ structure under which the reward is larger than the penalty is
necessary for the follower to acquire a costly signal, which is a preceding condition for the revelation
of truthful information. The reward should be large enough to outweigh the incentive of free-riding
2
on existing public information. However, an excessively large reward is also not optimal because it
raises the incentive to di¤erentiate herself too much. Third, when the penalty is greater than the
reward, both players’ forecasts are always the same regardless of whether or not the information is
costly to acquire. However, the answers to the issue of the truthfulness of revealed information are
di¤erent in both cases. If the information is exogenous, the truthfulness of revealed information is
not guaranteed, although possible, because the follower exhibits herding if a di¤erent signal from the
leader is observed. If the information is endogenous, the truthfulness of revealed information cannot
be expected at all because an agent does not observe her costly signal and instead just imitates the
leader. This is intuitive because even if a di¤erent signal from the leader is observed after paying a
cost, she would deviate from it and exhibit herding.
The topic of the agents’strategic information revelation has been addressed in several papers which
adopt the so-called forecasting contest model. The forecasting contest model refers to the situation in
which the market commits, ex-ante, to a particular payo¤ structure. (For examples, see Bernhardt,
Campello and Kutsoati (2004), Laster, Bennett and Geoum (1999), Ottaviani and Sorensen (2005),
and Ottaviani and Sorensen (2006a).)2 We build on these previous works. The departing point is that
we pay attention to the strategic behavior, even at the stage of information acquisition, by assuming
the presence of costly private information. We study the existence of the equilibrium where the agent
announces informative forecasts after an investment in private information. Our …ndings imply that
if we are interested in the truthfulness of information of which the acquisition is costly, we have to
pay more attention to the issue of whether or not an agent invests in costly information, rather than
the issue of the ignorance of the acquired information through herding and anti-herding.
The remainder of this paper proceeds as follows. Section 2 reviews related literature. Section 3
introduces the model. Section 4 considers the case in which information is given exogenously, which
is a necessary prelude to the analysis of Section 5. Section 5 considers the case in which information
is costly to acquire. Section 6 is a discussion, and Section 7 contains concluding remarks.
2
Related Literature
There are some papers that deal with the topic of the agents’strategic behavior when private information is costly to acquire.3 Yang (2011) and Burguet & Vives (2000) are related papers that also
consider the costly private information, but in di¤erent contexts from my model. Both models use
n-players sequential prediction setting which has been widely adopted in the literature of herding and
informational cascades. The most distinctive point from my model is that neither model considers the
payo¤ externality. In my model, on the other hand, the competitive environment agents face causes
both the informational- and the payo¤ externality. Thus, their models do not address the topic of
the e¤ects of competitive environment on the agent’s decision of acquiring a costly signal, which is a
main question of my model.
2
For a more detailed discussion regarding this approach, see Ottaviani and Sørensen (2006a and 2006b).
In Kultti and Miettinen (2006) and Kultti and Miettinen (2007), agents pay a cost for some information. In
their models, however, the agent pays for information about their predecessors’ actions, given that their own private
informative signals are given for free.
3
3
Yang (2011) considers a scenario where each investor decides whether to maintain or withdraw
the investment when he has a chance to acquire a costly private signal informative to this decision.4
Her model proposes that the choice of the second investor in‡uences the e¢ ciency of herd behavior.
For example, if the second investor acts di¤erently than his predecessor, the follower infers that the
precision of the second investor’s private signal is signi…cant and so exhibits herding without acquiring
his private signal. This herding is more likely to be e¢ cient because it is based on relatively precise
information. If the second investor acts in the same manner as his predecessor and the precision
of the follower’s signal is not signi…cant, he exhibits herding which is more likely to be ine¢ cient.
As only the informational externality is considered in her model, there are some distinctions in the
analysis and the result. For example, in Yang (2011), players do not need to consider the decisions
of successors and so the backward induction is not used in the analysis. The possibility of deviation
from the predecessor when a private signal is not acquired is also not necessary to consider.5
In Burguet & Vives (2000), in predicting a random variable, an agent has to decide how much e¤ort
to devote to acquiring costly private information and how much to rely on public information. The
main question asked in their model is how the costly information a¤ects the informational externality
and learning. The model proposes that agents acquire too little private information when it is costly
because agents do not consider the positive e¤ect the private information they gather has on the
followers. This result is due to only the informational externality, without the payo¤ externality, being
considered in their model. Their model also proposes that there should be a subsidy in acquiring
costly private information to give an incentive to do so, as my model also proposes. However, the
explicit analysis regarding it is missing. My model attempts to characterize the condition of the
payo¤ structure under which an agent acquires costly private information.
The papers presented herein share the similar feature with my model in that they study the
topic of the agents’ strategic information revelation in the context of a forecasting contest model.
Laster, Bennett and Geoum (1999) analyze the forecaster’s strategic incentive to di¤erentiate in the
winner-takes-all simultaneous contest where both accuracy and publicity matter. They show that
di¤erentiation can be derived even when all agents have identical information. The forecaster, who is
biased toward publicity, tends to deviate more from the consensus in order to minimize the possibility
of sharing publicity. Ottaviani and Sorensen (2005) consider the winner-takes-all contest where all
agents forecast simultaneously. In an extreme case, if only the forecaster with the lowest (highest)
error is rewarded (punished), the incentive to di¤erentiate increases (decreases). The result of a
numerical simulation says that the more convex (concave) the prize structure, the greater (less) the
incentive to di¤erentiate. Ottaviani and Sorensen (2006a) consider the winner-take-all simultaneous
contest in which relative accuracy matters. In their model, the agents excessively seek to distinguish
themselves by weighting their private information heavily. Bernhardt, Campello and Kutsoati (2004)
analyze the e¤ects of both absolute and relative accuracy on the forecaster’s strategic information
4
Each investor’s information cost, which is private information, is not identical and it determines the precision of the
signal. It is assumed that the followers anticipate the lowest value of the signal cost that satis…es the conditions under
which the predecessors are willing to acquire their private signals. This determines how the followers infer the precision
of the predecessor’s signal.
5
This makes the follower always believe that the second investor observed his costly signal for sure if the second
investor acts di¤erently than his predecessor. This is not true if the payo¤ externality is considered.
4
revelation by analyzing the last forecaster in the sequence. They show that if the payo¤ structure is
convex (concave), the last analyst biases his report in the direction of private information (consensus).
None of these models explicitly assume that private information is costly to acquire. Hence, the
analysis of the stage of information acquisition in a sequential model is not covered. In addition, the
main result of previous models is that the unbiased forecast or truth-telling of private information
cannot be expected if the payo¤ structure is either convex or concave. We show that this is not true
if private information is costly to acquire. The convex payo¤ structure is necessary to induce an
investment and truth-telling of costly private information.
The topic of truthfulness in providing information has also been addressed in the framework
of reputational cheap-talk. In this scenario, the market evaluates the agents’ types while using all
available information ex post, such as the realized true state and the issued forecasts. The critical
point where the reputational cheap-talk model diverges from the forecasting contest model is that the
market is either unable or unwilling to commit ex ante to a particular evaluation rule.
Scharfstein and Stein (1990) consider a reputational cheap-talk model in which the smart agent
observes a conditionally correlated signal, while the dumb agent observes only the noisy signal. From
the follower’s standpoint, taking the same action as the leader is more likely to result in being
evaluated as the smart type, resulting in herding. E¢ nger and Polborn (2001) show that, even in a
setting similar to that used by Scharfstein and Stein (1990), anti-herding can be derived if relative
accuracy is important. The recent paper by Ottaviani and Sorensen (2006b) analyzes the e¤ects
of reputational concern on the truthful information revelation using a more general model where
the binary private signal structure is generalized. Swank and Visser (2008) deal with the topic of
endogenous information acquisition in the context of the reputational cheap-talk model. They …nd an
equilibrium in which the leader does not acquire information, and instead, shifts the task of acquiring
information to the follower.
In the reputational cheap-talk model, what concerns an agent is how he is perceived and evaluated
by others. In Swank and Visser (2008), for example, the equilibrium strategy depends on how much an
agent cares about his reputation, and the critical value of information cost is a function of the market’s
prior belief that an agent is a smart type. The approach of the forecasting contest model, adopted
in my model, is appropriate in analyzing the e¤ect of the payo¤ structure on an agent’s strategic
behavior. For example, in my model, the equilibrium strategy depends on the quality of information,
the payo¤ structure, and the information cost. Speci…cally, the critical value of information cost is a
function of the reward and the penalty.
3
Model
There are two players A and B, i 2 fA; Bg, who provide forecasts about the unknown true state,
w 2 fH; Lg, where H and L are mutually exclusive. For both players, the prior probability of each
state is Pr(w = H) = Pr(w = L) = 12 . Before making a forecast, each player has the opportunity to
observe a private signal, i 2 = fh; lg, which is correlated with the true state. If players choose to
observe i , the draws of the signals are conditionally independent, given the true state. The signal,
5
i,
partially reveals information about the true state in the following way:
Pr(
i
= wj w) = pi and Pr(
i
6= wj w) = 1
pi
where pi 2 12 ; 1 . Here, pi measures the precision of player i’s signal, i , so it can be interpreted as
information quality. As pi approaches 12 , the signal becomes less informative about the true state;
and as pi approaches 1; the signal becomes more informative about the true state. Throughout this
paper, we assume both players are homogeneous in that pA = pB = p. The information quality, p, is
public information.
Player i0 s action, ai 2 i = fh; lg, represents a forecast about the true state. If ai = h (ai = l),
this denotes that i0 s forecast is w = H (w = L). The ordering of both players’ actions is decided
exogenously. Without loss of generality, we assume that A is the leader and B is the follower. With
these sequential actions, B has a chance to observe A’s action before taking her own action.
Player i’s ex-post gross payo¤, _ i (ai ; aj ; w), is de…ned by the following table where > 1; > 1;
and 6= :6
w
aB = w
aA = w
1; 1
aA 6= w
;
aB 6= w
;
1; 1
Table 1: Ex-post payo¤ structure
Each player’s gross payo¤ is determined after both players have announced their forecasts and
the true state is revealed. Throughout the paper, we consider the case in which the true state is not
a¤ected by the forecasts.
This payo¤ structure addresses the competitive environment of the two players. Suppose that both
players act identically. Then, if their actions reveal the true state correctly, both players earn +1, and
if not, both earn 1: On the other hand, if both players take di¤erent actions, the player who takes
the correct action gets > 1 and the incorrect player gets
< 1: In other words, if an agent’s
forecast turns out to be correct, the other agent’s identical forecast causes a negative externality
because a good reputation or reward would be divided. On the other hand, if an agent’s forecast
turns out to be wrong, the other agent’s identical forecast causes a positive externality because a bad
reputation or penalty would be shared. However, as ai = w or ai 6= w cannot be veri…ed in advance,
uncertainty is embedded within the model. We assume that 6= . Hence, our case is either >
or < where the …rst (second) denotes the case in which the payo¤ structure is biased toward the
reward (penalty).7
6
In the table, ai = (6=) w is not the description of each player’s strategy. It denotes the accuracy of announcements
which is determined after the realization of the true state w.
7
The reward for a successful performance and the penalty for a disappointing performance can be asymmetric,
depending on the type of task assigned. Sometimes, a contract may contain clauses that explicitly detail the asymmetry
between reward and penalty. Although not, the agent may subjectively interpret the task as inherently producing the
asymmetry between reward and penalty. For example, if a given task is relatively easy (hard), agents may infer that
there is little (much) to gain and much (little) to lose. That means, if the task is relatively easy (hard), the agent may
evaluate her situation as the one in which the payo¤ structure is biased toward the penalty (reward).
6
In this model, we assume that A is always given, exogenously, to A. On the other hand, for B, we
consider both cases; i) the private signal, B , is given exogenously; and ii) it is costly to acquire B .
In the latter case, B must pay the information cost in order to observe B . Below are the de…nitions
used throughout this paper.8
De…nition 1
Truthful revelation: When player i observes her signal i , if ai = i ; we say
that i is revealed truthfully.
De…nition 2
Herding: Suppose that B observes her signal B . When B =
6 aA , if aB 6= B ;
we say that B exhibits herding.
De…nition 3
Anti-herding: Suppose that B observes her signal B . When B = aA , if
aB 6= B , we say that B exhibits anti-herding.
De…nition 4
Imitation & Deviation: When B does not observe B , if aB = aA ( aB 6= aA );
we say that B imitates (deviates from) A.
As A’s signal A is private information, she can strategically decide whether to follow A or not
in announcing her forecast aA . As to this A’s strategic behavior, we restrict our attention only to
the separating equilibrium. This restriction can be justi…ed by the argument that we are mainly
interested in the case where there is credible public information the follower B can free ride on. In
our setting, this denotes the case where B can infer A’s private signal perfectly. As two types are
considered for A and each type has two actions to take, B can infer A’s private signal A perfectly in
any separating equilibrium if it exists. On the other hand, if a pooling equilibrium exists, A cannot
be inferred from aA because one type of A deviates from A . If the public information is not credible
in this sense, then it may give B more incentive to acquire costly private signal B . In other words,
the case where B can infer A should be analyzed by priority because that is the most adverse setting
in inducing B’s costly investment on B . If B is acquired even in this setting, probably it would
be more likely to occur in some less adverse cases. Thus, we focus on the existence of separating
equilibrium.9
4
Exogenous Information
In this section, we assume that the private signal, i , is given exogenously to each player - a necessary
prelude to the analysis of the case with endogenous information acquisition. By comparing the results
from both cases, we can highlight the novel …ndings of the case where the information acquisition is
endogenous.
The timing of the game of this case is as follows:
T1) Nature decides the true state w. Both players observe the ex-post payo¤ structure and the
information quality p.
8
Note that we assume that A acts as the leader and B acts as the follower.
Nevertheless, the existence of a pooling equilibrium where A cannot be inferred perfectly should not be excluded.
Its existence should be checked if our interest is to characterize every equilibrium.
9
7
T2) A observes A . Then, she announces a forecast after deciding whether or not to truthfully
reveal A .
T3) B observes B and aA . Then, she announces a forecast after deciding whether or not to
truthfully reveal B .
T4) The true state w is revealed, and each player earns a payo¤.
Now consider each player’s strategy. A’s pure strategy is sA :
! A , where
= fh; lg and
A = fh; lg. That is, given A , A decides whether or not to truthfully reveal the signal. For B,
sB :
= fh; lg and i = fh; lg for i 2 fA; Bg. B’s pure strategy di¤ers from
A ! B , where
A’s because B has a chance to observe aA before announcing her own forecast.
The equilibrium concept used is the perfect Bayesian equilibrium. Note that, since A is private
information, whether A = aA or A 6= aA cannot be veri…ed. Let 2 [0; 1] be B’s belief that A
truthfully reveals A . Then, the strategy pro…le s = fsA ; sB g and constitute a Perfect Bayesian
equilibrium if each player’s expected payo¤ is maximized, given , the other …rm’s strategy, and,
especially, is consistent with sA in terms of Bayesian updating. As mentioned, in characterizing the
equilibrium, we restrict our attention only to the separating equilibrium.
Our game can be represented by the following game tree.
< Figure 1 here >
In our setting, there are two types of A, i.e., A 2 fh; lg, and each type has two actions to take,
i.e., aA 2 fh; lg. As B observes her own signal B 2 fh; lg, she faces one of the following two cases:
B = aA and B 6= aA . B’s equilibrium strategy should describe whether or not B follows B in each
of these two cases. Thus, we have to consider following four strategies: i) aB = B regardless of B ,
ii) aB = B if B = aA and aB 6= B if B 6= aA (Herding), iii) aB 6= B if B = aA and aB = B if
aA 6= B (anti-herding), iv) aB 6= B always.
The leader A does not know which signal B 2 fh; lg is observed by the follower B in the subgame.
Thus, A’s posterior belief should be Pr( w; B j A ) and A’s expected payo¤s depending on B’s strategy
can be calculated from
E
E
A ( aA
A ( aA
=
6=
Aj A)
=
Aj A)
=
X
B 2fh;lg
X
X
Pr( w;
B j A ) A (aA
=
A ; aB ; w)
Pr( w;
B j A ) A (aA
6=
A ; aB ; w)
w2fH;Lg
X
B 2fh;lg w2fH;Lg
(1)
B ; A =h)
A =hjw) Pr(w)
where Pr( w; B j A = h) = Pr(w;
= P Pr( B ; Pr(
. Here, E A ( aA = A j A ) is A’s
Pr( A =h)
A =hjw) Pr(w)
w2fH;Lg
expected payo¤ when she follows A , and E A ( aA 6= A j A ) is the one when she deviates from A .
8
Next, given
E
E
B
2 fh; lg and aA , B’s expected payo¤ is derived from
B
B
( aB =
( aB 6=
B j aA ; B )
=
B j aA ; B )
=
Pr( ~A ;
X
Pr( wj ~A ;
B) B
(aA ; aB =
B ; w)
Pr( wj ~A ;
B) B
(aA ; aB 6=
B ; w)
(2)
w2fH;Lg
X
w2fH;Lg
jw) Pr(w)
and ~A denotes the inferred A according to B’s
B jw ) Pr(w)
belief, . For example, given aA = h, if B believes that A is truthful ( = 1), she infers that A = h,
but if not ( = 0), she infers that A = l. Here, E B ( aB = B j aA ; B ) is B’s expected payo¤ when
she follows B , and E B ( aB 6= B j aA ; B ) is the one when she deviates from B .
where Pr( wj ~A ;
B)
=
P
w2fH;Lg
B
Pr( ~A ;
Lemma 1
1) In the game tree, the cases I and IV are symmetric and the cases II and III are symmetric.
2) Given B’s same best response, the strategy aA = A strictly dominates the strategy aA 6= A .
Proof of Lemma 1
In the appendix.
A
Then, we can show that there exists a separating equilibrium where each type
in announcing aA , and moreover it is the unique separating equilibrium.
Proposition 1
The equilibrium where aA =
A
A
2 fh; lg follows
is the unique separating equilibrium.
Proof of Proposition 1
In the appendix.
According to Proposition 1, the leader A truthfully reveals her signal A . This implies that B
can perfectly infer A by observing aA . Then, we can focus on deriving B’s best response knowing
that B assigns zero probability to the possibility that A deviates from A . In round 2, B faces one of
the following two cases: B = aA or B 6= aA . For each case, we compare two equations of (2). The
analysis yields the following result.
Proposition 2
p
+ + +1
.
Suppose that i is given exogenously to both players. Let p = ( +1)
1
1
1) Suppose > : There exists a critical value p 2 2 ; 1 such that if p 2 2 ; p , B announces
a di¤ erent forecast from A by exhibiting anti-herding and if p 2 (p ; 1), B reveals B truthfully.
2) Suppose < : Then, B always announces an identical forecast to A by exhibiting herding.
Proof of Proposition 2
In the appendix.
9
As Proposition 2 says, B’s best response depends on both the asymmetry in the reward and
penalty and the information quality as follows. First, suppose that the reward is greater than the
penalty, i.e., > : In order to earn ; B must correctly forecast and, moreover, be the only one
who does so. For example, if aB = aA , even if the forecast turns out to be correct, only +1 (< ) is
earned. Hence, B has an incentive to di¤erentiate herself from A by announcing a di¤erent forecast.
However, always announcing a di¤erent forecast is not optimal because it also entails the risk of being
penalized alone. For example, if it turns out that aB 6= w; then B’s payo¤ can be
(< 1), which
is the worst outcome. Hence, B should balance the incentive to di¤erentiate and the incentive to
correctly forecast, which depends upon the quality of the information.
If the quality of the information is relatively low, i.e., p 2 21 ; p ; B always announces the opposite
forecast from A. That is, if B = A , B deviates from her signal (anti-herding) but if B 6= A , B
reveals her signal truthfully. Although both players observe the same signal, B has only a weak belief
in the accuracy of the signal because the quality of the information is relatively low. Thus, B is more
biased toward earning alone and the incentive to di¤erentiate dominates. Then, B always wants
to announce a di¤erent forecast from A, yielding anti-herding when B = A . On the other hand,
suppose that the quality of the information is relatively high, i.e., p 2 (p ; 1). Even though B still has
an incentive to di¤erentiate, she also has a relatively strong belief in the accuracy of her signal, due
to the relatively high quality of the information. Hence, the incentive to di¤erentiate is dominated,
and therefore, she truthfully reveals her signal without herding or anti-herding.
Second, suppose that the penalty is greater than the reward, i.e., < . Then, B always forecasts
identically to A. That is, if B = A , B reveals her signal truthfully but if B 6= A , B deviates
from her signal (herding). If both agents observe con‡icting signals, the ex-post probability for each
state is :5. In this situation, making identical forecasts yields the expected payo¤ E B = 0, whereas
producing di¤erent forecasts yields a negative expected payo¤ because the penalty is greater than the
reward. Intuitively, by making a forecast identical to A, B can prevent the worst outcome, in which
she gets B =
, even if her forecast turns out to be wrong. In other words, she can be free from
the concern that she alone will be penalized.
It is a relatively well known notion that if the reward is greater than the penalty, the agent
has an incentive to di¤erentiate, and, in the opposite case, the agent has an incentive to share the
blame. In the present model, both herding and anti-herding arise due to asymmetry in the reward
and the penalty. Notably, agents are assumed to observe the conditionally independent signals with
the same precision. Hence, in the present model, the assumption that the better-informed agents
observe correlated signals, the usual assumption in reputational cheap talk models, is not necessary
for deriving herding or anti-herding.10
Both A and B observe signals with the same precision. Therefore, no one’s information should
outweigh the other’s. However, as is well known, a sequential ordering can suppress the follower’s
truthful revelation of private information. Then, especially for the case in which a sequential ordering
is inevitable, it would be worthwhile to explore the condition under which the follower’s truthful
10
Ottaviani and Sorensen (2000) show that, even in the context of the reputational herding model, conditionally
correlated signals are not necessary for herding to result.
10
revelation of her private signal can be guaranteed.
In Proposition 1, if < ; B’s truthful revelation is not guaranteed because B exhibits herding
for all p 2 12 ; 1 . It is essential that the penalty should not be greater than the reward. Even when
the reward is greater than the penalty, however, B truthfully reveals B only if p 2 (p ; 1).
Corollary 1
1) @p @( ; ) > 0 and lim p ( ; ) = 1
2) lim p ( ; ) =
!
1
2
!1
Proof of Corollary 1
In the appendix.
The result @p @( ; ) > 0 implies that the large reward is not optimal to induce B’s truthful revelation
of B . The parameter set of p for which B exhibits anti-herding increases because the incentive to
di¤erentiate works too much. The result lim p ( ; ) = 1 shows that the excessive reward induces B
to exert anti-herding for all p 2
1
2; 1
!1
. On the other hand, the result lim p ( ; ) =
!
1
2
demonstrates
that as ! , in the limit, B reveals B truthfully for all p 2 12 ; 1 . Hence, B’s truthful revelation of
1
B is guaranteed for all p 2 2 ; 1 . This shows that, in order to maximize the parameter set of p for
which B truthfully reveals B , the di¤erence between and should be small as much as possible.
Indeed, it can be shown that if = , there exists an equilibrium in which B reveals her private
signal truthfully for all p 2 21 ; 1 .11 Hence, when a sequential ordering is inevitable, the symmetry
between the reward and the penalty is the optimal payo¤ structure for inducing the follower’s truthful
revelation of the informative private signal.
5
Costly Information Acquisition
In this section, the model is extended to include the case in which the follower B must pay an
information cost in order to observe her own signal. That is, whether or not to observe B is B ’s
endogenous decision problem. However, we maintain the assumption that A can observe A for free,
so we can limit our focus on B’s strategic decision about acquiring costly private information when
she knows that the signal revealed by A is always correlated with the true state.
As B is costly to acquire, B’s net payo¤ is
B
(aA ; aB ; w) = _ B (aA ; aB ; w)
where is an indicator function, such that
is the gross payo¤ de…ned in Table 1.
= 1 if
B
c
is acquired; and
= 0 if not. Here, _ i (ai ; aj ; w)
11
When = , if B = aA , B always truthfully reveals B (from (A1)); and if B 6= aA , B is indi¤erent between
truthful revelation and exhibiting herding (from the proof of Lemma A.2). Then if we let z = Pr(aB = B ) when
B 6= aA , there exists a mixed strategy equilibrium in which A always truthfully reveals A and B truthfully reveals
B with probability z 2 [0; 1]. This equilibrium can be con…rmed from (A3) and (A4), which show that A’s strictly
dominant strategy is to reveal A truthfully regardless of B’s truthfulness in revealing B .
11
The timing of the game in this scenario is as follows:
T1) Nature decides the true state w. Both players observe the ex-post payo¤ structure, the quality
of information p, and the information cost c.
T2) A observes A . Then, she announces a forecast after deciding whether or not to reveal A
truthfully.
T3) B observes aA . Then, she makes a decision of whether or not to observe the costly signal B .
If she observes B , she also decides whether to reveal it truthfully. If she does not observe B , she
decides whether to imitate or deviate from aA . Then, she announces her forecast.
T4) The true state w is revealed and each player earns the payo¤.
A’s pure strategy is the same as in the case in which information is given exogenously. B has
an additional decision: whether or not to acquire costly B . Thus, sB : A ! B
B where
i = fh; lg and B = fObserve B and pay c, Don’t observe B g. The corresponding game tree is as
follows.
< Figure 2 here >
We continue to focus on the separating equilibrium. The analogous reasonings used in the previous
section yield that the cases I and IV are symmetric and the cases II and III are symmetric. It can
also be checked that, for B’s same best response, the strategy aA = A strictly dominates the strategy
aA 6= A . These result in that the unique separating equilibrium is the one where aA = A .
Proposition 3
Suppose that B is costly to acquire, i.e. c > 0. The equilibrium where aA =
separating equilibrium.
A
is the unique
Then, we can focus on deriving B’s equilibrium strategy knowing that B assigns zero probability
to the possibility that A deviates from A .
The analysis procedure is as follows. In following, ~A denotes the inferred A according to B’s
belief = 1. For the case in which B is observed, B’s best response regarding whether or not to
reveal B truthfully can be derived from (2). Then, B’s expected payo¤ of acquiring costly B can
be derived from the following:
E
B
(aB ) =
X
B 2fh;lg
X
Pr( w;
w2fH;Lg
~
B j A ) A (aA ; aB ; w)
c
(3)
Here, note that B’s posterior belief should be Pr( w; B j ~A ) because even B herself does not know in
advance which signal B 2 fh; lg will be observed when she makes a decision of observing B . The
realization of aB , given B , follows the best response derived in (2). On the other hand, if we consider
12
the case in which she does not observe
E
B
(aB = aA ) =
B,
her ex-ante expected payo¤s are:
X
Pr( wj ~A )
A (aA ; aB
= aA ; w)
(4)
Pr( wj ~A )
A (aA ; aB
6= aA ; w)
(5)
w2fH;Lg
E
B
(aB 6= aA ) =
X
w2fH;Lg
where (4) is the expected payo¤ when B imitates aA and (5) is the expected payo¤ when she deviates
from aA . Then we compare (3), (4), and (5) to derive B’s equilibrium strategy when c > 0. The
analysis yields the following result.
Proposition 4
( +1)
)( +1)
,p=
Suppose that B is costly to acquire, i.e. c > 0. Let c = ( +1)(
2
(
+
+2)
p
(
)+ (
)(
4c)
:
and p^ =
2(
)
Case 1: When > .
1) Suppose c > c . If p 2 12 ; 2+1++ , B deviates from aA without observing
1+
2+ +
; 1 , B imitates aA without observing
2) Suppose 0 < c < c .
aA without observing B ; if
if p 2 (^
p; 1) ; B imitates aA
Case 2: When < .
1) B always imitates aA
p
( +1)( +1) c(
B
and if p 2
B.
Then, there exist p; p^ 2 21 ; 1 such that if p 2 12 ; p , B deviates from
p 2 (p; p^) ; B acquires the costly signal B and reveals it truthfully, and
without observing B .
without observing
B.
Proof of Proposition 4
In the appendix.
For simplicity, the following notations will be used: d denotes B’s strategy to deviate from aA
without observing B , m denotes B’s strategy to imitate aA without observing B , s denotes B’s
strategy to observe B and reveal it truthfully, and B denotes B’s strategy:
First, consider the case in which > . If c > c , then B 2 fd; mg. This represents a situation
where a su¢ ciently high information cost outweighs the expected gain of observing a costly signal.
Hence, B makes use of the existing information A . Whether she imitates or deviates from aA depends
upon the information quality. If it is relatively high, B = m and if not, B = d because a relatively
high (low) p means that A has the greater probability of being correct (incorrect).
If 0 < c < c ; then B 2 fd; s; mg: Interestingly, the acquisition of costly information is not
monotone with respect to the information quality. The costly information is acquired only if its
quality is moderately accurate. If the quality of the information is su¢ ciently high (low), B has a
strong belief for that A = w ( A 6= w). Thus, although the information cost is not su¢ ciently high,
the expected net gain of observing B is less than the expected payo¤ of making use of the existing
information. Hence, B = m ( B = d). That is, if the information quality is extreme, it strongly
signals whether A is likely to be correct or incorrect. Hence, compared to acquiring a costly private
13
)
,
signal, making use of A attains a greater expected payo¤. Meanwhile, if the information quality
is intermediate, i.e., p 2 (p; p^) ; it does not strongly signal whether A = w or A 6= w: Hence, the
expected net gain of observing B is greater than that of making use of the existing information.
Therefore, B has an incentive to observe B even though it is costly, so B = s.
Second, suppose that < : Then, B = m. Here, it should be noted that B is not acquired,
not because it is costly to acquire, but because the payo¤ structure is biased toward the penalty.
Recall that, when B is exogenous and so c = 0; B always exhibits herding if < . Thus, B has
no incentive to acquire a costly private signal because, even if it is a di¤erent signal from what the
leader observed, she would ignore it and exhibit herding.
Remark 1
Suppose that B is costly to acquire, i.e. c > 0. Then, the follower B exhibits neither herding nor
anti-herding for the acquired private signal.
The procedure for deriving Proposition 4 also demonstrates that if B observes costly B , she always
reveals it truthfully. This is an intuitive result. As B is costly to acquire, before making a decision
on whether or not to observe B , B compares the expected gain that will result from observing the
costly B and the expected gain from making use of A . B observes B only if the former is greater
than the latter. Hence, if B observes the costly signal, there is no incentive to ignore it regardless of
either B = aA or B 6= aA . If she had any incentive to ignore B , she would have no reason to pay
a cost in order to observe it. In this way, the introduction of costly private information raises the
issue of "free-riding" rather than the issue of "herding or anti-herding". That is, if the acquisition of
information is endogenous and we are interested in the truthfulness of revealed information, the main
question should be whether or not an agent will invest in a costly information, rather than whether
or not the acquired information will be ignored through herding or anti-herding.
Corollary 2
The summary of qualitative comparative static results can be presented as follows.
c
p^
p
+
+
+
< Table 1: The e¤ects of ; ; c on p^; p when
c
c
+
+
+ (if
(if
< Table 2: The e¤ects of ;
Proof of Corollary 2
14
>
and 0 < c < c >
> 3 + 2)
< 3 + 2)
on c when
>
>
In the appendix.
In Proposition 4, the critical values of the information quality are identi…ed as a function of the
reward, the penalty and the information cost, i.e., p^ = p^ ( ; ; c) and p = p ( ; ; c). According to
Corollary 2, when > and 0 < c < c , as increases or decreases, i) the parameter set of p
where B deviates from aA without observing B increases, and ii) the parameter set of p for which B
imitates aA without observing B decreases. This result is intuitive because the reward works for the
incentive to di¤erentiate and the penalty works for the incentive to avoid the worst outcome. It also
says that, as expected, the parameter set of p for which the costly information is acquired decreases
as the information cost c increases. For c = c ( ; ), the increase in reward always increases c , so
it is more likely that the parameter set of c for which the truthful equilibrium is derived increases.
On the other hand, the e¤ect of the change in the penalty depends on whether or not the reward is
su¢ ciently larger than the penalty. If the reward is su¢ ciently large, even if the penalty increases,
c still increases. However, if not, c decreases as the penalty increases. Hence, it is more likely that
the information cost outweighs the incentive to acquire information.
Proposition 4 says that, despite the costly acquisition of the private signal and the availability
of existing free public information, there exists an equilibrium in which the costly signal is acquired
and revealed truthfully. In order to induce that ideal equilibrium, at the least the penalty should
not be greater than the reward. That is, the payo¤ structure under which the reward is greater than
the penalty is a necessary condition for the follower’s forecast to be informative. Even when the
reward is greater than the penalty, however, a truthful equilibrium exists as long as the prevailing
public information is of moderate quality. In other words, the acquisition of the costly signal is not
monotone with respect to the information quality.
Corollary 3
Suppose that B is costly to acquire, i.e. c > 0 and > .
1) As ! , in the limit, B always imitates aA without observing B for all c > 0.
2) As ! 1, in the limit, B always deviates from aA without observing B for all c > 0.
Recall that when B is given exogenously for free, the optimal payo¤ structure for inducing
B’s truthful revelation of B is the symmetric reward and penalty. Meanwhile, in Proposition 4,
! 12 . Hence, as ! , in the limit, B always imitates aA without
lim c ! 0 and lim 2+1++
!
!
observing B and so B’s forecast is not informative at all. This implies that the optimal payo¤
structure when B is given exogenously is no longer optimal when it is costly to acquire B . Intuitively,
the reward should be greater than the penalty, at least to a certain degree where the bene…t outweighs
the costly investment on a signal. Nevertheless, the excessively large reward is also not optimal for
driving B’s informative forecast. In Proposition 4,
lim c =
!1
B
+ 1, lim
!1
1+
2+ +
= 1, lim p = 1, lim p^ = 1
!1
!1
This implies that, if ! 1, in the limit, the follower always deviates from aA without observing
for all c > 0. The excessively large reward makes the incentive to di¤erentiate work too high.
15
Then, the follower has no incentive to acquire her own private information: she would ignore it, opting
to announce a di¤erent forecast as the leader. If she had any incentive to ignore B , she would have
no reason to pay a cost in order to observe it. In other words, if we let k =
, the optimal k
which maximizes the parameter set of p for which the follower’s forecast is informative should be that
0 < k < 1.
6
6.1
Discussion
The case where the information quality, p, is not known
We have assumed that the information quality p is public information, which a¤ects the follower’s
strategic behavior because it signals the precision of the public information. Then, what would happen
if p is not known? Given that pA = pB = p, if the information quality p is not known, agents know
that they observe the signals of the same precision, but do not know how precise the signals are. It
is not hard to check i) A’s optimal strategy and ii) B’s optimal strategy when the penalty is greater
than the reward ( > ), of this case. Recall that when p is public information, i) the leader A’s
optimal strategy is to follow his signal A for all p 2 12 ; 1 , and ii) the follower B’s optimal strategy
when > is to imitate aA without observing B for all p 2 21 ; 1 . For both cases, the information
quality p does not a¤ect the optimal strategy, implying that there should be no change in the optimal
strategy even when p is not public information. However, B’s optimal strategy when the reward is
greater than the penalty ( < ) is a¤ected if p is not public information. To analyze this case, we
…rst have to de…ne the distribution from which p is drawn. Then, player i’s expected payo¤ can be
R1
expressed by E i = 1 i (p) dF (p) where p 2 12 ; 1 and F (p) is the distribution.
2
As a simple motivating example, consider the case where F (p) is uniform distribution. As B does
not know the precision of the signal, B’s expected payo¤ is a function of the payo¤s, and so B’s
optimal strategy depends on how large the reward is. As the reward gets larger (smaller), B has more
(less) incentive to di¤erentiate herself from A. If the reward is extremely larger than the penalty, the
incentive to di¤erentiate always dominates. Hence, B deviates from A’s action aA without observing
B . If the reward is not signi…cantly larger, B would like to observe B and reveal it truthfully, as
long as the information cost is not too high. As the reward is not su¢ ciently large, B wants to be
balanced between two incentives of di¤erentiating herself and making a correct forecast. Observing
B is a compromise of these two incentives. However, if the information cost is too high and acquiring
B is not an option, then B’s optimal strategy depends on how large the reward is. If the reward is
slightly larger than the penalty, then B imitates aA because not su¢ ciently large reward induces B to
be biased toward avoiding the risk of being penalized alone. On the other hand, if the reward is larger
than the penalty to a certain extent, then she deviates from aA because the incentive to di¤erentiate
dominates.
6.2
The case where the leader’s signal is also endogenous
The current model assumes that only the follower B’s signal B is endogenous. As mentioned, the
assumption that A is given exogenously can be justi…ed by the reasoning that we would like to focus
16
on the strategic decision of acquiring costly private signal when there is credible public information.
However, considering the case where both agents’signals are endogenous would be a very meaningful
extension which should be explored in studying the topic of endogenous private signal. It brings an
interesting question of how the follower responds when she knows that the leader’s announcement
is not informative but still a¤ects her payo¤, which in turn a¤ects A’s decision of observing costly
signal. The analysis done in the current model is a necessary prelude to analyze this case.
The backward induction can be used in our setting to consider this case. Note that A should make
a decision of observing A 2 fh; lg while she does not know which signal will be observed. What A
knows is that she should follow the signal A , i.e., aA = A , if it is observed, regardless of A . (Here,
we can use the intuition that there is no deviation from the signal which is costly to acquire.) B also
knows this and she has a prior belief that Pr ( A = h) = Pr ( A = l) = 21 . In that sense, both cases of
A = h and A = l are ex-ante symmetric to both players. Then, A’s decision of observing the costly
signal does not depend on A . The information quality, the information cost, and the payo¤s are all
needed to identify the condition under which A observes A . Those are public information, so B is
informed of all of those. Thus, B can identify whether or not A observed A , and so the backward
induction can be used to derive the equilibrium. The equilibrium can be characterized in this way.
We …rst derive B’s best response when A makes a randomized announcement without observing A .
Note that we already know B’s best response when A observes A . Then, A can foresee B’s best
responses contingent on her decision of observing of A , and uses those in her decision of whether or
not to observe A .
It is not easy to foresee clearly what would happen in this case if the entire analysis is not done
in detail. However, although possible to be done, it is not tractable to do an analysis in our current
setting. The followings are what we can expect using the results derived in the current model. First,
if the penalty is greater than the reward, there should exist no equilibrium where both players’
announcements are informative. We already know that B does not observe her signal in any case if
A is observed. B may have an incentive to observe B when A was not observed, which depends
on cA ? cB where ci = ci ( ; ; p) is the critical value of player i’s information cost above which
i cannot be observed. When A was not observed, if cA > cB , no player’s announcement can be
informative because both players have the same information cost, so c > cA > cB . We can expect
the equilibrium where B observes B when A was not observed only if cB > c > cA . Next, if the
reward is greater than the penalty, we know that B has an incentive to observe B even when A was
observed. Thus, we can expect the equilibrium where both players’announcements are informative.
Once again, it should depend on cA ? cB . When A was observed, if cA > cB > c or cB > cA > c, B
would be willing to observe B , and so both players’announcements would be informative. In other
cases, B will imitate or deviate from A depending on the quality of information without acquiring
B . In this way, the most important task to be done to characterize the equilibrium of this case would
be to identify ci = ci ( ; ; p).
17
7
Concluding Remarks
In this paper, we study how the agent’s strategic behavior in regard to the acquisition and revelation
of costly private information is a¤ected by the payo¤ structure when there is public information on
which the agent can free-ride. The ideal situation is the one in which the agent invests in a costly
private signal without relying on public information and reveals it truthfully. If the penalty is greater
than the reward, then the agent only cares about avoiding the penalty. Hence, the follower does not
acquire costly private information, and so her forecast is not informative. If the reward is greater than
the penalty, on the other hand, as long as the existing public information is moderately accurate, the
agent acquires costly private information. Moreover she reveals it truthfully without exerting herding
or anti-herding. Hence, the payo¤ structure under which the reward is greater than the penalty is
a necessary condition for this equilibrium. However, an excessively large reward is also not optimal
because it makes the incentive to di¤erentiate work too high, yielding no investment in costly private
information.
Some results of the present model depend on the assumption that both agents observe the same
quality signal. Deriving the equilibrium in which the follower’s forecast is informative would be well
worth considering, especially under this assumption, because no one’s information should be ignored
if both players are of the same ability. One potential extension regarding this point is to consider
heterogeneous agents who observe the signals with di¤erent precision. We also consider the case in
which the market commits ex-ante to a particular payo¤ structure. Although the payo¤ structure
given in this model can be interpreted as a short-cut for the relative reputational concerns, the direct
incorporation of it may provide the answers to the question of how an agent’s incentive to acquire
costly information is a¤ected by reputational concern. These extensions await future work.
18
8
Appendix
8.1
Proof of Lemma 1
1)
Consider the cases I and IV of the game tree. Here, it should be noted that B’s strategy should
describe whether or not B follows B in each of two cases of B = aB and B 6= aB . Using (1), it can
be checked that E A ( aA = hj A = h) = E A ( aA = lj A = l) for B’s same strategy. So, the cases I
and IV are symmetric for A.
Next, using (2), we can show that
E
B
( aB =
B j aA
= h;
B
= h) = E
B
( aB =
B j aA
= l;
B
= l)
E
B
( aB 6=
B j aA
= h;
B
= h) = E
B
( aB 6=
B j aA
= l;
B
= l)
E
B
( aB =
B j aA
= h;
B
= l) = E
B
( aB =
B j aA
= l;
B
= h)
E
B
( aB 6=
B j aA
= h;
B
= l) = E
B
( aB 6=
B j aA
= l;
B
= h)
Thus, the cases I and IV are symmetric even for B. This symmetry for both players implies that
each player’s optimal strategy in both cases should be the same. Likewise, the cases II and III are
symmetric, implying that each player’s optimal strategy in both cases should be the same.
2)
Use (1) in the following.12 If B exhibits herding when
E
A ( aA
=
If B exhibits anti-herding when
B
E
If B follows
B
E
A ( aA
=
Aj A)
E
A ( aA
= aA and follows
Aj A)
E
A ( aA
6=
6=
B
B
6= aA and follows
Aj A)
when
Aj A)
B
when
B
= aA ,
2>0
6= aA ,
= ( + ) (2p
1) > 0
truthfully always,
A ( aA
=
Aj A)
E
A ( aA
6=
Aj A)
= (2p
Thus, given B’s same best response,always the strategy aA =
the strategy aA 6= A is a strictly dominated strategy.
8.2
= 4p
B
1) (
A
p + p + 1) > 0
is a strictly dominant strategy and
Proof of Proposition 1
Consider a separating strategy: "aA = h if A = h and aA = l if A = l." Given this strategy, B
believes that A = h if aA = h and A = l if aA = l. Suppose that, as one example, B’s best response
when aA = h is "always aB = B regardless of B ". In this case, by the symmetry, B’s best response
when aA = l should also be "always aB = B regardless of B ". Now, if the type A = h deviates
12
We exclude the case where B always deviates from
B,
because it is a strictly dominated strategy for B.
19
from this strategy and selects aA = l, B believes that A = l, and her best response is "aB = B
regardless of B ". We know that, by Lemma 1, 2), E A ( aA = A j A ) > E A ( aA 6= A j A ). Hence,
the type A = h has no incentive to deviate from aA = h. Likewise, the type A = l has no incentive
to deviate from aA = l. This should hold for other B’s best responses. Hence, the separating strategy:
"aA = h if A = h and aA = l if A = l." is an equilibrium strategy.
Next, consider another separating strategy: "aA = l if A = h and aA = h if A = l." As one
example, if B’s best response when aA = l is always aB = B regardless of B , this should be the same
even when aA = h. If A = h deviates from this strategy and selects aA = h, still B’s best response is
aB = B regardless of B . Then, by Lemma 1, 2), E A ( aA = A j A ) > E A ( aA 6= A j A ). Hence,
A = h deviates from aA = l and instead selects aA = h. Thus, "aA = l if A = h and aA = h if
A = l." is not an equilibrium strategy. This result holds for other B’s best responses.
Thus, "aA = h if A = h and aA = l if A = l." is the unique separating equilibrium.
8.3
Proof of Proposition 2
Without loss of generality, assume that
Suppose that B = aA . From (2),
E
B
(aB =
B)
=
X
B
= h. (The following result should hold even when
Pr( wj ~A = aA ;
B) B
(aA ; aB =
B ; w)
=
w2fH;Lg
E
where E
B (aB
6=
=
X
Pr( wj ~A = aA ;
B) B
w2fH;Lg
(aA ; aB 6=
B ; w)
=
= l.)
1
2p + 1
(1 p)2
p2
2p2 2p + 1
is B’s expected payo¤ when he follows his signal B and E B (aB 6= B ) is the
Pr( ~A ; B jw) Pr(w)
. Then,
one when he deviates from his signal. Here, Pr( wj ~A ; B ) = P
~
w2fH;Lg Pr( A ; B jw ) Pr(w)
B
(aB =
B)
2p
2p2
B
E
B)
B (aB
=
B)
E
B (aB
6=
B)
=
(
)p2 + 2(1 + )p (1 + )
(2p2 2p + 1)
The denominator is 2p2 2p + 1 > 0 for all p 2 21 ; 1 . Now, let f (p) (
)p2 + 2(1 + )p (1 + ).
First, if < , then f (p) > 0 for all p 2 21 ; 1 . So always E B (aB = B ; w) > E B (aB 6= B ; w).
Second, if > , we can check the followings: 1) f (p) is a strictly concave function, 2) f (p) attains
2(1+ )
2(1+ )
( +1)( +1)
> 0, 3) f 21 = 4 < 0, and
the maximum value at p = 2(
) > 1 where f 2(
) =
(
)
4) f (1) = 1 + > 0. Then, for p 2 21 ; 1 , f (p) is a monotone increasing function. Also there exists
p 2 21 ; 1 such p
that p 7 p =) f (p) 7 0. Therefore, p 7 p =) E B (aB = B ) 7 E B (aB 6= B )
+1
+ + +1
where p =
:
Lemma A.1
p
+ + +1
Suppose that B = aA . In following, p = +1
.
1
1) If
> ; there exists a critical value p 2 2 ; 1 such that if p 2
herding and if p 2 (p ; 1) ; she reveals her signal truthfully.
2) If < , she reveals her signal truthfully for all p 2 12 ; 1 .
20
1
2; p
, she exhibits anti-
Next, suppose that
E
B
(aB =
6= aA . From (2),
B
B)
X
=
Pr( wj ~A = aA ;
B) B
(aA ; aB =
B ; w)
=
2
w2fH;Lg
E
B (aB
6=
B)
X
=
Pr( wj ~A = aA ;
B) B
w2fH;Lg
(aA ; aB 6=
B ; w)
=0
The comparison yields the following result.
Lemma A.2
Suppose that B 6= aA . Then for all p 2
< , B exhibits herding.
1
2; 1
, if
> , B reveals her signal truthfully and if
From Lemmas A.1 and A.2, the following is derived.
Lemma A.3
p
+ + +1
.
In following, p = ( +1)
1
1) Suppose that > . For p 2 2 ; p , if B = aA , she exhibits anti-herding and if
she reveals B truthfully. For p 2 (p ; 1), she reveals B truthfully regardless of aA .
2) Suppose that < . For all p 2 21 ; 1 , if B = aA , she reveals B truthfully and if
she exhibits anti-herding.
B
6= aA ,
B
6= aA ,
This yields Proposition 2.
8.4
Proof of Corollary 1
p
p
( + +2) 2 ( +1)( +1)
( + 1) p
. Here, ( + + 2) 2 ( + 1) ( + 1) > 0 because
2
( +1)( +1)(
)
2
p
p
2
( + + 2)
2 ( + 1) ( + 1) = (
)2 > 0 where + + 2 > 0 and 2 ( + 1) ( + 1) > 0.
p
+ +
+ 1 = 1 and lim (
) = 1. Then,
So, @p @( ; ) > 0. Also, for p , lim ( + 1)
1)
@p ( ; )
@
=
1
2
!1
@ (( +1)
p
+ +
+1)
!1
by L’Hospital’s theorem, lim p ( ; ) = lim
= 1.
@
!1
!1
p
2) For p , lim ( + 1)
+ +
+ 1 = 0 and lim (
) = 0. Then, by L’Hospital’s
!
!
p
p
2
1 +1p2 ( +1)( +1)
1 +1p2 ( +1)
theorem, lim p ( ; ) = lim
= 12 .
=
2
2
2
!
8.5
( +1)( +1)
!
( +1)
Proof of Proposition 4
Assume that B observes her costly signal. Then, her best response is the same as that of the case
where B is given exogenously (Lemmas A.1 and A.2.). Next, assume that B does not observe her
signal. Then, she should decide whether to imitate or deviate from aA given the belief Pr( wj ~A ).
Then, from (4) and (5),
E
B
(aB = aA ) =
X
Pr( wj ~A = aA )
w2fH;Lg
21
A (aA ; aB
=
A ; w)
= 2p
1
(A1)
E
B
(aB 6= aA ) =
X
Pr( wj ~A = aA )
A (aA ; aB
w2fH;Lg
6=
A ; w)
=
p + (1
p)
(A2)
and
E
B (aB
= aA )
E
Then, p ?
? ;
1+
2+ + =) (A1) ? (A2). Here,
1+
1
>
2+ + ? 2 . Therefore, when
(A1) < (A2). On the other hand, if
B (aB
6= aA ) = p(2 +
+ )
1+
2+ +
it can be checked that i)
; if p 2
1+
2+ +
(1 + )
< 1 for p 2
; 1 ; (A1) > (A2) and if p
1
2; 1
< ; (A1) > (A2) for all p 2
.
Lemma A.4
Suppose that B does not observe her signal B .
1) Suppose > : If p 2 2+1++ ; 1 ; she imitates A’s action and if p 2
from A’s action.
2) Suppose < : Then she always imitates A’s action.
1
2 ; 1 and ii)
2 21 ; 2+1++
1+
1
2 ; 2+ +
if
;
, she deviates
Using Lemmas A.1, A.2 and A.4, we derive B ’s strategic decision in regards to observing her
costly signal and being truthful in revelation.
1) When >
Case 1-1) When p 2
If p 2
observing
1+
2+ +
1+
2+ +
;1
; 1 , the strategy B can take is either to imitate aA or reveal
B
truthfully after
B.
E
B
(aB =
B)
=
X
w2fH;Lg
2
= p (
E
B
X
Pr( w;
B 2fh;lg
) + p(
X
(aB = aA ) =
~ = aA )
Bj A
+ 2)
1
Pr( wj ~A = aA )
B (aA ; aB
=
B ; w)
(A3)
c
B(
) = 2p
1
(A4)
w2fH;Lg
where (A3) is the expected payo¤ when B observes her costly signal and (A4) is the expected payo¤
Pr(w; B ;~A )
=
when she imitates A’s action without observing her own signal. In (A3), Pr( w; B j ~A ) = Pr ~
( A)
Pr( B ;~A jw) Pr(w)
P
.
~
w2fH;Lg Pr( A jw ) Pr(w)
Then,
E B (aB = B ) E B (aB = aA ) = p2 (
) + p(
) c
We denote that h(p)
p2 (
) + p(
) c. Then, the following points can be checked: 1)
h(p) is a strictly concave function and it attains the maximum value at p = 21 , 2) h(0) = c < 0;
3) h(1) =
c < 0; 4) h
c>
1
2
h
1
2
4
, h
1
2
=
1
4
(
)
< 0. Then, for all p 2
c; and 5) h
1+
2+ +
1+
2+ +
=
(
(
)( +1)( +1)
( + +2)2
c: From 4), if
; 1 , (A3) < (A4). On the other hand, if c <
> 0. Then, as h(p) is a monotone decreasing function for p 2
decides h(p) ? 0. If
)( +1)( +1)
( + +2)2
< c, h(p) < 0 for all p 2
22
1+
2+ +
1+
2+ +
;1 , h
1+
2+ +
4
;
? 0
; 1 . On the other hand, if
(
)( +1)( +1)
( + +2)2
h(p) < 0. Here, note that
(
c>
)( +1)( +1)
,
( + +2)2
that if p 2
1
2 ; 1 such
( +1)(
)( +1)
.
4 >
( + +2)2
(A4) for all p 2 12 ; 1
> c, there exists p^ 2
1+
2+ +
(A3) <
1+
2+ +
that if p 2
; p^ , h(p) > 0 and if p 2 (^
p; 1),
Hence, the above can be summarized as follows: If
(
)( +1)( +1)
,
( + +2)2
there exists p^ 2
p
(
)+ (
; p^ ; (A3) > (A4) and if p 2 (^
p; 1) ; (A3) < (A4). Here, p^ =
2(
Lemma A.5
Suppose that >
1) Suppose c >
(
(
1+
2+ +
and p 2
)( +1)( +1)
:
( + +2)2
)( +1)( +1)
:
( + +2)2
. If c <
; 1 . In following, p^ =
(
)+
p
(
2(
)(
)
4c)
1
2; 1
)(
)
such
4c)
.
.
Then, B deviates from A’s action.
2) Suppose c <
Then there exists p^ 2 2+1++ ; 1 such that if p 2
observes her signal and reveals it truthfully and if p 2 (^
p; 1) ; B imitates A’s action.
1+
2+ +
; p^ ; B
Case 1-2) When p 2 21 ; p
If p 2 12 ; p , the strategy B considers is either to observe her signal or deviate from aA without
observing her signal. Her best response in this case can be derived easily from the following reasoning.
Suppose that she observes her signal. In this case, if B = aA ; then she exhibits anti-herding and
if B 6= aA ; she reveals her signal truthfully. Thus, always aB 6= aA . Also, if she does not observe
her signal, she deviates from aA , which yields that aB 6= aA . That is, whether or not she observes
her signal, always it is induced that aB 6= aA . Then, B has no incentive to observe her costly signal.
Thus, in this case, her best response is to deviate from aA without observing B .
Lemma A.6
Suppose that >
and p 2
1
2; p
: Then, B deviates from A’s action.
Case 1-3) When p 2 p ; 2+1++
In this case, the strategy B can take is either to reveal
from aA without observing B . Then,
E
B
(aB =
B)
X
=
w2fH;Lg
= p2 (
E
B
(aB 6= aA ) =
X
X
Pr( w;
B 2fh;lg
) + p(
B
~ = aA )
Bj A
+ 2)
Pr( wj ~A = aA )
truthfully after observing it or deviate
1
B (aA ; aB
w2fH;Lg
B (aA ; aB
=
B ; w)
(A5)
c
6= aA ; w) =
p( + )
(A6)
and
E
B
(aB =
B)
E
B
(aB 6= aA ) = p2 (
) + p (2 + 2)
c
1
We denote that g (p)
p2 (
) + p (2 + 2) c
1. Then, the following can be veri…ed:
1) g (p) is a strictly concave function. 2) g (p) attains the maximum value at p = 22 +2
2 > 1; 3)
g (p = 0) < 0; 4) g p =
1
2
can be checked that, if c >
Hence, if c >
(
1+
. It
2+ +
1+
> 0.
2+ +
)( +1)( +1)
,
( + +2)2
< 0, and 5) g (p = p ) < 0. Now consider the value of g p =
(
)( +1)( +1)
,
( + +2)2
)( +1)( +1)
,
( + +2)2
g (p) < 0 and if c <
(A9) < (A10) for all p 2
23
(
p ; 2+1++
)( +1)( +1)
,
( + +2)2
g p=
. Also, if c <
(
p; 2+1++
there exists p such that if p 2 (p ; p) ; (A5) < (A6) and if p 2
p
( +1)( +1) c(
)
( +1)
p=
:
Lemma A.7
Suppose that >
(
and p 2 p ; 2+1++
. In following, p =
( +1)
p
)( +1)( +1)
1) Suppose c >
: Then, B deviates from A’s action.
( + +2)2
+1)
2) Suppose c < ( ( )(+ +1)(
: Then, there exists p 2 p ; 2+1++
+2)2
deviates from aA and if p 2 p; 2+1++ ; B observes B and reveals it
; (A5) > (A60). Here,
( +1)( +1) c(
)
.
such that if p 2 (p ; p) ; B
truthfully.
Then, Lemmas A.5, A.6 and A.7 yield the following result.
Lemma A.8
Suppose > :
)( +1)
: If p 2 2+1++ ; 1 ; B imitates A’s action and if p 2 21 ; 2+1++ ;
1) Suppose c > ( +1)(
( + +2)2
she deviates from A’s action.
)( +1)
2) Suppose 0 < c < ( +1)(
: If p 2 12 ; p ; B deviates from A’s action, if p 2 (p; p^) ; she
( + +2)2
observes her costly signal and reveals it truthfully, and if p 2 (^
p; 1) ; she imitates A’s action.
2) When <
Suppose < . Then, B ’s best response can be derived easily from following reasoning. When B
observes B , if aA = B ; B reveals her signal truthfully and if aA 6= B ; she exhibits herding. Also if
B does not observe B , B always imitates aA . Thus, whether or not she observes her costly signal, it
is derived that aA = aB . Therefore, B has no incentive to observe her costly signal.
Lemma A.9
Suppose that < . Then, B always imitates aA without observing
B.
Finally, Lemmas A.8 and A.9 yield Proposition 4.
8.6
Proof of Corollary 2
1) First, p^ is determined from p2 (
from the implicit function theorem,
@ p^
=
@
@ p^
=
@
@ p^
=
@c
)+p (
@F
@
@F
@P
@F
@
@F
@P
@F
@c
@F
@P
) c = 0. If we let F =
=
p (1 p)
(2p 1) (
)
=
p (p 1)
(2p 1) (
)
=
(2p
24
1
1) (
)
>0
<0
<0
p2 (
)+p (
) c,
Second, p is determined from p2 (
) + p (2 + 2) c
p (2 + 2) c
1, from the implicit function theorem,
2)
@c
@
@c
@
< 0 if
= ( + 1)2
@p
@
=
@p
@
=
@p
@c
=
3
+2
( + +2)3
@G
@
@G
@P
@G
@
@G
@P
@G
@c
@G
@P
> 0. Also,
=
=
=
@c
@
1 = 0. If we let G = p2 (
2(
(p 1)2
>0
p + p + 1)
2(
p2
<0
p + p + 1)
2(
1
>0
p + p + 1)
= ( + 1)2
< 3 + 2.
25
3 2
.
( + +2)3
So,
@c
@
> 0 if
)+
> 3 + 2 and
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27
T
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III
AH
T
θB=ｌ
Nature
Player B
аA=l
Player A
аA=h
Nature
θB=h
AH
Player B
θB=h
I
T
θB=l
θA=h
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Nature
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θB=ｌ
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θA=l
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AH
Player B
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Player B
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θB=l
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<Figure 1>
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θB=ｌ
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Player B
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Player A
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Player B
Nature
T
AH
Player B
O
D
θB=h
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аA=h
Player A
<Figure 2>
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θA=l
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аA=h
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