This note studies a version of the Stackelberg model in which the Leader has more information about demand than the Follower. We show that there exists a unique D1 equilibrium and that this equilibrium is perfectly revealing. We also give a full characterization of the equilibrium in terms of the posterior beliefs of the Follower and show under which condition there is first mover disadvantage.Separating equilibria;signalling games;Stackelberg competition

Janssen, M.C.W.

Maasland, E.

English

Research Papers in Economics

Center
for
Economic Research
ON THE UNIQUE D1 EQUILIBRIUM
IN THE STACKELBERG MODEL WITH ASYMMETRIC INFORMATION
By Maarten C.W. Janssen and Emiel MaaslandON THE UNIQUE D1 EQUILIBRIUM
IN THE STACKELBERG MODEL WITH ASYMMETRIC INFORMATION
*
Maarten C.W. Janssen
† Emiel Maasland
‡
November 1997
Abstract
This note studies a version of the Stackelberg model in which the Leader
has more information about demand than the Follower. We show that there
exists a unique D1 equilibrium and that this equilibrium is perfectly reveal-
ing. We also give a full characterization of the equilibrium in terms of the
posterior beliefs of the Follower and show under which condition there is
first mover disadvantage.
Keywords: Separating equilibria, signalling games, Stackelberg competition.
JEL Classification: C72, C73, D82, L10
* We thank Eric van Damme and Santanu Roy for helpful comments on earlier versions of the paper.
† Dept. of Economics/Micro, Erasmus University, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands.
‡ CentER for Economic Research, Tilburg University,P.O. Box 90153,5000 LE Tilburg, The Netherlands.1
1. Introduction
The Stackelberg model in which the Leader has some information about demand that the
Follower does not have has been first studied by Gal-Or (1987). She showed that the
model has many perfect Bayes-Nash equilibria, depending on the specification of the out-
of-equilibrium beliefs. Moreover, she demonstrated that unlike the Stackelberg model with
perfect information, there are equilibria such that the Leader makes less profit than the
Follower. She obtains this result by making specific assumptions about the nature of
uncertainty.
In this note we assume that the out-of-equilibrium beliefs satisfy the D1 requirement as
introduced by Banks and Sobel (1987) and Cho and Kreps (1987). We will show that
there exists a unique D1 equilibrium which is perfectly revealing. We will also
characterize the equilibrium strategies and provide a necessary and sufficient condition
under which the Leader’s profit is smaller than the Follower’s. Our result is completely
independent of any distributional assumptions concerning the type of uncertainty.
The note makes use of results obtained by Malaith (1987) and Ramey (1996). Ramey
(1996) extends the analysis of Cho and Sobel (1990) to signalling games with a continuum
of types. Under some appropriate assumptions, he shows that any D1 equilibrium must be
separating. The Stackelberg model with asymmetric information is a signalling model with
a continuum of types. As some of the assumptions made by Ramey (1996), in particular
Assumption 1, do not hold in our case we provide a more intuitive proof of the fact that
any D1 equilibrium must be separating in the special case of our model. Malaith (1987)
shows that under a set of regularity conditions, there exists a unique separating
equilibrium in signalling games with a continuum of types. We basically show that the
Stackelberg model with asymmetric information satisfies the assumptions imposed by him.
Section 2 briefly presents the model. The sequential equilibrium concept and the D1
criterion are defined in Section 3. Results and their proofs are given in Section 4. Section
5 concludes with some comments.
2. The Model
We consider a Stackelberg model with two firms, a Leader and a Follower. Demand is
given by the linear inverse demand function p = a - b(qL+qF), where p is the price and qL
and qF are, respectively, the output chosen by the Leader and the Follower. The Leader2
and the Follower choose their output level so as to maximize profits. The profit functions
of the Leader and the Follower are given by pL(qL,qF,a) and pF(qL,qF,a), respectively. The
output choice of the Leader is observed by the Follower before it makes an output choice
itself. The value of the intercept a is known to the Leader, but unknown to the Follower.
Before observing the output choice of the Leader, the Follower thinks that a is drawn from
some continuous probability distribution with support [aL,aH], where aL > 0. Hence, the
strategies of the Leader and Follower can be written as qL(a), respectively, qF(qL). As the
results are independent of the particular shape of the probability distribution, we do not
make any further assumptions about it. Without loss of generality we assume b to be equal
to 1. The above model is assumed to be common knowledge among the players.
3. Sequential Equilibrium and the D1 Criterion
Let the players’ equilibrium strategies be given by and let the qL(a) and qF(qL),
Follower’s belief about a conditional on observing qL be denoted by the distribution
function µ(aq L ). Moreover, let â(qL) be the expectation of this distribution. If is qL(a)
strictly monotonic, then the strategy is revealing and â(qL) is the inverse of on the qL(a)
relevant domain. The triple is a sequential equilibrium if the {qL(a),qF(qL),µ (a qL)}
following three conditions hold:3
(1) qL(a) Î argmax
qLÎ
pL(qL,qF(qL),a) for all a Î [aL,aH];
(2) qF(qL) Î argmax
qFÎ õ
ó
aH
aL
pF(qL,qF,a)dµ (a qL) for all qLÎ ;
( 3 )a )i f q L Î range qL and õ
ó
{a qL(a) qL}
dµ(a) > 0, then
µ (a qL) is calculated using Bayes Rule;
b) if qL Î range qL and õ
ó
{a qL(a) qL}
dµ(a) 0, then
µ (a qL) is any distribution with the property that
supp µ (a qL) Î cl {a qL(a) qL};
c) if qL Ï range qL, then µ (a qL) is unrestricted.
The above definition is the standard notion of sequential equilibrium applied to the present
context. The full-support assumption in (3) b) is invoked only in establishing that
sequential equilibria satisfying the D1 criterion must be separating.
The D1 criterion imposes restrictions on the out-of-equilibrium beliefs of the Follower. In
particular, let be the equilibrium profit of the Leader pL(a) ºp L ( q L( a ),qF(qL),a)
observing a. Fix qL Ï range Suppose there is a nonempty set A Ì [aL,aH] such that qL.
the following holds: for all ã Ï A, there is an a Î A such that pL(qL,qF,ã) ³ (ã) pL
implies that pL(qL,qF,a) > A sequential equilibrium satisfies the D1 criterion if, and pL(a).4
only if, supp µ
*(aq L )ÌAfor all qL Ï range Intuitively, the Follower observing an qL.
out-of-equilibrium quantity qL is restricted to place zero posterior weight on a type ã
whenever there is another type a that has a stronger incentive to deviate from the
equilibrium, in the sense that type a would strictly prefer to deviate for all qF that would
give type ã a weak incentive to deviate.
4. Analysis
In this section we show that there exists a unique D1 equilibrium. Without imposing
restrictions on the reaction of the Follower we first, however, show that a profit
maximizing strategy for the Leader is non-decreasing.
Lemma. "qF(qL): if qL(a) is a best response to qF(qL), then qL(a) is non-decreasing.
Proof. Suppose the statement in the lemma is not true and that there exist two points a’
and a’’ where a’<a’’, such that Since is a profit maximizing choice given a’ qL>qL . qL
and is a profit maximizing choice given a’’, it follows that:
1 qL
pL(qL a ) qL(a qL qF(qL)) ³ qL (a qL qF(qL )) pL(qL a );
pL(qL a ) qL (a qL qF(qL )) ³ qL(a qL qF(qL)) pL(qL a ).
Multiplying the first inequality by -1 and adding the two inequalities up yields
(qL qL)(a a ) ³ 0.
As and a’’ > a’, this leads to a contradiction. So we conclude that qL(a) is non- qL>qL
1 If qF(qL) would be a mixed strategy, the same argument applies when substituting
EqF(qL) for qF(qL) and EpL for pL.5
decreasing.
Q.E.D.
It is clear that we cannot get any additional results without imposing some restrictions on
the out-of-equilibrium beliefs of the Follower. We believe, however, that some
specifications of the out-of-equilibrium beliefs are not very reasonable. In particular, when
the model is common knowledge, the Follower can also infer the content of the Lemma,
namely that a rational Leader’s strategy is non-decreasing. It seems reasonable that the
out-of-equilibrium beliefs should be consistent with this fact in the following sense: if in
equilibrium the Leader produces some for all values of a in some interval [a’,a’’], qL
then the Follower should infer from observing that the intercept is smaller than or qL e
equal to a’. From the proof of Proposition 1 below it becomes clear that this is essentially
what the D1 requirement amounts to in the context of our model.
Proposition 1. In any D1 equilibrium the strategy is strictly monotonic increasing. qL(a)
Proof. The proof consists of two parts. First, we show that if is not strictly qL(a)
monotonic increasing on [a,aH] for some a > aL, then it cannot be continuous. Second, we
show that when the out-of-equilibrium beliefs satisfy Criterion D1 can neither be qL(a)
discontinuous nor be constant on an interval starting from aL. Hence, the equilibrium
strategy must be strictly increasing.
(1) Suppose is continuous and not strictly increasing on [a,aH] for some a > aL. qL(a)
Then there exist a’ > aL, a’’ and such that We show qL qL(a) qL for all aÎ[a ,a ].
that a firm that observes a’ is strictly better off by producing slightly less.
pL(qL a ) (a
qL
2
ˆ a(qL)
2
)qL ( )6
pL(qL e a ) (a
qL e
2
ˆ a(qL e)
2
)(qL e)( )
As there exist e such that ( ) is strictly larger than ( ). Hence, a ˆ a(qL)>a >ˆ a(qL e),
firm observing a’ will deviate.
(2) Suppose then that is discontinuous and not strictly increasing. Then there exist qL(a)
a’, a’’ and such that and "e>0: If qL qL(a) qL for all aÎ(a ,a ) qL(a e)<qL.
is continuous at a’ we can use the argument under (1) to show that the Leader qL(a)
observing a’ will deviate. Let us then consider the case that is discontinuous at a’. qL(a)
For small enough e is not on the equilibrium path. We first show that for any d>0, qL e
the D1 criterion implies that after observing such an the Follower should place zero qL e
posterior weight on any a strictly larger than a’+d. For any a Î (a’,a’’), the equilibrium
pay-offs are given by
pL(a) (a qL
ˆ a(qL) qL
2
)qL.
The pay-offs when deviating depend on the reaction of the Follower and are given by
pL(qL e a) (a qL e qF)(qL e).
It is beneficial to deviate if, and only if, the first expression is smaller than the second,
i.e., if and only if
ˆ a(qL) qL
2
qL > qF(qL e) 2qLe e(a e).
The equation reveals that for any e > 0 the larger a, the smaller the maximum value of qF
for which it is beneficial to deviate. Hence, if we fix in the definition of the D1 criterion7
A to be equal to (a’,a’+d] for some small positive d, then it is clear from the above that
the following holds: for all ã > a’+d, there is an a in (a’,a’+d] such that pL(qL,qF,ã)
³ (ã) implies that pL(qL,qF,a) > Hence, the D1 criterion requires that supp pL pL(a).
µ
*(aq L )£a’+d.
Next, we focus on the Leader who observes a’+d, where d is small. Using the above
argument, we know that for small enough d and e, Hence, there ˆ a(qL)>a d³ˆ a(qL e).
exist e and d such that
p(qL e a d) (a d
qL e
2
ˆ a(qL e)
2
)(qL e)>p ( q La d ) ( a d
q L
2
ˆ a ( q L )
2
) q L .
Thus, given the D1 requirement it is beneficial for the Leader who observes a’+d to
deviate. By substituting aL for a’ a similar argument can be made to show that qL(a)
cannot be constant on an interval starting from aL. This concludes the proof of Proposition
1.
Q.E.D.
In the next proposition we demonstrate that there exists a unique revealing equilibrium of
the Stackelberg model with asymmetric information. We then characterize it in terms of
the Follower’s conditional expectation of the intercept. Unfortunately, an analytical
expression for qL(a) does not exist.
Proposition 2. The Stackelberg model with asymmetric information has a unique D1
equilibrium.
Proof. Proposition 1 shows that a D1 equilibrium must be separating. Here, we will show
that there is also a unique separating equilibrium. Mailath (1987) shows that in a class of8
models that satisfy certain regularity conditions and an initial value condition, there is a
unique separating equilibrium. It is easily shown that the model analyzed here satisfies the
regularity conditions (1) - (5) of Malaith (1987). The initial value condition holds in our
model if qL(aH)=aH/2. We demonstrate that this is the case as qL(aH) cannot be smaller or
larger than aH/2.
(1) Let the strategy of the Leader be such that qL(aH)<aH/2. Then the Follower will
respond to qL(aH) by producing (aH-qL(aH))/2. If the Leader deviates and sets qL=aH/2, the
posterior expectation of the Follower will be such that â(aH/2)£aH. It follows that produ-
cing qL=aH/2 results in a higher profit when a=aH. Hence, qL(aH) cannot be smaller than
aH/2.
(2) Let the strategy of the Leader be such that qL(aH)>aH/2. Then the Follower will
respond by producing (aH-qL(aH))/2. The profit to the Leader in this case is equal to
pL=(aH-qL-(aH-qL)/2)qL<aH
2/8. If the Leader deviates to aH/2, the expectation of the
Follower will be â(aH/2)<aH, yielding a best response of qF=(â(aH/2)-aH/2)/2. This implies
that the profit to the Leader by deviating is equal to pL=(aH-aH/2-(â(aH/2)-aH/2)/2) aH/2,
which due to the fact that â(aH/2)<aH is larger than aH
2/8. Hence, qL(aH) cannot be larger
than aH/2.
Q.E.D.
We are now in the position to characterize the unique D1 equilibrium. We first determine
the Follower’s best response. This is easily seen to be
qF(qL)
â(qL) qL
2
,
provided qL £ â(qL). Hence, the profit of the Leader who observes a is equal to
(a qL
â(qL) qL
2
)qL (a
â(qL)
2
qL
2
)qL.
The first-order condition for profit maximization by the Leader is, therefore,
In a separating equilibrium, â(qL(a))=a and, hence, the first-order condition simplifies to
a 1
2
â (qL(a))qL(a)
â(qL(a))
2
qL(a) 0.
Solving this differential equation yields
a(qL)
2
1
2
a (qL)qL qL 0.9
where c is a constant.
a(qL) 2qLlnqL cqL,
As qL(aH)=aH/2, it follows that c=2+2ln(aH/2). Substituting this expression for c yields
a(qL) 2qL(1 ln
aH
2qL
). (1)
The increasing function a(qL) is concave on the interval [qL(aL),aH/2], where qL(aL)>0.
Consequently, the inverse function qL(a), which is the decision rule of the Leader, is
convex on [aL,aH]; see Figure 1. As qL(a) is convex and it is easily limaL®0 qL(aL) 0,
seen that qL(a)<a/2 for aÎ[aL,aH). Hence, on the interval [aL,aH) is strictly smaller qL(a)
than the equilibrium quantity in the full information model and the equilibrium quantity of
the Follower is strictly larger.
Figure 1
Now that we have characterized the equilibrium strategy of the Leader in terms of its
inverse function, it is interesting to investigate the conditions under which there is a first
mover disadvantage. We will say that there is an ex ante first mover disadvantage if given
the distribution of a, the expected profit of the Leader is smaller than the expected profit
of the Follower. We will say that there is an ex post first mover disadvantage if for a
given realization of a, the profit of the Leader is smaller than the profit of the Follower.
As we have not made any assumptions regarding the distribution of a on the interval10
[aL,aH], we investigate the scope for ex post first mover disadvantages.
Proposition 3. There is an ex post first mover disadvantage if, and only if, a<3aH/(2e
½).
Proof. It is clear that pL < pF if, and only if, qL < qF. From equation (1) it is clear that this
is the case if, and only if, qL <½ ( q L +2qLln(aH/(2qL))), or ½ < ln(aH/(2qL)). From (1) it
follows that ln(aH/(2qL))=(a-2qL)/(2qL), so that qL < qF if, and only if, 3qL(a)<a . Any a £
aH can be written as (1+g)aH/e
g for some g³0. From (1) again it then follows that qL =
aH/(2e
g) if, and only if, a = (1+g)aH/e
g. Hence, 3qL(a)<aif, and only if, 3aH/(2e
g)<
(1+g)aH/e
g Ûg>½ .A sais decreasing in g, there is an ex post first mover disadvantage,
if a <3 a H/(2e
½).
Q.E.D.
It is clear that ex post first mover disadvantage for some values of a is a necessary
condition for ex ante first mover disadvantage. From the above it can be inferred that if
there is enough probability mass on small enough values of a, there will also be ex ante
first mover disadvantage. Indeed, simulation results in which a is uniformly distributed on
[aL,aH] show that there is indeed scope for ex ante first mover disadvantage if the ratio of
aH/aL is large enough.
5. Concluding Remarks
The analysis of this note shows that there exists a unique D1 equilibrium in the
Stackelberg model with asymmetric information. This result is, among other things, of
interest to the literature on role choice (see, e.g., Mailath, 1993 and Daugethy and
Reinganum, 1994). In that literature it is frequently assumed that there are just two or
three possible states of demand that are sufficiently distinct from each other. This
assumption is made in order to guarantee a unique equilibrium in the subgame in which
the informed player moves first. This note basically argues that such an assumption is not
needed, because by restricting the out-of-equilibrium beliefs in an appropriate way, there
exists a unique D1 equilibrium even if the uncertainty about demand follows a continuous
distribution.
We concentrated on uncertainty about the intercept of the inverse demand function.
However, it turns out that an analogous analysis can be made for the case of uncertainty11
about the slope. Of course, in the latter case, the equilibrium strategy of the Leader is
strictly decreasing (instead of increasing) in the value of the slope parameter.
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Asymmetric Information Acquisition and Behavior in Role Choice Models: An Endogenously Generated Signaling

D1 Signaling Equilibria with Multiple Signals and a Continuum of Types,’

Endogenous Sequencing of Firm Decisions,’

First Mover Disadvantages with Private Information,’

Incentive Compatibility in Signaling Games with a

Strategic Stability and Uniqueness in Signaling

On the Unique D1 Equilibrium in the Stackelberg Model with Asymmetric Information

On the Unique D1 Equilibrium in the Stackelberg Model with Asymmetric Information

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