European Summer Symposium in Economic Theory
Gerzensee, 28 June to 9 July 1999
Mediators and Mechanism Design:
Why Firms Hire Consultants
Kay Mitusch and Roland Strausz
We gratefully acknowledge financial support from the Review of Economic Studies. We would also like to thank
Studienzentrum Gerzensee for their generosity in hosting this meeting.
The views expressed in this paper are those of the author(s) and not those of the funding organization(s) nor of CEPR, which takes
no institutional policy positions
Mediators and Mechanism Design: Why
Firms Hire Consultants
3
3
Kay Mitusch and Roland Strausz
May 27, 1999
Abstract
This paper oers a new type of explanation for economic institutions as playing the
role of mediators in the sense of Myerson (1985) to facilitate communication in contracting settings with ex ante asymmetric information and limited commitment. It
derives necessary and sucient conditions under which mediation is strictly helpful
and provides a straightforward, yet general intuition for this result. As an application of our idea we explain the widely observed use of consultants during the
restructure of rms.
Keywords: mechanism design, communication equilibrium, imperfect commitment
JEL Classication No.: D82, C72
3 Free
University Berlin, Dept. of Economics, Boltzmannstr.20, D-14195 Berlin (Germany); email
addresses: [email protected] and [email protected], respectively. We thank
Helmut Bester, Roman Inderst, seminar participants at Autonoma University in Barcelona and the
"Quatschgruppe 2" for helpful comments and suggestions.
2
1
Introduction
This paper studies the role of mediators in a principal{agent problem with ex ante hidden
information when the commitment power of the principal as contract designer is limited.
Already Myerson (1985, 1986) and Forges (1986) observed the potentially benecial role
of mediators in, what they considered, games of communication. This observation has
important but hitherto unnoticed consequences for the theory of contracting with ex
ante hidden information. Since a major purpose of contract theory is to explain real{life
economic institutions as part of optimal contractual arrangements, it rst of all raises the
question whether mediators should be an integral part of the analysis of optimal contracts.
Excluding mediators from the analysis leads to a rather narrow denition of optimality,
because the obtained optimal contracts might be improved upon by mediators.1 Second
and more compelling, since contracting parties may benet from mediators, one may
expect the existence of real{life economic institutions as reecting the role of mediators.
This paper addresses both these questions.
The rst question concerns the fundamental methodology of the theory of mechanism
design with limited commitment. We feel that this important question has received
too little discussion in the literature.2 Far from settling the issue, this paper merely
stresses its importance and oers two contributions to it. First, we derive necessary and
sucient conditions for mediation to be helpful. This is important as in settings in which
mediators are not helpful the question whether to in{ or exclude mediators in the analysis
is irrelevant.3 The conditions we derive have an intuitive interpretation: a mediator is
helpful if and only if the probability of conict between the principal and the agent is
positive but not too large, i.e. has to be strictly smaller than one.
Second, we provide a straightforward, but general intuition why mediation is benecial
in contracting settings with limited commitment. Since in general one cannot exclude
that mediation aects results, such an explanation may be helpful in judging the role of
mediators without providing a fully{edged analysis. One may explain the benecial role
of mediators in mechanism design problems with limited commitment as follows. The
problem for the principal in a setting without commitment is that she may not want
to obtain all information, since her myopic treatment of it may make it too expensive
1 The
question is also relevant if one is only interested in the set of implementable allocations. For
instance, Crawford and Sobel (1982) study information transmission in a similar model to ours, but
disregard the possibility of mediation.
2 An indirect reference to this problem may be found in Fudenberg and Tirole (1992, p.344) where
they raise it as a fundamental question about the interpretation of an extensive form of a game.
3 For instance in the standard setting with full commitment the principal can contractually commit
to any behavior of the mediator and, consequently, mediators do not aect results.
3
to induce agents to supply it.4 The principal must therefore nd some way to induce
her agents to reveal their information partially. This she achieves when she receives
the agents' messages in some stochastic way such that a message does not uniquely
identify the agent's type and some degree of pooling occurs. Therefore, in a setting
without commitment the extent to which a principal can induce a stochastic mixture of
messages is important. In this respect a mediator is helpful. Without a mediator an
agent must perform the randomization himself and must therefore be indierent between
the allocations that are induced by the messages he uses. This is not required when a
mediator is present, since he may perform the randomization on part of the agents. With
a mediator agents must only prefer the mixing pattern that is designed for them, but
need not be indierent between the allocations over which the randomization occurs.
The second point this paper makes is to introduce a new explanation for economic
institutions as fullling the role of a mediator.5 By modeling our principal{agent problem
as a restructuring problem of a rm, we show how one may explain the wide{spread use
of consultants as playing the role of benecial mediators. More specically, we consider a
principal{agent problem in which a rm's management, the principal, has to choose a restructuring decision and the rm's department, the agent, has private information which
is relevant for the optimal restructuring choices. To ensure a successful restructure management should therefore provide explicit incentives to induce disclosure. This however
may be a dicult task. As information in real{life is a rather fuzzy concept and hard to
verify by third parties, it will be dicult and costly to write explicit incentive contracts
that condition transfers on the exact revelation of information. Implicit contracts may
oer a solution to this problem, but a troubled rm, whose very survival is at stake, will
nd it hard to maintain the reputation that is required to support implicit agreements.
We will therefore assume that information transmission is non{contractible.
The fact that information is non{contractible may lead to major distortions in the
disclosure of private information. Parties will be reluctant to reveal their information, as
guarantees that they will not be hurt by their disclosure cannot be given. For instance,
members of a department may fear for their jobs if they reveal information that indicates
that their department should better be closed. Even if there exist restructuring options
that, based upon the private information, would improve both the department's and the
rm's general situation, the ailing department would not want to disclose it, as it fears
that once management has the information, it will not implement this option but go for
4 Full
disclosure may even be innitely costly, i.e. impossible, for the principal (e.g. Laont and Tirole
1988).
5 Our view of a mediator stands therefore in stark contrast to the existing literature on mediation
which has always been uncomfortable with the crucial role he plays in sustaining the equilibrium outcome
and has mainly concentrated on ways to dispense with him (e.g. Forges 1988, Ben-Porath 1998).
4
a full closure of the department.
Exactly here lies the role for the consultant. Since his objectives dier from those
of management, the members of the organization may be less afraid that he abuses the
disclosed information to hurt them. However, since consultants give only non{binding
recommendations to management, their usefulness in mitigating the non{veriability of
information is not self{evident. Based upon the proposed recommendation, management
will namely try to deduce the private information in order to improve on the consultant's
plan. Of course, this behavior of management undermines the delegation value of the consultant, since the rational members of the organization will anticipate it. Consequently,
the consultant will only be useful if he is able to circumvent this problem. He must oer
recommendations from which management may not deduce information that it can use
to hurt the agents who disclose it. Due to management's learning behavior consultants
will not be able to mitigate the non{veriability of information completely. The role of a
consultant is therefore not based upon a standard third party argument of delegation as
a means of self{commitment, but is much more sophisticated. It coincides with the role
of mediators in the sense of Myerson (1985).
Clearly, this paper focuses only on one aspect of a consultant's job. That of facilitating internal communication. We abstract from other important factors which may
contribute to a demand for consultants, such as improved analytical skills, expertise,
and psychological factors. Nevertheless, we like to stress that our specic explanation
of a consultant is consistent with the, at rst sight, peculiar and hitherto unexplained
fact that consultancy rms are paid in a cost plus fashion. One would expect rms to
oer consultants incentive based contracts, comprising shares or stock options in order
to circumvent potential problems of moral hazard. However, in practice incentive based
payments are a rarity. Given our explanation, the reason is clear. The usefulness of a
consultant depends crucially on the fact that his main objective is not to maximize the
rm's value, but rather to maintain his reputation as a fair mediator. Oering incentive
payments such as stock options would align the consultant's incentives with those of the
rm and undermine his credibility.6
The remainder of the paper is organized as follows. In the next section we model the
restructuring problem of the rm and derive some basic properties. We then analyze the
problem as a standard mechanism design problem and show that the solution depends
crucially on the full commitment possibility of the mechanism designer. In Section 4 we
6 The
purpose of this paper is not to investigate the credibility of the consultant. Instead we analyze a
static, non{repeated environment in which a consultant is indierent between the restructuring outcomes
and has a weak incentive to be a fair mediator. We conjecture that our static model may be embedded
into a repeated framework in which his incentives to act as a fair mediator may be made more explicit.
5
analyze the restructuring problem without any commitment, but direct communication
between rm and department. In Section 5 we introduce the consultant and show how
he may benet the principal. We examine his role more generally in Section 6. Section
7 illustrates our analysis and results in a straightforward example. Finally, Section 8
concludes and points out other settings in which mediators may be helpful. Longer
proofs are relegated to the appendix.
2
Model and Preliminaries
Consider a rm that consists of a management, the principal, and a department, the
agent, which has private information i 2 I. Management has to adopt a restructuring
option y 2 IR. The eect of the restructuring decision y depends on the private information of the department. For simplicity, we assume that the department's information
may only take on two values, i.e. I = f1; 2g. With probability 1 0 the department
possesses the information 1 and with probability 2 (0; 1) its private information is 2.
We suppose that both management and the department have each some preferred
option y 2 IR and the farther the restructuring option is from this preferred option the
more they dislike it and increasingly so. We capture this idea by assuming that the
players have Von{Neumann Morgenstern utility functions which are strictly concave and
attain a maximum on IR.7 The utility functions depend on the private information of
the agent. We write the principal's utility function as V (y) when the agent has the
private information i = 1; 2. Similarly, we denote agent i's utility function as U (y). For
technical reasons we assume that the utility functions are well dened over IR and three
times continuously dierentiable.
Moreover, we adopt a monotonicity condition concerning the agent's utility functions:
i
i
U1(y) < U2(y) for all y 2 IR:
0
0
The condition implies that the preferred restructuring options of the agents
y
a
i
arg max U (y)
y
i
exhibit the ordering y1 < y2 .
The monotonicity condition further implies that if one agent is indierent between
two distinct allocations y1 and y2 , the other agent must have a strict preference. More
generally:
a
a
Lemma 1 Let y1 < y2 .
If U1 (y1)
7 The
U1(y2 )
then
model is similar to Crawford and Sobel (1982).
U2(y1 ) < U2 (y2):
6
If U2 (y1)
then
U2(y2 )
U1(y1 ) > U1 (y2):
Proof: The statement follows directly from
U1(y2 ) 0 U1(y1 ) =
Z
y2
y1
U1 (y)dy <
Z
0
y2
y1
U2(y)dy = U2(y2 ) 0 U2 (y1 ):
0
Q.E.D.
We will not make explicit assumptions about the relation between V1 (y) and V2 (y),
except that the preferred restructuring options
y
p
i
arg max V (y):
i
y
exhibit the order y1 < y2 .
Finally, we will concentrate on the case where the principal would preferably want to
implement more extreme restructuring options than either of the agents. Taken together
the three ordering assumptions imply8
p
p
y1 < y1 < y2 < y2 :
p
a
a
p
This paper explores the idea that management has diculty to commit to some implementation function. Instead of taking the standard assumption that a principal can
perfectly commit herself to any implementation function before communication takes
place, we concentrate on the other extreme that the principal is not able to commit herself at all.9 As a consequence the implemented restructuring option will in the end only
depend on the beliefs of management concerning the private information of its department. Since there exist only two possible types of private information, these beliefs are
fully described by some p 2 [0; 1], representing the probability that the agent is of type
2. Given a belief p, management will implement the decision
y(p) arg max (1 0 p)V1(y) + pV2 (y):
y
The following lemma gives already some indication about the possible restructuring
outcome and will be helpful in the subsequent analysis.
8 To
9 I.e.
avoid a lengthy case by case study we concentrate on this ordering.
the principal is also unable to commit to any form of conditional payments.
7
Lemma 2 For any belief p 2 [0; 1] management will choose a restructuring plan y(p),
which lies in between y1 and y2 , i.e.
p
p
y1
p
y(p) y2 :
p
Moreover, y (p) > 0.
0
Before starting the analysis, it will be helpful to introduce a few more denitions. We
say that the interests of agent i are compatible with those of the principal, if and only
if U (y ) U (y ) (j 6= i). We will refer to such an agent i as a compatible agent. If
agent i is not compatible, i.e. U (y ) < U (y ), we say that the principal and agent i have
conicting interests.
p
i
i
p
i
j
p
i
p
i
i
j
Lemma 3 There is at least one compatible agent.
Proof: With y1 = y1 and y2 = y2 it follows directly from Lemma 1 that if one agent is
not compatible, the other one is.
Q.E.D.
p
p
In the following we assume that the interests of agent 1 are compatible with those
of the principal.10 If in addition agent 2's interests are also compatible, there does not
exist a meaningful incentive problem and the principal is able to extract all information
regardless of her commitment possibilities. We will therefore assume that agent 2's
interests are incompatible with those of the principal, such that a possible conict of
interest exists. The parameter represents the probability of such a conict. In order
to arrive at a more intuitive classication of the ex ante probability of conict, it will be
convenient to introduce two threshold levels 1 > 0 and 2 > 0. Let > 0 be such that
i
U (y( )) = U (y1 ):
p
i
i
i
Since agent 2 is incompatible, these values are well dened and the monotonicity condition
implies 0 < 1 < 2 < 1. We shall also use the following classication of the ex ante
probability of conict. We say that the ex ante probability of conict is small if < 1
and that the ex ante probability of conict is large if > 2 .
3
The Full{commitment Benchmark
As a benchmark and for future reference we analyze rst the rm's restructuring problem
in the more standard setting that management can commit to a mechanism before it asks
10 This
assumption is without loss of generality since due to Lemma 3 at least one agent is compatible.
Were this only agent 2 then we may "mirror" our problem by redening the restructuring options as
y 0 = 0y and exchange the roles of agent 1 and 2.
8
its department for information. With full commitment the rm's restructuring problem is
a standard mechanism design problem for which the classical revelation principle applies.
The optimal contract may therefore be found in the set of direct mechanisms that induce
the department to reveal its information truthfully.
Consequently, an optimal contract is the solution to the following problem, where
inequalities (1) and (2) represent the two incentive compatibility constraints.11 12
;
max (1 0 )V1(y1 ) + V2 (y2 )
y 1 ;y2
U1 (y1 ) U1(y2 )
s.t.
(1)
U2 (y2) U2 (y1 ):
(2)
Disregarding the incentive constraints the solution is (y1 ; y2 ) = (y1 ; y2 ) and coincides
with the solution under symmetric information. However, since agent 2's interests are
not compatible, this solution is not feasible. Therefore, at least one incentive constraint
binds at the optimum. As may be expected, the following proposition shows that this
will be the incentive constraint of the incompatible agent, i.e. agent 2. Lemma 1 then
implies that, at the optimum, agent 1's incentive constraint is slack.
fc
fc
p
p
Proposition 1 Under full commitment the incentive constraint of agent 2 binds at the
optimum and the optimal full commitment contract (y1 ; y2 ) exhibits y1 < y1 < y2 < y2 .
fc
fc
fc
p
fc
p
The proposition states that the optimal y2 is smaller than y2 . Taking as starting
point the symmetric information solution (y1 ; y2 ) this is intuitive, since lowering y2 from
the point y2 relaxes agent 2's incentive compatibility constraint. At rst sight it may be
surprising that it is optimal to set y1 below y1 . This becomes clearer when we consider
that a y1 lower than y1 also relaxes agent 2's incentive constraint. Of course, a y1 lower
than y1 reduces the principal's utility from a truthful revelation of agent 1. However,
since y1 is the optimal restructuring choice under agent 1, i.e. V1 (y1 ) = 0, this loss is
only of the second order. The relaxed incentive constraint on the other hand represents
a rst order gain.
fc
p
p
p
p
fc
p
p
p
p
0
p
Before turning to the analysis under non{commitment, we stress two important features of the optimal full commitment contract. First, the options prescribed by the
optimal menu are suboptimal ex post. Since the department reveals itself perfectly management will have an ex post incentive to implement the options y1 or y2 rather than the
p
11 There
p
are no individual rationality constraints, as we assume that management must choose some
option and no outside option exists. Alternatively, the outside options are so low that they will not be
binding, i.e. agent i's outside option is smaller than Ui (y2p ).
12 As we use the full{commitment case only as a benchmark we disregard stochastic menus.
9
prescribed options y1fc or y2fc , respectively. The credibility of management's commitment
is therefore crucial. If the department doubts it and expects that management will not
keep its promises, the ex ante incentives to report truthfully are destroyed.
Second, the optimal full{commitment contract commits management to a restructuring option y1 < y1p . Lemma 2 shows however that there does not exist a belief for management that would lead to such a restructuring option. Therefore, when management
has no possibility to commit itself, it would never take this option. Non{commitment
will therefore make it impossible to attain the full{commitment solution in Proposition
1. As this result is independent of whether management uses a consultant, it shows that
we cannot expect the consultant to mitigate completely the limitations due to a lack of
commitment of management.
4 Direct Communication: No Consultant
As discussed in the introduction, full commitment of management is perhaps too stark an
assumption. It may be seen as a modeling extreme that, in reality, is only partially met.
To study the eect of such limited commitment we adopt in the remainder of the paper
the other extreme that management cannot commit itself at all. The lack of commitment
implies that management cannot propose a menu of restructuring options, such as a direct
mechanism, from which the department may pick its preferred option. Management may
nevertheless want to communicate with its department in the hope that this leads to
some revelation of information. Without a consultant such communication takes place
directly between management and department. The direct communication game is as
follows:
1. Management sets some message space M for the agent.
2. The agent announces a message m 2 M .
3. Management updates its belief.
4. Management chooses a restructuring option y .
We apply the solution concept of Perfect Bayesian Equilibrium (PBE) to this direct communication game. Such an equilibrium species a message space M , an announcement
strategy i for the agent, a belief p = (p1 ; : : : ; p M ) of the principal, and an implementation strategy y = (y1; : : : ; y M ). That is, if the agent sends the message m 2 M , the
principal's belief that the agent is of type 2 is pm , which induces her to implement restructuring option ym . Since we are interested in the question whether the principal can
j
j
j
j
10
do strictly better with a consultant than without, we will concentrate on the PBE that
yields the principal the highest utility.
Due to a generalized revelation principle proven in Bester and Strausz (1998), we may
without loss of generality assume that the message space corresponds to the set of types,
i.e. M = I = f1; 2g. This implies that the agent eectively announces some type i.
Consequently, we may represent a strategy of agent i by some i 2 [0; 1] which denotes
the probability that the agent announces that he is of type 1. We may furthermore
assume without loss of generality that 1 > 0, 2 < 1, and 1 2 .13
Thus we will look for a PBE with M = f1; 2g. The combination (1 , 2 , p1, p2 , y1, y2 )
constitutes a Perfect Bayesian Equilibrium if it satises the following three conditions:
1. The agent's announcement strategy is optimal given the principal's implementation
strategy, i.e.
Ui (y1 ) + (1 0 )Ui (y2 ):
i Ui (y1 ) + (1 0 i )Ui (y2 ) = max
(3)
2. The principal's belief is Bayes' consistent with the agent's strategy, whenever possible. This implies that
p1 = p(1; 2 ) and p2 = p(1 0 1; 1 0 2 );
with
p(x; y ) y
:
x(1 0 ) + y
(4)
Note that since 1 2 (0; 1] and 2 2 [0; 1) both p(1 ; 2) and p(1 0 1 ; 1 0 2) are
well{dened.
3. The principal's implementation strategy is optimal given her belief p, i.e.
yi = y(pi ):
In equilibrium, the agent's strategy combination (1 ; 2) yields the principal the utility
V (1 ; 2 ) = (1 0 )[1 V1 (y(p(1; 2 ))) + (1 0 1 )V1 (y (p(1 0 1 ; 1 0 2 )))]
+ [2V2(y (p(1 ; 2 ))) + (1 0 2 )V2 (y(p(1 0 1 ; 1 0 2 )))]:
13 This
implies that agents tell the truth with a strict positive probability, but, in contrast to the
standard revelation principle, it may be optimal for the principal to let the agents lie with some positive
probability. For more details see Bester and Strausz (1998).
11
The principal's utility is increasing in 1 and decreasing in 2 .14 This reects the intuitive
fact that more information is better for the principal. Since the principal's utility depends
on the degree of information the agent reveals, it will be helpful to distinguish between
the following ve types of equilibria:
1. A full revelation equilibrium in which the agent's type is perfectly revealed: 1 = 1,
2 = 0.
2. A non{revelation equilibrium in which the agent's announcement does not reveal
anything: 1 = 2.
3. A partial revelation equilibrium in which the announcement of each agent reveals
some, but not all information: 1 < 1, 2 > 0, and 1 6= 2 .
4. A type 1 partially full revelation equilibrium that leads to a full revelation of agent
1 with positive probability, but not of agent 2: 1 < 1 and 2 = 0.
5. A type 2 partially full revelation equilibrium that leads to a full revelation of agent
2 with positive probability, but not of agent 1: 1 = 1 and 2 > 0.
As indicated in the previous section a full revelation equilibrium does not exist. In
such an equilibrium the principal chooses y1p and y2p , which leads agent 2 to pool with
agent 1 rather than revealing himself truthfully. As is familiar from the literature on
cheap talk, a non{revelation equilibrium always exists, but yields the principal the lowest
payo of all equilibrium types. The principal is therefore better o in either of the
remaining types of equilibria. Any of these involves at least one message that reveals
the agents only partially, which requires that both agents use this message with positive
probability. The agent, who uses also the other message, is actively mixing over the two
messages and must therefore be indierent between the allocations which they induce.
Due to the monotonicity assumption, the two agents cannot be indierent between two
dierent allocations at the same time. Therefore, a partial revelation equilibrium will not
exist. For a similar reason the type 2 partially full revelation equilibrium does not exist.
Such an equilibrium would imply y1 < y2 = y2p and since the outcome y2p is the worst
outcome to agent 2, he will strictly prefer message 1 to message 2. The only remaining
candidate is an equilibrium of type 4 which leads to a full revelation of type 1 with
positive probability. This equilibrium implies y1 = y1p < y2 and requires that agent 1 is
indierent between the allocation y1p and a dierent allocation y2. Due to the concavity of
envelope theorem yields dV=d1 = (1 0 )(V1 (y1 ) 0 V1 (y2 )) 0 and dV=d2 = 0(V2 (y2 ) 0
V2 (y1 )) 0. The sign follows due to 1 2 , which implies y1 = y (p(1; 2 )) < y2 = y (p(1 0 1 ; 1 0 2 ))
in equilibrium.
14 The
12
the agent's utility function there exists exactly one restructuring option y2 > y1p for which
this indierence obtains. It equals y (1 ). The equilibrium therefore exists if there exists
a mixing behavior of agent 1 which leads to a belief 1. Bayesian consistent updating
implies that this is the case if and only if < 1. We therefore arrive at the following
proposition.
Proposition 2 The optimal Perfect Bayesian Equilibrium with direct communication
exhibits the following structure:
1. If the ex ante probability of conict is small, i.e. < 1 , the optimal PBE is a
partially full revelation equilibrium in which agent 1 reveals himself with probability
(1 0 )=((1 0 )1 ) > 0. The equilibrium outcome is y1 = y1p < y2 = y (1) < y2p.
2. If the ex ante probability of conict is not small, i.e. 1 , the optimal PBE is a
non{revelation equilibrium and the outcome is y ( ).
The proposition shows that non{commitment not only makes it more dicult for the
principal to induce information revelation, it may actually make it impossible. Only if
the ex ante probability of a conict of interest is small, is management able to extract
information from the agent. But also in the informative equilibrium there remain three
sources of ineciencies as compared to the commitment case. First, the allocation y1
is suboptimally high, as without commitment it is not possible for the principal to implement a restructuring option y < y1p . Second, there is a "stochastic misallocation",
since agent 1 misrepresents his type with positive probability. Third, the allocation y2
is suboptimally low, i.e. y2 < y2fc . The latter two ineciencies have the same origin.
In order to induce agent 1 to mix, the principal must make him indierent between the
two equilibrium outcomes. Stochastic misallocation is therefore a necessary feature for
information revelation under non{commitment and in contrast to the full{commitment
case agent 1 rather than agent 2 is made indierent in equilibrium.
5 Mediated Communication: Benecial Consultants
In the previous section management communicated directly with its department in order to obtain information. In this section we assume that the communication between
management and department is channeled through a consultant.15 In analyzing the role
15 In
contrast to standard models of third{party delegation, the principal in our framework is unable to
delegate the nal implementation decision. It should be clear that this makes the role of the third party
much weaker. In line with standard literature on delegation, we assume that there exist no possibilities
of collusion.
13
of the consultant we will rst take a pragmatic approach, which enables us to derive in
a straightforward way sucient conditions under which the consultant is strictly helpful.
In section 6 we will then characterize the consultant more generally and show that the
sucient condition is also necessary. Moreover, the next section derives conditions under
which the outcomes obtained in this section are indeed optimal.
The consultant's basic role is straightforward. After being hired he communicates with
the department and subsequently proposes a recommendation to management. Consistent with empirical observation, we assume that the communication between department
and consultant occurs in private and is unobservable to management. Since management hires the consultant, we assume that he acts in its interest. We capture this by
modeling that upon hiring the consultant, management prescribes him a proposal rule
that describes exactly which recommendation the consultant has to give given what was
revealed during the communication stage.
For the consultant to be of any help and in the spirit of the revelation principle,
it makes sense to restrict our attention to incentive compatible rules, which induce the
department to reveal its information and induce the principal to follow the recommendation. A further natural restriction to focus on are proposal rules where the number of
possible proposals coincides with the number of types. In this section we will therefore
concentrate on incentive compatible 2{proposal rules. Figure 1 represents such a proposal rule graphically. The rule works at follows. If the agent claims he is of type i, the
consultant recommends the restructuring option rj with probability ij .
Seen from an informational perspective the proposal rule garbles the truthful revelation of agent i by recommending r1 with probability i1 and r2 with probability
i2 = 1 0 i1 . Such garbling prevents the principal from identifying the agent's type and
may help to make the consultant's recommendation incentive compatible with respect
to the principal. Without loss of generality we assume 11 21 , i.e. the recommendation r1 (weakly) indicates that the agent is type 1, while the recommendation r2 is more
indicative that he is type 2.
A 2{proposal rule P is incentive compatible with respect to the agent if it oers him
a weak incentive to reveal his true type, i.e.
11 U1(r1 ) + 12 U1(r2 ) 21 U1 (r1) + 22 U1 (r2)
21 U2 (r1 ) + 22 U2 (r2 ) 11U2 (r1 ) + 12U2 (r2 ):
Due to i2 = 1 0 i1 this reduces to
U1 (r1) U1 (r2 ) and U2 (r2) U2 (r1 )
(5)
respectively. Note that the incentive constraints coincide with the incentive compatibility
conditions (1) and (2) of the full commitment framework.
14
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11
1
2
21
12
22
r1
0 21
0 11
r2
Figure 1: The consultant's 2{proposal rule
An incentive compatible proposal rule P should also induce the principal to follow
the consultant's recommendation. For a recommendation rj this is the case if
r1 = y(p(11 ; 21)) and r2 = y (p(12 ; 22 ));
(6)
where the function p(:; :) ensures Bayes' consistent beliefs and is given by (4).
Given these incentive constraints we may write the choice of the optimal incentive
compatible 2{proposal rule by the following maximization problem:
V (P ) (1 0 )f11V1 (r1) + (1 0 11 )V1 (r2 )g
+ f21V2(r1 ) + (1 0 21 )V2 (r2)g
s.t.
(5) and (6):
max
11 ;21 ;r1 ;r2
In order to derive the optimal 2{proposal rule it is helpful to introduce the following
denition of informativeness. We say that an incentive compatible recommendation pair
(rk ; rl ) is more informative than an incentive compatible recommendation pair (rk ; rl ) if
the distance jrk 0 rl j is larger than the distance jrk 0 rl j. Incentive compatibility motivates
the denition, since the distance is larger only if the recommendations rk and rl are more
informative about the agent's type.
0
0
Lemma 4 The principal's utility is increasing in the informativeness of an incentive
compatible recommendation pair (r1 ; r2).
Lemma 4 shows that our notion of informativeness is consistent with the intuitive
idea that more information is better for the principal. Yet, this result is not obvious,
since increasing informativeness has both a positive and a negative eect. Obviously, a
more informative recommendation is benecial, since it enables the principal to tailor
her restructuring options more accurately. A negative eect, however, is caused by the
need for incentive compatibility. If a pair (rk ; rl ) is more informative than a pair (rk ; rl ),
then incentive compatibility requires that both agents induce the recommendation rk
0
15
less often. That is, also the agent, for which recommendation rk is indicative. This is a
negative eect. Still, by Lemma 4, the positive eect outweighs the negative one.
Having established that more informative, incentive compatible recommendations are
benecial, we are able to derive the optimal 2{proposal rule in the case that the ex ante
probability of conict is not high.
Proposition 3 For < 2 the optimal incentive compatible 2{proposal rule is characterized by (y1; y2 ) = (y1p ; y(2 )). At the optimum, the incentive constraint of agent 2 is
binding and agent 1 is perfectly revealed with probability (2 0 )=((1 0 )2) > 0.
The optimal 2{proposal rule resembles in an important aspect the optimal contract
under direct communication. In both equilibria agent 1 is revealed with positive probability in which case the restructuring option y1p is implemented. An important dierence
is, however, that in contrast to the case without a consultant, the incentive constraint of
agent 2 rather than of agent 1 is binding.16 This dierence implies that with a consultant
the probability with which agent 1 is revealed is larger. Consequently, the principal is
able to extract more information when she employs the consultant. This insight yields
the following proposition.
Proposition 4 A sucient condition for the principal to benet from the consultant is
that the ex ante possibility of conict is not too large, i.e. < 2.
We may explain the benecial eect of the consultant by referring to the equilibrium
requirement (3) of direct communication and the incentive constraints (5) of mediated
communication. Recall that the principal's utility is increasing in the amount of information that is revealed. Due to the limited commitment the principal is, however, unable
to induce full information revelation and can induce at most a partial revelation of information. This requires that messages need to be used in some stochastic way. If there
is no consultant, then this mixing has to be done by the agents themselves. Equilibrium
requirement (3) implies quite naturally that the mixing agent has to be indierent between the allocations over which he mixes. This is not the case when the principal uses
a consultant. In this case the consultant performs the mixing and an agent only has to
prefer his mixing package over the mixing package of the other agent. This requirement
yields the incentive constraints (5), which are weaker than the equilibrium requirement
(3).
16 Note
that the consultant equilibrium has this important aspect in common with the optimal full{
commitment contract.
16
6
Mediated Communication: Optimal Consultants
In this section we take a more general approach toward the consultant. This will clarify
that his role is exactly that of a mediator in the sense of Myerson (1985). Moreover, it
allows us to show that the sucient condition which we identied in the previous section
is also a necessary condition for the principal to benet from a consultant.
In the previous section we introduced the idea of a proposal rule and restricted attention to incentive compatible 2{proposal rules. We may however consider a much more
general notion of proposal rule in that the rule prescribes the following. First, it species
a message space M1 from which the agent has to pick a message. Second it species a
message space M2 from which the consultant sends a message to the principal. Third, it
species the probability with which the consultant sends a message m2 2 M2 when the
agent sent the message m1 2 M1 . A proposal rule P may therefore be written as a tuple
(M1 ; M2 ; ) where maps M1 into a probability distribution over M2.
The game between the principal and the agent when a consultant is available runs as
follows:
1. The principal announces publicly the consultant's proposal rule P = (M1 ; M2 ; ).
2. The agent sends a message m1 2 M1 to the consultant. The message is communicated in private such that the principal does not observe it.
3. The consultant sends the message m2 2 M2 according to the probability distribution
(m1 ) to the principal.
4. The principal updates her beliefs and decides which project to implement.
Note that management's choice of a proposal rule P at stage 1 induces a proper subgame as of stage 2. A Perfect Bayesian Equilibrium of this subgame describes for each
type of agent an announcement strategy, which may be represented by a probability distribution over the set M1, and an implementation strategy for the principal that describes
which restructuring option the principal chooses given the consultant's message. In principle also the principal's strategy may involve randomization. Last, a Perfect Bayesian
Equilibrium describes a belief function p for the principal, which represents the belief of
the principal given that the consultant sent a message m2 2 M2 . A Perfect Bayesian
Equilibrium has to satisfy three requirements: 1) the agent's announcement strategy is
optimal given the principal's implementation strategy; 2) the principal's belief is Bayes'
consistent with the agent's strategy, whenever possible; and 3) the principal's implementation strategy is optimal with respect to the belief function p.
17
Having described the consultant and the corresponding game more generally, it should
now be clear that his role is equivalent to that of a mediator in the literature on communication. The following lemma expresses an important result of this literature.
Lemma 5 Without loss of generality management may restrict its attention to proposal
rules that are (Bayesian) incentive compatible, i.e. induce the principal to follow the
consultant's recommendation and induce the agent to report his type truthfully.
Lemma 5 is a version of the revelation principle and is given without proof.17 It
shows that one may assume without loss of generality that the optimal proposal rule
uses a message space M1 = f1; 2g for the agent and the message space M2 = IR for the
consultant. That is, the restriction in the previous section that the consultants makes an
actual restructuring proposal rather than sending some general message is without loss
of generality. Also the restriction to incentive compatible proposal rules does not restrict
generality. In contrast, on the basis of the revelation principle the assumption that the
rule has only two possible outcomes seems less innocuous.
Due to the revelation principle we have only to consider incentive compatible proposal
rules P = (f1; 2g; IR; 1 ; 2) with i a distribution function over IR. To circumvent
measure{theoretic considerations we study the slightly smaller class of proposal rules
that randomize over a nite, but arbitrarily large number of recommendations in IR.
That is, we consider proposal rules of the form P = (f1; 2g; R; 1; 2 ) with R IR nite
P
and i 2 IR+R and jR=1 ij = 1 for i = 1; 2. Without further loss of generality we adopt
the following ordering assumption
j
j
j
j
2(j +1) 1j
1(j +1)2j
(7)
for all j = 1; 2; : : : ; jRj 0 1.
As we already saw in the previous section, an incentive compatible proposal rule
entails two dierent forms of incentive compatibility. First, the recommendations must
be incentive compatible in the sense that the principal has no strict incentive to diverge
from the consultant's proposal. For a recommendation rj this obtains if
rj = y(p(1j ; 2j ))
(8)
where p(:) is given by (4) and ensures Bayes' consistent updating. We call a recommendation rj for which equality (8) holds incentive compatible. Note that since y (p) is
increasing, the ordering condition (7) implies that an incentive compatible proposal rule
exhibits rj rj +1 for all j = 1; 2; : : : ; jRj 0 1.
17 For
details see Myerson (1985, 1986) and Forges (1986).
18
Second, the proposal rule must be incentive compatible in the sense that an agent
does not have a strict incentive to misreport his type. A proposal rule P is incentive
compatible with respect to agent 1 if
X
j U1(rj ) j
1
X
j U1 (rj ):
2
j
(9)
A proposal rule P is incentive compatible with respect to agent 2 if
X
j
j U2(rj ) 2
X
j
j U2 (rj ):
1
(10)
A proposal rule P is incentive compatible, if all its recommendations are incentive
compatible and if it is incentive compatible with respect to both agents. As is well known,
the need for incentive compatibility puts restrictions on the set of implementable proposal
rules.
Lemma 6 If an incentive compatible proposal rule P induces some revelation of information then
1. the ex ante probability of conict may not be large, i.e. < 2 .
2. if one agent's incentive constraint holds with equality, the other one's is satised
with strict inequality.
3. there must be recommendations rj 2 R such that y( ) < rj < y2p .
Since Proposition 4 shows that for < 2 there exists a contract under which the
principal is better o employing the consultant, the rst result of Lemma 6 establishes
that the sucient condition < 2 is also a necessary condition for the consultant to be
helpful. We therefore arrive at the main result of this paper.
Theorem 1 The principal does strictly better with a consultant if and only if the ex ante
possibility of conict, , is positive but not too large, i.e. 0 < < 2 .
Allowing general proposal rules complicates the analysis of the optimal rule. The
problem that arises is that the principal may select a proposal rule that uses more than
two recommendations. Such additional recommendations lead to an articial randomness
of the proposal rule which may relax incentive constraints.18 This extends to our setting
with limited commitment and prevents a simple characterization of the optimal contract.
Instead we obtain the following partial characterization of the optimal proposal rule.
18 This
is a well{known result from the theory of mechanism design. See for instance Fudenberg and
Tirole (1988, p.307).
19
Proposition 5 If < 2 an optimal proposal rule has the following properties:
(i) Agent 2's incentive constraint binds.
(ii) Agent 1 is revealed with strictly positive probability, i.e. r1 = y1p .
(iii) All recommendations except for r1 are more indicative of agent 2 than of agent
1; more precisely, rj > maxfy( ); y2a g for all j > 1.
Reecting the fact that stochastic mechanisms may be optimal, we cannot exclude
that the optimal proposal rule uses more than two recommendations. However, a standard approach in implementation theory is to derive sucient conditions on the risk
attitudes of the parties that render stochastic mechanisms suboptimal. Naturally, these
conditions regard the parties whose incentive constraints bind at the optimum. Since in
the present model there exist incentive constraints with respect to the agent as well as
to the principal, the conditions involve the risk attitudes of both parties. The following
proposition shows that if the principal's utility function exhibits decreasing absolute risk
aversion (Vi000 0) and the combined absolute risk aversion of principal 1 and agent 2
concerning the restructuring option y is large enough the optimal rule does not involve
articial randomness.19
Proposition 6 A sucient condition for a 2{proposal rule to be optimal is that Vi000 0
for i = 1; 2 and for all y 2 (y2a ; y2p) it holds that
U200 (y)
V100(y )
+2 0
U20 (y)
V1 (y)
V10 (y)
0 V (yp) 0 V (y) :
1
1
1
(11)
Under the conditions stated in Proposition 6 the optimal incentive compatible 2{
proposal rule that we characterized in Lemma 3 is generally optimal. Its actual form,
however, depends on the ex ante probability of conict . Similarly, the optimal contract
without a consultant also depends on this parameter. Both contracts seem to exhibit a
discontinuity, since the type of contract changes from an information revealing into an
unrevealing contract at the threshold level 2 , respectively 1 . To investigate this issue
more closely let V NC ( ) denote the principal's maximum payo if there is no consultant,
and V C ( ) if there is a consultant. The next proposition shows that the principal's payo
function does not exhibit a discontinuous jump at the respective threshold levels. This
observation leads to the conclusion that the amount of information revelation induced
by each of these contracts decreases continuously as the ex ante probability of conict
approaches the respective threshold level.
19 Note
that the conditions are satised for any function U2 if V1000 = 0 and V2000 0.
20
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..................................................................................................................................................................................................................................................................................
V C ( )
V NC ()
0
1 = 0:4
2 = 0:8 1 (a)
(b)
Figure 2: An implementation example
Proposition 7 Suppose Vi000 0 for both i = 1; 2 and (11) holds for all y 2 (y2a ; y1p ).
Then the function V NC ( ) is linear on the interval (0; 1 ) and continuous at = 1 .
Likewise, the function V C ( ) is linear on the interval (0; 2 ) and continuous at = 2 .
7
An example
In this section we illustrate our results with a straightforward example. Let the agent's
utility function satisfy U1 (y ) = 0(y 0 2)2 and U2(y ) = 0(y 0 4)2 . These utility functions
are strictly concave and satisfy the monotonicity condition as U10 (y ) = 4 0 2y < U20 (y) =
8 0 2y for all y 2 IR. Let the principal's utility be V1 (y ) = 0y 2 and V2 (y ) = 0(y 0 10)2 .20
First order conditions yield the preferred restructuring options:
y(p) = 10p; y1p = 0; y2p = 10; y1a = 2; y2a = 4:
Consequently, the equilibrium outcome lies in the interval [0; 10] and we may restrict our
attention to this interval. Figure 2(a) illustrates the agent's utility functions graphically.
Note that since U2 (0) > U2 (10), there exists a conict of interest between agent 2 and
the principal. Consequently, the principal cannot induce both agents to reveal their
information fully. In such an equilibrium the principal would choose y1 = y1p and y2 = y2p
and agent 2 would be strictly better o not to tell the truth.
If the principal does not use a consultant, there exist at most two distinct equilibrium
outcomes. The non{revelation equilibrium yielding the outcome y () always exist. A
possible second equilibrium is a partially full revelation equilibrium with y13 = y1p = 0
and y23 = y(p2) = 4. In this equilibrium agent 1 mixes between the two messages and
must be indierent between the two outcomes; see Figure 2(a). If both equilibria exist,
the principal will prefer the latter, more informative equilibrium. The existence of the
20 Note that the utility functions satisfy the conditions of Proposition 6, i.e. a 2{proposal rule is optimal.
21
more informative equilibrium will depend on the parameter . In order for y23 = 4 to be
an equilibrium the principal must have the belief p2 = 2=5. Since in this equilibrium we
have 2 = 0, the belief p2 = 2=5 can only be Bayes' consistent if 1 = (2 0 5)=(2 0 2).
As 1 must itself be larger than 0, the partially full revelation equilibrium only exists
when < 2=5. The optimal PBE outcome without a consultant is therefore the partially
full revelation outcome (y13 ; y23) = (0; 4) for < 2=5 and the non{revelation equilibrium
y1 = y2 = 10(1 0 ) for 2=5.
If the principal employs a consultant, the consultant's candidate equilibrium proposal
is (r13; r23) = (0; 8). In such an equilibrium agent 2 is made indierent between the two
outcomes; see Figure 2(a). For this to be an equilibrium, the principal must have the
beliefs p2 = 4=5, if she is to follow the consultant's proposal r23 = 8. This requires that
1 = (4 0 5 )=(4 0 4), which is only feasible if < 4=5. Hence, for < 4=5 the
outcome (r13; r23) = (0; 8) is sustainable with consultant. Note that this equilibrium is
more informative than any equilibrium that exists without the consultant. Hence, the
principal is better o.
This is made more explicit in Figure 2(b), where we plotted the principal's payo functions without a consultant, V NC ( ), and the payo function with a consultant, V C ().
It shows that for any < 4=5, the principal's payo is larger when she employs a consultant. For 4=5, the curves V 1 () and V 2( ) coincide and the consultant is not
helpful.
8
Conclusion
This paper explored the idea that consultants during a rm's restructure play the role
of benecial mediators in communication. The idea may be applied to many other existing intermediaries. For instance, one may see an independent regulatory agency as a
mediator between a privately informed industry and a government which is susceptible
to a ratchet eect.21 Our argument may also explain the use of arbitrators in bargaining
situations between privately informed parties (e.g. wage bargaining between a union and
an employer, or peace negotiations between countries). Parties in such bargaining situations have often a long term relationship which does not end after the completion of the
current bargaining process. If parties fear that a revelation of private information during
the current bargaining process places them at a disadvantage in future bargaining confrontations, the revelation of information is hindered in the same sense as in our model.
Parties may then decide to employ a mediator in the form of an arbitrator to overcome
21 The
ratchet eect is crucially linked to limited commitment. For more details see Laont and Tirole
(1988) and Strausz (1997).
22
this problem.
In our analysis we obtained some intuitive limits on the benecial role of communication mediators. Obviously, if there is no potential conict of interests between the
parties, they can communicate eciently even without a mediator. On the other hand, if
the potential conict of interests is too strong, a mediator is unable to mitigate the problem of information revelation associated with limited commitment. Our analysis shows
that a mediator is only benecial if there is a large enough possibility that the objectives
of the parties somehow coincide. Still, the resulting outcome could be improved upon if
the party with limited commitment could also transfer its decision right to the mediator.
Yet, in many real{life situations this is not feasible.
Clearly, the consultant's role as a mediator has been rather limited. He is a machine
rather than an independent economic agent who chooses his actions to maximize his own
utility. This may be seen as problematic since there exist collusive pressures once the
consultant has obtained private information. In exchange for a small bribe management
may ask the consultant to reveal more information than the proposal rule prescribes. We
conjecture that the consultant may resist these pressures in a repeated version of our
static model in which in each period a dierent rm needs a restructure and pays a fee
for the consultant's services. In such a setting the consultant may fear that accepting
collusion proposals may lead agents to distrust him as a fair mediator and deprive him
from future fees.22
Appendix
Proof of Lemma 2
Suppose y(p) < y1p then (1 0 p)V1 (y (p)) + pV2(y (p)) < (1 0 p)V1 (y1p ) + p(V2 (y1p ) 0 (V2 (y1p ) 0
R p
V2 (y (p)))) = (1 0 p)V1(y1p ) + p(V2 (y1p ) 0 yy(1p) V20 (y)dy) < (1 0 p)V1 (y1p ) + p(V2(y1p )), where
the rst inequality follows from the fact that y1p maximizes V1(y ) and the second follows
since V20(y ) > 0 for y < y1p , due to the concavity of V2 . Consequently, y (p) does not
maximize the expression (1 0 p)V1 (y ) + pV2(y ). Similarly, one may show that y (p) > y2p
is not possible.
To prove the second statement note that y(p) is implicitly dened by the rst order
condition (1 0 p)V10 (y (p)) + pV20 (y(p)) = 0. Dierentiating w.r.t. p yields @y=@p =
22 More
specically, we have in mind that management's utility is in fact the rm's prot and is
a publicly observable, but imperfect signal about the consultant's action. We conjecture that a folk
theorem in the sense of Fudenberg, Levine, and Maskin (1994) may be obtained to sustain the truthful
behavior of the consultant.
23
[V10(y ) 0 V20 (y )]=[(1 0 p)V100(y ) + pV200(y )]. Note that for y 2 [y1p ; y2p ] one has V20 (y ) 0 and
V10 (y) 0 with at least one strict inequality. The numerator is therefore negative. Due
to concavity of V1 and V2 the denominator is also negative. This implies @y=@p > 0.
Proof of Proposition 1
Recall that at least one agent's incentive constraint must be binding at the optimum.
Lemma 1 then implies that the other agent's IC has slack. Suppose agent i's IC binds,
i.e. y13 y23 are such that
Ui (y13 ) = Ui (y23 ):
(12)
Equation (12) denes y23 as an implicit function of y13 for the range y13 < yia . Denote this
implicit function as y2i (y1 ). Concavity of Ui , y1 < yia , and y2i (y1) > yia imply @y2i =@y1 < 0.
Likewise, (12) dene the implicit function y1i (y2) for y2 > yia . Also, @y1i =@y2 < 0.
Suppose yj3 yjp with j 2 f1; 2g and fj; 0j g = f1; 2g then y0i j (yj3 ) < y0p j , due
to Ui (y1p ) > Ui (y2p ) and the concavity of Ui . It therefore holds that V00 j (y0i j (yj3)) > 0
and Vj0(yj3) 0. A small reduction in yj3 and a corresponding increase in y03 j such
that (12) remains satised increases management's utility. Hence, yj3 yjp cannot be
optimal. Obviously, y23 > y1p and we have established that the optimal contract exhibits
y13 < y1p < y23 < y2p .
To show that the incentive constraint of agent 2 is binding, suppose by contradiction
that agent 1's IC binds, i.e. (1) is satised with equality. Lemma 1 then implies that
(2) is strictly satised. Now consider a raise in y23 . For a small enough raise, inequality
(2) remains satised. As established, an optimal contract satises y13 < y1p < y23 < y2p,
which implies V20(y23) > 0 and U10 (y23 ) < 0, the small raise in y23 increases the principal's
utility, while rendering (1) satised with strict inequality. Hence, (y13 ; y23 ) cannot have
been optimal.
Proof of Proposition 2
We rst show non{existence of equilibria of the types 1, 3, and 5.
In a full revelation outcome necessarily (1 ; 2 ) = (1; 0). Management's beliefs must
therefore satisfy p1 = p(1; 0) = 0 and p2 = p(0; 1) = 1 and implement y1 = y (1) = y1p
and y2 = y (0) = y2p , which due to U2 (y2p ) < U2 (y1p ) contradicts (3). A full revelation
equilibrium does therefore not exist.
A partial revelation equilibrium does not exist, since in such an equilibrium i 2 (0; 1)
for both i = 1; 2 and y1 6= y2. By (3) this would require Ui (y1 ) = Ui (y2) for both i = 1; 2.
Lemma 1 shows this is not possible.
Also a type 2 partially full revelation equilibrium does not exist. Such an equilibrium
exhibits 1 = 1 and 0 < 2 < 1. Consequently, p1 = p(1; 2 ) 2 (0; 1) and p2 =
24
p(0; 1 0 2 ) = 1, which implies y1 2 (y1p ; y2p) and y2 = y (0) = y2p . Moreover, due to (3)
0 < 2 < 1 requires U2 (y1) = U2 (y2p ). However, such a y1 2 (y1p ; y2p ) does not exist, because
U2 (y2p ) < U2 (y1p ) and the concavity of U2 implies U2 (y2p ) < U2 (y) for all y 2 [y1p ; y2p).
Hence, only two equilibrium candidates are left. The non{revelation equilibrium,
which always exists in the form (1 ; 2 ; p1 ; p2 ; y1 ; y2) = (; ; ; ; y( ); y( )) with 2
(0; 1), and the partially full revelation equilibrium. Obviously, the latter yields the principal a higher payo. However, a type 1 partially full revelation equilibrium exists if
and only if < 1 . This follows from the observation that in a type 1 partially full
revelation equilibrium 0 < 1 < 1 and 2 = 0. Consequently, p1 = p(1; 0) = 0 and
p2 = p(1 0 1 ; 1) < 1, which implies y1 = y (0) = y1p and y2 2 (y1p ; y2p ). Moreover, due
to (3) 0 < 1 < 1 requires U1 (y2 ) = U1 (y1p ) and thus y2 = y(1). This in turn requires
p2 = p(1 0 1 ; 1) = 1 and hence, by (4), 1 = (1 0 )=[(1 0 )1 ]. However, 1 must be
non{negative and, in order to have some information revelation, must dier from 2 = 0.
Therefore, an informative equilibrium requires < 1 . For its existence it remains to
be checked that 2 = 0 satises the incentive constraint (3) of agent 2; this follows from
Lemma 1.
Proof of Lemma 4
Consider a proposal rule P with some recommendation rk such that rk 2 (y1p ; y2p ). Dene,
for > 0 but small, the proposal P kl () as the following transformation of P :
1k () 1k 0 1l 2k ( ) 2k 0 2l rk () y(p(1k (); 2k ()))
1l ( ) 1l + 1l 2l () 2l + 2l rl ( ) = rl
1j () 1j
2j () 2j
rj () = rj for j 6= k; l.23
The transformation is structured in such a way that if the recommendations of the
original proposal P are incentive compatible then this also holds for the recommendations
in P kl ( ). Moreover, if rises, the pair (rk ( ); rl ) becomes more informative, since jrk ( ) 0
rl j is increasing in due to Sign(@rk =@) = Sign(1l 2k 0 1k 2l ). It suces to show that
dV (P kl )=d > 0. To see this note rst that, since rule P is incentive compatible it holds
for any rj 2 R
(1 0 pj )V10 (rj ) + pj V20 (rj ) = 0;
(13)
where pj = p(1j ; 2j ). Moreover the concavity of V1 and V2 imply that
Sign(rj 0 y) = Sign((1 0 pj )V10 (y) + pj V20 (y ))
(14)
Using (13) one obtains
dV (P kl ())
= (1 0 )1l [V1 (rl ) 0 V1 (rk )] + 2l [V2 (rl ) 0 V2 (rk )]
d
23 We
prove the property for general proposal rules which may have more than two recommendations.
25
= (2l + (1 0 )1l )
Z
rl
rk
(1 0 pl )V1 (y ) + pl V2 (y ) dy > 0:
0
0
where the sign follows from (14).
Proof of Proposition 3
Since agent 2 is incompatible, full revelation is not possible so that at least one incentive
constraint must be binding. By Lemma 1 at most one incentive constraint is binding when
some information revelation occurs. Since Ui (y1p ) > Ui (y2p ) for both i, a binding incentive
constraint implies that in an equilibrium with some information revelation r1 < r2 < y2p
and thus 21 < 11 < 1.
Now suppose an incentive compatible proposal rule P is such that agent 1's incentive
constraint is binding, i.e. U1 (r1 ) = U1 (r2 ) and U1(r2 ) < 0. Consider the proposal rule
P 21 () as dened in the proof of Lemma 4. Note that for > 0 small enough the proposal
rule P 21 () is feasible and prescribes the incentive compatible recommendations r1( ) = r1
and r2 ( ) > r2 . We now show that the proposal rule P 21 () with > 0 remains incentive
compatible with respect to both agents. Recall that P 21(0) = P is such that the incentive
compatibility constraint of agent 2 is slack. Due to continuity the constraint remains slack
for a proposal rule P 21( ) with > 0 small enough. Note furthermore that for > 0
one has r2 () > r2 , and since U1 (r2 ) < 0, one obtains U1 (r1 ) = U1 (r2 ) > U1(r2 ()).
Therefore, if P is incentive compatible, then also P 21 () for > 0 small enough. Since
by Lemma 4 the principal's utility increases with > 0, a proposal rule P = P 21 (0) for
which U1(r1 ) = U1(r2 ) cannot be optimal.
Now suppose P is such that the incentive constraint of agent 2 is binding and 21 > 0.
Consider the proposal rule P 12( ) with > 0. Using the same argument as above, P 12 ()
remains incentive compatible for > 0 small enough and increases the principal's utility.
Hence, a proposal rule P with a binding incentive constraint of agent 2 and 21 > 0 cannot
be optimal. We therefore conclude that the optimal 2{proposal rule P is characterized by
a binding incentive constraint of agent 2 and 21 = 0. This implies (y1 ; y2) = (y1p ; y (2 ))
which in turn implies p(1 0 11; 1) = 2 and hence, by (4), 11 = (2 0 )=[(1 0 )2 ].
However, 11 must be non{negative and, in order to have some information revelation,
must dier from 21 = 0. Therefore, an informative equilibrium requires < 2 . (See
also statement (i) of Lemma 6 below.)
0
0
Proof of Lemma 6
If an incentive compatible proposal rule (f1; 2g; R; 1 ; 2) induces information revelation
P
P
then there is some rj 2 R with 1j 6= 2j . Since 1j = 2j = 1, there exist rj ; rk 2 R
with 1j > 2j and 1k < 2k . Incentive compatibility of a recommendation rj implies
Sign(2j 0 1j ) = Sign(rj 0 y( )), i.e. rj and rk satisfy rk > y () > rj .
26
If 2 it follows U2(y()) U2(y1p).
Consequently, due to the concavity of U2,
2j > 1j implies U2 (rj ) < U2 (y ()), while 2j < 1j implies U2(rj ) > U2 (y( )). It
P
P
P
therefore follows that j (2j 0 1j )U2 (rj ) = j maxf2j 0 1j ; 0gU2 (rj ) 0 j maxf1j 0
P
P
2j ; 0gU2 (rj ) < j maxf2j 0 1j ; 0gU2 (y ()) 0 j maxf1j 0 2j ; 0gU2 (y ()) = 0; contradicting the incentive compatibility condition (10). Therefore if an incentive compatible
rule P induces information revelation, then neccesarily < 2.
To prove the second statement note rst that for i = 1; 2:
0
1
jRj
jRX
j01 X
j
j
X
X
@
1k 0 2k A [Ui (rj ) 0 Ui (rj +1)]:
(1j 0 2j )Ui (rj ) =
j =1
j =1
k =1
k =1
Due to (7), one has jk=1 1k 0 jk=1 2k > 0 for all j = 1; : : : ; jRj 0 1. Hence, by the
monotonicity assumption,
1
jRj
jRX
j01 0 X
Z r
j
j
X
X
j +1
@
(1j 0 2j )U1 (rj ) = 0
1k 0 2k A
U10 (y ) dy
rj
j =1
j =1
k =1
k =1
0
1
jRX
j01 X
Z r
jRj
j
j
X
X
j +1
@
> 0
1k 0 2k A
U20 (y) dy = (1j 0 2j )U2(rj ):
rj
j =1
j =1
k =1
k =1
Therefore, if (9) holds with equality, (10) is satised with strict inequality and vice versa.
For the third statement, note that if < 2 then for all rl < y( ) it holds U2 (rl ) U2 (y( )). Now suppose that for all rk > y( ) it holds rk = y2p , then U2 (rk ) < U2 (y1p )
P
P
for all k such 2k > 1k . It follows j (2j 0 1j )U2 (rj ) = j maxf2j 0 1j ; 0gU2 (rj ) 0
P
P
P
j maxf1j 02j ; 0gU2 (rj ) <
j maxf 2j 01j ; 0gU2 (y ( ))0 j maxf1j 02j ; 0 gU2 (y ( )) =
0; contradicting the incentive compatibility condition (10). Therefore if an incentive
compatible rule P induces information revelation, then there exists an rj such that
y () < rj < y2p .
P
P
Proof of Proposition 5
Since < 2 it follows from Proposition 4 that the optimal proposal rule induces some
amount of information revelation. Since full revelation is not possible some, and by
Lemma 6 one, incentive constraint is binding at the optimum.
For some rk ; rl 2 R with rk 2 (y1p ; y2p ) consider the transformation P kl as dened in
the proof of Lemma 4. In order to evaluate the impact of the transformation on the
incentive constraints, dene the functions
jRj
jRj
X
X
fikl ( ) [1j ( ) 0 2j ( )]Ui (rj ()) 0 [1j 0 2j ]Ui (rj )
j =1
j =1
= (1l 0 2l )[Ui (rl ) 0 Ui (rk ( ))] + (1k 0 2k )[Ui (rk ()) 0 Ui (rk )]:
(15)
Now consider the derivative of fikl () evaluated at = 0:
dfikl (0)
= (1l 0 2l )[Ui (rl ) 0 Ui (rk )] + (1k 0 2k )Ui0(rk )rk0 (0);
d
(16)
27
where
Sign (rk0 (0)) = Sign(1l 2k 0 1k 2l ) = Sign(rk 0 rl ):
(17)
For statement (i), suppose by contradiction that P is such that agent 1's IC binds.
An incentive compatible proposal rule P that induces information revelation contains
a pair (rk ; rl ) such that y2p > rk > y( ) > rl , by statement 3 of Lemma 6. Incentive
compatibility implies 1k < 2k , 1l > 2l , while (17) implies rk0 (0) > 0.
If U1 (rk ) U1(rl ), it follows, due to rk > rl , that rk > y1a. Consequently, U10 (rk ) < 0.
These properties imply df1kl (0)=d > 0, which means that agent 1's IC (9) remains satised
for small > 0. Agent 2's IC is also satised since, for = 0, it has slack. Hence, there
exists a > 0 for which the transformation P kl () is feasible. By Lemma 4 such a P is
not optimal.
Now consider the case U1 (rk ) > U1(rl ), which due to rk > rl implies rl < y1a . If there
exists an " > 0 such that for all 2 (0; ") it holds that f1kl () 0, then P is not optimal
by the above argument. If such an " does not exist, then f1kl () < 0 for suciently close
to zero. But then there exists a ^ > 0 such that f1kl (^) = 0. This follows from continuity
of f1kl () and the fact that there exists a 0 > 0 such that U1(rk (0 )) = U1 (rl ), which
implies f1kl ( 0) > 0. The transformation P kl (^) is feasible because, due to f1kl (^) = 0,
agent 1's IC still holds in equality, which, by Lemma 6, implies that also agent 2's IC is
satised. Since ^ > 0, Lemma 4 implies that P is not optimal. It therefore cannot be
optimal to have agent 1's IC bind. Hence, at the optimum agent 2's IC binds.
For the remainder of the proof consider an incentive compatible proposal rule P that
induces some revelation of information and for which agent 2's IC binds. For statement
(ii), let now rk < y( ) < rl , which implies 1k > 2k , 1l < 2l , and, by (17), rk0 (0) < 0.
Assume by contradiction that rk > y1p .
If U2 (rk ) U2(rl ), it follows, due to rk < rl , that rk < y2a. Consequently, U20 (rk ) > 0.
These properties imply df2kl (0)=d < 0, which means that agent 2's IC (10) is satised
for small > 0. For > 0 but small also agent 1's IC remains satised, since, for = 0,
it has slack. Hence, there exists a > 0 for which the transformation P kl () is feasible.
Lemma 4 implies that P is not optimal.
Now consider the case U2 (rk ) > U2(rl ), which due to rk < rl implies rl > y2a . If there
exists an " > 0 such that for all 2 (0; ") it holds that f2kl () 0, then P is not optimal
by the above argument. If such an " does not exist, then f2kl ( ) > 0 for suciently
close to zero. But then there exists a ^ > 0 such that f2kl (^) = 0. This follows from
continuity of f2kl () and the fact that there exists a 0 > 0 such that U2(rk ( 0 )) = U2(rl ),
which implies f2kl (0 ) < 0. The transformation P kl (^) is feasible because agent 2's IC
still holds in equality which, by Lemma 6, implies that agent 1's still has slack. Since
^ > 0, Lemma 4 implies that P is not optimal. Thus it is never optimal to have an
28
rk 2 (y1p ; y ()). Since there must be an rk < y (), we conclude that an optimal incentive
compatible recommendation rule exhibits r1 = y1p and rj y () for all j > 1.
For statement (iii), consider an optimal P , i.e. P exhibits r1 = y1p and rj y () for
all j > 1. Now consider the transformation P k 1( ) with k > 1. Since 21 = 0, it follows
df2k1 (0)
= 11 [U2 (y1p ) 0 U2 (rk )] + (1k 0 2k )U20 (rk )rk0 (0);
(18)
d
and since rk > r1 = y1p it holds rk0 (0) > 0. Now if y( ) y2a and P contains an
rk 2 [y( ); y2a ], then U20 (rk ) 0 and 1k 2k . Moreover, U2(y1p ) < U2 (rk ) and it follows
that (18) is negative. Hence, for some small > 0 the transformation P k1 () is feasible
and yields the principal more. Consequently, P is not optimal. If y2a < y( ) and P
contains an rk = y (), then 1k = 2k . Since < 2 implies U2 (y( )) > U2 (y1p ), it
follows that (18) is negative and P is suboptimal.
Proof of Proposition 6
We must prove that an incentive compatible proposal rule with jRj > 2 and rj 6= rk for
all k 6= j cannot be optimal. Suppose by contradiction that such a P is optimal, then
by Proposition 5 it satises r1 = y1p and maxfy (); y2a g < r2 < r3 y2p. Moreover, since
agent 2's incentive constraint binds at the optimum, it holds that r2 < y(2 ) and there
must also exist an r3 > y (2 ). This implies U2 (r2 ) > U2 (y1p ) > U2 (r3 ).
Denote by P (2 ; 3 ) the proposal rule which results from a joint transformation P 21(2 ),
P 31 (3 ), as dened in the proof of Lemma 4. Write as V (2 ; 3 ) the principal's payo associated with P (2 ; 3). Its partial derivative with respect to j evaluated at 2 = 3 = 0
is
@V (0; 0)
= (1 0 )11[V1 (y1p ) 0 V1(rj )]:
@
j
The principal's marginal gain from a joint transformation with 3 = 3(2 ) = 02 with
> 0 is therefore
dV (2; 3 (2 )) @V (0; 0) @V (0; 0)
=
0 @ d2
@2
3
2 =0
p
= (1 0 )11 [V1(y1 ) 0 V1 (r2) 0 (V1 (y1p ) 0 V1 (r3 ))]:
(19)
To evaluate the impact of a marginal change of P (0; 0) on the incentive constraint of
agent 2 dene
F (2 ; 3) = f221 (2 ) + f231 (3 ):
Recalling (18) and using 3 (2 ) = 02 the total derivative of F (2; 3 (2 )) evaluated at
2 = 0 is:
dF (0; 0)
@F (0; 0) @F (0; 0)
=
0 @ d2
@2
3
29
=
11 [U2(y1p ) 0 U2 (r2 )] 0 (22 0 12)U2 (r2 )r2(0)
0 (11[U2(y1p) 0 U2(r3)] 0 (23 0 13)U2(r3)r3(0))
= 11 [U2(y1p) 0 U2(r2)] 0 U2(r2)y (p2)p2 p2 0 0 [U2(y1p) 0 U2(r3)] 0 U2(r3)y (p3)p3 p3 0 (20)
equation follows from rj (0) = y (pj )pj (0) and pj (0)(2j 0 1j )=11 =
0
0
0
0
0
0
where the last
pj (pj 0 )= with
0
0
0
0
0
0
pj (j ) p(1j (j ); 2j (j )):
A marginal change of P satises incentive compatibility of agent 2 if (20) is negative, i.e.
if
[U (yp) 0 U2(r2)] 0 U2(r2)y (p2)p2 p 0 :
2 1p
[U2(y1 ) 0 U2(r3)] 0 U20 (r3)y0(p3)p3 p 0
By (19) a marginal change of P (weakly) increases the principal's utility if
V1 (y1p ) 0 V1 (r2 )
:
V1 (y1p ) 0 V1 (r3 )
Thus, a marginal change of P which (weakly) increases the principal's utility while leaving
the rule incentive compatible exists, if
V1 (y1p ) 0 V1 (r2 ) [U2(y1p ) 0 U2(r2 )] 0 U20 (r2 )y 0(p2)p2 p 0
(21)
V1 (y1p ) 0 V1 (r3 ) [U2(y1p ) 0 U2(r3 )] 0 U20 (r3 )y 0(p3)p3 p 0
where rj = y(pj ). Since U2(r2) > U2(y1p) > U2(r3) and U 0(rj ) < 0 < y0(pj ) and p3 > p2 >
, condition (21) is strictly satised for any if and only if
D (p2) D (p3 )
(22)
with
V1 (y1p ) 0 V1(y (p))
D (p) 0
:
(23)
U (y(p))y0 (p)p2
0
0
2
3
2
3
2
Therefore, if D(p) is weakly increasing a proposal with jRj > 2 is not optimal. Straightforward calculations yield that the derivative of D(p) is larger or equal to zero when
U200 (y(p)) 2y 0(p) + py 00(p)
V10(y (p))
+
0
(24)
U20 (y(p))
py 0 (p)2
1V (y(p))
with
1V (y) V1(y1p) 0 V1(y):
The denition of y(p) implies 2y0(p) + py00(p) = (2V100(y(p)) 0 py0(p)b(y(p)))y0(p)=a(y(p))
and py0(p) = V10(y(p))=a(y(p)) with
a(y ) (1 0 p)V100 (y) + pV200 (y ) < 0;
b(y ) (1 0 p)V1000 (y ) + pV2000(y ):
30
Hence, dropping the dependence on p we may rewrite (24) as
U200(y ) 2V100 (y) b(y )
V10 (y)
+
0
0
U20 (y) V10 (y) a(y)
1V (y) :
Now if V1000(y) 0 and V2000(y) 0 then b(y) 0 and (25) is satised if
(25)
U200 (y )=U20 (y) + 2V100(y )=V10 (y ) 0V10 (y)=1V (y ):
Proof of Proposition 7
If there is no consultant, the principal's payo for < 1 is
"
!
#
1 0 1 0 p
NC
V ( ) = (1 0 )
V (y ) + 1 0
V (y( )) + V2(y (1))
1 (1 0 ) 1 1
1 (1 0 ) 1 1
= 10 V1(y1p) + (1 0 1) V1(y(1)) + V2(y(1))
(26)
1
1
which is linear in . For 1 the payo is V NC () = (1 0 )V1(y()) + V2(y()).
Since (26) converges to this payo as approaches 1, the function V NC () is continuous
at 1. For the case with a consultant the proof runs analogously.
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metrica 56