Modern Economy, 2010, 1, 51-58
doi:10.4236/me.2010.11005 Published Online May 2010 (http://www.SciRP.org/journal/me)
Direct Mechanisms, Menus and Latent Contracts
Gwenaël Piaser1,2
1IPAG Business School, Paris, France
2Luxembourg School of Finance, Luxembourg, Luxembourg
Received February 17, 2010; revised March 20, 2010; accepted March 30, 2010
Abstract
In common agency games, one cannot characterize all equilibria by considering only direct mechanisms. In
an attempt to overcome this difficulty, Peters [1] and Martimort and Stole [2] identified a class of indirect
mechanisms (namely, menus) which are able to characterize every equilibrium. Unfortunately, menus are
difficult to handle, and several methodologies have been proposed in the literature. Here, it is shown that,
even if authors consider menus rather than simpler mechanisms, many equilibria described in the literature
could have been characterized by direct incentive compatible mechanisms. Use of more sophisticated me-
chanisms was not necessary in these cases.
Keywords: Common Agency, Revelation Principle, Delegation Principle, Direct Mechanisms, Menus, Latent
Contracts
1. Introduction
The restriction to direct incentive compatible mecha-
nisms is a cornerstone of contract theory. It provides a
simple and elegant method for characterizing arbitrary
equilibria in any principal-agent model, even with very
complex communication between the players. Because of
its tractability, the principal-agent model has been very
successful, and it has revitalized many economic fields:
Regulation, redistribution, insurance and others.1 Multi-
agent games have provided the basis for auction theory
and the theory of the provision of public goods.
Unfortunately, the restriction to direct incentive com-
patible mechanisms causes some loss of generality in
multi-principal games. Intuitively, simple contracts fail
to be general because the structure of the game involves
endogenous information. For a principal, relevant infor-
mation includes not only the type of the agent (for exam-
ple his/her willingness to pay in a case of a duopoly) but
also the message that the agent sends to other principals;
the message sent sets a particular agreement between a
principal and the agent, which could modify the agent’s
willingness to pay for the products of other principals.
A strategy for overcoming this limitation is to give up
the concept of “direct mechanism” or any of its gener-
alizations, and consider the Taxation Principle. This prin-
ciple was introduced by Hammond [4], Guesnerie [5]
and Rochet [6], and states that there is no loss of general-
ity in considering menus, or nonlinear prices. Peters [1]
and Martimort and Stole [2] show that an equivalent of
the Taxation Principle (they call it Delegation Principle)
makes it possible to characterize any equilibrium of any
common agency game. The problem with this approach
is that the concept of menu is large for common agency
games, and, even if it simplifies the game, equilibria re-
main hard to characterize. To reach tractable problems,
other ad hoc assumptions are added to restrict the menu
set.
The present paper does not question the validity of the
differing further assumptions made in the literature. We
welcome assumptions (differentiability or continuity) if
they allow ready characterization of equilibria in this
class of games. The cost of these assumptions is proba-
bly a loss of generality.2 Nevertheless, the author does
not believe that restrictions invalidate the results ob-
tained with menus. The methodologies used to find a
fixed-point in common agency games in which menus
are allowed are criticized. The present paper shows that,
in almost all models of the common agency literature,
equilibria characterized by menus could have been char-
acterized by direct mechanisms. The basic intuition is
that menus can characterize a large set of equilibria be-
cause a principal, by using a menu, can create sophisti-
cated rewards.
2In common agency games, some equilibria may be sustained by dis-
continuous menus; see Laffont and Tirole [7] chap. 17.
1See Laffont and Martimort [3] for a complete survey.

G. PIASER 52
N
Given menus, it might seem that analysis of common
agency games is simply a matter of computation. Unfor-
tunately, though the use of menus may be helpful in this
class of game, it does not permit ready characterization
of equilibria. Below, it is argued that common method-
ologies used in the literature characterize only a re-
stricted set of pure strategy equilibria. Let us now con-
sider the “latent contract” concept, which gives insight
into the main result.
The next section presents a basic common agency
model. Section 3 defines direct mechanisms and menus.
Section 4 introduces the concept of latent contracts. In
section 5 some examples are presented from the litera-
ture. Section 6 sets out conclusions.
2. The Model
Consider a scenario in which there are a number of prin-
cipals (indexed by ) contracting with
one agent (denoted by index 0). The agent’s type is
drawn from a compact set having a probability dis-
tribution

1,...,in

(.)
F
that is common knowledge. The princi-
pal also makes an action: He has to decide which
allocation i to implement. The implemented allo-
cation is observable and contractible. This means that a
principal can write a contract which specifies his chosen
allocation. Let us make the stronger assumption that a
principal is not able to contract on a probability distribu-
tion over . The payoff to principal is represented
by the von Neumann-Morgenstern utility function
i
i
y
i
Y
Y
i
1
: ...
i
VY Y
and for the agent the payoff is represented by the function
1
:...
i
UY Y
Preferences could be more general; the restriction to
von Neumann-Morgenstern utility functions is not criti-
cal for any of the following results, but merely makes the
model simpler. Moreover, applications in the literature
invariably consider this class of preferences. The princi-
pals compete through mechanisms. Each principal’s
mechanism is a map from i
M
to , where
i
Yi
M
is the
message space. Each set i
M
(for any principal i) is
compact, and each map i

is measurable. For clarity,
we require any mechanisms i

to be such that the im-
age set

i

i
M
 is a compact set. We denote by i

the set of all available map i

to principal when his
message space is
i
i
M
. We denote by
iN
i
M
M


 the
collection of the chosen message space, and

is the
collection of . Denote by the decision
in that the agent gets by sending the message to
the principal i. We explicitly assume that the rule
i


i
m

i

i
y
i
Yi m


.
i
 is enforceable. Once a principal has announced a
mechanism, he commits himself to respect his own rule;
if he receives a message , he cannot choose a deci-
sion different from
i
m


ii
m  . Finally, since the sets


ii
M
 are compact, it follows that

arg

ii
m
 
,
,i i
ii
max σmθ
mm M
i
Mi
Uσ,




 (1)
The agent’s pure strategy is to choose a message
i
mi M

for each principal. Hence, a pure strategy for
the agent is a map 0:σΘ Σ
M

. We denote by
the collection of all these possible pure strategies and by
0
Σ
0i

 the message sent to principal . i
Given the type set , the common agency game is
the array:
Θ
   


.,,
Mi i
iN iN
=θF

0
,,Σ,U.,,V.ΓΘΣ θ (2)
3. Direct Mechanisms and Menus
In the game
M
Γ
we have made only made standard
assumptions concerning the sets i
M
. The message
spaces may be quite complex. We can simplify the game
by considering direct mechanisms. For each principal the
message space i
M
is given, and coincides with the
agent’s type space . We thereby restrict the strategy
spaces of the principals.
Θ
The strategy of principal is the map i:
ii
Y




i
0
Σ

;
we let be the strategy space for principal , and
be the collection of all such strategy profiles. The
strategy of the agent is then a map , and denotes
the collection of all such maps by
i
Σ

Σ

0
σ

0i

 the message sent
to principal . i
Given , the common agency game induced by di-
rect mechanism is the array:
Θ


  


.,,
i
iN
iN
θF

0
,,Σ,
i
U

.θ,,V.Γ=ΘΣ

 (3)
Direct mechanisms have an obvious appeal—the mes-
sage spaces are simple and given. But, to be useful, we
need more than a simplification of the message space. To
apply the traditional principal-agent methodology, we
need also a definiton of the concept of incentive com-
patibility in our context.
Definition 1 A collection of strategies


**
0
,
iiN




is an incentive compatible equilibrium of the game
Γ


if it satisfies the following two conditions:

G. PIASER 53




**
0
,, ,...,
iiN
θΘσ σθ=θθ

  (4)
and

*
0
,,
ˆˆ ,,
ii iii
iN
ifissuch that


 

 
 
 (5)
then is not a profitable deviation.
ˆi
σ
When is it possible to restrict attention to direct incen-
tive compatible mechanisms? In other words, is it always
the case that, for any equilibrium of the game
M
 , there
exists a incentive compatible equilibrium of the game
such that the two equilibria are outcome equivalent?
For common agency games this is not so (see, for in-
stance, Peck [8], Martimort and Stole [2]). In games with
multiple principals, equilibria may exist whose outcomes
cannot be supported in equilibrium in the corresponding
direct mechanism game.
Γ

Peters [1] and Martimort and Stole [2] show that even
if one cannot restrict attention to direct incentive com-
patible mechanisms, a modified version of the Taxation
Principle applies. This principle states that, without loss
of generality:
1) One can restrict the set of message space and con-
sider the sets of all compact subsets of rather than
i
Y
i
M
.
2) The map is the identity over the chosen subset
of .
i
σ
i
Y
Given , the common agency game induced
by menu is the array
iN
Y= Y


    

0
,,,.,,.,,
Yi i
iN iN
Γ=ΘZΣUθVθF


. (6)
where 0 Σ is defined as and
0
Σ*
0i

as . Where
*
0i
σ
i
Z
denotes the set of all compact subsets of , and
is a generic element ofi
i
Yi T
Z
. We will use the obvious
notatio i T n .
iN
T=


Theorem 1 The three following statements can be es-
tablished:
1) For every equilibrium


**
0
σ,σ of the game
M
Γ
there exists an outcome equivalent equilibrium


**
0
T,

of the game Y
Γ
.
2) For any equilibrium

**
0
T,


of the game Y
Γ
there exists an outcome equivalent equilibrium


**
0
,σσ
of the game
M
Γ
.
3) For any incentive compatible pure strategy equilib-
rium of the game , there exists a pure
strategy equilibrium of the game

**
0
σ,σ


Θ
Γ

M
Γ
such that the two
equilibria are outcome equivalent.
The two first statements have been shown by Peters [1]
and Martimort and Stole [2]. The third is a result from
Peters [9]. The last statement of the theorem is very gen-
eral, and it does not rely on any assumption about the
action space available to the agent. It applies to all com-
mon agency models in the literature to date.
4. Latent Contracts
This section demonstrates how “latent contracts” can
help to characterize a larger set of equilibria.3 By latent
contract or or latent decision is meant any decision
reachable by the agent but never implemented at equilib-
rium, whatever the type of the agent. For example, if a
principal uses a incentive compatible direct mechanism,
this mechanism does not involve any “latent contract”.
This former mechanism is a map from the type set to
the decision set , denoted . By definition, for any
decision in the image set , there is a type
Θ
i
Yi σ

i
y


i
σΘ


θΘ

such that


ii
σθ=y
 .
Definition 2 We say that a menu contains latent
decisions if, given the strategies of the other players
and
i
T
-i
T
*
0

,
*
0i
,,
ii
yTθΘi
y
(7)

  
From this definition, we can reach the following theo-
rem:
Theorem 2 Consider the game Y
Γ
and a pure
strategy equilibrium. If principals offer menus without
latent decisions, then there exists an output equivalent
incentive compatible equilibrium in the direct mecha-
nisms game Θ

.
Proof Consider an equilibrium of the
game

***
0 ii
T,T,σ


Y
Γ
. We wish to construct an output equivalent
equilibrium in the game Θ

. By assumption, menus


ii
T,T
 do not involve latent decisions. One can con-
sider the agent’s equilibrium best reply *
0

, which is,
for every collection of menu , a function from set TZ
TΘ

to the set . We can construct unambiguously
the following direct mechanisms denoted
T
*
i

:
 
**
0
,, ,
iiii
iN θΘσθ=TTθ 

 

, (8)
where the mapping *
0i

is defined as above. We have
3Latent contracts were introduced by Hellwig [10], and the concept is
widely used in the literature on foundations of competitive equilibrium;
see for example Bisin et al. [11].

G. PIASER 54
Σ
constructed the strategies of the principals
in the game . Let us construct the agent’s best reply,
denoted . For all and for all , we de-
note by the image set of the mapping

**
ii
σ,σ


Σ

Θ


σΘ
0
σ

i

σ
θΘ
i

.
Define the best reply mapping of the agent as follows:
  
Θ,θ 

σ

**
00 i
σ,θ=,,Θ 
 
Σθ
 (9)
For all collections of direct mechanism Σ




, we
have






*
0
12
,N
Uσ θ
hh Θ
,
ii
iN
h


iN

,σσ 
12
,hh
σ
,..., n
argmax
h


,
N
Θθ
θ>U

θ=


,...,
ii
h
h



(10)
Suppose not:



*
0
,
,,
n
i i
iN
Θ
Uσσ σθθ


 ,
(11)
By construction,

 


,,
iN



*
0ii
iN
Uσσ σσΘ θ

 
*
0ii
σ,

θθ=U . Con-
sequently,
  





*
0,,
i
iN
θ



ii

i
U>U h 
iN
Θθ

 (12)
Since by construction we have


i
Θ,i iN σ 

σh

θ
,
we generate a contradiction.
Moreover, we have:

, ,...,θΘ σθ=θ *
0
,σ
 (13)
because, by definition for all and for every prin-
cipal ,
θΘ
i


*
,
i
=Tθ
ii
Σ

**
0

i
θ
. Hence the candidate equi-
librium is incentive compatible.
Supppose that principals play , and principal
deviates toward σ (all other players keep their
strategies). The agent’s bast play is then .
The “no latent decision assumption” implies that
*
σ
i

,θ

**
,
ii



0



**
0
,,
ii ii
iN Tsuch thatTθ   *
i
,=yy θΘ 
(14)
Hence


*
i
=T *
i
,iN 

*
0
σ

(15)
Using the definition of , we can state that



i
** **
00
,,iN,
,, ,,
i
ji ijii
Σ
θ=ΘTθ

 

 


,
jN
(16)
Under the “no latent decision” condition:

*
*
,,
ii
ii
iNy ThΘ
such thath=y

 
 (17)
Hence, for all θΘ

, ther

e exists

 
N
ii
hθ,hθΘ
such that:
  







 

*
\
****
00 \
,
,,, ,,
iij jjN i
iiijiijN i
hθhθ
=ΘTθΘTθ

 










(18)
Moreover, by definition
 




**
0iijii
hθ,h θσ,,θ 


.
Suppose that for principal the deviation is strictly
profitable:
i










**
0
***
0
,, ,
,, ,
ijiijN
ijiijN
VθθdF θ
>V θθdF θ








(19)
Equation (19) can now be rewritten as







 


**
0
***
0
,,,
,,,
ijiijN
ijiijN
VΘTθθdF θ
>VT TθθdF θ







(20)
which is a contradiction. We conclude that

***
0
,,
ii



 
is an equilibrium of the game Θ
Γ
. By construction, this
equilibrium is output equivalent to the equilibrium


** *
0ii
T,T,

.
Consider now the following example:
Example 1 Each principal () must make a de-
cision (
1,2i=
1
i
y
or 2
i
y
with ). The corresponding
payoffs are given by the following matrix:
1,2i=
1
2
y 2
2
y
1
1
y (2,2,3) (0,3,1)
2
1
y (1,0,1) (1,1,2)
where the first element in each cell refers to the payoff of
Principal 1, the second element to the payoff of Principal
2, and the last element to the agent’s payoff.
If we consider that principals are using direct mecha-
nisms (which are take-it or leave-it offers since informa-
tion is complete), there is only one pure strategy equilib-
rium: Principal 1 plays 2
1
y
and principal 2 plays 2
1
y
.
Agent’s payoffs are not relevant, since the the agent
plays no role.
If principals are allowed to use menus, so that they of-
fer subsets of


12
ii
y,y , then there are two pure strategy

G. PIASER 55
equilibria. In the first equilibrium, principal 1 offers the
degenerated menu and the principal 2 offers the
menu . In the second equilibrium, principal 1 of-
fers the menu and principal 2 offers the menu
. The agent chooses

2
1
y

12
11
y

2
2
y

2
2
y

y,

1
2
y, 1
1
y
from principal 1 and
1
2
y
from principal 2. The outcome is finally
implemented. The outcome

11
12
y,y



2

1
1
y

12
11
y
11
1
y

y,
y, cannot be sup-
ported by an equilibrium if principals use only direct
mechanisms. If principal 1 offers , the direct me-
chanism is not the best reply for principal 2. He
gets more by offering . The outcome can
be implemented because menus and

1
2
y

y

2
2

11
12
y,y



2
2
y
1
2
y,
embed latent decisions: 2
1
y
and 2
2
y
are not chosen by
the agent, but they are crucial because they prevent de-
viations.
Let us consider a second example.
Example 2 The type of the agent is with prob-
ability , and with probability . Pay-
offs are given by the following matrices:
1
θ
1-
1
p=1/22 θ1 p
1
θ
1
2
y 2
2
y
1
1
y
2
1
y
(2, 2, 3) (0, 3, 1)
(1, 0, 1) (1, 1, 2)
2
θ
1
2
y 2
2
y
1
1
y
2
1
y
(0, 1, 2) (1, 0, 1)
(0, 3, 1) (2, 2, 3)
This game has an equilibrium in the menu game. Each
principal proposes the menu


12
ii
y ,
1
θ
1,2y, i=
1
1
; the agent
chooses if his type is , and chooses
if his type is. Since the set of possible me-
nus is very small, we can check that for principal 1 no
deviations (which are the singletons) are profitable. The
same holds for principal 2. The outcome can also be
supported as an equilibrium in the direct mechanism
game. For principal 1 the former strategy can be repro-
duced in the following way: he plays

11
12
y,y


2
1
y,

2
2
y2 θ
y
if the agent
sends the message , and plays
1
θ2
1
y
if the agent an-
nounce . Principal 2 plays the same strategy (
2
θ1
2
y
if
and
1
θ2
2
y
if ). The agent can reach any cell by
misreporting his type. By analogy with the menu game, it
is best for the agent to announce his real type. Using a
similar argument, one can check that the strategies de-
scribed are also best replies for the principals.
2
θ
In the preceding example, the menus

and

12
11
y,y


12
22
yy,
1
1
do not embed latent decisions. For principal 1,
y
(resp. 2
1
y
) is chosen when the agent’s type is
(resp. ). Similarly, for principal 2, item
1
θ
2
θ1
2
y
is chosen
if the agent is of type , and
1
θ2
2
y
is chosen when the
type is 2. Menus do not embed latent decisions, so that
the equilibrium can be sustained by direct mechanisms.
Moreover, at equilibrium, the agent reveals his true type.
θ
Consider a last example taken from the literature.
Example 3 (Biais and Mariotti [12]) There are two
principals (indexed by ), and their decision spac-
es are
1,2i=


12
Y =0,1

Y= . A generic decision is
denoted by
=Y


q
θ
t,
-qt
. The two principals have the same
utility functions , where is the information of
the agent,
θ


0,1θ. The distribution function of
over
θ


0,1 is denoted by
F
. Agent preferences are
represented by the utility function , where the
variable
-tλθq


0,1λ; this is common knowledge. The
agent is constrained to accept contracts

and

11
t,q


2
q
2
t, such that 12
1qq+

. Assume that


Eθ<
λ
.
Then an equilibrium exists in which each principal offers
the menu



 
θM =qt,Ysuchthat
q=
q =tE
1
. If the
two principals offer this menu, the agent (whatever his
type) will choose the allocation and


t=E θ. If
one principal deviates, and offers a unique contract


t,q

, then
1) If


t,q

 is below the line

tqE

, the agent
will accept this contract and the deviating principal
makes zero profit, as he did at equilibrium. The deviation
is not profitable.
2) If


t,q

 is above the line

tqE


1q
, the agent
will always accept the contract offered. Whatever his
type, the agent will buy a quantity q

 from the
other principal. Since

tq E


, the deviating prin-
cipal makes losses.
This argument can easily extended to any kind of
menu. If we now look at the best direct mechanism
against the menu
M
, it is obvious that the degenerated
mechanism gives the contract



,1E 

to every type.

G. PIASER 56
It is also clear that, if one principal plays that mechanism,
it is not the best reply for the other principal to play that
same mechanism. He should offer the contract


t,q

,
where tq


 . This contract is accepted by the agent
only when his type is 1


2
θ
1
θ
, in which case it provides
positive profit to the principal. The equilibrium charac-
terized by Biais and Mariotti [12] is efficient, so that if
we ignore it, we may reach wrong conclusions.
5. Applied Common Agency Models
Let us now focus on examples taken from the literature.
To characterize equilibria in the set of menus is not a
trivial exercise. Martimort [13] and Martimort and Stole
[2] have introduced a sophisticated methodology.
1) They consider that principal i uses direct mecha-
nisms to reply to the menus and to the agent’s
strategy.
*
1
σ

11/ 2p=
*
i
T
2) From the best direct mechanism, one can deduce a
menu.
3) If this is done for every principal, and if each prin-
cipal is playing the menu derived from the best direct
mechanism, we finally obtain an equilibrium.
At equilibrium, menus do not involve latent decisions;
each item is chosen by some agent. This method pro-
vides no gain over the traditional method; any equilib-
rium characterized using this methodology can be char-
acterized by the simple use of direct mechanisms. Here is
an example that shows how this methodology fails to
characterize any equilibrium of a common agency game.
Example 4 The type of the agent is with prob-
ability , and with probability 1. Pay-
offs are given by the following matrices:
1
θ
1
p
1
2
y 2
2
y
1
1
y
2
1
y
(2, 2, 3) (0, 3, 1)
(1, 0, 1) (1, 1, 4)
2
θ
1
2
y 2
2
y
1
1
y
2
1
y
(4, 2, 2) (2, 3, 0)
(1, 0, 1) (1, 1, 5)
This common agency game has one pure strategy
equilibrium. The first principal (P1) plays the menu
get an utility of 3 and the principal 2 plays the
menu

1
11
y,y

2

1
2
y
, and gets an expected utility of 2. Clearly,
the first principal has no profitable deviation (he gets his
maximum payoff). The second principal has two possible
deviations in the menu game:

and

. If
the second principal plays or the menu

2
2
y


12
22
y,y

2
2
y


2
2
y
1
2
y,

2
the agent will always choose the item in princi-
pal’s 1 menu. Hence, deviating gives to principal 2 an
expected utility of 1.

1
y
Using the Martimort-Stole algorithm we cannot char-
acterize the equilibrium described above. If principal 2 is
playing the menu


1
2
y, the unique best reply of princi-
pal 1 is to play the direct mechanism (with incentive
compatible revelation of type):


1
θ


1
2
=y
11 1
θ= 
Playing this mechanism is equivalent to playing the
menu


1
1
y. The best reply to the menu


1
1
y for prin-
cipal 2 (in the set of direct mechanisms) is to play the
mechanism


2
θ


2
2
=y
21 2
θ= 
or equivalently the menu


2
2
y, and not the menu


2
1
y.
By construction, menus characterized by the Marti-
mort-Stole algorithm do not embed latent decisions; they
support equilibria which can also be supported by direct
mechanisms.
This methodology has been used, explicitly or implic-
itly, in several papers: Biais et al. [14] Calzolari [15],
Laffont and Pouyet [16], Martimort and Stole [17] and
Khalil et al. [18], and in several other unpublished papers:
Olsen and Osmundsen [19], Diaw and Pouyet [20] or
Calzolari and Scarpa [21]. These authors are able to cha-
racterize regular and realistic equilibria. Nevertheless,
these equilibria could have been characterized using the
standard methods of mechanisms design. Moreover, the
authors may not succeed in characterizing all of the equi-
libria of the communication game.
When Martimort and Stole [17] consider a complete
information version of their game (i.e., when


1Θ

;
roughly speaking, when their model is qualitatively sim-
ilar to our first example) by using menus, they are able to
characterize equilibria that could not have been charac-
terized by direct mechanisms. (Direct mechanisms are
take-it or leave-it offers in that case.) They do not then
use the methodology presented above. Hence for com-
plete information games, their analysis of their model is
invaluable.4
Martimort [22] proposes an original methodology. He
proposes focusing on direct mechanisms, and extends the
4They use the term “singleton contracts” instead of take-it or leave-it
offers.

G. PIASER 57
type set: the agent can report a type belonging to the set
, with , . At equilibrium, whatever his
type is, the agent is reporting his true type, but the fact
that he can report
Θ
ΘΘ
ΘΘ 
\

 
, (a type which does not ex-
ist) and get an outcome that cannot be reached if he re-
ports any , extends his possible strategies. The
message is never sent at equlibrium, but the possibil-
ity of reporting an “absurd type” makes some deviations
of the other principals unprofitable, as in examples 1 and
3. Clearly, this methodology is able to characterize equi-
libria that cannot be characterized if we restrict our at-
tention to direct mechanisms. Sending the message is
clearly a “latent decision”.
θΘ
θ

θ

Let us reconsider our last example.
The equilibrium can be also characterized in the fol-
lowing way. The first principal plays the direct mecha-
nism


12
112 11
:σθ,θ,θy,y





1
11 1
1
12 1
2
11
=
=
=
σθ y
σθ y
σtθy








where is an absurd type. θ

The second principal plays the direct mechanism
 
12
21222
:σθ,θ,θy,y




1
21 2
1
22 2
=
=
σθ y
σθ y







The direct mechanisms and constitute an
equilibrium. The best strategy for the agent is to reveal
his type. (We do not describe the agent’s strategy, as it is
very long and is not necessary.) The first principal has no
profitable strategy; he gets his maximum payoff in each
state of nature. For the second principal there are many
possible direct mechanisms. But the second principal
cannot get a payoff greater than 2: in every state of na-
ture, if is implemented, the agent will report the
type to the first principal 2 (whatever his real type)
and the second principal will get a payoff of value 1.
Thus, principal 2 has no profitable deviations. If princi-
pal 1 plays the direct mechanism
1
σ 2 σ 
2
2
y
θ

Thus, principal 2 has no profitable deviations. If prin-
cipal 1 plays the direct mechanism



22
112 11
:σθ,θy,y 


1
11 1
1
12 1
=
=
σθ y
σθ y







then the second principal has a profitable deviation; play-
ing the mechanism



2
21 2
2
22 2
=
=
σθ
y
σθ
y







gives him a payoff of value 3. The weakness of this ap-
proach is that there is no theory of how to determine the
set and of how to construct the mechanism for
values of which are not in . Indeed, Martimort [22]
characterizes the equilibrium using the Martimort-Stole
algorithm.
Θ

1
σ
θ Θ
6. Conclusions
Almost all of the literature on common agency with in-
complete information focuses on equilibria that can be
characterized by direct mechanisms.5 Some papers ex-
plicitly apply the Revelation Principle even if it is not
applicable. By doing this they may characterize only a
subset of all equilibria, and miss some realistic equilibria.
Other articles use different mathematical tools and more
complex mechanisms, but without characterizing a larger
set of equilibria. We still lack a simple, general, system-
atic approach for characterizing all of the equilibria of a
large class of common agency games. The complexity of
the existing methodology (menu or extended types) in-
dicates that this will be a demanding task.
7. Acknowledgements
I am grateful to Andrea Attar, Eloisa Campioni and Piero
Gottardi for their comments and suggestions. I thank also
François Boldron, Cyril Hariton, Sylvain Latil, Patrick
Leoni and Martin Meier and seminar participants at
CORE and at the European Economic Association Con-
gress in Amsterdam for their help and comments. All
errors are mine.
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exception.

G. PIASER 58
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