We consider Common Agency games of moral hazard and we suggest that there is only a very weak support for the standard restriction to take-it or leave-it contracts.Menus, Common Agency.

Andrea Attar

Gwenaël Piaser

Nicolas Porteiro

English

Research Papers in Economics

D  S  E 
Working Paper  
ISSN: 1827/336X 
A note on Common
Agency models of
models of moral hazard
Andrea Attar
Gwenael Piaser
Nicolàs Porteiro
Dipartimento  Scienze Economiche 
Department 
of Economics
Ca’ Foscari University of
Venice 
No. 36/WP/2006Working Papers  
Department of Economics  
Ca’ Foscari University of Venice  
No. 36/WP/2006 
ISSN 1827-336X 
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  Department of Economics 
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A note on Common Agency models of moral hazard 
 
 
Andrea Attar 
 IDEI, Toulouse, and Università di Roma, La Sapienza 
 
Gwenaël Piaser 
University of Venice 
 
Nicolás Porteiro 
 Universidad Pablo de Olavide 
 
 
 
 
 
Abstract 
We consider Common Agency games of moral hazard and we suggest that there is only a 
very weak support for the standard restriction to take-it or leave-it contracts. 
 
Keywords  
Menus, Common Agency. 
 
JEL Codes 
D82 
 
 
 
 
 
 
Address for correspondence: 
Gwenaël Piaser 
Department of Economics 
Ca’ Foscari University of Venice 
Cannaregio 873, Fondamenta S.Giobbe 
30121 Venezia - Italy 
Phone: (++39) 041 234 91 22 
Fax: (++39) 041 2349176 
e-mail: piaser@gmail.com 
  
This Working Paper  is published under the auspices of the Department of Economics of the Ca’ Foscari University of Venice. Opinions 
expressed herein are those of the authors and not those of the Department. The Working Paper series is designed to divulge preliminary or 
incomplete work, circulated to favour discussion and comments. Citation of this paper should consider its provisional character. 1 Introduction
When Common Agency games of complete information are considered, it is well known that restricting
principals to the use of simple take-it or leave-it offers implies a loss of generality. If principals were
allowed to propose menus over the relevant alternatives, there could be equilibrium outcomes that could
not be supported by simple offers.1
Inparticular,Peters[2003]identiﬁesasetofrestrictionsonplayers’preferences,called"no-externalities
assumption", guaranteeing that every pure strategy equilibrium allocation of a Common Agency game
with menus can in fact be supported at (pure strategy) equilibrium in a simpler game where principals’
strategies are restricted to be take-it or leave-it offers. Interestingly, these conditions are satisﬁed by a
large class of games.2
In a related work, Attar et al. [2006] argue that these conditions are in fact not sufﬁcient in a moral
hazard scenario. The present paper hence proposes an alternative set of conditions to establish the result.
These conditionsare quite restrictive, since they require the agent’s best-reply to be always single-valued.
Nonetheless, they cannot be easily weakened as we show with a counter-example.
Many recent researches consider a framework where multiple principals compete in the presence of
moral hazard: Bernheim and Whinston [1986], Bisin and Guaitoli [2004], Ishiguro [2005], Martimort
[2004], Parlour and Rajan [2001], Attar et al. [2005]. Most of these papers limit the analysis to take-it or
leave-it offers strategies. Our research suggests that this restriction may be problematic.
The note is organized as follows. Section 2 presents the reference model, Section 3 provides our
Theorem and the related counter-example.
2 Common Agency under moral hazard
We refer to a scenario where there are a number of principals (indexed by j ∈ N = {1,...,n}) contracting
with one agent (denoted by index, 0). We assume that the agent takes an effort e from the set E and
obtains an allocation yj ∈Yj from principal j. The effort choice is not contractible.
The payoff to principal j ∈ {1,...n} is represented by the von Neumann-Morgensternutility function
Vj : Õ
k∈N
Yk×E →
R+.
1See, among many others, the two inﬂuential works of Peters [2001] and Martimort and Stole [2002]. These papers also show
that menus can efﬁciently replace any sort of indirect communication mechanism (this is usually referred to as the Delegation
Principle).
2Peters [2003] provides several examples.
1For the agent the payoff is represented by the function
U : Õ
k∈N
Yk ×E →
R+.
Principals compete through menus. A menu for principal j, say Mj, is a subset of the set of feasible
offersYj.
We denote by Sj the set of all menus available to principal j, and by S the collection of all the Sj
sets. The agent chooses an item yj from each menu and an effort e. For every collection of menus
(M1,M2,...,MN), the strategy for the agent is hence a map s0 : M → M ×E, with M = ×i∈NMi, and S0
is the corresponding agent’s strategy set. The Common Agency game with moral hazard we refer to is
given by:
G =
￿
S,S0,u(.,e),(vj)j∈N
￿
.
We assume that all the relevant spaces satisfy standard regularity assumptions (for a general discus-
sion, the reader can refer to Peters [2001] and Peters [2003]). We focus our attention on the pure strategy
Subgame Perfect Equilibria (SPE) of the game G.
We now study Common Agency games where principal strategies are restricted to be take-it or leave-
it offers: the menu offered by every principal is a singleton. A strategy for principal j in this simpler
game is the offer ˜ yj ∈Yj; we let ˜ Sj be the strategy space for principal j and ˜ S = ×j∈N ˜ Sj.
The strategy of the agent is then a map ˜ s0 : ˜ S → E, and ˜ S0 denotes the collection of all such maps.
The Common Agency game induced by take-it or leave-it offers is then:
˜ G =
￿
˜ S, ˜ S0,U(.),(Vj(.))j∈N
￿
.
Peters [2003] identiﬁes a set of sufﬁcient conditions on players preferences guaranteeing that every
equilibrium outcome of the menu game G can be supported as an equilibrium in the game ˜ G. In the pure
moral hazard framework that we introduced here, these conditions (referred to as the "no-externalities
assumption") can be stated in the following way:
For every principal j ∈ N:
A. There exists a functionWj :Yj ×E →
R+ such that for all (y1,y2,...,yn) ∈Y, and for all e ∈ E
Vj(y1,y2,...,yn,e) =Wj(yj,e)
B. For any closed subset B ⊂Yj there is a yj ∈ B such that
2U(yj,y−j,e) ≥U(y′
j,y−j,e)
for all y′
j ∈ B, y−j ∈Y−j, and e ∈ E.
That is: each principal’s utility only depends on his own action and on the agent’s effort; in addition,
the agent has a weak preference ordering over the actions of every single principal that is independent of
her effort choice and of the other principal’s offers.
Unfortunately, these conditions turn out not to be sufﬁcient if moral hazard is explicitly considered:
Attar et al. [2006] show that whenever the single agent is indifferent over alternative outcomes, there can
be equilibrium allocations of the menu game that cannot be sustained through simple offers even if the
"no-externalities assumption" holds. If the single agent is indifferent among alternative outcomes, then
her strict preference order over each principal’s offers may depend on other principals’ proposals, even
though the weak preference order does not. Importantly,such a preference reversal can be exploited with
menus in a way that is not available through direct mechanisms.
We proposehere an alternativeapproachto establish the result, eliminatingany source of indifference
on the agent’s side. By doing so, we are able to generalize the no-externalities assumption in one dimen-
sion, as it is no longer necessary to assume that there are no direct externalities among principals (part A
of Peters’ original condition).
3 A no-externality Theorem with moral hazard
We ﬁrst of all restrict the analysis to a speciﬁc class of games, that we denote G .
Deﬁnition 1 A Common Agency game G belongs to the class G if, for every j ∈ N, for every y−j ∈Y−j,
U (yj,y−j,e)  =U
￿
y′
j,y−j,e′￿
, ∀yj ∈Y,y′
j ∈Y,e ∈ E,e′ ∈ E, with (yi,e)  =
￿
y′
i,e′￿
Hence, everything else equal, the agent is never indifferent to a change in the offer made by any of
the principals.
This condition is generically satisﬁed in the payoff space of the agent whenever ﬁnite games are
considered.3 Now, it turns out that for every Common Agency game belonging to G , the original no-
externality condition (B) is in fact sufﬁcient to provide a proof:
Theorem 1 Consider anygameG in theclassG . Ifthe preferencesof theagentsatisfy condition(B),then
for every outcome that can be supported as a pure strategy SPE of the game G, there is a pure strategy
equilibrium of the direct mechanism game ˜ G that implements the same outcome.
3Importantly, when the relevant decision sets are not ﬁnite (as it is the case in many applications), indifference is a common
feature of economic models (for more precise deﬁnitions of genericity in games, see Anderson and Zame [2001]).
3Proof. Take the game G. For every given array of menus (Mj,M−j) we let the best reply of the agent be:
[y∗,e∗] = argmax
(y∈M,e∈E)
U (y,e). (1)
We now consider a SPE of the game G, that we denote
￿￿
M∗
j
￿
j∈N
,s∗
0
￿
, and we let (y∗,e∗) be its corre-
sponding equilibrium outcome.
The candidate for equilibrium in the game ˜ G will therefore be the array
￿￿
˜ y∗
j
￿
j∈N
, ˜ s∗
0
￿
, where for
every j:
˜ y∗
j = y∗
j (2)
and ∀ (˜ yj, ˜ y−j) ∈ ˜ S:
˜ s∗
0 = argmax
e∈E
u(˜ yj, ˜ y−j,e). (3)
Oneshouldnoticethatthisstrategyproﬁleinducesthesameoutcomeasthatcorrespondingto
￿￿
M∗
j
￿
j∈N
,s∗
0
￿
in the game G.
We then have to show that, if we consider the game ˜ G, none of the players has a proﬁtable deviation.
First, we look at the single agent: if principals are offering
￿
˜ y∗
j, ˜ y∗
−j
￿
, then ˜ s∗
0 = e∗ is by construction her
best reply.
Consider now the behavior of principals in the game ˜ G: let assume that principal j deviates to ˜ y′
j. For
the deviation to be proﬁtable it should be:
Vj
￿
˜ y′
j, ˜ y∗
−j, ˜ e′￿
>Vj
￿
y∗
j,y∗
−j,e∗￿
where ˜ e′ is such that:
˜ e′ = argmax
(e∈E)
U
￿
˜ y′
j,y∗
j,e
￿
.
Importantly, the deviation was already available in the original game G. Furthermore, the best reply
in terms of effort to the deviation ˜ y′
j is going to be the same in the game G and in ˜ G. If we denote
￿
y
′
−j,e
′￿
= argmax
(y−j∈M−j,e∈E)
u
￿
˜ y′
j,y−j,e
￿
the best reply of the agent that, in the game G, is induced by the deviation ˜ y′
j, then it must be ˜ e′ = e
′
.
This last result can be shown by contradiction. Let us consider the game G and suppose that ˜ e′  = e
′
.
Now, given separability, the agent will not modify her choices y∗
−j in the menus M−j offered by the
non-deviating principals. If ˜ e′  = e
′
, then one should have:
4u
￿
˜ y′
j,y∗
−j, ˜ e′￿
 = u
￿
˜ y′
j,y∗
−j,e
′￿
that cannot be true, otherwise either ˜ e′ or e′ is not a maximum. Hence, letting ˜ y′
j be a proﬁtable
deviationin the game ˜ G contradictsthe assumptionof(y∗,e∗) beinga (purestrategy)equilibriumoutcome
of the menu game G.
Importantly,no restriction on principals’ preferences has been explicitly introduced. That is, once we
restrict the analysis to Common Agency games in the class G , condition (A) is not needed to prove the
theorem.
The conditions we have introduced are indeed quite strong and they are not satisﬁed in any of the
existing works of Common Agency under moral hazard. 4 Trying to weaken them is anyway not a trivial
exercise. In particular, we introduce a counter-example showing that taking the agent’s effort choice to
be single-valued is not enough to get the result. In other words, the restriction to the class G turns out to
be critical.
We provide in what follows an example of a Common Agency game with moral hazard where no-
externalityconditions(A)and(B) aresatisﬁedandtheagentoptimaleffortchoiceis always single-valued.
We show that even in this case our Theorem 1 cannot be established.
That is, we consider games in the class G′:
Deﬁnition 2 A Common Agency game G belongs to the class G′ if, for every j ∈ N,
∀(yj,y−j) ∈Y, ∀e ∈ E,∀e′ ∈ E, e  = e′ =⇒U(yj,y−j,e)  =U(yj,y−j,e′).
Common agency games in the class G′ exhibit the property that the agent’s effort choice is unique for
every array of principals’ offers. Now, consider the following example, where two principals (P1 and P2)
contract with one agent in a pure moral hazard scenario. P1 can take three decisions a, b or c and P2 can
choose between A, B andC. The agent can select her non-contractible level of effort in the set {e′,e′′}.
If the agent chooses effort e′, the correspondingpayoffs are given by the matrix:
A B C
a (1,1,0) (1,2,0) (1,0,0)
b (4,1,1) (4,2,5) (4,0,5)
c (3,1,5) (3,2,5) (3,0,5)
4We essentially require that the optimal effort selected by the agent should be independent of the allocations proposed by
principals. In addition, no indifference is allowed. This implies that the effort choice cannot alter in any relevant way the payoffs’
distribution. We should also remark that M. Peters has recently proposed the introduction of a further restriction to the original
no-externalities assumption (see Peters [2005]). He argues that taking the set of optimal choices of the agent from the menu of each
principal to be independent from her effort choice and from the other principals’ offers is a sufﬁcient condition to reestablish the
result. Unfortunately, even this last condition can be hardly applied to standard moral hazard environments.
5where the ﬁrst element in each cell refers to the payoff of P1, the second one to the payoff of P2 and the
last one to that of the agent. Payoffs associated to the choice of e′′ are:
A B C
a (0,3,4) (0,1,4) (0,0,4)
b (1,3,4) (1,1,4) (1,0,4)
c (2,3,4) (2,1,4) (2,0,4)
The payoffs corresponding to the optimal effort choice of the agent are summarized in the matrix:
A B C
a (0,3,4) (0,1,4) (0,0,4)
b (1,3,4) (4,2,5) (4,0,5)
c (3,1,5) (3,2,5) (3,0,5)
This gamebelongsto the class G′, and(A) and(B) are satisﬁed. However,the outcome(4,2,5) cannot
be sustainedas an equilibriumif principalsare restrictedto the use of take-it orleave-it offers. If Principal
1 plays b, then Principal 2 will always prefer to play A rather than B.
The outcome (4,2,5) can indeed be supported at equilibrium in a game where principals are allowed
to use menus. In particular, the menus M∗
1 = {b,c} for P1 and M∗
2 = {B} for P2 lead to a pure strategy
equilibrium where (4,2,5) is implemented. If P1 offers the menu {b,c} and P2 offers the degenerate
menu {B}, then the agent will pick the element {b} from the ﬁrst menu and exert the effort e′. It is
straightforward to show that there are no other menus giving P1 a higher payoff. It is also clear that
P2 has no incentive to play the degenerate menus {A} and {C}: as a consequence we cannot ﬁnd any
proﬁtable deviation for P2 either.
Hence, the standard approach to model strategic competition in the presence of a single agent who
is taking a non-contractible effort appears to be signiﬁcatively limited. If principals are allowed to offer
menus in a standard moral hazard environment, then one can typically generate new equilibrium out-
comes. 5
Acknowledgments
This noteis an outcomeofa researchstartedwhenall theauthorswereworkingat CORE. Thus,we would
like to thank all CORE people for their unique support. We also thank E. Campioni for her extremely
useful comments on a previous draft of this work.
AndreaAttargratefullyacknowledgestheﬁnancialsupportoftheMarieCurieEIFprogramno.515589/2005.
All responsibility remains to the authors.
5One should notice that there is always a (weak) rationale for the use of take-it or leave-it offers. Theorem 1 in Peters [2003]
shows that every equilibrium outcome of this simple game survives when principals are allowed to use menus.
6References
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A note on Common Agency models of moral hazard

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