In a repeated interaction between a principal and two agents with inter-agents externalities and asymmetric information, we show that optimal decentralization within the organization is limited to the first period and across agents.Delegation, Hierarchy, Asymmetric information

Axel Gautier

Dimitri Paolini

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 4 
  
Delegation, externalities and organizational design 
  
 
 
Axel Gautier  
Université de Liège 
Dimitri Paolini  
CRENoS and DEIR, University of Sassari
Abstract 
In a repeated interaction between a principal and two agents with inter-agents externalities and asymmetric 
information, we show that optimal decentralization within the organization is limited to the first period and across 
agents.
We would like to thank Francis Bloch, Bernard Caillaud, Jacques Crémer, Luca Deidda, Tito Pietra, A. Piggins, Jacques Thisse, seminar 
participants in CRENoS, IRES and Liège and an anonymous referee for their comments. The second author thanks Italian MIUR for financial 
support. 
Citation: Axel Gautier and Dimitri Paolini, (2009) ''Delegation, externalities and organizational design'', Economics Bulletin, Vol. 29 no.4 pp. 
2683-2692. 
Submitted: Sep 07 2009.   Published: October 28, 2009. 
 
     1. Introduction
In multi unit organizations, with externalities (the choices made by one
unit aect the prot of the others), two key factors will drive the task allo-
cation problem: externalities and asymmetric information. In the absence of
externalities, there is no need to coordinate agents' choices and prot is max-
imized in a fully decentralized structure where the agents have all the power.
In the absence of asymmetric information, there is no reason to delegate and
prot is maximized in a centralized organization where the principal keeps
all the power. In an organization where there are both externalities and
asymmetric information, benets and costs are associated with delegation.
In a single period interaction, delegation is benecial because the decider
has superior information (Dessein, 2002). With repeated interaction, dele-
gation has an additional benet: the principal can improve her knowledge
of the agents' information by observing their past decisions (Gautier and
Paolini, 2007). Delegation is a learning process: when the agent uses his
private information to make better decision, the principal revises her beliefs
by observing the agent's choice and she can improve her decisions.
We model a two-period interaction between one principal supervising two
agents (units). At each period, a project must be implemented in each unit.
At time zero, the principal chooses the process of decision-making for the
following periods. Projects are transferable control actions (Aghion et al.,
2002): projects cannot be contracted out but control is contractible. Each
agent has a piece of private information and he exerts an externality on the
other. In this context, the questions we raise in this paper are which decisions
should be decentralized and when?
To answer the second question ('when?'), we build up on Gautier and
Paolini (2007) that have shown that, in a repeated set-up, an agent discloses
his private information when he receives control over the rst decision. We
show that this result continues to hold true in a multi agents setting where
there are multiple sources of information. Hence, optimal decentralization
within the organization remains limited to the rst period. The answer to
the rst question ('which'?) is the main contribution of this paper. We show
that delegation is limited across agents. Even though the principal acquires
information, delegation remains costly because the agents do not take the
externalities into account. Moreover, the benet of an additional piece of
information decreases because information is partly redundant. Hence, the
number of delegated decisions depends on the quality of the signal produced
1and on the cost of producing it. The quality of the signal depends on the
correlation between the information of the agents. The cost of producing
a signal, which is the cost of delegating decisions, depends on the external-
ity exerted by the agent that receives the control over decisions. Following
that, symmetric agents could be treated dierently in the organization if one
signal, obtained by the delegation of a decision to one agent, brings enough
information to the principal.
2. The Model
There are three players, the principal and two agents (units) l = A;B,
and two periods t = 1;2. Before the rst period, the principal decides who
will have the power to choose the project (dl
t) in unit l at time t. In each
period a project is implemented by both units.
Each agent l has private information represented by a state parameter l
drawn out of a set  = f1;2g with 1 < 2 and = 2   1.
The principal only knows the prior distribution of (A;B) over   ,
which is represented by a joint probability distribution fv11;v12;v21;v22g
where vij = prob(A = i;B = j). The correlation between 's can be mea-
sured by  = jv11v22   v12v21j. To simplify, we assume that: v11 = v22 = vii
and v12 = v21 = vij: Then, =jvii   1
4j and it takes a value between 0 (inde-
pendence) and 1
4 (perfect correlation).
The projects dier in just one dimension and there is a continuum of
possible decisions: dl
t 2 (0;+1): Each unit is a prot center. Depending
on l, there is one project maximizing prot in unit l. The prot in unit l
when a project dl
t is implemented at time t is equal to ldl
t  
(l dl
t)2
2 . The
unit's prot is maximized for dl
t = l + l and it decreases with the distance
between the preferred project and the actual one.
In addition, unit l exerts an externality on unit k. The externality is
measured by a parameter , identical for the two units. A decision dl
t taken
in unit l reduces the prot in unit k by dl
t.
The prot in unit l is thus:

l = 
ld
l
1  
(l   dl
1)2
2
+ 
ld
l
2  
(l   dl
2)2
2
  (d
k
1 + d
k
2);
with l 6= k.
The agents maximize the prot in their unit while the principal is inter-
ested in the maximization of the total prot (maximized for dl
t = l+l ).
23. Decisions of the principal and the agents
Each decision dl
t is made by the person to whom the principal has al-
located the right to decide. When the principal must decide on a project
dl
t, her decision depends on the information she has. Let us represent her
information by a distribution of beliefs ij over   . The decision dl
t that
maximizes total prot is:
d
l
t = 
l + E(
l j ij)   ;
where E(l j ij) denotes the expected value of l conditional on beliefs ij.
When agent l must decide on a project dl
t, he must take into account
that the principal will revise her beliefs after observing dl
t. This changes the
agents' prot if the principal can make a decision after the agent i.e. only if
dl
1 is delegated and dA
2 or dB
2 is not. If the principal cannot use the agent's
information, the agent chooses his preferred project dl
t = (l+l): In all other
cases, the agents and the principal play a signaling game and we must search
for equilibria. Usually, signaling games have multiple equilibria. We use Cho
and Kreps (1987) intuitive criterion (IC) to rene the set of equilibria. We
can establish that:
Proposition 1 Under delegation of dl
1, the only equilibrium that survives IC
is the least costly separating (LCS) equilibrium.1
Proof. See appendix.
Gautier and Paolini (2007) have proved this result in the one principal -
one agent case. Here, we show that the structure of preferences is such that
the result extends to the one principal - two agents case. That is, whatever
the principal's knowledge of the state parameters, there always exist one state
in which the agent would be better o if he manages to transmit the true
information to the principal i.e. the single crossing property holds true for
all possible information structure. Consequently, a separating equilibrium
always exists. Moreover, the IC selects the most ecient one among the
non-empty set of separating equilibria. This means that, if the principal
delegates dl
1 to agent l, she learns l and improves her knowledge of k. Both
second period decisions are thus based on a more accurate information.
To transmit information to the principal, the agents must sometimes take
a suboptimal decision to make their information transfer credible i.e. the
1The Riley (1979) outcome.
3decisions that maximize the agent's payo may not be incentive compatible.
Our corollary claries that point.
Corollary 2 If 2  2, the LCS equilibrium is dl
1() = l + l.
Notice that, whenever this condition does not hold true, the principal
optimally retains control over all decisions (proposition 3 hereafter).
4. Optimal organization
Given that there is an externality between the two units, any form of
delegation has a cost for the principal because there is no coordination in
the project choices. Nevertheless, delegation benets the principal because
(i) the decisions of the agents are based on better information than those
of the principal and (ii) the principal improves her knowledge of the state
parameter after observing delegated decisions. In this section, we derive the
optimal organization.
Starting from a fully centralized organization. The principal bases her
decisions on the expected value of l rather than on its true value. But the
principal internalizes the externalities imposed by one agent to the other.
Under centralization, the decisions taken by an uninformed principal are:
^ d
l
1 = ^ d
l
2 = 
l + (vii + vij)
l
1 + (vij + vii)
l
2    (1)
Next, consider partial delegation where the principal delegates dl
1 to
agent l and retains control over the remaining decisions. If dl
1 signals l
to the principal (and we know form proposition 1 that it is the case), the
principal becomes informed about l but also improves her information about
k, k 6= l if there is correlation between the information. Under the condition
of corollary 2, the optimal decision under partial delegation are:
d
k
1 = ^ d
k
1 (2)
d
l
1() = 
l + 
l (3)
d
k
2 = 
k + E(
k j 
l)    (4)
d
l
2() = 
l + 
l    (5)
With one signal obtained by delegating dl
1, the principal implements the
rst best in period 2 for agent l and bases her decision dk
2 on a better infor-
mation than the prior distribution of k.
4With full delegation, the decisions implemented in unit l are given by
(3) and (5) and the decisions implemented in unit k are given by
d
k
1() = 
k + 
k (6)
d
k
2() = 
k + 
k    (7)
Let us denote by C, PD and D, the principal's prot under central-
ization (C), partial delegation (PD) and full delegation (D). Starting from
the prot C, we can dene the marginal benet of delegation. The marginal
benet of delegating a rst project to agent l is equal to Mb(1) = PD C.
Simple computation gives that delegating one project amounts to a change
of total prot equals to Mb(1) =  2 + 1
2(1 + 82)2.
If the principal delegates a second project to agent k, the principal be-
comes informed about l and k. We can thus dene the marginal benet of
delegating a second decision as Mb(2) = D  PD =  2 + 1
2(1 82)2:
Hence, if Mb(1) > 0, the total prot increases if dl
1 is delegated to agent
l. Likewise, if Mb(2) > 0, both rst period projects must be delegated to the
agents. Clearly, Mb(1)  Mb(2), that is the marginal benet of delegating
a rst project is higher than the marginal benet of delegating a second one
because the information contained in a second signal is partially redundant
(at least for  > 0). Therefore, if Mb(1) > 0 > Mb(2), the principal optimally
delegates to only one agent.
Finally, notice that under the condition of our corollary, we have Mb(3);
Mb(4) < 0, which means that second period delegation would never be opti-
mal. We thus have established the following:
Proposition 3 The optimal organization is: centralization for 2 
22
1+82,
delegation of dl
1 to agent l for 2 2 [
22
1+82;
22
1 82] and delegation of dA
1 and
dB
1 to A and B for 2 
22
1 82.
The striking result of this paper is the optimality of limited delegation
in a repeated context. Delegation is limited to the rst period (Gautier
and Paolini, 2007) and across agents. Agents can be treated asymmetrically
within the organization. The principal selects one delegate that will be re-
sponsible for the production of the signal and keeps control over the decisions
concerning the other agent.
The quality of the signal produced by the agent depends on the degree
of correlation. Mb(1) is increasing in : when the information are correlated
5the value of a unique signal is higher. Conversely, Mb(2) decreases in  :
the informational content of the second signal decreases with the degree of
correlation. Figure 1 illustrates the optimal organizational structure as a
function of the correlation parameter. Asymmetric treatment of the agents
is more likely when the agents have correlated information. Note also that
in a one period model, there is no learning associated with delegation and
agents will be treated symmetrically.
In this model, we have analyzed a dynamic task allocation problem with
externalities and asymmetric information. An alternative mechanism could
be that the agents communicate to the principal and the principal chooses the
actions. Crawford and Sobel (1982) show that communication is not perfectly
informative. In a message game both agents reveal their private information
to the principal who then takes all decisions if 42  2. Clearly there exists
a parameter set where communication fails and delegation is optimal. This
conrms Dessein (2002) who recognizes that "Delegation is typically a better
instrument to use the local knowledge of the agent than communication."
-
6

1/4 0
2
22
42
42
3
22
1 82
22
1+82
Mb(1) < 0
Mb(1) > Mb(2) > 0
Mb(1) > 0 > Mb(2)
Figure 1: Optimal organization
Appendix: Proof of proposition 1 and corollary 2
In a two-players game, IC selects the Riley outcome if there are only two
states and if the payo function satises the single crossing condition. To
prove proposition 1, we replicate the argument (used in Gautier and Paolini,
2007) in this three-players game.
Each agent when he receives control over d1 plays a signaling game with
the principal. But, two versions of this game must be considered: in the
6rst, the other agent k discloses his private information and thus, at t = 2,
the principal knows k. In the second, the principal does not learn k either
because the agent decided not disclose information or because he did not
receive control over dk
1.
In this game played between l and the principal, a Bayesian equilibrium
is a triple (d
1;d
2;) where:
(BE1) 8 2 , d
1() 2 argmaxd1 l(d1;d
2;)
(BE2) 8d
1, d
2() 2 argmaxd2
P
 (jd1)P(d1;d
2;)
(BE3) The posterior beliefs (jd1) are consistent with the Baye's rule.
We apply to the set of Bayesian equilibria the Cho and Kreps (1987) intuitive
criterion. A Bayesian equilibrium fails the intuitive criterion if:
(IC1) The equilibrium payo of the agent in one state of the world (i) is
greater with the equilibrium strategy than with any other strategy.
(IC2) It exists a strategy ~ d1 such that the equilibrium payos in the other
state of the world (j) are smaller than those with the strategy ~ d1 once the
principal is convinced that ~ d1 could not have been chosen by the agent in
state i.
(1) Suppose that agent k truly reveals his private information if he con-
trols dk
1. Then, whatever the choice of dl
1 by agent l, he will not be able
to change the principal's belief about k. In this game, the standard sort-
ing condition is satised; This is sucient to kill all the pooling and the
separating equilibria but the Riley outcome.
To be more explicit, consider rst the set separating equilibria. In a
separating equilibrium d
2() = l + ; an equilibrium is separating if d
1()
satises the incentive constraint:

l(d

1();d

2();)  
l(d

1(~ );d

2(~ );) (8)
The equilibrium is supported by pessimistic out-of-equilibrium beliefs:
(1jd1 6= d
1(2)) = 1. Then, d
1(1) = l + 1 and the set of d(2) that
satises the constraint (8) is the set D of separating equilibria. Applied to
the set D, the intuitive criterion renes all the out-of-equilibrium beliefs:
(2jd1 2 D) = 1. Hence, a rational agent selects the decision dl
1(2) that
maximizes his prot under constraint (8) i.e selects his preferred decision
within D. And the only separating equilibrium that survives the intuitive
criterion is the least costly separating equilibrium (Riley outcome). Last,
note that l + 2 2 D if 2  (k)2.
Consider any pooling equilibrium where d
1(1) = d
1(2). In state 2, the
agent is worse-o than when he can signal his type. Then, we can associate
7with any pooling equilibrium a decision ~ d1 such that:
(IC1) In state 1, the agent prefers the pooling equilibrium to ~ d1, whatever
the beliefs associated with ~ d1
(IC2) In state 2, the agent prefers ~ d1 to the pooling equilibrium if the prin-
cipal is convinced that (1j~ d1) = 0.
If in state 1 the agent never deviates to ~ d1, the intuitive criterion imposes
that the beliefs associated with ~ d1 change to (2j~ d1) = 1. Consequently, the
agent will quit the pool in state 2 and no pooling equilibria will survive the
intuitive criterion.
(2) Suppose that agent k does not disclose his private information at
period one either because he does not control dk
1 or because he plays a pooling
equilibrium. In this case, disclosing the value of l changes the principal's
beliefs on both l and k. We must then identify the state i in which the
agent has no incentive to hide his private information (if it exists).
If agent l plays a pooling equilibrium, the second period decisions are:
dk
2 = k +
1+2
2   , k = A;B. If, in state i, the agent manages to signal
his type, the second period decisions change to: dl
2 = l + i    and dk
2 =
k+viii+vijj , i;j = 1;2. We must show that there exists a state i 2 
in which the agent has would prefer to inform the principal. Replacing the
decisions in the prot functions, we can show that such a state always exists.
Then, we can use the same reasoning as above to eliminate all the pooling.
If agent l plays a separating equilibrium, the second period decisions are:
dl
2(i) = l + i    and dk
2 = k + viii + vijj   . If, in state i, the agent
deviates, he changes the second period decisions to: dl
2 = l + j    and
dk
2 = k + viji + viij   , i;j = 1;2. We must show that there exists at
most one state i 2  in which such a deviation is protable. Replacing the
decisions in the prot functions, we can show that indeed, deviating cannot
be protable in both states. Hence, we have our standard sorting conditions.
We can thus nd the set of separating equilibria and apply the same reasoning
as above to select the most ecient one.
Acknowledgement
We would like to thank Francis Bloch, Bernard Caillaud, Jacques Cr emer,
Luca Deidda, Tito Pietra, A. Piggins, Jacques Thisse, seminar participants
in CRENoS, IRES and Li ege and an anonymous referee for their comments.
The second author thanks Italian MIUR for nancial support.
8References
[1] Aghion, P., Dewatripont, M. and P. Rey (2002) "On Partial Contracting"
European Economic Review 46, 745-753.
[2] Cho, I. K., and D. Kreps (1987) "Signaling Games and Stable Equilibria"
Quarterly Journal of Economics 12, 179-221.
[3] Crawford, V. and J. Sobel (1982) "Strategic Information Transmission"
Econometrica 50, 1431-1452
[4] Dessein, W. (2002) "Authority and Communication in Organizations"
Review of Economic Studies 69, 811-838.
[5] Gautier, A., and D. Paolini (2007) "Delegation and Information Revela-
tion" Journal of Institutional and Theoretical Economics 163, 574-597.
[6] Riley, J. (1979) "Informational Equilibrium" Econometrica 47, 331-59
9

Delegation, externalities and organizational design

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