Chapter 3
The principal-agent problem
The principal-agent problem describes a class of interactions between two
parties to a contract, an agent and a principal. The legal origin of these terms
suggests that the principal engages the agent to act on his (the principal’s)
behalf. In economic applications, the agent is not necessarily an employe
oftheprincipal. Infact,whichoftwoindividualsisregardedastheagent
and which as the principal depends on the nature of the incentive problem.
Typically, the agent is the one who is in a position to gain some advantage
by reneging on the agreement. The principal then has to provide the agent
with incentives to abide by the terms of the contract.
We divide principal-agent problems into two classes: problems of hidden
action and problems of hidden information. In hidden-action problems, the
agent takes an action on behalf of the principal. The principal cannot observe
the action directly, however, so he has to provide incentives for the agent to
choosetheactionthatisbestfortheprincipal.Inhidden-informationprob-
lems, the agent has some private information that is needed for some decision
to be made by principal. Again, since the principal cannot observe the agent’s
information, he has to provide incentives for the agent to reveal the infor-
mation truthfully. We begin by looking at the hidden-action problem, also
known as a moral hazard problem.
3.1 The model
For concreteness, imagine that the principal and the agent undertake a risky
venture together and agree to share the revenue. The agent takes some
1
2 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM
action that a?ects the outcome of the project. The revenue from the venture
is assumed to be a random function of the agent’s action.
Let A denote the set of actions available to the agent with generic element
a. Typically, A is either a finite set or an interval of real numbers. Let S
denote a set of states with generic element s. For simplicity, we assume that
the set S is finite. The probability of the state s conditional on the action a
is denoted by p(a,s). The revenue in state s is denoted by R(s) ≥ 0.
The agent’s utility depends on both the action chosen and the consump-
tion he derives from his share of the revenue. The principal’s utility depends
only on his consumption. We maintain the following assumptions about
preferences:
? The agent’s utility function u : A×R
+
→ R is additively separable:
u(a,c)=U(c)?ψ(a).
Further, the function U : R
+
→ R is C
2
and satisfies U
0
(c) > 0 and
U
00
(c) ≤ 0.
? The principal’s utility function V : R→ Ris C
2
and satisfies V
0
(c) > 0
and V
00
(c) ≤ 0.
Notice that the agent’s consumption is assumed to be non-negative. This
is interpreted as a liquidity constraint or limited liability. The principal’s
consumption is not bounded below; in some contexts this is equivalent to
assuming that the principal has large but finite wealth and non-negative
consumption.
3.2 Pareto e?ciency
The principal and the agent jointly choose a contract that specifies an action
and a division of the revenue. A contract is an ordered pair (a,w(·)) ∈ A×W,
where W = {w : S → R
+
} is the set of incentive schemes and w(s) ≥ 0 is
the payment to the agent in state s.
Suppose that all variables are observable and verifiable. The principal
and the agent will presumably choose a contract that is Pareto-e?cient.
This leads us to consider the following decision problem (DP1):
max
(a,w(·))
X
s∈S
p(a,s)V (R(s)?w(s))
3.3. INCENTIVE EFFICIENCY 3
subject to
X
s∈S
p(a,s)U(w(s))?ψ(a) ≥ ˉu,
for some constant ˉu.
Proposition 1 Under the maintained assumptions, a contract (a,w(·)) is
Pareto-e?cient if and only if it is a solution to the decision problem DP1 for
some ˉu.
Suppose that (a,w(·)) is Pareto-e?cient. Put ˉu equal to the agent’s pay-
o?.Bydefinition, the contract must maximize the principal’s payo? subject
to the constraint that the agent receive at least ˉu. Conversely, suppose that
the contract (a,w(·)) is a solution to DP1 for some value of ˉu. If the contract
is not Pareto-e?cient, then there must be another contract that yields the
same payo? totheprincipalandmoretotheagent. Butthenitmustbe
possible to transfer wealth to the principal in some state, contradicting the
optimality of (a,w(·)).
Suppose that the sharing rule satisfies w(s) > 0 for all s.Thenoptimal
risk sharing requires:
V
0
(R(s)?w(s))
U
0
(w(s))
= λ,?s.
These are sometimes referred to as the Borch conditions. If the action a
belongs to the interior of A and if the functions p(a,s) and ψ(a) are di?er-
entiable at a,then
X
s∈S
p
a
(a,s)[V (R(s)?w(s))?λU(w(s)] + λψ
0
(a)=0.
3.3 Incentive e?ciency
Now suppose that the agent’s action is neither observable nor verifiable. In
that case, the action specifiedbythecontractmustbeconsistentwiththe
agent’s incentives. A contract (a,w(·)) is incentive-compatible if it satisfies
the constraint
X
s∈S
p(a,s)U(w(s))?ψ(a) ≥
X
s∈S
p(b,s)U(w(s)) ?ψ(b),?b.
4 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM
A contract is incentive-e?cient if it is incentive-compatible and there does
not exist another incentive-compatible contract that makes one party bet-
ter o? without making the other party worse o?. We can characterize the
incentive-e?cient contracts using the following decision problem (DP2):
max
(a,w(·))
X
s∈S
p(a,s)V (R(s)?w(s))
subject to
X
s∈S
p(a,s)U(w(s))?ψ(a) ≥
X
s∈S
p(b,s)U(w(s))?ψ(b),?b,
and
X
s∈S
p(a,s)U(w(s))?ψ(a) ≥ ˉu.
Proposition 2 Under the maintained assumptions, a contract (a,w(·)) is
incentive-e?cient only if it is a solution of DP2 for some constant ˉu.A
contract that solves DP2 is incentive-e?cient if the participation constraint
is binding for every solution.
The proof of the “only if” part is similar to the Pareto e?ciency argument.
If (a,w(·)) is a solution to DP2 and is not incentive-e?cient, there exists an
incentive-e?cientcontractthatgivestheprincipalthesamepayo? and the
agent a higher payo?. But this contract must be a solution to DP2 that
strictly satisfies the participation constraint.
The assumption of a uniformly binding participation constraint is restric-
tive: see Section 3.7.1 for a counter-example.
ThisDPcanbesolvedintwostages.First,foranyactiona,computethe
payo? V
?
(a) from choosing a and providing optimal incentives to the agent
to choose a.CallthisDP3
V
?
(a)=max
w(·)
X
s∈S
p(a,s)V (R(s)?w(s))
subject to
X
s∈S
p(a,s)U(w(s)) ?ψ(a) ≥
X
s∈S
p(b,s)U(w(s))?ψ(b),?b,
X
s∈S
p(a,s)U(w(s)) ?ψ(a) ≥ ˉu.
3.4. THE IMPACT OF INCENTIVE CONSTRAINTS 5
Note that U(·) and V (·) are concave functions. A suitable transformation of
this problem (see Section 3.10) is a convex programming problem for which
the Kuhn-Tucker conditions are necessary and su?cient.
Once the function V
?
is determined, the optimal action is chosen to max-
imize the principal’s payo?:
a
?
∈ argmaxV
?
(a).
The advantage of the two-stage procedure is that it allows us to focus on
the problem of implementing a particular action. As we have seen, DP3 is
(equivalent to) a convex programming problem and hence easier to “solve”
and it turns out that many interesting properties can be derived from a study
of DP3 without worrying about the optimal choice of action.
3.3.1 Risk neutrality
An interesting special case arises if the principal is risk neutral. In that
case, maximization of the principal’s expected utility, taking a as given, is
equivalent to minimizing the cost of the payments to the agent. Thus, DP3
can be re-written as
min
w(·)
X
s∈S
p(a,s)w(s))
subject to
X
s∈S
p(a,s)U(w(s))?ψ(a) ≥
X
s∈S
p(b,s)U(w(s)) ?ψ(b),?b,
X
s∈S
p(a,s)U(w(s))?ψ(a) ≥ ˉu.
3.4 The impact of incentive constraints
What is the impact of hidden actions? When does the imposition of incentive
constraints a?ect the choice of contract?
If one of the parties to the contract is risk neutral, it is particularly easy
to check whether the first best can be achieved, that is, whether an incentive-
e?cient contract is also Pareto-e?cient.
Risk neutral principal
6 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM
Suppose, for example, that the principal is risk neutral and the agent is
(strictly) risk averse, i.e., U
00
(c) < 0. The Borch conditions for an interior
solution imply that w(s) is a constant for all s. Inthatcase,theagent’s
income is independent of his action, so in the hidden action case he would
choose the cost-minimizing action. Thus, the firstbestcanbeachievedwith
hidden actions only if the optimal action is cost-minimizing.
Risk neutral agent
Suppose that the agent is risk neutral and the principal is (strictly) risk
averse, i.e., V
00
(c) < 0. Then the Borch conditions for the first best imply
that the principal’s income R(s)?w(s) is constant, as long as the solution is
interior. This corresponds to the solution of “selling the firm to the agent”,
but it works only as long as the agent’s non-negative consumption constraint
is not binding.
In general, there is some constant ˉy such that
R(s)?w(s)=min{ˉy,R(s)}
and
w(s)=max{R(s)? ˉy,0}.
Both parties risk averse
More generally, if we assume the first best is an interior solution and maintain
the di?erentiability assumptions discussed above, the first-order condition for
the first best is
X
s∈S
p
a
(a,s)[V (R(s)?w(s))?λU(w(s)] + λψ
0
(a)=0.
and the first-order (necessary) condition for the incentive-compatibility con-
straint is
X
s∈S
p
a
(a,s)[U(w(s)] ?ψ
0
(a)=0.
So the incentive-e?cient and first-best contracts coincide only if
X
s∈S
p
a
(a,s)V (R(s)?w(s)) = 0.
3.5. THE OPTIMAL INCENTIVE SCHEME 7
Example: Suppose that there are two states s =1,2 and R(1) <R(2) and
let p(a) denote the probability of success (s =2). At an interior solution,
the necessary condition derived above is equivalent to
p
0
(a)[V (R(2)?w(2)) ?V (R(1)?w(1))] = 0
or
R(2)?R(1) = w(2)?w(1),
assuming p
0
(a) > 0. This allocation will not satisfy the Borch conditions
unless the agent is risk neutral on the interval [w(1),w(2)].
Note that there may be no interior solution of the problem DP3 even
under the usual Inada conditions. See Section 3.7.2 for a counter-example.
3.5 The optimal incentive scheme
In order to characterize the optimal incentive scheme more completely, we
impose the following rstrictions:
? The principal is risk neutral, which means that if two actions are equally
costly to implement, he will always prefer the one that yields higher
expected revenue.
? There is a finite number of states s =1,...,S and the revenue function
R(s) is increasing in s.
? Monitone likelihood ratio property:Thereisafinite number of actions
a =1,...,A and for any actions a<b,theratiop(b,s)/p(a,s) is non-
decreasing in s. We also assume that the vectors p(b,·) and p(a,·)
are distinct, so for some states the ratio is increasing. The expected
revenue
P
s∈S
p
a
(a,s)R(s) is increasing in a.
Now consider the modified DP4 of implementing a fixed value of a:
V
??
(a)=max
w(·)
X
s∈S
p(a,s)V (R(s)?w(s))
subject to
X
s∈S
p(a,s)U(w(s))?ψ(a) ≥
X
s∈S
p(b,s)U(w(s))?ψ(b),?b<a,
X
s∈S
p(a,s)U(w(s))?ψ(a) ≥ ˉu.
8 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM
The di?erence between DP4 and the original DP3 is that only the downward
incentive constraints are included.
Obviously, V
??
(a) ≥ V
?
(a). Suppose that V
??
(a) >V
?
(a). This means
that the agent wants to choose a higher action than ainthemodifiedproblem.
But this is good for the principal, who will never choose a if he can get a
better action for the same price. Thus,
max
a
V
?
(a)=max
a
V
??
(a).
Thus,wecanusethesolutiontothemodified problem DP4 to characterize
the optimal incentive scheme.
Theorem 3 Suppose that a ∈ argmax V
?
(a). The incentive scheme w(·) is
a solution of DP4 if and only if it is a solution of DP3.
3.6 Monotonicity
Many incentive schemes observed in practice reward the agent with higher
rewards for higher outcomes, i.e., w(s) is increasing (or non-decreasing) in s.
It is interesting to see when this is a property of the theoretical optimal in-
centive scheme. Assuming an interior solution, the Kuhn-Tucker (necessary)
conditions are:
p(a,s)V
0
(R(s)?w(s))?λp(a,s)U
0
(w(s))?
X
b<a
μ
b
{p(a,s)?p(b,s)}U
0
(w(s)) = 0
or
V
0
(R(s)?w(s))
U
0
(w(s))
=
?
λ +
X
b<a
μ
b
?
1?
p(b,s)
p(a,s)
?
!
.
By the MLRP, the right hand side is non-increasing in s, so the left hand
side is non-increasing, which means that w(s) is non-decreasing.
3.7 Examples
There are two outcomes s =1,2,whereR(1) <R(2),andtwoprojects
a =1,2 represented by the respective probabilities of success 0 <p(1,2) <
p(2,2) < 1. The costs of e?ort are ψ(1) = 0 and ψ(2) > 0. The agent’s utility
3.7. EXAMPLES 9
function U(·) is assumed to satisfy U(0) = 0 and the reservation utility is
ˉu =0.
The inferior project can be implemented by setting w(s)=0for s =1,2.
Suppose the principal wants to implement a =2. The constraints can be
written as
(IC) (1?p(2,2))U(w(1)) + p(2,2)U(w(2))?ψ(2) ≥
(1?p(1,2))U(w(1)) + p(1,2)U(w(2))
which simplifies to
(p(2,2)?p(1,2))(U(2)?U(1)) ≥ ψ(2)
and
(IR) (1?p(2,2))U(w(1)) + p(2,2)U(w(2))?ψ(2) ≥ 0.
In order to satisfy the (IR) constraint, consumption must be positive in at
least one state. This implies that the expected utility from choosing low
e?ort is strictly positive:
(1?p(1,2))U(w(1)) + p(1,2)U(w(2)) > 0,
so if the (IC) constraint is satisfied, the (IR) constraint must be strictly
satisfied:
(1?p(2,2))U(w(1)) + p(2,2)U(w(2))?ψ(2) > 0.
Thus, if (w(1),w(2)) is the solution to the optimal contract problem, the
(IR) constraint does not bind. The principal’s problem can then be written
as:
min
w
(1?p(2,2))w(1) + p(2,2)w(2)
s.t. (w(1),w(2) ≥ 0
(p(2,2)?p(1,2))(U(w(2))?U(w(1))) ≥ ψ(2).
Then it is clear that a necessary condition for an optimum is that w(1) = 0.
So the optimal contract for implementing a =2is (0,w
?
(2)),wherew
?
(2)
solves the (IC):
(p(2,2)?p(1,2))U(w
?
(2)) = ψ(2).
The payment w
?
(2) needed to give the necessary incentives to the manager
will be higher:
10 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM
? the higher the cost of e?ort ψ(2);
? the smaller the manager’s risk tolerance (as measured by U(w(2)) ?
U(0));
? the smaller the marginal productivity of e?ort (as measured by p(2,2)?
p(1,2)).
To decide whether it is optimal to implement high or low e?ort, the prin-
cipal compares the profit from optimally implementing each level of e?ort.
The maximum profitfromlowe?ort is
(1?p(1,2))R(2) + p(1,2)R(1).
The maximum profitfromhighe?ort is
(1?p(2,2))R(1) + p(2,2)R(2)?p(2,2)w
?
(2).
So high e?ort is optimal if and only if
(p(2,2)?p(1,2))(R(2)?R(1)) ≥ w
?
(2),
that is, the increase in expected revenue is greater than the cost of providing
managerial incentives.
3.7.1 Optimality and incentive-e?ciency
Suppose there are two states s =1,2,twoactionsa =1,2 and the reservation
utility is ˉu =0. The principal and the agent are both risk neutral. The other
parameters of the problem are given by
R(1) = 0 <R(2)
ψ(1) = 0 <ψ(2)
0 <p(1,2) <p(2,2).
The action a =1is optimally implemented by putting
w
1
(s)=0,?s.
The action a =2is optimally implemented by putting
w
2
(s)=
?
0 if s =1
ψ(2)/(p(2,2)?p(1,2)) if s =2.
3.7. EXAMPLES 11
The payo? to the principal from each action is
V
?
(a)=
?
p(1,2)R(2) if a =1
p(2,2)(R(2)?ψ(2)/(p(2,2) ?p(1,2))) if a =2.
Suppose the parameter values are chosen so that V
?
(1) = V
?
(2).Thenthe
contract (a,w(·)) = (1,w
1
(·)) solves DP1 for the reservation utility ˉu =0
but is not incentive e?cient, because the agent is better o? with the contract
(a,w(·)) = (2,w
2
(·)).
3.7.2 Boundary solutions
In the preceding example, we note that the agent’s payo?is zero in state s =0
whichever action is implemented. It might be thought that this boundary
solution is dependent on risk neutrality but in fact boundary solutions for
optimal incentive scheme are possible even if U
0
(0) = ∞ ,forexample,for
the utility function U(c)=c
α
where 0 <α<1. In this case, U(0) = 0 so,
taking the other parameters from the previous example, the optimal incentive
scheme for a =1is still
w
1
(s)=0,?s.
For a =2the optimal incentive scheme is
w
2
(s)=
?
0 if s =1
U
?1
(ψ(2)/(p(2,2)?p(1,2))) if s =2.
This example provides a good illustration of the dangers of simply assuming
an interior solution.
3.7.3 Local incentive constraints
In many problems, convexity implies that one only has to consider local
deviations in order to characterize an optimum. The analogous principle
in principal-agent problems is to check only local incentive constraints. For
example, if a =1,...,Aanditisdesiredtoimplementanactiona then one
would only check the neighboring constraints a ? 1 and a +1(or in the
case where only downward constraints are considered, one would look at the
constraint between a and a?1 only). There is in general no reason to think
that this method will produce the right answer: there may well be non-local
constraints that are binding at the optimum. For example, suppose that
12 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM
there are two states s =1,2 and three actions a =1,2,3. The principal and
the agent are both assumed to be risk neutral and the reservation utility is
ˉu =0. The other parameters are as follows:
R(1) = 0 <R(2)
ψ(1) = 0 <ψ(2) = ψ(3)
0 <p(1,2) <p(2,2) <p(3,2).
The optimal incentive scheme to implement a =3is
w
3
(s)=
?
0 if s =1
(ψ(3)/(p(2,2)?p(1,2))) if s =2.
Because a =2has the same cost but lower probability of success than a =3,
the agent will never be tempted to choose a =2as long as the payment in
state s =2is positive; but he may well be tempted to choose a =1if the
payment in state s =2is too low. Thus, the incentive constraint between
a =1and a =3will be binding but the incentive constraint between a =3
and a =2will not.
To ensure that the local constraint was su?cient, we would need to impose
the following inequality on the parameters:
p(3,2)?p(2,2)
ψ(3)?ψ(2)
≤
p(2,2)?p(1,2)
ψ(2)?ψ(1)
.
This is, in e?ect, an assumption of diminishing returns to scale: the marginal
product of e?ort as measured by the ratio of the change in the probability of
success to the change in cost is declining. In more general problems, stronger
conditions are needed to ensure that only local incentive constraints bind.
See, for example, the discussion of the first-order approach in Stole (2001).
3.7.4 Participation constraints
3.8 The value of information
Theprincipalmayobservesomeinformationthatisrelevanttotheagent’s
action in addition to the revenue from the project. We can incorporate this
possibility in the current setup by assuming that the state is an ordered pair
s =(s
1
,s
2
) ∈ S
1
×S
2
and that the revenue is a function R(s
1
) of the first
3.9. MECHANISM DESIGN 13
component. Then s
2
is a pure signal of the action a.Thefirst-order condition
for an interior solution to DP4 is
V
0
(R(s
1
,s
2
)?w(s
1
,s
2
))
U
0
(w(s
1
,s
2
))
=
?
λ +
X
b<a
μ
b
?
1?
p(b,s
1
,s
2
)
p(a,s
1
,s
2
)
?
!
The state s
2
gives information about the action of the principal if the like-
lihood ratio p(b,s
1
,s
2
)/p(a,s
1
,s
2
) varies with s
2
for some fixed s
1
.Inother
words, all relevant information should be reflectedintheagent’spayment.
3.9 Mechanism design
The principal-agent problem is a special case of the general problem of mech-
anism design, that is, designing a game form that will implement a desired
outcome as an equilibrium of the game. Suppose there is a finite number of
agents i =1,...,I,eachofwhomhasatypeθ
i
∈ Θ
i
and chooses an action
a
i
∈ A
i
. There may also be a set of actions a
0
∈ A
0
chosen by the mechanism
designer. Let Θ =
Q
I
i=1
Θ
i
and A =
Q
I
i=0
A
i
anddenoteelementsofΘ and
A by θ and a respectively. An agent’s utility is given by u
i
(a,θ),thatis,
u
i
: A×Θ→ R.
An agent’s type is private information, but the distribution of types p(θ)
is common knowledge, as are the sets Θ
i
and A
i
and the utility functions u
i
.
The mechanism designer faces two problems: how to get the agents to
reveal their information truthfully and how to get them to choose the “right”
actions. The general form of a mechanism contains two stages: in the first
agents are asked to send messages to the planner and in the second the
planner sends instructions to the agents. Let M
i
denote the space of messages
available to agent i and let M =
Q
I
i=1
M
i
.LetM
0
denote the planner’s
message space and f : M → M
0
denote the decision rule chosen by the
planner. Then each agent has to choose a strategy (σ
i
,α
i
), where σ
i
: Θ
i
→
M
i
and α
i
: M
0
× Θ
i
→ A
i
.Givenf we have a well-defined game with
players is i =1,...,I, strategy sets {Σ
i
}
I
i=1
and payo? functions {U
i
}
I
i=1
,
where U
i
: Σ→ R is defined by
U
i
(σ,α,θ)=u
i
(α(f ?σ(θ),θ),θ).
A Bayes-Nash equilibrium for this game is a strategy profile (σ
?
,α
?
) such
that, for every agent i,
E [U
i
(σ,α,θ)|θ
i
] ≥ E [U
i
((σ
i
,α
i
),(σ
?i
,α
i
),θ)|θ
i
],?θ
i
,?(σ
i
,α
i
).
14 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM
A mechanism (f,M) is called a direct mechanism if M
i
= Θ
i
for i =
1,...,I and M
0
= A. In other words, agents’ messages are their types and
the planner’s message is the vector of desired actions. For any agent i,the
truthful communication strategy in a direct mechanism is a communication
strategy σ
i
such that
σ
i
(θ
i
)=θ
i
,?θ
i
.
Similarly, in a direct mechanism, an action strategy α
i
is truthful if
α
i
(a)=a
i
,?a ∈ A.
The Revelation Principle allows us to substitute direct mechanisms for gen-
eral mechanisms and restrict attention to truthful strategies.
Theorem 4 (RevelationPrinciple) Let (σ,α) be a Bayes-Nash equilibrium
of the mechanism (f,M). Then there exists a Bayes-Nash equilibrium (?σ,?α)
of the direct mechanism (
?
f,Θ) such that (?σ
i
, ?α
i
) are truthful for every i and
the outcomes of the two equilibria are the same:
α?f ?σ =?α?
?
f ? ?σ.
Proof. Put
?
f = α?f ?σ.
Although the proof is trivial, this result o?ers a great simplification of
the problem of characterizing implementable SCFs. A SCF is a function
f : Θ → A that specifies an outcome for every state of nature θ.Wecan
think of the SCF f as a collection of decision rules (f
0
,f
1
,...,f
I
),onefor
each agent i.TheSCFf is incentive-compatible if, for every i, truth-telling
is optimal and the decision rule f
i
is optimal, assuming that every other
agent j tells the truth and follows the decision rule f
j
,thatis,
E [u
i
(f(θ),θ)|θ
i
] ≥ E
h
u
i
3
a
i
,f
?i
(
?
θ
i
,θ
?i
),θ
′
|θ
i
i
.
Theorem 5 The direct mechanism (f,Θ) has a truthful equilibrium if and
only if f is an incentive-compatible SCF.
Remark 1 The theorem suggests that we can “implement” f using a direct
mechanism, but the direct mechanism may have other equilibria. Full im-
plementation requires that every equilibrium of the mechanism used have the
same outcome. For this it may be necessary to use a general mechanism.
Most of the implementation literature is taken up with this problem of try-
ing to eliminate unwanted equilibria, either by using fancier mechanisms or
stronger solution concepts.
3.10. NON-CONVEXITY AND LOTTERIES 15
Remark 2 The principal-agent problem is a special kind of mechanism de-
sign problem. In the problem described earlier, there are two “agents” (the
principal and the agent both being economic agents in the eyes of the mech-
anism designer). The agent chooses an action a ∈ A, the principal has no
action to choose, and the mechanism designer chooses the incentive scheme
w(·) ∈ W. Since there is no private information about types, the SCF is an
incentive-e?cient allocation f =(a,w(·)) and the direct mechanism has a
truthful equilibrium in which the agent chooses the correct value of a.Even
in this simple context we can see the problem of multiple equilibria at work.
Typically, the incentive scheme is chosen so that the agent is indi?erent be-
tween a and some other action b. It would be an equilibrium for the agent
to choose b even though this would not be as good for the principal. We can
use this example to illustrate how a more complex mechanism and a stronger
equilibrium concept helps resolve this di?culty. Suppose that the principal
is told to choose the incentive scheme and that the agent chooses his action
after observing the incentive scheme. The appropriate solution concept here
is subgame perfect equilibrium: the agent should choose the best response
(action) to any incentive scheme and not simply the one that is chosen in
equlibrium. Clearly, the truthful equilibrium of (a,w(·)) remains a SPE of
this game but (b,w(·)) does not. If the principal anticipates that the agent
will choose b under the incentive scheme w(·) and if the principal prefers a
to b, then he will choose an alternative ?w(·) which is very close to w(·) but
makes the agent strictly prefer a to b.Thus,(b,w(·)) is not a SPE.
Remark 3 The sequential game described above, in which the principal of-
fers an incentive scheme and the agent responds optimally to any scheme
o?ered, is closer to the original formulation of the principal-agent problem
than the decision problems analyzed above. We have taken an approach much
closer to the Revelation Principal, in which we focus exclusively on the truth-
ful equilibria. Within the context of mechanism design, we can see that both
approaches are closely related.
3.10 Non-convexity and lotteries
The principal-agent problem as stated earlier is not a convex programming
problem because the feasible set defined by the incentive constraints is not
16 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM
convex:
X
s∈S
p(a,s)U(w(s))?ψ(a) ≥
X
s∈S
p(b,s)U(w(s))?ψ(b),?b
The concave function U(·) appears on both sides of the inequality. However,
a simple transformation suggested by Grossman and Hart converts this into a
convex programming problem. Let C(u)=U
?1
(u) for any number u. C(·) is
convex because U(·) is concave and we can write the implementation problem
equivalently as
min
u(·)
X
s∈S
p(a,s)V (R(s)?C(u(s)))
subject to
X
s∈S
p(a,s)u(s)?ψ(a) ≥
X
s∈S
p(b,s)u(s)?ψ(b),?b,
X
s∈S
p(a,s)u(s)) ?ψ(a) ≥ ˉu.
Because the incentive scheme u(s) is written in terms of utility rather than
consumption, the incentive constraints are linear in the choice variables and
hence the feasible set is convex.
This trick works because of the additive separability of the utility func-
tion.Ingeneral,thiswillnotworkandwearestuckwithahighlynon-convex
problem. One general solution to non-convexities is to introduce lotteries.
Let the incentive scheme specify a probability distribution W(c,s) over non-
negative consumption levels c conditional on the state s and let the utility
function take the general form u(c,a). The incentive constraint is then writ-
ten as
X
s∈S
[p(a,s)?p(b,s)]u(c,a)W(dc,s) ≥ 0,?b.
Expected utility is linear in probabilites, so once again the incentive con-
straints define a convex feasible set of distributions W(·). Lotteries are not
simply a solution to a technical problem (non-convexity). They can also
increase welfare.
Note that even if the implementation problem does not include non-
convexities because of the additive separability of preferences, the global
principal agent problem may do so because the cost function ψ is non-convex.
Although each action a can be implemented e?ciently with a non-stochastic
incentive scheme, there may be a gain from randomizing over the action a.
