2002. 26
Principal-Multiagent Relationships with Costly Monitoring

Yasuyuki Miyahara

Principal-Multiagent Relationships with Costly Monitoring

Yasuyuki Miyahara†
Graduate School of Business Administration
Kobe University
2-1 Rokkodai-cho Nada-ku Kobe 657-8501, JAPANmiyahara@rose.rokkodai.kobe-u.ac.jp

This Version: October 17, 2002

Abstract
This paper analyses the principal and multi-agent relationships with costly monitoring. When monitoring is costly, the choice of monitoring is a strategic variable.
Randomization of actions induces the incentive for monitoring and the principal
can extract a relevant information about actions from agents by communicatingwith the agents. If moniroring cost is sufficiently small, then first-best outcomes areapproximately implementable. Moreover, under certain conditions, we construct a
contract to approximately implement the desired action profile as a unique sequential equilibrium with first-best outcome.

Keywords: Communication, Contract, Costly monitoring.
JEL Classification Numbers: C72†

I am very thankful to Hisao Hisamoto, and Hideo Suehiro for useful comments. Of course, any
remaining errors are my own.

1.

Introduction

Monitoring is one of the important activities in various organizations. To know what isgoing on in the organization can improve the efficiency of production. For example, job
rotation is applied in many Japanese firms. It enables the workers to know what theircolleagues are doing. Managers often delegate monitoring activity to workers.
In the present paper, we consider the costly monitoring situation. The principal (manager) can not monitor the agents, but each agent (worker) can perfectly monitor the opponent if he pays a cost. If he does not pay a cost, he can not obtain any information
about the opponent. And, we assume that monitoring is private. Each agent can not
know whether he is monitored by the opponent and his observed information is private.
This situation has not been analysed. In our costly monitoring situation, since the principal is assumed not to enforce the agents to monitor each other, the principal needs
to design the incentive contracts while she consider additional incentive constraint thatagents voluntarily monitor the opponent’s action.
In the literature on principal and many agents relationships, several papers have studied the situation where agents take unobservable actions and have characterized the optimal incentive contracts. And it has become apparent that, in many situations, thefirst-best outcome is not attainable. The second-best outcome is attainable, that is, efficiency losses arise. See Holmström (1982) and Mookherjee (1984). On the other hand,when agents can perfectly observe opponents’ actions at no coste, it has been shown that
efficiency losses can be improved: Ma (1988) and Itoh (1993). In particular, Ma (1988)
has shown that the first-best result can be established when each agent (worker) can monitor the opponents’ actions at no cost and communication is available. In his analysis,
costless mointoring is crucial.
Our model relaxes the perfect monitoring assumption in Ma (1988) and Itoh (1993).
The present paper closely relates to Ma (1988) in that, though the agent has to pay

1

a cost, each agent can perfectly monitor the opponent and communication is possible.1
There are two distinctions between Ma (1988) and ours. One of them is with regard tomonitoring technologies as stated above. The other is with regard to the timing of communication. We assume the simultaneous communication that each agent simultaneously
announces a message. Moreover, we focus on a simple communication that each agent
uses only two messages. The principal has only to know whether her desired actions arechosen or not, in order to (virtually) implement an action profile. On the othere hand,
Ma (1988) uses sequential communication that agent 2 announces a message after knowing agent 1’s announced message. Though sequantial communication is often useful for
unique implementaion, we do not have to rely on it.
Our question is whether (approximate) first-best outcome is attainable in the costly
monitoring situation. It is shown that there exists no contract that exactly implements thefirst-best action at less than the second-best cost. In other words, only the second-best
outcome is attainable, even if communication is possible, when we focus on is a contract
under which the first-best action is chosen with probability one in equilibrium. That isshown in section 3.
However, if we permit the agents to randomize their actions, approximate first-best
outcome is attainable when monitoring costs are sufficiently small. Approximate first-best
outcome means that the first-best action profile is chosen with almost probability one andthe expected implementation cost is almost the first-best. And, we say that the first-best
action profile is virtually implementable if there exists a contract under which the firstbest action profile is chosen with almost probability one in equilibrium. In the costly
monitoring situation, randomization of actions is necessary to give each agent incentive
to monitor the opponent. In the perfect monitoring situation, agents do not have torandomize actions since it is not needed to consider incentive for monitoring.
1

Itoh (1993) assumes that the principal can not communicate with the agents. He shows that the
principal can implement any action profile with less costs under agent side contracting than under noside contract. However, in his setting, first-best outcome is not attainable.

2

We can show that the first-best action profile is not only virtually but also uniquely
implementable at alomost the same cost as the first-best. Firstly, as the benchmark, we
provide a contract that induces approximate first-best outcome, where the agents’ wages
are not contingent on public output but only on messages. Then, it will be apparent
that, given the contract, the agents do not choose the principal’s preferred equilibrium.
The agents may choose a bad equilibrium for the principal but it is Pareto superior
equilibrium in the agent’s view point. Next, we consider wages contigent on output aswell as messages, and for the first step for unique implementability, we provide a contractthat induces an approximate first-best outcome and have its equilibrium which Paretodominates any other equilibrium. Lastly, under certain conditions, we present a contract
which induces a unique sequential equilibrium with approximate first-best outcome.
The paper proceeds as follows. In section 2, we set up the model. In section 3, we
discuss a contract which virtually implements first-best actions when monitoring is costly
and the wages are not contingent on outputs but messages. Section 4 presents a contract
under which first-best action profile is virtually implementable in coalition-proof Nashequilibrium. In section 5, we introduce a contract that uniquely and virtually implements
first-best action profile in sequential equilibrium.
2.

The Model

We consider a principal and two agents model with the following timing2 :Monitoring Realization of output

Contract
6
r

r

6
r

r

6
r

r

?

?

?

ActionCommunication

Payment-

Figure 1. Timing
2

Although, in the present paper, two agents case is analised, some modification induces the same
results in the case of more than two agents.

3

In the contract stage, the principal offers the agents an incentive contract. If any
agent rejects the offer, the game ends and all parties receive their reservation utility
levels. Without loss of generality, we normalize each agent’s reservation utility level tobe zero. If both of the agents accept the contract, the game goes to the next stage. The
contract specifies the wage schedule, sk , and the set of messages, Mk , available to agent k.
Agent k’s wage is contingent on verifiable variables. Hence, contracts are represented by{sk (·), Mk }k=1,2 .
In the action stage, each agent simultaneously chooses an effort level, ak , from the finite
set Ak . The set, A1 (resp. A2 ), has I (resp. J) elements. That is, A1 = {a11 , · · · , aI1 } (resp.
A2 = {a12 , · · · , aJ2 }). Agent k’s behavioral strategy in the action stage is to choose probability distribution over Ak . That is, we allow agents to mix actions. Denote πk ∈ ∆(Ak )as agent k’s mixed action, where ∆(Ak ) is the set of the probability distributions over Ak .
In the monitoring stage, each agent can privately monitor the opponent’s action before
the output is realized. The set of choice in monitoring stage is denoted by G = {γ, φ},where γ denotes monitoring and φ denotes no monitoring. The monitoring strategy isdefined by the mapping gk : Ak → ∆G, where ∆G is the set of probability distributions
over G. To monitor the opponent costs γ (> 0).3 In the following discussion, we focus on
the situation where γ is sufficiently small. If agent k monitors the opponent, he knows
perfectly which action the opponent has chosen. However, whether he has monitored or
not can not be known by anyone else. For example, when agent 1 chooses effort level ai1 ,he monitors the opponent, and observes aj2 , then (ai1 ; aj2 ) is private information. If agent k
does not monitor the opponent, he can not obtain any additional information at all. Then(ak ; φ) is his private information.
(ak ; a−k ) and (ak ; φ) are said to be agent k’s state. We denote agent k’s state by ω̂k .
It is the information he obtains before communication stage. The set of agent k’s stateis denoted by Ω̂k ≡ Ak × Ωk , where Ωk ≡ A−k ∪ {φ}. Before the communication stage,3

Though we denote γ both the decision of monitoring and monitoring cost, that will be not confusing.

4

agent k observes the ωk ∈ Ωk ≡ A−k ∪ {φ}, and (ak ; ωk ) is his private information, i.e. hisstate.
In the communication stage, each agent simultaneously and publicly announces amessage, mk , from the finite set Mk , which is specified in the accepted contract. In thefollowing analysis, we can induce the desirable results by restricting the messages available
to each agent to R and P , i.e., Mk = {R, P } for all k = 1, 2. The reporting strategy fk foragent k is defined by the mapping from agent k’s state to the set of his messages. Thatis, fk : Ω̂k → ∆(Mk ), where ∆(Mk ) is the set of probability distributions over Mk .
Finally, in the payment stage, the principal pay the agents the wages {sk }k=1,2 according to the accepted contract when the output x ∈ X realizes. X is the set of possible
outputs. The probability distribution of output x conditional on action pair (a1 , a2 ) isdenoted by p(x|a1 , a2 ). We assume that this distribution has full support, that is, for each
(a1 , a2 ) ∈ A1 × A2 and each x ∈ X, p(x|a1 , a2 ) > 0. x could be multi-dimensional. Assume that the number of possible realizations of x is L, that is, |X| = L. We assume thatrealized output is publicly observable (to the principal and all the agents) and verifiable(to court).
Each agent’s utility is a additively separable with respect to wage, action, and monitoring: uk (sk ) − Ck (ak ) − γk . uk (·) is the agent k’s utility of wage. Ck (ak ) represents
the disutility, or cost, of taking action ak . Agent 1’s disutility function C1 (·) is strictlyincreasing, that is, C1 (ai1 ) is strictly in i. And assume C1 (ai1 ) > 0 for each i ∈ {1, . . . , I}.
Similarly, C2 (aj2 ) is increasing in j and C2 (aj2 ) > 0 for each j ∈ {1, . . . , J}. γk is monitoringcost. If agent k monitors the opponent, then γk = γ. Otherwise, γk = 0.
We assume that each agent is risk averse. That is, uk (·) is continuous, strictly increasing, concave, and unbounded. Note that these assumptions imply that the inverse
function hk (·) = u−1
k (·) exists and that, for any v ∈ R, there exists s ∈ R such thatuk (s) = v.

5Agent k’s wage sk is paid contingent on verifiable variables. Here, output x andpublicly announced message pair (m1 , m2 ) are verifiable. Hence, the wage is a function ofx and m = (m1 , m2 ), that is, sk (x, m) is paid to agent k when x and m is realized.
A strategy of agent k is defined by σk = (πk , gk , fk ). Given strategy σ = (σ1 , σ2 ),agent k’s expected utility is defined by
Uk (σ) = E [uk (sk (x, m)) − Ck (ak ) | σ ] .
The principal is risk neutral with utility function V (x) −P

k=1,2 sk , where V (x) is theprincipal’s private benefit from output x. And the principal’s expected utility is definedbyB(σ) = E Xp(x|a) V (x) −

x∈X



X
k=1,2



 ¯¯
sk (x, m) ¯¯ σ  .
In the first-best situation where the agents’ actions are observable and verifiable, theprincipal can force the agents to choose any action pair (a1 , a2 ) by guaranteeing exactly
their resevation utilities. Then, the minimum cost to induce ak is hk (Ck (ak )). The firstbest action pair is the one that maximizes the principal’s expected utility
"

E

X

#

p(x|a)V (x) −x∈XX

hk (Ck (ak )).

k=1,2And, we donote it by ae = (ae1 , ae2 ). The first-best outcome is the one in which the principal
implements action pair ae at the first-best cost h1 (C1 (ae1 )) + h2 (C2 (ae2 )). We impose anassumption for non-trivial situation.
Assumption 1: There exists unique (ae1 , ae2 ) that satisfies Ck (aek ) > Ck (a1k ) for eachk = 1, 2.
In the second-best situation, that is, each agent can not observe the opponent andcommunication is not possible, wages paid to agents are contingent only on outcomes. A
contract {sk (·)}k=1,2 implements ae without communication if and only if it satisfies the6incentive compatibility and individual rationalityXp(x|ae )uk (sk (x)) − Ck (aek ) ≥x∈XXp(x|aik , ae−k )uk (sk (x)) − Ck (aik ),∀i, and ∀k = 1, 2,

x∈X

and
X

p(x|ae )uk (sk (x)) − Ck (aek ) ≥ 0,

∀k = 1, 2.

x∈X
e
SB e
The second-best cost to induce ae is denoted by hSB
1 (a ) + h2 (a ), which is equal to
e
SB e
hSB
1 (a ) + h2 (a ) = mins1 (·),s2 (·)

X

p(x|ae )h1 (u1 (s1 (x))) +X

p(x|ae )h2 (u2 (s2 (x))).

x∈X

x∈X

e
SB e
e
Under assumption 1 and risk-averse agents, it holds that hSB
1 (a )+h2 (a ) > h1 (C1 (a1 ))+h2 (C2 (ae2 )). In the second-best situation, it is possible that ae is never implementable.
3.

BenchmarkMa (1988) analyzes the perfect monitoring siutation, which are closely related to ours.
In his model, each agent can perfectly observe the opponents’ action at no cost, andthe principal can not do at all. Then, there exists a contract to implement the firstbest action pair at the first-best cost. Consider the following contract in which wages
are contingent only on messages. If agent 1 announces message R, then agent 2 is paidh2 (C2 (ae2 )). If agent 1 announces P , agent 2 is paid d, where d < h2 (C2 (a12 )). Payments
to agent 1 are analogously defined. Then, it is easy to show that strategy of choosing
aek with probability one and announcing R if the oppnents chooses ae−k and P otherwise
forms a perfect equilibrium.
The above contract exhibits the unsatisfactory aspect that a bad equilibrium for the
principal exists. That is, strategy of choosing a1k and announcing always R also constitutea perfect equilibrium. In order to solve that problem, Ma (1988) introduced a sequential
message reporting communication. And, he showed that when wages are contingent on
output as well as messages, under a certain condition, there exists a contract that uniquely
implements the first-best action pair at the first-best cost in subgame perfect equilibrium.

7

There is a prominent distinction between the perfect and costly monitoring situation.
In the perfect monitoring situation, there exists a contract to implement the first-best
action pair as an equilibrium in which the first-best action pair is chosen probability
one, in other words, it is exactly implementable at the first-best cost. However, in the
costly monitoring, there exists no contract under which the first-best action pair is exactly
implemented at even near first-best cost. We provides the following proposition.
Proposition 1: For any positive γ, if the first-best action pair is exactly implementable,
then its implementation cost is the second-best one.
Suppose that there exists a contract such that (ae1 , ae2 ) is chosen with probability
one in equilibrium. Then, each agent does not monitor the opponent. Since, on theequilibrium path, agent k believes that the opponent chooses ae−k with probability one, hecan knows which action has chosen without monitoring the opponent. Hence, to monitor
the opponent is not a best response for each agent. Therefore, if (ae1 , ae2 ) is surely chosenon the equilibirum path, agent k reaches the state (aek ; φ) with probability one. Thatis, if agent unilaterally deviated from the first-best action, he could not be detected by
the opponent. The message reported by agent is not valuable since it does not convey
additional information his opponent’s at all. The discussion implies that contract which
exactly implement ae is, essentially, one without communication. Consequently, if ae isexactly implementable, the implementation cost is the second-best one.
In the present paper, we allow agents to mix actions and focus on the situation where
monitoring cost γ is sufficiently small. And, We examine whether it is possible that, inthe equilibrium, the first-best action pair is chosen with high probability and the expected
payment to the agents is approximately equal to the first-best cost.
Definition 1: We say that â is virtually implementable if, for any positive δ, there existsγ(> 0) and a contract under which there is an equilibirum strategy {(πk , gk , fk )}k=1,2 suchthat |πk − π̂k | < δ for k = 1, 2, where π̂k is the vector such that π̂k (âk ) = 1 and π̂k (ak ) = 0

8

for all ak 6= âk .4
Assume that the wages are not contingent on outputs but only on messages. From
the discussion on proposition 1, if the wages are not contingent on outputs, the firstbest action pair is never exactly implementable. On the other hand, the first-best action
pair is virtually implementable with almost the same costs as in the first-best situation.
Consider the contract in which there are two available messages for each agent and wages
are contingent only on messages. It is summarized by table 1.
R

PR

C1e + α1 , C2e + α2

0, C 2

P

C 1, 0C 1, C 2
Table 1.

Hereafter, we use the corresponding utility payment instead of explicit monetary payment. In each cell, agents’ utility payments are written. For example, agent 1’s utility
payments are v1 (R, R) = u1 (s1 (R, R)) = C1 (ae1 )+α1 = C1e +α1 ，v1 (P, P ) = u1 (s1 (P, P )) =C1 (a11 ) = C 1 ，and v1 (R, P ) = u1 (s1 (R, P )) = 0.
Given the above contract, there is a following type of equilibrium strategy σ̂ = (σ̂1 , σ̂2 ).




e



 1 − δk (γ) if ak = akσ̂k : π̂k (ak ) = 







δk (γ)

if ak = a1k

,





e



 γ if ak = akĝk (ak ) = 






 φ otherwise





e
e
e



 R if ω̂k = (ak ; a−k ), (ak ; φ)fˆk (ω̂k ) = 






P

otherwiseδk (γ) will be defined below.

4| · | represents the standard Euclidian norm.

9

.

,

Following this strategy, agent k mixes between aek and a1k . If he chooses aek , then hemonitors the opponent, and otherwise, he does not. And only if his state is (aek ; ae−k ) and(aek ; φ), then he announces R, and otherwise, P .
Proposition 2: Suppose that wages are contingent only on announced messages. Then,
there exists a contract that virtually implements (ae1 , ae2 ) with almost the same cost as in
the first-best.
Proof: We show that σ̂ is a sequential equilibrium strategy. Since the discussions aresame for each agent, we show that the above strategy is a best response for agent 1 givenσ̂2 . First, show that if each agent follows the equilibirum strategy, agent 1 has an incentive
to mix between ae1 and a11 , that is, he is indifferent between ae1 and a11 .
(1 − δ2 (γ)) · (C1e + α1 ) + δ2 (γ) · C 1 − C1e − γ = C 1 − C 1 = 0.
The first equality implies that ae1 and a11 are indifferent and the second equality implies
that the reservation utility is guaranteed in the equilibrium. Arranging the above equation
yieldsδ2 (γ) =α1 − γ,∆e1 + α1where ∆e1 = C1e − C 1 .
Define that α1 is a function of γ as following.
α1 (γ) = γ κ ,where 0 < κ < 1.

Hence,
δ2 (γ) = γ κ

1 − γ 1−κ= γ κ · D1 (γ).
∆1 + γ κ

When 0 < γ ≤ γ̄, it is satisfied that 0 < D(γ̄) ≤ D1 (γ) < 1/∆1 . Then, δ2 (γ) > 0 is
satisfied for any γ such that 0 < γ < γ̄.
If each agent follows the equilibrium strategy, to choose ai1 such that i 6= 1, e is not abest response since agent 1’s expected utility is C 1 − C1 (ai1 ) < 0. Therefore, it has been

10

shown that, for any γ (0 < γ < γ̄), π̂1 is a best response. Then, limγ→0 α1 (γ) = 0, andlimγ→0 δ2 (γ) = 0. That is, π̂2 (ae2 ) tends to one as γ goes down to zero. And these implies
that the implementation cost tends to the first-best one.
Secondly, the incentive in the monitoring stage is examined. Agent 1’s future expectedutility after choosing ae1 is, when he monitors,Ê1γ = (1 − δ2 (γ)) · (C1e + α1 ) + δ2 (γ) · C 1 − γ.
When he does not monitor, he will obtain at mostÊ1φ = (1 − δ2 (γ)) · (C1e + α1 ).
Then, for sufficiently small γ, it is satisfied thatÊ1γ − Ê1φ = γ κ (C 1 · D1 (γ) − γ 1−κ ) > 0.
Thus, when agent 1 chose ae1 , to monitor the opponent’s action is a best response. Thatis, ĝ1 (ae1 ) = γ. On the other hand, after choosing ai1 such that i 6= e, when he dose notmonitor, his expected utility is C 1 , and when he does, C 1 − γ. Hence, when choosing ae1 ,not monitoring is a best response. That is, ĝk (ai1 ) = φ for all i 6= e.
Finally, the best responses in the communication stage are examined. Define that
µ1 (ω̂1 ; σ̂)(R) is the belief of agent 1 induced according to the equilibrium strategy σ̂.
That is, it is agent 1’s belief at state ω̂1 that agent 2 would announce R. The belief at
each state is as follows.
µ1 ((ae1 ; ae2 ) ; σ̂)(R) = 1,(1)µ1 ((ae1 ; φ) ; σ̂)(R) = 1 − δ2 (γ),(2)µ1 (ω̂1 ; σ̂)(R) = 0,∀ω̂1 6= (ae1 ; ae2 ), (a1e ; φ)(3)

From (1), at state (ae1 ; ae2 ), agent 1 believes that agent 2 would announce R. Then, R is astrict best response for agent 1. From (2), at state (ae1 ; φ), agent 1 believes that agent 2

11

would announce R with almost probability one when γ is sufficiently small. Then, R isa strict best response for agent 1. From (3), at each state other than (ae1 ; ae2 ), agent 1
believes that agent 2 would announce P . Then, P is a strict best response for agent 1.
The above discusstions are true to agent 2. Q.E.D.
Although the above contract viutually implements (ae1 , ae2 ), the unsatisfactory aspectof the contract is that there exists a bad equilibrium, σ b , such that each agent chooses a1k
and always announces R without monitoring. Equilibrium σ b may be plausible since σ bPareto-dominates σ̂ from the agents’ view point. However, σb is bad for the principal. In
the following sections, we propose desirable contracts to virtually implement the first-best
action pair as a plausible equilibrium with approximate first-best outcome.
4.

Collusion-Proof Equilibirum

In this section, we adopt coalition-proof Nash equilibrium defined by Bernheim, Peleg, andWhinston (1987) as the equilibrium concept. In the two-agent case, coalition-proof Nashequilibria are the outcomes that is not Pareto-dominated by any other Nash equilibrium.
Hence, we suppose that the agents can collusively choose preferable one of equilibria whichis induced by the given contract. We say that an action pair is virtually implements incoalition-proof Nash equilibrium if there exists a contract virtually implements an action
pair and its equilibrium is coalition-proof Nash equilibrium.
The wages are contingent not only on the messages but also on the outputs. The
following condition 1 is assumed. Its role will be clear below.
Condition 1:
P (τ ) 6= P (ae1 , ae2 ),∀τ ∈ ∆(A\(ae1 , ae2 )),where P (a1 , a2 ) = (p(x1 |a1 , a2 ), · · · , p(xL |a1 , a2 )) and P (τ ) =

P

j
j
i
i
i,j P (a1 , a2 ) · τ (a1 , a2 ).

Condition 1 implies that the probability distribution P (ae1 , ae2 ) can not be induced
when any agent deviates the first-best action. That is, the following system has a unique12

solution in τ .
XXi

τ (ai1 , aj2 ) · p(xl |ai1 , aj2 ) = p(xl |ae1 , ae2 ),∀xl ∈ X,j

τ (ai1 , aj2 ) ≥ 0,∀(ai1 , aj2 ) ∈ A1 × A2 .

The system has L equations and I × J inequalities. If L ≥ I × J, condition 1 is genericallysatisfied. Condition 1 is equivalent to the following5 :Condition 10 ： There exists a vector û ≡ (û(x1 ), · · · , û(xL )) such that
P (ai1 , aj2 ) · û > 0,∀(i, j) 6= (e, e) and P (ae1 , ae2 ) · û1 < 0.

When there exists a vector û satisfied with condition 10 , for any scholar ν > 0, vector û0 = ν · û also satisfies condition 10 . Therefore, |P (ai1 , aj2 ) · û| can be made to bearbitrarilly small or large.
Consider the following contract.
R

P

R

C1e + α1 , C2e + α2

C 1 − ε1 (i, j) , C2e + α2 + ε2 (j, i)P

C1e + α1 + ε1 (i, j) , C 2 − ε2 (j, i)

C1 , C2

We define εk (i, j) = P (aik , aj−k ) · û. In this contract, for example, when (P, R) is
announced and x is realized, the utility-payment for agent 1 is v1 ((P, R), x) = C1e +α1 + û(x). The expected utility-payments are written in the above matrix given action
pair (ai1 , aj2 ). The interesting aspect of this contract is as follows: When (ae1 , ae2 ) is chosen,
announcing R is the dominant strategy for each agent in the communication stage. Andif agent k chooses an action other than aek , P is dominant.
We describe all the equilibrium. The iterative elimination of strictly dominated strategies reduces the normal-form representation as follows.

[Table 2 Here]5

See Fan (1956).

13

(aek , R) is the strategy that agent k chooses aek and announces R without monitoring.
Similarly, (a1k , P ) is the strategy that he chooses a1k and announces P without monitoring.
(aek , γ) is the strategy that he chooses aek and announces R if he observe ae−k and P
otherwise. Each strategy does not specify moves at the information set which can not bereached since the unspecified moves do not affect Nash equilibrium.
The process of the elimination of strictly dominated strategies is as follows. Consider
agent 1. (ae1 , P ) and (a11 , R) are strictly dominated by (a11 , P ). And, (ai1 , ·) such that i 6= e
is strictly dominated by (a11 , P ) if γ is sufficiently small and |ε(i, j)| is appropriately small.
Finally, (ae1 , γ) dominates (aek , ·) other than (aek , R). Similarly, for agent 2, the elimination
process is same. Consequently, the above matrix is induced.
In this reduced normal form game, there are only three equilibrium outcomes.
(1) σ: Take (a1k , P ) with probability one.
(2) σ̄: Mix (a1k , P ) and (aek , γ).
(3) σ ∗ : Mix (a1k , P ), (aek , R), and (aek , γ).
Apparently, σ is a Nash equilibrium. In this equilibrium, each agent’s expected utility
equals to zero. Let Σ be the set of strateies that is payoff equivalent to σ. Then, eachstrategy in Σ is also Nash equilibrium. In set Σ, there is unique sequential equilibriumsuch that agent k chooses a1k and always announces P without monitoring.
In case (2), Nash equilibrium strategy is σ̄k = (1 − δk (γ))[(aek , γ)] + δk (γ)[(a1k , P )],for each k = 1, 2, where δk (γ) is the same as defined in section 3. In this equilibrium,
each agent’s expected utility equals to zero and the first-best action pari vurtually implementable at almost the same cost as the first-best. Let Σ̄ be the set of strateies thatis payoff equivalent to σ̄. Then, each strategy in Σ̄ is also Nash equilibrium. In set Σ̄,there is unique sequential equilibrium. Agent k chooses aek with probability 1 − δk (γ), anda1k with δk (γ). In the monitoring stage, when he chooses aek , he monitors the opponent.

14

When he chooses aik such that i 6= e, he does not monitor the opponents. In the communication stage, he announces R at the states (aek ; ae−k ), and P otherwise. The proof isalmost the same as in section 3 and omitted.
In case (3), Nash equilibrium strategy is given asσk∗ = (1 − xk (γ) − yk (γ)) · [(aek , γ)] + xk (γ) · [(aek , R)] + yk (γ) · [(a1k , P )],
where

³

xk (γ) =

´

³

∆e

k
αk 1 − ε−kγ(e,1) − γ 1 + ε−k (e,1)∆ek + ε−k (e, 1) + αk´,yk (γ) =γ.
ε−k (e, 1)Let Σ∗ be the set of strateies that is payoff equivalent to σ ∗ . Then, each strategy in Σ∗ isalso Nash equilibrium. In set Σ∗ , there is unique sequential equilibrium. Agent k chooses
aek with probability 1−yk (γ), and a1k with yk (γ). In the monitoring stage, when he chooses
xk (γ)
k (γ)−yk (γ), and does not with 1−y. When he chooses
aek , he monitors with probability 1−x1−y
k (γ)
k (γ)

aik such that i 6= e, he does not monitor the opponents. In the communication stage, heannounces R at the states (aek ; ae−k ) and (aek ; φ), and P otherwise.
If we put αk = γ κ , where 0 < κ < 1, then xk (γ) > 0 and yk (γ) > 0 hold. And it is heldthat limγ→0 xk (γ) = 0 and limγ→0 yk (γ) = 0. That is, (ae1 , ae2 ) is virtually implementablewith almost the same cost as the first-best. In the equilibrium, each agent obatains the
expected utility of Rk (γ) = yk (γ) · (∆e−k + αk + ε(1, e)) > 0. And, limγ→0 Rk (γ) = 0. Thatis, σ ∗ Pareto-dominates σ and σ̄.6
Cnsequently, we obtains the following proposition 3.
Proposition 3: Suppose that Condition 1 holds. There exists a contract under which(ae1 , ae2 ) is virtually implementable in coalition-proof Nash equilibrium with almost thesame cost as the first-best.

6

In the equilibrium under σ ∗ , agents obtain strictly positive expected utility. If each utility-payment isreduced by Rk (γ) for each m and x, agents’ expected utility can be equivalent to the reservation utilities
without disturbing equilibrium. Then, under σ and σ̄, each agent obtains negative expected utility.

15

5.

Unique Implementation

In this section, we construct a contract that implements the first-best actions virtuallyand uniquely in sequential equilibrium.
P (a11 , τ2 ) 6= P (a11 , ae2 ),Condition 2:∀τ2 ∈ ∆e2 \τ2e ,where τ2e is the mixed action such that τ2e (ae2 ) = 1 and τ2e (a2 ) = for each a2 6= ae2 .
This condition implies that, if agent 1 commits a11 , agent 2 can not induce the sameprobability distribution by deviating from ae2 as P (a11 , ae2 ). This condition is genericallysatisfied if L ≥ J. Therefore, both conditions 1 and 2 are generically satisfied if L ≥ I ×J.
Condition 2 is equivalent to the following:
Condition 20 : There exists a vector ũ2 ≡ (ũ2 (x1 ), · · · , ũ2 (xL )) such that
P (a11 , ae2 ) · ũ2 − C2e > P (a11 , aj2 ) · ũ2 − C2 (aj2 ),∀j 6= e.

That is, when agent 1 commits the least cost action a11 , there exists a vector ũ2 thatmakes agent 2 strictly prefer to taking ae2 . If there exists such a vector ũ2 , then, for any
schalar η ∈ R, û2 = (ũ2 (x1 ) + η, · · · , ũ2 (xL ) + η) satisfies condition 20 .
Let Ũ2 be the set of ũ2 that satisfies condition 20 . We denote ũ2 (i, j) = P (ai1 , aj2 ) · ũ2 .
Then, there exists a ũ2 ∈ Ũ2 such that the following maximization problem has a uniquesolution ĵ.
max ũ2 (e, j) − C2 (aj2 ).
j6=eSuppose that the maximization problem has multiple solutions. Say, ĵ and ξ are thesolutions. Then, if C2 (aĵ2 ) > C2 (aξ2 ) without loss of generality, P (ae1 , aĵ2 ) 6= P (ae1 , aξ2 ) mustbe held.7 Hence, there exists a xl such that p(xl |ae1 , aĵ2 ) > p(xl |ae1 , aξ2 ). If we consider avector û2 such thatû2 = (ũ2 (x1 ), · · · , ũ2 (xl−1 ), ũ2 (xl ) − δ, ũ2 (xl+1 ), · · · , ũ2 (xL )),If P (ae1 , aĵ2 ) = P (ae1 , aξ2 ), ĵ can not be the solution since P (ae1 , aĵ2 )· ũ2 = P (ae1 , aξ2 )· ũ2 . This contradicts
the assumption that ĵ is a solution of the maximization problem.
7

16

then it is possible to make ĵ only the solution. In fact, there exists a δ̄ such that, forany δ ∈ (0, δ̄), ĵ is unique solution ot the maximization problem given û2 . Condition 20
implies that Ũ2 is a set to be L-dimensional, open, and non empty. If ũ2 ∈ Ũ2 , then, forsome δ > 0, û2 ∈ Ũ2 . The discussions below are assumed to be ĵ = 1 without loss ofgenerality.
Consider the following contract.8
RPR

C1e + α1 , C2e + γ

C 1 − Kε1 (i, j) , C2e + ε2 (j, i) + γ

P

C1e + α1 + ε1 (i, j) , Z

C 1 , ũ2 (i, j)

Assume that ũ2 (i, j) > Z holds for any i, j, and ũ2 (e, 1) − C 2 = 0. And take K(> 0)
so that −∆e1 − Kε1 (e, e) > α1 > 0. In fact, those are possible under condition 1 and 2.
For agent 1, the best respose property in the communication stage is the same as thecontract in the previous section. On the other hand, for agent 2, when (ae1 , ae2 ) is chosen,
announcing R is the best response in the comminication stage if agent 1 announces R,
and P is the best response if agent 1 announces P . When agent 2 chooses an action otherthan ae2 , P is dominant strategy.
Table 3 below is the reduced normal form game induced by eliminating all strictly
dominated strategies. For agent 1, only three strategies survive after the elimination.
However, for agent 2, there are many strategies. Each strategy is defined as in theprevious section.

[Table 3 Here]The process of elimination of strictly dominated strategies for agent 1 is as same asin the previous section. Hence, (ae1 , R), (ae1 , γ), and (a11 , P ) survive. Examine strictly
dominated strategies for agent 2 after eliminating all strictly dominated strategies foragent 1. Strategies (aj2 , ·) such that j > e are strictly dominated by (ae2 , γ) if γ is sufficiently
8

As in the previous section, expected utility-payments given (ai1 , aj2 ) are written in each cell.

17

small. And (ae2 , R) is stricly dominated by (a12 , P ). Because, when agent 2 takes (ae2 , R),
he obtains the expected utility γ if agent 1 takes (ae1 , R) or (ae1 , γ), and Z − C2e if (a11 , P ).
However, (ae2 , P ) is not strictly dominated though (ae2 , R) is strictly dominated. And foreach 1 ≤ j < e, (aj2 , R) is strictly dominated by (aj2 , P ). Consequently, table 3 is induced.
Note that (a11 , P ) is chosen with positive probability in any equilibrium. Suppose that(a11 , P ) is not chosen in an equilibrium. Then, (a21 , P ) is unique best response for agent 2
since agent 1 chooses (ae1 , R) and/or (ae1 , γ). Given agent 2’s strategy (a12 , P ), agent 1’s
best response is (a11 , P ). Contradiction.
First, show that, in equilibrium, (ae1 , R) can not be chosen with positive probability.
Suppose that (ae1 , R) is chosen with positive probability. From the above discussion,
agent 1 must randomize between (ae1 , R) and (a11 , P ). Let U1 (σ1 , σ2 ) be expected utility
given strategy (σ1 , σ2 ). There must exist σ2 such that U1 ((ae1 , R), σ2 ) = U1 ((a11 , P ), σ2 ) = 0
since U1 ((a11 , P ), σ2 ) = 0 for any σ2 . That is,σ(ae2 , P )[−∆e1 − Kε1 (e, e)] + σ(ae2 , γ)α1 +X

σ(aj2 , P )[C 2 − C2 (aj2 ) − Kε1 (e, j)] = 0.

j<eSince U1 ((ae1 , R), (ae2 , P )) > 0, U1 ((ae1 , R), (ae2 , γ)) > 0, and U1 ((ae1 , R), (aj2 , P )) < 0 for all
j < e, it is satisfied thatPjj<e σ(a2 , P ) > 0.

And, when α1 is sufficiently small, there

exists j(< e) such that σ2 (aj2 , P ) > T · α1 , where T > 0, for any σ2 that satisfies the above
equation. Then,
U1 ((ae1 , γ), σ2 ) − U1 ((ae1 , R), σ2 ) =X

σ(aj2 , P )Kε1 (e, j) − γ.

j<eWhen α1 = γ κ , where 0 < κ < 1, the right hand side of the equation is negative forsufficiently small γ. That is, U1 ((ae1 , γ), σ2 ) > U1 ((ae1 , R), σ2 ). Hence, there does not exist
equilibria such that both (ae1 , R) and (a11 , P ) are chosen with positive probablity.
Next, suppose that we eliminated (ae1 , R) from table 3. Then, if agent 2 think thatagent 1 would choose (ae1 , γ) or (a11 , P ), each (aj2 , P ) such that 1 < j < e is never chosen
if γ is sufficiently small, since it is strictly dominated by (ae2 , γ).

18

[Table 4 Here]
Hence, if, in the reduced normal form game of table 4, there exists unique equilibrium, then that implies that the game induced by the contract has unique equilibrium.
Consequently, we obtains the following proposition.
Proposition 4: Suppose that condition 1 and 2 hold. Then, (ae1 , ae2 ) is virtually anduniquely implementable in sequential equilibrium with almost the same cost as the firstbest.
Proof: In table 4, pure strategy equilibrium does not exist. Mixed strategies are candidates for the equilbrium. Examine whether (ae2 , P ) would be chosen with positive probability in equilibirum. To choose (ae2 , P ) with positive probability, it is necessary to be
indifferent between (ae2 , P ) and (a12 , P ). However, U2 (σ1 , (ae2 , γ)) > U2 (σ1 , (ae2 , P )) holds
for σ1 such that U2 (σ1 , (ae2 , P )) = U2 (σ1 , (ae2 , P )). Therefore, mixing (ae2 , P ) and (a12 , P )can not be equilibrium strategy.
Consequently, there is unique Nash equilibrium outcome. Equilibrium strategy σ̃ =(σ̃1 , σ̃2 ) is given byσ̃1 = (1 − δ̃1 (γ))[(ae1 , γ)] + δ̃1 (γ)[(a11 , P )],whereδ̃1 (γ) =γ> 0,ũ2 (1, e) − ũ2 (1, 1) − ∆e2andσ̃2 = σ̄2 .
Let Σ̃ be the set of strateies which is payoff equivalent to σ̃. Then, each strategy inΣ̃ is also Nash equilibrium. In set Σ̃, there is unique sequential equilibrium: Agent kchooses aek with probability 1 − δ̃k (γ), and a1k with δ̃k (γ), where δ̃k (γ). In the monitoringstage, when he chooses aek , he monitors the opponent. When he chooses aik such that19

i 6= e, he does not monitor the opponents. In the communication stage, he announces R
at the states (aek ; ae−k ), and P otherwise.
In equilibrium σ̃, the first-best action pair is chosen with almost probability one when
γ is sufficiently small. And each agent obtains utility-payment Cke with almost probability
one. That is, the first-best outcome is approximately attainable. Q.E.D.
Let R̃2 (γ) be expected utility agent 2 obtains in equilibrium σ̃. It is possible thatR̃2 (γ) 6= 0, that is, individual rationality condition is not bind or not satisfied. If eachutility-payment is reduced by R̃2 (γ) for each m and x, agent 2’s expected utility can beequal to the reservation utility without disturbing equilibrium.
References[1] Bernheim, D., B. Peleg, and M. Whinston (1987), “Coalition-Proof Nash Equilibria
I. Concepts,” Journal of Economic Theory, 42, 1–12.
[2] Fan, K.(1956), “On Systems of Linear Iqualities,” in “Linear Inequalities and Related
Systems” (H. W. Kuhn and A. W. Tucker, Eds.), Princeton, NJ.
[3] Holmström, B.(1982), “Moral Hazard in Teams,” Bell Journal of Economics, 13,
324–340.
[4] Itoh, H.(1993), “Coalitions, Incentives, and Risk Sharing,” Journal of Economic
Theory, 60, 410–427.
[5] Ma, C.(1988), “Unique Implementation of Incective Contracts with Many Agents,”
Review of Economic Studies, 55, 555–571.
[6] Mookerjee, M.(1984), “Optimal Inceitive Schemes with Many Agents,” Review of
Economic Studies, 51, 433–446.

20

Discussion Paper
No.
2001･1

AuthorTakashi Hatakeda
Nobuyuki Isagawa

2001･2Michiyasu Nakajima
Takeshi Mizuguchi
Katsuhiko Kokubu
Yasushi Onishi
Koji Okubayashi

2001･3
2001･4

2001･5
2001･6
2001･7

2001･8

2001･9Katsuhiko Kokubu
Akihiro Noda
Yasushi Onishi
Tomomi Shinabe
Kunio Miyashita
Akinobu Sakashita
KOKUBU,
Katsuhiko
NASHIOKA, ErikoDAIKUHARA, Rie
KOKUBU,
Katsuhiko
KURASAKA,
TomokoShuzo Murata

2001･11

Makoto Yano
Fumio Dei
Mahito Okura

2001･122001･13

Mahito Okura
Shuzo Murata

2001･14

Koji Okubayashi
Toshinori Takashina

2001･15
2001･16

Takuji Hara
Shuzo Murata

2001･17

Norio Kambayashi

2001･18
2001･19
2001･20

Yuichi Fukuta
Kazuhiro Tanaka
Kentaro Nobeoka
Kazuhiro Tanaka

2001･21

Mahito Okura

2001･22
2001･23

Kazuhiro Tanaka
Mahito Okura

2001･24

Katsuhiko Kokubu
Akihiro Noda
Yasushi Onishi
Tomomi Shinabe
Akira Higashida

2001･10

Title
Stock Price Behavior Surrounding Repurchase
Announcements:
Evidence from Japan
Material Flow Cost Accounting of IMU (October 2000)

Date
1／2001(in Japanese)

1／2001

Japanese Manufacturers Without Factories:
Cases of Sony, Matsushita, Misumi, People
Determinants of Environmental Report Publicationin Japanese Companies

1／2001

Logistics Strategy of Japanese Port Management
Issues and Method of Organizational Culture Studies:
Functionalism Approach
Environmental Accounting in Japanese Companies as ofNovember 2000

(in Japanese)

2／20013／2001(in Japanese)

3／20012／2001

Corporate Environmental Accounting: A Japanese
Perspective

The Comparative Study of Japanese & U.S. Companies(6) The Health & Long Term Care-The Management
Education
A Trade Model with Vertical Production Chain andCompetition Policy in the Downstream Sector
Risk Classification: A Study of the Tradeoff between
Quantity and Quality of Insured
Is the Separating Equilibrium Really Desirable?
The Comparative Study of Japanese & U.S. Companies(5) The Conflict Among Companies
Research Report on the relations between there-employment process and former job career at thecompulsory retirement of workers of a firm
The Social Shaping of Medicines
The Comparative Study of Japanese & U.S. Companies(7) Toward Restructuring Japanese Corporations
Cultural influences on IT usage among workers:a UK-Japanese comparison
A Test for Rational Bubbles in Stock Prices
Toward the Effective Reform of Japanese Corporate
Governance
Structural Changes in Top Management―The Effect ofCorporate Governance Reform in Japan
In the Separating Equilibrium Really Desirable ?
―revised version of No.2001･12―
Introduction to Corporate Governance
The Role of Some Insurance Contracts in a Subjective
Equilibrium Model with Observable Effort
Environmental Disclosure of Japanese companies

3／2001(in Japanese)

4／2001

12／2000(in Japanese)

5／2001(in Japanese)(in Japanese)

5／20016／2001(in Japanese)

7／2001(in Japanese)(in Japanese)

7／20017／20017／2001(in Japanese)

7／20017／2001(in Japanese)

7／2001(in Japanese)

8／2001(in Japanese)(in Japanese)

8／20018／2001(in Japanese)

8／2001

Discussion Paper
No.
2001･25

2001･26
2001･27

AuthorKatsuhiko Kokubu
Tomomi Shinabe
Akira Higashida
Yasushi Onishi
Akihiro Noda
Katsuhiko Kokubu
Eriko Nashioka
Masakazu Takagi

2001・28 Mahito Okura
2001・29 Atsushi Takao

2001・30 Mahito Okura
2001・31 Masakazu Takagi
2001・32 Norio Kambayashi
Harry Scarbrough2001・33 Fumitoshi Mizutani
Shuji Uranishi
2001・34 Mahito Okura

2001・35 Tsutomu Harada
2001・36 Nobuyuki Isagawa

2001・37 Nobuyuki Isagawa
2002・1

Nobuyuki Isagawa

2002・2

Nobuyuki Isagawa

2002・3Mahito Okura

2002・4

Elmer Sterken
Ichiro Tokutsu
Mahito Okura
Mahito Okura
Nobuyuki Isagawa
Koji Okubayashi
Toshinori Takashina

2002・5
2002・6
2002・7
2002・8

2002・9Hajime Shimizu

2002・10 Nobuyuki Isagawa
2002･11

Nobuyuki Isagawa

Title
Environmental Reporting of Japanese companies(in Japanese)

Date
8／2001

Environmental Accounting of Japanese companies(in Japanese)

8／2001Elementary Study of East Asian Corporate andManagement System
Vertical Differentiation in the Insurance Market
A Study on the Penetration Process of Modern Insurance
by Evolutionary Economics―Why do the Modern
Insurance and the Premodern Mutual Society coexist?―
An Essay in the Economics of Post-loss Minimisation:
An Analysis of the Effectiveness of the Insurance Law
and Clauses
Co-operation between Hanshin(Osaka-Kobe) Area andSouth-east Asia-Finding business opportunities of mutual
benefits
Cultural influences on IT use amongst factory managers：
A UK-Japanese comparison
The Post Office vs. Parcel Delivery Companies：
Competition Effects on Costs and Productivity
An Essay in the Economics of Post-loss Minimisation:
An Analysis of the Effectiveness of the Insurance Law
and Clauses―revised version of No.2001・30―
An Examination of IT Paradox in Japan
Open-Market Repurchase Announcements, Actual
Repurchases, and Stock Price Behavior in Inefficient
markets
Corporate Financial Strategy and Stock Price Behavior ina Noise Trader Model with Limited Arbitrage
Cross Holding of Shares, Unwinding Cross Holding of
Shares, and Managerial
Entrenchment
A Theory of Stock Price Behavior following Repurchase
Announcements
An Equilibrium Analysis of the Insurance Market
with Vertical Differentiation
What are the determinants of the number of bankrelations of Japanese firms?
Review Article：Adverse Selection in Insurance Market
Welfare Effect of Firm Size in Insurance Market
Does Investment Horison Affect Investment Strategy ?
Research Report of Careers and Employment Process ofWorkers over 60 years old :a Case of a Japnaese
Manufacturing Company
On the Existence of Tax Equilibrium : Incomplete Market
with Capital Income Tax
Cost of Financial Distress and Debt Restructuring :Debt Forgiveness and Equity-for-Debt Swap
Open-Market Repurchase Announcements, Actual
Repurchases, and Stock Price Behavior in Inefficient
Markets―revised version of No.2001･36―

9／2001(in Japanese)(in Japanese)

9／20019／20019／2001(in Japanese)

9／2001

10／2001
10／2001
11／2001(in Japanese)

11／2001
12／2001
12／2001(in Japanese)

1／ 2002(in Japanese)

1／ 20022／20023／2002(in Japanese)(in Japanese)(in Japnaese)

3／20023／20023／20024／2002(in Japnaese)

4／2002(in Japnaese)

4／20025／2002

Discussion Paper
No.
2002･12

AuthorKenji Kutsuna
Richard Smith

2002・13 Kunio Miyashita
2002・14 Hajime Shimizu
2002・15 Hajime Shimizu
2002・16 Koji Okubayashi
2002・17 Fumitoshi Mizutani
Shuji Uranishi
2002・18 Kazuhisa Otogawa
2002・19 Atsuo Takenaka
2002・20 Tsuneo Nakano

2002・21 Tsuneo Nakano
2002・22 Fumitoshi Mizutani
Shuji Uranishi
2002・23 Atsushi Takao
Mahito Okura
2002・24 Fumitoshi Mizutani
2002・25 Fumitoshi Mizutani
Takuya Urakami
2002・26 Yasuyuki Miyahara

Title
Why Does Book Building Drive Out Auction Methodsof IPO Issuance?
Evidence and Implications from Japan
International Logistics and Modal Choice
Incomplete markets with capital income tax and public
goods
On the possibility of Pareto improving of tax-equilibrium
China-Japan Comparison of Work Organization
The Post Office vs. Parcel Delivery Companies:
Competition Effects on Costs and Productivity―revised version of No.2001･33―
Earnings Forecast and Earnings Management of JapaneseInitial Public Offerings Firms
The Classification of Foreign R&D Units
The Dutch East India Company and its Corporate
Governance―revised version of No.2000･40―
The British East India Company and its Corporate
Governance
Privatization Effects on TFP Growth and Capital
Adjustments
Some Notes on the Privatization of Postal Life Insurancein JapanPrivately Owned Railways’ Cost Function, Organization
Size and Ownership
A Private-Public Comaprison of Bus Service Operators
Principal-Multiagent Relationships with Costly
MonitoringDate
5／2002(in Japnaese)(in Japnaese)

6／20026／20026／20027／20027／20028／2002(in Japanese)(in Japanese)

8／20028／2002(in Japanese)

8／20028／2002(in Japanese)

9／20029／20029／2002
10／2002