Moral Hazard in Sequential Teams

Roland Strausz�

Free University of Berlin

December 3, 1996

Abstract

This paper considers a team in which production takes place sequentially and

in which agents observe the actions taken by previous agents. We show that for

such teams sharing rules exist which are balanced and induce e�cient production

as the unique equilibrium outcome. This in contrast to team structures studied

by Holmstr�om (1982) in which agents act simultaneously. The sharing rule which

induces e�cient production is simple, intuitive and robust to noise, sabotage, and

collusive behavior. It induces e�cient production even when agents obtain imper-

fect information about previous actions.

JEL Classi�cation: D82, L23

Keywords: Teams, moral hazard, unique implementation.

Discussionpaper Nr. 1996/27, FU-Berlin

�FU Berlin, Boltzmannstr. 20, D-14195, Berlin, Germany. I would like to thank the following people:

Helmut Bester, Anette Boom, Matthias Dewatripont, Sjaak Hurkens and John Reimers. This paper has

bene�ted from the Oberwesel lectures 1995 and 1996 and I would especially like to thank Jacob Glazer,

who unknowingly gave me the idea of this paper. Furthermore I would like to thank the participants of

the "Quatsch Gruppe" at the FU-Berlin. E-mail: Strausz@ccmailer.wiwiss.fu-berlin.de

1

2

1 Introduction

The seminal article Holmstr�om (1982) considers the problem of team production with

moral hazard. Holmstr�om's paper shows that in a setting with risk neutral agents no

sharing rules exist which are balanced and induce the team to produce e�ciently. This

result has a strong implication for teams, as the only rengotiation-proof sharing rules

are those which are balanced. Consequently a team by itself cannot achieve e�cient

production. Holmstr�om further shows that non-balanced sharing rules exist which induce

e�cient production and argues that by contracting with a non-productive outside party

a team is able to commit to non-balancing sharing rules. Teams which contract with

outside parties have therefore an advantage over teams which do not contract with

outside parties. Since a natural interpretation of the outside party is that of the owner

of the team, Holmstr�om o�ers an explanation of the commonly observed separation of

ownership and labor based on reasons of e�ciency.

Holmstr�om assumes that production takes place simultaneously, i.e. every teammem-

ber chooses his e�ort while being ignorant of the choices of his fellow team members.

In reality, however, production often takes place sequentially. A typical example is con-

veyor belt production, where a team of workers assembles a product in distinct stages

and where each worker is responsible for a certain prodcution stage only. In such a

setting a worker may observe the actions taken by previous workers and condition his

actions on these observations. Also the co-authoring of acadamic papers is an example

of sequential production. During the process of writing authors have a good idea about

how much e�ort each participant is putting into the common project and can make ther

own level of e�ort depend on these observations.1 Sequential settings are therefore richer

than the simultaneous production settings considered by Holmstr�om (1982).

This papers shows that the result that balanced sharing rules cannot induce e�cient

prodcution depends crucially on Holmstr�om's assumption that team members choose

their e�ort simultaneously. In a setting in which production takes place sequentially

there do exist balanced sharing rules which uniquely implement e�cient production. By

1Note that the use of a third party as a budget breaker does not seem standard practice in co-

authoring papers.

3

sequential team production we mean that agents act one after the other and that, even

though the actions are not veri�able, an agent observes which actions his predecessors

have chosen.2

To contrast our result to Holmstr�om's we will �rst recall the basic problem of team

production with moral hazard. Controlling team production becomes problematic when

the agent's e�ort levels are not directly contractible and are in some way or another

substitutes. In this case �nal output is the only veri�able variable, but cannot be used

as an inference for determining exactly which agent delivered which e�ort level. Much

like in the standard principal-agent problem the issue of moral hazard arises and team

members must be given incentives to exert e�ort.

Holmstr�om's result, that there exists no balanced sharing rule which tailors incentives

in such a way that e�cient production results, is remarkable. The result is, however,

quite intuitive when one considers di�erentiable sharing rules. If a team member shirks

he bene�ts fully from the reduction in his cost of e�ort, but is only partly a�ected

by the reduction in output caused by his shirking. More striking is the fact that the

possibility to impose monetary punishments, i.e. using non-continuous sharing rules,

does not lead to �rst best implementation. Due to the fact that �nal output cannot

be used as an inference, an agent who shirks cannot be identi�ed and, therefore, not

be punished. The only way punishments can be given is at random, but due to the

balanced budget constraint the collected fees must be redistributed to the other agents.

Risk neutrality then ensures that the redistribution of punishments nulli�es the e�ect of

random punishments.3

This paper shows that if team members produce sequentially then a sharing rule

exists which is budget neutral and induces �rst-best behavior as the unique equilibrium

2A setting in which agents act sequentialy, but do not observe the others actions is strategically

equivalent to a situation in which the agents act simultaneously.3As shown by Rasmusen (1987) the assumption of risk neutrality is important. When agents are

risk-averse then schemes which use random punishments may induce �rst best behavior. The reason is

that in utility terms a random punishment is not nulli�ed when redistributed. Although agents receive,

in expectation, the same amount in money, the utility derived from it is lower. The balanced budget

constraint requires that payments are balanced, not the utilities.

4

outcome. The problem of a sequential team is much similar to that of Holmstr�om's

simultaneous team. Even though team members observe e�ort levels, it is not possible

to contract on these observations, as they are not veri�able by the enforcing party. Just

like in Holmstr�om's setting contracts can therefore only be made contingent on �nal

output. The fact that e�ort is observable internally makes it nevertheless possible to

implement e�cient production. The idea is to device a sharing rule which makes an

agent's action dependent on his observations about the e�ort levels chosen by previous

agents. Such a sharing rule induces agents to signal their information by in uencing �nal

output in a particular way. Final output may then be used to identify and punish the

shirking team member. We show that this idea is feasible under the restriction that the

sharing rule is balanced. The sharing rule we present induces a quite natural behavior

of the team members: A team member works e�ciently when he has not observed any

shirking and refuses to perform any e�ort when he observed that some previous agent

shirked.

Although many actual production settings have sequential features, the literature on

agency theory with sequential production is very modest. The present paper comes

closest to Banerjee and Beggs (1989). They study a non-team agency problem of a

principal whose two agents act in a sequential order. The �rst agent does not a�ect

production, but in uences only the cost function of the second agent, who then produces

some output. They show that implementation of e�cient actions is only possible if the

game is sequential, i.e. if the second agent observes the �rst agent's action. Although

the issues which are analyzed are di�erent and the nature of the scheme di�ers from

ours, also their scheme induces the second agent to behave di�erently when he observes

that the �rst agent shirked.

2 The Model

Consider n agents who exert e�ort sequentially in order to produce some output y 2 IR+.

We number the agents such that the �rst agent to perform e�ort is agent 1, the second

is agent 2, etc. Output y depends on the e�ort of the agents: y = y(e1; : : : ; en), where

5

ei 2 IR+ is the e�ort level of agent i. Output is strictly increasing in each agent's e�ort,

but marginally decreasing: @y=@ek > 0, @2y=@e2k< 0.

The agents are risk-neutral and dislike e�ort. We assume that their utility functions

are separable in wealth and e�ort. The e�ort costs ck of agent k depend on agent k's

own e�ort level and, possibly, on the e�ort level chosen by previous agents:

ck = ck(e1; : : : ; ek):

Given a payment I and the e�ort levels feig agent k's utility is

uk = I � ck(e1; : : : ; ek):

We make the standard assumption that e�ort costs are increasing in the agent's own

e�ort ek and marginally so, i.e. @ck=@ek � 0 and @2ck=@e2k� 0. In the production game

each agent has to choose a non-negative amount of e�ort. An e�ort of e = 0 shall be

interpreted as no e�ort, by which we mean that an agent does not incur any costs if he

does not perform any e�ort, i.e. ck(e1; : : : ; ek�1; 0) = 0, and that if no agent performs

e�ort an output of zero is produced, i.e. y(0; : : : ; 0) = 0. In order to guarantee that it is

e�cient for every agent to perform a strictly positive level of e�ort, we assume that the

marginal cost of e�ort at e = 0 is zero, i.e. @ck=@ek jek=0= 0.

We further assume that a team member's e�ort has a (weakly) positive external e�ect

on the cost of e�ort of subsequent agents.4 This is guaranteed by the following condition:

Assumption 1 For every l < k the cost function ck : IRk

+! IR+ satis�es the following

condition.@ck

@el(e1; : : : ; ek) � 0:

E�ort may therefore have two positive e�ects. First, it increases �nal output and,

second, it lowers e�ort costs of subsequent agents. Note that assumption 1 also holds

for cost functions which are independent of the e�ort levels of previous agents as in

4Appendix B analyzes the more general case in which e�ort may have arbitrary e�ects on the other

team members' cost functions.

6

Holmstr�om's original paper. In this case e�ort does not in uence the cost of e�ort of

other team members and has only the positive e�ect that it raises production.

We make the following assumptions about the informational structure. Only the

�nal output y is veri�able and contractible. The e�ort level ek is not veri�able, but

observable by agent k and all agents after k. This assumption models the sequential

character of the team. Note that an agent needs to observe all previous e�ort levels if

he is to know his cost and production function. The assumption that agents observe all

previous e�ort levels further implies that there does not exist asymmetric information

between the team members before or during the production process. This allows us to

use subgame perfectness as implementation concept. (The assumption that an agent

observes all the e�ort levels of previous agents is relaxed in section 5.)

3 Implementing E�cient Production

Since �nal output is the only veri�able variable, a sharing rule I can only be made

contingent on �nal output y and not on the individual e�ort levels feig. It has, therefore,

the form I � (I1(y); : : : ; In(y)). and is balanced if for all possible realizations of y the

following condition holds:

nXk=1

Ik(y) = y: (1)

E�cient production takes place at those e�ort levels for which the surplus S � y�P

ck

is maximized. For convenience we assume such a maximizer e� = (e�1; : : : ; e�

n) exists and

is unique. A su�cient condition for uniqueness is that the surplus function S(e) is

strictly concave in e. The �rst order conditions are then necessary and su�cient for the

maximum e�, i.e.@y

@ek(e�1; : : : ; e

�

n) =

nXi=k

@ci

@ek(e�1; : : : ; e

�

i);

for all k = 1; : : : ; n. We de�ne e�cient output as the output y� � y(e�1; : : : ; e�

n) produced

by the e�cient e�ort levels e�.

The central question of this paper is whether there exists a budget neutral sharing

7

rule I which induces the team members to perform e�ciently. Moreover, we would like

to obtain that a sharing rule exists for every individual rational division of the e�cient

output y�. The question we ask is therefore whether a budget neutral sharing rule I

exists which implements e�cient production e� and for which the equilibrium outcome

is such that a team member k receives a share y�k. The division is individual rational

when y�k� ck(e

�

1; : : : ; e�

k) � 0. Note that under our assumptions a positive surplus from

team production is guaranteed and that an individual rational division fy�ig of the output

y� is always possible.

Before presenting a sharing rule which induces e� as the unique equilibrium outcome

we �rst introduce some further notation. Let the constant yk represent the output which

is produced, when the �rst k agents perform at their e�cient e�ort level and the last

n� k agents do not perform any e�ort:

yk � y(e�1; : : : ; e�

k; 0; : : : ; 0):

Note that 0 = y0 < y1 < y2 < : : : < yn = y�. Furthermore, let c�kbe the cost of e�ort of

agent k when he and all agents before him choose their e�cient e�ort level, i.e.

c�k� ck(e

�

1; : : : ; e�

k):

Last, let u�kbe the payo� of agent k when all agents choose their e�cient e�ort levels

and the members share the �nal output according to fy�ig, i.e.

u�k� y�

k� c�

k:

Since the division of the output fy�ig is individual rational it follows u�

i� 0 for all

i = 1; : : : ; n.

Consider the following payment sharing rule for agent 1

I1(y) �

8>>>>><>>>>>:

y �P

n

i=2u�i

if y < y1

y�1 if y1 � y � y�

y�1 + (y � y�) if y > y�

8

: : : -

0 y1 y2 y3 yn�2 yn�1 y�

I1(y) y�P

n

i=2u�i

y�

1y�

1+y� y

�

I2(y) u�

2y� y�

1�

Pn

i=3u�i

y�

2y�

2

I3(y) u�

3y�y�

1�y�

2�

Pn

i=4u�i

y�

3y�

3

.

.

.

.

.

.

.

.

.

.

.

.

In�1(y) u�n�1 y�

Pn�2

i=1y�i�u�n

y�n�1

y�n�1

In(y) u�

n y�Pn�1

i=1y�i

y�

n

Figure 1: Scheme I.

and for agent k > 1:

Ik(y) �

8>>>>><>>>>>:

u�k

if y < yk�1

y �P

k�1

i=1 y�k�P

n

i=k+1 u�

iif yk�1 � y < yk

y�k

if y � yk

Note that the sharing rule satis�es the balanced budget constraint.

Figure 1 illustrates the sharing rule. The scheme sets two thresholds, yk�1 and yk, for

an agent k. If �nal output lies below the threshold yk�1 then agent k receives a payment

u�k. When �nal output exceeds the threshold of yk agent k receives the payment y�

k. If

�nal output lies in the interval [yk�1, yk) then agent k receives the output y in excess of

the payments to the other agents. This amount is lower than u�kand acts as a punishment.

The structure of the sharing rule for agent 1 di�ers in so far that he obtains all output

in excess of the e�cient level y�.

Proposition 1 Let assumption 1 be satis�ed then for any individual rational division

fy�ig of the e�cient output y� there exists a balanced sharing rule I = (I1(y); : : : ; In(y))

which uniquely implements e�cient production e�.

The proof of the proposition consists of several steps and is relegated to the appendix.

The intuition behind the sharing rule is nevertheless extremely straightforward. As

already mentioned the problem with team production is that �nal output is the result

of multiple unveri�able actions and that actions are in some way or another substitutes.

9

This implies that if some agent shirks, he cannot be identi�ed or punished. In a sequential

game, in which a shirker is observed by fellow team members, the problem is that the

observations cannot be veri�ed by parties who enforce contracts. The enforcing party

can therefore not directly identify the shirker. By using sharing rule I the team can

circumvent this problem. The rule induces team members to work e�ciently as long

as they observe that no other team member has shirked. As soon as an agent observes

that some other agent before him shirked, he will not perform any e�ort. This means

that when agent k shirks then some �nal output y results which will lie in the interval

[yk�1; yk). Final output can then be used to expose agent k as the shirker and punish

him accordingly.

To make the sharing rule work at least three conditions have to be met. The �rst

condition is that the choice of an agent not to perform any e�ort, when he observes that

some team member shirked, must be rational. To see that this condition is met by rule

I, consider the last agent n who observes that shirking has taken place. He then knows

that intermediate production is below yn�1. This implies that if he does not perform any

e�ort, he receives a payment of u�nwithout incurring e�ort costs. This is the maximum

payo� he can hope for and not producing any e�ort is an optimal response. Given this

behavior of agent n and assumption 1 it is also optimal for agent n � 1 to perform

no e�ort as soon as she observes shirking. Continuing this argument shows that under

sharing rule I agents will stop performing e�ort as soon as they observe shirking.

Note that assumption 1 is important in this respect.5 It excludes the possibility that

it may be optimal for an agent to overwork in order to compensate for the shirking

of previous agents. That this might be a problem for the rule I becomes clear by

considering the following. Let agent k be the �rst agent who shirks, i.e. he chooses

ek < e�kand let ~el be the e�ort level for agent l > k who undoes the shirking, i.e.

y(e�1; : : : ; e�

k�1; ek; 0; :::; 0; ~el; 0; : : : ; 0) = yl. Since output is increasing in e�ort it follows

5A sharing rule which uniquely implements e�cient production when assumption 1 does not hold is

presented in appendix B.

10

that ~el > e�l. If assumption 1 did not hold it could happen that

y�l� cl(e

�

1; : : : ; e�

k�1; ek; 0; :::; 0; ~el) � y�

l� c�

l; (2)

even though ~el > e�l. In this case it is optimal for agent l to choose el = ~el instead of

choosing el = 0. Under assumption 1 inequality (2) will never hold and consequently

agent l always receives a payo� less than u�lif she overworks to compensate the shirking

of previous agent(s).

A second condition for rule I to work is that a team member has to be kept from

shirking given that his shirking is detected. This means that a potential shirker must

receive a lower payo� from shirking than what he receives from performing at his e�cient

level. Since shirking implies that subsequent agents do not perform any e�ort, it follows

that if agent k is the �rst agent to shirk then �nal output will indeed lie in the interval

[yk�1; yk). According to sharing rule I agent k then receives the �nal output y, but has

to pay all agents before him y�iand all agents after him u�

i. This implies that all agents

other than agent k get their prearranged payo� from e�cient team production. Since e�

is the unique maximizer of the surplus S, any output y < y� will not be large enough to

cover the payments of a potential shirker k to other agents, while leaving him a payo�

of more than u�k.

Last, the sharing rule should not induce agents to try to in ict punishments on other

agents by overworking. Since overworking will always involve e�ort costs higher than c�k

and payments do not exceed y�k, overworking will always lead to a payo� of less than u�

k.

Sharing rule I implements e�cient production under the threat of punishments. Even

though no punishments are given in equilibrium they may cause problems when schemes

are required to satisfy limited liability of the agents. This may be considered as a

disadvantage of the rule. We argue, however, that the punishments are not that high

as compared to their use in standard auditing literature, where often only unbounded

punishments lead to �rst best implementation (e.g. Baron and Besanko (1984) and

Border and Sobel (1987)). Since the upperbound of punishments is y�, the sharing rule

I satis�es limited liability if the limited liability levels are at �y�.6

6The upperbound is approached in quite speci�c settings in which one agent by himself generates

11

Moreover, we may defend punishments in scheme I on fairness grounds. Punishments

are namely such that if a team member k shirks then all other team members l 6= k still

receive their pre-arranged payo� u�l. Punishments are therefore fair, in so far as they are

derived from the principle that shirking should not take place at the expense of other

team members.

It may, at �rst sight, be surprising that the punishments in sharing rule I are indeed

high enough to keep a team member from shirking. This is due to the fact that what

the rule implements is exactly the e�cient output y�, which per de�nition can most

e�ciently be produced by the e�cient e�ort levels.

The sharing rule I has many appealing features. It is simple and induces straight-

forward behavior of the team members in equilibrium. Moreover, it imposes bounded

punishments and implements e�cient production in a unique way. Without studying

the issue explicitly we mention that the rule is also robust against collusive behavior,

albeit without side payments.

4 Sabotage

Under sharing rule I an agent k chooses his e�cient e�ort level when he observes that

none of the previous agents has shirked. In this case shirking is suboptimal, since it

induces subsequent agents to choose an e�ort level of zero. The choices e = 0 would lead

to a �nal output that lies between the thresholds yk�1 and yk and, consequently, agent k

is punished. It seems therefore important that agent k does not have the possibility to

destroy some of the output and in ict a punishment on a previous agent. We interpret

the destruction of output as sabotage and study in this section whether sabotage upsets

the outcome of the rule I.

We can model the possibility of sabotage by allowing agents to choose negative e�ort

the complete surplus, while incurring almost no costs and pre-arranged shares are such that this agent

receives only his reservation utility. E.g. let c�

n� 0 and let pre-arranged shares fy�

ig be such that

u�

n= 0, (i.e. y

�

n= c

�

n� 0) Furthermore, let the production function be such that agent n produces

almost all output, (i.e. yn�1 � 0). Under these circumstances the last agent receives a payo� of

yn�1 �P

n�1

iy�

i� cn(:;0) � �y

� when he is the �rst to shirk and chooses en = 0.

12

levels with the interpretation that a negative e�ort level destroys output. That is, we

extend the assumption @y=@ek > 0 over the interval ek < 0. Sabotage has constant

or decreasing marginal returns, i.e. @2y=@e2k� 0 for ek < 0. To model the fact that

sabotage is costly and increasingly so, the agents' cost functions must be convex in the

agent's own e�ort and obtain a unique minimum of zero at e = 0. We may further

extend assumption 1 over negative e�ort levels. Note that the assumption implies that

the e�cient e�ort level e� will not involve sabotage, i.e. sabotage is not e�cient.

Referring to sharing rule I it becomes immediately clear that sabotage does not a�ect

the equilibrium outcome, since by destroying output no agent can gain a higher payo�.

Note that also here the extended version of assumption 1 is important. It excludes

the possibility that by sabotage an agent makes it easier for some subsequent agent to

produce and makes it attractive for him to compensate the sabotage and shirking.

Proposition 2 Scheme I is robust to the possibility of sabotage.

5 Observational Requirements

Until now we assumed that each team member observes the e�ort level of all previous

agent and showed that this is su�cient for the subgame perfect implementation of e�-

cient production. For some environments, however, this may seem a stark assumption

which is di�cult to be met. A natural question is therefore whether this observational

requirement is also a necessary condition for scheme I to induce e�cient production. In

this subsection we study the existence of an equilibrium outcome with e�cient produc-

tion when observations about previous actions are imperfect. We will concern ourselves

only with imperfect observations which are caused by the lack of observation and not by

exogenous uncertainty. By this we can avoid the need to introduce moves of nature.

In order to concentrate on the e�ects of imperfect information on e�cient implemen-

tation in teams, we let the agent's cost function depend only on his own e�ort. This

circumvents the problem that due to the unobservability of some action an agent does

not know his own cost function. The unobservability of an agent's own cost function

13

would be a rather non-standard in agency models.

The introduction of imperfect observations implies that the game induced by scheme I

becomes an extensive game with imperfect information. Such games require a di�erent

implementation concept from subgame perfectness. Instead we use perfect Bayesian

implementation.

We will study two speci�c observational structures:

Setting 1: A team member k observes only the intermediate state of production after

his predecessor has taken his e�ort level and before he has to take his own e�ort level,

i.e. team member k observes yk � y(e1; : : : ; ek�1; 0; : : : ; 0).

Setting 2: Each team member (except for the �rst agent) observes only the e�ort level

of his predecessor.

Note that setting 1 the reduced observability poses a problem when e�ort levels are

imperfect substitutes, i.e. when there does not exist a production function ~y : IR+ ! IR+

such that ~y(P

i ei) = y(e1; : : : ; en). In this case di�erent combinations of e�ort levels

may lead to the same state of intermediate production, but result in quite di�erent

intermediate production functions. In setting 1 an agent k must therefore be sure that

when he faces an intermediate level of exactly yk = yk that this level of intermediate

production was indeed reached by the e�ort levels (e�1; : : : ; e�

k�1) and not by some other

combination.

Proposition 3 Consider setting 1 and let jN j > 2. Then scheme I induces an extensive

game with imperfect information for which e�cient production is a perfect Bayesian

equilibrium outcome.

Proof: We give the equilibrium strategies and the system of beliefs which support this

equilibrium. Let a team member k's belief, �k, about previous e�ort levels depend on

his observation of the intermediate output, yk, and satisfy the following.

�k(e1 = e�1; : : : ; ek�1 = e�k�1jyk = yk�1) = 1

�k(e1 < e�1; : : : ; ek�1 < e�k�1jyk < yk�1) = 1

14

�k(e1 = e�

1; : : : ; ek�2 = e

�

k�2; ek�1 = ~ek�1jyk > yk�1) = 1;

with ~ek�1 such that y(e�1; : : : ; e

�

k�2; ~ek�1) = yk. (Note that ~ek�1 > e�

k�1.) Consider the

following strategy �k for a team member k > 1: �k = e�

k if yk = yk�1, �k = 0 if yk < yk�1

and �k = maxf0; ~ekg, with ~ek such that y(e�1; : : : ; e�

k�1; ~ek) = yk, yk > yk�1. It is easy

to see that the beliefs �k with the strategies �k and the strategy �1 = e�

1 for agent 1

constitute a perfect Bayesian equilibrium with e�cient production as the equilibrium

outcome.

Q.E.D.

In setting 1 a team member observes only the intermediate state of production, yk,

and not the individual actions of previous agents. In order to guide his choice of e�ort

he therefore forms a belief about these actions which are consistent with his observation

yk. The equilibrium outcome will depend on how these beliefs are formed. The scheme

I induces e�cient production when a team member k believes the following. If the

intermediate production yk conforms with the target level yk�1 of agent k � 1, i.e. the

state of production which would have occurred when no previous players shirked, then

agent k does indeed believe that nobody shirked. Agent k can therefore safely choose his

e�cient e�ort level, since he believes that this will result in an intermediate production

yk, which induces also agent k + 1 to believe that no shirking has occurred. When the

intermediate production lies below the target level of agent k � 1, then agent k believes

that all agents shirked and therefore that if he had to reach his target level, he must

choose an e�ort level exceeding e�k which would yield a payo� of less than u�

k. It is better

for him to choose ek = 0 as subsequent agents using the same reasoning will choose

also e = 0. Agent k therefore believes to attain a payo� of u�k by choosing e = 0. If

intermediate production lies above the target level of agent k � 1, then agent k must

conclude that some agent overworked. If agent k actually believes that only the last

agent overworked, while all other agents chose their e�cient e�ort level, then agent k

can safely choose an e�ort level below his e�cient e�ort level. It is then optimal for him

to choose an e�ort level by which he reaches exactly his target level yk if possible and

zero otherwise. For an agent l before k it does therefore not pay to overwork as it results

15

in a payo� of y�l �cl(el) which is smaller than the payo� he would have gotten by playing

the equilibrium.

Proposition 4 Consider setting 2 with jN j > 2. The scheme I induces an extensive

game with imperfect information for which e�cient production is a perfect Bayesian

equilibrium outcome.

Proof: We give the strategies and the system of beliefs that support this equilibrium.

Let a team member k's belief, �k, about previous e�ort level depend on his observation

of the e�ort level, ek�1, in the following way. �k(e�

1; : : : ; e�

k�2jek�1 � e�

k�1) = 1 and

�k(0; : : : ; 0jek�1 < e�

k�1) = 1. The strategy �1 = e�

1 and �k = e�

k if ek�1 = e�

k�1,

�k = maxf0; ~ekg, with ~ek such that y(e�1; : : : ; e

�

k�2; ek�1; ~ek) = yk, if ek�1 > e�

k�1, and

�k = 0 if ek�1 < e�

k�1 are an equilibrium given the belief system f�kg and the belief

system f�kgk is Bayesian consistent with these strategies.

Q.E.D.

In setting 2 scheme I induces e�cient production if agents interpret the e�ort level

of the previous agent as a signal. An e�ort level of exactly e�

k�1 signals that nobody has

shirked and that it is safe for agent k to choose his e�cient e�ort level and signal to

agent k + 1 that no shirking has occurred. Agent k interprets an e�ort level which is

below e�

k�1 as an indication that not only agent k � 1 shirked, but also all agents before

him. It is therefore optimal for agent k to choose an e�ort of zero. This also signals to

agent k + 1 that shirking has taken place. When agent k observes an e�ort level which

exceeds e�k�1, he believes that all agents before k � 1 have chosen their e�cient e�ort

levels. If possible, it is then optimal for him to choose an e�ort level by which he reaches

exactly his target level yk, otherwise he chooses an e�ort of zero. This would signal to

agent k + 1 that all previous agents shirked and induces her not to perform any e�ort.

However, agent k is not bothered by this as he believes that output has reached his

target level yk anyway, which guarantees him a payment y�k. For agent k � 1 choosing

an e�ort which exceeds e�k�1 is therefore not optimal given the beliefs and strategies of

others.

16

Note that the equilibria in proposition 3 and 4, which sustain e�cient production

as the equilibrium outcome, are not unique. The concept of Perfect Bayesian Equilib-

rium is too weak to achieve a unique equilibrium outcome. One can show, however,

that equilibria which do not lead to �rst best outcomes require rather peculiar out-o�-

equilibrium-beliefs. We conjecture, that the Cho and Kreps' (1987) intuitive criterium

su�ces to obtain e�cient production as the unique equilibrium outcome.

6 Noisy Production

The model we have studied so far does not exhibit exogenous uncertainty. An interesting

question is whether �rst best implementation is also possible under uncertainty. Scheme I

worked on the principle that if one agent shirks, then all subsequent agents will choose an

e�ort level of zero. The �nal output can then be used to uniquely identify the shirker and

punish him appropriately. When output is noisy �nal output cannot uniquely determine

who shirked. Scheme I seems, therefore, to depend crucially on the fact that output is

deterministic.

In order to investigate whether our result depends crucially on the absence of noise

we adopt the assumption of complete observation and introduce noise in a similar way

to Banerjee and Beggs (1989). Let �nal output x depend on e�ort and a noise term "

which is realized after every agent has chosen his e�ort level, i.e.

x(e; ") = y(e) + ":

We assume that it is common knowledge that E["] = 0 and that " is distributed over

the interval [��;�] with density function f("), i.e. f(") = 0 for all " 62 [��;�]. Note

that the model is a straightforward extension of the basic model of section 3. It remains

a model of symmetric information. Moreover, the introduction of noise in an additive

form leaves the �rst best actions e� unchanged and enables us to make a meaningful

comparison between the former and present model.

We want to analyze a model, in which noise plays only a moderate role in determining

�nal output. We say that noise is a relatively small determinant of �nal output when

the following assumption is satis�ed:

17

Assumption 2: The parameter � satis�es

� < mink2f1;:::;n�1g

8<:(y

� � yk)�nX

i=k+1

c�

i

9=; :

The assumption implies that the maximum possible noise is less than the e�cient

surplus which is created by any group of last k agents. Since the e�cient surplus is

always strictly positive, the assumption will always be satis�ed for � small enough.

Note that if the team members' cost functions do not depend too much on other team

members e�ort level then every team member's e�cient e�ort level creates a surplus (i.e.

yk � c�

k > yk�1) and assumption 2 is equivalent to demanding that � is smaller than the

surplus which is created by the last agent, i.e. � < (y� � yn�1)� c�

n.

Consider the sharing rule I with for agent 1

I1(x) =

8>>>>><>>>>>:

x�Pn

i=k+1 u�

i if x < y1 + �

y�

k if y1 + � � x � y� + �

y�

k + (x� y�) if x > y

� + �

and for agent k > 1:

Ik(x) =

8>>>>><>>>>>:

u�

k if x < yk�1 + �

x�Pk�1

i=1 y�

k �Pn

i=k+1 u�

i if yk�1 + � � x < yk + �

y�

k if x � yk + �

The idea behind the scheme is similar to that of scheme I and di�ers only in the

target levels. It tries to induce the team members to stop working as soon as they

observe that somebody has shirked. The problem with noisy output is, however, that it

is more di�cult to identify a shirker. For instance, if �nal output x lies in the interval

(y1��; y1+�) it is no longer clear whether agent 1 or agent 2 shirked. It could be that

agent 1 shirked a little and agent 2 chose the e�ort level of zero in order to identify this.

Or it could be that agent 1 performed at his e�cient e�ort level and agent 2 decided to

shirk and chose an e�ort level close to zero. Scheme I deals with this by simply punishing

agent 1 even though it is not clear that he was indeed the shirker. It is clear that this

18

will prevent agent 1 from shirking, but might make it pro�table for agent 2 to shirk. As

the range (y1 � �; y1 + �) is not too large, this is not the case.

Proposition 5 Let assumption 1 and assumption 2 be satis�ed then for any individual

rational division fy�i g of the e�cient output y� there exists a balanced sharing rule I

which uniquely implements e�cient production e�.

Proof: See appendix A.

Q.E.D.

Proposition 5 shows that when noise is relatively small, �rst best implementation is

still possible. It shows, moreover, that scheme I is robust to noise in the sense that

scheme I is the limit case of the scheme I when noise disappears, i.e.

lim�!0

I = I:

Proposition 6 Scheme I is robust to noise.

7 Conclusion and Implications

This paper showed that in teamwork environments with sequential production schemes

exist which are balanced and induce e�cient production in equilibrium. This is in con-

trast to settings in which team production takes place simultaneously. We have argued

that sequential environments di�er from simultaneous ones in that agents may observe

the actions taken by previous agents. Although these observations are not veri�able by

a court and cannot be contracted on, balanced sharing rules exist which induce agents

to condition behavior on their observations. An agent's action may therefore in uence

the action taken by subsequent agents. Rational agents will take this dependence into

account when choosing actions. Using this we constructed a balanced sharing rule that

tailors the dependence in such a way that it becomes optimal for agents to choose their

e�cient e�ort levels. The sharing rule is remarkably simple and induces an equilibrium

behavior which is rather intuitive. We have shown that it is robust to sabotage and

19

noise and also induces e�cient production in environments in which agents observe only

a subset of previous e�ort levels.

The paper has a straightforward implication for organizing production in teams. In

reality the production structure will be an endogenous variable and its choice is up to the

team. This paper shows that if organizing production sequentially is a viable option, then

a team can circumvent the use of a budget breaker by setting up a sequential production

structure. Whether the team opts for sequential production will then depend on the

trade-o� between the cost of structuring production sequentially rather than horizontally

and the cost of engaging a third party as the budget breaker.

Given the existence of such a trade-o� this paper also o�ers the basis for an empirical

test to verify Holmstr�om's claim, that the wildly observed separation of ownership and

labor is due to reasons of e�ciency. If the claim is correct then one should expect to see

more separation in situations where it is more costly to organize production sequentially.

Apart from the aforementioned observation that in academics people do not use third

parties as budget breakers when writing joint papers, we are regrettably not aware of

the availability of empirical data to conduct such a test.

An important appealing feature of the scheme is that in settings with complete infor-

mation it implements e�cient production in a unique way. This implies that the paper

may also contribute to the principal multiple-agent theory (eg. Mookherjee (1984) and

McA�ee and McMillan (1991)). This literature typically assumes that the agents take

their actions simultaneously. As shown by Mookherjee (1984) such settings are often

plagued by a multiplicity of equilibria.7 This problem was addressed by Ma (1988) who

shows how a principal may use ex post message games �a la Moore and Repullo (1988)

in order to uniquely implement actions in the principal multiple-agent setting. He as-

sumes that actions are chosen simultaneously, but are ex-post observable by the agents.

The present paper shows that in sequential environments with an information structure

similar to Ma (1988) simple direct schemes exist which uniquely implement e�cient so-

lutions. This paper, therefore, indicates that sequential production may alleviate the

7The non-balanced scheme presented in Holmstr�om (1982) for example supports e�cient production

as a Nash equilibrium outcome, but it is not the unique Nash equilibrium outcome.

20

multiple equilibria problem.

Appendix A

Proof of proposition 1:

We prove that scheme I uniquely implements e�cient production as claimed by propo-

sition 1. Formally the scheme I induces a sequential game G with n agents in which

agent k has the payo� function uk(e) = I(y(e))� ck(e1; : : : ; ek). Our claim is that e� is

the unique subgame perfect equilibrium outcome of this sequential game. To show this

we �rst de�ne the subgame Gk(e1; : : : ; ek) as the subgame starting from agent k + 1 in

which the �rst k agents have chosen (e1; : : : ; ek).

Lemma A.1 For all ek 2 IR+ with k < n and en 6= e�nthe following condition holds

y(e�1; : : : ; e�

k�1; ek; 0; : : : ; 0)�k�1X

i=1

y�i�

nX

i=k+1

u�i� ck(e

�1; : : : ; e�

k�1; ek) < u�k

Proof: We prove by contradiction. Suppose not, then

y(e�1; : : : ; e�

k�1; ek; 0; : : : ; 0)�k�1X

i=1

y�i�

nX

i=k+1

u�i� ck(e

�1; : : : ; e�

k�1; ek) � u�k:

It then follows

y(e�1; : : : ; e�

k�1; ek; 0; : : : ; 0)� ck(e�1; : : : ; e�

k�1; ek) �k�1X

i=1

y�i+

nX

i=k+1

u�i+ u�

k

, y(e�1; : : : ; e�

k�1; ek; 0; : : : ; 0)� ck(e�1; : : : ; e�

k�1; ek) �k�1X

i=1

y�i+

nX

i=k

(y�i� c�

i)

, y(e�1; : : : ; e�

k�1; ek; 0; : : : ; 0)� ck(e�1; : : : ; e�

k�1; ek) � y� �nX

i=k

c�i

, y(e�1; : : : ; e�

k�1; ek; 0; : : : ; 0)� ck(e�1; : : : ; e�

k�1; ek)�k�1X

i=1

c�i� y� �

nX

i=1

c�i:

This contradicts that e� is the unique maximizer of y(e)�P

n

k=1ck(e1; : : : ; ek).

Q.E.D.

21

Lemma A.1 guarantees that if every agent chooses an e�ort level of zero after he

observes that some agent has shirked, then shirking is not pro�table. A shirker k is then

sure to receive a utility lower than the utility he obtains by not shirking. Of course

shirking will only be prevented if a potential shirker rationally believes that all agents

after him will indeed choose an e�ort level of zero.

Lemma A.2 If el < e�l^ : : : ^ ek�1 < e�

k�1 ^ then it holds for all ek > 0 that

y(e�1; : : : ; e�

l�1; el; : : : ; ek;0; : : : ; 0)�k�1X

i=1

y�i�

nX

i=k+1

u�i� ck(e

�1; : : : ; e�

l�1; el; : : : ; ek) < u�k:

Proof: De�ne �ek � 0 such that y(e�1; : : : ; e�

k�1; �ek; 0; : : : ; 0) = y(e�1; : : : ; e�

l�1; el; : : : ; ek;0; : : : ; 0),

if such an �ek � 0 exists. Otherwise de�ne �ek = 0. Since y is a monotonic increasing

function in all ei's it holds that �ek < ek. It now follows that

y(e�1; : : : ; e�

l�1; el; : : : ; ek; 0; : : : ;0)�k�1X

i=1

y�i�

nX

i=k+1

u�i� ck(e

�1; : : : ; e�

l�1; el; : : : ; ek) �

y(e�1; : : : ; e�

k�1; �ek; 0; : : : ; 0)�k�1X

i=1

y�i�

nX

i=k+1

u�i� ck(e

�1; : : : ; e�

l�1; el; : : : ; ek) < (3)

y(e�1; : : : ; e�

k�1; �ek; 0; : : : ; 0)�k�1X

i=1

y�i�

nX

i=k+1

u�i� ck(e

�1; : : : ; e�

k�1; �ek) < u�k; (4)

where (4) follows from lemma A.1 and (3) from assumption 1 and the fact that �ek < ek.

Q.E.D.

The lemma shows that if a team member observes that some sequence of previous

agent shirked then performing positive e�ort and this way in icting the punishment on

himself yields a payo� of less than u�k. In the light of lemma A.1 this is not a surprising

result.

Lemma A.3 If e is a subgame perfect equilibrium outcome with y(e) < y� then e must

be such that for all l with yl > y(e) it holds that el = 0.

Proof: If not, then the last agent with el > 0 could have decreased her e�ort level to zero

and so reduce her cost of e�ort without decreasing her payment u�l. Under assumption

22

1 a decrease of el increases the cost of e�ort for all agents after l. Agents after l would

therefore only increase their e�ort level if their increased e�ort would lead to a higher

payment, i.e. if �nal output is increased to y�. This of course would bene�t agent l only

more, since then also her payment is increased from u�lto y�

l.

Q.E.D.

Lemma A.4 Let (~el; : : : ; ~ek) be such that ~ei < e�ifor all i = l; : : : ; k. Then the subgame

Gk(e�1; : : : ; e�

l�1; ~el; : : : ; ~ek) has the unique subgame perfect equilibrium outcome ek+1 =

: : : = en = 0.

Proof: We prove by induction. Consider the subgame Gn�1(e�1; : : : ; e�

l�1; ~el; : : : ; ~en�1).

Since ~ei < e�iit follows that y(e�

1; : : : ; e�

l�1; ~el; : : : ; ~en�1; 0) < yn�1. The e�ort

choice en = 0 yields agent n the payo� u�n. We now show that any en > 0

results in a payo� lower than u�n. Consider the e�ort choice en > 0 such that

y(e�1; : : : ; e�

l�1; ~el; : : : ; ~en�1; en) < yn�1. Since such an en results in a payment u�nat

strictly positive e�ort cost, agent's n's net payo� is smaller than u�n. If agent n chooses

an e�ort level en such that y(e�1; : : : ; e�

l�1; ~el; : : : ; ~en�1; en) 2 [yn�1; y�) then agent n's pay-

o� is y(e�1; : : : ; e�

l�1; ~el; : : : ; ~en�1; en)�P

n�1i=1

y�i�ck(e

�1; : : : ; e�

l�1; ~el; : : : ; ~en�1; en). Applying

lemma A.2 it follows that this is smaller than u�n, if such an en is to exist at all. If agent

n chooses an e�ort level en such that y(e�1; : : : ; e�

l�1; ~el; : : : ; ~en�1; en) � y� then he receives

a payo� y�n�cn(e

�1; : : : ; e�

l�1; ~el; : : : ; ~en�1; en). Note that if such an en exists, it necessarily

holds that en > e�n. Consequently cn(e

�1; : : : ; e�

l�1; ~el; : : : ; ~en�1; en) > cn(e�1; : : : ; e�

n�1; en) >

cn(e�1; : : : ; e�

n�1; e�n) and agent n's payo� is smaller than u�

n. For agent n it is therefore

optimal to choose en = 0 in the game Gn�1(e�1; : : : ; e�

l�1; ~el; : : : ; ~en�1).

Left to prove is the induction step that Gk�1(e�1; : : : ; e�

l�1; ~el; : : : ; ~ek�1) has as the

unique subgame perfect equilibrium outcome ek = : : : = en = 0 given that the subgame

Gk(e�1; : : : ; e�

l�1; ~el; : : : ; ~ek) with ~ek < e�khas as the unique subgame perfect equilibrium

outcome ek+1 = : : : = en = 0. Since ek+1 = : : : = en = 0 is the unique subgame perfect

equilibrium outcome of Gk(e�1; : : : ; e�

l�1; ~el; : : : ; ~ek) when ~ek < e�kit follows that an e�ort

level ek < e�kyields agent k a payo� u�

k� ck(e

�1; : : : ; e�

l�1; ~el; : : : ; ~ek�1; ek). This expression

has a maximum u�kfor ek = 0. Now consider ek � e�

k. If this e�ort choice results in a �nal

23

output y(e) < yk�1 then agent k receives a payment u�kwhich yields him a payo� of less

than u�kdue to the positive e�ort costs. If the e�ort choice ek � e�

kleads to a �nal output

y(e) 2 [yk�1; yk) then ek+1 = : : : = en = 0 (lemma A.3). It follows from lemma A.2 that

agent k's payo� is less than u�k. If agent k chooses an ek > e�

ksuch that a �nal output

y(e) > yk results then agent k receives a payo� y�k� ck(e

�1; : : : ; e�

l�1; ~el; : : : ; ~ek�1; ek). But

since ck(e�1; : : : ; e�

l�1; ~el; : : : ; ~ek�1; ek) > c�k, this is also smaller than u�

k. For agent k it is

therefore optimal to choose ek = 0, which leads to the outcome ek = : : : = en = 0.

Q.E.D.

From lemma A.4 it follows directly that as soon as shirking occurs, then subsequent

agents will choose an e�ort level of zero. The following lemma shows that such behavior

of agents prevent a potential shirker from actually shirking.

Lemma A.5 The subgame Gk(e�1; : : : ; e�

k) has the unique subgame perfect equilibrium

outcome (ek+1; : : : ; en) = (e�k+1

; : : : ; e�n).

Proof: We prove by induction. Consider the subgame Gn�1(e�1; : : : ; e�

n�1) then agent

n's payo� is y(e) �P

k 6=n u�k� cn(e) for en < e�

n. An e�ort level en � e�

nyields the

payo� y�n� cn(e), which has the maximum u�

nfor en = e�

n. The equilibrium outcome of

Gn�1(e�1; : : : ; e�

n�1) is therefore en = e�n.

Now consider the subgame Gk�1(e�1; : : : ; e�

k�1) given that the unique subgame perfect

equilibrium outcome of the subgame Gk(e�1; : : : ; e�

k) is (ek; : : : ; en) = (e�

k; : : : ; e�

n). This

implies that the e�ort level ek = e�kyields agent k a payo� y�

k� ck(e

�1; : : : ; e�

k) = u�

k.

Since agent k's payment is at most y�kan e�ort level ek > e�

kyields a payo� y�

k�

cn(e�1; : : : ; e�

k�1; ek) < u�k. An action ek < e�

kleads to the subgame Gk(e

�1; : : : ; e�

k�1; ek)

which has the unique subgame perfect equilibrium outcome ek+1 = : : : = en = 0, with

y(e) 2 [yk�1; yk). The associated payo� is therefore y(e)�P

l6=k u�l�ck(e

�1; : : : ; e�

k�1; ek) <

u�k. The e�ort level which leads to the highest payo� for agent k is unique and equals e�

k.

The subgame Gk�1(e�1; : : : ; e�

k�1) has therefore the unique subgame perfect equilibrium

outcome (ek; : : : ; en) = (e�k; : : : ; e�

n).

Q.E.D.

24

We are now able to prove the main proposition.

Proposition A.1 The scheme I = (I1; : : : ; In) induces a game with n players for which

the e�ort levels e� is the unique subgame perfect equilibrium outcome.

Proof: Consider agent 1. If agent 1 chooses e1 < e�1 then due to lemma A.4 an output

y(e) < y�1 results and agent 1 receives the payo� y(e)�P

n

i=2 u�i � c1(e1) which according

to lemma A.1 is smaller than u�1. Choosing e1 = e�

1induces e�cient production (lemma

A.5) and yields agent 1 the payo� y�1 � c1(e�1) = u�1. Choosing e1 > e�1 leads either to an

output y(e) � y� or to an output y(e) > y�. In the former case agent 1 receives the payo�

y�1�c1(e1) < y�1�c1(e�1) = u�1. If a �nal output y(e) > y� results, then e2 = : : : = en = 0.

If not, then the last agent with a positive e�ort level, say agent l, could decrease his e�ort

by some � > 0 and so reduce his e�ort cost while maintaining the payment y�l . Since

e� maximizes the expression y(e) �P

ci(e1; : : : ; ei) and ck(e1; : : : ; ek�1; 0) = 0 it follows

that y(e�)� c1(e�1) > y(e�)�

Pci(e

�1; : : : ; e�i ) > y(e1; 0; : : : ; 0)� c1(e1). This implies that

y�1+(y(e1; 0; : : : ; 0)�y�)� c1(e1)) < y�� c1(e�1) = u�1. We conclude that the e�ort choice

e1 = e�1 yields agent 1 the maximum payo� u�1 and the game G has therefore the unique

subgame perfect equilibrium outcome (e1; : : : ; en) = (e�1; : : : ; e�n).

Q.E.D.

Proof of proposition 5:

Proof: Because the proof of proposition 5 is analogous to that of proposition 1 we will

only show that if agent k observes that agent k�1 is the �rst agent to shirk and if agent

k anticipates that following agents choose not to work if he chooses ek < e�k then agent

k will choose ek = 0.

Let agent k � 1 be the �rst agent to shirk. This means that ek�1 < e�k�1, while

(e1; : : : ; ek�2) = (e�1; : : : ; e�k�2). This implies that y(e1; : : : ; ek�1;0; : : : ; 0) < yk�1 Now

consider agent k. Given that all agent after him choose an e�ort of zero if he chooses

ek = 0 we obtain that ek = 0 yields a �nal output x < yk�1 + �. Agent k's payo� is

therefore u�k.

25

It is obvious that choosing ek > e�k does not make sense for agent k. Independent of

what subsequent agents choose, the choice will yield agent k a payment of at most y�k

while, due to assumption 1, his e�ort costs will exceed c�k. Agent k would be better o�

choosing ek = 0, which yields u�k.

If agent k chooses an ek 2 (0; e�k) then we have three cases to consider. De�ne

~y � y(e1; : : : ; ek; 0; : : : ;0). Since ek < e�k it follows that ~y < yk and that x < yk + � for

all realizations of ".

Case i) If ek is such that ~y � yk�1 + 2� then

Uk =

Z~y+�

~y��(x�

k�1Xi=1

y�i �nX

i=k+1

u�i )g(x)dx� ck(e1; : : : ek) (5)

=Z �

��(~y + "�

k�1Xi=1

y�i �nX

i=k+1

u�i )f(")d"� ck(e1; : : : ; ek) (6)

= ~y �k�1Xi=1

y�i �nX

i=k+1

u�i + E["]� ck(e1; : : : ; ek) < u�k; (7)

with g(x) the density function of the random variable x.

Case ii) If ek is such that ~y � yk�1 then x = ~y + " < yk�1 + � and agent k's utility is

u�k � ck(e1; : : : ; ek) < u�k.

Case iii) If ek is such that ~y 2 (yk�1; yk�1 + 2�) then there exists a t 2 (��;�) such

that ~y + t = yk�1 + �. Agent k's utility is then

Uk =

Z t

��u�kf(")d"+

Z �

t(y + "�

k�1Xi=1

y�i �nX

i=k+1

u�i )f (")d"� ck(e1; : : : ; ek) (8)

<

Z t

��u�kf(")d"+

Z �

t(yk + ��

k�1Xi=1

y�i �nX

i=k+1

u�i )f (")d" (9)

=Z t

��u�kf(")d"+

Z �

t(yk + ��

Xi 6=k

y�i +nX

i=k+1

c�i )f (")d" (10)

=

Z t

��u�kf(")d"+

Z �

t(yk + �� (y� � y�k) +

nXi=k+1

c�i )f(")d" (11)

=

Z t

��u�kf(")d"+

Z �

t(u�k + �� (y� � yk �

nXi=k

c�i ))f(")d" (12)

�

Z t

��u�kf(")d"+

Z �

tu�kf(")d" = u�k; (13)

where inequality (13) follows from assumption 2.

We therefore obtain that the action ek = 0 leads to the highest payo� for agent k.

26

Note that if all team members before agent k have produced at their e�cient e�ort

level then agent k is in fact indi�erent between performing the e�ort e�k or no e�ort at

all. Given the equilibrium behavior of subsequent team members both actions would

lead to a payo� of u�k. That in equilibrium agent k has to choose e�k becomes clear by

considering that if agent k chooses ek = 0 then it would have been strictly better for

agent k � 1 to overwork by an " > 0. This argument shows that the unique equilibrium

outcome is that agents choose their e�cient e�ort levels.

Q.E.D.

Appendix B

If assumption 1 does not hold then penalties need to be harsher in order to induce �rst

best behavior. In this case consider the following payment scheme I0 for agent k < n:

I 0k(y) =

8><>:

y�k � y� + y if yk�1 � y < yk

y�k otherwise

and for agent n

I0n(y) =

8><>:

y�n � y� + y if y > yn�1

y�n otherwise

Scheme I0 di�ers from scheme I in so far that it if agent k shirks it also gives team

members after agent k the payment y�k instead of u�k. Punishments are therefore higher

and the scheme will not be robust to sabotage.

Lemma B.1 For all ek 6= e�k it holds that

y�k � y� + y(e�1; : : : ; e�k�1; ek; 0; : : : ;0)� ck(e

�1; : : : ; e

�k�1; ek) < u�k:

Proof: If not then this implies that there exists a ek 6= e�k such that

y(e�1; : : : ; e�k�1; ek; 0; : : : ; 0)� ck(e

�1; : : : ; e

�k�1; ek) � y� � ck(e

�1; : : : ; e

�k):

27

But then

y(e�1; : : : ; e�k�1; ek; 0; : : : ; 0)� ck(e

�1; : : : ; e�k�1; ek) (14)

�k�1Xl=1

cl(e�1; : : : ; e

�l )�

nXl=k+1

cl(e�1; : : : ; e

�k; ek; 0; : : : ; 0) (15)

= y(e�1; : : : ; e�k�1; ek;0; : : : ; 0)� ck(e

�1; : : : ; e

�k�1; ek)�

k�1Xl=1

cl(e�1; : : : ; e

�l ) (16)

� y� � ck(e�1; : : : ; e

�k)�

k�1Xl=1

cl(e�1; : : : ; e

�l ) (17)

> y� �nX

l=1

cl(e�1; : : : ; e�l ): (18)

This contradicts the fact that e� maximizes y(e)�Pn

l=1 cl(e1; : : : ; el).

Q.E.D.

Proposition B.1 The scheme I 0 = (I 01; : : : ; I0n) induces a game with n players for which

the e�ort levels e� is the unique equilibrium outcome.

Proof: The scheme I induces a sequential game G(I0) with n agent in which agent k has

the payo� function uk(e) = I 0(y(e))�ck(e1; : : : ; ek). De�ne the subgame G0(e1; : : : ; ek) as

the subgame starting from agent k+1 in which the �rst k agents have chosen (e1; : : : ; ek).

Say that agent 1 chooses e1 < e�1. Then the subgame G01(e1) has as the unique

subgame perfect equilibrium outcome e2 = : : : = en = 0. By applying lemma B.1 it

follows that agent 1's payo� from e1 < e�1 is strictly less than u�1. Also for e1 > e�1 it

is immediate that agent 1's payo� is strictly less than u�1. We conclude that agent 1's

payo� is strictly less than u�1 for any e1 6= e�1.

Consider the subgame G01(e�1), because y(e�

1;0; : : : ; 0) = y1 and because the unique

subgame perfect equilibrium outcome in the subgame G02(e

�1; e2) with e2 < e�2 is e3 =

: : : = en = 0, also agent 2's payo� is strictly less than u�2 for any e2 6= e�2 in the subgame

G01(e�1).

By continuing this argument one may show that for every agent k with k < n the

action ek 6= e�k yields a payo� strictly less than u�k in the subgameG0k�1(e

�1; : : : ; e

�k�1). Now

consider the subgame G0n�1(e

�1; : : : ; e

�n�1). This game is a one-person decision problem.

28

For any en � 0 agent n's payo� is y�n � y� + y � cn(e�1; : : : ; e�n�1; en). Maximizing this

expression yields as its maximum u�n at en = e�n. The e�ort level e�n is the unique best

choice of agent n.

Given that e�n is the unique best response to (e�1; : : : ; e�n�1) and that in the sub-

game G0n�2(e

�1; : : : ; e�n�2) agent n� 1 receives strictly less than u�n�1 for all en�1 6= e�n�1.

The unique subgame perfect outcome of the subgame G0n�2(e

�1; : : : ; e

�n�2) is (en�1; en) =

(e�n�1; e�n). Continuing this argument shows that (e�k; : : : ; e

�n) is the unique subgame per-

fect equilibrium outcome of the subgame G0k�1(e

�1; : : : ; e

�k�1). Consequently (e�1; : : : ; e

�n)

is the unique subgame perfect equilibrium outcome of the game G(I 0).

Q.E.D.

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