Optimal Information Asymmetry and Team Performance Measurement
Naomi R. Rothenberg
Naval Postgraduate School
August 2010
ABSTRACT: This paper studies the effect of agents’ access to pre-decision information,
depending on whether only group performance is rewarded or individual performance is
rewarded. Pre-decision information only affects agents’ incentives, which in turn depend on the
type of performance evaluation system. With public information and group performance
measures, the principal prefers a perfect signal if the high productivity state is more informative,
and otherwise prefers no information. With group measures, only team success is rewarded, but
cooperation is more costly in the higher productive states when the high state is more
informative, so the principal prefers to be able to distinguish the states. With individual
performance measures, the principal can replicate the group performance measure or induce
competition, and in both cases prefers no information. A private signal exacerbates the control
problem, and generally, with either performance evaluation system, the principal prefers
uninformed agents. However, if agents can become at least slightly informed, then with
individual performance measures the principal is indifferent to the level of information conveyed
in the signal. Individual performance measures better balance the incentive effects of private
information. With group measures, by definition the principal is restricted to a cooperative
performance based pay and always prefers the lowest level of information.
Keywords: Pre-decision information, teams, performance measurement, principal-agent
JEL codes: M41, D82, M52, J33
1. Introduction
An important consideration in management accounting is how to choose the quality of
information managers have about productivity. While better information can improve decisionmaking, in a decentralized organization managers with private access to these information
systems have better information about their performance measure, which exacerbates an already
existent control problem. Even when the owner also observes this information, from a strict
incentive point of view having more information about the productivity state may not always be
beneficial. Further, incentive effects are complicated when managers work in a team
environment and their performance is evaluated jointly. This paper considers the incentive
effects of information used by managers working in a correlated environment and whose
performance is measured jointly, either as a group or individually.
There is a long line of literature that has studied the effect of private information in a setting
with a single worker, with much of the focus on incentives and communication. This paper
considers the effect of private pre-decision information on teams, because team production
creates more complex performance measurement issues than individual production. With teams,
managers are either rewarded for success when coworkers do not perform well, (i.e. relative
performance evaluation or RPE), or are rewarded for success when coworkers also perform well,
(i.e. joint performance evaluation or JPE) and the reward system affects incentives differently.
With joint performance evaluation workers may cooperate more but free-riding can increase
(Alchian and Demsetz, 1972; and Holmström, 1982). Relative performance evaluation promotes
competition, although collusion can be a problem (i.e. Mookherjee, 1984), and can also
discourage cooperation (see Lazear 1995 for a discussion). While previous work focused on the
advantages and disadvantages of the different types of performance measures, this paper
1
considers how incentives involving either cooperation or competition stemming from the
performance evaluation system affect the optimal amount of productivity information.
I use a principal-multi-agent model, where two risk-neutral agents with limited liability work
on a team. A team in this setting means agents’ productive environment is correlated and each
agent’s performance optimally depends on the other agent’s performance. Each agent’s
productive output depends on both his own effort and a random state of nature reflecting the
productivity state that is common to both agents. An information system provides a signal about
the productivity state after contracting but before agents make their productive choices. In both
possible productivity states, the principal wishes the agents to provide effort, so as to increase the
chance of high productive output. Thus the information provided by the signal does not help
make productive decisions, but rather only affects the incentives provided by the principal.
Agents’ incentives to work are affected by the performance measurement system in place and
whether agents have private access to pre-decision productivity information or whether the
principal also observes the signal. With a group (i.e. aggregate) performance measure, agents’
individual output is indistinguishable, and only joint performance evaluation (JPE) is possible,
where agents are rewarded when the entire team is successful. With individual performance
measures, the principal can not only distinguish between agents’ output but can use the output of
both agents’ output for contracting and the principal can optimally use relative performance pay
(i.e. RPE), where an agent’s pay depends not only on his own productive output, but the his team
member’s productive output as well.
When the signal is public and the high productivity state is more informative than the low
state, then with group performance measures, the principal prefers a perfect signal but with
individual performance measures, the principal prefers an uninformative signal. With group
2
measures, because agents’ output is indistinguishable, the principal is restricted to using JPE, the
cooperative solution, and with the high state more informative, the expected payment in the high
productivity state is higher and the principal prefers to be able to distinguish the state. However,
with individual performance measures, the principal has more options and can either use the
competitive solution, RPE, or can replicate the group solution with JPE. So when the high state
is more informative, with individual measures, the principal can switch to the lower cost RPE.
With the low state more informative, the principal uses JPE with both group and individual
performance measurement systems and prefers an uninformative signal.
When the agents privately observe the productivity signal, then the control problem is
exacerbated, but the effects differ depending on the performance measure available. With group
performance measures, only JPE is optimal and with the agents privately informed, the principal
prefers that the signal is uninformative. However, with individual performance measures, JPE,
RPE, and a combination of JPE and RPE, where an agent is paid when his output is high, but the
amount depends on the other agent’s output, are all possible optimal contracts. If the signal can
be completely uninformative, then with individual measures, the principal prefers no information
and either JPE or RPE is optimal, depending on which state is more informative. However, if the
signal is bounded below and agents can become at least slightly informed (that is a completely
uninformative signal is ruled out), then the combination of JPE-RPE contract is optimal, and the
principal and agents are indifferent to the amount of information conveyed in the signal. With the
combination JPE-RPE contract, expected profits are independent of the level of information in
the signal. However, with group performance measures, the principal prefers the agents are the
least informed as possible, but with individual measures, because of the richer set of performance
measures available, the payments are independent of the amount of information in the signal.
3
These results demonstrate that the incentive effects of the information conveyed in the
productivity signal can be mitigated depending on the performance evaluation system.
This paper is related to work that studies settings where the agent has private pre-decision
information, starting with Christensen (1981), who shows that the principal is not always better
off with an informed agent, which was followed by Baiman and Evans (1983) and Penno (1984),
who study the value of communication when the agent has private pre-decision information. The
focus in these models is on communication, and in this paper, I do not allow for communication
between the agents and the principals, but rather focus on the impact of the agent’s information
on incentives through the performance measurement system. However, Penno (1984) also
focuses on restricting information to the agent and shows that the principal may prefer to
partition the information system, so as to provide some, but not perfect information, to the agent.
More recently, Rajan and Sauoma (2006), who also allow for communication, study the effect of
the amount of information asymmetry and show that the principal prefers either a perfectly
informed or perfectly uninformed agent. Baiman and Sivaramakrishnan (1991) use an
unrestricted performance measure and Bushman et al (2000) use an imperfect performance
measure of firm value and both also restrict communication from a privately informed agent and
demonstrate the value of agent’s information. This paper considers a team setting, where agents
can have asymmetric access to information, and focuses on the effect of the performance
measure on whether agents should be restricted from productivity information.
Other reasons to restrict information to agents include preventing agents from colluding as
demonstrated by Feltham and Hoffmann (2008), and motivational concerns of leaders sharing
their information with a subordinate in non-contracting setting as shown by Blanes i Vidal, et al
(2007) and Komai, et al (2007) extend Hermalin (1998). While I also find that sometimes
4
information should be restricted to agents, I find that it depends the performance measurement
system.
2. Model
The model, adopted from Che and Woo (2002), consists of a risk neutral principal and two
risk neutral agents, 1 and 2, who have limited liability. Each agent contributes unobservable,
costly effort to a team project. The agent’s effort can be high or low, e ∈ {0, 1}, and has a cost
ce, where c > 0. Agent i’ s individual output, xi is also binary and can be good (xi = H) or bad (xi
= L). The probability distribution of xi depends on agent i’s choice of effort as well as a common
shock, θj, j ∈ {L, H}, with each state equally likely.
The probability distribution over x is assumed to display first-order stochastic dominance
in both the agent’s effort and in the signals of productivity. Notationally, 1 > Pr( x Hi / ei = 1, θH) =
pH > Pr( x Hi /ei = 0, θH) = qH and 1> Pr( x Hi /ei = 1, θL) = pL > Pr( x Hi /ei = 0, θL) = qL > 0. Firstorder stochastic dominance in the signals of productivity implies that θH is a more productive
state than θL, or pH ≥ pL and qH ≥ qL. I also assume that the marginal productivity in θH is more
than the marginal productivity in θL, or (pH - qH) > (pL > qL) > 0. The ex-ante probability that the
agent will produce the high output if he chooses high effort is Pr( x Hi /ei = 1) = .5pH + .5 pL and if
he chooses low effort is Pr( x Hi /ei = 0) = .5qH + .5 qL.
There is a signal about productivity (i.e. about θ) which is observed after contracting but
before agents choose their level of effort. The signal is denoted by !ˆ , and its accuracy is
captured by the parameter, s. Generally two cases are considered: one where the signal can be
completely uninformative, and one where the signal can be somewhat informative. Either s ∈ [.5,
5
1], where s = .5 means no information about θ and s = 1 means perfect information about θ, or s
∈ [ s , 1], where s > .5, which means that there is some information conveyed in the signal. If the
low productivity state occurs, or θ L, then with probability s the low signal, !ˆL will be observed,
and if the high productivity state occurs, or θH, then with probability s the high signal, !ˆH will be
observed.
The timeline is as follows. With the performance measurement system in place, the principal
chooses s, the level of information that agents can access that informs them of the productivity
state. Then the principal offers each agent a contract, which each agent may reject in which case
the game ends, or the agents accept the contract and join the firm. Nature determines the value of
θ, the signal !ˆ is observed and agents simultaneously choose their effort. Finally the
performance evaluation system produces a performance measure of the team’s production.
The control system either produces an aggregate (i.e. group) performance measure of the
team project, x1 + x2 (where individual contributions cannot be distinguished) or it produces an
individual measure of an agent’s performance, xi. In either case, the output of the control system
is available for contracting. In the case of group performance measures, output can be high,
medium, or low, where a high team output means that x1 = x2 = H, medium output means either
x1 = H and x2 = L or x1 = L and x2 = H and low team output is when x1 = x2 = L. In the case of
individual performance measures, the principal can distinguish between and use both agents’
output in contracting with a particular agent. I also assume that the principal cannot commit to
the use of the agent’s communication of !ˆ .
The principal uses the performance measure produced by the control system on which to base
payments to the agents. Let wi i
x ,x k
denote the payment to agent i if the principal can use individual
6
performance measures and can base payments on both agent i’s output and agent k’s output (for i
! k). With individual performance measures, there are four possible realizations. Let w Ii ≡
i
i
i
i
( wHH
, wHL
, wLH
, wLL
) denote the vector of payments to agent i. If only group measures are
available, then wGi ≡ ( wHi , wMi , wLi ). Note that payments with group performance measures are
i
i
the same as with individual performance except that wHL
= wLH
. Given both agents working hard,
the chance that the output of both agents is high is Pr(xHH/ei = 1, ek= 1) = .5(pH 2 + pL2), the
chance that the output of one agent is high and the other is low is Pr(xHL/ei = 1, ek= 1) = Pr(xLH/ei
= 1, ek= 1) = .5[pH(1 - pH)+ pL(1 - pL)], and the chance that both agents’ output is low is Pr(xLL/ei
= 1, ek= 1) = .5[(1 - pH)2 + (1 - pL)2].
Throughout the paper, I assume that work is sufficiently valued by the principal, so that the
principal always wishes both agents to work hard. The principal’s problem is to minimize
expected payments to the agents, subject to the agents’ participation constraints that ensure the
agent will at least earn his reservation wage, which I normalize to zero, and incentive
compatibility constraints that ensure the agent prefers high effort to low effort. The principal’s
problem is as follows:
Program P
Min
! Pr(x j / ei = 1,ek = 1)wi (x j )
i
w !0
j
i
k
i
s. t. !j Pr(x j / e = 1,e = 1)w (x j ) " c # 0
(IR)
! Pr(x j / ei = 1,e k = 1,"ˆn )wi (x j ) # c $
j
! Pr(x j / ei = 0,e k = 1,"ˆn )wi (x j ), n = L, H
j
(IC)
7
Note that the program solved by the principal with group measures is the same as with individual
i
i
measures, but with the additional constraint that wHL
= wLH
. In the following section, I analyze
the model, varying the performance measure available.
3. Analysis
3.1 Public Signal
In this section, the principal and the agents observe the signal about productivity. The
purpose of studying the principal’s problem in the absence of private information about
productivity is to ascertain the optimal amount of information with no private information. This
section also serves as a benchmark for the subsequent sections determine the effect of the agents’
private information about productivity to the principal. I solve the principal’s problem with
individual and with group performance measures given the principal’s observation of θ and
determine the optimal level of s. Finally, I compare the optimal amount of information under
both performance measurement systems.
Denote the vector of payments to agent i dependent on !ˆ as wi( !ˆn ), n = L, H. The
principal’s problem with observation of the productivity signal is as follows:
ˆ
Program P!
Min
! ! Pr(x j / ei = 1,e k = 1,"ˆn )wi ("ˆn , x j )
i
w !0
n
j
s. t. ! ! Pr(x j / ei = 1,ek = 1,"ˆn )wi ("ˆn , x j ) # c $ 0
n
(IRT)
j
! Pr(x j / ei = 1,e k = 1,"ˆn )wi ("ˆn , x j ) # c $
j
! Pr(x j / ei = 0,e k = 1,"ˆn )wi ("ˆn , x j ), n = L, H
(ICT)
j
With group measures, only the agents’ aggregate output or the team’s output is tracked. If
both agents produce the high output, then the performance measurement system produces a high
8
signal, if one agent produces the high output and the other agent produces the low output, then
the system produces a medium signal, and if both agents produce the low output, the system
produces a low signal. The principal’s problem with group performance measures is to solve
ˆ
Program P! , with payments wG (!ˆn ) = [ wH (!ˆn ) , wM (!ˆn ) , wL (!ˆn ) ], n = L, H.
With group performance measures, the solution involves binding incentive constraints. The
optimal contract involves joint performance evaluation (JPE) where each agent is paid only if
both produce the high outcome. Payments are w( !ˆH ) = (
(
c
, 0, 0) and w( !ˆL ) =
spH x + (1 ! s)pL y
c
, 0, 0), where x = (pH – qH) and y = (pL – qL). The expected cost is [spH2 +(1 –
spL y + (1 ! s)pH x
s)pL2]wH( !ˆH ) + [spL2 +(1 – s)pH2]wH( !ˆL ).With a team setting, tacit collusion among agents may
be a problem. In the agents’ subgames, where both agents choose effort simultaneously given the
contract payments, because the incentive constraint is binding, one agent will be indifferent
between working hard and shirking, given the other agent is working hard. A second equilibrium
is when both agents choose low effort. However, the equilibrium with both agents working
dominates the shirking equilibrium. Thus, tacit collusion among the agents is not a problem in
this setting. 1
With group measures, the principal’s only incentive compatible option is to pay the risk
neutral agents when the entire team is successful (hereafter referred to as joint performance
evaluation, or JPE). The principal can use the information of the correlated environment to
determine payments, but because the principal cannot discern each agent’s individual
performance, he cannot exploit the correlation completely and use relative performance
1
See Demski, et al (1988) for analysis of the tacit collusion problem with multi-agents who are risk neutral and have limited
liability.
9
evaluation, when an agent is rewarded for his high output but only when the other agent has a
low output. With group measures it is impossible for the principal to know which agent produced
the low outcome.
Because the principal also observes the productivity signal, the payments are tailored to the
signal. That is, when the principal and the agents observe the high (low) productivity signal, the
payment is based inversely on the marginal productivity of the high (low) state. So, the payment
based on the high productivity signal is lower than the payment based on the low productivity
signal. This is because with higher marginal productivity, the control problem is not as severe
and it is easier to motivate the agents. Higher marginal productivity in a productivity state is not
as strong a condition as informativeness, and ensures that the states differ, which is when
information about productivity will be meaningful.
Further, because the principal and agents all publicly observe the signal, not only is the
payment dependent on s, but the probability of the agents’ output is also dependent on s.
Determining the optimal level of s involves examining the expected payments. Whether or not
the expected payment in the high state is more than expected payment in the low state depends
on whether or not the high state is more informative about the agents’ actions than the low state.
If the high state is more informative than the low state, then the expected payment given the high
signal will be more than the expected payment given the low signal. This is just an application of
informativeness principle in Holmstrom (1979).
Also, if the high state is more informative than the low state, then the expected payment for
the high (low) productivity signal is increasing (decreasing) in s. However, with s = .5, there is
no difference in the expected payments. Therefore, whether or not the principal prefers the signal
to be completely uninformative or perfectly informative depends on which state is more
10
informative about agents’ actions, as the following lemma states.
Lemma 1: If the principal and the agents observe the productivity signal, then with group
performance measures, the principal prefers s* = 1 if
qL
q
< H , otherwise the principal prefers
pL
pH
s* = .5.
Proof: All proofs are in the Appendix.
The optimal amount of pre-decision information with group performance measures depends
on how informative x is about agents’ effort in the productivity state, that is whether x more
informative in the low state or the high state. With x more informative in the high productivity
state, then the principal prefers a perfect signal about productivity. The higher expected payment
is based on the state with the higher marginal productivity, so it pays for the principal to be able
to perfectly distinguish between the states. However, if x is more informative in the low
productive state, then the expected payment based on the low productivity increases as s
increases, but because of the lower marginal productivity in that state, it increases more than the
decrease in the expected payment tied to the high productive state. In that case, the principal
prefers no information and the payments are the same for both signals.
Another way to see the reason that sometimes the principal prefers a perfect signal and
sometimes a completely uninformed signal is that with perfect information about the productivity
state, the payments to the agent are tailored to the state, while with no information, the payments
are based inversely on the average of the marginal productivity of both states. So whether or not
the principal is better or worse off with a perfect productivity signal involves comparing the
average marginal productivity to the actual marginal productivity, state-by-state. In the high
productivity state, having perfect information is advantageous to the principal because the
average marginal productivity is lower than the marginal productivity in the high state, making
11
the expected payment more with no information. However, in the low productivity state, the
opposite occurs and it is better for the principal to not be informed. Which of these effects
dominates depends on which state is more informative. If output is more informative in the high
productivity state, then the advantage with perfect information outweighs the cost and vice versa.
Next I turn to a performance measurement system that measures each agent’s individual
output. With a correlated environment, the principal finds it beneficial to base payments on both
agents’ output. With individual performance measures, the principal’s problem is to solve
ˆ
Program P! but with the full set of payments, or w I (!ˆn ) = [ wHH (!ˆn ) , wHL (!ˆn ) , wLH (!ˆn ) ,
wLL (!ˆn ) ], n = L, H. Like the solution with group performance measures, the solution also
involves binding incentive constraints. The optimal contract involves either joint performance
evaluation (JPE), which is when agents are paid a bonus when both agents are successful, or it
involves relative performance evaluation (RPE), which means one agent is paid a bonus when he
obtains the high output and the other agent produces the low output. With JPE, the solution is the
same as with group performance measures, but with RPE, the solution is as follows. Payments
are wI( !ˆH ) = (0,
c
c
, 0, 0) and wI( !ˆL ) = (
,
s(1 ! pH )x + (1 ! s)(1 ! pL )y
s(1 ! pL )y + (1 ! s)(1 ! pH )x
0, 0) and the expected cost is [spH(1 – pH)+(1 – s)pL(1 - pL)]wHL( !ˆH ) + [spL(1 - pL) +(1 – s)pH(1 –
pH)]wHL( !ˆL ).
Checking the agents’ subgames, the problem with RPE is that the (work, work) equilibrium
is Pareto-dominated by the (work, shirk) and (shirk, work) equilibria. The agent who works
when the other agent shirks is better off than if the other agent works, and the latter agent is
indifferent between working and shirking. One way that the team equilibrium of (work, work)
can be implemented with RPE as suggested by Ishiguro (2002) is to add a small amount, say ε, to
12
the payment wHL, making the IC constraint slack. Then, the RPE contract is qualitatively optimal
in that it is the least expected cost than any other contract; however, there are an infinite number
of RPE contracts, so there is not a single optimal contract. 2
By comparing the expected payments with JPE to the RPE, if s = 1 then there is no difference
between expected payments with JPE or with RPE, but if .5 ≤ s < 1, then JPE is optimal when
qL
q
> H , otherwise RPE is optimal. With at least some error in the signal, the payment
pL
pH
depends inversely on the marginal productivity of both states, with the weights dependent on the
amount of error in the signal. In the high productivity state, even though with both RPE and JPE
the same amount of weight is put on the wrong state, the payment is higher with JPE. The entire
team is more likely to produce the high output in the high state, so the cost is higher than with
RPE, when it is not as likely for one agent to have a high output and the other agent the low
output. However, in the low productivity state, RPE is more costly than JPE and which effect
dominates depends on which state is more informative.
It turns out that the principal’s preference for the amount of error in the productivity signal
does not depend on whether JPE or RPE is optimal. The following Lemma describes the
principal’s preference for the level of s with individual performance measures.
Lemma 2: If the principal and the agents observe the productivity signal, then with individual
performance measures, the principal always prefers s* = .5.
If
qL
q
> H then the optimal contract is JPE, which is the same as with group performance
pL
pH
measures, and from Lemma 1, the optimal s is s* = .5. If
qL
q
< H then RPE is optimal, and the
pL
pH
2
Here and in the rest of the paper, I will assume that ε is added to the payment where needed to make (work, work) and Paretodominating equilibrium, but ε will be suppressed.
13
principal prefers s* = .5. With RPE, if the high productivity state occurs then no information is
preferred to any informative signal, however the opposite is true in the low productivity state.
Because RPE is only optimal when the high productivity state is more informative, then the
effect with no information in the high state dominates the effect in the low state.
With individual performance measures, the principal has more payment options than with
group performance measures because the agents’ output can be distinguished. Thus the principal
never prefers a perfect productivity signal with individual measures. The following proposition
compares the optimal information with group performance measures to the optimal information
with individual performance measures.
Proposition 1: If the principal is informed about the productivity signal, then if
qL
q
< H , with
pL
pH
group performance measures, the principal prefers a perfect productivity signal (i.e. s* = 1), but
with individual performance measures, the principal prefers that the signal be uninformative (i.e.
s* = .5). If
qL
q
> H , then there is no difference between individual and group performance
pL
pH
measures and the principal prefers an uninformative signal (i.e. s* = .5).
When the productivity signal is public, then either the principal prefers a perfect signal or a
completely uninformative signal, but it depends on the performance measurement system and in
which state the agents’ output is most informative. With perfect information, the principal can
distinguish between the states, and with JPE must pay a higher payment in the more productive
state. Therefore, having no information is only preferred if this effect dominates, which occurs
when the high productivity state is more informative about the agents’ actions. With group
performance measures, the principal is restricted to only using JPE, so either perfect information
or no information is preferred. However, with individual performance measures, the principal
14
has more freedom in paying the agents, and with RPE the principal makes the higher payment
compared to JPE in lower state, so if the high state is more informative, then RPE is preferred.
Regardless of whether the principal uses RPE or JPE with individual performance measures, an
uninformative signal is preferred. In the next section, I analyze the principal’s preferences when
the agents are privately informed.
3.2 Private Information
In this section agents privately observe the signal about productivity. With private
information, agents are better informed about the performance measure, which exacerbates the
control problem. In the setting here, there are no productive benefits from observing the signal
because agents are motivated to work hard regardless of the productivity state. It may be
tempting to conclude that the agent should always be uninformed, that is the signal should
always be uninformative. However, in the previous section, when the principal also observed the
productivity signal, the control problem was not exacerbated by agents’ private information and
still the principal did not always prefer a perfect productivity signal.
I first consider group performance measures, and just as with an informed principal, only the
agents’ aggregate output or the team’s output is tracked. The principal’s problem is to solve
Program P, with payments wG = (wH, wM, wL). With two incentives constraints, the solution can
involve either one binding constraint or both binding constraints. If only one constraint is
binding, the binding constraint must be based on the low productivity signal, (IC - !ˆL ). To see
why, assume first that the constraint based on the high productivity signal, (IC - !ˆH ) is binding.
Then wG = (
c
, 0, 0). However, with these payments, the constraint based on the
spH x + (1! s) pL y
low productivity signal, (IC - !ˆL ) will not be satisfied, with any s > .5. The reason is that the
15
payment based on the higher productivity is too low to motivate the agent to work hard when the
low productivity signal is received. The lower the productivity, the more costly it is to motivate
the agent because his productivity has a smaller effect on the outcome compared to a state with
higher productivity. Further, both incentive constraints cannot be binding because the group or
aggregate performance measurement system does not provide a rich enough set of performance
measures. The optimal contract thus involves one binding constraint, the constraint based on the
low productivity signal, (IC - !ˆL ). Payments are wG = (
c
, 0, 0), and expected
spL y + (1! s) pH x
( pL2 + pH2 )c
cost to the principal is
. Also like when the principal is informed, tacit
spL y + (1! s) pH x
collusion in the agents’ subgames is not a problem in this setting. 3
Compared to when the principal also observes the productivity signal, the uninformed
principal cannot tailor the payments to the signal, but must make one payment dependent only on
the agents’ aggregate output. As long as the signal is informative (i.e. for any s > .5), an
inefficiency arises because the uninformed principal must make a higher payment when the
agents privately observe the high productivity signal. If the principal also observes the signal,
then he could lower the payment based on the marginal productivity in the high state.
By inspection, the expected payment with group performance measures and an uniformed
principal is increasing in s. Unlike when the principal is informed, only the payment depends on
the error in the signal, s. Because the principal does not observe the signal, the probability of
making the payment is independent of s. The following proposition formally states the results
when the principal is uninformed with group performance measures.
3
See Demski, et al (1988) for analysis of the tacit collusion problem with multi-agents who are risk neutral and have limited
liability.
16
Proposition 2: If the agents privately observe the productivity signal, then with group
performance measures, the principal always prefers the agents to be uninformed (i.e., s* = .5).
With group performance measures, because of the inefficiency with an uninformed principal, or
the overpayment when the high productivity state occurs, the potential benefit from having an
informed signal is lost. Even if s was bounded below by s > .5, with group measures and a
private productivity signal, the principal would always prefer that the agents be as least informed
as possible, that is s* = s . One caveat is that the principal prefers uninformed agents because the
signal does not help the agent make better productive decisions, but rather only exacerbates the
control problem. In other settings the improvement in productive choices may be beneficial
enough to override the costs of an exacerbated control problem (i.e., Penno 1984). With no
productive benefits, because of the performance measure’s restricted nature, there is no benefit to
an informative signal, and therefore the principal prefers that the agents be completely
uninformed.
Next I consider the performance measure system measures each agent’s individual output as
opposed to group performance measures, where an individual agents’ output is indistinguishable.
This serves to compare the optimal amount of productivity information with individual measures
of performance to the results with group performance measures.
The principal’s problem is to solve Program P with the payments wI ≡ (wHH, wHL, wLH, wLL).
With a richer set of possible performance measures, the principal’s problem is more complex
than with group performance measures. Even with one constraint binding with individual
measures it is possible for either JPE or RPE to be optimal, as when the principal was informed.
However, with privately informed agents, it is possible that both incentive constraints are
binding, and in that case a combination of JPE and RPE is optimal, where a payment is made
17
when an agent’s output is high, but the amount also depends on the other agent’s output. Unlike
the situation with group performance measures, the solution to the principal’s problem depends
on which state is more informative, as well as the level of s. The following lemma states how
these conditions affect the optimal contract.
Lemma 3: If the agents are privately informed, with individual performance measures if
qL
>
pL
qH
pL x
and 0.5 < s <
, then the optimal contract involves JPE and only the low
pL x + pH y
pH
productivity signal incentive constraint, IC - !ˆL , is binding, otherwise a combination of JPE and
RPE is optimal, and both incentive constraints are binding. If
qL
q
< H and 0.5 < s <
pL
pH
pH [x( pL x + pH y) + y( pH x + pL y)]
2( pH x ! pL y)( pL x + pH y)
pH [x( pL x + pH y) + y( pH x + pL y)]2 ! 4xy( pH x ! pL y)( pL x + pH y)
, then the optimal contract
2( pH x ! pL y)( pL x + pH y)
involves RPE and only the high productivity signal incentive constraint, IC - !ˆH , is binding,
otherwise a combination of JPE and RPE is optimal and both incentive constraints are binding.
Unlike group performance measures, with individual performance measures, there are many
possible optimal contracts. Possibilities include JPE, RPE, and a combination of both where an
agent is paid when his output is high, but the amount depends on the other agent’s output. If the
low productivity state is more informative than the high productivity state about the agents’
effort, or if
qL
q
> H , then JPE is optimal with the incentive constraint on the low productivity
pL
pH
signal binding as long as s is not too big, or as long as agents are not too informed. The solution
18
is, the same as with groups, w Ii = (
c
, 0, 0, 0). With s small enough, there are two
spL y + (1! s) pH x
options for payments, one involving JPE and one involving RPE, and if the high productivity
state is not as informative the low productivity state, then just like when the principal is
informed, it doesn’t pay to induce competition when agents are privately informed. However, for
higher values of s, (i.e.
pL x
≤ s ≤ 1) then a combination of JPE and RPE is optimal and
pL x + pH y
both incentive constraints are binding. That solution is w Ii =
(
c ! wHL [s(1! pL ) y + (1! s)(1! pH )x c( pH x ! pL y)
,
, 0, 0). An agent is paid when his output is
spL x + (1! s) pH y
xy( pH ! pL )
high, but the amount depends on the other agent’s output, where more is paid if the other agent’s
output is low compared to if the other agent’s output is high. Note that the payment wHL is
independent of s while the payment wHH depends not only on s but also depends on wHL.
If the likelihood ratios are such that
qL
q
< H , then the high productivity state is more
pL
pH
informative than the low productivity state about the agents’ effort, and again similar to when the
principal is informed, RPE is optimal with the incentive constraint on the high productivity
signal binding as long as s is not too big, or as long as agents are not too informed. The solution
with RPE is w Ii = (0,
c
, 0, 0). However, if s is big enough, then just
s(1! pH )x + (1! s)(1! pL ) y
like when the low state is more informative, the same combination of RPE and JPE is optimal.
Turning to the optimal s, when s is unbounded, or s ∈ (.5, 1) with JPE, there is no difference
between group performance measures and individual performance measures which means that
the optimal s is s*= .5. Further, with RPE, the principal also prefers the agents to be completely
19
uninformed. If agents’ information can range from completely uninformed to perfectly informed,
i.e. s ∈ (.5, 1), then the following proposition formally states the optimal amount of information
from the principal’s perspective.
Proposition 3: If s ∈ (.5, 1) and the agents are privately informed, with individual performance
measures, the principal always prefers the agents to be uninformed (i.e., s* = .5).
When the information system is unbounded between completely uninformed and perfectly
informed, then with both group and individual performance measures, the principal prefers the
agents remain completely uninformed. This is not surprising, given that the information does not
improve agents’ productive decisions but rather only exacerbates the control problem by giving
the agents more information about their performance measure.
However, suppose that there is a lower bound on s, denoted by s , such that s > .5, which
means that agents become slightly informed after contracting. This is the situation where s = .5 is
precluded and agents learn a little something about productivity after joining the firm. From
Lemma 3 the optimal contract differs depending on the likelihood ratios and the level of s. With
JPE or RPE the principal will prefer that agents be as uninformed as possible, or s* = s .
However, with the combination contract of RPE and JPE, then the principal is indifferent to the
amount of information conveyed in the productivity signal, as the following proposition states.
Proposition 4: If s ∈ ( s , 1) and the agents are privately informed, with individual performance
measures, if
qL
q
pL x
> H and if
< s < 1, the principal and the agents are indifferent to
pL x + pH y
pL
pH
the level of s, otherwise the principal prefers s* = s . If
qL
q
< H and if
pL
pH
20
pH [x( pL x + pH y) + y( pH x + pL y)]
2( pH x ! pL y)( pL x + pH y)
pH [x( pL x + pH y) + y( pH x + pL y)]2 ! 4xy( pH x ! pL y)( pL x + pH y)
< s < 1, the principal and the
2( pH x ! pL y)( pL x + pH y)
agents are indifferent to the level of s, otherwise the principal prefers s* = s .
From Lemma 3, the two cases when the principal is indifferent to the amount of information
the agents possess occur the optimal contract involves a combination of JPE and RPE. It turns
out that the payments are independent of s and because the principal is uninformed the
probability of a given output also does not depend on s, (i.e. the agents’ information). The
surprising result is that with individual performance measures, for any amount of productivity
information above a given level, the expected payments are independent of s and therefore do not
vary as s increases or as agents become more informed. So, the principal and the agents are
indifferent in terms of the control problem about how informed the agents become. This is stark
contrast to group performance measures, where the only possible performance measure is JPE
for all s and where the lowest possible s or the least amount of information is preferred by the
principal.
4. Conclusion
This paper studies the incentive effect of pre-decision information on team production and
performance evaluation. With public information, with group performance measures and only
joint performance evaluation, the principal prefers either a completely perfect signal or an
uninformative signal, depending on which productivity state is more informative. However, with
individual performance measures, the principal has more leeway in which to reward agents, and
prefers an uninformative signal. When the signal is private, with group measures, the information
21
only exacerbates the control problem and so the principal prefers the agents be as uninformed as
possible. However, with individual performance measures,
The results suggest that the design of an information system has incentive effects that differ
depending on the type of the performance measurement system and whether the information is
also monitored by the principal. Further, this paper has important implications for other
contracting settings, such as supply-chain settings.
22
Appendix
Proof of Lemma 1: With group performance measures and a public signal, the expected
payment is WG(s) =
∂ W G /∂ s =
[spH2 + (1 ! s)pL2 ]c [spL2 + (1 ! s)pH2 ]c
+
. With s continuous, the derivative is
spH x + (1 ! s)pL y spL y + (1 ! s)pH x
c( pH2 ! pL2 )[spH x + (1 ! s)pL y] ! ( pH x ! pL y)[spH2 + (1 ! s)pL2 ]c
+
[spH x + (1 ! s)pL y]2
c( pL2 ! pH2 )[spL y + (1 ! s)pH x] ! ( pL y ! pH x)[spL2 + (1 ! s)pH2 ]c
, which can be simplified as:
[spL y + (1 ! s)pH x]2
c( pH y ! pL x)pH pL
c( pL x ! pH y)pH pL
+
2
[spH x + (1 ! s)pL y]
[spL y + (1 ! s)pH x]2
(1)
The expected payment WG(s) is increasing in s if and only if (1) > 0, or qL/pL > qH/pH. WG(s) is
decreasing in s if and only if (1) < 0, or qL/pL < qH/pH.

Proof of Lemma 2: With individual performance measures and a public signal, the expected
payment with RPE is WI(s) =
[spH (1 ! pH ) + (1 ! s)pL (1 ! pL )]c
+
s(1 ! pH )x + (1 ! s)(1 ! pL )y
[spL (1 ! pL ) + (1 ! s)pH (1 ! pH )]c
, and with JPE the expected payment is as above with group
s(1 ! pL )y + (1 ! s)(1 ! pH )x
performance measures. The proof first establishes when JPE is optimal and when RPE is
optimal, and then the optimal s* is established under both contracts. In order to determine the
optimality of JPE and RPE, it is easier to compare expected payments state-by-state. First, given
[spH2 + (1 ! s)pL2 ]c [spH (1 ! pH ) + (1 ! s)pL (1 ! pL )]c
ˆ
the signal ! H , JPE is optimal if
<
, which is
spH x + (1 ! s)pL y
s(1 ! pH )x + (1 ! s)(1 ! pL )y
[spL2 + (1 ! s)pH2 ]c
ˆ
true if qL/pL > qH/pH. Given the signal ! L , JPE is optimal if
<
spL y + (1 ! s)pH x
[spL (1 ! pL ) + (1 ! s)pH (1 ! pH )]c
, which is also only true if qL/pL > qH/pH. If qL/pL > qH/pH, then
s(1 ! pL )y + (1 ! s)(1 ! pH )x
23
JPE is optimal and from Lemma 1, s* = .5, but if qL/pL < qH/pH, then RPE is optimal. To find s*
under RPE, the derivative is ∂WI/∂s =
c[ pH (1 ! pH ) ! pL (1 ! pL )][s(1 ! pH )x + (1 ! s)(1 ! pL )y]
[s(1 ! pH )x + (1 ! s)(1 ! pL )y]2
[(1 ! pH )x ! (1 ! pL )y][spH (1 ! pH ) + (1 ! s)pL (1 ! pL )]c
+
[s(1 ! pH )x + (1 ! s)(1 ! pL )y]2
c[ pL (1 ! pL ) ! pH (1 ! pH )][s(1 ! pL )y + (1 ! s)(1 ! pH )x]
[s(1 ! pL )y + (1 ! s)(1 ! pH )x]2
c[spL (1 ! pL ) + (1 ! s)pH (1 ! pH )][(1 ! pL )y ! (1 ! pH )x]
, which can be simplified as:
[s(1 ! pL )y + (1 ! s)(1 ! pH )x]2
c(1 ! pH )(1 ! pL )( pH y ! pL x)
c(1 ! pH )(1 ! pL )( pL x ! pH y)
+
2
[s(1 ! pH )x + (1 ! s)(1 ! pL )y]
[s(1 ! pL )y + (1 ! s)(1 ! pH )x]2
(2)
The expected payment WG(s) is increasing in s if and only if (2) > 0, which is true if qL/pL <
qH/pH.

Proof of Proposition 1: This follows from Lemmas 1 and 2.

Proof of Proposition 2: With group performance measures and a public signal, the expected
( pL2 + pH2 )c
payment is W (s) =
, and because pLy < pHx, it is easy to see that the expected
spL y + (1! s) pH x
G
payment is increasing in s.

Proof of Lemma 3: With individual performance measures and a public signal, the proof is split
into two main sections. In each section, all possible contracts are considered that satisfy the
constraints, and the optimal contract is determined by comparing the expected payments under
each possible contract.
1. Assume qL/pL > qH/pH. There are three possibilities to consider: the incentive constraint under
!ˆL is binding, both incentive constraints are binding, and the incentive constraint under !ˆH is
24
binding. If only IC- !ˆL is binding, then there are two possible contracts: JPE, with expected
payments of
( pL2 + pH2 )c
[ pL (1 ! pL ) + pH (1 ! pH )]c
or RPE with expected payments of
.
s(1 ! pL )y + (1 ! s)(1 ! pH )x
spL y + (1! s) pH x
Comparing expected payments, JPE is optimal if and only if .5 < s <
pL x
. Next, if both
pL x + pH y
incentives constraint are binding, then the solution is a combination of JPE and RPE, with
expected payments of
( pL2 + pH2 )c{xy( pH ! pL ) ! ( pH x ! pL y)[s(1! pH )x + (1! s)(1! pL ) y]}
+
[xy( pH ! pL )][spL x + (1! s) pH y]
[ pL (1 ! pL ) + pH (1 ! pH )]c( pH x ! pL y)
. Comparing expected payments between JPE and the
xy( pH ! pL )
combination of JPE and RPE, JPE will be preferred if and only .5 < s <
pL x
. If only ICpL x + pH y
!ˆH is binding, then the only possible contract that also satisfies IC- !ˆL is an RPE contract, with
expected payment of
[ pL (1 ! pL ) + pH (1 ! pH )]c
. It is sufficient to show that the combination
s(1 ! pH )x + (1 ! s)(1 ! pL )y
of JPE and RPE will be preferred to RPE based on IC- !ˆH for all possible s. Comparing expected
payments, the combination JPE and RPE contract is less costly than RPE based on IC- !ˆH if and
only if:
s2(pHx - pLy)(pLx + pHy) - spH[(pLx + pHy)x + (pHx - pLy)y] + pH2xy < 0
(3)
Setting the inequality in (3) equal to zero and solving the quadratic equation yields two roots,
!b ± b 2 ! 4ac
, with a = (pHx - pLy)(pLx + pHy), b = pH[(pLx + pHy)x + (pHx - pLy)y] and c =
2a
!b + b 2 ! 4ac
!b ! b 2 ! 4ac
pH xy. Checking the roots,
> 1, and
is < .5 if qL/pL > qH/pH,
2a
2a
2
which is true. Also, the function on the left-hand side of (3) is decreasing in s. To see why, take
25
the derivative of the left-hand side and setting it less than zero, which yields s <
pH x( pL x + pH y) + pH y( pH x ! pL y)
. This is clearly true because the right-hand side is greater
( pH x ! pL y)( pL x + pH y)
than one. Therefore, for all s ∈(.5, 1), the inequality in (3) holds and the combination of JPE and
RPE is preferred to RPE based on IC- !ˆH . Collecting the results, if .5 < s <
is the optimal contract, and if
pL x
, then JPE
pL x + pH y
pL x
< s ≤ 1, then the combination of JPE and RPE is the
pL x + pH y
optimal contract.
2. Assume qL/pL < qH/pH, which means that
pL x
< .5 and JPE can not be the optimal
pL x + pH y
contract. In addition, RPE based on a binding IC- !ˆL cannot be optimal because then IC- !ˆH will
not be satisfied. So, there are only two possible contracts: a combination of JPE and RPE with
both incentive constraints binding, and RPE with only IC- !ˆH binding. From above, if the
ineaulity in (3) holds, then the combination of JPE and RPE is the optimal contract. If qL/pL <
qH/pH, then
∈(
!b ! b 2 ! 4ac
is > .5. Therefore, the inequality in (3) holds only for s
2a
!b ! b 2 ! 4ac
, 1), otherwise RPE with only IC- !ˆH binding is the optimal contract.
2a

Proof of Proposition 3: If qL/pL > qH/pH, then with individual performance measures, from
Lemma 3 and Proposition 2, the principal will prefer JPE and s* = .5. If qL/pL < qH/pH, then for
lower levels of s, from Lemma 3, the principal prefers RPE with only IC- !ˆH binding. To see why
the principal prefers s* = .5, take the derivative of the expected payment, which is
26
![ pL (1 ! pL ) + pH (1 ! pH )]c[(1 ! pH )x ! (1 ! pL )y]
, which is positive (i.e., the expected payment
[s(1 ! pH )x + (1 ! s)(1 ! pL )y]2
is increasing in s), because with qL/pL < qH/pH, (1- pL)y > (1-pH)x.
Proof of Proposition 4: From Lemma 3 if qL/pL > qH/pH and s ∈(
qH/pH and s ∈(

pL x
, 1) or if qL/pL <
pL x + pH y
!b ! b 2 ! 4ac
, then the optimal contract is a combination of JPE and RPE with
2a
both incentive constraints binding. The expected payment is
( pL2 + pH2 )c{xy( pH ! pL ) ! ( pH x ! pL y)[s(1! pH )x + (1! s)(1! pL ) y]}
+
[xy( pH ! pL )][spL x + (1! s) pH y]
[ pL (1 ! pL ) + pH (1 ! pH )]c( pH x ! pL y)
, and is independent of s. The payment wHL is by
xy( pH ! pL )
inspection independent of s, but for wHH, it becomes evident after examining the derivative of the
payment. The derivative of the payment is
wHL
!c( pH x ! pL y)
[spH x + (1 ! s)pL y]2
[(1 ! pH )x ! (1 ! pL )y][spH x + (1 ! s)pL y] ! [s(1 ! pH )x + (1 ! s)(1 ! pL )y]( pH x + pL y)
. With
[spH x + (1 ! s)pL y]2
some algebra and substituting for wHL, then the derivative is zero.

27
References
Alchian, A. and H. Demsetz, 1972. “Production, Information Costs, and Economic
Organization”. American Economic Review 62(5): 777-795.
Baiman, S. and J. Evans (1983) “Pre-Decision Information and Participative Management
Control Systems”. Journal of Accounting Research 21(2): 371-395.
Baiman, S. and K. Sivaramakrishnan (1991) “The Value of Private Pre-Decision Information in a
Principal-Agent Context”. The Accounting Review 66(4): 747-766.
Bushman, R., R. Indjejikian, and M. Penno. (2000). “Private Predecision Information,
Performance Measure Congruity, and the Value of Delegation”. Contemporary Accounting
Research 17(4): 561-587.
Blanes i Vidal, J. and M. Möller (2007). “When Should Leaders Share Information with Their
Subordinates?” Journal of Economics & Management Strategy 16(2): 251-283.
Che, Y. and S. Woo (2001). “Optimal Incentives for Teams”. The American Economic Review
91(3): 525-541.
Christensen, J. (1981). “Communication in Agencies”. The Bell Journal of Economics
12(2):661-674.
Demski, J., D. Sappington and P. Spiller (1988). “Incentive Schemes with Multiple Agents and
Bankruptcy Constraints”. Journal of Economic Theory 44:156-167.
Feltham, G. and C. Hoffmann (2008) “Impact of Alternative Reporting Systems in Multi-Agent
Contracting”. Working Paper, University of British Columbia and University of Mannheim.
Hermalin, B. (1998). “Toward an Economic Theory of Leadership: Leading by Example”.
American Economic Review, 88 (5): 1188-1206.
Holmström, B. 1979. “Moral Hazard and Observability” The Bell Journal of Economics, 10 (1):
28
74-91.
Holmström, 1982. “Moral Hazard in Teams”. The Bell Journal of Economics, 13 (2): 324-340.
Ishiguro, S. (2002) “Optimal Incentives for Teams: Comment.” The American Economic Review
92 (5): 1711-1712.
Komai, M., M. Stegeman, and B. Hermalin (2007) “Leadership and Information”. American
Economic Review 97(3): 944-947.
Lazear, Edward P. 1995. Personnel Economics. The MIT Press.
Mookherjie, D. (1984). “Optimal Incentive Schemes with Many Agents”. Review of Economic
Studies 51(3): 433-446.
Penno, M. (1984). “Asymmetry of Pre-Decision Information and Managerial Accounting”.
Journal of Accounting Research 22(1): 177-191.
Rajan, M. and R. Sauoma (2006). “Optimal Information Asymmetry”. The Accounting Review
81 (3): 677–712.
29