Objective versus Subjective PerformanceIndicators in Incentive Contracts∗

(preliminary and incomplete)

Madhav V. Rajan†

Stefan Reichelstein‡

September 2006

∗We are grateful to Anil Arya, Judson Caskey, Rick Young, and workshop participants at Berke-ley, Michigan and Ohio State for helpful suggestions on an earlier draft of this paper.

†Graduate School of Business, Stanford University, mrajan@gsb.stanford.edu‡Graduate School of Business, Stanford University, reichelstein@stanford.edu

1 Introduction

Theoretical and empirical studies on managerial incentives have increasingly pointed

to the role of subjective, non-verifiable performance indicators. Murphy and Oyer

(2003) note that nearly two-thirds of the companies in their sample base bonuses, in

part, on subjective assessments of individual performance. In contrast to the explicit

incentive provisions derived from objective and verifiable information (e.g., audited

financial data), subjective information can be used to generate implicit incentives.1

In particular, earlier studies have considered long-term (infinite horizon) settings and

shown the existence of equilibria in which an agent will exert effort in each period

and the principal rewards favorable subjective information.2

Discretionary bonus pools are an alternative instrument for creating implicit in-

centives in a short-term (one-period) setting. Under a bonus pool arrangement, the

principal commits to pay out a certain amount which frequently varies with the real-

ization of available objective performance indicators. Bonus pools, especially in the

form of annual bonus plans, are widely used in practice. 3 However, the share of the

bonus pool given to a particular agent is determined by an implicit contract which

amounts to a “promise” contingent on the realization of the non-verifiable subjective

signal. In effect, the principal threatens to withhold parts of an agent’s bonus pool in

1Throughout this paper we suppose that objective information is verifiable and contractible. Incontrast, subjective information is viewed as non-verifiable for contracting purposes. As discussedfurther below, such information may be available to all parties or only to the principal.

2The papers by Bull (1987), Baker el al. (1994), Pearce and Stacchetti (1998) and Levin (2003)fall into this camp.

3Murphy and Oyer (2003) provide a wealth of descriptive information on such plans, using arelatively large proprietary dataset. More illustratively, consider the following excerpt from the WaltDisney Company’s description of its compensation plans for fiscal 2005: “.. The size of the pool offunds from which bonuses may be awarded to corporate executives other than those named in theSummary Compensation Table as well as business segment executives and other eligible employeeswill depend upon performance against financial goals and other measures established at the outset ofthe fiscal year. .. For the Company’s business segments, 50% of the bonus pool determination will bebased upon performance against segment-level financial goals, 30% will be based on other, segment-level performance factors and 20% will be based on the Company’s overall performance against theCompany performance goals described above under “Setting Company performance goals.” As inthe past, actual bonus awards to individual eligible employees will remain subject to the overalldiscretion of the Committee.”

1

case of unfavorable subjective information. To make such an arrangement credible,

the amount withheld must be paid to other agents either inside or outside the agency.

This paper examines the use of objective and subjective performance indicators

in optimal incentive contracts. Due to their non-verifiability, subjective signals en-

tail an additional agency cost which derives from the budget balancing constraint

inherent in bonus pools. In a one-agent setting, the principal must be prepared to

“burn money” by diverting parts of the bonus pool to a third party. With multi-

ple agents, bonus pools generally result in inefficient risk sharing since the budget

balancing constraint makes it necessary to tie an agent’s compensation to subjective

metrics reflecting on the performance of other agents.4 These additional costs of sub-

jective information make it essential to examine whether the standard predictions on

the value and relative use of multiple information signals, as obtained in settings of

verifiable information, do extend to subjective information.

A well-known result in agency theory, due to Holmstrom (1979), is that an ad-

ditional signal is valuable for contracting purposes if and only if it is incrementally

informative in a statistical sense, given the initial signal. We find that this charac-

terization generally does not carry over to subjective signals. Provided the objective

signal is sufficiently “strong” relative to its subjective counterpart, the principal will

be better off ignoring the subjective signal altogether. In other words, the budget bal-

ancing costs associated with subjective information may result in a “corner solution”

rather than merely a reduced weight on the subjective metric.5

4Several branches of the economics literature have studied the implications of budget balancingconstraints for incentives in organizations. Beginning with the design of public choice mechanisms(Green and Laffont, 1979), subsequent studies have examined the balancing constraints of partner-ships, e.g., Holmstrom (1982), Huddart and Liang, (2003, 2005) and Baliga and Sjostrom (2005).Budget balancing is generally not an issue in principal-agent models where the principal is theresidual claimant, unless the presence of subjective information makes it essential and costly forthe principal to commit to a balanced mechanism, i.e., a bonus pool, in order to make an implicitcontract credible.

5In deriving this result, we confine attention to incentive contracts based on the objective andsubjective signal realizations. MacLeod (2003) has shown that if the agent observes another signalthat is correlated with the subjective signal received by the principal, then it may be advantageousto employ “message sending” games in which the agent’s payoff is determined according to messagessent by both parties. We discuss the class of admissible mechanisms in more detail in Section 2

2

Incentive contracts observed in practice frequently appear to be less differentiated

than the optimal contracts suggested by agency theory. In particular, contracts used

in practice frequently seem to entail compression in the sense that a range of different

signal realizations is effectively pooled into the same outcome. In a one-agent setting

with subjective information only, MacLeod (2003) has established a compression re-

sult showing that an optimal bonus pool arrangement will pay the entire amount to

the agent unless the worst possible subjective outcome materializes.6 Thus, subjective

information and its associated contracting costs will lead to a dichotomous incentive

contract which treats most outcomes as “acceptable” but punishes extremely unfa-

vorable outcomes.

When the principal can rely on both subjective and objective signals to gauge

the agent’s unobservable effort, we obtain a “super-compression” result: for all but

the lowest possible objective outcome, the subjective metric is ignored and therefore

the entire bonus pool corresponding to the particular objective outcome is paid out

to the agent. Furthermore, and consistent with MacLeod (2003), the bonus pool

corresponding to the lowest possible objective outcome is also paid out in full to the

agent, except when the subjective metric assumes the worst possible outcome. As a

consequence, the principal will be forced to divert money to a third party only in the

“extreme” event that both the objective and the subjective metric reveal the most

unfavorable outcome.

Our finding that the subjective information is used in a “lexicographic fashion”

has several implications. First, the result is broadly consistent with the traditional

notion of conditional variance investigation.7 Accordingly, the principal seeks to

obtain additional (subjective) information only in case the primary objective metric

below.6This results holds under the familiar MLRP condition which guarantees that for ordinary ob-

jective signals the agent’s compensation would be monotonically increasing in the observed signal.7In management accounting textbooks, this is often related to the philosophy of management

by exception, which refers to the investigation and gathering of additional information on variancesor deviations from plan when, and only when, exceptional results are observed. See, for example,Maher et al. 2006, Chapter 16.

3

reveals an outcome below some given threshold level. Secondly, in the event that the

subjective metric is relevant at all, there is no need to determine its exact realization

other than to check that the outcome is not the most unfavorable.

When the principal contracts with a group of agents, it is no longer necessary to

divert money to third parties. Extending an earlier result (Rajan and Reichelstein,

2006), we show that optimal incentive contracts based on both subjective and ob-

jective performance indicators amount to proper bonus pools: the principal retains

discretion for distributing a budgeted amount(which varies with the realization of the

objective metrics) among the participating agents based on the available subjective

information. In sharp contrast to the single-agent case, informative subjective signals

will always be valuable and furthermore optimal incentive schemes will not be com-

pressed, that is, the number of distinct compensation levels for each agent is generally

equal to the number of signal realizations.

The very nature of bonus pools implies relative performance evaluation (RPE)

with regard to the subjective metrics. For instance, two symmetric agents will ceteris

paribus be paid the same regardless of whether the subjective outcomes for both

are favorable or unfavorable. At the same time, each agent’s share of the bonus pool

increases if his own subjective outcome remains unchanged but that of the other agent

declines. Somewhat surprisingly, we also establish a need for relative performance

evaluation with regard to the objective metrics, despite our assumption that the

agents’ signals exhibit stochastic independence. Intuitively, such forms of RPE result

as a consequence of two interacting properties: the bonus pools are monotonically

increasing in each objective metric and, at the same time, the principal seeks to

punish an agent exhibiting consistently poor performance indicators. Starting with

unfavorable subjective outcomes, the balancing requirement of bonus pools therefore

implies that if one agent’s objective signal becomes unfavorable, another agent must

receive higher compensation as a consequence.

The analysis in this paper is most closely related to Baiman and Rajan (1995),

MacLeod (2003) and Rajan and Reichelstein (2006). MacLeod’s paper considers a

4

single agent with whom the principal can contract only on the basis of subjective infor-

mation. We establish that in the presence of objective signals, subjective information

may not be valuable for contracting purposes. Furthermore, if such information is

used at all, the resulting contracts will be super-compressed by not paying out the

full bonus pool only under extreme conditions. Baiman and Rajan (1995) examine

a two-agent setting and demonstrate that proper bonus schemes are valuable for the

principal in order to incorporate a subjective (and hence unverifiable) signal. Our

analysis goes further by establishing the optimality of proper bonus pools in the

presence of both objective and subjective information. Our earlier work (Rajan and

Reichelstein, 2006) examined the relative weights placed on objective versus subjec-

tive information in the context of a LEN framework. A major restriction of linear

contracts, however, is that they cannot capture the relative performance evaluation

issues that are central to the present study.

The remainder of the paper is organized as follows. The next section provides

necessary and sufficient conditions for subjective information to be valuable. Com-

pression of optimal incentive schemes is examined in Section 3. We extend the model

to multiple agents and issues of RPE in Section 4. Conclusions and directions for

future work are presented in Section 5.

2 Value of Subjective Information

We begin our analysis with the simplest possible setting in which the principal seeks

to motivate an agent to take a given action, ah. This action is more costly for the

agent than “shirking”, i.e., to take a less productive action al. To provide incentives,

the principal can rely on both an objective metric, x, and a subjective metric, y.

We think of the objective metric as financial information which can be verified to

third parties and therefore is contractible. In contrast, the subjective signal is not

verifiable. This signal may also not be observed by the agent, possibly because it

5

reflects direct observations by the principal or informal reports from other sources.8

In the simplest case both signals are binary, that is either signal realization is

“high” or “low”. Naturally, a high outcome suggests greater effort on the part of the

agent in the sense that:

ph ≡ Prob[x = xh|a = ah] > pl ≡ Prob[x = xh|a = al]

and,

qh ≡ Prob[y = yh|a = ah] > ql ≡ Prob[y = yh|a = al].

One way for the principal to make credible use of her subjective information is to

specify a bonus pool for each of the two possible objective outcomes.9 If the subjective

information is indicative of shirking on the part of the manager, the principal can

choose to redirect some or all of the budgeted amount to an outside third-party,

such as a charity. Since the owner is ex-post indifferent as to the recipient of the

bonus pool, the presumption is that she will follow the course of action that is most

efficient from an ex-ante standpoint. For a given objective outcome, x, the principal’s

implicit contract with the agent therefore specifies how a fixed amount of money will

be divided between the agent and the third party depending on the observed signal.

We denote the bonus pools corresponding to the two states by wh and wl, respec-

tively. Furthermore, the principal “promises” compensation payments shh, shl, slh, sll

which satisfy the inequalities wh ≥ max{shh, shl} and wl ≥ max{slh, sll}. Any differ-

ence between the bonus pool wj and the actual compensation payment sjk is paid to

the outside third party. The risk-averse agent is assumed to have additively separa-

ble preferences over wealth, U(·) and cost of effort, represented by e(·). His expected

utility, exclusive of the cost effort, is denoted by:

8We discuss this aspect in more detail below.9Our earlier work (Rajan and Reichelstein, 2006) focused exclusively on multi-agent settings. In

that context, we used the term fixed payment scheme for arrangements where an outside third partyserves as a potential “money sink” and referred to bonus pools as fixed payment schemes which arebalanced across the set of participating agents. In contrast, such arrangements will be called properbonus pools in Section 4 below.

6

E[U(sjk)|ah] ≡ [U(shh) ·qh +U(shl) · (1−qh)]ph +[U(slh) ·qh +U(sll) · (1−qh)](1−ph).

The principal’s optimization problem then becomes:

P1: minwj ,sjk

[wh · ph + wl · (1− ph)]

subject to:

(i) E[U(sjk)|ah]− e(ah) ≥ U ,

(ii) E[U(sjk)|ah]− e(ah) ≥ E[U(sjk)|al]− e(al),

(iii) for all j, k: wj − sjk ≥ 0.

The class of admissible incentive mechanisms deserves comment at this stage.

When y is non-verifiable information available to both the principal and the agent,

the parties can conceivably enter into a contract specifying payments as a function

of messages about the information received. MacLeod (2003, Proposition 4) shows

that if both parties observe the non-verifiable signal y, then it is possible to construct

a message game and a corresponding outcome function (that involves the use of a

third party) so as to achieve second-best performance in equilibrium.10 On the other

hand, the signal y may be subjective in the sense that only the principal observes

it. Applicable examples include direct observations or informal reports from various

sources. In that case, message games cannot sustain any outcome beyond those

attainable by bonus pools. This follows from the observation that by the Revelation

Principle equilibrium messages correspond to truthful reporting. Yet, if the principal

is to report her observation truthfully, her payoff must be the same for all realizations

of the signal.

10It should be noted, however, that such mechanisms are necessarily afflicted by multiple equi-librium problems. Any pair of messages that forms an equilibrium in one state (agent’s action andsignal realization) also forms an equilibrium in any other state.

7

Thus the class of mechanisms considered in P1 entails no loss of generality if the

signal y is genuinely subjective information that is accessible only to the principal

but not the agent. An alternative justification for excluding message-based games is

that the very factors that make the information contained in y non-verifiable, also

makes it prohibitively costly to write a contract which specifies outcomes depending

on the parties’ reports concerning the non-verifiable information.

A well-known result in the agency literature due to Holmstrom (1979) is that given

some signal, x, an additional signal, y, is valuable for contracting purposes if and only

if y informative (in a statistical sense) over and above the signal x. In the context of

our model, we note that if hypothetically y were a verifiable and contractible signal,

it would clearly be valuable due to the assumed stochastic independence of the two

signals. In the following discussion we shall say that the subjective signal y is not

valuable conditional on the objective outcome x = xj, if the solution to P1 is such

that sjh = sjl = wj. Accordingly, we define the subjective signal as not valuable if it

is not valuable for any j ∈ {l, h}.It is readily seen that the subjective signal is valuable provided it is sufficiently

informative. In particular, suppose that the probabilities ph, pl and ql are held fixed

yet qh → 1. The principal can then achieve an approximation of the first-best solution

by ignoring the objective metric for compensation purposes and by offering the agent

a fixed payment plus a bonus pool. That amount is paid to the agent only if the

subjective metric results in a favorable outcome. The magnitude of the bonus pool

can be chosen so as to satisfy the incentive compatibility constraint while the fixed

payment can be chosen to meet the participation constraint. As qh → 1, the result-

ing contract entails no risk for the agent and therefore approximates the first-best

solution.

In stating the following result, we adopt the notation: Q ≡ 1−ql

1−qh and P ≡ 1−ph

1−pl

and V (z) ≡ ddz

U−1(z).11

11All proofs are in the Appendix.

8

Proposition 1 The subjective metric is valuable if and only if:

Q− pl

ph · PQ− 1

<V

(U + e(ah)(1−pl)−e(al)(1−ph)

(ph−pl)

)

V(U − e(ah)pl−e(al)ph

(ph−pl)

) . (1)

Proposition 1 says that the standard reasoning of informative signals being valu-

able does not apply to subjective information. In order to use such information

credibly, the principal incurs an additional cost represented by the need to divert

parts of the bonus pool to a third party with probability 1 − qh. The nature of this

cost is such that under the conditions identified in Proposition 1 the principal will

prefer a “corner solution” which ignores the subjective signal entirely rather than

lower the weight placed to this signal.

To interpret the inequality in (1), we note that the left-hand side represents a

measure of the relative strength of the two signals. Both sides are always greater

than one since V (·) is an increasing function. Furthermore, the right-hand side is

independent of both qh and ql. Holding ph and pl fixed, the inequality in (1) will

not be met if the subjective signal is of relatively poor quality in the sense that qh

is close to ql, leading Q to approach one. Consistent with our observations above,

the subjective metric will be valuable as qh → 1 since then the left-hand side in (1)

approaches one, while the right-hand side remains fixed at some value exceeding one.

3 Compression of Incentive Contracts

One of the continuing challenges for contract theory is that many of the incentive

contracts observed in practice tend to be simplistic relative to the predictions made

by principal-agent models. One explanation for this discrepancy is that the existing

models usually do not account for the costs of writing, understanding and imple-

menting more“complex” contracts. An intuitive measure of contract complexity is

9

the number of contingencies specified by the contract.12 Accordingly, a simpler con-

tract tends to pool a larger set of input variables (signal realizations) and assigns them

the same outcome (compensation level). From that perspective, a dichotomous bonus

scheme would be the simplest possible incentive scheme. One obvious advantage of

such a scheme is that it may be unnecessary to verify the exact signal realizations.

A major result in MacLeod’s (2003) analysis is that with only subjective informa-

tion, the optimal incentive scheme will be “compressed.” Specifically, the agent always

receives the entire bonus amount except when the lowest possible outcome is realized,

in which case the bonus pool is entirely diverted to the third party. This result, which

only requires the familiar monotone likelihood ratio property (MLRP), highlights the

incremental cost of subjective information. If the subjective information were in fact

verifiable for contracting purposes, the principal would find it optimal to employ a

monotone increasing compensation scheme. In contrast, with subjective information

the entire bonus pool must be paid out anyhow, and therefore the cheapest incentive

provision is a “stick” approach which punishes only extremely low performance.

An immediate question then is whether similar compression results hold in set-

tings where the principal has access to both objective and subjective performance

indicators. Proposition 1 above identifies conditions under which the optimal incen-

tive scheme will indeed be compressed. The reason is that, if the inequality (1) does

not hold, the optimal incentive scheme entails only two distinct levels of compensa-

tion even though there are four different (and informative) signal realizations. We

first focus on the binary action and binary outcome setting introduced in the previ-

ous section and then present a generalized model with finitely many different signal

outcomes and a continuum of possible action choices.

Observation 1: The solution to P1 is such that the subjective metric has no value

whenever the objective outcome is favorable, that is, wh = shh = shl.

12Earlier literature has formalized related notions of costly contracting; see, Dye (1985), Melumadet al. (1997) and Laffont and Martimort (2001) on this point.

10

To establish this claim, supposed to the contrary that an optimal incentive scheme

has the property that wh = shh > shl.13 The principal could then choose the following

variation of the incentive scheme. Given the outcome (xh, yl), the agent is paid shl+∆,

where ∆ is such that shl +∆ < shh, and at the same time the payments slh and sll are

reduced by ε1 and ε2, respectively. If the bonus pool corresponding to the low objective

outcome is lowered to max{slh−ε1, sll−ε2}, the principal’s fixed payout is unchanged

for the high objective outcome but lower for the low objective outcome. Therefore

any such variation leaves the principal better off provided the agent’s incentive and

participation constraints are still met. It will be convenient to define:

U(slh)− U(slh − ε1) ≡ ∆U1.

and

U(sll)− U(sll − ε2) ≡ ∆U2.

It is readily verified that for any given ∆, there exist corresponding ε1 and ε2 such

that the agent’s incentive and participation constraints are met provided:

(1− pl)

(1− ql) · pl[∆U1 · ql + ∆U2 · (1− ql)] >

(1− ph)

(1− qh) · ph[∆U1 · qh + ∆U2 · (1− qh)].

Since ∆U1 and ∆U2 can assume any positive values, depending on the choices of

ε1 and ε2, the above inequality can always be satisfied unless both:

(1− ph)

(1− qh) · ph· qh >

(1− pl)

(1− ql) · pl· ql

and

(1− ph)

(1− qh) · ph· (1− qh) >

(1− pl)

(1− ql) · pl· (1− ql).

13Obviously, there is no reason for the principal to waste money by setting both wh > shh andwh > shl.

11

Yet, the latter inequality would contradict the monotone likelihood requirement

that ph > pl. Thus, we conclude that in the context of our binary setting the principal

will always ignore the subjective metric provided the objective metric assumes the

favorable outcome.14

At first glance the result in Observation 1 may seem counterintuitive. Suppose

the subjective metric is perfect in the sense that qh = 1 > ql. A fixed bonus pool (the

magnitude of which is independent of the objective outcome) will then achieve the

first-best. In contrast, it may seem that the principal exposes the agent to unwar-

ranted risk by using a bonus pool only in case x = xh, as implied by Observation 1.

We note, however, that such an incentive scheme can be structured so that the agent

is precisely reimbursed for the targeted level of effort if either the objective outcome

is favorable or otherwise at least the subjective outcome is favorable. The bonus pool

corresponding to this mixed outcome can be chosen so that the agent is effectively

deterred from shirking and in equilibrium is not exposed to any risk.

To generalize our findings in connection with Observation 1, we now consider a

more general model structure in which the agent’s effort is chosen from a continuum

[a, a] and there are n possible outcomes both for the objective and the subjective

metric. As before, we maintain the assumption that, conditional on a, the two signals

are statistically independent. Specifically:

Prob[x = xj, y = yk|a] = pj(a) · qk(a).

Throughout our analysis both pj(a) and qk(a) are assumed to satisfy the familiar

MLRP condition.15 We continue to adopt, with no loss of generality, the cost min-

14A natural question at this point is why the above construction cannot be applied similarly to thebonus pool corresponding to the low objective outcome. In particular, it is natural to ask whetherit would be cheaper to lower the bonus pool, wl, but preserve incentive feasibility by increasing theagent’s compensation sll. Straightforward algebra shows that provided ph > pl such a variationcannot satisfy both the incentive compatibility and the participation constraint.

15The density pj(a) satisfies the MLRP condition if pj(a′)pj(a) is monotone decreasing in j for all

a′ < a.

12

imization approach first used by Grossman and Hart (1983). That is, the principal

seeks to implement a given action ao at minimal cost:

P2: minwj ,sjk

n∑j=1

wj · pj(a0)

subject to:

(i) E[U(sjk)|a0]− e(a0) ≥ U

(ii) a0 ∈ arg maxa{E[U(sjk)|a]− e(a)}

(iii) for all j, k, wj − sjk ≥ 0.

To characterize the solution to Program P2, we assume that the first-order ap-

proach is valid. Earlier literature has specified a variety of sufficient conditions en-

suring the validity of this approach (e.g., Jewitt 1988, Sinclair-Desgagne 1994). Let

pja(a) denote the derivative of pj(a) with regard to a 16

Assumption (A1): The first-order approach is valid, i.e., the incentive constraint

in P2 can equivalently be represented as:

d

da{

n∑j=1

n∑

k=1

U(sjk) · pj(ao) · qk(ao)} − e′(ao) = 0. (2)

In the absence of the objective metric, the solution to P2 would be such that

the entire bonus pool is paid out entirely unless the lowest possible outcome y1 is

realized. One intuitive explanation of this result, which was derived by MacLeod

(2003, Proposition 6) is as follows. Suppose there are three possible outcomes (n = 3)

and the agent’s compensation is monotonically increasing in the outcome, with the

highest level of compensation equal to the bonus pool. Consider now a variation

16The validity of the first-order approach is assumed here for Propositions 2-4. One sufficientcondition for the validity of this approach in the context of our model is that, like in MacLeod(2003), the agent’s action determines a convex combination of two given densities.

13

of this incentive scheme which lowers the bonus pool but pays the agent more in

the intermediate state. Such a variation obviously results in a lower compensation

expense (bonus pool) to the principal. Furthermore, it can be shown that any such

variation still implements the targeted action and satisfies the participation constraint

provided the MLRP condition is satisfied. Thus the principal prefers to compress the

compensation scheme for favorable outcomes.

In the following analysis, we measure the degree of compression of an incentive

scheme by the number of distinct compensation levels received by the agent. If both

metrics were in fact objective and verifiable, then optimal incentive contracts would

“generally” entail no compression in the sense that the range of sjk would be n2.17

On the other hand, if the subjective metric is not valuable for a particular realization

of the objective metric, xj, there will be partial compression since the range of sjk is

reduced by n− 1 values. Formally, we define |sjk| to be the size of the range of {sjk}.

Proposition 2 The subjective metric has no value unless the objective metric as-

sumes the lowest possible outcome. The optimal incentive scheme for P2 satisfies

|sjk| ≤ n + 1, such that |suk| = 1 for 2 ≤ u ≤ n and |s1k| ≤ 2.

Proposition 2 represents a “super-compression” result. First, the finding that

the subjective metric is never of value unless x = x1 directly extends our finding in

Observation 1 above. Any bonus pool arrangement corresponding to xj for j > 1

could be improved upon by raising the compensation payments corresponding to all

realizations of the subjective metric other than yn. At the same time the principal

could lower the entire bonus pool corresponding to x1. Thus, the range of {sjk} is at

most 2 · n− 1.

If the subjective metric is used at all when x = x1, then there is further com-

pression since the bonus pool, w1, is paid out in full to the agent for all subjective

17In particular, the only setting in which the number of distinct payoff levels would fall below n2

is if there were two separate output vector realizations for which the sum of the likelihood ratios atthe optimal action happened to coincide. Moreover, even in this situation, the principal would needto know the realizations of both signals in order to compute the payoff level for the agent.

14

outcomes unless y = y1. As a consequence, the principal diverts money to a third

party only in the “unlikely” event that both metrics result in the lowest possible out-

come. The effective range of {sjk} is thus at most n+1. An immediate implication of

this finding is that the presence of objective measures leads to “super-compression”

in the use of information for performance evaluation. In particular, when only subjec-

tive signals are available for contracting, the principal reduces n signals to 2 possible

payoffs (MacLeod (2003)). With both objective and subjective measures, there are

at most (n + 1) payoffs out of n2 possible signal realizations, which, for any n > 1,

represents a greater relative level of compression (as n+1n2 < 2

n).

Our finding in Proposition 2 predicts a lexicographic use of the subjective metric.

This result is of significance from a transaction cost perspective which recognizes

that collecting and verifying the particular realization of a signal is costly. Our

characterization of optimal contracts is consistent with the traditional management

concept of conditional variance investigations; see, for instance, chapter 16 in Maher et

al. (2006). The general idea is that a supervisor collects additional information only if

a first investigation yields an outcome below a certain threshold level. Principal-agent

models have found it relatively difficult to rationalize such policies.18

4 Multiple Agents

It is natural at this point to ask whether the above findings extend to settings in which

the principal contracts with a group of agents. When multiple agents participate

in the same bonus pool, their compensation levels must become interdependent. To

examine the implications of subjective information for relative performance evaluation

18Agency models that have analyzed variance investigation policies include Baiman and Demski(1980), Dye (1986) and, more recently, Fagart and Sinclair-Desgagne (2004). In these models, theprincipal commits in advance to incur a cost to generate an additional signal, y, on agent effort,based on the realization of an initially informative signal, x. While it is generally optimal to committo a bang-bang lower-tailed investigation policy, Lambert (1985) and Young (1986) demonstrate thatthis result depends crucially on the assumed utility function for the agent, as well as the correlationstructure across the two signals.

15

(RPE), we focus on a setting in which the signals are stochastically independent across

agents and therefore their compensation schemes would be independent of each other

if all signals were objective and verifiable. Specifically, we extend the preceding

single-agent analysis by assuming that for each agent i, 1 ≤ i ≤ 2, the subjective and

objective signals are conditionally independent:19

Prob[xi = xji , yi = yk

i |ai] = pji (ai) · qk

i (ai).

Bonus pools can now be indexed to the objective outcome (xj1, x

u2) for 1 ≤ j, u ≤

n. Accordingly, we use the notation wju. Agent i′’s compensation scheme takes

the form si(xj1, y

k1 , x

u2 , y

v2). The principal seeks to design implicit incentive schemes

that minimize the expected bonus pool compensation subject to the constraints that

the targeted effort levels constitute a Nash-equilibrium and in that equilibrium the

participation constraints are met.

P3: minwju,si(·)

n∑j=1

n∑u=1

wju · pj1(a

01) · pu

2(a02)

subject to: for 1 ≤ i, ı ≤ 2,

(i) E[Ui(si(xj1, y

k1 , x

u2 , y

v2)|ao

i , aoı )− ei(a

oi ) ≥ Ui

(ii) aoi ∈ arg max

ai

{E[Ui(si(xj1, y

k1 , x

u2 , y

v2)|ai, a

oı )− ei(ai)}

(iii) wju − s1(xj1, y

k1 , x

u2 , y

v2)− s2(x

j1, y

k1 , x

u2 , y

v2) ≥ 0 for all 1 ≤ j, k, u, v ≤ n.

To characterize the solution to P3, we shall again assume that the first-order

approach is valid.20

19Throughout this section, we shall confine attention to a setting with two agents in order toeconomize on notation. As discussed below, the extension of our results to an arbitrary number ofagents is relatively straightforward.

20As noted in connection with Proposition 2, the approach is valid if each distribution function issuch that an agent’s effort determines a linear combination of two given distributions.

16

Assumption (A2): The first-order approach is valid, i.e., for each agent i the

incentive constraint in P3 can equivalently be represented as:

∂

∂ai

E[Ui(si(xj1, y

k1 , x

u2 , y

v2)|ao

i , aoı )− e

′i(a

oi ) = 0. (3)

One feasible solution to the principal’s problem in P3 is to replicate the single-

agent solution. Intuitively, the advantage of this solution is that neither agent is

burdened with unnecessary risk resulting from the variation in the signals that pertain

to the other agent’s action. On the other hand, the principal can potentially use each

agent as a “budget breaker” by interlinking the two compensation schemes. A bonus

pool is said to be proper if the inequality in (iii) in P3 is satisfied as an equality for

all outcomes. Thus, a proper bonus pool never results in money being diverted to a

third party.

Proposition 3 Given (A2), a solution to P3 specifies proper bonus pools for all

objective outcomes, (xj1, x

u2).

This result extends our earlier finding in Rajan and Reichelstein, (2006, Propo-

sition 1) to settings with both objective and subjective performance indicators. On

balance, it is less expensive for the principal to accept the inefficient risk sharing

associated with proper bonus pools rather than wasting money that is paid to ex-

ternal parties. Put differently, if each agent becomes a budget breaker for the other

agent, the compensation schemes will become riskier yet average total compensation

will be lowered. Proposition 3 also “validates” the approach in Baiman and Rajan

(1995) who show that a proper bonus pool is an instrument for utilizing subjective

information so as to improve upon incentive schemes based on verifiable signals. The

result is also consistent with recent experimental work by Fisher et al. (2005), who

demonstrate the benefits of having a formula-based total bonus pool and allowing the

employer discretion to then allocate it across employees.

We next establish certain monotonicity properties of optimal compensation schemes.

17

Proposition 4 Any solution to P3 is such that:

(i) The bonus pools wju are increasing in each of the objective metrics xj1 and xu

2 .

(ii) Each agent’s compensation is increasing in both his objective and subjective

metric.

(iii) Ceteris paribus, each agent’s compensation is decreasing in the other agent’s

subjective metric.

The first finding in Proposition 4 is consistent with the observation that in practice

the magnitude of bonus pools is usually tied to underlying financial (and verifiable)

metrics such as earnings per share, ROA, or Economic Value Added (EVA).21 The

second finding shows that Proposition 1 does not extend to multiple agents. Once

another agent can serve as a “budget breaker”, the subjective metric is always valuable

for contracting purposes, regardless of the objective outcome. Finally, we find that

the principal must engage in relative performance evaluation (RPE) with regard to

the subjective metrics. Because of the balancing constraints, two symmetric agents

will be paid the same regardless of whether both subjective metrics reveal favorable or

unfavorable outcomes. The bonus pool creates effective incentives by offering “large”

rewards to an agent whose subjective outcome is favorable in contrast to that of the

other agent.

Since the presence of a second agent reduces the cost of using subjective infor-

mation, it is natural to revisit the compression result reported in Proposition 2. To

that end, it suffices to show that “in general” there is no compression, i.e., we can

find parameter values such that each agent’s compensation scheme assumes exactly

as many distinct values as there are different outcomes.

Observation 2: With two (or more) agents, optimal compensation schemes entail

no compression.

21In Rajan and Reichelstein (2006) the size of the bonus pools is independent of the realizationsof the objective outcome. This property does not emerge optimally but as a direct consequence ofconfining attention to linear incentive schemes.

18

To demonstrate this result, it suffices to check that if, for example, the agents are

symmetric with square-root utility functions and there are three possible outcomes for

each metric, then for “generic” probability values, each agent’s compensation scheme

is fully differentiated in the sense that |si(xj1, y

k1 , x

u2 , y

v2)| = 34 = 81.

Proposition 4 leaves open the question of RPE with regard to the objective metrics.

Clearly, such forms of RPE would not be desirable absent any subjective information

since all signals are stochastically independent. On the other hand, it is conceivable

that the need for RPE inherent in any bonus pool arrangement effectively spills over

to the objective metrics.22 We show next that the monotonicity established in Propo-

sition 4 does not extend to the other agent’s objective metric.

Corollary: In any optimal incentive scheme, one agent’s compensation cannot be

monotonic in the other agent’s objective outcome.

Proposition 4 and the above corollary can be illustrated by the numerical example

provided in Table 1. The example is generated in a binary setting where each manager

can choose between two effort levels and the principal wishes to implement the high

effort level. Moreover, the objective and subjective measures are also assumed to have

binary support. Note that the findings in Proposition 4 are evident in the numerical

example - the bonus pool increases in the objective measures, and the bonus paid to

each agent is monotone increasing in his own objective and subjective metrics, and

decreasing in the other agent’s subjective metric. However, the interaction of the two

types of metrics implies that agent 1’s compensation is non-monotone in agent 2’s

objective outcome, and varies in subtle ways. In particular, the change in manager 1’s

compensation in manager 2’s objective measure, x2, depends on the reference point

with regard to y2. We recall that by Proposition 4 the magnitude of the bonus pool

is increasing in x2. If y2 = yh2 and 2’s objective outcome drops from xh

2 to xl2, then

22For the linear incentive schemes considered in Rajan and Reichelstein (2006), the magnitude ofthe bonus pool is constant and for given subjective outcomes, each agent’s compensation dependsonly on his own objective metric.

19

Table 1: Optimal Bonus Pools and Allocation

Bonus Poolswhh 98.20 whl 80.06 wlh 80.06 wll 61.09

Agent 1’s Bonusshhhh1 49.10 shhhl

1 54.10 shhlh1 44.09 shhll

1 49.10

shlhh1 44.11 shlhl

1 70.26 shllh1 40.03 shlll

1 68.36

slhhh1 35.95 slhhl

1 40.03 slhlh1 9.80 slhll

1 11.70

sllhh1 30.54 sllhl

1 52.16 slllh1 8.93 sllll

1 30.54

Agent 2’s Bonusshhhh2 49.10 shhhl

2 44.09 shhlh2 54.10 shhll

2 49.10

shlhh2 35.95 shlhl

2 9.80 shllh2 40.03 shlll

2 11.70

slhhh2 44.11 slhhl

2 40.03 slhlh2 70.26 slhll

2 68.36

sllhh2 30.54 sllhl

2 8.93 slllh2 52.16 sllll

2 30.54

both agents receive lower compensation as the bonus pool shrinks. On the other

hand, if x2 drops from xh2 to xl

2 when y2 = yl2, then agent 2 is showing consistently

low performance. It is efficient for the principal to punish such outcomes, but to

do so within the constraints of a bonus pool, agent 1 must be paid more. Thus, we

find that RPE results in both positive and negative affiliations between the outcome

of one agent’s objective metric and another agent’s compensation. Our finding has

implications for the likelihood of detecting relative performance evaluation in practice

by the use of standard empirical tests.

20

Parameters

ah 3 al 0 U1 = U21γsγ γ 0.5 e(ah) 3

ph 0.7 pl 0.3 qh 0.7 ql 0.3 e(al) 0

The numerical example in Table 1 also motivates our next line of inquiry in this

paper. In particular, it is easily shown via calculation that the optimal incentive

scheme presented in Table 1 suffers from a severe implementation problem. It is of

course true that the solution to P3 yields each manager his reservation utility of 10

in the case where each manager exerts high effort. However, if agent 2 chooses low

effort, then agent 1’s utility playing high effort is actually 0.33 lower than the utility

he gets if he plays low effort. A symmetric analysis applies to agent 2. Thus both

agents have an incentive to deviate to low effort anticipating the other does so; in

this event, they each realize a higher utility (11.52) than in the obedient equilibrium

(10). Thus, both choosing high effort is a Nash equilibrium that is strictly dominated

by another Nash equilibrium where both choose low effort.

One possible solution to this implementation problem is to augment the agents’

strategy sets to incorporate a message-sending game in addition to their choice of

effort levels (see, e.g., Ma (1988). In this scenario, the agents are in effect asked

to report on each other’s performance. This approach, however, has been criticized

for the complexity of the resulting game constructions. Moreover, the games require

the agents to issue messages and assume that the principal can commit to payoffs as

a function of such reports. An alternative approach that does not impose such re-

quirements (and may be more expensive as a consequence) involves strengthening the

incentive constraints in order to implement the high effort choice as a dominant strat-

egy (see, e.g., Demski and Sappington (1984)). Intuitively, given the independence of

the agents, one would expect that the efficient dominant strategy incentive scheme in

effect reverts to giving each agent his optimal second-best scheme. Surprisingly, this

turns out not to be the case.

21

Proposition 5 The optimal incentive scheme that implements high effort as a domi-

nant strategy equilibrium is a bonus pool that is proper almost everywhere, with the one

possible exception being the case where all four metrics record their lowest realization.

Table 2 shows the optimal bonus pool arrangements under the same assumptions

as in Table 1, but with the addition of the constraint that the high effort level has to

be implemented as a dominant strategy. The main result of interest from Table 2 is

that sllll1 + sllll

2 = 46.62 < 55.65 = wll, while every one of the 15 other signal vector

realizations results in a proper bonus pool. In particular, the bonus pool is fully paid

out for all cases except the lowest one.

To conclude this section, we conjecture that our findings in Propositions 3, 4 and

5 extend directly to an arbitrary number of agents. With regard to the corollary

following Proposition 4, while the result is likely true for any number of agents, the

precise pattern of RPE for a group of n agents remains open for further investigation.

5 Concluding Remarks

To make credible use of subjective performance metrics in short-term incentive con-

tracts, a principal must commit to paying out an amount which is independent of

the actual realization of the subjective metrics. Such bonus pool arrangements are

costly for the principal and therefore impact the demand for and use of subjective

performance indicators. When contracting with a single agent, it may be optimal to

ignore an incrementally informative though non-verifiable signal altogether. If such

a signal is valuable for contracting purposes, we predict a lexicographic relation: the

subjective metric is relevant only if the objective metric results in an unfavorable

outcome and even then the role of the subjective metric is only to punish very poor

performance. This “super-compression” result is broadly consistent with the general

principle of conditional variance investigations.

22

Table 2: Optimal Bonus Pools and Compensation with Dominant Strategy Implemen-tation

Bonus Poolswhh 97.31 whl 82.41.38 wlh 82.41 wll 55.65

Agent 1’s Bonusshhhh1 48.65 shhhl

1 53.72 shhlh1 43.59 shhll

1 48.65

shlhh1 45.49 shlhl

1 71.00 shllh1 41.20 shlll

1 71.00

slhhh1 36.92 slhhl

1 41.20 slhlh1 11.41 slhll

1 11.41

sllhh1 27.83 sllhl

1 47.95 slllh1 7.70 sllll

1 23.31

Agent 2’s Bonusshhhh2 48.65 shhhl

2 43.59 shhlh2 53.72 shhll

2 48.65

shlhh2 36.92 shlhl

2 11.41 shllh2 41.20 shlll

2 11.41

slhhh2 45.49 slhhl

2 41.20 slhlh2 71.00 slhll

2 71.00

sllhh2 27.83 sllhl

2 7.70 slllh2 47.95 sllll

2 23.31

23

When the principal seeks to create incentives for a group of agents, the cost

associated with subjective information is reduced. Optimal bonus pool arrangements

will result in risk sharing that is inefficient, relative to the benchmark setting of

objective and verifiable information. However, there is no longer the dead-weight

cost of having the principal commit to diverting money to third parties for certain

contingencies. The presence of both subjective and objective signals for multiple

agents gives rise to a complex pattern of relative performance evaluation. Holding

the objective metrics fixed, an agent will be paid more if the subjective performance

indicators of other agents decline (bonus pool arrangement). At the same time, we find

that an agent’s compensation will increase if ceteris paribus another agent’s objective

outcome becomes unfavorable, provided the other agent’s subjective metric is also

unfavorable. The opposite will happen when the other agent’s results are mixed, that

is, the unfavorable objective outcome is balanced by a favorable subjective outcome.

From a theoretical perspective, there are multiple promising avenues for extend-

ing the analysis presented in this paper. As mentioned in the Introduction, earlier

literature on contracting with non-verifiable information has focused primarily on the

repeated interactions between a principal and a single-agent. It would be natural

to examine the performance of bonus pools in a multi-period setting depending on

the exact commitments made by the principal, e.g., the principal retains discretion

in how much is being paid out in a given period, but makes a commitment to some

aggregate payout over a certain time horizon. Such a multi-period framework could

also could be used to address notions of collusion among the agents and influence

activities towards the principal.23

It would also be natural to broaden the present framework so as to include super-

visors who can collect, at a personal cost, subjective information about the agents’

23In our one-period model, the principal is indifferent about carrying out the terms of the implicitcontract. This indifference may give rise to influence activities by the agents. While such activi-ties may cancel each other in equilibrium, they may nonetheless represent a hidden cost of bonuspool arrangements. In a multiperiod framework, the principal’s indifference will be supplanted byreputation concerns.

24

productive contributions. The principal would then want to delegate the provisions

of incentives to the better-informed supervisors, who, in turn, will need to be given

incentives to collect information. This would lead naturally to a theory of corpo-

rate governance, where the board of directors (and the compensation committee in

particular) acts as the intermediary between owners and managers. At issue then is

the structure of the compensation arrangements for the board which motivate the

gathering of subjective information to be used in the compensation of management.

These and other interesting extensions await future research.

25

A Appendix

Proof of Proposition 1: We proceed by first characterizing the solution to program

P1. Let λ and µ denote the multipliers for constraints (i) and (ii) respectively. Denote

by βjk the multiplier for the constraint wj − sjk ≥ 0 (there are 4 such inequalities

embedded in constraint (iii)). The first-order conditions for the optimal solution with

regard to wh and wl are:

−ph + βhh + βhl = 0 (4)

−(1− ph) + βlh + βll = 0 (5)

Similarly, the first-order conditions for the optimal solution with regard to shh, shl, slh

and wl are:

λ · ph · qhU ′(shh) + µ[ph · qh − pl · ql]U ′(shh)− βhh = 0 (6)

λ · ph · (1− qh)U ′(shl) + µ[ph · (1− qh)− pl · (1− ql)]U ′(shl)− βhl = 0 (7)

λ · (1− ph) · qh · U ′(slh) + µ[qh(1− ph)− ql(1− pl)]U ′(su)− βlh = 0 (8)

λ · (1−ph) · (1−qh)U ′(sll)+µ[(1−ph) · (1−qh)− (1−pl) · (1−ql)]U ′(sll)−βll = 0 (9)

Note that by (6) and (7), ql < qh.

26

βhh

ph · qh · U ′(shh)= λ+µ

[1− pl · ql

ph · qh

]> λ+µ

[1− pl · (1− ql)

ph · (1− qh)

]=

βhl

ph · (1− qh)U ′(shl)(10)

It is clear that wh = max{shh, shl}. To show that shh ≥ shl, suppose that the

opposite were true. Then, shh < shl = wh ⇒ βhh = 0 (by complementary slackness),

which in turn would imply βhl = ph (from (6)). But from (A7), βhh = 0 ⇒ βhl < 0,

which is a contradiction. We must therefore have wh = shh ≥ shl. Similarly, we can

use (5), (8), and (9) to show that wl = slh ≥ sll.

To complete the first part of the proof, we now establish that wh = shh = shl.

Again, suppose not, so that wh = shh > shl. But this implies that βhl = 0 and in

turn (from (7) and (9)) that:

0 =βhl

ph · (1− qh)U ′(shl)= λ + µ

[1− pl · (1− ql)

ph · (1− qh)

]

> λ + µ

[1− (1− pl) · (1− ql)

(1− ph) · (1− qh)

]

=βll

(1− ph) · (1− qh)U ′(sll).

however, that would imply βll < 0, which is impossible. Using these results, we now

re-cast program P1 as follows:

minwh,wl,δ

[ph · wh + (1− ph) · wl]

subject to:

ph · U(wh) + (1− ph) · qh · U(wl) + (1− ph) · (1− qh)[U(wl)− δ]− e(ah) ≥ U (11)

27

ph · U(wh) + (1− ph) · qh · U(wl) + (1− ph) · (1− qh)[U(wl)− δ]− e(ah) ≥pl · U(wh) + (1− pl) · ql · U(wl) + (1− pl) · (1− ql)[U(wl)− δ]− e(al).

Since both (11) and(12) are satisfied as equalities, it must be the case that:

uh = U(wh) = U +e(ah) · (1− pl)− e(al) · (1− ph)

(ph − pl)− δ · (1− ph) · (1− pl) · (qh − ql)

(ph − pl)(12)

and

ul = U(wl) = U− e(ah) · pl + e(al) · ph

(ph − pl)+δ · p

h · (1− pl)(1− ql)− pl · (1− ph)(1− qh)

(ph − pl)(13)

The above program can thus be reduced to the following unconstrained optimiza-

tion:

Z(δ) ≡ minδ≥0

ph · U−1(uh) + (1− ph) · U−1(ul)

where U−1(·) is the inverse of the agent’s utility function, and uh and ul are given by

the expression in (12) and (13), respectively. Differentiating, we obtain:

d

dδZ(δ) = −ph · V (uh)

[(1− ph) · (1− pl) · (qh − ql)

(ph − pl)

]

+(1− ph)V (ul)

[ph · (1− pl) · (1− ql)− pl · (1− ph) · (1− qh)

(ph − pl)

], (14)

where we continue to use the notation (introduced in the text) where V (·) represents

the derivative of U−1. Further,

28

d2

dδ2Z(δ) = ph · V ′(uh)

[(1− ph) · (1− pl) · (qh − ql)

(ph − pl)

]2

+(1− ph) · V ′(ul)

[ph · (1− pl)(1− ql)− pl · (1− ph)(1− qh)

(ph − pl)

]2

> 0.

Therefore a necessary and sufficient condition for subjective information to have

value (i.e., for the optimal δ to be positive) is that ddδ

Z(δ)|δ=0 < 0, i.e., the right-hand

side of (14) is less than zero at δ = 0. This requires:

ph · (1− pl) · (qh − ql) · V(

U +e(ah) · (1− pl)− e(al) · (1− ph)

(ph − pl)

)>

[ph · (1− pl)(1− ql)− pl · (1− ph)(1− qh)

]V

(U +

e(al) · ph − e(ah) · pl

(ph − pl)

)

or, equivalently:

ph·(1−pl)·(1−ql)−pl·(1−ph)·(1−qh)ph·(1−pl)·(qh−ql)

<V

(U + e(ah)·(1−pl)−e(al)·(1−ph)

(ph−pl)

)

V(U − e(ah)·pl−e(al)·ph

(ph−pl)

).

Collecting terms and using the notation for p and Q introduced in the text, we obtain:

Q− pl

ph · PQ− 1

<V

(U + e(ah)(1−pl)−e(al)(1−ph)

(ph−pl)

)

V(U − e(ah)pl−e(al)ph

(ph−pl)

) ,

which is inequality (1).

Proof of Proposition 2: To solve program P2, we follow the notation in the proof

of Proposition 1 and denote by λ be the multiplier for constraint (i), and by βjk the

multiplier for the restriction wj−sjk ≥ 0. Furthermore µ will denote the multiplier for

29

constraint (ii), as expressed in its first-order form in (2). We then have the following

first-order conditions for the variables wj and sjk :

−pj(a0) +n∑

k=1

βjk = 0 (15)

λ · U ′(sjk) · pj(a0) · qk(a0) + µ · U ′(sjk)[pj(a0) · qka(a0) + qk(a0) · pj

a(a0)]− βjk = 0.

Here, pja(·) denotes the derivative of pj(·) with respect to a. It follows that:

βjk

U ′(sjk) · pj(a0) · qk(a0)= λ + µ

[qka(a0)

qk(a0)+

pja(a

0)

pj(a0)

]. (16)

To first see that sj,k+1 ≥ sj,k, suppose not, i.e., sj,k+1 < sj,k ≤ wj ⇒ βj,k+1 = 0

(by complementary slackness). But this implies, using (16) and MLRP, that

0 =βj,k+1

U ′(sj,k+1) · pj(a0) · qk+1(a0)= λ + µ

[qk+1a (a0)

qk+1(a0)+

pja(a

0)

pj(a0)

]> λ + µ

[qka(a0)

qk(a0)+

pja(a

0)

pj(a0)

]

=βjk

U ′(sjk) · pj(a0) · qk(a0)

However that is impossible since βjk > 0.

We next show that for all realizations xj > x1, the subjective measure is ignored.

Again, suppose not, i.e., there is some xj > x1 such that, for some k, sjk < wj. Then

this βjk = 0, which, again from MLRP and (A14), would imply:

0 = λ + µ

[qka(a0)

qk(a0)+

pja(a

0)

pj(a0)

]> λ + µ

[qka(a0)

qk(a0)+

pj−1a (a0)

pj−1(a0)

]

=βj−1,k

U ′(sj−1,k) · pj−1(a0) · qk(a0).

30

contradicting that βjk must be non-negative.

Finally, we demonstrate that when x = x1, the subjective metric is used only if

y = y1. Suppose to the contrary that there exists yk > y1 such that s1k < w1. This

implies that β1k = 0, or;

0 =β1k

U ′(s1k) · p1(a0) · qk(a0)= λ + µ

[qka(a0)

qk(a0)+

p1a(a

0)

p1(a0)

]>

λ + µ

[qk−1a (a0)

qk−1(a0)+

p1a(a

0)

p1(a0)

]=

β1,k−1

U ′(s1,k−1)p1(a0) · qk−1(a0),

resulting again in a contradiction to the requirement that β1,k−1 ≥ 0. We have thus

proved that subjective metrics are used only when x = x1 and y = y1, as was to be

demonstrated.

Proof of Proposition 3: Consider program P3, with the incentive compatibility

constraints in (ii) replaced by their first-order approach representations. We use λi

and µi to represent the multipliers for agent i’s individual rationality and IC con-

straint, respectively, and γjkuv as the multiplier for the bonus pool constraint when

the realized outcome is (xj1, y

k1 , x

u2 , y

v2).

To represent the first-order condition for s1(·), we denote the derivative of pj1(a)

by pj′1 (a). Therefore:

λ1 · U ′1(s1(·)) · pj

1(a01) · qk

1(a01) · pu

2(a02) · qv

2(a02)

+µ1 · U ′1(s1(·)) · pu

2(a02) · qv

2(a02)[p

j1(a

01) · qk′

1 (a01) + pj′

1 (a01) · qk

1(a01)]− γjkuv = 0,

or

γjkuv

pj1(·) · qk

1(·) · pu2(·) · qv

2(·)· 1

U′1(s1(·))

= λ1 + µ1

[qk′1 (·)qk1(·)

+pj′

1 (·)pj

1(·)

](17)

31

Similarly, the first-order condition for s2(·) reduces to:

γjkuv

pj1(·) · qk

1(·) · pu2(·) · qv

2(·)· 1

U′2(s2(·))

= λ2 + µ2

[qv′2 (·)qv2(·)

+pu′

2 (·)pu

2(·)]

(18)

To show that proper bonus pools are optimal for all verifiable outcomes, suppose

not, i.e., assume there is some realization (xj1, y

k1 , x

u2 , y

v2) for which

s1(·) + s2(·) < wju.

This implies that γ jkuv = 0 or, using (17), that

λ1 + µ1

[qk′1 (a0

1)

qk1(a

01)

+pj′

1 (a01)

pj1(a

01)

]= 0. (19)

But (19) is independent of u and v, so this must hold for all realizations of (xu2 , y

v2),

implying in turn that (from (17)),

γ jkuv = 0 for all u, v. (20)

Similarly, using the right hand side of (18), γ jkuv = 0 implies that

λ2 + µ2

[qv′2 (a0

2)

qv2(a

02)

+pu′

2 (a02)

pu2(a

02)

]= 0.

This equation is independent of j and k, and so it must hold for all realizations

of (xj1, y

k1), implying (from (18)) that

γjkuv = 0 for all j, k. (21)

Repeating the preceding arguments yields:

γ jkuv = 0 for all k, v. (22)

Finally, the first-order condition for the bonus pool amount wju is:

32

pj1(a

01) · pu

2(a02) =

∑

k

∑v

γ jkuv. (23)

Clearly, (23) cannot hold if (22) holds and therefore we have established that

proper bonus pools are optimal for all objective outcome realizations.

Proof of Proposition 4: From Proposition 3, we know that proper bonus pools are

optimal. Let wju denote the bonus pool when the verifiable outcomes are xj1 and xu

2 .

With a slight abuse of notation in the specification of densities, we can then rewrite

program P3 as:

minwju,s1(·)

∑j

∑u

wju · pj1(·) · pu

2(·)

subject to:

∑j

∑

k

∑u

∑v

U1(s1(xj1, y

k1 , x

u2 , y

v2))p

j1(·)qk

1(·)pu2(·)qv

2(·)− e1(a01) ≥ U1

∑j

∑

k

∑u

∑v

U2(wju − s1(x

j1, y

k1 , x

u2 , y

v2))p

j1(·)qk

1(·)pu2(·)qv

2(·)− e2(a02) ≥ U2

∑j

∑

k

∑u

∑v

U1(s1(xj1, y

k1 , x

u2 , y

v2))p

u2(·)qv

2(·)[pj1(·)qk′

1 (·) + pj′1 (·)qk

1(·)]− e′1(a

01) = 0

∑j

∑

k

∑u

∑v

U2(wju−s1(x

j1, y

k1 , x

u2 , y

v2))p

j1(·)qk

1(·)[pu2(·)qv′

2 (·)+pu′2 (·)qv

2(·)]−e′2(a

02) = 0

Using the same notation for multipliers as in the proof of Proposition 3, we obtain

the following first-order condition for s1(·):

λ1U′1(s1(·))pj

1(·)qk1(·)py

2(·)qv2(·)− λ2U

′2(w

ju − s1(·))pj1(·)qk

1(·)pu2(·)qv

2(·)+µ1U

′1(s1(·))pu

2(·)qv2(·)[pj

1(·)qk′1 (·) + pj′

1 (·)qk1(·)]

−µ2U′2(w

ju − s1(·))pj1(·)qk

1(·)[pu2(·)qv′

2 (·) + pu′2 (·)qv

2(·)] = 0.

33

It will be convenient to rewrite this condition as:

U′1(s1(·))

U′2(w

ju − s1(·))=

λ2 + µ2

[qv′2 (·)qv2 (·) +

pu′2 (·)

pu2 (·)

]

λ1 + µ1

[qk′1 (·)qk1 (·) +

pj′1 (·)

pj1(·)

] (24)

The first-order condition for wju can similarly be simplified to:

∑

k

∑v

U′2(w

ju − s1(·))[λ2 + µ2

[qv′2 (·)qv2(·)

+pu′

2 (·)pu

2(·)]]

qk1(·)qv

2(·) = 1 (25)

To show that wju is monotone increasing in j, consider an arbitrary u and arbitrary

j > j. Note that the left hand side of (25) is independent of xj1, other than the U

′2

term. Therefore, it cannot be the case that

wju − s1(xj1, y

k1 , x

u2 , y

v2) < wju − s1(x

j1, y

k1 , x

y2, y

v2)

for all k and v, since (25) would then be violated for either wju or wju. So, there

must exist some {yk1 , y

v2} such that

wju − s1(xj1, y

k1 , x

u2 , y

v2) ≥ wju − s1(x

j1, y

k1 , x

u2 , y

v2) (26)

But j > j and MLRP together imply, from (24), that:

U′1(s1(x

j1, y

k1 , x

u2 , y

v2))

U′2(w

ju − s1(xj1, y

k1 , x

u2 , y

v2))

<U′1(s1(x

j1, y

k1 , x

u2 , y

v2))

U′2(w

ju − s1(xj1, y

k1 , x

y2, y

v2))

(27)

In light of (26), this can hold only if

U′1(s1(x

j1, y

k1 , x

y2, y

v2)) < U

′1(s1(x

j1, y

k1 , x

y2, y

v2)) ,

or,

s1(xj1, y

k1 , x

y2, y

v2) > s1(x

j1, y

k1 , x

y2, y

v2) (28)

34

Summing the inequalities in (26) and (28) yields: wju > wju, which is the desired

result.

To show that wju is monotone increasing in u, note that the equality in (24) can

be substituted into (25) to yield:

∑

k

∑v

U′1(s1(·))

[λ1 + µ1

[qk′1 (·)qk1(·)

+pj′

1 (·)pj

1(·)

]]qk1(·)qv

2 = 1 (29)

Other than the U′1(·) term, the above expression is independent of xu

2 . So, consider

an arbitrary j, and arbitrary u > u. Then, in order for (29) to be met for both wju

and wju, there must exist some {yk1 , y

v2} such that

s1(xj1, y

k1 , x

u2 , y

v2) ≥ s1(x

j1, y

k1 , x

u2 , y

v2) (30)

But from (24), u > u and MLRP imply that

U′1(s1(x

j1, y

k1 , x

u2 , y

v2))

U′2(w

ju − s1(xj1, y

k1 , x

u2 , y

v2))

>U′1(s1(x

j1, y

k1 , x

u2 , y

v2))

U′2(w

ju − s1(xj1, y

k1 , x

u2 , y

v2))

.

Given (30), this inequality can hold only if

U′2(w

ju − s1(xj1, y

k1 , x

u2 , y

v2)) < U

′2(w

ju − s1(xj1, y

k1 , x

u2 , y

v2)),

i.e.,

wju − s1(xj1, y

k1 , x

u2 , y

v2) > wju − s1(x

j1, y

k1 , x

u2 , y

v2) (31)

Adding (30) and (31) then yields wju > wju, completing the proof that the bonus

pool is monotone increasing in objective indicators.

To examine the behavior of each manager’s compensation as a function of the

realized subjective metrics, consider (24). For any given {xj1, x

u2}, note that the left-

hand side of (24) is strictly decreasing in s1(·). From MLRP, the right-hand side is

strictly decreasing in yk1 , and strictly increasing in yv

2 . Together, this implies that

35

manager 1′s payoff increases in his own subjective metric and decreases in the other

manager’s subjective metric. Since we have a proper bonus pool, the analogous result

for manager 2 follows immediately.

Finally, manager 1’s payoff is increasing in his objective metric because for arbi-

trary k, u, v, and j > j we have:

U′1(s1(x

j1, y

k1 , x

u2 , y

v2))

U′2(w

ju − s1(xj1, y

k1 , x

u2 , y

v2))

<U′1(s(x

j1, y

k1 , x

u2 , y

v2))

U′2(w

ju − s1(xj1, y

k1 , x

u2 , y

v2))

<U′1(s1(x

j1, y

k1 , x

u2 , y

v2))

U′2(w

ju − s1(xj1, y

k1 , x

u2 , y

v2)).

Here, the first inequality follows from MLRP and j > j, using (24), and the second

inequality from wju > wju. Comparing the first and last expressions immediately

yields

s1(xj1, y

k1 , x

y2, y

v2) > s1(x

j1, y

k1 , x

y2, y

v2).

A parallel argument yields the same result for agent 2.

Proof of Corollary: We shall first show that agent 2’s compensation cannot be

monotonic increasing on agent 1’s objective outcome. Suppose not, i.e., suppose that

for every realization of (u, v), agent 2’s compensation is monotonic increasing on agent

1’s objective outcome. Then, ∀1 ≥ j > j ≥ 0,∀k,

s2(xj1, y

k1 , x

u2 , y

v2) > s2(x

j1, y

k1 , x

u2 , y

v2).

Then (25) will be violated for either wju or wju. The same reasoning applies and it

can be shown that agent 2’s compensation cannot be monotonic decreasing on agent

1’s objective outcome. Symmetrically, agent 1’s compensation cannot be monotone

on agent 2’s objective outcome, as (29) will otherwise be violated.

36

Proof of Proposition 5: In a binary action setting, with ali < ah

i , suppose that

the principal wants to implement the high action ahi as a dominant strategy. Then,

solving the appropriate modification of P3, the first-order conditions for the bonus

pool and compensations can be derived as below:

pj1(a

1h) · pu

2(ah2) =

∑

k

∑v

γjkuv (32)

γjkuv

pj1(a

h1)q

k1(a

h1)p

u2(a

h2)q

v2(a

h2)· 1

U′1(s1(·))

= λ1+

[µ1 + α1

pu2(a

l2)q

v2(a

l2)

pu2(a

h2)q

v2(a

h2)

] [1− pj

1(al1)q

k1(a

l1)

pj1(a

h1)q

k1(a

h1)

]

(33)

γjkuv

pj1(a

h1)q

k1(a

h1)p

u2(a

h2)q

v2(a

h2)· 1

U′2(s2(·))

= λ2+

[µ2 + α2

pj1(a

l1)q

k1(a

l1)

pj1(a

h1)q

k1(a

h1)

] [1− pu

2(al2)q

v2(a

l2)

pu2(a

h2)q

v2(a

h2)

]

(34)

Now, suppose that there is some vector {j, k, u, v} 6= {1, 1, 1, 1} for which the

bonus pool is not paid out in full. With no loss of generality, assume that j > 1. By

complementary slackness, this indicates that γjkuv = 0, and therefore that the rhs of

both (33) and (34) equal zero. But, for a nontrivial incentive problem, it must be the

case that αi+µi > 0. From MLRP, we know then that the rhs of (33) strictly increases

in j and k, and the rhs of (34) strictly increases in u and v. In turn, this implies that

γ jkuv < 0, where j = j − 1, which is impossible. We thus have a contradiction.

37

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