Project Estimate Padding

8/6/2019 Project Estimate Padding

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MANAGEMENT SCIENCEVol. 52, No. 9, September 2006, pp. 13451358issn 0025-1909 eissn 1526-5501 06 5209 1345

informs

doi10.1287/mnsc.1060.0532 2006 INFORMS

Project Assignments When Budget

Padding Taints Resource AllocationAnil Arya

The Ohio State University, 2100 Neil Avenue, Columbus, Ohio 43210, [email protected]

Brian MittendorfYale School of Management, 135 Prospect Street, New Haven, Connecticut 06520, [email protected]

This paper shows that rotation programs can be an effective response to concerns of employee budgetpadding. Rotation programs naturally create a portfolio of assignments for each manager, and the result-ing diversification can reduce the downside of resource rationing. In particular, the production versus rentstrade-off linked with adverse selection problems can be more efficiently carried out when the firm faces twomanagers with average information advantages, rather than one with a large advantage and one with a smalladvantage. Roughly stated, rotation of project assignments is a way of smoothing information across managers.

On the other hand, if a firm places a premium on treating different types of projects in distinct ways, specializedassignments can be preferred due to the ability to confine project types to individual managers.

Key words : adverse selection; budget padding; job rotationHistory : Accepted by Stefan Reichelstein, accounting; received November 11, 2004. This paper was with the

authors 2 months for 2 revisions.

1. IntroductionHolistic (broad) work practices such as rotation pro-grams are utilized by a variety of industries for a widearray of employees. Conventional wisdom suggeststhat periodically switching employee work assign-

ments fosters learning due to increased exposure todifferent facets of a firms business. Rotation pro-grams can also appease workers by providing taskvariety, enhancing socialization, assisting in manage-ment and executive development, and amelioratingthe effects of career plateaus (Sites-Doe 1996, p. 86).

In this paper, we present an additional factor thatfavors rotationone that is rooted in the issue ofadverse selection. Adverse selection problems entaila production-rents trade-off, the outcome of whichdepends critically on the extent to which managersenjoy information advantage. Rotation programs cre-ate a portfolio of assignments for each manager;

the resulting diversification can take the sting outof the firms requisite resource rationing. In particular,the downside of production distortions can be lim-ited when the firm faces two managers with averageinformation advantages rather than one with a largeadvantage and one with a small advantage. Roughlystated, rotation of project assignments is a way ofsmoothing information across managers.

This paper employs a variant of the two-periodresource allocation model in Antle and Fellingham(1990) wherein project cost in each period is privately

known to a manager.1 Despite costs being uncor-related across periods, Antle and Fellingham showmerits to using memory-based contracts in conduct-ing the production-rents trade-off. The key insight isthat a firm may find it easier to elicit truthful bud-gets from a manager in the first period by exploit-ing his uncertainty regarding costs in the subsequentperiod. Making use of the managers residual uncer-tainty necessitates memory in contractingthe firmssecond-period decision is contingent on the man-agers first-period report.

This paper extends the theme to the assignmentof different types of projects among multiple man-agers, finding that benefits of a memory contract aremost pronounced when the firm exploits diversifi-cation through broad project assignments. In otherwords, when it is worthwhile to connect independentperiods in evaluating managers, it is also worthwhile

to mix independent assignments among managers.Under a memory contract, the firm engages in pro-duction cuts in only the extreme event of costs beinghigh in both periods. When project types differ withrespect to their probability distributions over costs,assigning diverse projects to each manager reduces,in aggregate, the probability of the extreme event.

1 The Antle and Fellingham paper has illustrious predecessorswhich detail control issues in capital budgeting (Antle and Eppen1985, Harris et al. 1982). Antle and Fellingham (1997) and Rajanand Reichelstein (2004) provide excellent reviews of this literature.

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Arya and Mittendorf: Project Assignments When Budget Padding Taints Resource Allocation1346 Management Science 52(9), pp. 13451358, 2006 INFORMS

That is, if the probability of high cost is p for oneproject type and q for another project type, undermemory contracts, production is forgone with prob-ability p2 for one agent and q2 for the other withspecialization, and with probability pq for each agentwith rotation. Because p2+q2 > 2pq, rotation is accom-

panied by a smoothing of production distortions.Of course, rotation also comes with a cost. If thefirm benefits by dealing with distinct types of projectsin separate ways (say, the memory contract is optimalfor only one project type), mixing project assignmentscan preclude such differential treatment. In such cir-cumstances, specialization has the edge.

The vast literature on adverse selection stresses thatthe problem of information asymmetry is larger thanthat of mere distribution of wealth. In the contextof the linear adverse selection model of Antle andFellingham (1990), the point is made forcefully: distor-tions take the form of complete stoppage of otherwise

valuable production. Moreover, distortions are opti-mally confined to extreme tails introducing a non-linearity in an otherwise linear game. This paperscontribution is in showing how such nonlinearitiesimpact a firms project assignment decisions and, inparticular, how a firm can utilize rotation programsto smooth distortions. In effect, the production versusrents trade-off both influences and is influenced byproject assignments.

This papers key themes are further reinforced by three extensions. First, we study a modifica-tion wherein each manager receives early informa-tion on all projects assigned to him. The timing

change implies that manager uncertainty is no longeravailable for the firm to exploit. Antle et al. (1999)and Arya and Glover (2001) show that while thisdisables memory-based contracts, another contractthat bundles project decisions comes into play. The

bundling benefits in these single manager models can be attributed to a sample-size effect: by the centrallimit theorem, the more the number of independentobservations (projects), the thinner the tails of thetotal cost distribution which are afflicted by produc-tion cuts and information rents.2

The previous statement hints that the firm may

turn to rotation programs with even greater gusto inthe case of early information arrival. This is becausethe diversification effect applies at both ends of thetotal cost distribution. At the high-cost end, foregoneproduction is smoothed as before. Now, there arealso gains at the low-cost tail because information

2 In the extreme case of perfect negative correlation in project costs,the tails are completely eliminated thus removing all inefficiencies;after all, with bundled projects and 1 correlation, the managerenjoys no information advantage over the firm as far as total costsare concerned.

rents too are smoothed. This paper presents necessaryand sufficient conditions for specialization and rota-tion each to be optimal that are consistent with thisintuition.

A second extension addresses the circumstancewherein the principal receives additional information

either about a particular type of project or about aspecific agent. The nature of information has a pro-found effect on the value of rotation programs. Forexample, if the principal learns the cost of a particu-lar project type, specialization is the preferred course.After all, assigning one (informationally) challengingproject to each manager, as is done under rotation,eliminates the firms ability to use the memory-basedcontract, forcing her to evaluate each project piece-meal. In contrast, specialization, by keeping the chal-lenging project type confined to one manager, allowsfor the use of a memory-based contract.

When the principals additional information is

agent specific (she learns the cost of projects under-taken by one manager), rotation can again be optimal.The issue here is that if the principal makes assign-ments so as to maximize her surplus with one agent,she may be left with the worst assignment as far ascontracting with the other agent is concerned. Rota-tion is a middle-of-the-road approach which enablesher to obtain an average level of surplus from eachagent.

In our model, project outcomes are unaffected bymanagerial proficiency (or any other differential trait).This assumption allows us to hone in on the diver-

sification effect of rotation. A final extension relaxesthis assumption by modeling one of the managers asbeing more skilled (in particular, producing a higherrevenue). With differentially skilled agents, if the prin-cipal has to reject a project, she would much rather it

be the project of the less-skilled agent. Now, the abil-ity of specialization to confine a project type to oneagent is more attractive because it can ensure that the

better-skilled agent is matched to the project type thatis most often undertaken. This factor can reduce, butnot necessarily eliminate, the desirability of rotation.

As noted previously, there exist several other expla-nations for rotation programs. While rotation is com-

monly justified on the grounds of generating morewell-rounded or more content employees, effects onemployer knowledge have also been detailed. Forone, rotation has been explained as a means throughwhich employee-specific results can be separatedfrom task-specific results (Ortega 2001). That is, withrotation, it is much harder to fool an employer byattributing successes to acumen if everyone assignedto the same job also achieves success. Job rotation isalso a means of insulating employees so they aremore diligent in executing and assessing current tasks


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Arya and Mittendorf: Project Assignments When Budget Padding Taints Resource AllocationManagement Science 52(9), pp. 13451358, 2006 INFORMS 1347

(Arya and Mittendorf 2004, Milgrom and Roberts1992). In effect, with rotation, employees recognizethat any potential ratcheting of performance stan-dards or undesired consequences of honesty will be aconcern for the rotated-in employee, not for them.

In a moral hazard agency setup, Holmstrom and

Milgrom (1991) show a preference for assignmentsthat provide each manager with similar-type jobs.A specialized approach serves to confine bad mea-sures (pressing incentive problems) to one manager.As Demski (1994, p. 576) writes, the problem with

broader assignments is that bad measures mightdrive out good measures. On the other hand, com-plementarities among tasks can point to a shift awayfrom specialized assignment (Zhang 2003). In ouradverse selection setting, even in the absence ofcomplementarities, diversity in assignments proves

beneficial by smoothing distortions associated withemployee information advantages.

2. ModelIn this paper, we employ a variant of the Antle andFellingham (1990) two-period model of resource allo-cation. A firm consists of a risk-neutral principal(budget center), who makes project assignment andfunding decisions, and two risk-neutral agents (divi-sion managers), A and B, who are responsible forproject implementation. In each period, two projectsare available. If implemented, each project yields rev-enue of X. However, there is uncertainty over the pre-cise project cost. The cost of each project is either cL

or cH, cL < cH; projects are positive net present value(NPV), X > cH. In each period, the probability that theproject cost is cH equals p for one project and q for theother project. Any two project costs are independent,i.e., costs are uncorrelated within and across periods.Without loss of generality, assume that p > q > 0. Asa convenient label, we refer to the projects as p-typeand q-type, respectively.

At the outset, the principal commits to an assign-ment policy and to project approval/funding rules.Given there is one p-type project and one q-typeproject in each period, the assignment question is:Should each agent be assigned the same type of

project in each period, or should each agent be rotatedto a different type of project in the second period?

At the beginning of period t, t = 1 2, agent i,i = A B, privately learns cit , the cost of the projectassigned to him in that period. Because costs areuncorrelated, there is no reason for relative projectevaluation, and each agents project approval andfunding decision depends only on his own reports.The principals first-period project approval and fund-ing decisions can be conditioned on the agentsperiod 1 cost report ci1, while her second-period

Figure 1 Timeline

The principal

signs contracts

with agents

which specify

project assign-

ments and

project approval/funding rules.

Agent i is paid

ym, and the

principal

receives xm.

i

i

Agent i is paid

ymn, and the

principal

receives xmn.

i

i

Agent i privately

learns the cost

of his assigned

period 2 project

and submits

report c2 = cn.i

Agent i privately

learns the cost

of his assigned

period 1 project

and submits

report c1 = cm.i

decisions can be conditioned on the agents period 1and period 2 reports ci1 c

i2.

Notationally, if ci1 = cm, the principal transfers yim to

agent i in period 1. In return, she receives xim, xim =X

if she approves the agents project, and 0 otherwise.If ci1 = cm and c

i2 = cn, the principal transfers y

imn to

the agent in period 2. In return, she receives ximn,ximn =X if she approves the agents period 2 project,and 0 otherwise.

The principals utility is

i=A B ximy

im+x

imny

imn;

she consumes project revenues less funds transferred.Agent is utility is yim cjx

im/X+ y

imn ckx

imn/X, when

ci1 = cj, ci2 = ck, c

i1 = cm, and c

i2 = cn; the agent consumes

slack, the funding above cost. The timeline in Figure 1summarizes the sequence of events.

Without loss of generality, Agent A is assignedthe p-type project and Agent B is assigned theq-type project in period 1. Specialization correspondsto Agent A (Agent B) being reassigned the p-type(q-type) project in period 2. Rotation corresponds toAgent A (Agent B) being switched over to the q-type(p-type) project in period 2.

The principals project approval and funding terms

for Agent A under rotation are determined by solv-ing program (P). In (P), the principal maximizesher expected utility subject to the following con-straints. First, the truth-telling constraints (TT1 andTT2 ensure that Agent A has incentives to report hisperiod 1 and period 2 project costs truthfully. By theRevelation Principle (Myerson 1979), requiring truth-telling is without loss of generality. Second, the lim-ited liability constraints (LL1 and LL2 require thatfunds transferred by the principal are sufficient tocover investment costs. (When it causes no confusion,as in (P), we suppress the agent superscript.)

Program (P)

Maxxj xjk0Xyj yjk

pxH yH+ qxHH yHH+ 1 q

xHL yHL+ 1 pxL yL

+ qxLH yLH+ 1 qxLL yLL

st yjk ckxjk /X yjn ckxjn/X jkn (T T2)

yj cjxj/X+ qyjH cHxjH/X

+ 1 qyjL cLxjL /X


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ym cjxm/X+ qymH cHxmH/X

+ 1 qymL cLxmL/X j m TT1

yj cjxj/X 0 j LL1

yj cjxj/X+ yjk ckxjk/X 0 j k LL2

Agent Bs terms under rotation are obtained sim-ply by replacing p and q in the program with qand p, respectively. Replacing only q with p in (P)yields Agent As terms under specialization; similarly,replacing only p with q yields Agent Bs terms underspecialization. Finally, comparing the principals sur-plus in the specialization case with that in the rotationcase provides the preferred assignment rule.

3. ResultsWhen the principal knows project costs (the first-bestsetting), project assignment and funding decisions are

trivial. Because the model assumes that agents haveno influence on the revenues or costs of the projects,any assignment rule will work. Moreover, becauseX > cH, the projects are always accepted and fundedat actual cost. In other words, the model rules outthe typical reasons for and against rotation: the firmdoes not directly benefit either from an agent hav-ing a diverse set of on-the-job experiences or from anagent developing specialized expertise. The absenceof the typical issues allows a crisp characterization ofthe informational role of assignment policies.

In particular, the information asymmetry that nat-urally arises in decentralized organizations can influ-

ence assignment choice. Project assignment becomesa question of how information should be split amongthe agents both within and across periods. With thisin mind, we examine the principals expected surplusunder each assignment rule.

3.1. SpecializationOne possibility is to foster agent specialization,assigning the p-type project in each period toAgent A, and the q-type project to Agent B. In thiscase, because the two agents have different assign-ments, the principal may treat them differently. But,the same options are available for each agent. The

relevant budgeting options entail offering either (i) arationing contract in each period, (ii) a memory-basedcontract, or (iii) a slack contract in each period (Antleand Fellingham 1990, p. 16).

Under two-fold rationing, denoted (R R), the prin-cipal funds the agents project in each period if andonly if the agent reports that the projects cost is cL.

3

That is, xL = X, yL = cL, xmL = X, and ymL = cL, and

3 The first (second) element in denotes the contract governingthe agents period 1 (period 2) project.

all other variables are 0. A rationing contract elim-inates agent information rents (there is no tempta-tion to pad the budget), but requires the principal toreject projects sometimes. At the other extreme is two-fold slack, denoted (S S), under which there is noproduction distortion (the project is always accepted)

but agent rents are maximal because the agent ispaid cH in each period. That is, xj= xmn =X and yj=ymn = cH.

(RR and (SS are myopic contractsthe prin-cipals decisions in each period depend only on theagents report in that period. Sometimes, however, amemory-based contract, denoted M, can be optimal.Under M , the principals second-period decisionsdepend on the agents first-period report in the fol-lowing manner: xL = xH = X, yL = yH = cH, xHL = X,yHL = cL, xHH = yHH = 0, xLL = xLH = X, and yLL =yLH=C, where C, the expected production cost of theagents period 2 project, equals pcH+ 1pcL in con-

tracting with Agent A and qcH+ 1 qcL in contract-ing with Agent B. In effect, the agent is offered theslack contract in the first period. In the second period,he is offered the rationing contract if his first-periodreport is cH and the expected-cost contract if his first-period report is cL. Note, in contrast to RR andSS, M is middle-of-the-road in that it involves

both production cuts (which occur if cost is cH in bothperiods) and payment of information rents (whichoccurs if cost is cL in the first period).

The memory contract satisfies truth-telling. To seethis, consider the agents second-period reportingincentives. Under a rationing contract there is no rea-

son for the agent to pad, and under the expected-cost contract, his second-period report is not used atall, satisfying T T2. Now, consider the agents first-period reporting incentives. Because the agent doesnot know period 2 costs when submitting his period 1report, under both rationing and the expected-costcontract, the agents expected rents are 0. This, cou-pled with the fact that the agents period 1 report doesnot affect the decision/funding in period 1, impliesthat T T1 is satisfied.

The memory contract also satisfies limited liabil-ity. The only delicate case here is when costs are cLin period 1 and cH in period 2. While period 2sexpected-cost payment by itself is not sufficient toimplement the project, it is sufficient given the carryforward of cH cL from period 1. This is not to saythat the carry forward of funds is what is importantin M; setting yL = C and yLm = cH gives an equiv-alent characterization under which limited liabilityconstraints are satisfied period by period. The criti-cal feature of M is that it is tailored to exploit theagents uncertainty over the second-period cost.

Which of the three contracts is preferred dependson the NPV foregone by production cuts relative to


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the potential gain from limiting the agents informa-tion rents. In particular, the principals expected sur-plus with Agent A under each of these contracts is

under RR 21 pX cL (S1)

under M X cH+ p1 pX cL+ 1 p

X pcH 1 pcL (S2)

under SS 2X cH (S3)

Replacing p with q in the above equations (denotingthe new equations by S1S3) provides the prin-cipals expected surplus in contracting with Agent B.Comparing S1S3 and S1S3, then yields theprincipals optimal contract under specialization withAgent A and Agent B, respectively. This solution ischaracterized in Observation 1. In presenting this andthe papers other results, the ratio X cH/X cL

proves convenient. The ratio, bounded by 0 and 1, isincreasing in X and, hence, referred to as the revenueratio.

Observation 1. Under specialization, the princi-pals optimal contract with each agent is as follows:

Agent A

R R M S S X cHX cL0 11 p

1+ p1 p

1 p

1 p1 p

Agent B

R R M S S X cHX cL0 11 q

1+ q1 q

1 q

1 q1 q

The observation is in line with intuition. Rationingis most beneficial when production cuts are not tooexpensive (the revenue ratio is small), slack whensuch cuts require sacrificing large NPV, and memoryfor the in-between scenario.

Note that Agent Bs cutoffs are shifted to the rightof Agent As. Intuitively, production cuts in eachperiod under a rationing contract occur with prob-

ability p with Agent A and with probability q withAgent B. p > q implies the expected NPV forgoneis greater with A than with B, and so rationingis employed for a smaller set of parameters withAgent A. The fact that Agent As cutoffs entail lessrationing is consistent with the empirical observa-tion that weaker divisions are often given more lee-way in investment decisions (e.g., Bernardo et al. 2006and Rajan et al. 2000). Here, Agent As projects areexpected to cost more, and the firm reacts by beingmore lenient in project approval.

3.2. RotationThe principals other option is to rotate each agentto a different type of project in the second period,so Agent A performs a p-type project followed bya q-type project, while Agent B performs a q-typeproject followed by a p-type project. Under rotation,

two-fold rationing, memory, and two-fold slack con-tracts can again come into play. In addition, becausean agent takes a different type of project in eachperiod, the principal may also choose to offer a dif-ferent myopic contract in each period, i.e., contractsSR and RS are also possibilities.4 The prin-cipals surplus under each of these contracts withAgent A is

under RR 1 pX cL+ 1 qX cL (R1)

under M X cH+ p1 qX cL+ 1 p

X qcH 1 qcL (R2)

under SS 2X cH (R3)

under SR X cH+ 1 qX cL (R4)

under RS 1 pX cL+X cH (R5)

Interchanging p and q in the above equations (de-noting these equations by R1R5) provides theprincipals expected surplus in contracting with AgentB. Comparing R1R5 and R1R5 then yieldsthe principals optimal contract under rotation withAgent A and Agent B, respectively. This solution ischaracterized in Observation 2.

Observation 2.

Under rotation, the principalsoptimal contract with each agent is as follows:

Agent A

R R M S S X cHX cL0 11 p

1+ q1 p

1 q

1 q1 p

Agent B

R R R S M S SX cHX cL0 1Min

1 p Max

1q1+p1q

, Max

1 q

1q1+p1q

Min

1 q p

qp1q

1p1p1q

A few comments on the observation are in order.First, because p > q, the myopic contract that entailsrationing with p-type and slack with q-type is domi-nated (from the principals perspective) by a contractunder which slack governs the p-type project andrationing governs the q-type project. Thus, contracts

4 Much as in Antle and Fellingham (1990), it is straightforward toconfirm that all other memory-based contracts under rotation aredominated by one of the myopic contracts.


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(R S) for Agent A and (S R) for Agent B are domi-nated, and, hence, do not appear in the observation.

Additionally, one might have expected that simplyinterchanging p and q in Agent As cutoffs wouldyield Agent Bs cutoffs. This is not the case becausewhile the slack with p-type and rationing with

q-type contract is dominated by the memory con-tract for Agent A, this is not necessarily the casefor Agent B. The reason the (S R) contract is dom-inated by M for Agent A is because: (i) under

both arrangements slack is offered in period 1, and(ii) M outperforms (S R) in period 2 because Mentails a weighted average of rationing and first-best.5

In contrast, under (R S) for Agent B, rationing isoffered in period 1 (slack is offered in period 1 underthe memory contract), thereby making the contractsnoncomparable.

Consistent with the above, without the (R S) con-tract in the mix, the cutoffs for Agent B should mir-

ror those for Agent A. This is the case for sufficientlylarge q q p2/1 p1 p, when the (R S) intervalin Observation 2 becomes empty and the cutoffs par-allel those for Agent A. Intuitively, in contracting withAgent B if q is large, rationing in period 1 but notperiod 2 is less attractive, and RS is dominated byeither RR or M. This line of argument also sug-gests that if q were small, (R S) becomes an attractiveoption. As it turns out, if q is sufficiently small q 2p 1/p, rationing with the q-type is so desirablethat the M interval disappears for Agent B. And, forintermediate q-values, all four intervals are nonempty.

Also, if the principal offers a memory contract to

only one agent, Agent A will be the recipient. A mem-ory contract to either agent introduces the same pro-duction distortionthe period 2 project is rejectedwith probability pq. However, the principals share ofthe surplus is different across agents. Agent A, whoundertakes a p-type project in the first period, earnsexpected rents of 1 pcH cL under M . In con-trast, Agent B, who undertakes a q-type project inthe first period, earns rents of 1 qcH cL underM. Because p > q, the principal pays less informa-tion rents to Agent A, leading to the dominance ofAgent As memory contract. Further, because the prin-cipals surplus under the M contract to Agent Ais higher than under the (R S) contract to Agent B(equivalently, (S R) contract to Agent A), the Minterval for Agent A subsumes both the (R S) and theM intervals for Agent B.

While the results under rotation and specializationhave related cutoffs, the precise values are differ-ent. It is this difference that leads to the principals

5 Recall that the agent is offered rationing or the expected-cost con-tract in period 2. Under the latter, the principals payoff, X C, isfirst-best.

preference for particular assignment policies, an issuewe turn to next.

3.3. Specialization vs. RotationObviously, if myopic contracts are optimal for bothagents under any assignment rule, that rule cannot

be (strictly) preferred by the principal. After all, theperformance of myopic contracts in one regime cansurely be duplicated in the other regime. Hence, anecessary condition for a particular assignment ruleto be preferred is that under this rule at least oneagent is offered the memory contract.

Consider the case where both agents are offeredmemory contracts under specialization. In this case,the principal forgoes production with probability p2

in dealing with Agent A (who is in charge of thetwo p-type projects) and with probability q2 in deal-ing with Agent B (who is in charge of the two q-typeprojects). A shift to rotation with memory contracts

yields valuable production gains: now production iscut with probability pq for each agent. As 2pq is lessthan p2 + q2, the principal would rather give up pro-duction in two medium-sized tails of size pq each thanin a thick tail of p2 and a thin tail of q2. Proposition 1confirms this intuition.

Proposition 1. If the optimal contracts under spe-cialization prescribe memory for each agent, rotation is

preferred.

Proof. The principals surplus under specializationif M is offered to Agent A and Agent B is S2 +S2. A shift to rotation with M contracts offered

to each agent yields a surplus of R2 + R2. Tak-ing the difference, the principals surplus under rota-tion exceeds that under specialization by p q2 X cH > 0.

The use of memory-based contracts effectivelyinduces a demand for diversification by the principal.By diversifying assignments across agents, the princi-pal is able to limit production cuts.

However, the results are quite different when onlyone agent is offered the memory contract. Supposethat under rotation, Agent A is offered the memorycontract and Agent B is offered the two-fold rationingcontract. In this case, the principal is better off opt-

ing for specialization and offering M to Agent Aand (R R) to Agent B. Under Agent As memorycontract, production cuts occur with probability pqunder rotation and increase to p2 under specialization.Under Agent Bs two-fold rationing contract, produc-tion cuts occur in period 1 with probability q undereither assignment; in period 2, cuts occur with proba-

bility q under specialization and p under rotation. Thenet effect of these factors is that specialization reducesexpected production cuts by p q p2 pq = 1pp q > 0. This force is at work in Proposition 2


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which, analogous to Proposition 1, presents sufficientconditions for specialization to be preferred.

Proposition 2. If the optimal contracts under rotationprescribe memory for one agent and two-fold rationing orslack for the other agent, specialization is preferred.

Proof. The principals surplus under rotation ifM is offered to Agent A and (R R) is offered toAgent B is R2+ R1. A shift to specialization withanalogous contracts yields a surplus of S2 + S1.Taking the difference, the principals surplus underspecialization exceeds that under rotation by 1 p p qX cH > 0.

The principals surplus under rotation if M isoffered to Agent A and (S S) is offered to Agent Bis R2 + R3. A shift to specialization with (S S)offered to Agent A and M offered to Agent B yieldsa surplus of S3 + S2. Taking the difference, theprincipals surplus under specialization exceeds that

under rotation by qp

qX

cH > 0.

The two propositions provide sufficient conditionsunder which each assignment rule outperforms theother. The next proposition extends this thinking toidentify necessary and sufficient conditions for eachassignment rule to be optimal. The conditions areexpressed as a function of the revenue ratio. Roughlystated, if the revenue ratio is in an intermediate range,memory-based contracts are desirable, and these con-tracts are more effective when agent assignments arerotated. In effect, rotation can be viewed as a way ofcreating a portfolio of project types for each agent,with portfolios giving rise to meaningful diversifica-

tion gains. On the other hand, assigning a portfolioof project types to an agent means that dissimilartypes are forced to be regulated by the same kind ofcontract, i.e., a myopic contract for one type and amemory-based contract for the other type is no longeran option. Thus, when the principal prefers using acombination of memory-based and myopic contracts,specialization is the way to go.

Proposition 3. The preferred project assignment pol-icy is as follows:

Either Specialization Rotation Specialization EitherX cHX cL0 k1 k2 k3 k4 1

where

k1 =1 p

1+ p1 p

k2 = Min

1 q

1+ pp q + q1 p

1 p

1 1 pp q

k3 = Max

1 q

1+ qp q

1 p

1 p+ q2p q

k4 =1 q

1 q1 q

Proof. The proposition follows from a comparisonof the optimal contracts presented in Observations 1and 2 under each region of the X cH/X cL val-ues, as detailed in (1)(3) below.

(1) Assume that the preferred assignment pre-scribes myopic contracts to both agents. In this case,

the contracts to consider involve either rationing forboth types, slack for both types, or slack for p-typeand rationing for q-type. The last of these cannot

be part of the preferred arrangement. Recall that theM contract for Agent A under rotation dominatesthis choice (i.e., (S R) is not in Observation 2). Givenp > q, rationing for both types is the preferred route ifand only if it is optimal for an agent who is assignedthe p-type project in each period (Agent A under spe-cialization). The cutoff k1 then follows from Obser-vation 1it equals the first cutoff used in Agent Asevaluation. Similarly, slack for both types is the pre-ferred route if and only if it is optimal for an agent

who is assigned the q-type project in each period(Agent B under specialization). The cutoff k4 then toofollows from Observation 1it equals the second cut-off used in Agent Bs evaluation.

(2) From the previous step, the optimal arrange-ment in (k1 k4 makes use of the memory contract forat least one agent. From Proposition 1, if it is optimalto offer memory contracts to both agents, then spe-cialization is dominated by rotation. Hence, for spe-cialization to be undominated, it must prescribe thememory contract for only one agent. From Observa-tion 1, under specialization, there are only two suchcontract possibilities:

(i) M is offered to Agent A and (R R) is offeredto Agent B, or

(ii) (S S) is offered to Agent A and M is offeredto Agent B.Similarly, from Observation 2 and Proposition 2, thereare again only two contract possibilities under whichrotation can be undominated:

(iii) M is offered to both Agents A and B, or(iv) M is offered to Agent A and (R S) is

offered to Agent B.The principals surplus in (i) equals S2 + S1; in(ii) equals S3+ S2; in (iii) equals R2+ R2; andin (iv) equals R2+ R5.

(3) From the previous two steps, the optimalassignment rule in (k1 k4 is a matter of comparingthe four alternatives listed in (2). In (k1 k2, the prin-cipals surplus is maximum under (i). This followsfrom the fact that the surplus in (i) is greater than thatin (ii) in this region, and the first term in k2 equatesthe principals surplus in (i) with (iii), while the sec-ond term equates (i) with (iv). Similarly, in (k3 k4,the principals surplus is highest under (ii). This fol-lows from the fact that the surplus in (ii) is greaterthan that in (i) in this region, and the first term in k3


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equates the principals surplus in (ii) with (iv), whilethe second term equates (ii) with (iii). This leaves justthe (k2 k3 interval. From the above characterizationof k2 and k3, it follows that in this interval (i) and (ii)are dominated by either (iii) or (iv). So, in this inter-val, rotation is optimal, and the optimal contract can

be either (iii) or (iv).This completes the proof of the proposition. Intuitively, the assignment policies are equivalent

for extreme values of the revenue ratio. At the lowend, all four projects are accepted only for reportsof cL; for low ratio values, the principal pulls therationing trigger to discipline budget padding irre-spective of the assignment policy. At the high end, allfour projects are always accepted and funding of cHis provided for each; for high ratio values, productioncuts are too expensive to initiate under any assign-ment policy.

However, for intermediate revenue ratio values, the

outcomes under the assignment regimes diverge. Forrevenue ratio values just above the lowest region,a move away from rationing toward more produc-tion (via a memory contract) becomes useful forthe p-type projects, but rationing remains critical forthe q-type projects. This is because rationing with thep-type projects is more expensive for the principalaproject is rejected when its cost is high, an event thatoccurs more often under the p-type than the q-typeproject. By keeping project types separate, specializa-tion allows the principal to use a myopic contract forone project type and a memory contract for the otherproject type.

Similarly, for ratio values slightly below the high-est region, a move away from slack toward rentcutting (via a memory contract) becomes useful forthe q-type projects, but slack remains critical for thep-type projects. Again, specialization is well suitedfor this tact. As revenue ratio values move furtheraway from the extremes, the appeal of disparate treat-ment through specialization becomes less important,and the diversification gains that come from rotation

become paramount. Hence, rotation is the preferredassignment in the middle region of revenue ratio val-ues in the proposition.

This comparison highlights a subtle role of projectassignment rules. When a firm has a set of projects toparcel out to different employees, it is implicitly alsodeciding how to parse out information advantagesto the agents. If isolating information advantagesis important, then specialization has added appeal.On the other hand, if smoothing information advan-tages across agents is useful, rotation policies proveworthwhile.

Note that we use the term project for agentassignments. The use of this terminology is intendedto convey the discretion retained by the principal in

deciding whether or not to undertake the venture.Rotation programs are also associated with jobs ortasks for which implementation is not in question.Although there is a distinction between the terms, ourresults can readily be extended to such cases. Say thatinstead of outright rejection, the principal can request

a more routine implementation of the task/job underwhich there is no uncertainty in costs but revenuesare substantially reduced. As long as the principalhas access to such less efficient alternatives, similarresults apply as above. In fact, if the net revenuesfrom this alternative equal zero, the principals prob-lem is equivalent to that studied in the main setup.

A feature of our model is that project costs areuncorrelated eliminating any obvious reasons forlinkages across projects or agents. Relaxing thisassumption can give rise to additional forces that mayinfluence project assignments. Suppose, for example,that projects of the same type have perfectly corre-

lated costs. In this case, under specialization, withno agent uncertainty to exploit, the principals onlyoptions are to offer two-fold rationing or slack. Theperformance of these contracts can surely be repli-cated under rotation. In fact, rotation is strictly pre-ferred under the following contract. In period 1, eachagent is offered the best myopic contract. In period 2,projects are always accepted and each agents fund-ing equals the other agents reported cost in period 1.In effect, rotation is a way of guaranteeing agents thattheir reports will not be used against them in thefuture; the reports only affect the rotated-in agent.6

Thus, with correlated costs it may be somewhat easier

to derive a demand for rotation.

4. Extensions

4.1. Early Learning by the AgentsThe previous section of this paper shows that gainsfrom interdependent project decisions, as undermemory-based contracts, can be amplified by the useof rotation programs. Memory-based contracts exploitagent uncertainty over second-period costs to confineproduction losses to tails of the total cost distribution,and diversification through rotation programs servesto minimize this distortion by smoothing it across

agents.Given this role of rotation, one might worry that a

reliance on residual agent uncertainty may make rota-tion benefits rather fragile. That is, if agents can findmeans to accelerate their learning, the use of memory(and thus rotation) may be rendered moot. However,it turns out that while memory-based contracting isindeed disabled by early learning, the diversification

6 Similar insulating benefits of rotation are highlighted in Aryaand Mittendorf (2004) and Milgrom and Roberts (1992).


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benefits that come from rotation continue to survive,indicating their durability.

In a variant of the Antle and Fellingham (1990)model, Antle et al. (1999) and Arya and Glover (2001)show that interdependent project decisions (bundling)can also be optimal even in the absence of agent

uncertainty. Bundling makes use of the principalsimproved information over total costs. With multipleindependent draws from the same underlying distri-

bution, the likelihood of observing extremes (all lowor all high) is small. By the central limit theorem,as the number of observations increases, the distribu-tion of the sum of observations approaches normal-ity, a distribution characterized by thin tails. Whenthe observations apply to the same agent (i.e., comefrom multiple projects a manager controls), the thintails can make it easier for the principal to overcomethe agents incentive constraints.7 We next extend thisanalysis by demonstrating that bundling gains can

also be amplified when combined with across-agentdiversification achieved via rotation.Continue with the same model as before but with

one change. Assume that each agent learns the cost ofboth projects under his auspices immediately follow-ing the project assignment. Now the memory-basedcontract is no longer feasibleif project 1 cost is cLand project 2 cost is cH, the agent gains by reportingcH for both projects.

Early learning by the agent, however, has a silverlining. The principal can now condition his project 1decision not just on the agents report on project 1

but also on his report on project 2. In particular, the

principal can lump the two project decisions togetherto determine funding in the following manner. If theagent reports total costs are cL + cH or less, bothprojects are adopted and the agent is provided fund-ing of cL+ cH; else, both projects are rejected. Denotethis bundling contract by (B.

As with M in the previous section, B is a middle-ground contract which keeps both agent rents andproduction cuts in play. Also similar to M, B rel-egates production cuts to the upper tail of the costdistribution. B also brings the advantage of relegat-ing agent rents to the lower tail. That is, projects arerejected only when both costs are high, and rents are

paid only when both costs are low.The principals surplus with Agent A with the

bundling contract under specialization and rotation,respectively, is

B under specialization 1 p22X cL cH

B under rotation 1 pq2X cL cH

7 The general theme of linking otherwise independent problemsto discover agents preferences is developed in Jackson andSonnenschein (2005).

The contract possibilities under specialization androtation are the same as before, except that M isreplaced by B contract (Antle et al. 1999, Arya andGlover 2001). The counterpart to the two observationsthen is as follows. Under specialization, the princi-pals optimal contract with each agent is

Agent A

R R B S S X cHX cL0 11 p

1+ p

1 p2

1+ p2

Agent B

R R B S S X cHX cL0 11 q

1+ q

1 q2

1+ q2

While the exact cutoffs are obviously changed, they

are in much the same spirit as in the previous sectionincluding the fact that Agent Bs cutoffs are staggeredto the right. In contrast, under rotation, the princi-pals optimal cutoff with Agent A and Agent B arenow identical. This change from the previous sectionoccurs because the bundling contract for each agentyields the same surplus for the principal. Intuitively,with information arrival over time not an issue, thesequence in which an agent performs the p-typeproject and the q-type project is unimportant. The fol-lowing summarizes the principals optimal contractwith Agents A and B under rotation:

SR if Agent A

R R B R S if Agent B SS X cHX cL0 11 p1q

1 pqMin

1p

p

1 pq

1+ pq

Max

1 q

1 pq

1+ pq

The analogous counterpart to Proposition 1 holds.If the optimal contracts under specialization pre-scribe bundling for each agent, rotation is preferred.In the previous section, production distortions weresmoothed by rotation. In this setting, rotation alsoprovides the benefit of smoothing agent informationrents. To elaborate, with rotation, the high-cost tailof the total cost distribution is shrunk by probabilityp2 + q2 2pq = p q2, so production distortions are

decreased by (p

q

2

2X

2cH. The low-cost tail isalso reduced by (pq2, and information rents are cutby (p q2cH cL. The net effect of the two benefitsis (p q22X cL cH > 0.

8

We next present necessary and sufficient conditionsin the early information arrival setup for each assign-ment rule to win out. The difference from Proposi-tion 3 is only that there is now an intermediate region

8 Proposition 2 is, of course, vacuous in the early information arrivalsetting because under rotation it is never optimal to treat agentsasymmetrically.


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of indifference wherein slack for p-type and rationingfor q-type projects is optimal. (Recall that in the previ-ous section, under rotation, the memory contract forAgent A dominated such a choice.)

Proposition 4. The preferred project assignment pol-icy under batch processing is as follows:

Either Specialization Rotation Either Specialization EitherX cHX cL0 k1 k

2 k

3 k

4 k

5 1

where

k1 =1 p

1+ p

k2 =1 p1+ p 2q

1+ p2 2pq

k3 = Min

1 p

p

1 2pq+ q2

1+ 2pq q2

k

4=

Max1 2pq+ q

2

1+ 2pq q2

1 q

1+ q

k5 =1 q2

1+ q2

Proof. The proposition follows from a compari-son of the optimal contracts under each region of theX cH/X cL values, as detailed in (1)(3) below.

(1) k1 equals the first cutoff used in characterizingAgent As optimal contract under specialization. Forsuch small revenue ratio values, rationing is optimalfor both types regardless of assignment. k5 equals thesecond cutoff used in characterizing Agent Bs opti-mal contract under specialized assignment. For suchlarge revenue ratio values, slack is optimal for bothtypes.

(2) Given the previous step, the optimal arrange-ment in k1 k

5 is one of the following:(i) Myopic contracts with slack for p-type and

rationing for q-type, leading to equivalent payoffsunder each assignment rule. In this case, the princi-pals surplus is 2X cH+ 1 qX cL.

(ii) Specialization under which B is offered toAgent A and (R R) is offered to Agent B. The princi-pals surplus is 1p22XcLcH+21qXcL.

(iii) Specialization under which (S S) is offeredto Agent A and (B is offered to Agent B. The princi-pals surplus is 2X cH+ 1 q

22X cL cH.(iv) Rotation under which (B is offered to both

agents. The principals surplus is 21 pq2X cL cH.

(3) From the previous steps, the optimal assign-ment rule in (k1 k

5 is a matter of comparing the fouralternatives listed in (2). The cutoffs k2, k

3, and k

4

summarize this comparison. In (k1 k

2, the principalssurplus is maximum under (ii), with k2 equating theprincipals surplus in (ii) with (iv). In (k3 k

4, the prin-cipals surplus is maximum under (i). The first term

in the minimization expression of k3 equates the prin-cipals surplus in (i) with (iv); the second term in themaximization expression of k4 equates the principalssurplus in (i) with (iii). If q < 2p 1, these are the

bounds that come in play. However, if q 2p1, thisinterval becomes empty consistent with the notion

that for large q-values, the principal may choose notto ration with q-types. Hence, the common term to k3and k4 equates the principals surplus in (iii) with (iv).In the remaining intervals, (k2 k

3) and (k

4 k

5), theabove construction ensures that (iv) and (iii) are opti-mal, respectively.

This completes the proof of the proposition. Note that an optimal policy of rotation comes with

the ancillary effect of treating agents in a symmetricfashion. The move to dole out equal information rentsis not driven by the principals desire to exhibit fair-ness and keep influence activities in check. Rather, it isa consequence of work practices designed to smooth

production and rent distortions across agents.

4.2. Additional Information for the PrincipalIn the model, we assumed that the principal receivesno information about project costs other than thatrevealed to her by the agents. In some circumstances,the principal may obtain additional information (viaauditing, for example) that can alleviate concerns ofagent misreporting. In 4.2.1, we assume that theprincipal learns the cost of a particular project type.In 4.2.2, we assume that the principals additionalinformation relates not to a particular project type butrather to a particular agent. As we see next, the nature

of the principals information can greatly influenceher preferred assignment rule.

4.2.1. Information Pertaining to a ParticularProject Type. Suppose that the principal learns thecosts of the p-type projects at the same time theassigned agent learns the costs. Given this informa-tion, the principal can write the first-best contractwith the agent assigned the p-type project: she paysthe agent the actual cost of implementing the project,and her surplus equals XpcH 1pcL. As the nextproposition states, when information asymmetry per-tains to only one project type, the principal does not

benefit from rotating agent assignments. (If the princi-pal learns the cost of the q-type projects, the preferredproject assignment policy is as below with q replacedwith p.)

Proposition 5. If the principal learns the cost of thep-type project, the preferred project assignment policy is as

follows:

Either Specialization Either X cHX cL0 11 q

1+ q1 q

1 q

1 q1 q


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Proof. When the principal observes the cost of thep-type projects, the optimal contract prescribes theassigned agent be paid the actual project cost. Underrotation, because each agent undertakes one p-typeproject, the principal is constrained to dealing withthe q-type project with the rationing or the slack con-

tract. That is, under rotation, each agent is offereda myopic contract. Thus, rotation cannot be strictlypreferred. In contrast, under specialization, the prin-cipal assigns the p-type projects to the same agent,and is able to offer the M contract to the agentassigned the q-type projects. Hence, specialization isstrictly preferred if and only if it is optimal to use theM contract to deal with this project type. Observa-tion 1 conducts precisely this exercise (for Agent B)and, so, the cutoffs in Proposition 5 are the same asthe cutoffs in Observation 1.

Intuitively, when uncertainty associated with a par-ticular project type is removed, the issue of creating

a portfolio of project types for each agent becomesmoot, and rotation is no longer worthwhile.

4.2.2. Information Pertaining to a ParticularAgent. Suppose that the principal learns the cost ofprojects run by one agent, say Agent B. That is, theprincipal can monitor Agent Bs activities eliminat-ing his ability to pad budgets. On the other hand, asin the previous section, Agent A is unconstrained inhis reporting.9 As the next proposition states, whenthe principals information is agent specific ratherthan project specific, the principal may again opt forrotation.

Proposition 6. If the principal learns the cost of projects implemented by Agent B, the preferred projectassignment policy is as follows:

Specialization Rotation Specialization X cHX cL0 11p

1p+q2pMin

1

1+q

1p1p1q

Proof. The optimal project assignment and con-tract for Agent A under specialization and rotation arepresented below in (1) and (2), respectively. (Agent Bis assigned the remaining project in each period and,of course, paid actual cost.) Comparing the principalssurplus under specialization and rotation, as detailed

in (3), yields the preferred assignment policy.(1) Under specialization, one of the following

arrangements can be optimal:(i) Agent A is assigned the q-type projects, and

offered the (R R) contract.

9 The setup in 4.2.2 can be interpreted in other ways. For exam-ple, agents may differ inherently in their propensity to misreport:Agent B is honest while Agent A is self-interested. Also, the prin-cipal may herself be one of the agents. That is, much as a teamleader, the principal both assigns projects to others and undertakesprojects herself.

(ii) Agent A is assigned the q-type projects, andoffered the M contract.

(iii) Agent A is assigned the p-type projects, andoffered the M contract.

(iv) Agent A is assigned the p-type projects, andoffered the (S S) contract.

The only other two possibilities are as in (i) and (iv),except project assignments of the agents are switched.These arrangements are dominated. In particular, if(R R) is offered to Agent A, and he is assigned thep-type projects, the principals surplus is decreasedfrom that in (i) by 2p qX cH > 0. Similarly, if(S S) is offered to Agent A, and he is assigned theq-type projects, the principals surplus is decreasedfrom that in (iv) by 2p qcH cL > 0.

(2) Under rotation, one of the following arrange-ments can be optimal:

(i) Agent A is assigned a p-type project in thefirst period, a q-type project in the second period, and

offered the (R R) contract.(ii) Agent A is assigned a p-type project in the

first period, a q-type project in the second period, andoffered the M contract.

(iii) Agent A is assigned a p-type project in thefirst period, a q-type project in the second period, andoffered the (S S) contract.

Recall that the M contract yields a higher surplusfor the principal when the agent undertakes a p-typefollowed by a q-type project, rather than in the reversesequence. This dictates the assignment choice in (2).

(3) A necessary condition for rotation to be pre-

ferred is that the arrangement in 2(ii) is optimal. Thisis because the principals surplus in 1(i) is greater thanin 2(i) by (p qX cH > 0. Also, the principalssurplus in 1(iv) is greater than in 2(iii) by p q cH cL > 0. The necessary and sufficient conditionfor rotation to be preferred is, thus, obtained by com-paring the principals surplus in 2(ii) with that in 1(i)through 1(iv). The principals surplus in 2(ii) exceedsthat in 1(iii) by pp qX cH > 0. Equating theprincipals surplus in 2(ii) with that in 1(i) yields thetransition point from Specialization to Rotation.Equating the principals surplus in 2(ii) with that in1(ii) and 1(iv) yields the two terms that comprise the

transition point from Rotation to Specialization.This completes the proof of the proposition. Intuitively, when the revenue ratio is sufficiently

small, the principals surplus under rationing witha q-type project is not much less than her first-bestsurplus with this project type. Hence, scrutiny of theq-type projects is not so vital to the principalsheassigns this project type to Agent A under specializa-tion. At the same time, the assignment of the p-typeprojects to the monitored agent, Agent B, ensures thatreporting concerns with the p-type are alleviated as


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well. Much the same intuition applies when the rev-enue ratio is sufficiently large. In this case, the princi-pals surplus under slack with a p-type project is notmuch less than her first-best surplus with this projecttype. Hence, it is the p-type projects that are assignedto Agent A, while reporting on the q-type projects is

kept in check by assigning them to Agent B.For intermediate values of the revenue ratio, eitherspecialization or rotation can win out. The princi-pals surplus is highest under the M contract if itis offered to the agent who is assigned the q-typeprojects. The principals surplus is lowest under thefirst-best contract if it is offered to the agent who isassigned the p-type projects. If the principal prefersthe combination of these extremes, specialization isoptimal. However, if the principal prefers to smooththe extremes, rotation is the only feasible option.Roughly stated, rotation is a means of ensuring thatthe principal earns a moderate level of surplus from

each agent.

4.3. Managerial SkillIn our model, projects are implemented by (ex ante)identical agents. If this were not the case, there may

be additional reasons for favoring a particular assign-ment rule. To consider this issue, assume that agentsdiffer in their abilities. In particular, Agent A is bet-ter skilled and produces project revenue of X + , > 0; the less-skilled agent, Agent B, produces X as

before.10 Note that, with differentially skilled agents,if the principal has to reject a project, she would muchrather it be Agent Bs project so she at least does

not lose the extra in revenue. With this additionalforce in play, there is a stark reversal in two previ-ous results. First, even with myopic contracts, therecan be a strict preference for a particular assignmentrule. Second, rotation is not guaranteed to be pre-ferred even with memory-based contracts. The follow-ing observation states these results.

Observation 3. (i) If the optimal contracts underrotation prescribe two-fold rationing for each agent,specialization is preferred.

(ii) If the optimal contracts under rotation prescribememory for each agent, and > 1/qp qX cH,specialization is preferred.

(iii) If the optimal contracts under specializationprescribe memory for each agent, and < 1/q p qX cH, rotation is preferred.

The observation is intuitive. In part (i), under rota-tion, the skilled agent undertakes a p-type projectand a q-type project, and so the expected -benefitis 1 p+ 1 q. Under specialization, the prin-cipal can offer (R R) to each agent but assign the two

10 Equivalently, the skilled agent is better at cutting costs; the cost ofeither project type is reduced by when implemented by Agent A.

q-type projects to the better-skilled agent, obtaining ahigher expected -benefit of 21 q.

Part (ii) of the observation highlights how the-effect can undermine the diversification benefit ofrotation. Recall that with = 0 and memory-basedcontracts, rotation is sure to outperform specializa-

tion because projects are rejected with probability 2pqrather than p2 + q2. However, when > 0, special-ization provides an added benefit of utilizing theskilled agent more often. In particular, by assign-ing the q-type projects to Agent A under special-ization, the principal loses the -benefit only withprobability q2; she loses the -benefit with a higherprobability pq under rotation. Thus, for sufficientlylarge , the benefit of specialization to restrict theskilled agent to the project type more often imple-mented outweighs the diversification gains associatedwith rotation. Part (iii) of the observation applies atsmall valuesat = 0, this is simply Proposition 1.

As one might expect, identifying necessary and suf-ficient conditions for each assignment rule to be pre-ferred for all -values is cumbersome. However, toillustrate the key points developed above, we nextpresent the necessary and sufficient conditions in thecase

X cHX cL

> Max

1 p

1 p1 p

1 q

1+ q1 q

This lower bound on the revenue ratio ensures thatunder specialization, (S S) is the optimal contract tooffer the agent assigned the p-type projects and rulesout (R R) for the agent assigned the q-type projects.Proposition 7 confirms the intuition in Observation 3

that as increases, the principals preference shiftsfrom rotation to specialization.

Proposition 7. When agents differ in their abilities,the preferred project assignment policy is as follows:

Rotation Specialization Either X cHX cLMax

1p1p1p

Max 1q

1+qpq f1,

1q

1q1q1

1q

1+q1q

1p1p+q2pq

f2

where

f1 =pq21+ p q

1+ qp qcH cL1+ qp qpq

if

Project Estimate Padding

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