All-or-Nothing Payments�

Bo Cheny

January 30, 2012

Abstract

We develop a general principal-agent framework to study optimal incentiveschemes where agents are hired to work on multiple heterogenous and inter-dependent projects. The incentive schemes can be based on output measures,interpreted as the principal�s payo¤s, as well as input measures, regarded asobservation of some of the agents�e¤orts. We identify that a unifying featureof the optimal incentive schemes, called all-or-nothing payments, arises in threenatural senarios of the general framework: unobservable inputs, veri�able in-puts, and observable but unveri�able inputs. Our framework and results embedand generalize several previous studies on multitask principal-agent problemswith a limited liability constraint.Keywords: All-or-nothing payments; Moral hazard; Multitask incentive

contracts; Veri�ability.JEL classi�cation: C70, D23, D86, M52

1 Introduction

One of the central problems organizations typically face is how to design optimal in-centive schemes in the presence of multiple aggregated and disaggregated performancemeasures.1 Additional complications that commonly arise in such complex contractdesign problems include veri�ability issues of di¤erent measures, multiple tasks, andthe possibility of relative performance incentive schemes with multiple agents. We

�I am grateful to two anonymous referees and an associate editor for various enlightening andconstructive suggestions. I am also grateful to Hideshi Itoh, Patrick Schmitz, Larry Samuelson, ZaifuYang, and Rui Zhao for helpful discussions, and to seminar and conference participants at GETA2009, Kyoto University, and Yokohama National University for useful comments. All remainingerrors are mine.

yDepartment of Economics, Southern Methodist University, 3300 Dyer Street, Suite 301,Umphrey Lee Center, Dallas TX 75275-0496 (email: bochen@smu.edu; tel.: +1 214 768 2715;fax: +1 214 768 1821).

1See Raith (2008) and references therein, for various illustrations and examples.

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take up such issues and analyze the design of optimal incentive schemes in a generalmultitask agency framework where the principal employs several agents and observesmultiple performance measures for each agent. In several natural contracting sce-narios, arising from the availability and veri�ability of the performance measures, we�nd that a common crucial feature linking all the optimal incentive compensationsis a bonus scheme that rewards only a high level of performance on all the tasks anagent performs. We term such a feature as �all-or-nothing payments�.Speci�cally, we consider a contracting environment between a principal and sev-

eral risk-neutral agents with limited liability, each working on a project that consistsof multiple activities or tasks. There are two key elements in our framework. First,the multiple (heterogenous) tasks for an agent are interdependent in the sense thatthe agent�s e¤orts in the multiple activities are complementary for the principal. In-deed, in reality, complementarities among the tasks may be the �rst reason for taskclustering and may arise by sharing resources among tasks, the economy of scale orother informational or technological synergy.2 The second key element is the exis-tence of multiple performance measures: The principal may compensate an agent onboth veri�able output performance measures and input performance measures. Theoutput measures may be pro�ts, sales revenue, the security of an entire system ofcomputer networks, while the input measures can range from veri�able informationsuch as standard routines, working hours or service quality from customer satisfac-tion surveys, to subjective observation such as supervisors� subjective evaluations,dedication or support for the organization, etc.In such a multitask multi-agent framework, we characterize the principal�s opti-

mal incentive schemes in three natural contracting environments: unobservable inputmeasures,3 veri�able input measures, and unveri�able but observable input measures.We �nd that in all three optimal incentive schemes, the crucial element is an all-or-nothing payment scheme, where the principal rewards the agents only if an �allsuccess�outcome (on both of the input and output measures) is observed.For illustration of the framework and the results on optimal incentive schemes,

consider the problem where a business owner, who hires several salespersons, designsthe best compensation scheme. Each salesperson�s job involves multiple activitieswhich are complements for the owner�s overall pro�t: searching and landing newaccounts, up-selling existing customers, technical support and customer service, etc.A salesperson�s compensations can be based on the pro�t he generates for the owner,and possibly on some observable input measures, such as hours worked, the numberof new accounts landed, period reviews, etc. Our results provide a rationalizationfor the common use of an optimal compensation in a general environment: Theoptimal incentive scheme takes the form of a straight commission-like payment scheme� a salesperson is only paid a positive reward after a high pro�t level, as well as

2See MacDonald and Marx (2001) for interesting real-world examples on such complementarity.3This is equivalent to the contracting environment where input measures are observable but

unveri�able, and a contract based on relative performance of the agents is not feasible.

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satisfactory input performance measures mentioned above.4

This paper is most closely related to Laux (2001), Zhao (2008) and Chen (2010).Laux (2001), an important precursor to this study, �rst formally shows that in amultitask with limited liability context, there are gains from multitasking. The keyinsight is that task clustering allows to punish agents for failures on some tasks by low-ering the rewards on successful tasks, thus slackening the limited liability constraint.One contribution of this paper is to extend Laux�s insight to a general setting wheretasks/activities can be asymmetric and interdependent.5 With a more general frame-work, Laux�s insight can also be understood and interpreted in a broader sense (seeSection 4 for a detailed discussion).Laux�s somewhat speci�c setting, however, does not allow for input monitoring.

Analyzing a similar setting but with input monitoring, Zhao (2008) �nds that theprincipal is strictly better o¤ by conditioning the wage scheme only on the noisyoutput signals, as long as not all e¤ort choices are observable, hence an �all-or-nothingmonitoring�result. Lifting a limitation on the incentive schemes, Chen (2010) showsthat (in Zhao�s setting) with veri�able outputs and inputs, the optimal contractthat exploits the risk-neutrality of the agent (with limited liability) to its full extentindeed uses both measures and entails a lower wage cost. Another contribution of thisstudy is to extend the result in Chen (2010) to a general framework with asymmetricinterdependent tasks and multiple performance measures.Most importantly, this study unites the results in Laux (2001), Zhao (2008) and

Chen (2010) by providing a framework that embeds and generalizes all these models.In particular, our framework permits to study what happens if input measures areobservable but not veri�able. We show that the rent squeezing insight, which Lauxformulated when inputs are not observable, not only extends to the case of observableand veri�able inputs (Zhao (2008) and Chen (2010)), but also to a situation whereinputs are observable but not veri�able.The remainder of the paper is organized as follows: Section 1.1 provides additional

literature review. Section 2 introduces the model. Our results on the principal�soptimal contracts for the three cases mentioned previously are presented in Section3. Section 4 contains concluding remarks. All proofs are relegated to an Appendix.

1.1 Additional Related Literature

This paper straddles two strands of the literature: multitask agency problems andrelative performance incentive schemes. The �rst formal study on multitask agencyproblems was Holmström and Milgrom (1991), where (task-speci�c) linear piece rates

4Admittedly, designing the best compensation plan for salespeople in reality is much more com-plicated than the environment considered in this paper and has been a highly controversial topic(see the review article Bergen, Dutta and Walker (1992)).

5In our setting, the outputs are supermodular and the e¤ort costs are submodular in the agents�e¤ort choices, which generalizes the settings in the papers above.

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are studied and it is found that low-powered incentive schemes can be optimal whenthere are di¤erences in measurement accuracy on the tasks.6 Bond and Gomes (2009)study a class of multitask agency problems with symmetric and independent tasksand continuous e¤ort choices. With two contracting constraints (the budget con-straint and the monotonicity constraint), the authors characterize the optimal incen-tive schemes, which are generally �fragile�, and a new source of allocative ine¢ ciencyacross tasks is also identi�ed.7 While some similarities exist, our setting and that inBond and Gomes (2009) do not overlap completely as our setting allows asymmet-ric and interdependent tasks. Moreover, we consider the issues of veri�ability andmultiple performance measures, which are absent in Bond and Gomes (2009).8

An additional complementary literature on multitasking, instead of taking taskassignment as given, studies optimal task assignment/division. This includes thejob design model in Holmström and Milgrom (1991), the project allocation issue inLaux (2001), optimality of task separation in Dewatripont and Tirole (1999), jointor individual accountability for performance in Corts (2007), and bene�cial e¤ects oflabor division in Ratto and Schnedler (2008).On the literature of relative performance incentive schemes, a classic reference is

Lazear and Rosen (1981) who show that with risk-neutral agents, both rank-order in-centive schemes and individual-based incentive schemes allocate resources e¢ ciently.Subsequent studies have shown that correlation across agents in the randomness af-fecting output signals enables the principal to gain by basing an agent�s reward onother agents�outputs.9 In another set of articles on contracting with multiple agents(Bhattacharya (1983), Malcomson (1984, 1986)), it was shown that an additionalattractive feature of rank-order tournaments is their correction of a potential moralhazard of the principal when agents� output performances are not veri�able. Incharacterizing the optimal incentive schemes with observable but unveri�able inputs(Section 3.3), we use a similar idea to design a relative performance contract thatuses multiple performance measures with mixed veri�ability and we identify anotherequivalence result. We postpone a more detailed discussion on comparisons with the

6The multitasking literature has somewhat evolved since the 1990s, mostly happening in ac-counting (e.g., Feltham and Xie (1994), Datar, Kulp and Lambert (2001)) and labor economics(e.g., Baker (2000, 2002), Schnedler (2008)). Other less related papers include Holmström and Mil-grom (1994), Itoh (1991, 1994) and MacDonald and Marx (1991). Dewatripont, Jewitt and Tirole(2000) provide a survey on various contributions of multitask agency theory and applications.

7Speci�cally, with multiple tasks and a continuous e¤ort choice on each task, an e¤ort vectoris incentive compatible i¤ a local incentive constraint and an extremal incentive constraint holdsimultaneously. Hence, small shocks on fundamentals (e.g., e¤ort costs) may result in a total collapseof the agent�s e¤ort. This �fragility� contrasts with the standard contracting problem where onlylocal incentive constraint bind. E¤ort misallocation arises when the agent works only on a subsetof tasks while the principal prefers the agent to �diversify�his e¤ort choices among all the tasks.

8Raith (2008) studies the problem of designing optimal compensations when multiple performancemeasures exist, from an angle where the agent has private information on his productivity.

9See, e.g., Holmström (1979, 1982), Green and Stokey (1983), Nalebu¤ and Stiglitz (1983), andMookherjee (1984).

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literature on relative performance measures to Section 3.3.

2 Model

Consider a situation where a principal is endowed with multiple identical and inde-pendent one-time projects. The principal can hire m agents (m 2 N, m � 2), one foreach project, to complete these projects. The basic elements of each principal-agentrelationship are described as follows:E¤ort and Production: Each project contains n > 1 heterogenous and possiblyinterdependent tasks, which are collected in the set N = f1; 2; :::; ng. For each task,the agent makes the choice of an e¤ort level, 0 or 1 (representing shirk and workrespectively). For notational simplicity, denote a generic e¤ort choice of the agent asE � N , indicating that the agent works in all tasks in E, and shirks in all tasks inNnE. Given this, an e¤ort vector that features working (resp., shirking) in all tasksis simply N (resp., ?).Given e¤ort vector E � N , output signal y, which can be interpreted as pro�ts,

sales revenues or customer satisfaction surveys, is generated stochastically accordingto the probability distribution P (yjE), where y 2 Y . Y , being the set of all signals,is �nite and contains at least two elements. In our setting, we allow the case where aseparate output signal is available for each task, as well as the scenario where thereis an aggregate output signal for all n tasks. We assume that distribution P has fullsupport, or P (yjE) 2 (0; 1) for all y and E. In addition, conditional on e¤ort choices,the output signals are independent across agents.Performance Measures: The principal can use several performance measures tomonitor the agent�s e¤ort. The �rst is the standard output-performance measure �the output signal y, which throughout the paper is assumed to be veri�able. Besidesthe imperfect output signals, an input-performance measure may also be available� the principal can costlessly and perfectly observe some, but not all, e¤ort levelsin the n tasks, where the number of such tasks (called observable tasks hereafter)is k, k 2 f1; 2; :::; n� 1g. For the analysis in the sequel, we consider cases wherethe observation of the k e¤ort levels is veri�able (i.e., standard routines, administra-tive/clerical tasks or simply working hours), as well as cases where such observationhas the nature of �I know it only when I see it�and is thus di¢ cult and costly tobe veri�ed to a court (i.e., professionalism, dedication, or subjective evaluation of asupervisor). Hereafter, we will use terminology �output signals� to denote �outputperformance measures�, and �perfect observation on the k e¤ort choices�to denote�input performance measures�.Payo¤s: In our setting, both the principal and the agent are risk-neutral. We alsoassume that the agent has a reservation utility level of zero and is protected bylimited liability, i.e., payments from the agents to the principal are never feasible.10

10The case of a generic reservation utility level u can be easily adapted without qualitatively

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For simplicity, we assume that the principal�s objective is to motivate the agents toexert e¤ort on all n tasks at a minimal wage payment.11 Given a wage w, interpretedas incentive payments, the agent�s payo¤ is

u (w; E) = w � C (E) ;

where C (E) represents the agent�s cost of exerting e¤ort E, and C : 2N ! R. Weassume that the cost function C (E) is submodular, i.e., for any i; j 2 N , i 6= j, andE � Nn fi; jg,12

0 < C (E [ fi; jg)� C (E [ fjg) � C (E [ fig)� C (E) : (1)

We normalize the cost of shirking in all tasks to be zero, or C (?) = 0. Our speci-�cation of C (E) allows for cases where costs in exerting e¤ort in di¤erent tasks areasymmetric or non-linear, subsuming the symmetric and linear e¤ort cost speci�ca-tion in Laux (2001) and Zhao (2008) as a special case. Submodularity of C (E) canbe intuitively interpreted as that e¤orts are complements for each agent as workingin one task helps reducing the cost of exerting e¤ort in another task.Assumptions on P (yjE): Denoting the signal pro�le that features an overall suc-cess as s, the output function P (sjE) is assumed to be supermodular for any E � N .Formally, the function P (sjE) is supermodular in E if and only if

P (sjE [ fi; jg)� P (sjE [ fjg) � P (sjE [ fig)� P (sjE) � 0; (2)

for all i; j 2 N , i 6= j, and E � Nn fi; jg. In our setting, �the function P (sjE)having increasing di¤erences�is equivalent to P (sjE) being supermodular.13 Noticethat supermodularity is only imposed on P (sjE) and not on P (yjE) for y 6= s.Importantly, our speci�cation of P (sjE) allows for symmetric and independent tasks(as in Laux (2001), Bond and Gomes (2009), and Zhao (2008)), as well as asymmetricand interdependent tasks.14 Observe that supermodularity of P (sjE) can be intu-itively interpreted as that the agent�s e¤ort levels in the tasks are complements for

a¤ecting our results.11Although natural and important in our setting, this assumption also implicitly implies that e¤ort

costs be invariably small compared to the project pro�ts. In addition, maintaining this assumptionprevents us from analyzing issues such as the agent�s e¤ort mis-allocation (as in Holmstrom andMilgrom (1991) and Bond and Gomes (2009)). If we adopt a standard cost-and-bene�t analysis forthe principal, then implementing partial e¤ort might be optimal, but then one might argue why theprincipal would assign all n tasks to a single agent (such an argument is admittedly speci�c for oursetting and not useful in general). We discuss on this issue further in Section 4.12As the discussion on supermodularity, submodularity here is equivalent to C (E) having decreas-

ing di¤erences, and an equivalent de�nition can be given similar to that of P (sjE).13For a function de�ned on a product of ordered set, supermodularity is equivalent to increas-

ing di¤erences (Vives (2000)). An alternative but equivalent de�nition is P (sjE) + P (sjE0) �P (sjE \ E0) + P (sjE [ E0), for all E;E0 � N .14In Bond and Gomes (2009), the principal�s output function can be concave, linear or convex in

the number of successful tasks. However, the tasks are symmetric and importantly independent, inthe sense that an e¤ort level in one task does not a¤ect the probability of success of another task.

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the principal � a high e¤ort in one task, while inducing success in that task morelikely, also increases the probability of success of another task, ceteris paribus.In addition to the supermodularity condition in (2), we assume that signal s is

also the state with the highest likelihood ratio for the probability functions P (yjN)and P (yjK), where K � N denotes the e¤ort choice where the agent only works inall k observable tasks:

(i)P (sjN)P (sj?) >

P (yjN)P (yj?) ; and (ii)

P (sjN)P (sjK) >

P (yjN)P (yjK) for all y 2 Y , y 6= s. (3)

The maximum likelihood ratio assumption is common in the literature (e.g., Innes(1990), Laux (2001), Zhao (2008) and Bond and Gomes (2009)). It is worth notingthat we typically do not require both (i) and (ii) in (3) to hold simultaneously:15

When the principal only uses output signals in a contract, the relevant binding in-centive constraint is shirking from N to ? � in this case, (i) should hold so thata positive wage is paid only after output signal s; On the other hand, when bothoutput signals and e¤ort observation in the k observable tasks can be used to moti-vate e¤ort N , the relevant binding constraint is shirking from N to K (this is truebecause of (1) and (2)), in which case (ii) should hold so that output signal s is againthe most informative signal in detecting deviation. Our results (except Proposition 3with wage comparisons) can thus be obtained with a weaker version of (3), namely,the maximum likelihood ratios are obtained at di¤erent output signals.Finally, we describe the time and information structure in the principal�s con-

tracting problem. The principal �rst proposes the wage schemes to the agents, whocan then accept or reject the o¤er(s). If at least one agent rejects, the game ends.Otherwise, the agents simultaneously make e¤ort choices. The resulting noisy out-put signals and the e¤ort levels on the observable tasks are observed, along with theimplied wage payo¤s to the agents. We assume that the proposed wage schemes andthe realized wage payments to the agents are veri�able.

3 All-or-Nothing Payments in Optimal Contracts

We now characterize the principal�s optimal contracts. Three natural scenarios areconsidered in the characterization: (1) unobservable e¤ort observation (on the ktasks), (2) observable and veri�able e¤ort observation, and (3) observable but un-veri�able e¤ort observation. For the �rst two cases, it su¢ ces to consider individualoptimal contracts, and we consider an optimal contract that is based on the agents�relative performances in the last case. Importantly, we show that a unifying featurethat ties all the optimal contracts together is an all-or-nothing payment scheme.

15We present a two-task example in Section 3.2 to further illustrate the two conditions (2) and(3). In particular, conditions (i) and (ii) in (3) can arise simultaneously.

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3.1 Optimal Contract with Unobservable Inputs

Consider �rst the case where the input measures are not observable to the principal,hence only the noisy output signals are relevant in designing the optimal contract.Notice that this case is equivalent to the scenario where the observation on the k e¤ortchoices is observable but unveri�able when the principal can only o¤er individualcontracts.The principal�s contracting problem is to minimize the expected wage payment,

based only on the output signals, subject to the limited liability and the incentiveconstraints so that the agent chooses e¤ort N , as described by (P) :

(P)

maxw(y)

�P

y2Y P (yjN)w (y)

s:t:w (y) � 0; 8 y 2 Y; (LL)Py2Y

P (yjN)w (y)� C (N) �Py2Y

P (yjE)w (y)� C (E) ; 8E � N: (IC)

Here w (y) is the wage payment given output signal y; (LL) the limited liabilityconstraint; and (IC) the incentive constraint indicating that the agent prefers e¤ortN to any other e¤ort. The main di¢ culty in solving (P) lies in that the incentiveconstraint (IC) contains a total of (2n � 1) individual constraints, representing vari-ous ways that the agent can deviate from the e¤ort N . To circumvent the di¢ culty,we �rst consider a reduced program where (LL) is replaced by an extremal deviationconstraint where the agent prefers N to ?. We then show that the same optimalscheme solves both program (P) and the reduced program.16 Proposition 1 presentsthe optimal contract.

Proposition 1 Suppose that the e¤ort observation on the k tasks is not available.The principal�s optimal contract is to o¤er the agent wage scheme �w (y), which solvesprogram (P):

�w (y) =

(C(N)�C(?)

P (yjN)�P (yj?) ; if y = s;0, otherwise.

(4)

The intuition of Proposition 1 is as follows: Given that the likelihood ratio P (yjN)P (yj?)

is maximized when y = s, signal s is hence the most informative state in detectingshirking from �N�to �?�� here, as mentioned before, we only require condition (i)in (3). The principal should then pay the agent a positive amount only after seeingsignal s. Given such a wage scheme, the agent�s payo¤ from an e¤ort pro�le E is:

u (�w; E) =

�C (N)� C (?)P (sjN)� P (sj?)

�P (sjE)� C (E) ;

16This is a standard approach in the literature (Laux (2001), Zhao (2008) and Bond and Gomes(2009)). We follow Bond and Gomes (2009) in calling this an extremal incentive constraint.

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which is supermodular in E, or the agent views the e¤ort choices in di¤erent tasksas complements under �w (y) in (4). As a result, the agent�s goal is perfectly alignedwith that of the principal and exerting e¤ort in all n tasks is hence a best response.As mentioned, the rent squeezing insight and the all-or-nothing payment scheme

being optimal are �rst obtained in Laux (2001) in a similar multitask agency settingwith symmetric independent tasks and constant linear e¤ort costs.17 Proposition 1generalizes these results by showing that the interesting features carry over to theoptimal contract in a more general multitask contracting environment where taskscan be interdependent and asymmetric.

3.2 Optimal Contract with Observable and Veri�able Inputs

We now turn to the principal�s optimal contracting problem when both the outputsignals and e¤ort observation on the k tasks are observable and veri�able. Speci�cally,the principal�s least cost contract, which can now explicitly depend on both the outputsignals and the veri�able e¤ort observation, solves program (P 0):

(P 0)

maxw(y)

�P

y2Y P (yjN)w (y)

s:t:

w (y) � 0; 8 y 2 Y; (LL)Py2Y P (yjN)w (y)� C (N) �

Py2Y P (yjK)w (y)� C (K) ; (IC1)P

y2Y P (yjN)w (y)� C (N) � 0: (IC2)

Recall that K is the e¤ort choice where the agent works only in the k observabletasks. The incentive system resembles a forcing contract where the agent is punishedseverely (a zero wage) if he shirks in any of the k tasks, which is feasible as the e¤ortobservation in the k tasks is now veri�able. Moreover, it su¢ ces for the principal todeter the deviation of shirking in all (n� k) unobservable tasks (IC1), for a similarreason as in Proposition 1. As (IC1) might be binding, we only require condition (ii)in (3) to hold for this section. Condition (IC2) ensures a non-negative expected wagepayment for the agent when he chooses e¤ort N .The principal�s optimal contract derived from the contracting problem (P 0) is

presented in Proposition 2.

Proposition 2 Suppose that both the output signals and the e¤ort observation on ob-servable tasks are veri�able. The principal�s optimal contract with both performancesis to o¤er the agent a positive wage w (y) only if y = s and the agent works in all kobservable tasks, where

w (s) =

(C(N)�C(K)

P (sjN)�P (sjK) ; ifP (sjN)P (sjK) �

C(N)C(K)

;C(N)P (sjN) ; if

P (sjN)P (sjK) >

C(N)C(K)

:(5)

17Also see Proposition 1 in Zhao (2008) for a similar result, but with a di¤erent proof.

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In the optimal contract with veri�able inputs, the principal uses the harshestpunishment whenever she observes a shirk in an observable task: a zero wage forall y�s. When the agent indeed works in all k observable tasks, we distinguish thefollowing two cases in obtaining the optimal wage scheme:First, if it holds that P (sjN)

P (sjK) �C(N)C(K)

, which happens when the output signals arenoisy or not very correlated, or k is small, then only the incentive constraint (IC1)is binding: Since now monitoring using both output signals and observation on thek tasks is not e¤ective or precise enough, the agent has to receive some positive rentso as to be motivated to exert e¤ort in all n tasks.Second, if we have P (sjN)

P (sjK) >C(N)C(K)

(when the output signals are precise or highlycorrelated, or k is large), then incentive constraint (IC2) is the only relevant incentiveconstraint. In this case, the output signals and the perfect observation in the kobservable e¤ort choices are jointly powerful enough to incentivize the agent to exerte¤ort in all n tasks at a zero rent.Observe that constraint (IC2) in (P 0) can also be regarded as the agent�s partici-

pation constraint. This constraint is not binding in Laux (2001) or Zhao (2008). Inour general multitask setting, however, this condition may be binding as intuitivelythe rent left to the agent can be pushed very low (indeed, zero) with the additionalveri�able e¤ort observation and interdependent tasks, which generates informationaland/or technological synergy among the tasks.As described, multitask agency problems with partial e¤ort observation have also

been analyzed in Laux (2001), Zhao (2008) and Chen (2010), with the assumptionof symmetric independent tasks with constant linear costs. Laux (2001) considers ascenario where the agent works on N type A projects (with veri�able noisy outputsignals only) and M type B projects (with veri�able perfect e¤ort observation only)with independent tasks. Proposition 2 is more general in several aspects: First,our principal observes the agent e¤ort choices in the k observable tasks, which alsoa¤ect the overall output signals in our setting. In particular, Proposition 2 applies toscenarios where there is only an aggregate output signal for all (M +N) tasks. This isimportant for cases where the output signal (e.g., sales revenues) is generated by theagent�s e¤orts in an integrated way such that it is di¢ cult to single out the speci�c�contribution�of each individual task. Second, our setting allows for asymmetric andinterdependent tasks. As in Chen (2010), Proposition 2 can also be applied to showthat if the principal is not restricted to using a forcing contract as in Zhao (2008),then the principal is better o¤ under the optimal wage scheme w (y) in (5) than thewage scheme �w (y) (in (4)), which only depends on the output signals. The followingcorollary illustrates this:

Corollary 1 The principal�s expected wage payment in the optimal contract withmixed performances (Ew) is lower than that in the optimal contract with outputperformance (E �w): Ew �E �w. In particular, Ew <E �w holds under one or both ofthe following conditions: (a) P (sjN)

P (sjK) >C(N)C(K)

and (b) strict supermodularity of P (sjE)

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or strict submodularity of C (E).18

Two brief remarks are in order for Corollary 1: First, conditions (a) and (b) can beintuitively interpreted as that there is indeed informational or technological synergyamong the tasks or between the k observable tasks and the (n� k) unobservable tasks.Hence, the optimal contract with mixed performances is strictly better. Second, atan intuitive level, the simple wage comparison result is not surprising and comesfrom the fact that in a moral hazard setting, additional informative signals enablethe principal to better address the agent�s incentive problem. This is also consistentwith the standard results in the previous moral hazard literature.19

We now present a two-task example to further illustrate our setting and our resultsobtained so far.

An Example: Consider a one-agent setting where the project has two identical butinterdependent tasks. For each task, the agent chooses C (work) or D (shirk), and forexpositional convenience, let e¤ort costs be symmetric and linear: The cost for workon a single task is c > 0 and shirk is costless. Suppose that for each task i 2 f1; 2g, theoutput signal yi takes two possible values, yi 2 Y = fs; fg, representing �success�and�failure�respectively, and for both tasks, we have P (sjC) = P (f jD) = q 2

�12; 1�,

indicating that for each task, output signal s (resp., f) occurs more likely when theagent works (resp., shirks) on the task. We further assume that the (linear) correlationbetween the two output signals y1 and y2 is always �, which can be interpreted as thespillover e¤ects of an e¤ort in a task has on the outcome of the other task. Finally,we assume that in addition to y1 and y2, the principal can also observe the seconde¤ort choice but not that in the �rst task.We �rst evaluate the supermodularity condition (2) and the maximum likelihood

condition (3). Note that by the above speci�cation, the probability distributionP (yjE) can be expressed as (E 2 fCC;CD;DC;DDg and y 2 Y 2):

s fs q2+�q (1�q) (1��) q (1�q)f (1��) q (1�q) (1�q)2+�q (1�q)

P (yjCC)

s fs (1��) q (1�q) q2+�q (1�q)f (1�q)2+�q (1�q) (1��) q (1�q)

P (yjCD)Fig 1. Signal Distributions Conditional on E = CC and E = CD:

The output signal distributions P (yjDC) and P (yjDD) can be de�ned analogously.Given P (yjE), one can readily verify that if q > 1

2and � � 0, strict supermodularity

18Strict supermodularity of P (sjE) (resp., submodularity of C (E)) holds if inequality (2) (resp.,inequality (1)) is strict for all pairs i; j in N . Clearly, this is stronger than necessary for the purposeof Corollary 1, which remains valid if such inequality in (1) or (2) holds for at least one pair of tasks.We present condition (b) in its current form only for expositional convenience.19See, e.g., Holmström (1979), Gjesda (1982), Grossman and Hart (1983), and Kim (1995).

11

of P (sjE) and the maximum likelihood ratio condition hold.20

We next consider optimal individual contracts in this simple setting. The princi-pal�s optimal expected wage payments in the optimal output-based contract and theoptimal contract with mixed performances can be calculated as, respectively,

E �w (y) =2c � P (sjCC)

P (sjCC)� P (sjDD) and Ew (y) =(

c�P (sjCC)P (sjCC)�P (sjDC) , if

P (sjCC)P (sjDC) � 2

2c, if P (sjCC)P (sjDC) > 2

:

Notice that E �w (y) > Ew (y) given strict supermodularity of P (sjE). In addition, ifP (sjCC)P (sjDC) > 2, equivalently (1� �) (1� q) <

13, the optimal mixed-performance contract

speci�es an expected wage 2c. This implies that if the synergy between the outputsignals is large enough (� close to 1) and/or if output signals are precise enough (qclose to 1), the optimal mixed-performance contract induces e¤ort E = CC at a zerorent for the agent.

3.3 Optimal Contract with Observable and Unveri�able In-puts

We now analyze optimal incentive schemes when the principal observes both perfor-mance measures, but only the noisy output signals are veri�able. In such a scenario,the observation on the k e¤ort choices cannot be used in an individual contract in aself-enforcing way: When incentive schemes depend on unveri�able performance mea-sures, a moral hazard problem arises � The principal will always ex post misrepresentan agent�s performance to reduce �nal wage payments, as such misrepresentation can-not be veri�ed and denied by a court. Such ex post opportunistic behavior from theprincipal would render the contract in Proposition 2 infeasible.The literature has put forward two main approaches where such observable but

unveri�able performance can still be exploited e¤ectively. The �rst approach is to letthe two parties arrange their transactions repeatedly so as to design a self-enforcingrelational contract : If both parties engage in repeated transactions, myopic behaviorsuch as the one mentioned above will be costly, as each party knows such behaviorwill be detected and punished in the future.21 This approach is clearly not feasible inour one-shot setting. The other approach is found in the literature of moral hazardwith multiple agents, which suggests that when interacting with multiple agents, theprincipal can design a contract that is based on the agents� relative performances(Bhattacharya (1983) and Malcomson (1984, 1986)). If the incentive scheme is basedon the rankings of the agents�performances, the potential adverse incentives of the

20Speci�cally, we have P (sjCC)� P (sjE) � P (sjE)� P (sjDD) ; where E = DC or CD. More-over, P (sjCC)P (sjDD) �

P (yjCC)P (yjDD) and

P (sjCC)P (sjDC) �

P (yjCC)P (yjDC) hold for all y 2 Y

2. In particular, condition �q > 12

and � � 0�is imposed mainly due to the maximum likelihood condition. Supermodularity in thissetting merely requires �1 � 4q (1� q) (1� �)�.21See Kvaløy and Olsen (2009) and references therein on this line of research.

12

principal disappears as the total compensation paid to the agents is independentof the �nal performance outcome, leaving no �nancial gain for the principal to fal-sify reporting the performance rankings. Following this insight, we design a relativeperformance contract. Notice that our relative performance contract importantly em-ploys both the veri�able output signals and the unveri�able e¤ort observation, whichis di¤erent from usual rank order contracts in the literature.We �rst describe the design of the relative performance contract, which speci�es

a game for the agents. The key issue of the design lies in carefully specifying theagents�payo¤s in the game so that the contract engenders a Nash equilibrium wherethe agents choose high e¤ort in all n tasks. First, given our setting with risk-neutralagents, the principal can without loss of generality specify two rewards in the relativeperformance contract, w and l, where w is the reward for the winner(s) and l for theloser(s). With the limited liability constraint, we can hence set l = 0. The incentivescheme is then designed in two steps as follows:

1. Agents with output signal s are �rst singled out as potential recipients of thereward w. As long as there are at least two agents whose realized output signalpro�les are s, the principal has to incur a wage payment w (the principal paysnothing if there is at most one agent with output signal s).

2. Among the agents with output signal s, prize w is awarded to the agent with the�best�e¤ort choice in the k observable tasks � �best�in the sense of exertinge¤ort on the largest number among the k tasks. In case of a tie, the winner ofthe prize w is chosen with equal probability.22

The principal�s problem here is to �nd the minimal reward w so as to ensure theexistence of a Nash equilibrium where all agents choose e¤ort N . This is describedby the following program

�PR�:

�PR� maxw �w

s:t:w � 0; (LL)Ui (N ;N; � � � ; N ;w) � 0;8i (IC 0)Ui (N ;N; � � � ; N ;w) � Ui (E;N; � � � ; N ;w) 8i;8E � N; (IC 00)

where Ui (E;N; � � � ; N ;w) is agent i�s expected utility given w and e¤ort pro�le(N; � � � ; N) of the other agents, when i chooses e¤ort E;

Ui (E;N; � � � ; N ;w) (6)

=

� Pm�1j=1

P (sjE)j+1

�m�1j

�P (sjN)j (1� P (sjN))m�j�1 � w � C (E) , if K � E;

0; otherwise,

22With risk-neutral agents, one can equivalently design the scheme by giving the winning agentsequal shares of the prize w in case of a tie.

13

where m is the number of agents (m � 2, m 2 N) and �K � E� implies thatwhen choosing E, agent i works at least in all the k observable tasks. In derivingUi (E;N; � � � ; N ;w), notice that agent i, given (N; � � � ; N), receives positive payo¤sif and only if (1) i works at least in the k observable tasks, and (2) i�s realized outputsignal is s, together with at least one other agent.We now present the key result of this subsection: the optimal relative performance

contract solving�PR�, again featuring all-or-nothing payments, is as good as the

optimal individual contract identi�ed in Proposition 2 for the principal:

Proposition 3 Assume that only the output signals are veri�able. Then for theprincipal, the optimal relative performance contract solving

�PR�is ex ante equivalent

(in expected wage payments) to the optimal individual contract (w1 (y) ; w2 (y)) in (5),the latter being only feasible when both the output signals and the e¤ort observationon the k tasks are veri�able.

We present several remarks for Proposition 3:First, in our setting, when e¤ort observation is unveri�able, an enforceable op-

timal individual contract is the one characterized in Proposition 1. Together withCorollary 1, Proposition 3 shows that a contract based on relative performance evalu-ations can indeed outperform optimal individual contracts when veri�ability of directe¤ort observation is an issue. This is consistent with the previous literature on thatrank-order contracts can correct the moral hazard issue of the principal. The novelaspect of Proposition 3 is to show that in a general multitask setting with multipleperformance measures, some being veri�able, a relative performance scheme can beusefully adapted to employ the veri�able aspects of the performances, so as to decreasethe principal�s expected incentive payments.Following this point, Proposition 3 suggests one potential motivation for common

practices such as outstanding employee awards frequently used in organizations: Someaspects of employees�e¤ort are observable, but may also be subjective and di¢ cult toverify such as professionalism, dedication to the job, support for the organization, etc.In such environments, a simple and yet e¤ective way to employ such information isaugmenting the individual performance scheme with relative performance evaluationsso that incentive payments depend on both the absolute individual performances andan ordinal ranking of the unveri�able and subjective evaluations. Such a scheme isself-enforcing and cost e¤ective to the principal, especially when a strong synergyexists among the tasks and the observable e¤ort is important for the principal.Second, the crucial feature of our relative performance scheme is that the principal

pays reward w if and only if there are two or more agents having output signal s, andsuch a payment is made independently of the agents�e¤ort choices in the k observabletasks. While identifying two or more output signal s is veri�able and feasible in oursetting, such a speci�cation may seem awkward. The reason for such a speci�cationis the following: The optimal contract in Proposition 2 is especially e¤ective whenP (sjN)P (sjK) >

C(N)C(K)

, as the agent is left with zero rents. If one were to specify �w will

14

be awarded when there is at least one agent with output s�, every agent wouldthen choose no e¤ort at all since e¤ort ? always yields a small but positive payo¤,give our full-support assumption on P (sjE).23 Notice that the possibility of sucha pro�table deviation under zero rents also implies that our relative performancescheme dominates the scheme with paying a reward regardless of output signals,i.e., when output signals are unveri�able (also see our fourth remark below). Onepossible and natural interpretation of our speci�cation, however, is that the principalcan stipulate in the contract that whenever the prize w is awarded, at least two agentswill be chosen as the recipients. This is indeed commonly seen in practice.Third, evidently, our relative performance contract is not the unique contract to

implement the result of Proposition 2: Since the agents choose e¤ort simultaneously,the key incentive issue is that e¤ort N a best response for each agent, if all theother agents choose N . As our agents are risk-neutral, one can consequently designdi¤erent relative performance schemes to achieve the same outcome (for example, onecan design a scheme with multiple levels of non-zero prizes). Our relative performancecontract is, in our opinion, attractive as it is simple and it achieves the best possibleoutcome for the principal in our setting.Fourth, veri�ability of the output signals is important for the equivalence result in

Proposition 3. Indeed, if both output signals and perfect input observations are un-veri�able, one can similarly construct a relative performance scheme that is based onordinal rankings of the agents�performances, but the principal now has to pay rewardw regardless of the �nal outputs of the agents. One can show that the correspondingrank order contract is dominated by the one characterized in Proposition 3 in thatthe principal pays a higher expected reward when the outputs are unveri�able. Thisstands in contrast to Lazear and Rosen (1981), but is consistent with the �ndings inGreen and Stokey (1983) on that a relative performance incentive scheme may usethe available information in a rather crude way.24 For detailed analysis on this, seethe working paper Chen (2011).Finally, it is important to note that one potential problem of our relative perfor-

mance contract is that our contract design does not necessarily implement the e¤ortpro�le (N; � � � ; N) uniquely: there could be other Nash equilibria where the agentsexert e¤ort in some or none of the n tasks. This issue is not new and has been doc-umented in the literature.25 Our general framework, while elegantly characterizes a

23Observe that in this case, Ui (?;N; � � � ; N ;w) =Pm�1

j=1P (sj?)j+1

�m�1j

�P (sjN)j (1� P (sjN))m�j�1�

w � C (?), which is always positive given the full-support assumption and C (?) = 0.24In Green and Stokey (1983), in the absence of a common shock, tournaments are dominated

by individual contracts as they introduce extra randomess in the (risk-averse) agents� paymentswhich now depend on other agents� idiosyncratic shocks. In our setting with risk-neutral agentsand limited liability, rank-order contracts are dominated as they ignore some cardinal feature ofoutput signals. While the two settings are not completely comparable, both results illustrate thatrank-order contracts make ine¢ cient use of information.25See Mookherjee (1984) and Demski and Sappington (1984) for various agency examples in

correlated environments on this multiplicity problem. Ma, Moore and Turnbull (1988) and Ma

15

general environment for all-or-nothing payment schemes, is mathematically inconve-nient to fully describe the complete set of equilibria: The supermodular and submod-ular speci�cations are meant to capture a general setting where only the extremalincentive constraint is binding, so as to induce an all-or-nothing payment result. Inproving Proposition 3, a similar phenomenon arises, but now the optimal reward wcarries a non-trivial probability component (constant � in the proof). To character-ize the full set of Nash equilibria under this reward, one inevitably encounters thecomplex task of evaluating the change in probability (compared to �) of obtainingw relative to the change in e¤ort costs, where our non-separable speci�cations giveus little hope in disentangling such technical complexity.One possible argument for the multiplicity issue is that in our symmetric setting,

it su¢ ces to treat �everyone exerting full e¤ort� as a focal point. However, suchan argument may also rest on shaky ground without a corresponding equilibriumselection theory.26 This being said, a possible approach to mitigate the problem ofagents coordinating on a wrong equilibrium is to introduce discrimination among theex ante identical agents: The principal divide the reward evenly among the agentsonly if all the agents work in the observable tasks and obtain output s. When agentscoordinate on partial e¤orts in the k tasks (and obtain output s), the principal givesreward w to a single agent (and the speci�cation of such agents can even depend onthe speci�c mis-coordination so that no single agent regards (partial) shirking as adominant strategy).27

4 Conclusion and Discussion

Designing the right incentive plans for organizations is complicated by the existence ofmultiple performance measures, multiple activities involved in a speci�c job, and thepossibility of using relative performance schemes when there are multiple employees.This paper provides a general multitask agency framework to analyze the principal�soptimal incentive schemes with the presence of multiple agents and multiple perfor-mance measures with di¤erent veri�ability issues.In our framework, the principal�s output is supermodular and an agent�s e¤ort

cost is submodular in the e¤ort choices of the agent. Such a general speci�cationapplies to scenarios where an agent�s various activities contribute to a single outputin an integrated and complementary way so that identifying the speci�c contribution

(1988) demonstrate that the principal can however use more subtle contractual arrangements (e.g.,adding extra innocuous strategies, appointing some agent to monitor the other agents) to alleviatethe multiplicity problem. A similar issue arises in relational contracts as repeated interactions makemultiple implicit contracts sustainable.26Indeed, as remarked by a referee, there is probably little hope to argue in favor of the equilibrium

if agents obtain zero rents in the equilibrium (when P (sjN) =P (sjK) > C (N) =C (K)).27Such a discrimatory policy may seem arbitrary, but it could be a motivation for ranks and

hierarchies among (identical) agents. See Winter (2004) for an interesting illustration of this idea.

16

of e¤ort on an individual activity to the principal�s overall objective is di¢ cult. Moreimportantly, our framework enables us to systematically vary the characteristics ofthe e¤ort observation in analyzing the principal�s optimal incentive schemes.Under this framework, we characterize the principal�s optimal contracts in three

natural scenarios: unobservable inputs, observable and veri�able inputs, and observ-able but unveri�able inputs. A common feature arising in the optimal contracts is an�all-or-nothing payment� incentive scheme. Our result contributes to the literatureon multitasking agency problems with limited liability by embedding and generalizingseveral existing models (Laux (2001), Zhao (2008) and Chen (2010)).Although our results are obtained in a fairly general framework, a remaining nat-

ural question is then �how far the proposed set-up resembles a �maximal domain�for all-or-nothing payments?�It is without doubt that risk-neutrality (with limitedliability) of the agents is crucial for our results so that the risk-sharing issue is irrel-evant. What is less clear is the role of supermodularity of pro�ts and submodularityof costs. We divide our discussion on this in two parts.First, with respect to the advantage of combining tasks (though this is not our

main focus and as mentioned, has already been pointed out in Laux (2001)), super-modularity and submodularity are not both necessary to realize �gains from mul-titasking�: Supermodularity of pro�ts and submodularity of costs make the tasksnatural complements for both the agent and the principal. It is hence intuitive tocombine the tasks in an optimal individual contract. However, it is clear from theproof of Proposition 1 that if either supermodularity or submodularity (but not both)fails, as long as the synergy or complementarity of pro�ts (resp., e¤ort costs) is strongenough to dominate the countering e¤ect of decreasing returns to scale in e¤ort costs(resp., pro�ts), then the same result arises.28 On this aspect, though it seems that wehave merely provided a more general framework for achieving gains from multitask-ing, our result makes it clear that complementarities, informational or technological,of the tasks are indeed crucial for combining tasks to a single manager � the previ-ous literature identi�es the informational synergy for the principal, while this paperidenti�es a general form of complementarity for both parties.Second, from a technical point of view, supermodularity of pro�ts and submod-

ularity of e¤ort costs are again not necessary for an all-or-nothing payments result:When either condition (or both) fails, it may no longer be true that only the extremalincentive constraint is binding. Nevertheless, all-or-nothing payments may still arisewhen the principal pays the agent a positive wage only in the state of outcome thatis the most informative in detecting shirking from N to the preferred e¤ort of theagent (i.e., the corresponding binding incentive constraint). A similar argument goes

28Our result is hence not a consequence of some knife-edge phenomenon. This is important forthe submodularity assumption on e¤ort costs: A common approach in the multitask literature isagents regarding e¤orts on various tasks as substitutes: a higher e¤ort on one task crowds out thaton other tasks. Our results are thus robust to small crowding-out e¤ects of e¤orts, especially whenthe complementaries of the tasks are strong for the principal.

17

for the general case where the principal adopts a standard cost-and-bene�t approachin determining the optimal e¤ort choice to implement � though in this case, a newissue of why in our setting assigning these n tasks to the same agent in the �rst placewould emerge.As we have argued above, supermodularity of pro�ts and submodularity of e¤ort

costs are not indispensable in driving our results. Nevertheless, we have chosen towork exclusively with these speci�cations in this paper, as in our opinion such aframework prescribes an intuitive economic environment, is mathematically elegantto work with, and is importantly consistent with the previous literature.

Appendix

Proof of Proposition 1. Consider �rst the following reduced program with (IC)being replaced by the extremal incentive constraint where the agent prefers N to ? :

�Py� maxw(y)

�P

y2Y P (yjN)w (y)

s:t:w (y) � 0; 8 y 2 YP

y2Y P (yjN)w (y)� C (N) �P

y2Y P (yj?)w (y)� C (?) :

Construct the following Lagrangian, where � (resp., � (y)) is the Lagrange multi-plier of the reduced incentive (resp., limited liability) constraint in program

�Py�:

L (w (y) ;�; � (y)) = �X

y2YP (yjN)w (y) + � (y)w (y)

+�hX

y2Y(P (yjN)� P (yj?))w (y)� C (N) + C (?)

i:

The �rst-order necessary condition for w (y) can be derived as:

�P (yjN) + � (y) + � (P (yjN)� P (yj?)) = 0: (A� 1)

First, it is easy to show that � > 0: If not, then (A � 1) implies � (y) > 08y.Complementary slackness � (y)w (y) = 0 and (C (N)� C (?)) > 0 further implythat the extremal incentive constraint is violated:Next, suppose that � (y) = 0 (and hence w (y) > 0) for some y 6= s and � (s) > 0.

Applying (A� 1) for signal y and signal s respectively, we have:29

P (sjN)P (sj?) <

�

�� 1 =P (yjN)P (yj?) ,

which is in contradiction to (3). It follows that w (y) > 0 only if y = s. Hence, theoptimal wage scheme for program

�Py�is:

�w (y) =

(C(N)�C(?)

P (sjN)�P (sj?) ; if y = s;0, otherwise.

29Note that � = 1 yields P (yj?) = 0, contradicting the full-support assumption on P .

18

To show that the wage scheme �w (y) also solves program (P), it su¢ ces to showthat �w (y) implies thatP

y2Y P (yjN) �w (y)� C (N) �P

y2Y P (yjE) �w (y)� C (E) ; 8 E � N:

Or equivalently,

P (sjN)[C(N)�C(?)]P (sjN)�P (sj?) � C (N) � P (sjE)[C(N)�C(?)]

P (sjN)�P (sj?) � C (E) ;8 E � N: (A� 2)

Recursively applying the inequalities (1) and (2) (supermodularity of P (sjE) andsubmodularity of C (E)), we have (denote l as the cardinality of E, jEj = l):30

P (sjN)� P (sjE)n� l � P (sjE)� P (sj?)

l; (A� 3)

C (N)� C (E)n� l � C (E)� C (?)

l: (A� 4)

Therefore,

[P (sjN)� P (sjE)]P (sjN)� P (sj?) [C (N)� C (?)] �

n� ln

[C (N)� C (?)] � C (N)� C (E) ;

which veri�es (A� 2) after rearranging the terms.

Proof of Proposition 2. We distinguish two cases:

Case 1 P (sjN)P (sjK) �

C(N)C(K)

:

Recall that K is the e¤ort choice where the agent only works in the k observabletasks. Given P (sjN) � C(N)

C(K)P (sjK), we now show that the optimal wage scheme is

derived using only (IC1):

30For expositional convenience, let E = f1; 2; :::; lg and NnE = fl + 1; l + 2; :::; ng. Then

C (N)� C (E)= C (N)� C (Nn fng) + C (Nn fng)� C (Nn fn; n� 1g) + :::+ C (Nn fn; :::; l + 2g)� C (E)� (n� l) [C (Nn fn; n� 1; :::; l + 2g)� C (E)]= (n� l) [C (E [ fl + 1g)� C (E)] ;

while at the same time,

C (E)� C (?) = C (E)� C (En flg) + C (En flg)� C (En fl; l � 1g) + :::+ C (f1g)� C (?)� l [C (E)� C (En flg)] :

Finally, C (E [ fl + 1g)�C (E) � C (E)�C (En flg) (again, by submodularity) implies that (A�4)is true. Inequality (A� 3) can be established in a similar fashion.

19

From (IC1), using a similar argument as in the proof of Proposition 1, the optimalwage scheme can be calculated as:

w0 (y) =

(C(N)�C(K)

P (yjN)�P (yjK) ; if y = s;0, otherwise.

(A� 5)

It is easy to show thatP

y2Y P (yjN)w (y) � C (N) � 0, or (IC2) is satis�ed withsuch w0 (y) when P (sjN)

P (sjK) �C(N)C(K)

.

Case 2 P (sjN)P (sjK) >

C(N)C(K)

:

Given P (sjN) > C(N)C(K)

P (sjK), the optimal wage scheme is determined only by(IC2), from which, the wage scheme can be calculated as:

w00 (y) =

(C(N)�C(?)P (yjN) ; if y = s;0 , otherwise:

(A� 6)

It is easily veri�ed that (IC1) is satis�ed under w00 (y).Finally, the incentive constraints where the agent does not shirk in any of the

k observable tasks can be dispensed with, by specifying a zero wage upon such anobservation, as e¤ort observation is veri�able. We conclude that the wage schemew (y), de�ned in (5) is the optimal wage scheme.

Proof of Corollary 1. The principal�s expected wage payments under �w (y) andw (y) can be written as, respectively,

E �w =P (sjN) [C (N)� C (?)]P (sjN)� P (sj?) ;

Ew =

(P (sjN)[C(N)�C(K)]P (sjN)�P (sjK) ; if P (sjN)

P (sjK) �C(N)C(K)

;

C (N) ; if P (sjN)P (sjK) >

C(N)C(K)

:

When P (sjN)P (sjK) �

C(N)C(K)

, then supermodularity of P (sjE) and submodularity of C (E)(or equivalently, (A� 3) and (A� 4)) directly imply that

C (N)� C (?)C (N)� C (K) �

k

n� k �P (sjN)� P (sj?)P (sjN)� P (sjK) , or E �w � Ew:

In addition, if the inequality (1) or (2) holds for at least one pair of tasks i; j in N ,then C(N)�C(?)

C(N)�C(K) >P (sjN)�P (sj?)P (sjN)�P (sjK) , or E �w > Ew.

Next, when P (sjN)P (sjK) >

C(N)C(K)

, we have E �w > Ew as P (sjN)P (sjN)�P (sj?) > 1 by our full

support assumption on P .

20

Proof of Proposition 3. We �rst simplify the constraints (IC 0) and (IC 00) in�PR�, starting with the simpler (IC 0), which is equivalent to

(IC 0) : w � C (N)

P (sjN)� ; (A� 7)

where

� =m�1Xj=1

1

j + 1

�m� 1j

�P (sjN)j (1� P (sjN))m�j�1 :

To interpret �, observe that ��m�1j

�P (sjN)j (1� P (sjN))m�j�1�is the probability

that there are j (among (m � 1)) agents having output signal s when all the other(m� 1) agents choose N , while � 1

j+1�indicates that agent i will share the reward w

with such j agents when i�s output is also s.Constraint (IC 00), on the other hand, involves multiple incentive constraints:First, given that all the other agents choose N , if i shirks in some (or all) of the

k observable tasks, i�s payo¤ is zero even with output s: If i is the only agent withoutput s, the principal pays 0 to the agents given our design; if another (or more)agent also obtains output s, i�s payo¤ is again zero because of her incompetent e¤orton the k tasks. Hence, (IC 0) is su¢ cient to encompass the cases where i (partially)shirks in the k tasks.Consider now the set of constraints in (IC 00) where agent i at least works in all

the k observable tasks: K � E � N . In this case, agent i�s payo¤ is:

Ui (E;N; � � � ; N ;w) =m�1Xj=1

P (sjE)j + 1

�m� 1j

�P (sjN)j (1� P (sjN))m�j�1w � C (E)

= �P (sjE)w � C (E) :

Since � is a constant to i, as in the proof of Proposition 1, it su¢ ces to consideran extremal constraint �Ui (N ;N; � � � ; N ;w) � Ui (K;N; � � � ; N ;w)�, which yields

w � C (N)� C (K)[P (sjN)� P (sjK)]� . (A� 8)

Summarizing, the incentive constraints (IC 0) and (IC 00) reduce to two simpleinequalities (A� 7) and (A� 8).As � is a constant, we can then again discuss two familiar cases:

1. If P (sjN)P (sjK) �

C(N)C(K)

, then only (A� 8) is binding. Hence, the principal�s expectedwage payment, denoted as E ~wR is:

E ~wR =~P [C (N)� C (K)]

� [P (sjN)� P (sjK)] ;

21

where ~P is the probability that �two or more agents have realized output s�:

~P =mXj=2

�m

j

�P (sjN)j (1� P (sjN))m�j .

With a simple change of the upper and lower bounds of summation, constant� can be equivalently written as

� =mXj=2

1

j

�m� 1j � 1

�P (sjN)j�1 (1� P (sjN))m�j

=1

m

mXj=2

�m

j

�P (sjN)j�1 (1� P (sjN))m�j :

Therefore,

E ~wR =~P [C (N)� C (K)]

� [P (sjN)� P (sjK)] = mP (sjN)C (N)� C (K)P (sjN)� P (sjK) = mEw:

2. If P (sjN)P (sjK) >

C(N)C(K)

, then (A�7) is the relevant constraint. The principal�s expectedwage payment is:

E ~wR =~PC (N)

�P (sjN) = mC (N) = mEw:

We conclude that the optimal relative performance contract is ex ante equiva-lent (in terms of expected wage payments) to the optimal individual contracts inProposition 2 from the principal�s point of view.

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