Itoh ('91) - Incentives to Help

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Incentives to Help in Multi-Agent SituationsAuthor(s): Hideshi ItohSource: Econometrica, Vol. 59, No. 3 (May, 1991), pp. 611-636Published by: The Econometric SocietyStable URL: http://www.jstor.org/stable/2938221.

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Econometrica, Vol. 59, No. 3 (May, 1991), 611-636

INCENTIVES TO HELP IN MULTI-AGENT SITUATIONSBY HIDESHI ITOH1

This paper concerns moral hazard problems in multi-agent situations where coopera-tion is an issue. Each agent chooses his own effort, which improves stochastically theoutcome of his own task. He also chooses the amount of "help" to extend to other agents,which improves their performance. By selecting appropriate compensation schemes, theprincipal can design a task structure: the principal may prefer an unambiguous division oflabor, where each agent specializes in his own task; or the principal may prefer teamworkwhere each agent is motivated to help other agents. We provide a sufficient condition forteamwork to be optimal, based on its incentive effects. We also show a nonconvexity ofthe optimal task structure: The principal wants either an unambiguous division of labor ora substantial teamwork.KEYWORDS: Principal-agent relationships, moral hazard, multiple tasks, team produc-tion, incentives to help.

1. INTRODUCTIONTHIS PAPER CONCERNS moral hazard problems in multi-agent situations wherecooperation is an issue. We consider a situation where each agent can allocatehis effort to various production activities called tasks. Tasks are assumed to be"independent" of each other: the outcome of each task depends on an exoge-nous random variable which is stochastically independent of the random vari-ables affecting the other tasks; and revenues from each task only depend on theoutcome of that task. Relative performance evaluation therefore does not give areason for the wage schedule to an agent to depend on the outcome of the tasksassigned to the other agents. (See Baiman and Demski (1980), Green andStokey (1983), Holmstrom (1982), Lazear and Rosen (1981), Mookherjee (1984),and Nalebuff and Stiglitz (1983).)We focus on incentives of agents to "help" each other. Each agent chooseshis own effort level which improves stochastically the outcome of the task forwhich he is mainly responsible. Agents also choose the amount of "help" toextend to other agents which improves the outcomes of their tasks. Theprincipal, who cannot observe the effort chosen by each agent, designs wageschedules contingent on outcomes. By selecting appropriate wage schedules, theprinciple can design a task structure: The principal may prefer a specialized taskstructure, where each agent is inclined not to help other agents and specializesin his own task. In this case, by the assumption of independent tasks, each agentcan be treated completely separately. The principal however may choose anonspecialized task structure, called teamwork, in which agents are motivated

1 This paper is based on a chapter in my doctoral dissertation submitted to Stanford University,March, 1988. I am grateful to Masahiko Aoki for inspiring me to this research, to David Baron,George Mailath, John McMillan, Dilip Mookherjee, Mark Wolfson for helpful comments, andespecially to a co-editor and two anonymous referees for their very careful comments as well aseditorial assistance, and to David Kreps for his constructive suggestions and constant encourage-ment.611

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612 HIDESHI ITOHto help each other. In this second case, the principal has to make the wagecontract to an agent contingent on the outcomes of the other agents' tasks.Since the outcome of each task is affected by both the own effort of the agentassigned to that task and the helping effort by the other agents, this nonspecial-ized task structure leads to team production.2In our model, team production endogenously appears as a result of the wageschemes chosen by principal. This research should be distinguished from theprevious literature that analyzes how disadvantages of team production, namelythe unobservability of the marginal contribution of each agent and the resultingfree-rider problem, are resolved in situations where team production exoge-nously exists because of the nature of tasks or production processes (Alchianand Demsetz (1972), Holmstrom (1982), McAfee and McMillan (1986)).

Situations where each agent allocates effort to multiple tasks have beenanalyzed by Drago and Turnbull (1987, 1988) and Lazear (1989). In these papersparticular forms of compensation schemes are exogenously imposed. Drago andTurnbull (1987) and Lazear (1989) consider tournaments and show that whenagents play effort choice subgames noncooperatively, positive helping effortdoes not occur in equilibrium. In addition, Drago and Turnbull (1987) show thatthis is also true under noncompetitive promotion schemes such as a quotascheme, and then they consider cases where agents can reciprocate via bindingagreements. Lazear focuses on "negative helping effort," called sabotage, andshows that pay compression between the winner and the loser may be prefer-able in order to reduce sabotage by agents. Drago and Turnbull (1988) examinewhether agents efficiently allocate given total effort between own effort andhelping effort under two particular linear compensation schemes; an individualpiece rate scheme and a group piece rate scheme. They find that depending onwhether the agents behave noncooperatively or cooperatively, individual piecerate schemes may lead to "inefficient under-cooperation," while group piecerate schemes may cause "inefficient over-cooperation."Rather than specifying some particular form of wage schedules, I follow theliterature in the "standard" agency models with hidden action (but withouthidden knowledge), in particular, Grossman and Hart (1983), Mookherjee(1984), and Rogerson (1985), and examine the general incentive problem in thesituation explained above.3Although this approach usually does not yield manysharp predictions, the logic behind the problem will be clarified. We analyze ahidden action model of the relationship between a principal and two risk averseagents who select efforts noncooperatively.4 Because of the independenceacross tasks, one would only find in the model that the wage schedule to anagent is contingent on the outcomes of the two tasks in order to give him an

2According to Alchian and Demsetz (1972), team production exists when several inputs to aproduction process are utilized by more than one individual and the product is different from thesum of the outputs from inputs separately used by each individual.3Recently, Holmstrom and Milgrom (1989) have started to study related issues of multitaskagents in the linear model developed by Holmstrom and Milgrom (1987).4 The case where the agents can collude via side contracts is analyzed in Itoh (1990).



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MULTI-AGENT SITUATIONS 613incentive to help the other agent: When the principal wants neither agent tochoose positive helping effort, it is strictly better to treat them completelyseparately.

The main purpose of this paper is to examine what factors affect theprincipal's decision to induce team production through interdependent incen-tive schemes rather than to treat the agents separately through individual-basedschemes. To this end, we utilize the first-order approach5 and examine how amarginal change of the optimal independent contract in the direction ofteamwork, by making the payments to an agent contingent on the outcome ofthe other agent's task as well, affects the principal's welfare. There are twoquestions associated with this. First, is such a marginal change in fact able toinduce an agent to help the other agent? Second, if so, does this actually workto the principal's advantage? Or could it be that help has undesirable incentiveeffects on the other agent that outweigh whatever direct advantages it mighthave?The answer to the first question is yes when the agent's marginal disutilitywith regard to helping effort is zero at zero help. This case is likely to apply ifagents have task specific preference. Teamwork is then always optimal when theproblem is risk sharing only. However, the existence of the moral hazardproblem and the resulting strategic interaction among agents may lead theprincipal to adopt individual-based schemes because teamwork actually works tohis disadvantage. We provide, as an answer to the second question posed above,a sufficient condition under which this is not the case and teamwork actually isoptimal. The condition represents the effects of introducing "a small amount ofteamwork" on the principal's welfare through the incentive compatibility con-straints. In particular, teamwork is optimal if own effort and helping effort arecomplementary so that an agent responds to an increase in help from the otheragent by increasing his own effort. Even in cases where an agent's optimalresponse to an increase in help is to reduce his own effort (the case offree-riding), teamwork is optimal if the resulting decrease in own effort reducesthe costs of inducing him to work appropriately on his own task sufficiently.Such a case holds when an agent's task is so monotonous that the marginalproductivity of own effort decreases or the marginal disutility of own effortincreases drastically as he raises his own effort level.If tasks are similar and agents only care about the total amount of effort, theyare reluctant to provide even a small amount of help because the marginaldisutility of helping effort at zero help is positive. Then the answer to the firstquestion turns out to be no. That is, perturbing the optimal independent wageschedule in the direction of teamwork by a small amount cannot elicit positivehelping effort. A large perturbation is required to induce any help, and with riskaverse agents this has a substantial cost in terms of inefficient risk sharing. Put

5Unlike in the one-agent model, the monotone likelihood ratio property (MLRP) and theconvexity of the distribution function condition (CDFC) are not sufficient for the first-orderapproach to be valid in our model. We will present a sufficient condition which consists of MLRP, ageneralization of CDFC, and a condition on the agents' utility functions for income.



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614 HIDESHI ITOHdifferently, if we decompose the problem as is done in Grossman and Hart(1983), the cost function for implementing various levels of effort increasesdiscontinuously as we move from zero to positive help. Since the expectedrevenue function is continuous, a little bit of help will always be suboptimal.Thus, we fail to obtain sufficient conditions for teamwork to be optimal byexamining the local necessary conditions which must be satisfied by the optimalindependent contract. However, this may partially explain why the personnelmanagement of the firm is extreme in the sense that we observe either a strictdivision of labor through individual wage schemes or substantial teamworkthrough team-based wage schemes. Note that the existence of multiple agents isnot at all necessary for this nonconvexity result. The result applies whenever asingle agent has options of allocating his effort to multiple tasks.

While the paper focuses on theoretical issues of incentive problems due tohidden action of the agents working on multiple tasks, the central questionasked in the paper is discussed in various recent literature in economics andmanagement: U.S.-Japan comparisons in organization structures and personnelpolicies (Aoki (1988), Kagono et al. (1985), Lincoln and McBride (1987));management of American high performance companies (Waterman (1987));work design for individuals and for groups (Hackman and Oldham (1980)); arecent research on productivity performance in American manufacturing indus-tries (Dertouzos et al. (1989)). The current paper is intended to be a steptoward the economic analysis of teamwork and optimal task design.The rest of the paper is organized as follows. In Section 2, the model ispresented. Some preliminary results are obtained in Section 3. Sections 4 and 5are the main part of the paper. In Section 4, a sufficient condition for teamworkto be optimal is given. The nonconvexity result is presented in Section 5. Section6 is concluding remarks.

2. THE MODEL

We consider the relationship between one principal and two agents n = 1,2.Extension to the general case of N agents is straightforward, and will be brieflydiscussed in the final section. Since task assignment is not our concern, weassume that agent n is exogenously assigned to a technologically well-defined,"independent" task n because, for example, only he has some necessaryexpertise for that task or both agents are equally capable. Task n is indepen-dent of task k (k 0 n) in the sense that the outcome of task n, which is arandom variable, is stochastically independent of the outcome of task k.6 Forsimplicity, we assume that the outcome of each task is either success (S) orfailure (F). Let pf be the probability of success in task n, and Pi' be the jointprobability that the outcomes of task n and task k are i and j, respectively(i, j E - {S, F}). For example, PsF =p'(1 _pk)Each agent simultaneously chooses a level of effort, which jointly determinesthe probability distribution of the outcomes of the tasks. The effort level chosen

6 Unless otherwise noted, whenever n and k appear, it is assumed that k 0 n holds.



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MULTI-AGENT SITUATIONS 615by agent n is a two-dimensional variable en = (an, bn)E &x _ [0, A] x [0, B],where an is the effort expended on his own task, called own effort, and bn is forthe task occupied by agent k, called agent n's helping effort for agent k.7Agentn's own effort and agent k's helping effort jointly affect the probability distribu-tion of the outcome of task n: the success probability of task n is a function ofthe input (an, bk), and is written as pn(an, bk). The joint probability Pij is hencethe function of (en, ek) and is written as Pi (en, ek). We assume that for eachn, pn(.) is twice continuously differentiable with regard to its arguments, anddenote the partial derivatives by pn, Pnb' and so on. We adopt the followingassumptions on pn(.).

ASSUMPTION1: For each n, (i) pn(a, b)> 0 for all (a, b); (ii) pn(a, b)> 0for all (a, b), with strict inequality if a > 0; (iii) pn(O, ) > pk(0, 0); and (iv)pn(a, b) E (0, 1) for all (a, b).Part (ii) gives the reason b is called helping effort. It is plausible to assumethat agent k, who is not assigned to task n, can increase the productivity of taskn through various kinds of help. Agent k's doing some routine work for agent nmay help the latter improve his productivity by concentrating on more sophisti-cated work in task n. Or agent k may possess skills or knowledge which, whilenot essential to task n, are still valuable at that task. Note that (i) and (ii) are

equivalent to the strict monotone likelihood ratio property (strict MLRP) in ourtwo-outcome model. Thus by Milgrom (1981), observation of success in task n is"more favorable than" observation of failure: a helping effort by agent k (or anown effort by agent n) fixed, the posterior distribution of the own effort of agentn (the helping effort of agent k, respectively) given observation of success intask n dominates the posterior probability distribution given observation offailure in the sense of the first-order stochastic dominance. Part (iii), with anassumption on agents' utility functions stated shortly, excludes the case wherethe input into a task is only the helping effort from the agent assigned to theother task. One extreme, but possible case in which this assumption holds is thatagent n's own effort is required in order for the other agent's help to improvethe productivity of task n, that is, pg(0, b) = 0 for all b. The final part (iv) statesthat there is no moving support.Let Tij be the total revenue earned by the principal when the outcomes oftask 1 and task 2 are i and j, respectively. The expected total revenue, denotedby R(e1, e2), is then given as R(el, e2) = FEijPi1(e1, e2)wr1.To focus on endoge-nous formation of team production, we adopt an inessential assumption that theprincipal'srevenue is additively eparable n tasks:Tij = ,T + 7j where Tin isthe revenue from task n when its outcome is i = S, F with wTr'> w'. Theexpected total revenue of the principal is then written as R(e1, e2) = Hl(a1, b2)+ H2(a2, bl) where H' is the expected revenue from task n defined byH'(an,a bk) = wF + p (an,a bk )( 7 n)

7For simplicity, we take A > 0 and B > 0 sufficiently large so that we ignore the case whereeffort levels reach these upper bounds.



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616 HIDESHI ITOHThe principal selects a wage schedule for each agent. She (the principal) hasthe ability to commit herself to that schedule. The outcome of each task isassumed to be publicly observable, so that the wage schedule can be contingent

on the outcomes of the two tasks. Denote by wj the wage paid to agent n whenthe outcomes of task n and task k are i and j, respectively. The wage schedulefor agent n is a four-dimensional vector w' = (w's, W'F, wF, wFF). The (vonNeumann-Morgenstern) utility function of agent n is defined on V"x Mx 97where V" is an interval (w,oo). We further assume that the utility function istwice continuously differentiable and has the additive form Vn(w)- Gn(en)where Vn is strictly increasing and (weakly) concave. That is, the agents areassumed to be risk neutral or risk averse. Concerning the disutility of effort, wedenote the partial derivatives by Gan,Ganb,nd so on. The assumptions on thesederivatives are as follows:

ASSUMPTION 2: For each n, (i) GQn(a,b) > 0 for all (a, b), with strict inequalityif a > 0; (ii) Ggn(a,b)> 0 for all (a, b), with strict inequality if b > 0; (iii)Gan(O,0) < Gn(0, 0); and (iv) Gn(v) is strictly convex.

Assumption 2 states that the agents are generally effort averse and that theirdisutility rises at an increasing rate as they work harder. And in order to induceagents to exert positive own effort levels, it is assumed that when neither owneffort nor helping effort is positive, the marginal disutility of helping effort is atleast as large as that of own effort. The most important part of Assumption 2 isthat given a positive own effort, we allow a case where a marginal disutility ofhelping effort at zero help is strictly positive as well as a case where it is zero.The former case occurs, for example, when an and bn represent the time agentn allocates to his own task and the other task and the disutility of work dependsonly on the total amount of time he works; Gn(a, b) = Gn(a + b). Then given apositive own effort, a small amount of help increases the agent's working timeand increases his disutility of effort. The latter case may occur when the agentshave task specific disutility for each task because, for example, tasks are of verydifferent types. An example is Gn(a, b) = Kn(a) + L (b) with K,I(O)= LI (O) = 0.For the wage schedule wn and the effort combination (en, ek), letUn(Wn, en, ek) be the expected utility of agent n, defined by Un(wn, en, ek) =EE1Pijn(en, ek)Vnf(w;j) Gn(en). The principal is assumed to be risk neutral andto choose wage schedules (w', w2) and effort levels (el, e2) to maximize herexpected residual profits. The principal's Original Problem, denoted by (OP), isgiven as follows:

max R(e1,e2) - EPi(el,e2)(w11j+w ) subject tow ,w ,ei,e2 i j(OP) (NIC) Un(Wn, en, ek) > Un(Wn, el, ek) for all el and n = 1,2,

(PC) U (w , en, ek) > Uj for n = 1,2,



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MULTI-AGENT SITUATIONS 617where U' is the reservationutility level of agent n. The constraints(NIC),calledNash incentivecompatibility onstraints, tate that, givenwage schedules,the effortchoice of the agents forms a Nash equilibrium.The last constraints(PC) are the participation onstraints.FollowingGrossmanand Hart(1983),we can decompose problem(OP) intotwo parts: (i) the ImplementationProblem which solves, given effort levels(e1,e2), the wage schedules that minimize the expectedwage paymentsof theprincipalto the agents subject to (NIC) and (PC); (ii) the EffortSelectionProblemwhichfinds the effortlevels that maximize he principal's xpectednetprofits.Formally,these problems are described as follows. Let v1'= Vn(wj) and

=n(J'(). Instead of wn, we can regard SS=(v, VSF, VF,S VFF)neY foreach n as the choice variableof the principal.We sometimescall Vn(as well aswn) a wage schedulefor agent n. Let hnbe the inverse functionof the utilityfunctionon income Vn.Then if the principalwants to implementeffortlevels(e1,e2), she has to solve the followingproblem(IP):(IP) min EPi(e1, e2)(h1(v) + h2(vj)) subject o (NIC) and (PC).v,V i JNote that all the constraintsare linearin v/. Thus, if agentsare risk aversesothat hn is strictlyconvex for all n, problem(IP) for each effort combination(e1,e2) is a standardconvexprogramming roblemwith an infinitenumberoflinearconstraints. f the feasible set in (IP) is nonempty, he solution (v1,v2) to(IP) exists. This can be provedby following Grossmanand Hart (1983) andMookherjee 1984).We call the solution the optimalwage schedules or (e1,e2).Let C(el, e2) be the minimum cost to implement (e1, e2), which is the optimalvalue of problem(IP) for that effort pair. Then the Effort Selection Problem(EP) solves(EP) maxR(e1, e2) - C(e1, e2).

el,e2The existence of the solution to (EP), called the second-bestefforts,can beshownby using argumentsanalogousto those in Grossman and Hart (1983).Throughout he paper, it is assumedthat the second-bestefforts are not theleast costlyones (en = (0, 0) for all n). In addition,we supposethat the principalwants each agentto work at least on his own task.ASSUMPTION 3: The solution to (EP) satisfies an > Ofor n = 1,2.We make this assumption n order to simplifythe expositionin the paper,particularlyn Section 4, as well as to focus on incentives of mutualhelp: itexcludesthe case in whichthe principalwantsboth agentsto workonly on onetask (a2 = b1 = 0 or a1 = b2 = 0). However, the results in this paper hold withoutthis assumption: nlyminormodifications rerequiredas we will comment aterin Section4. One sufficient ondition orAssumption3 to be true is Gn(0,b) = 0



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618 HIDESHI ITOHfor all b and n, becausethen the marginalbenefit of increasingown effortfromzero outweighsthe marginalcost, regardlessof help, and hence the principalwants each agentto workat least a little bit on his own task.

3. PRELIMINARY RESULTSAs a benchmarkor the analysis,we define the first-best olutionwhich is theeffortcombination he principalwants to implementwhen she can observe theeffort choice of each agent. To ensure that the principalcan implement anyeffort level by some wage schedule that guarantees the agent exactly hisreservationutility level, we assume the following:8ASSUMPTION 4: For each n, (i) limww Vn(w)= - oo;and (ii) Un + Gn(en)Efor all en.Define the first-bestcost functionCnFB(*)or agent n by CFnB(en) = h (Un +Gn(en)).The first-bestsolution solvesthe followingproblem FB):

(FB) maxR(el, e2) -CF1B(elj) -CFB(e2).el,e2Let en = (an*bn*) e the first-besteffort of agent n. A few characteristics f thefirst-best olutionwill be found in the next section. In particular,we will obtaina necessaryandsufficientconditionfor the first-besthelpingeffortfor an agentto be positive,by examining irst-order onditions.In the second-bestsituationwhere the effort of each agentis noncontractible,the principalutilizes the observation f outcomes n order to provideeachagentwith incentivesto choose desirableeffort levels. To induce each agentto workon his owntask,hiswageschedulemust be contingenton its outcome.Similarly,the wage to agent n must depend on the outcomeof task k if the principalwants agent n to choose a positivehelpingeffort for agent k. Generally,thisincentiveeffect distortsthe optimal risksharingbetween the principaland eachagent, and therebythe principalincurs a loss due to the unobservability feffort.As in the standardone-agentmodel with moralhazard,however,when all theagentsare riskneutral,there is no such loss and the principalcan achieve thefirst-bestsolution.This is obviouswhen the first-bestsolutionhas zero help forboth agents. The principal then has an independent relationship with eachagent,and hence she can induce the first-bestown effort by payingeach agentthe wholemarginal evenuefrom his task. Note that the principal an attainthiswithoutbearing any risk by subtracting he expected net profit from task n,Hn a,*, O) CnB(a*, 0), from the payment to agent n.The argument xtends to the case in which the first-best olution has positivehelp for some agent. Supposebn* 0. The principal s now able to induce the

8Assumption 4 (ii) holds automatically if the agents' utility-of-income functions Vn are un-bounded above.



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MULTI-AGENT SITUATIONS 619first-besteffort by payingthe marginal otal revenuesfrom both tasksto agentn. And she subtracts the fixed amount R(e , e2 ) - CnB(e*) from the payment.9Under this scheme, however,the principal pays the whole marginalrevenuefrom task k to both agents, and therebyshe prefersthe outcomeof task k to befailure ratherthan success.The principalhence has to share some riskwithriskneutralagents.10

PROPOSITION 1: Suppose that all the agents are risk neutral. Then theprincipalcan achieve the first-best solution by an interdependentlinear scheme which payseach agent the whole marginal increases of the revenuesfrom all the tasks.Now suppose that at least one of the agents, say, agent n, is risk averse. If the

principalwants him to select positive help, his wage schedulemust dependonthe outcome of task k. The converse s also true:If the helpingeffortof agentnis zero, then the optimalwage scheduleshould be independentof the outcomeof task k. This is an applicationof the so-called"sufficient tatistics heorem" nmulti-agent ituationsby Holmstrom 1982)and Mookherjee 1984).Thisresultprovides he sufficientand(partial)necessaryconditionfor optimalpaymentsofan agent to be independentof the other agent's performance.The sufficientcondition for agent n's contract is that given ek' for all en, een,iJ(en edlP1nj(e,, k) is independent of j (i, j = S, F). That is, the outcome of task n is asufficient tatisticfor the effortchoiceby agent n. In our model,this ratio is ofthe form

Pin(an, bk)Pjk ak S bn)(1) Pin(an,bk) j(ak, bn)where Pn( ) pf(.) and Pp(.) = 1 pfn( ). When our attentionis confined tothe subset of feasible efforts in which help is equal to zero (bn = bn= 0), (1) isindependentof j. Thus, the individual-basedwage scheme to agent n is betterfrom the risk sharingpoint of view.

PROPOSITION 2: Suppose that agent n is risk averse. Then the optimal wageschedule of agent n is contingent on the outcome of task k if and only if bn>09 By following McAfee and McMillan (1986), we will be able to generalize the optimality of thesimilar linear wage schedule to the case where hidden knowledge as well as hidden action exists.When hidden knowledge exists, because of the information rents due to private information held bythe agents, the first-best solution is not achievable. And the principal does not give the agents all ofthe marginal increases in the total profits in order to extract some of their information rents.10One might ask whether there is any other wage scheme implementing the first-best solutionwith no risk imposed on the principal. The answer is no, at least in our model of two possible

outcomes. The proof is presented in earlier versions of the paper. The intuition goes as follows. Inorder to induce the agents to choose the first-best inputs to task k, the principal has to pay thewhole marginal expected revenue from that task to each agent. However, when the principal givesagent n the whole marginal expected revenue from task k and she does not incur any risk, themarginal expected revenue from task k paid to the other agent must be zero. Then the principalcannot provide agent k with an incentive to choose a positive effort for his own task. She thereforemust bear some risk when she wants to induce help even though the agents are risk neutral. Thiscontrasts with the case of a relationship with one agent.



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620 HIDESHI ITOHProposition2 shows that the only reason in our modelwhy the wagescheduleof a riskaverseagent depends on the outcome of the taskassigned o the otheragent is to give the former agentan incentiveto help the latter. The reasonforthe principal o select interdependentwage schemes in our model is thereforedifferentfrom that in the literatureon relativeperformance valuation isted inSection 1. This literature considers the situation in which the productionfunctionis separablein effort and the outcomes of tasks are correlated.Therelativeperformance valuation s then valuablesince the principalcan use theperformanceof an agent as a signal of the effort choice by the other agent.There it is typicalto find that the paymentto one agent is decreasing n theother'sperformance.There are howevercases where the reversewouldbe true.For example,consider two salesmenattemptingto sell the same goods in acommonterritory.Suppose that there are not manypotentialcustomers n thatarea so that the outcomesof the salesmenarehighlynegatively orrelatedgiventheir effort choice. Then salesman n will be rewardedmore the better is theperformanceof salesman k, because a better outcome of salesman k conveysthe information hat salesman n workedharder. On the other hand, in oursetting where the outcomes are independent given effort choice, a betteroutcomeof salesmank does not provide any information oncerning he owneffort of salesman n, but does convey some informationabout salesman n'shelping effort.

4. INDUCED TEAMWORK AS AN OPTIMAL TASK STRUCTUREThe main objectiveof the paper is to obtain conditions under which theprincipalmotivates he agents to help each other in the equilibrium,hat is, theprincipalendogenouslycreates teamwork.By Proposition1, when both agentsare riskneutral,the principalwantsthemto help one another f andonlyif thefirst-bestsolution has positivehelpingeffortfor both agents. Teamwork n this

case therefore does not have any peculiarcost or benefit that does not existwheneach agent is paid as a functionof the outcome of his owntaskonly. Thuswe assumehereafterthatboth agentsare riskaverse.Suppose that each agent is contractedwith independently,with a wageschedulecontingentonlyon the outcomeof his own task. Since there existsnoincentive to select positivehelpingeffort under such an individual-based on-tract,we say that the principalchooses a specializedask structurewhere eachagent specializes n his own taskand is not involved n the othertask.Then thequestion we ask is: under what conditions does a marginal change of theoptimal independentscheme in the directionof teamwork,by making he wageschedule of an agent dependon the outcomeof the other agent'stask as well,improve he principal'swelfare?The analysis uses the first-orderapproach. Based on Mirrlees (1975),Rogerson (1985) shows that, in the standardone-agent model, the sufficientcondition for the first-orderapproachto be valid is the convexity of the



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MULTI-AGENT SITUATIONS 621distributionfunction condition (CDFC) along with the monotone likelihoodratio property (MLRP). In our model with production externalitiesbetweenagents, however,CDFCis not generallysufficient.The problem s that, thoughwe can generalizeCDFC for the joint probabilitydistributionof the outcomesof twotasks,thisgeneralizedcondition,alongwithnondecreasingwageschemes(generally derived from MLRP), does not guarantee the concavity of theexpected utility function of the agents. We have to find some sufficientcondi-tions that validatethe first-orderapproach o our model.We define the following Relaxed Problem (RP). Let Un be the partialderivativeof agent n's expected utility Un with regardto its argumentx. Thenin (RP), the principalchooses (w1,w2) and (e1,e2) such as a1 > 0 and a2> 0 tomaximize he objectivefunction in (OP) subjectto the participation onstraints(PC) and the following local Nash incentive compatibility constraints (LNIC) inplace of (NIC):(LNIC)

Uan(wn, ensek) = 0 and bnUbn,e(wn,efek) = 0 for n = 1,2.The next assumption on the joint probability distribution JF =(1-pf)(1-

pk) and the inversefunctionhnof Vn urnsout to be sufficient orvalidating hefirst-orderapproach.ASSUMPTION 5: For each n, (i) PFnF(en, k) is convex in en; and (ii) h ( ) isconvex.Part (i) is a generalizationof CDFC. From the economic point of view, thiscondition can be interpretedas some kindof stochasticallydiminishing eturns

to scale. Part(ii) can be best understoodby consideringHARA utilityfunctionswith - Vn'(w)/Vn"(w)= pw + 0. Then we can show that hn is convex if and onlyif p < 2. This implies that the coefficient of absolute risk aversion must notdecline too quickly.Examples nclude standardones such as the utilityfunctionwith a constant absoluterisk aversion correspondingo p =-I), the logarithmicutility function (p = 1), and the square root function (p = 2).11LEMMA 1: For (e1,e2) with an>O, n=1,2, (i) (w',w2) and (e1,e2) solve

(RP) if and only if they solve (OP); (ii) the optimal wage schedules are monotoneincreasing, that is, for each n, w iwF and w,>wn for i =S,F, with theequality if and only if bn = 0.1Alternatively we could assume that Psns(en,ek) is concave in en and hl(-) is concave forn = 1, 2, though utility functions satisfying the latter condition are less standard.



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622 HIDESHI ITOHThe proof is in the Appendix.Lemma 1 makes it possible to analyze theRelaxed Problem instead of the OriginalProblem.'2Because Gn() is strictlyconvex,Un(Wn *, ek) is alsostrictlyconcave,giventhat wn is monotone ncreas-

ing.To makethe argument impleandclear,we considerthe symmetric ituationin this section: the agents have the same utilityfunctionV(w) - G(a, b) and thesame reservationutility U and h = V-1; the successprobability f each task isgiven by the same functionp: Q/X 0 -* [0,1]; and the revenue from each taskwhen the outcome of the taskis i is wri.For the comparisonwiththe second-bestsolution,we providea necessaryandsufficientconditionfor positive helping effort to be optimal in the first-bestsituation.To simplify he analysisof this benchmark ase, we assume that theobjectivefunction in problem (FB) is strictlyconcave in (e1, e2). A sufficientcondition is that p( ) is concave,which is not implied by Assumption5. Thisensures the existence of the symmetric irst-bestsolution. Let e* = (a*, b*) bethe unique first-best olutionto problem FB) and e** = (a**, 0) be the solutionto problem (FB) subject o zerohelpingeffort.That is, a** is the first-bestowneffortlevel when the agentsare contractedwith independently.

ASSUMPTION 6: The objective function in problem (FB) is strictly concave in(el, e2).

PROPOSITION 3: Consider the symmetric situation as stated above. Then (i) ifa** = 0, the first-best solution e* satisfies a* = b* = 0, that is, e* = e**; (ii) ifa** > 0, then a* > 0, and the necessary and sufficient condition for e* =(a*, b*)to satisfy b* > 0 is given bypb(e ) Gb(e**)Pa(e**) Ga(e**)

PROOF: The first-orderconditionsfor a* and b*, which are necessaryandsufficient by Assumption 6, are given by PaGiS - ITF) < h'(U + G)Ga andPb(T7S - WF) ? h'(U+ G)Gb, with the equalities if a* > 0 and b* > 0, respec-tively. Similarly, e** satisfies the first inequality, with the equality if a** > 0.Part (i): If a** = 0, then by Assumptions 1 (iii) and 2 (iii), e** = (0,0) satisfiesthe first-order condition for b* as well as for a*, and hence e** = e*. Part (ii):Supposea** > 0. If a* = 0, then b* = 0 must hold. (The argument s the same

12 The reason to use the complementarity slackness conditions in (LNIC) is that it allows thesolution to (RP) to have zero helping effort: If we used Ub'= 0 instead, the independent wageschedule with zero helping effort could not be a solution in the" elaxed problem. The same problememerges in the standard one-agent problem: The relaxed problem and the doubly relaxed problemin Rogerson (1985) cannot have the least costly action as an optimal solution, though theimplementation of such an action is trivial in his agency model with one-dimensional effortvariables. Note also that for the same reason as above, without Assumption 3, we would have toreplace the first equation in (LNIC) by the complementarity slackness condition.



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MULTI-AGENT SITUATIONS 623as the proofof Part(i).) The least costlyeffortpairhence becomes the overallfirst-bestsolutionas well as the first-bestsolutionsubject to zero help. Contra-diction.Thus,a* > 0 must hold. For the necessitypart,supposePb/Pa < Gb/IGaat e**. Then by definition,e** satisfiesthe first-order onditions or a* and b*given above.This impliesthat e** is the uniquefirst-besteffort,which contra-dicts b* > 0. For the sufficiencypart, suppose instead b* = 0. Then by defini-tion, a* =a**, and e** satisfies the first-orderconditions given above. Then,however,Pb/Pa < Gb/Ga at e**, whichis a contradiction. Q.E.D.

The intuition is simple. If condition (2) holds, decreasinga smallamount ofown effortand increasinghelpingeffortfor a taskraisesthe successprobabilityof that task,whicheffect dominatesthe increase in disutility.Thus,the optimaleffortchoice has positive helpingeffort.Conversely, f the condition does nothold, then starting from the first-best independent contract, any marginalincrease in helpingeffortwould be less useful than the same increase in owneffort,whichby the definitionof the first-best ndependentcontract tself is toocostly.In particular,f Gb = 0 at zero helpingeffort,the conditionalwaysholdsso that the first-bestsolution is always eamwork. f the disutility ermdependsonly on the total amount of effort a + b, (2) becomes Pb(e**) > Pa(e**): thefirst-bestsolutionis teamwork f and only if helpingeffort is marginallymoreproductive hanown effortat the first-besteffortsubjectto zero help.

REMARK: Assumption6 is crucial n proving i) and the necessitypartin (ii):without it, one might have a* > 0, b* > 0 while a** = 0; or e** might satisfy thefirst-orderconditionswhile not globally optimal in the unrestricted irst-bestproblem.The sufficiencypart would hold withoutthe assumption,and hencethe discussionof local improvements ivenabovewould be still correct.

Returning to the second-bestsituation with risk averse agents, denote byWO= (W, wF) the optimal individual-basedwage schedule and by eo = (a0, 0)with a0> 0 the optimaleffort levels under that scheme.(The problem s trivialwhen a0= 0.) The utility-on-incomeevel corresponding o the wage w? isdenoted by vu. While the optimal independentcontract(w0, e0) may not beunique,supposefor simplicity hat the principaloffersthe same contract o bothagents.The following emmaprovidesa necessarycondition or offering wo,eO)to both agentsto be the second-bestsolution.The condition s derivedfrom theKuhn-Tucker onditionsof problem(RP). For each n, let Anbe the Lagrangemultiplierfor (PC), ,,n and {n be the multipliersfor the first and secondconstraints n (LNIC),respectively.Becauseof symmetry, hese multipliersarethe same for n = 1,2, and thereby we denote them by A, u, and 6.

LEMMA 2: Considerhe symmetricituation.If offering w0, eo) witha0> 0 toboth agents is the second-bestolution,there exist A> 0, ,u> 0, and 6 > 0 such



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624 HIDESHI ITOHthat(3) A pb(eO) Gb(e ) Gb(e0)

Ap(eO) - Ga(e)) Ga(e0)Pab(e ) Gab(e?) Pb(?) (Paa(e?) Gaa(eo)

r Pa(eo) Ga(e?) Pa( e?) Pa(eo) Ga(eo)o0.

PROOF: he proof is standard.Since (w?,eo) solve (RP), they satisfy thefollowingKuhn-Tucker ecessaryconditions:PaY0 + A(Pa8o - Ga) + A(Paas0 - Gaa) = 0,PbY + A(Pb8o-Gb)+ L(Pab8?-Gab)+ (-Gb) < ?

where y0 = (m, - w ) - (7rF - w?) and 80 = v - v?. Since a0 > 0 maximizes theagent'sexpectedutility,eo also satisfiesPaso= Gas

The first and the last of the three displayedequationsyieldY =p (Paaao - Gaa) = p IGaa- p Ga) > ?Pa Pa Pa

Substituting he last two equationsinto the second inequalityand dividingbyGa(e0)> 0 yield condition(3). The multipliersA and A are determinedby theKuhn-Tucker onditionsfor wo and wo as follows:pu=p(e0)(1 -p(e?))(pa(e0)) ([V?(ws)] - [V'(wF)] ) > O0A =p(e0)([V'(wo)] 1) + (1 -p(e0))([Vf(wF))] > 0. Q.E.D.

The next proposition,our firstmainresult, providesa sufficientconditionforteamwork o be optimalwhen agents'marginaldisutilitywith regardto helpingeffortis zero at zero helpingeffort.PROPOSITION: SupposeGb(a0, 0) = 0. Then the second-best olution s team-work f thefollowingconditionholds:

(4) Gaa(eO) + (Pab(e) Paa(e ))Ga(e0) + b o) Pa(eJJPROOF: y Lemma2, if the left-handside of (3) is strictly positive for allpositive A,,u, and 6, offering (w?,e?) to both agents cannot be the overallsecond-best.This impliesthatofferinganyotheroptimalindependentcontract,if any,to each agent(whetheror not both agentsare offered the samecontract)



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MULTI-AGENT SITUATIONS 625is also suboptimal, and hence the second-best solution must have positivehelping effort. Substituting Gb(e') = 0 and Gab(e0) = 0 (the second of whichfollows from Assumption 2 (ii) and the twice-continuous differentiability ofG( )), and dividing by pb(e0)/pa(e0) > 0 yield

(5) A + Gaa_eo__ Pab(e ) _Paa(eO) >0G(e + ( pb(e ) pa(eo)This holds for all A > 0 and pu> 0 if condition (4) holds. Q.E.D.

REMARK: This result does not rely on whether the second-best solution issymmetric (e, = e2) given that teamwork is optimal. It might be asymmetric suchthat both agents allocate more of their efforts to the same one of the tasks.Similarly, the result still holds without Assumption 3 if the extremely asymmet-ric case where both agents work only on one task is called teamwork as well asthe case of mutual help. Such asymmetric solutions might arise when Pab is veryhigh. It is not easy to obtain the exact conditions for the optimal teamworkcontract to be symmetric because of the nonconvexity of the implementationcost function C( ).13Note that by Proposition 3 when the marginal disutility of helping effort iszero at zero help, the first-best contract always induces positive helping effort(given a** > 0). Proposition 4 shows, however, that in the second-best situation,the principal may prefer zero helping effort unless (4) holds: There may be acase where the first-best solution is teamwork while the second-best is anindependent contract.Condition (4) represents the effects of introducing "a small amount ofteamwork" on the principal via the incentive compatibility constraints. Toclarify the condition, let (a, () be the optimal response functions of the agents,that is,

(a(e;w),fl(e;w)) = argmaxU(w,a',b',e)(a', b)where w is monotone increasing. Note that a(-) and fl8() are single-valuedbecause of the strict concavity of U(w, * e). At the optimal independentcontract (w?, eo), these response functions satisfy the following first-orderconditions:(6) Un (wo,,?ae ) = pa(a, 0) [ V(w?) - V(w?)] -Ga(a, 0) =0,

Ubnw0, a,f, eo) = -Gb(a, ) < 0.When Gb(a 0) = 0, increasing agent n's helping effort marginally from zerodoes not affect his own effort level. Then the only effect is on agent k's own

13 I am gratefulto MartinHellwigfor drawingmy attention to the issue of asymmetricolu-tions.



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626 HIDESHI ITOHeffort level. This effect can be obtainedfrom the firstequation n (6) as follows:(7) 0 0

Db (eOwO) Gaa -Paa[V - v]Pab/Pa

Gaa/Ga -Paa/PaBy (7) we obtainthe followingcorollaryof Proposition4.

COROLLARY 1: Suppose Gb(a0, 0) = 0. Then the second-best solution has posi-tive helping effort if(8) Pb(a O) +Pa(ao,o) OaO(0,;w) >0.

If condition(8) holds,the marginal ncreasein help inducedby the introduc-tion of a small amountof teamworkmakes the incentive compatibilitycon-straints ess stringent,andhence the principal s betteroff. The first termof theleft-hand side of (8) is the direct effect of marginalincreases in agent k'shelping efforton the successprobability f task n. The second termrepresentsthe indirect effect on the success probability hroughthe external effect ofmarginal ncreasesin agent k's help on agent n's own effort level. The formerdirect effectis alwayspositivesincehelpingeffort is productive.The latter effectis positive,in particular,f Pab> 0, whichmeans that the marginalproductivityof own effort is increasingin helping effort. We call this a complementaritycondition. Equation (7) shows that (when Gb(e0) = 0) the complementaritycondition is equivalentto the conditionthat, other effort levels fixed, whenagent k increaseshis helpingeffort,agentn adjustshis owneffortupward.Thisconditionholds, for example,when the success probability s multiplicativelyseparable in a and b; p(a, b) = q(a)r(b) with q( ) and r(-) increasing andconcave.Of coursethe complementarityonditionis far fromnecessary.Teamwork sclearly optimal when success probabilityis additively separable, that is,p(a, b) = q(a) + r(b) with q( ) and r(-) as above,so that there is no externalitybetween two agents' incentive problems.This is simply because the helpingeffort is productiveand inducinga small amount of help is costless. This alsoimpliesthat if there were onlya singleagent, the principalwouldalways nducehim to workon anothertask when his marginaldisutilityof effortinto the newtask is zero.

A more interestingresult is that even if an increase in an agent's helpingefforthas a negativeexternality n the other agent'sown effort level (the case offree-riding), ondition(8) can be satisfied.For example,supposethat own effortand helpingeffortare perfectsubstitutes;p(a, b) = q(a + b) with q(*) increas-ing and concave. Then becauseof the diminishing eturnsto total effort,eachagent reduces his own effort level when the other agent increases his help.



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MULTI-AGENT SITUATIONS 627However,this negativeeffect is not large enough.Since Pa =Pb and Paa Pbb,the only net effect in (4) is through Gaa/Ga. This term is positive so thatdac/db > -1: an increase n helpingeffortdoes not reducethe own effortby thesame amountbecausethe decreasein own effort reducesthe marginaldisutilityof own effort: teamworkcan reduce the cost of inducingeach agent to work on hisown task. Similarly, f Pab< 0 holds but Paa is negative enough, the principalmaybe betteroffby introducingeamworkbecausethe highermarginalproduc-tivityof own effort reduces the incentivecosts.The interpretationof the sufficientconditionis so far builton the effect ofteamworkon the incentivecompatibility onstraints.Supposeinstead(4) holdswith equality.This impliesthat the marginal hangeof the optimal ndependentcontract oward eamworkdoes not affectthe tightnessof the incentivecompat-ibility constraints.The principal is then better off since the participationconstraintsbecome less stringent:a small amount of help induced by themarginalchange of the independentscheme increasesthe success probabilitywhile it does not increase the disutilityof effort.The sufficientconditionfor teamwork o be optimalfails to hold only if anincreasein help decreasesthe marginalproductivity f own effort(Pab < 0) SOmuch that this negativeexternaleffect (a highdegreeof free-riding)dominatesthe other positive external effects through Paa/Pa and Gaa/Ga.It is well knownthatthe optimalwageschedulein the standardhidden actionmodel canbe interpreted n statistical erms.(See Hartand Holmstrom 1987).)Our resultcan also be interpreted rom the perspectiveof statistical nference.The sufficient condition (4) holds if Pab(e0)/Pb(e0) > paa(e0)/pa(e0). Whenpa(e0) =pb(e0), this is equivalent to the statement that helping effort affects theinference on agents'own effort choice more effectivelythan own effortsdo.14For example, if Pab> 0, increases in helping effort make the detection ofagents'own effort easier. Since Paa< 0, however, ncreasesin own effortmakethe detection more difficult.Even if Pab< 0 the conditionabove implies thatthe negativeeffectof helpingefforton the inferenceof own effort is at least assmall as that of own effort.Thus, as long as the marginalproductivity f owneffort and helpingeffort is the same, the better statisticalpropertyof helpingeffort is sufficient or teamwork o be optimal.

5. A NONCONVEXITYIN THE OPTIMALTASK STRUCTUREIn the previoussection,we haveobtaineda sufficientconditionfor teamworkto be optimalunderthe assumption hat the marginaldisutilityof helpingeffortis zero at zero help. What if it is strictlypositive as in the case where the

disutilityterm only depends on the total amount of effort? The result isdrasticallydifferent.It turns out that the optimalindependentcontract w?,e?)alwayssatisfiesthe necessarycondition n Lemma2. In the proofof the lemma,14 If Pa =Pb at eo, the inequalityabove is equivalentto d(pa/p)/db > d(pa/p)/da and

d(-Pa/(l -p))/db


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628 HIDESHI ITOHthe multipliersA and ,u are determinedby the Kuhn-Tucker onditions or wo?andw?. However,6 cannotbe specified romthe conditions.Thus, f Gb(e0) > O,by taking 6 sufficiently arge, condition(3) is always satisfied at (w?,eo). Inother words,there alwaysexists 6 suchthat (3) holds at (w?,e?); the left-handside cannotbe strictlypositivefor all 6 > 0.An implicationof this negativeresult is thatmarginalchangesof the optimalindependentcontracttoward teamworkcannot induce positive helpingeffort:By Proposition2, sucha marginal hangeis valuableonly if it inducesan agentto exert positive helping effort, and it in fact does when Gb(a0,0) = 0. WhenGb(a0,0) > 0, however,since the strict inequalityholds in the second equationin (6), the optimal response f3(e0;wo) of an agent does not respond to anymarginalchange from wo or eo. For example,considera small change of theindependent scheme of agent n, given by Wc '(E) wo? Evjnfor positive E andi,j E &. By choosing1J" such as nist 0 and 'F < 0, we can affect his marginalexpectedutilitywithregardto his helpingeffortbnwhen E is small:

Ub el o=pb(wS -WF) [ pV (Ws) + (1 -P)V '(w)] > 0where Ubn.,s the crosspartialderivativewithregard o bnand e. However,thesmall reward romchoosinga small amount of positivehelpingeffort is alwaysdominatedby the resulting ncreasein disutility.

The principalhence cannot induce a risk averse agent to providea smallamountof help unless his wage substantiallydepends on the outcome of theother agent'stask. Thus,we cannot obtain sufficientconditions or teamwork obe optimal by simplyexamining ocal conditions at the optimal independentcontract.We can howeverobtain a general,strongerresult that smallamountsof helping effort are always suboptimal from the principal's point of view. This isbecausea small amountof help increasesthe principal's evenueinfinitesimallywhile its implementation equiresdiscontinuously igherincentive costs for theprincipal han the costs of implementing ero help.PROPOSITION 5: Fix an> 0 and ek. If Ggn(an, )> 0, then C(e1, e2), the mini-mum expected cost to the principal of implementing (el, e2), is discontinuous at

bn =O as bn increases.PROOF:Let n = 1. We fix e2 and a, > 0, and let b, = E> 0 go to zero. Foreach E> 0, we have a unique optimal wage schedule W[n(E) that implements(e1(E), e2) at least costs where e1(E)= (a1, e). (The optimal scheme is uniquebecause the ImplementationProblem has a strictly convex objectivefunction

with linear constraints.)These wages WnJ(E) satisfy (PC) and (LNIC) so thatU = 0. Let vOJ(E)= Vn(w1j(EJ)).Then by Lemma 1, V V?(J)> Vn(E) and v/n(8)VF (E) hold for i, j = S, F, with strict inequalitiesfor n = 1. Lookingalong asubsequence f necessary,we can assumethat each w/j(E) convergesas E -O0, tosome w11E Yt. We claimthat(wj) implement e1(O),e2); fixinge2, if e1(O)s notagent l's best responseat wages wh.,then there is some strictlybetterresponse



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MULTI-AGENT SITUATIONS 629el, which would then be strictly better response to e2 at w2-(W)than e1(E) forsome (sufficiently small) E > 0, a contradiction. Similarly, we can show that e2must be a best response to e1(0) at wc2j.

Now if C() is continuous at E = 0, (WCJ)will be the optimal wage scheme for(e1(O), 2); by the continuity of wages and responses, the principal's net profitsare continuous as E -* 0. However, by continuity VCJ= Vj(W satisfyVS > vF and vC > VC for i, j = S, F,

and also satisfy (LNIC) with Ul = 0, that is,(9) Ubl(w, e1(0), e2) =pb(a2,0)[pl(al, b2)Sl + (1 -pl(al, b2))F1 ]

-Gl(al,0) = 0where a' = Vi -V F and 81= vl - Vl. By (9) and Gl(a1, 0) > 0, either 81 > 0or 81> 0 must hold: the optimal scheme of agent 1 for (e1(0), e2) must becontingent on the outcome of task 2. This contradicts Proposition 2. Q.E.D.

Figure 1 shows a graph of the implementation costs as a function of b1 = 8.When b1 > 0, however small, the principal requires imposing risk on agent1 through task 2 and this has the first-order effect under the assumptionGb(a1, 0)> 0. However, in the limit b1 = 0, there is no need for such risksharing.

COROLLARY: Fix an > 0 and ek and suppose Gn(a, 0) > 0. Then for suffi-ciently small 8 > 0, theprincipal's expectednet profits at bn= e are strictlysmallerthan those at bn= 0.PROOF:Let n = 1. By Proposition 5, C(a1, 8, e2) is discontinuously larger than

C(a1, 0, e2). On the other hand, R(e1, e2) is continuous at b1= 0 as b1 increases.Thus, for sufficiently small E > 0, R(a1, 1, e2) -C(a1, E, e2) < R(a1, 0, e2) -C(a1, 0, e2) holds. Q.E.D.

C (el (*), e2)

lmC e l bl),e2) FIUR4 4 C(el(O),e 2)

0- biFIGURE 1



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630 HIDESHI ITOHBy Corollary2, when an agent'smarginaldisutilityof zero help is strictlypositive,the principalwantshim to either specializein his own taskor providesubstantialhelp with the other agent.This resultmay partiallyexplainwhywe

observeeither strict individualwage schemesor significantly eam-basedwageschemes: intermediate cases, where wage schedules of workers depend onco-workers'performancesn a verylimited manner,are rarelyobserved.Note that thisresulthas nothingto do withmulti-agent ssues.The resultcanbe applied in situationswhere the principaldetermineswhether her agentshould workon a single taskor performmultipletasks.In fact, a similarresultholdseven for the standardone-agentmodelwhere the agent'seffortis chosenfroma closed interval[a, ii]. That is, if C(a) is the least cost of implementingaction a and if the agent is riskaverse(and other technicalconditionson thedistributionof outcomes and the disutilityof effort are met), C( ) increasesdiscontinuouslyat a. Note in this regard that Grossman and Hart (1983,Proposition1) show that C(-) will, in general,be lower semicontinuous.Weextend this, then, to give conditionswhereit is definitelydiscontinuousat a.As a finalremark,we note that the resultis similar o the nonconcavityn thevalue of information n Radner and Stiglitz (1984). They show that a smallamountof informationhas a negativemarginalnet value (as a smallamountofhelp for the principaln our setting)whenever he marginal ost of informationis strictlypositive.The difference s thatin theirmodel,thereis no discontinuityin the information ost functionat zero informationbecause theirmodel is notconcerned with incentive problems. The discontinuityarises in our modelbecauseof the existenceof the incentiveproblemdue to hiddenaction.We close this sectionwith a conjecture.Whenthe marginaldisutilitys alwayspositive,the first-best olution mayhavezero helpingeffort, dependingonlyonthe comparisonof the marginalproductivityand the marginal disutilityofhelpingeffortwiththoseof own effort.Thus,if the incentiveeffect discussed nthe previoussection is sufficientlyavorableto teamwork e.g., a high level ofcomplementarity),here may exist a case in whichthe first-bestsolutionis anindependentcontractwhile the second-bestsolutionis teamwork.

6. CONCLUDING REMARKSThe main objectiveof this paper was to find what determineswhether toinduce teamworkor unambiguousdivisionof labor. We found that there aretwo important actorsin determining he optimaltask structure rom incentivepointsof view:strategic nteractionbetweenagentsand their attitudestowardperformingmultipletasks.Althoughmanymodernproductionactivities nvolve

interdependenceamongand multipletasksperformedbyworkers, he resultsofthe paper suggest that the incentive effects of introducing hese features becarefullyevaluated.If an agent'smarginaldisutilityof performingan additional ask is zero, wewere able to obtain a sufficientconditionfor teamwork o be optimal.Team-workis optimal f each agentincreaseshis own effortresponding o an increasein help from the other agent. Teamworkcan also be optimal without such



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MULTI-AGENT SITUATIONS 631complementarity. Very often people free ride on others' help and reduce theirown effort. In such a case, more factors affect the task design problem. Supposethat each task is so monotonous and boring that the worker's marginal produc-tivity on that task declines or his marginal disutility increases drastically as heworks harder. If this positive effect of free riding dominates the negative effect,teamwork is still optimal because it reduces the cost of inducing each worker towork on his own task. This result corresponds to the assertion by behavioralscientists that job enlargement and enrichment can motivate workers to workhard.On the other hand, the situation becomes more complicated when agents arereluctant to provide even small amounts of help. For example, complementarityis no longer sufficient for teamwork to be optimal. We have to consider costs ofinducing an agent to perform multiple tasks, which are substantially larger thanthe incentive costs for the specialized agent. The result is the nonconvexity ofthe optimal task structure: The principal wants either a specialized structure ora substantial teamwork.Though not discussed in the main text, there is another important prob-lem associated with teamwork; the problem of collusion among agents. AsMookherjee (1984) points out, there are two kinds of collusion problems. Thefirst one comes from the multiplicity of Nash equilibria: Given wage schedules,there may exist another Nash equilibrium preferred by both agents to the onethe principal wants to implement. This is a serious problem which arises onlywhen teamwork is introduced in our model.'5 The other collusion problem isthat the agents may collude to choose some cooperative effort pair that isdifferent from the Nash equilibrium pair the principal attempts to implement.Introducing teamwork enables the agents to have full information about eachother's actions, and hence is likely to promote such collusion.16As an extension of our model, we briefly mention the case of more than twoagents. It is straightforward to extend our two-agent model to the case of Nagents: Agent n chooses, instead of a two-dimensional effort variable, anN-dimensional effort vector (ej ', en), where en is his own effort level and ek(k 0 n) is his helping effort level for agent k. The success probability of task n isa function of the inputs to that task, (en , en). The main results in this papercan be extended to this case. One important difference appears in the analysisof the sufficient condition for teamwork. When N > 3, whether or not agent nand agent k form a team generally depends on the effort choice by agent 1(1 0 n, k) as well: a "good" relation between agent n and agent k (e.g.complementarity between them) may be destroyed when agent 1 joins theirteam. Thus the sufficient condition for teamwork to be optimal is not as simpleand clear as the conditions in Proposition 4 and Corollary 1. If we assume that

15 One approach to this problem, due to Ma (1988), is to enlarge the strategy sets of the agents sothat the principal can implement what she prefers as a unique perfect Nash equilibrium of the effortchoice game by agents.16 However, this may not be a disadvantage of teamwork. Itoh (1990) shows that the principal, bydesigning incentive schemes appropriately, can be better off when agents collude. See alsoHolmstrom and Milgrom (1990) and Ramakrishnan and Thakor (1989).



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632 HIDESHI ITOHthe relation between two agents is independent of the third agent (which I thinkis not an unreasonable assumption), our sufficient conditions are still valid: Thecomparison between two agents is sufficient to determine whether they becomemembers of the same team to help one another. This independence assumptiontherefore has the following interesting implication on the optimal team size.Suppose that all the agents are identical. Then if they have zero marginaldisutility of helping effort at zero help, there is no particular cost of increasingthe size of the team. The principal chooses either N-size teamwork or no teamat all. However, if the agents care only about the total amount of effort so thattheir marginal disutility of help is not zero at zero help, increasing the size ofthe team raises the agent's marginal disutility of help, so that there may exist alimit on the team size.'7

In this paper, we focused on the incentive aspect of teamwork, following theliterature on the moral hazard in the principal-agent relationship. A shortcom-ing of this approach is that we cannot obtain many clear predictions on realmanagement policies.18 In particular, in our model, agents' abilities, characteris-tics, and task characteristics are common knowledge among all the relevantparties. Also we assumed away all dynamic aspects of task design problems. Inmore realistic settings incorporating learning, reputation, information asymme-tries, or firm-specific human capital accumulation, we will be able to pursue ourfurther understanding of teamwork and task structures.Department of Economics, Kyoto University Sakyo-ku, Kyoto 606, Japan

ManuscriptreceivedDecember 1988; final revision receivedApril, 1990.

APPENDIXIn this appendix, we obtain sufficient conditions for the first-order approach to be valid. Theprocedure of the proof is similar to that of Rogerson (1985). We first show that the wage schedule in

the solution of (RP) is monotone increasing: given an outcome of one task, the wage under successof the other task is at least as high as the wage under failure. Then we prove that the expectedutility of each agent is concave in his effort variables under the condition given in Assumption 5.We introduce another relaxed problem, denoted by (RP + ), in which the principal chooses(w , w2) and (el, e2) to maximize the objective function in problem (OP) subject to the participationconstraints (PC) and the following local Nash incentive constraints (LNIC +):(LNIC + )

Ua,(Wn,enf,ek) =0 forn= 1,2;Ub w, en,ek) 6 0 and bn *Ub(Wn enf,ek)=O forn=1,2.

The ct. traints (LNIC + ) are the standard local conditions for (NIC).17Our model might also offer an explanation of increasing returns to size from the incentiveviewpoint. Managers of two separate firms cannot share their tasks, and hence are contractedindependently. When the firms are merged, under some conditions, the owner would like to designnew interdependent contracts in order to induce them to share their tasks, and thereby increasingreturns to firm size might arise. I am grateful to George Mailath for suggesting this.18 In earlier versions of the paper, I obtained some properties of the optimal teamwork contract.



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MULTI-AGENT SITUATIONS 633For each n, let An be the Lagrange multiplier for (PC), An and (n be the Lagrange multipliersfor the conditions for an and bn in (LNIC) in the relaxed problem (RP), respectively. TheKuhn-Tucker necessary conditions for the optimal wage w/j is obtained as follows:

1 dPP (enl,ek)/danl liP'2(n,e)l(Al) = b nk)/n for i, i ELemma A-1 shows that the wage schedule for each agent is monotone increasing.

LEMMA A-1: Let (w, w2) be the optimal wage schedule of (RP) for (el, e2) such as an >0,n = 1,2. Then w' > wn and wni> w[n for i, j =S, F. In addition, the equality holds if and only ifbn = 0-

PROOF:For i, j = S, F,(A2) dPslda pn

dPjF/dan -paP =-pFnj pAP n Pb

pn pknln 1 p

If (l < 0, then by (Al) and (A2), wsnS wF1 must holdfor each j =5S,F. Then, pniSab pk(n_V)+ -p)V F-F) Ga6 a

for all an> 0, whichis a contradiction.Hence Mn> 0. Then w5nj>wFjfor all j =5S,F. Similarly,fbn>0? n must hold, so that w/i5> w/i for i =S, F. If bn=0O, hen by (Al), w/j5= w,'4 fori=S,F. Q.ED.1FThe next lemmashowsthat the solution o (RP) is in the constraint et of (RP + ).LEMMAA-2: Supposehat (w1 w2) andeA, e2) such as an > 0 for n = 1,T2solve (P). Then heysatisfy (LNIC +?).PROOF: Clearly it is sufficient to show Ub7 0. If bn> 0, by (LNIC), U,7 =0. Thus, (LNIC +) issatisfied.Next, supposebn= 0 and Ub, > 0. Thatis,

un = Pnk[kP n -sns5 ) + ( 1- _k) (V F-VF )n Ga]-G an< 0

Thus, either v~55>VSF or v5> VF> must hold. Then by Lemma A-i,Te5 >j>W for allm=iS, F.Thereforebn > , which is a contradiction. Thus, when bn= 0, Uj 6 0 must hold, so that (LNIC +)issatisfied. Q.E.D.LEMMAA-3: (w1u w2) and (el, e2) such as an > 0 for n = 1,2 solve (RP) if and onlyif they solve(RP+).PROOF:The "only ifs' part is trivial by Lemma A-2. Concerning the "i' part, supposewL , w2)andet , e2) solve (RP + ) while they do not solve (RP). Letis,2) and (e, e2) solve (RP). Denoteby UP(-) the principal's expected net profits. Then since (w, w2) and (e, e2) satisfy (LNIC), the



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634 HIDESHI ITOHoptimality of (w, w^2)and (l, e2) results in

up(w1, 2,ele2 > up(w1, 2,ele)However, by Lemma A-2, (wii,w2) and ( Y, ) satisfy (LNIC + ), so that they are in the constraintset of (RP + ). A contradiction. Q.E.D.

Based on Lemma A-1, Lemma A-4 provides sufficient conditions for the first-order approach tobe valid.LEMMA -4: Suppose that (w1, W2) solve (RP) for (el, e2) such as an > 0, n = 1,2. Thenfor eachn, the expected utility of agent n is strictly concave in en if either one of the following holds: (i)

PSS =pnpk is concave n en andh' is concave; ii) PFJ = (1 _pn)(1 _ k) is convex n en and hn isconvex.PROOF: Suppose that condition (i) holds. The expected utility of agent n at w' is given by

(A3) Un(wn,.)=VF +pn(.)(VnFVnF) +p(.)(VnS-VnF)+p n()pk(.)[(Vn -VnF) -(VnS-VFn)] -Gn(

When P~Sn pnpk is concave in en, clearlypn is concave n an and pk is concave in bn.And byVn- >0 and - Vn 0. iphner ih nLemmaA-1, F- VFF > 0 and F VFF > O. Thus,if the termwith SS =pnpk is concavein en,Un is strictly concave (because Gn is strictly convex). To show this, it is sufficient to show- VnF > Vn - VFn. By (Al) and (A2), we obtain

(A4) hn( ) -hnvFn) = hn( )-hF n FF) = nbnPb(pb + 1_pk)If the optimal solution has no positive helping effort by agent n, the right-hand side of (A4) is zero.Otherwise, the right-hand side is strictly positive. Then, since hn is an increasing function whenagent n is risk averse, it follows from (A4) and the hypothesis that hn is concave that VnS - VnF >vF -vFF.To show that condition (ii) is sufficient, let qm = 1 -pm and express the expected utility of agentn by using qfn and qk. Then follow the steps similar to the case of condition (i). Q.E.D.

LEMMA A-5: Supposeeither one of the conditions (i) and (ii) in Lemma A-4 holds. Then (w1, W2)and (el, e2) such as an > 0 for n = 1,2 solve (RP) if and only if they solve the original problem (OP).PROOF: The "only if' part is by Lemma A-4. The proof of the "if' part is similar to that ofLemma A-3. Q.E.D.The sufficiency of the condition (i) or (ii) in Lemma A-4 holds in the case of more than twooutcomes. Suppose that for n = 1, 2, there are Mn possible outcomes 1, , Mn of task n, and fori= 1, Mn, let 7rn be the revenue from task n when the outcome of task n is i with

17-n < ... 0 when the optimal helping effort level of agent n is zero. To showthis, we follow Rogerson (1985) and define the doubly relaxed problem, denoted by (DRP), which is



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MULTI-AGENT SITUATIONS 635the problem (RP) with the following constraints (DLNIC) in place of (LNIC):(DLNIC) U,(w',ef,ek)=O forn=1,2;

bn Ubn(w ,enl, ek) > 0 for n = 1,2.Note that the second line is not an equation. Then the Kuhn-Tucker necessary conditions for thesolution to (DRP) immediately lead to the multiplier (n for the second inequality in (DLNIC) isalways nonnegative. What must be proved is that the solution to (DRP) satisfies (LNIC).

LEMMA A-6: Suppose that (w1, w2) and (el, e2) such as an > 0 for n = 1, 2 solve (DRP). Thenthey satisfy (LNIC).PROOF: We have to show that (w , w2) and (el, e2) satisfy the second condition in (DLNIC) withequality. If (n > 0, this follows immediately from the complementarity slackness. Now suppose

(n = 0. Then the proof similar to that of Lemma A-1 shows that the wage schedule of agent n doesnot depend on the outcome of task k. Thus, if bn > 0,bn- Ubn = bn(-Gn)


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636 HIDESHI TOHKAGONO, T., I. NONAKA,K. SAKAKIBARA, AND A. OKUMURA (1985):Strategic vs. EvolutionaryManagement: A U.S.-Japan Comparison of Strategy and Organization. Amsterdam: North-Holland.LAZEAR, E. P. (1989):"Pay Equalityand IndustrialPolitics," Journalof PoliticalEconomy,97,

561-580.LAZEAR, E. P., AND S. ROSEN (1981): "Rank-Order Tournaments as Optimal Labor Contracts,"Journal of Political Economy, 89, 841-864.LINCOLN, J. R., AND K. McBRIDE (1987): "Japanese Industrial Organization in ComparativePerspective," Annual Review of Sociology, 13, 289-312.MA, C. (1988): "Unique Implementation of Incentive Contracts with Many Agents," Review ofEconomic Studies, 55, 555-571.McAFEE, R. P., AND J. McMILLAN (1986): "Optimal Contracts for Teams," mimeo, University ofWestern Ontario.MILGROM, P. R. (1981): "Good News and Bad News: Representation Theorems and Applications,"Bell Journal of Economics, 12, 380-391.MIRRLEES,J. (1975): "The Theory of Moral Hazard and Unobservable Behavior-Part I," mimeo,

Nuffeld College, Oxford.MOOKHERJEE,D. (1984):"OptimalIncentiveSchemeswith Many Agents,"Reviewof EconomicStudies, 51, 433-446.NALEBUFF, B. J., AND J. E. STIGLITZ(1983):"Pricesand Incentives:Towarda GeneralTheoryofCompensation and Competition," Bell Journal of Economics, 14, 21-43.RADNER, R., AND J. E. STIGLITZ 1984): "A Nonconcavity in the Value of Information," in M. Boyerand R. E. Kihlstrom, Bayesian Models in Economics. Amsterdam: North-Holland, 33-52.RAMAKRISHNAN,R. T. S., AND A. V. THAKOR(1989):"Cooperation ersusCompetition n Agency:Incentive Problems, Diversification, and Corporate Mergers," mimeo, Indiana University.ROGERSON,W.P. (1985): "TheFirst-OrderApproach o Principal-Agentroblems,"Econometrica,53, 1357-1367.WATERMANJR., R. H. (1987):The Renewal Factor: How the Best Get and Keep the CompetitiveEdge.

New York: Bantam Books.

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Itoh ('91) - Incentives to Help

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