Organizations engage in search whenever they perform nonroutine tasks, such as the definition and validation

Transcription

1 MANAGEMENT SCIENCE Vol. 5, No. 5, May, pp. 3 4 issn 5-99 eissn informs doi.7/mns..4 INFORMS Hierarhial Struture and Searh in Complex Organizations Jürgen Mihm, Christoph H. Loh INSEAD, 7735 Fontainebleau Cedex, Frane {jurgen.mihm@insead.edu, hristoph.loh@insead.edu} Dennis Wilkinson, Bernardo A. Huberman Soial Computing Lab, HP Labs, Palo Alto, California 9434 {dennis.wilkinson@hp.om, huberman@hpl.hp.om} Organizations engage in searh whenever they perform nonroutine tasks, suh as the definition and validation of a new strategy, the aquisition of new apabilities, or new produt development. Previous work on searh and organizational hierarhy has disovered that a hierarhy with a entral deision maker at the top an speed up problem solving, but possibly at the ost of solution quality ompared with results of a deentralized searh. Our study uses a formal model and simulations to explore the effet of an organizational hierarhy on solution stability, solution quality, and searh speed. Three insights arise on how a hierarhy an improve organizational searh: () assigning a lead funtion that anhors a solution speeds up problem solving; () loal solution hoie should be delegated to the lowest level; and (3) struture matters little at the middle management level, but it matters at the front line; front-line groups should be kept small. These results highlight the importane for every organization of adapting its hierarhial struture to its searh requirements. Key words: searh; omplexity; osillations; oordination; deentralized problem solving; hierarhy History: Reeived Deember 3, 7; aepted Deember 9, 9, by Olav Sorenson, organizations and soial networks. Published online in Artiles in Advane Marh 3,.. Introdution Large organizations need to solve problems that are omplex beause of multiple relevant tehnologies, globalizing markets, multiple interating business proesses, and ollaboration with external partners. In addition to stati omplexity, organizations often fae dynami environmental turbulene (Eisenhardt and Tabrizi 995). These hallenges demand that deisions with many parameters be taken quikly. Yet in the fae of suh irumstanes, deisions often annot be optimized (Simon 99). Rather, organizations engage in searh, or the generation of alternatives through the hange of a few (inremental searh) or many (radial searh) deision parameters, and then hoose the alternative that performs best. In large organizations, no single individual is able to grasp all deision parameters, so searh must be a distributed effort. Therefore, a entral hallenge of organizational design is to divide the omplex searh proess into manageable speialized tasks and to oordinate these tasks so that the firm reaps the benefits of onerted ation (Nadler and Tushman 997, Rivkin and Siggelkow 3, Siggelkow and Rivkin 5). In organization design, senior management alone is not apable of finding high-quality task solutions and must alloate tasks and delegate deisions, provide inentives, and struture ommuniation among the various organizational members eah with their respetive subpiee of the organization s overall problem (Siggelkow and Rivkin 5) to ahieve the best ombination of speed, solution quality, and risk of failure. The following example from engineering demonstrates how organizational issues influene the balane of these interrelated goals. In the semiondutor industry, the design of Intel s Itanium hip went around in a irle, finding itself in a nightmarish world where a hange to one module would ripple through the work of several hundred other people, leaving more problems in its wake (Hamilton, p. 9). The design finally onverged only after a highlevel manager froze several omponents, de fato entralizing the deision struture to improve solution speed, impliitly limiting omponent performane. This study examines how an organizational hierarhy an best be strutured to guide organizational searh. We use formal modeling to study the effets of hierarhial struture on problem-solving searh with respet to speed, solution quality, and failure risk. Previous work has observed that a hierarhy an improve organizational searh: entralized deision making at the top stabilizes searh, redues failure risk, and leads to faster deisions, whereas deentralized deision making inreases the solution quality (Rivkin and Siggelkow 3) and raises the organization s ability to ope with environmental hanges (Siggelkow and Rivkin 5). This work has illuminated important managerial trade-offs, but it has represented hierarhy 3

2 3 Management Siene 5(5), pp. 3 4, INFORMS Figure Organizational Hierarhy and Problem-Solving Searh CEO Behavioral aspets of hierarhy Strutural aspets of hierarhy Middle management Front-line management Front-line problemsolving workers Dept. Area Subproblem Dept. Subproblem Dept. Area Dept. Subproblem 3 Question: Horizontal oordination: Sequential or parallel work? Result: Sequential work is faster and produes higher quality. Question: Vertial oordination: How far to delegate deisions? Result: Delegate deisions all the way to front-line management. Question: What is the best depth/size of departments? Result: Politial struture in the middle does not matter. What matters is limiting the size of front-line working groups. only in its simplest form with one hief exeutive offier (CEO) and two workers and has foused on omparing entralization to deentralization. Important questions about how to struture hierarhies for searh remain unanswered. These questions involve deision-making proedures beyond the issues of entralization versus deentralization and the organization s formal struture. In this paper, we raise three questions that an organizational designer needs to onsider; the first two onern how deisions are made in a hierarhy, and the third onerns how the hierarhy itself should be strutured (see Figure ). First, how should the groups at a given layer of the hierarhy oordinate? Should they searh their respetive subproblems in parallel and then oordinate and adjust, or should they searh sequentially, with one lead funtion determining both a general diretion and onstraints for the others? Our results suggest that, in oordinating hierarhies, sequential deision making is more benefiial for searh. In this sense, status differenes among groups and a orresponding order in whih the groups influene a deision may be justified. Seond, how far should deision power be delegated: all the way down to the front line? We find that deision-making entralization is most effetive if delegated to front-line management: delegating the solution hoie to the lowest level of management generates most of the stability and speed benefits that a hierarhy offers. Third, how should a hierarhy be strutured? Beyond questions of ontrol loss, how does the size of the reporting groups influene the breadth versus speed of organizational searh? Our results suggest that hierarhial strutures at the middle management level matter little for the performane of the searh proess. It is at the front line that the grouping of problem-solving workers into departments matters most. Smaller groups have an easier time ahieving speed and stability. Our answers to these three questions ontribute to the organizational searh literature by illuminating the interations between organizational struture, hierarhial deision making, and the searh proess. As a seond, methodologial ontribution, this artile offers a formal searh model that allows for deriving an analytial haraterization of a general problem in distributed searh. The analytial model shows that, whenever alignment of subproblem deisions is less than perfet (for reasons of information overload or inentive onflits), problem-solving speed and stability will suffer in large organizations with deentralized deision making. In this ase, hierarhial entralization may speed up and improve the results of searh. The model omplements the widely used NK model in showing robustness of qualitative findings based on different miro deision strutures. We overview related work in, present the model in 3, and disuss analytial and simulation results in 4 and 5, respetively. Setion onludes, and the proofs are gathered in the appendix.. Organizational Hierarhy and Searh Searh is pervasive in organizations. Examples inlude finding and defining a strategy (Rivkin, Winter et al. 7), adapting to industry life yles (Doty et al. 993, Levinthal 997, Nelson and Winter 9), building organizational apabilities (Bruderer and Singh 99, Gavetti 5), and new produt development (Fleming and Sorenson, Mihm et al. 3, Yassine et al. 3). Searh may ontain elements of loal optimization by intelligent loal agents (e.g., Gavetti 5, Gavetti and Levinthal, Knudsen and Levinthal 7, Sorenson, Winter et al. 7), but in general, it is searh and not optimization that dominates omplex problem solving. For omplex problems, the searh proess annot be performed with regard to the whole problem at one

3 Management Siene 5(5), pp. 3 4, INFORMS 33 beause no ator or resoure has enough informationproessing apaity to onsider all aspets of the problem (Loh and Terwiesh 7, Simon 99, Van Zandt 999). However, the subproblems are typially not fully separable; their solutions depend on one another. As a result, organizations onsist of more or less tightly oupled groups (as in Cyert and Marh 93) that exeute speialized funtions while interating to produe organizational output. It is a entral hallenge for the organization to derive the benefit of dividing the overall searh problem into manageable tasks performed by speialized groups while oordinating those groups to obtain ohesive ation (Marh and Simon 95). Searh has been a topi in organization theory sine its infany (Cyert and Marh 93, Marh and Simon 95, Simon 99), but interest has surged with the introdution of formal models to organization theory notably the NK model (e.g., Levinthal 997), whih models ators as N nodes with K interdependenies; it was developed in physis as a spin glass model and then applied to evolutionary biology (Kauffman 993). Simulation work on the NK model omplemented by other models has shown that the deentralized searh of interdependent subproblems tends to slow and get bogged down as the problem s omplexity (the number of subproblems) inreases (Ethiraj and Levinthal 4, Huberman and Wilkinson 5, Mihm et al. 3): ongoing hoies in some groups make the requirements for other groups inherently unstable (Thomke 997, Van Zandt 999). Although these studies have identified managerial ations to mitigate the problem, they have not addressed the question of whether and how hierarhial organizational strutures an improve searh time and quality. A vertial hierarhy is the most ommon way of oordinating speialized groups and their separate deisions. There are other oordination mehanisms suh as liaisons, ross-unit groups, and informal networks but a vertial hierarhy is present in virtually every organization (Nadler and Tushman 997). Groundbreaking work on the role of hierarhies in searh (Rivkin and Siggelkow 3, Siggelkow and Rivkin 5) based on the study of NK searh landsapes suggests that fully entralized deision making stabilizes and speeds the searh proess but at the ost of performane. However, this work models hierarhy as onsisting of only one CEO and two managers, so it ould hardly examine the role of differentiated hierarhial strutures in searh. In our study, multiple players at several levels of a hierarhy engage in We take vertial hierarhy to mean that multiple individuals, or multiple groups, report to the same manager. searh while the hierarhy helps to integrate the subsolutions into an overall solution. In evaluating performane, we onsider not only the solution quality and onvergene time, as is ustomary for many NK simulations, but also the ability to onverge to a solution at all (see Mihm et al. 3, Yassine et al. 3). Our study fouses on searh, whih is an inherently dynami onept, and does not examine the ontrol loss aspet of hierarhies. Substantial work in that area an be found in the formal modeling literature (e.g., Bolton and Dewatripont 994, Child 94, Keren and Levhari 93, Moldoveanu and Bauer 4, Nadler and Tushman 997, Radner 993, Williamson 99, Van Zandt 999). 3. Model Setup We build a model of an organization that engages in deentralized searh to solve a omplex problem with many interdependent subproblems. Our model expliitly aptures two fundamental harateristis of omplex searh: () the loal, subproblem-level deisions may not be perfetly aligned toward an overall system optimum, beause no individual understands all the effets of her deisions on the system and beause there may be interest onflits, and () mutual updating and oordination may be delayed beause immediate broadasting of all subproblem deisions would ause information overload. As an example, imagine a produt innovation whose value depends on the quality of its assoiated market researh study, the engineering work, a new manufaturing proess, and a ompetent sales plan and exeution. All of these deision domains are interdependent (e.g., the sales plan depends on the features inluded, the feasible produt features depend on the manufaturing apabilities, and the manufaturing apabilities depend in turn on the design omplexity and sales volumes that justify investments). Furthermore, beause work progresses partially in parallel, information about the latest deisions in other domains is not always immediately available (e.g., while the sales team hanges projeted sales beause of new information about ustomer behavior, the manufaturing group may be building its apaity investment plan based on the previous projetion). We first show that large problems of this type systematially ause deision delays, exessive iterations, and instability (building on Mihm et al. 3). We then show how the presene of a hierarhy influenes deision making and interations among the deision makers. 3.. A Basi Searh Model Without Hierarhy An organization of N employees olletively produes an output of value P. The output value is a funtion of all the employees deisions, P h h N.To

4 34 Management Siene 5(5), pp. 3 4, INFORMS allow deentralized problem solving, the organization divides the overall problem into subproblems i of value P i, where problem i is delegated to one speialist deision maker i. For simpliity, suppose that eah subproblem P i (and thus eah employee) has one deision variable, h i, and that P i is bounded from above. Although it is not neessary for the analytial results, we assume for easier exposition that the overall performane is a sum of the individual funtions with relative importane weights i : 3 P = N i P i () i= If the subproblems were independent, that is, eah P i h i a funtion only of its own deision variable, then the overall system optimum would result from the deision makers individual optimizations. However, interdependene among the employees makes any subproblem performane a funtion of other subproblems: P i = P i h h N () We know from NK model researh that this struture an represent unlimited omplexity beause of the interdependenies in Equation (). A omplex value funtion typially has many loal optima. Beause no one in the organization understands this omplex problem in its entirety, it follows that no overall optimization is possible the organization must engage in searh. We first model searh in a flat organization (N interdependent but equal problem-solving employees, as in Mihm et al. 3). Then we show how a hierarhy hanges the searh. The organization starts with a previous solution, for example, the result obtained in the last searh, h h N. But this is a poor solution for the new problem, so eah employee goes off to work on her own subproblem. Two properties haraterize the deision-making proess that ensues. First, the individual employee i makes a deision aimed at optimizing the overall performane funtion P, onsidering everyone s interests but potentially disounting the importane of other subproblems ompared with her own subproblem (after all, a speialist is first and foremost evaluated by her own ompetene): the employee maximizes N j= b j i j P j h h N, where b i i = and the other b j i. In one benhmark Of ourse, employees may be responsible for multiple deision dimensions. However, it is often realisti to summarize the multiple dimensions in one omposite dimension. In addition, this simplifiation does not hange the qualitative results of the model. 3 The qualitative results of this model an be shown to hold for any omplex performane funtion P = f P P N ; a more general funtion merely ompliates the exposition. ase, all employees at fully holistially, b j i =, in the opposite benhmark, b j i = when i j; that is, eah employee i ats myopially (loally) beause of politial onflits or an inability to understand the onnetions to other subproblems. Reality is usually in between. Seond, the employee s deisions may be based on obsolete information about other subproblems. The employee annot take into aount any future reations by other employees to her deisions beause she laks the tehnial expertise. The employee takes the urrent status of other subproblems as given; we denote this with h j, j i. But information overload may prevent her from seeking by-the-minute updates from the other subproblems. Thus, the set h j, j i, reflets the deision status of the other subproblems at the last time i obtained an update from j. The employee maintains this assumption until she has another opportunity to be updated. Then the deision by employee i an be written as N h i = arg max b j i j P j h i h j (3) j= Updates our asynhronously after random time intervals: the employee learns the most reent deision status of the others partly by sheduled meetings of the entire team (strategy sessions, design or sales reviews, budget meetings). However, to a substantial degree, hange requirements also arise at random moments in time at unsheduled one-on-one events, suh as a risis phone all ( if you make this hange, you ll ause a risis for my subprojet ), an enounter in the hallway or at the offee mahine, or information aidentally overheard at an unrelated meeting (Allen 977). We reflet this partially unsheduled nature of hange requirements of small groups by having the update time points be independent Poisson proesses for eah deision maker, with exponentially distributed time intervals between updates (the reader may piture this as the next updating time being randomly drawn for the individual employee, independent of what happens to other employees). When deision maker i obtains an update, her problem (whih is driven by the interdependenies) hanges if any other employees have hanged their deisions (over a short enough time interval, only one other person, or omponent h j, will hange). Thus, employee i, in turn, must hange her deision h i, and this in turn hanges the onstraints for others when they obtain their updates later. Thus, the organization must ontinue to revisit all deisions until a fixed point is ahieved, a solution from whih no employee wants to hange. This onept of a asading adjustment proess has been well established in modeling

5 Management Siene 5(5), pp. 3 4, INFORMS 35 organizations (e.g., Hannan et al. 3a, b; Lounamaa and Marh 97). Poisson updates render evolution in the model asynhronous (updates of two employees never happen at exatly the same time). Asynhronous evolution is typial for soial systems unless there are extremely strit mehanisms for enforing oordination. 4 In summary, the organization is apaity onstrained in two ways. First, deision makers are speialized and apable of fully solving only their assigned subproblems. Seond, deision makers annot always be updated on all other subproblems (beause of information overload). The subsequent deisions by individual deision makers at Poisson time points (and therefore in random order), eah taking into aount past deisions of other deision makers, onstitutes an organizational searh through the spae of deisions h (the vetor of all deision variables h i ). The organization has arrived at a stable solution when all deision makers have ahieved their loal optima, given the solutions of the other subproblems, in other words, when no deision maker wants to hange her deision unilaterally (Loh and Terwiesh 7). In this stable state, deision makers are orretly updated about the overall solution status (h j = h j ), and the organization has settled in a loal optimum (whih is also a Nash equilibrium). Although an organization is not a demoray where everyone an vote on the final projet outome, the onsensus representation serves two purposes. First, it is a realisti approximation for flat (professional) organizations with partially autonomous ators. Seond, it funtions as a base ase to establish results for one extreme of a spetrum of entralization levels, whih we omplement later by introduing a hierarhy. 3.. The Curse of Complexity for Deentralized Problem Solving Problem searh performs well, independent of the size of the organization, if two onditions are simultaneously fulfilled. First, all employees at fully holistially, that is, in the interest of the whole and not just of their own subproblems (in the model, all b j i = ). Seond, updating is immediate: employees are always informed about the latest status of the other subproblems (in the model, the updating Poisson proesses have an infinite rate). Both onditions are unrealisti, beause employees do weigh their own subproblems more highly (owing to inentives and to lak of expertise about the other subproblems), and beause 4 Huberman and Glane (993) show that modeling a naturally asynhronous system with a synhronous mathematial proess may introdue distorting artifats, and Sorenson () stresses the importane of this result for organizations. instantaneous updating would result in information overload. Violating either ondition is suffiient to bog down problem solving: the deentralized searh of larger omplex problems takes progressively longer and may not even onverge to a solution. Intuitively the reason is that, beause of interdependene, an agent who hanges her deision may indue other agents to hange as well. The ritial question for problem-solving progress is whether potential loops of mutual influene die down as the agents work progresses over time or whether they amplify one another, leading to osillations and never-ending firefighting. We now show analytially that problemsolving osillations our whenever the first ondition (holisti behavior of the employees) is violated, even when updating is immediate. The ase of updating delays (violating the seond ondition) is analytially intratable; we onsider this violation in the simulations in 5. Reall that h is the vetor of the N agents deisions. Then, the problem-solving evolution that is, the hange of the deision status vetor over time, assuming immediate updates is desribed by h t + t = g h t (4) where g h t desribes the optimization that eah deision maker performs (see Equation (3)). Beause the solution status evolves asynhronously, only one agent i s omponent of the vetor funtion h t hanges over a very short time interval, t. Then, over suh a time interval, for agent i we an write g i h t = h i h t, and for other agents, g k i h t = h k t. In this way, we are able to examine inremental hanges in the problem-solving status for any point in time, t: h t + t h t = g h t h t. The system has reahed a fixed point if g h t h t = independent of whih player i makes a move over t: eah deision maker has found a loal optimum aording to Equation (3) and will therefore stik to that deision, given all others deisions. We assume that the problem has at least one fixed point; otherwise, it possesses no stable solution and failure to onverge is inevitable. The dynamis of a system are driven by its behavior around its fixed points. The question of whether a fixed point is stable and an be reahed is equivalent to evaluating whether, one the fixed point has somehow been found, the deisions of the deentralized agents drive the system to return to the fixed point when subjeted to (even very small) perturbations. Consider a fixed point h and a perturbation with random elements i entered about zero with some finite variane. Beause we are interested in what happens for an arbitrarily small perturbation,, most relevant

6 3 Management Siene 5(5), pp. 3 4, INFORMS problem-solving systems an be desribed by a Taylor series expansion: Define the partial Jaobian matrix J i, with J i ij = g i h / h j h=h, J i ii = g i h / h i h=h, J i jj =, and J i jk = for i j k j. The derivative g i h / h j expresses the interdependeny, whih is the effet that a small hange in j s deision has on the optimal deision of i. Let h n be the system state after n players have moved. Then, with the set i m v ontaining n elements, the problem-solving dynamis an be approximated by h n h J i J m J v where i m v N (5) The problem-solving system beomes unstable if mutual interations among subproblems enlarge a small perturbation from the fixed point. Here is where our first ondition beomes relevant: if all b j i =, then all employees reat to the perturbation in the same way, eah striving for the same optimum (fixed point) of P. Formally, the ross-partial derivatives g i h / h j h=h beome zero, and thus, for all i, the J i beome zero matries. Hene, no instability arises. However, if b j i < for i j, then the employees are not perfetly aligned: eah values her subproblem more than the others and strives in a slightly different diretion. Thus, the ross-partial derivatives beome nonzero, refleting the interdependenies among the subproblems. Moreover, these interdependenies annot be fully foreseen at the outset in a omplex problem. Interations and ontraditions among subproblems may arise at plaes that are neither antiipated nor desired (Thomke et al. 999). Thus, interdependenies g i h / h j h=h ontain random omponents. We may view these as being drawn when problem solving begins and then remaining onstant. Formally, then, we assume that the random interdependenies are drawn from any distribution with finite positive variane and that they are mutually independent. Given this struture, we an formally show the following result: Proposition. Suppose that N agents in an organization make deisions asynhronously and without information-updating delays, and suppose that eah employee values her subproblem higher than other subproblems (b j i < when i j in Equation (3));that is, system evolution an be represented by Equation (5) with random interdependenies as desribed previously. Then, as N grows, the probability that problem solving beomes unstable approahes (i.e., that a small random perturbation of the system around a solution is not dampened but is amplified arbitrarily). The appendix offers a formal formulation and the proof of Proposition, whose weak assumptions make it widely appliable. The proposition holds widely beause a yle of mutual dependenies beomes more and more likely as the number of subproblems grows, leading to self-reinforing yles of hanges and, hene, many iterations. Proposition implies that large organizational projets in flat organizations typially exhibit a ommon defiieny: long problem-solving yles that seem to lead nowhere, even though in priniple a solution should be possible. Management must take ation to stop the searh in spite of the suboptimal results that may follow. 4. Hierarhy and Problem-Solving Stability: Analytial Results In this setion, we show how introduing a hierarhy an hange the problem-solving dynamis. We ontinue with the ase of employees who value their subproblem higher than others, while reeiving instantaneous information updates. 4.. Problem Solving in a Hierarhial Struture The most obvious way in whih hierarhies an hange the deision-making proess is by introduing an organizational entity with the right to reverse others deisions. In that sense, hierarhies an foster the entralization of deision making. This is Rivkin and Siggelkow s (3) entralized hierarhy (whih we repliate in one variant of our model). However, a hierarhy also has the more subtle effet of influening ommuniation patterns among employees. In no organization of even moderate size does a deision maker ommuniate equally with all other deision makers. Employees are grouped into departments based on subproblem similarities and shared solution methods (Marh and Simon 95). Employees tend to oordinate and disuss subproblem solutions within their own department before talking to olleagues in other departments (Sosa et al. 4). Moreover, employees within a department work on related problems with shared solution methods, report to a ommon manager, and thus have a ommon identity. The performane of other departments may be viewed as less important than the performane of one s own department, partly beause the performane dimensions applying to other departments are less well understood and partly beause the other department may be viewed as the outgroup (Janis 97, Sosa et al. 4). Thus, multiple reasons push employees within departments to beome more insular, a widely observed phenomenon that is diffiult to avoid. In our model, this insularity has two effets. First, suppose employee i is in department. Then it is typially the ase that b k i b j i when k and j, refleting the aforementioned outgroup aversion that disounts the importane of other departments (at the ost of lower solution quality).

7 Management Siene 5(5), pp. 3 4, INFORMS 37 The seond effet of insularity is that P i / h k < P i / h j when k and j. This may reflet true modularity aross departments (as when a department groups the most losely related subproblems together), but it also reflets that a department buffers itself against ations by other departments. For example, the manufaturing group distrusts the quality of marketing s volume foreasts and so builds extra apaity: As long as they do not deviate too muh from the initial announement, we are prepared. Or perhaps marketing distrusts the quality of engineering s designs and buffers its volume foreast: As long as their speifiations do not deviate too far from what we had originally agreed, we ll be able to deliver the volume. By buffering of this type, one group redues the effet of deision hanges within a ertain range by another group on its own performane funtion (at a ost of lower solution quality). Mathematially, employee i s performane an be rewritten as P i h i h j h k ), where the influene of h k on P i is redued by disounting and buffering { h i = arg max b j i j P j h i h j h k j + } b k i k P k h i h j h k () k This equation formalizes how employee i s onsideration of the overall problem ontext is foused on neighboring subproblems in her own department; outside subproblems are disounted and their influene on one s own deisions is buffered away. 4.. Why Hierarhy Can Help Given these two potential effets of a hierarhy, the question is, how do they influene the organization s solution searh? Let us first onentrate on entralized deision making. Consider the simplest hierarhy with one manager and N employees. Suppose the manager is the entral deision maker, similar to Rivkin and Siggelkow (3); this means that eah employee i presents a solution P i as a suggestion to the manager, who then rejets or aepts it based on its ontribution to the whole problem P. Without further assumptions, we an now demonstrate the following: Proposition. If the organization makes deisions in a hierarhy with a entralized deision maker, then the solution quality P onverges monotonially to a final level. A formal statement and a proof of the proposition are given in the appendix. The entralized deision maker onstrains the interations among subgroups by allowing hanges only if they make the group s performane better. This has two impliations. First, the deision vetor h onverges to a stable solution in most ases (exept in ontrived situations where multiple solutions have the same performane value). Seond, even if the deisions h themselves do not onverge, the benefit of ontinued searh tapers off over time, whih enables the manager to establish termination riteria based on dereasing improvements. Thus, a entralized hierarhy dampens in a wide range of irumstanes, ensuring a viable solution in a reasonable time. But a hierarhy has other benefits as well. They arise beause interdependenies are treated more weakly aross department boundaries in Equation () via the lower importane weights b k i and the redued interdependenies P i / h k when i and k are in different departments. Proposition 3. Take the same problem-solving organization as in Proposition. Group the agents into nonoverlapping departments, and suppose that the importane of ross-group interdependenies (b k i and P i / h k )is disounted whenever k and i are in different departments. Then, for any organization with slow or nononverging problem solving, there is a hoie of departments and of interdependene importane suh that searh progress toward a solution beomes fast and stable. Proposition 3 states that dividing the organization into units and prioritizing the interdependenies within eah unit over interdependenies aross units redues problem-solving iterations. The reason is that the resulting group-myopi behavior redues the inidene of feedbak loops that indue the iterations desribed in Proposition, although negleting interdependenies may damage the solution quality. There are situations where a grouping (introduing departments) alone an ahieve stability without lost solution quality, espeially when the overall problem is easily modularized and naturally leads to weak interdependenies aross groups. However, there are also ases in whih ahieving stability requires fored prioritization and the neglet of some important rossdepartment interdependenies, whih severely limits solution quality. In many realisti ases, however, the interdependene redution required for stability is minor. 5 In summary, the analytial model suggests that managers who apply entralized deision making redue osillations and thereby foster stability and searh speed. In addition, a hierarhy an help an organization stabilize and speed up its problem- 5 For synhronous systems, quantitative formal statements for Gaussian matries an be made that a modest weakening of interdependenies redues the eigenvalues of the Jaobian matrix and leads to stability (Hogg et al. 99). For asynhronous systems, simulations suggest that a moderate weakening of ross-department interdependenies is also suffiient to establish stability.

8 3 Management Siene 5(5), pp. 3 4, INFORMS solving searh by downplaying ross-departmental interdependenies. Suh interdependenies are weakened by the natural behavior of the employees, who pay more attention to results in their own department than in other departments. This redues problemsolving osillations. Thus, what is often frustrating in terms of finding the best solution ( the other department does not listen to us ) atually has a strong advantage in terms of searh speed. In sum, hierarhy shapes the searh dynamis, even in the absene of ontrol loss and ompetene issues. 5. Simulation Results Although the the formal analysis in 4 offers a ausal theory of organizational dynamis in hierarhies, it annot answer some of our researh questions: What are the effets of the hierarhy s struture and of the vertial and horizontal oordination in it? We must resort to simulation for the answers to these questions. Likewise, our analytial results have established the searh aeleration benefits of a hierarhy when employees goals are imperfetly aligned (violation of the first ondition in 3.). Yet we must still demonstrate the benefit of a hierarhy when updating among subproblems is delayed (violation of the seond ondition in 3.). For this, too, we need simulations. A simulation model requires adding detail to the general model. 5.. The Simulation Model Consider a four-level hierarhy: 4 workers are grouped into six departments, whih in turn are strutured into two areas of three departments eah; the area managers report to the CEO. Suppose that groups at the same hierarhial level are of the same size, whih will allow us to fous on the fundamental effets of the hierarhy. All problem-solving expertise is onentrated at the bottom (the worker ) level: knowledge workers solve problems, and managers (at the department, area, and CEO level) manage but do not engage in atual problem solving. This simplifying assumption is in line with previous searh models. Moreover, we assume that the front-line workers at holistially in subproblem-level searh; that is, they use b j i = for all i j and do not redue interdependenies via safety buffers as in Equation (). We fous on this optimisti ase (as opposed to the myopi ase) beause previous studies (Mihm et al. 3, Rivkin and Siggelkov 3) have shown that fully myopi behavior by the problem-solving employees sarifies too muh solution quality. We now need to speify two domains of the simulation model. First, we use the flexibility offered by simulations to represent three additional dimensions of the hierarhy s deision-making struture that are not tratable in an analytial model. Seond, we need to speify detailed funtions P and P i to generate quantitative simulation results. Table summarizes three dimensions of how the organization oordinates aross levels and departments: (i) the level at whih the organization makes its deisions, (ii) the inentives of the managers above the front-line workers, and (iii) the way the organization sequenes problem-solving work aross departments. The first aspet of the deision-making struture the level at whih deisions are made depends on Table Three Dimensions of Deision Making in the Hierarhy Lous of deision making and inentives Centralized hierarhy with Centralized hierarhy with Ordering of problem solving myopi inentives for managers holisti inentives for managers Coordinating hierarhy Parallel: All groups (e.g.,departments) perform eah a solution iteration in parallel. The superior (e.g.,area manager) aepts the proposal if it improves the group s solution quality. The superior (e.g.,area manager) aepts the proposal if it improves the organization s overall solution quality. The superior (e.g.,area manager) heks for onsisteny with the other groups and fores another iteration if any group s solution is based on information that has beome obsolete in the last iteration and is thus inonsistent with other groups. Sequential: One lead group (e.g.,department) performs a solution iteration first. The next department takes this solution as the starting point for its iteration,and so forth. For eah group,the superior (e.g.,area manager) aepts the proposal if it improves the group s solution quality. For eah group,the superior (e.g.,area manager) aepts the proposal if it improves the organization s overall solution quality. Consisteny is guaranteed by the sequential way of proeeding. When all groups are done,the superior (e.g.,area manager) heks whether the first group s solution is onsistent with the subsequent deisions by the other groups and fores another iteration if the other groups have made hanges.

9 Management Siene 5(5), pp. 3 4, INFORMS 39 whether the hierarhy is entralized (ative) or oordinating. (Rivkin and Siggelkow 3 all the latter rubberstamping hierarhy, but oordinating aptures the spirit more aurately beause in our model the manager adds value by ensuring oordination.) In the entralized hierarhy, the manager takes the subproblem solutions that ome from the organizational level below as suggestions, and (onsistent with Proposition ) she aepts or rejets them depending on whether or not they improve the overall solution value of the manager s more aggregated goal funtion. In other words, the value of existing proposals an be tested, although optimization of the overall problem P remains elusive. In ontrast, the manager in the oordinating hierarhy does not make deisions diretly, instead ensuring oordination among the multiple subproblems: if one deision maker i s assumption about some other omponent deision j is wrong, that is, if the true h j h j (and thus the P j as pereived by employee i differs from the true P j ), then the manager ensures that one or both of the solutions are hanged until they are mutually ompatible. The entralized and the oordinating hierarhy are two extreme benhmarks; real organizations fall somewhere in between the two ases. For our seond dimension of deision making, we further divide the entralized hierarhy into two variants aording to the inentives affeting middle managers one and two levels below the CEO. We assume that front-line workers at holistially, but managers may well have politial agendas that are myopially foused on their organizational units (inentive onflits aross units tend to play a larger role at the management level than the front line). Thus, we differentiate between holisti inentives of the department managers (i.e., when they work also in the interest of the entire organization) and their individual inentives (when they disount the influene of subproblems outside their department and maximize the performane of their group possibly at the ost of other groups). The third aspet of the hierarhial deision-making struture is the order in whih the various departments obtain and oordinate their subproblem solutions. Sequential problem solving means that one lead funtion solves its subproblem first and gives its solution to the other groups at the same hierarhial level. Then the next department works on its problem, taking into aount the lead funtion s solution. When eah group has produed a solution proposal, the lead funtion reexamines its solution to identify whether it an do better after updating its assumptions about the other groups deisions, and so a seond round starts for all groups at this hierarhial level. This sequential problem solving ontinues until no group an make improvements with further hanges. The alternative oordination mode in a hierarhy is parallel, or onurrent, problem solving: departments at the same hierarhial level work in parallel, using as assumptions about the other departments their most reent information status. When all departments have obtained solutions to their subproblem areas, they exhange their respetive final status for example, in a system test. Based on this exhange, the departments update their assumptions and iterate in parallel until none sees any further need to hange its solution. Both sequential and simultaneous problem solving are widely observed in real organizations and often oexist. Sometimes, ertain subproblems are onsidered more important or ertain departments are more powerful and have more status than others (Terwiesh et al. ) in some ompanies, marketing has this status; in others, it is engineering or manufaturing. These funtions will tend to be lead funtions. But in other ases, subproblems are not prioritized and must proeed in parallel to meet deadlines. We onsider sequential and parallel problem solving as extreme benhmarks whose relative merits illuminate the priniples of oordinating deentralized work in a hierarhy. We now turn to the seond speifiation domain for the simulation namely, speifi value funtions P i. For the P i to be plausible, performane should never beome infinite, and a neighboring subproblem deision h j should influene both the optimal deision h i and its ahievable performane P i h i h j. The simplest form inorporating these demands is the following quadrati: 7 P i = [ d i ( h i N j i ) ] i jhj + I i j j i for i = N (7) Note that P i is not the true performane but rather worker i s optimization funtion, or the pereived performane based on the possibly obsolete (beause of updating delays) information h j about other subproblems. The form of P i is shown in Figure. The term in brakets is a quadrati funtion This is in ontrast to front-line deision makers whom we assume to at holistially. We are thus able to isolate the effet middle management attitudes have. 7 Mihm et al. (3) show how this funtional form an be viewed as a result of two Taylor series approximations of a general funtional form.

10 4 Management Siene 5(5), pp. 3 4, INFORMS Figure Subproblem Performane Funtion in the Simulation Model Performane P i Too low P i = [ d i (h i Σ ij h i ) + ] Π I ij Biasing influene ij of other deision maker Own deision h i Performane onstraint I ij from other deision maker Too high with one optimum in the deision variable h i, whose position is shifted linearly by the deision of agents j via the interdependene parameter ij ( If marketing inreases the number of segments to be served, h j, then manufaturing s optimal investment in platforms, h i, beomes larger ). Here, d i is a sale parameter that makes the term in parentheses omparable to the interept. The best ahievable performane of subproblem i is influened by agent j s deision via onstraints I ij that agent j puts on i s loal problem ( If marketing hooses a smaller market segment, h j, the best ahievable ost improvement, P i, is redued for the manufaturing department ). The I ij are ontinuous pieewise linear funtions: in a middle range, h j has an important (linearized) effet on P i ( Every additional dollar that your subproblem onsumes has a negative effet on my available budget ). If h j is extremely disadvantageous for subproblem i (and i s solution quality therefore low), then another hange in h j makes little differene ( If you absorb that muh budget, I annot ahieve minimal performane anyway; even if you save a few dollars, it does not help me ). Similarly, if h j is extremely advantageous for problem i, then a small hange makes little differene ( If you spend only so little, I have what I need, and a small inrease in your budget does not hurt me ). In the simulations, the interdependene parameters are randomly drawn from a uniform distribution, the onstraint interval limits from a uniform distribution, and the sale parameters d i from a standard lognormal distribution. It is notoriously diffiult to establish the robustness of results in searh models (see, e.g., Rivkin ). In our ase, the omputing effort behind the reported The assumption that loal performane has only one optimum is optimisti, reduing iterations beause the system does not jump among multiple loal optima of the agents. Multipliative omponent interations are widely used, e.g., in evolutionary biology, where the fitness of an organism is the produt of the ontributions of all geneti loi (Ewens 979, Lewontin 974). results exeeded a ombined 45 months of run time on a ontemporary PC, and another months of run time were spent on robustness tests. In addition to anhoring the base ases in the analytial results, we verified all reported experiments with sensitivity variations of key simulation parameters (suh as group size and ross-department effets). Although the quantitative results hange, the qualitative insights drawn from the simulations remain stable unless noted in the text. More details on robustness and the data itself are available from the authors. 5.. Anhoring Results The first set of simulation runs onnets our simulation model to the analytial models from 3 and 4 and also to previous work. It repliates the results from the analytial setion but with the assumption of imperfet alignment of front-line deision makers (Condition ) replaed by the assumption of delayed ommuniation (Condition ). Unless speifially indiated in the text, the organization has four layers with 4 front-line employees, and these employees behave holistially and listen to information from other departments (b k i = in Equation ()). We assess the effetiveness of the various hierarhy onfigurations on three performane dimensions. First, what is the probability that the searh proess onverges to a solution rather than diverging? Seond, given that problem solving does onverge, how long does it take the organization to settle on a solution? Third, what is the solution quality, the final value outome P? For all three dimensions, we show averages. Reported results are statistially signifiant at the 5% level. Figure 3 summarizes the set of results. Its rows represent the three performane dimensions, and its olumns represent the deision-making modes of Table. In the left olumn we assume that the hierarhy is oordinating only; in the enter olumn, the middle managers at individually and onsider only the performane of their own organizational units; the right-hand olumn represents managers with holisti inentives. Eah hart traks performane over the density of interdependenies and distinguishes performane between sequential and parallel problem solving. A density of % means that no subproblems are interdependent and a density of % that all subproblems are mutually interdependent; a higher density means that the problem is more omplex. Eah point in the urves represents the average of 5 simulation runs in whih the interdependeny strengths are repeatedly resampled as random variables (as desribed in 5.). If a graph seems to ontain only one urve, the seond urve is fully overed by the first.

11 Management Siene 5(5), pp. 3 4, INFORMS 4 Figure 3 The Curse of Deentralized Problem Solving and Sequential vs. Parallel Work Stability (perentage of reahing solution) Sequential oordinationlead funtion Parallel oordination Convergene time (millions) Solution quality (thousands) Density Density Density Coordinating hierarhy Centralized hierarhy with myopi inentives for managers Centralized hierarhy with holisti inentives for managers Two explanations help to interpret the graphs. First, the onvergene times assume reahing full onsensus and are thus unrealistially long (with one update and deision per hours on average, the entralized hierarhy with myopi inentives takes about, hours at 5% density, and the oordinating hierarhy takes, hours). In reality, management stops and fores a restart or freezes omponents muh sooner, satisfiing at the best solution found to date (as ourred in our opening example). We do not introdue freezing in the simulation beause any stopping time is arbitrary; the longer the onvergene to an equilibrium, the higher the performane loss of stopping the searh at any given time. Plotting the performane loss from a set of given stopping times produes qualitatively the same graph as the solution time shown here. The seond explanation addresses why the solution quality inreases with density. It does so beause the ross-subproblem interation onstraints I ij vary around their average value of (they are pieewise linear with a sensitive slope in the enter), and the employees, ating holistially and seeking overall system benefits, make deisions that tend to push interations into range where I ij >. (When the frontline workers at myopially, quality does not inrease with density.) The more interations employees have to play with (as density inreases), the more mutually benefiial solutions they are able to find, and thus system quality improves. This orresponds to the priniple in tehnology development that integrated systems have higher performane potential than modularized systems. However, further exploration of the effet is beyond the sope of this paper; here, we ompare solution quality aross deision modes but not aross different densities. The results in the graphs illustrate, first, the negative effet of density (whih implies omplexity) on problem-solving dynamis: as density inreases, stability and solution time deteriorate. Seond, onsistent with Propositions and and NK model results, we onfirm that a entralized hierarhy under all irumstanes helps to ahieve solution onvergene (Mihm et al. 3, Rivkin and Siggelkow 3, Siggelkow and Rivkin 5). In addition, entralization in the hands of myopially ating middle managers speeds up problem solving but at the prie of redued solution quality ompared with a oordinating hierarhy. However, redued quality holds only in the enter olumn with myopi managers, who take the (overall systemoriented) solution proposals from the front line and evaluate them with respet to their own organizational unit only. This allows them to settle on an overall solution quikly, but one of lower quality for the organization overall. However, when all managers are holistially oriented toward overall performane, the result hanges.