Hierarchical Structure and Search in Complex Organizations

Transcription

1 Submitted to Management Science manuscript MS R2 Hierarchical Structure and Search in Complex Organizations Jürgen Mihm, Christoph H. Loch INSEAD, Boulevard de Constance, 7735 Fontainebleau Cedex, France Dennis Wilkinson, Bernardo Huberman HP Labs, 5 Page Mill Road, Palo Alto, CA 9434 {dennis.wilkinson@hp.com, huberman@hpl.hp.com} Organizations engage in search whenever they perform nonroutine tasks, such as the denition and validation of a new strategy, the acquisition of new capabilities, or new product development. Previous work on search and organizational hierarchy has discovered that a hierarchy with a central decision maker at the top can speed up problem solving, but possibly at the cost of solution quality as compared with results of a decentralized search. Our study uses a formal model and simulations to explore the eect of an organizational hierarchy on solution stability, solution quality, and search speed. Three insights arise on how a hierarchy can improve organizational search: () assigning a lead function that anchors a solution speeds up problem solving; (2) local solution choice should be delegated to the lowest level; and (3) structure matters little at the middle management level, but front-line groups matter most and should be kept small. These results highlight the importance for every organization of adapting its hierarchical structure to its search requirements. Key words : Search, Complexity, Oscillations, Coordination, Decentralized Problem Solving, Hierarchy History : second revision. Introduction Large organizations need to solve problems that are complex because of multiple relevant technologies, globalizing markets, multiple interacting business processes, and collaboration with external partners. In addition to static complexity, organizations often face dynamic environmental turbulence (Eisenhardt and Tabrizi 995). These challenges demand that decisions with many parameters be taken quickly. Yet in the face of such circumstances, decisions often cannot be optimized (Simon 969). Rather, organizations engage in search, or the generation of alternatives through the change of a few (incremental search) or many (radical search) decision parameters, and then choose the alternative that performs best. In large organizations, no single individual is able to grasp all decision parameters, so search must be a collaborative eort. Therefore, a central challenge of organizational The authors thank Stephen Chick and Michael Vogt for helpful comments. We would like to extend our thanks to the associate editor and referees, who helped us to improve the paper considerably. We thank Michele Ibarra for providing us with ample computing resources.

2 2 Article submitted to Management Science; manuscript no. MS R2 design is to divide the complex search process into manageable specialized tasks and to coordinate these tasks so that the rm reaps the benets of concerted action (Nadler and Tushman 997, Rivkin and Siggelkow 23, Siggelkow and Rivkin 25). In organization design, senior management alone is not capable of nding high-quality task solutions and must allocate tasks and delegate decisions, provide incentives, and structure communication among the various organizational memberseach with their respective subpiece of the organization's overall problem (Siggelkow and Rivkin 25)in order to achieve the best combination of speed, solution quality, and risk of failure. The following example from engineering demonstrates how organizational issues inuence the balance of these interrelated goals. In the semiconductor industry, the design of Intel's Itanium chip went around in a circle, nding itself in a nightmarish world where a change to one module would ripple through the work of several hundred other people, leaving more problems in its wake (Hamilton 2). The design nally converged only after a high-level manager froze several components, de facto centralizing the decision structure so as to improve solution speed while implicitly limiting component performance. This study examines how an organizational hierarchy can best be structured to guide organizational search. We use formal modeling to study the eects of hierarchical structure on problemsolving search with respect to speed, solution quality, and failure risk. Previous work has observed that a hierarchy can improve organizational search: centralized decision making at the top stabilizes search, reduces failure risk, and leads to faster decisions, whereas decentralized decision making increases the solution quality (Rivkin and Siggelkow 23) and raises the organization's ability to cope with environmental changes (Siggelkow and Rivkin 25). This work has illuminated important managerial trade-os, but it has represented hierarchy only in its simplest formwith one CEO and two workersand has focused on comparing centralization to decentralization. Important questions about how to structure hierarchies for search remain unanswered. These questions involve decision making procedures beyond the issues of centralization versus decentralization and the organization's formal structure. In this paper, we raise three questions that an organizational designer needs to consider; the rst two concern how decisions are made in a hierarchy, and the third concerns how the hierarchy itself should be structured (see Figure ). First, how should the groups at a given layer of the hierarchy coordinate? Should they search their respective subproblems in parallel, and then coordinate and adjust, or should they search sequentially, with one lead function determining both a general direction and constraints for the others? Our results suggest that, in coordinating hierarchies, sequential decision-making is more

3 Article submitted to Management Science; manuscript no. MS R2 3 CEO Behavioral Aspects of Hierarchy Structural Aspects of Hierarchy Middle Management Front-line Management Front-line problemsolving Workers Dept. Mgr. Area Mgr. Sub-problem Dept. Mgr. Sub-problem 2 Dept. Mgr. Area Mgr. Dept. Mgr. Sub-problem 3 Question: Horizontal coordination: Sequential or parallel work? Result: Sequential work is faster and produces higher quality. Question: Vertical coordination: How far to delegate decisions? Result: Delegate decisions all the way to front-line management. Question: What is the best depth/size of departments? Result: Political structure in the middle does not matter. What matters is limiting the size of front-line working groups. Figure Organizational hierarchy and problem-solving search benecial for search. In this sense, status dierences among groups and a corresponding order in which the groups inuence a decision may be justied. Second, how far should decision power be delegated: all the way down to the front line? We nd that decision-making centralization is most eective if delegated to front-line management: delegating the solution choice to the lowest level of management generates most of the stability and speed benets that a hierarchy oers. Third, how should a hierarchy be structured? Beyond questions of control loss, how does the size of the reporting groups inuence the breadth versus speed of organizational search? Our results suggest that hierarchical structures at the middle management level matter little for performance of the search process. It is at the front line that the grouping of problem-solving workers into departments matters most. Smaller groups have an easier time achieving speed and stability. Our answers to these three questions contribute to the organizational search literature by illuminating the interactions between organizational structure, hierarchical decision making and the search process. As a second, methodological, contribution, this article oers a formal search model that allows for deriving an analytical characterization of a general problem in distributed search. The analytical model shows that, whenever alignment of subproblem decisions is less than perfect (for reasons of information overload or incentive conicts), problem solving speed and stability will suer in large organizations with decentralized decision making. In this case, hierarchical centralization may speed up and improve the results of search. The model complements the widely used NK model in showing robustness of qualitative ndings based on dierent micro decision structures. We overview related work in Section 2, present the model in Section 3, and discuss analytical and

4 4 Article submitted to Management Science; manuscript no. MS R2 simulation results in Sections 4 and 5, respectively. Section 6 concludes, and the proofs are gathered in the Appendix. 2. Organizational Hierarchy and Search Search is pervasive in organizations. Examples include nding and dening a strategy (Rivkin 2, Winter et al. 27), adapting to industry life cycles (Doty et al. 993, Levinthal 997, Nelson and Winter 982), building organizational capabilities (Bruderer and Singh 996, Gavetti 25), and new product development (Fleming and Sorenson 2, Mihm et al. 23, Yassine et al. 23). Search may contain elements of local optimization by intelligent local agents (e.g., Gavetti 25, Gavetti and Levinthal 2, Knudsen and Levinthal 27, Sorenson 22, Winter et al. 27), but in general, it is search and not optimization that dominates complex problem solving. For complex problems, the search process cannot be performed with regard to the whole problem at once because no actor or resource has enough information-processing capacity to consider all aspects of the problem (Loch and Terwiesch 27, Simon 969, Van Zandt 999). However, the subproblems are typically not fully separable; their solutions depend on one another. As a result, organizations consist of more or less tightly coupled groups (as in Cyert and March 963) that execute specialized functions while interacting so as to produce organizational output. It is a central challenge for the organization to derive the benet of dividing the overall search problem into manageable tasks performed by specialized groups while coordinating those groups to obtain cohesive action (March and Simon 958). Search has been a topic in organization theory since its infancy (Cyert and March 963, March and Simon 958, Simon 969), but interest has surged with the introduction of formal models to organization theorynotably the NK model (e.g., Levinthal 997), which models actors as N nodes with K interdependencies; it was developed in physics as a spin glass model and then applied to evolutionary biology (Kauman 993). Simulation work on the N K model complemented by other models has shown that the decentralized search of interdependent subproblems tends to slow and get bogged down as the problem's complexity (the number of subproblems) increases (Ethiraj and Levinthal 24, Huberman and Wilkinson 25, Mihm et al. 23): ongoing choices in some groups make the requirements for other groups inherently unstable (Thomke 997, Van Zandt 999). Although these studies have identied managerial actions to mitigate the problem, they have not addressed the question of whether and how hierarchical organizational structures can improve search time and quality. A vertical hierarchy is the most common way of coordinating specialized groups and their separate decisions. There are other coordination mechanismssuch as liaisons, cross-unit groups, and We take vertical hierarchy to mean that multiple individuals, or multiple groups, report to the same manager.

5 Article submitted to Management Science; manuscript no. MS R2 5 informal networksbut a vertical hierarchy is present in virtually every organization (Nadler and Tushman 997). Groundbreaking work on the role of hierarchies in search (Rivkin and Siggelkow 23, Siggelkow and Rivkin 25) based on the study of NK search landscapes suggests that fully centralized decision making stabilizes and speeds the search process but at the cost of performance. However, this work models hierarchy as consisting only of one CEO and two managers, so it could hardly examine the role of dierentiated hierarchical structures in search. In our study, multiple players at several levels of a hierarchy engage in search while the hierarchy helps to integrate the subsolutions into an overall solution. In evaluating performance, we consider not only the solution quality and convergence time, as is customary for many NK simulations, but also the ability to converge to a solution at all (see Mihm et al. 23, Yassine et al. 23). Our study focuses on search, which is an inherently dynamic concept, and does not examine the control loss aspect of hierarchies. Substantial work in that area can be found in the formal modeling literature (e.g., Bolton and Dewatripont 994, Child 984, Keren and Levhari 983, Moldoveanu and Bauer 24, Nadler and Tushman 997, Radner 993, Williamson 99, Van Zandt 999). 3. Model Setup We build a model of an organization that engages in decentralized search in order to solve a complex problem with many interdependent subproblems. Our model explicitly captures two fundamental characteristics of complex search: () the local, subproblem, level decisions may not be perfectly aligned toward an overall system optimum, because no individual understands all the eects of his decisions on the system and because there may be interest conicts, and (2) mutual updating and coordination may be delayed because immediate broadcasting of all subproblem decisions would cause information overload. As an example, imagine a product innovation whose value depends on the quality of its associated market research study, the engineering work, a new manufacturing process, and a competent sales plan and execution. All these decision domains are interdependent (e.g., the sales plan depends on the features included, the feasible product features depend on the manufacturing capabilities, and the manufacturing capabilities depend in turn on the design complexity and sales volumes that justify investments). Furthermore, as work progresses partially in parallel, information about the latest decisions in other domains is not always immediately available (e.g., while the sales team changes projected sales because of new information about customer behavior, the manufacturing group may be building its capacity investment plan based on the previous projection). We rst show that large problems of this type systematically cause decision delays, excessive iterations, and instability (building on Mihm et al. 23). We then show how the presence of a hierarchy inuences decision making and interactions among the decision makers.

6 6 Article submitted to Management Science; manuscript no. MS R2 3.. A Basic Search Model without Hierarchy An organization of N employees collectively produces an output of value P. The output value is a function of all the employees' decisions, P (h,..., h N ). In order to allow decentralized problem solving, the organization divides the overall problem into subproblems i of value P i, where problem i is delegated to one specialist decision maker i. For simplicity, suppose that each subproblem P i (and thus each employee) has one decision variable, h i, and that P i is bounded from above. 2 Although it is not necessary for the analytical results, we assume for easier exposition that the overall performance is a sum of the individual functions with relative importance weights α i : 3 P = N α i P i. () i= If the sub-problems were independent, that is, each P i (h i ) a function only of its own decision variable, then the overall system optimum would result from the decision makers' individual optimizations. However, interdependence among the employees makes any subproblem performance a function of other subproblems: P i = P i (h,..., h N ). (2) We know from N K-model research that this structure can represent unlimited complexity because of the interdependencies in Equation (2). A complex value function typically has many local optima. Because no one in the organization understands this complex problem in its entirety, it follows that no overall optimization is possiblethe organization must engage in search. We rst model search in a at organization (N interdependent but equal problem-solving employees, as in Mihm et al. 23). Then we show how a hierarchy changes the search. The organization starts with a previous solutionfor example, the result obtained in the last search, (h,..., h N). But this is a poor solution for the new problem, so each employee goes o to work on her own subproblem. Two properties characterize the decision-making process that ensues. First, the individual employee i makes a decision aimed at optimizing the overall performance function P, considering everyone's interests but still discounting the importance of other subproblems as compared with her own subproblem (after all, a specialist is rst and foremost evaluated by her own competence): the employee maximizes N j= b j,i α j P j (h,..., h N ), where b i,i = and the other b j,i [, ]. The ideal would be that all employees act fully holistically, b j,i =. The opposite (pessimistic) 2 Of course, employees may be responsible for multiple decision dimensions. However, it is often realistic to summarize the multiple dimensions in one composite dimension. In addition, this simplication does not change the qualitative results of the model. 3 The qualitative results of this model can be shown to hold for any complex performance function P = f(p,..., P N ); a more general function merely complicates the exposition.

7 Article submitted to Management Science; manuscript no. MS R2 7 assumption would be that b j,i = when i j, with each employee i acting myopically (locally) because of political conicts or an inability to understand the connections to other subproblems. Second, the employee's decisions may be based on obsolete information about other subproblems. The employee cannot take into account any future reactions by other employees to her decisions because she lacks the technical expertise. The employee takes the current 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 h j, j i, reect the decision 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 decision by employee i can be written as N h i = argmax b j,i α j P j (h i, {h j }). (3) j= Updates occur asynchronously after random time intervals: the employee learns the most recent decision status of the others partly by scheduled meetings of the entire team (strategy sessions, design or sales reviews, budget meetings). However, to a substantial degree, change requirements also arise at random moments in time at unscheduled one-on-one events, such as a crisis phone call (if you make this change, you'll cause a crisis for my subproject), an encounter in the hallway or at the coee machine, or information accidentally overheard at an unrelated meeting (Allen 977). We reect this partially unscheduled nature of change requirements of small groups by having the update time points be independent Poisson processes for each decision maker, with exponentially distributed time intervals between updates (the reader may picture this as the next updating time being randomly drawn for the individual employee, independent of what happens to other employees). When decision maker i obtains an update, her problem (which is driven by the interdependencies) changes if any other employees have changed their decisions. Thus, employee i, in turn, must change her decision h i, and this in turn changes the constraints for others when they obtain their updates later. Thus, the organization must continue to revisit all decisions until a xed point is achieved, a solution from which no employee wants to change. This concept of a cascading adjustment process has been well established in modeling organizations (e.g., Hannan et al. 23a,b, Lounamaa and March 987). Poisson updates render evolution in the model asynchronous (updates of two employees never happen at exactly the same time). Asynchronous evolution is typical for social systems unless there are extremely strict mechanisms for enforcing coordination. 4 4 Huberman and Glance (993) have shown that modeling a naturally asynchronous system with a synchronous mathematical process may introduce distorting artifacts, and Sorenson (22) stresses the importance of this result for organizations.

8 8 Article submitted to Management Science; manuscript no. MS R2 In summary, the organization is capacity constrained in two ways. First, decision makers are specialized and capable of fully solving only their assigned subproblems. Second, decision makers cannot always be updated on all other subproblems (because of information overload). The subsequent decisions by individual decision makers at Poisson time points (and therefore in random order), each taking into account past decisions of other decision makers, constitutes an organizational search through the space of decisions h (the vector of all decision variables h i ). The organization has arrived at a stable solution when all decision makers have achieved their local optima, given the solutions of the other subproblemsin other words, when no decision maker wants to change her decision unilaterally (Loch and Terwiesch 27). In this stable state, decision makers are correctly updated about the overall solution status (h j = h j ) and the organization has settled in a local optimum (which is also a Nash equilibrium). Although an organization is not a democracy where everyone can vote on the nal project outcome, the consensus representation serves two purposes. First, it is a realistic approximation for at (professional) organizations with partially autonomous actors. Second, it functions as a base case to establish results for one extreme of a spectrum of centralization levels, which we complement later by introducing a hierarchy The Curse of Complexity for Decentralized Problem Solving Problem search performs well, independent of the size of the organization, if two conditions are simultaneously fullled. First, all employees act fully holisticallythat is, in the interest of the whole and not just of their own subproblems (in the model, all b j,i = ). Second, updating is immediate: employees are always informed about the latest status of the other subproblems (in the model, the updating Poisson processes have an innite rate). Both conditions are unrealistic, since employees do weigh their own subproblems more highly (owing to incentives and to lack of expertise about the other subproblems), and since instantaneous updating would result in information overload. Violating either condition is sucient to bog down problem solving: the decentralized search of larger complex problems takes progressively longer and may not even converge to a solution. Intuitively the reason is that, because of interdependence, an agent who changes her decision may induce other agents to change as well. The critical question for problem-solving progress is whether potential loops of mutual inuence die down as the agents' work progresses over time or whether they amplify one another, leading to oscillations and never-ending reghting. We now show analytically that problem-solving oscillations occur whenever the rst condition (holistic behavior of the employees) is violated, even when updating is immediate. The case of updating delays (violating the

9 Article submitted to Management Science; manuscript no. MS R2 9 second condition) is analytically intractable; we consider this violation in the simulations in Section 5. Recall that h is the vector of the N agents' decisions. Then the problem-solving evolutionthat is, the change of the decision status vector over time, assuming immediate updatesis described by h(t + t) = g(h(t)), (4) where g(h(t)) describes the optimization that each decision maker performs (see Equation (3)). Since the solution status evolves asynchronously, only one agent i's component of the vector function h(t) changes over a very short time interval, t. Then, over such a time interval, for agent i, we can 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 incremental changes in the problem-solving status for any point in time, t: h(t + t) h(t) = g(h(t)) h(t). The system has reached a xed point if g(h(t)) h(t) = independent of which player i makes a move over t: each decision maker has found a local optimum according to Equation (3) and will therefore stick to that decision, given all others' decisions. We assume that the problem has at least one xed point; otherwise, it possesses no stable solution and failure to converge is inevitable. The dynamics of a system are driven by its behavior around its xed points. The question of whether a xed point is stable and can be reached is equivalent to evaluating whether, once the xed point has somehow been found, the decisions of the decentralized agents drive the system to return to the xed point when subjected to (even very small) perturbations. Consider a xed point h and a perturbation ɛ with random elements ɛ i centered about with some nite variance. Since we are interested in what happens for an arbitrarily small perturbation, ɛ, most relevant problemsolving systems can be described by a linearization: Dene the partial Jacobian matrix J i, with [J i ] ij = g i (h)/ h j h=h for i j, [J i ] ii = g i (h)/ h i h=h, [J i ] jj =, and [J i ] ji =. The derivative g i (h)/ h j expresses the interdependency, which is the eect that a small change in j's decision has on the optimal decision of i. Let h n be the system state after n players have moved. Then, with the set i, m,..., v containing n elements, the problem-solving dynamics can be described by h n h J i J m... J v ɛ where i, m,..., v {... N}. (5) The problem-solving system becomes unstable if mutual interactions among subproblems enlarge a small perturbation from the xed point. Here is where our rst condition becomes relevant: if all b j,i =, then all employees react to the perturbation in the same way, each striving for the same optimum (xed point) of P. Formally, the cross-partial derivatives g i (h)/ h j become zero and

10 Article submitted to Management Science; manuscript no. MS R2 thus, for all i, the J i become zero matrices. Hence, no instability arises. However, if b j,i < for i j, then the employees are not perfectly aligned: each values his subproblem more than the others and strives in a slightly dierent direction. Thus, the cross-partial derivatives become nonzero, reecting the interdependencies among the sub-problems. Moreover, these interdependencies cannot be fully foreseen at the outset in a complex problem. Interactions and contradictions among subproblems may arise at places that are neither anticipated nor desired (Thomke et al. 999). Thus, interdependencies g i (h)/ h j contain random components. We may view these as being drawn when problem solving begins and then remaining constant. Formally, then, we assume that the random interdependencies are drawn from any distribution with nite positive variance and that they are mutually independent. Given this structure, we can formally show the following result: Proposition : Suppose that N agents in an organization make decisions asynchronously and without information-updating delays, and suppose that each employee values his subproblem higher than other subproblems (b j,i < when i j in Equation (3)). That is, system evolution can be represented by Equation (5) with random interdependencies as described previously. Then, as N grows, the probability approaches that problem solving becomes unstable (i.e., that a small random perturbation of the system around a solution is not dampened but is amplied arbitrarily) after any number of steps L. The Appendix oers a formal formulation and the proof of Proposition, whose weak assumptions make it widely applicable. The proposition holds widely because a cycle of mutual dependencies becomes more and more likely as the number of subproblems grows, leading to self-reinforcing cycles of changes and hence many iterations. Proposition implies that large organizational projects in at organizations typically exhibit a common deciency: long problem-solving cycles that seem to lead nowhere, even though in principle a solution should be possible. Management must take action to stop the search in spite of the suboptimal results that may follow. 4. Hierarchy and Problem-Solving Stability: Analytical Results In this section, we show how introducing a hierarchy can change the problem-solving dynamics. We continue with the case of employees who value their subproblem higher than others, while receiving instantaneous information updates.

11 Article submitted to Management Science; manuscript no. MS R2 4.. Problem Solving in a Hierarchical Structure The most obvious way in which hierarchies can change the decision-making process is by introducing an organizational entity with the right to reverse others' decisions. In that sense, hierarchies can foster the centralization of decision making. This is Rivkin and Siggelkow's (23) centralized hierarchy (which we replicate in one variant of our model). However, a hierarchy also has the more subtle eect of inuencing communication patterns among employees. In no organization of even moderate size does a decision maker communicate equally with all other decision-makers. Employees are always grouped into departments based on subproblem similarities and shared solution methods (March and Simon 958). Employees tend to coordinate and discuss subproblem solutions within their own department before talking to colleagues in other departments (Sosa et al. 24). Moreover, employees within a department work on related problems with shared solution methods, report to a common manager, and thus have a common identity. The performance of other departments may be viewed as less important than the performance of one's own department, partly because the performance dimensions applying to other departments are less well understood and partly because the other department may be viewed as the outgroup (Janis 97, Sosa et al. 24). Thus, multiple reasons push employees within departments to become more insular, a widely observed phenomenon that is dicult to avoid. In our model, this insularity has two eects. First, suppose employee i is in department D. Then b k,i b j,i when k / D and j D. This reects the aforementioned outgroup aversion that discounts the importance of other departments (at the cost of lower solution quality). The second eect of insularity is that P i / h k < P i / h j when k / D and j D. This may reect true modularity across departments (as when a department groups the most closely related subproblems together), but it also reects that a department buers itself against actions by other departments. For example, the manufacturing group distrusts the quality of marketing's volume forecasts and so builds extra capacity: As long as they don't deviate too much from the initial announcement, we are prepared. Or perhaps marketing distrusts the quality of engineering's designs and buers its volume forecast: As long as their specications don't deviate too far from what we had originally agreed, we'll be able to deliver the volume. By buering of this type, one group reduces the eect of decision changes within a certain range by another group on its own performance function (at a cost of lower solution quality). Mathematically, employee i's performance can be rewritten as P i (h i, {h j }, {h k }), where the inuence of {h k } on P i is reduced by the buering: h i = argmax j D b j,i α j P j (h i, {h j }, {h k }) + k / D b k,i α k P k (h i, {h j }, {h k }). (6)

12 2 Article submitted to Management Science; manuscript no. MS R2 This equation formalizes how employee i's consideration of the overall problem context is focused on neighboring subproblems in her own department; outside subproblems are discounted and their inuence on one's own decisions is buered away Why Hierarchy Can Help Given these two potential eects of a hierarchy, the question is: How do they inuence the organization's solution search? Let us rst concentrate on centralized decision making. Consider the simplest hierarchy with one manager and N employees, similar to Rivkin and Siggelkow (23). Suppose the manager is the central decision maker; this means that each employee i presents a solution P i as a suggestion to the manager, who then rejects or accepts it based on its contribution to the whole problem P. Without further assumptions, we can now demonstrate the following: Proposition 2: If the organization makes decisions in a hierarchy with a centralized decision maker, then the solution quality P converges monotonically to a nal level. A formal statement and a proof of the proposition are given in the Appendix. The centralized decision maker constrains the interactions among subgroups by allowing changes only if they make the group's performance better. This has two implications. First, the decision vector h converges to a stable solution in most cases (except in contrived situations where multiple solutions have the same performance value). Second, even if the decisions h themselves do not converge, the benet of continued search tapers o over time, which enables the manager to establish termination criteria based on decreasing improvements. Thus, a centralized hierarchy dampens in a wide range of circumstances, ensuring a viable solution in a reasonable time. But a hierarchy has other benets as well. They arise because interdependencies are treated more weakly across department boundaries in Equation (6) via the lower importance weights b k,i and the reduced interdependencies P i / h k when i and k are in dierent departments. Proposition 3. Take the same problem-solving organization as in Proposition. Group the agents into non-overlapping departments, and reduce the importance of cross-group interdependencies (b k,i and P i / h k whenever k and i are in dierent departments). Then, for any organization with slow or nonconverging problem solving, there is a choice of departments and of interdependence importance such that search progress toward a solution becomes fast and stable. Proposition 3 states that dividing the organization into units and prioritizing the interdependencies within each unit over interdependencies across units reduces problem-solving iterations. The reason

13 Article submitted to Management Science; manuscript no. MS R2 3 is that the resulting group-myopic behavior reduces the incidence of feedback loops that induce the iterations described in Proposition, although neglecting interdependencies may damage the solution quality. There are realistic situations where a grouping (introducing departments) alone can achieve stability without lost solution quality, especially when the overall problem is easily modularized and naturally leads to weak interdependencies across groups. However, there are also cases in which achieving stability requires forced prioritization and the neglect of some important cross-department interdependencies, which severely limits solution quality. In many realistic cases, however, the interdependence reduction required for stability is minor. 5 In summary, the analytical model suggests that managers who apply centralized decision making reduce oscillations and thereby foster stability and search speed. In addition, a hierarchy can help an organization stabilize and speed up its problem-solving search by downplaying cross-departmental interdependencies. Such interdependencies are weakened by the natural behavior of the employees, who pay more attention to results in their own department than in other departments. This reduces problem-solving oscillations. Thus, what is often frustrating in terms of nding the best solution (the other department does not listen to us) actually has a strong advantage in terms of search speed. Hierarchy shapes the search dynamics, even in the absence of control loss and competence issues. 5. Simulation Results Although the the formal analysis in Section 4 oers a causal theory of organizational dynamics in hierarchies, it cannot answer some of our research questions: What are the eects of the hierarchy's structure and of the vertical and horizontal coordination in it? We must resort to simulation for the answers to these questions. Likewise, our analytical results have established the search acceleration benets of a hierarchy when employees' goals are imperfectly aligned (violation of the rst condition in Section 3.2). Yet we must still demonstrate the benet of a hierarchy when updating among subproblems is delayed (violation of the second condition in Section 3.2). For this, too, we need simulations. A simulation model requires adding detail to the general model. 5.. The Simulation Model Consider a four-level hierarchy: 48 workers are grouped into six departments, which in turn are structured into two areas of three departments each; the area managers report to the CEO. Suppose 5 For synchronous systems, quantitative formal statements for Gaussian matrices can be made that a modest weakening of interdependencies reduces the eigenvalues of the Jacobian matrix and leads to stability (Hogg et al. 989). For asynchronous systems, simulations suggest that a moderate weakening of cross-department interdependencies is also sucient to establish stability.

14 4 Article submitted to Management Science; manuscript no. MS R2 Ordering of Problem Solving Parallel Sequential All groups (e.g., departments) perform each a solution iteration in parallel. Then the superior (e.g., area manager) accepts the proposal if it improves the group solution quality. For each group, the superior (e.g., area manager) accepts the proposal if it improves the group solution quality. Then the superior (e.g., area manager) accepts the proposal if it improves the organization s overall solution quality. The superior (e.g., area manager) checks for consistency with the other groups and forces another iteration if any group s solution is based on information that has become obsolete in the last iteration and is thus inconsistent with other groups. 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 each group, the superior (e.g., area manager) accepts the proposal if it improves the organization s overall solution quality. Consistency is guaranteed by the sequential way of proceeding. When all groups are done, the superior (e.g., area manager) checks whether the first group s solution is consistent with the subsequent decisions by the other groups and forces another iteration if the other groups have made changes. Centralized Hierarchy with Myopic Incentives for Managers Centralized Hierarchy with Holistic Incentives for Managers Coordinating Hierarchy Locus of Decision Making and Incentives Table Three dimensions of decision making in the hierarchy that groups at the same hierarchical level are of the same size, which will allow us to focus on the fundamental eects of the hierarchy. All problem-solving expertise is concentrated 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 actual problem solving. This simplifying assumption is in line with previous search models. Moreover, we assume that the front line workers act holistically in subproblem-level search; that is, they use b j,i = for all i, j and do not reduce interdependencies via safety buers as in Equation (3). We focus on this optimistic case (as opposed to the myopic case) because previous studies (Mihm et al. 23, Rivkin and Siggelkov 23) have shown that fully myopic behavior by the problem-solving employees sacrices too much solution quality. We now need to specify two domains of the simulation model. First, we use the exibility oered by simulations to represent three additional dimensions of the hierarchy's decision-making structure that are not tractable in an analytical model. Second, we need to specify detailed functions P and P i in order to generate quantitative simulation results. Table summarizes three dimensions of how the organization coordinates across levels and departments: (i) the level at which the organization makes its decisions; (ii) the incentives of the managers above the front-line workers; and (iii) the way the organization sequences problem-solving work across departments. The rst aspect of the decision-making structurethe level at which decisions

15 Article submitted to Management Science; manuscript no. MS R2 5 are madedepends on whether the hierarchy is centralized (active) or coordinating. (Rivkin and Siggelkow (23) call the latter a rubberstamping hierarchy, but coordinating captures the spirit more accurately because in our model the manager adds value by ensuring coordination.) In the centralized hierarchy, the manager takes the subproblem solutions that come from the organizational level below as suggestions, and (consistent with Proposition 2) she accepts or rejects them depending on whether or not they improve the overall solution value of the manager's more aggregated goal function. In other words, the value of existing proposals can be tested, although optimization of the overall problem P remains elusive. In contrast, the manager in the coordinating hierarchy does not make decisions directly, instead ensuring coordination among the multiple subproblems: if the solution h i to one subproblem is inconsistent with the assumption behind the solution to another subproblem j (and thus the P j as perceived by employee j diers from the true P j ), then the manager ensures that one or both of the solutions are changed until they are mutually compatible. The centralized and the coordinating hierarchy are two extreme benchmarks; real organizations fall somewhere in between the two cases. For our second dimension of decision making, we further divide the centralized hierarchy into two variants according to the incentives aecting middle managers one and two levels below the CEO. We assume that front line workers act holistically, but managers may well have political agendas that are myopically focused on their organizational units (incentive conicts across units tend to play a larger role at the management level than the front line.) Thus, we dierentiate between holistic incentives of the department managers (i.e., when they work also in the interest of the entire organization) and their individual incentives (when they discount the inuence of subproblems outside their department and maximize the performance of their group possibly at the cost of other groups). The third aspect of the hierarchical decision-making structure is the order in which the various departments obtain and coordinate their subproblem solutions. Sequential problem solving means that one lead function solves its subproblem rst and gives its solution to the other groups at the same hierarchical level. Then the next department works on its problem, taking into account the lead function's solution. When each group has produced a solution proposal, the lead function re-examines its solution in order to identify whether it can do better after updating its assumptions about the other groups' decisions, and so a second round starts for all groups at this hierarchical level. This sequential problem solving continues until no group can make improvements with further changes.

16 6 Article submitted to Management Science; manuscript no. MS R2 The alternative coordination mode in a hierarchy is parallel, or concurrent, problem solving: departments at the same hierarchical level work in parallel, using as assumptions about the other departments their most recent information status. When all departments have obtained solutions to their subproblem areas, they exchange their respective nal statusfor example, in a system test. Based on this exchange, the departments update their assumptions and iterate in parallel until none sees any further need to change its solution. Both sequential and simultaneous problem solving are widely observed in real organizations and often co-exist. Sometimes, certain subproblems are considered more important or certain departments are more powerful and have more status than others (Terwiesch et al. 22)in some companies, marketing has this status; in others, it is engineering or manufacturing. These functions will tend to be lead functions. But in other cases, subproblems are not prioritized and must proceed in parallel in order to meet deadlines. We consider sequential and parallel problem solving as extreme benchmarks whose relative merits illuminate the principles of coordinating decentralized work in a hierarchy. We now turn to the second specication domain for the simulationnamely, specic value functions P i. In order for the P i to be plausible, performance should never become innite, and a neighboring subproblem decision h j should inuence both the optimal decision h i and its achievable performance P i (h i, {h j }). The simplest form incorporating these demands is the following quadratic: 6 [ ( P i = d i h i N j i ) 2 ] δ i,j h j + I i,j for i =,..., N. (7) Note that P i is not the true performance but rather worker i's optimization function, or the perceived performance based on the possibly obsolete (because of updating delays) information h j about other subproblems. The form of P i is shown in Figure 2. The term in brackets is a quadratic function with one optimum 7 in the decision variable h i, whose position is shifted linearly by the decision of agents j via the interdependence parameter δ ij (If marketing increases the number of segments to be served, h j, then manufacturing's optimal investment in platforms, h i, becomes larger). Here, d i is a scale parameter that makes the term in parentheses comparable to the intercept. j i 6 Mihm et al. (23) show how this functional form can be viewed as a result of two Taylor series approximations of a general functional form. 7 The assumption that local performance has only one optimum is optimistic, reducing iterations because the system does not jump among multiple local optima of the agents. Multiplicative component interactions are widely used, e.g., in evolutionary biology, where the tness of an organism is the product of the contributions of all genetic loci (Ewens 979, Lewontin 974).

17 Article submitted to Management Science; manuscript no. MS R2 7 P i = d i µ h i X ± ij h i 2 + Y Iij Pi Performance Performance constraint from other decision maker I ij Biasing influence ± ij of other decision maker too low Own decision h i too high Figure 2 Subproblem performance function in the simulation model The best achievable performance of subproblem i is inuenced by agent j's decision via constraints I ij that agent j puts on i's local problem (If marketing chooses a smaller market segment, h j, the best achievable cost improvement, P i, is reduced for the manufacturing department). The I ij are continuous piecewise linear functions: in a middle range, h j has an important (linearized) eect on P i (every additional dollar that your subproblem consumes has a negative eect on my available budget). If h j is extremely disadvantageous for subproblem i (and i's solution quality therefore low), then another change in h j makes little dierence (If you absorb that much budget, I cannot achieve minimal performance 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 change makes little dierence (If you spend only so little, I have what I need, and a small increase in your budget does not hurt me). In the simulations, the interdependence parameters are randomly drawn from a uniform [, ] distribution, the constraint interval limits from a uniform [, 2] distribution, and the scale parameters d i from a standard lognormal distribution. It is notoriously dicult to establish the robustness of results in search models (see, e.g., Rivkin 2). In our case, the computing eort behind the reported results exceeded a combined 45 months of runtime on contemporary PCs, and another 2 months of runtime were spent on robustness tests. In addition to anchoring the base cases in the analytical results, we veried all reported experiments with sensitivity variations of key simulation parameters (such as group size and crossdepartment eects). Although the quantitative results change, the qualitative insights drawn from

18 8 Article submitted to Management Science; manuscript no. MS R2 the simulations remain stable unless noted in the text. More details on robustness and the data itself are available from the authors Anchoring Results The rst set of simulation runs connects our simulation model to the analytical models from Sections 3 and 4 and also to previous work. It replicates the results from the analytical section but with the assumption of imperfect alignment (condition ) replaced by the assumption of delayed communication (condition 2). Unless specically indicated in the text, the organization has four layers with 48 front-line employees, and these employees behave holistically and listen to information from other departments (b k,i = in Equation (6)). We assess the eectiveness of the various hierarchy congurations on three performance dimensions. First, what is the probability that the search process converges to a solution rather than diverging? Second, given that problem solving does converge, how long does it take the organization to settle on a solution? Third, what is the solution quality, the nal value outcome P? For all three dimensions, we show averages. Reported results are statistically signicant at the 5% level. Figure 3 summarizes the set of results. Its rows represent the three performance dimensions, and its columns represent the decision-making modes of Table. In the left column we assume that the hierarchy is coordinating only; in the center column, the middle managers act individually and consider only the performance of their own organizational units; the right-hand column represents managers with holistic incentives. Each chart tracks performance over the density of interdependencies and distinguishes performance 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 complex. Each point in the curves represents the average of 5 simulation runs in which the interdependency strengths are repeatedly re-sampled as random variables (as described in Section 5.). If a graph seems to contain only one curve, the second curve is fully covered by the rst. Two explanations help to interpret the graphs. First, the convergence times assume reaching full consensus and are thus unrealistically long (with one update and decision per ten hours on average, the centralized hierarchy with myopic incentives takes about 2, hours at 5% density, and the coordinating hierarchy takes 2, hours). In reality, management stops and forces a restart or freezes components much sooner, satiscing at the best solution found to date (as occurred in our opening example). We do not introduce freezing in the simulation because any stopping time is arbitrary; the longer the convergence to an equilibrium, the higher the performance loss of stopping