Hierarchical Structure and Search in Complex Organizations

Submitted to

Management Science

manuscript MS-01261-2007-R2

Hierarchical Structure and Search in Complex Organizations Jürgen Mihm, Christoph H. Loch INSEAD, Boulevard de Constance, 77305 Fontainebleau Cedex, France {[email protected], [email protected]},

Dennis Wilkinson, Bernardo Huberman HP Labs, 1501 Page Mill Road, Palo Alto, CA 94304 {[email protected], [email protected]}

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: (1) 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

1.

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 1995). These challenges demand that decisions with many parameters be taken quickly. Yet in the face of such circumstances, decisions often cannot be optimized (Simon 1969). 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.

1

2

Article submitted to

Mihm et al.: Hierarchical Structure and Search Management Science; manuscript no. MS-01261-2007-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 1997, Rivkin and Siggelkow 2003, Siggelkow and Rivkin 2005). 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 2005)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 2001). 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 organiza-

tional 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 2003) and raises the organization's ability to cope with environmental changes (Siggelkow and Rivkin 2005). 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 1). 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

Mihm et al.:

Hierarchical Structure and Search Management Science; manuscript no. MS-01261-2007-R2

3

Article submitted to

CEO

Area Mgr.

Middle Management

Front-line Management

Area Mgr.

Dept. Mgr.

Dept. Mgr.

Dept. Mgr.

Dept. Mgr.

Front-line problemsolving Workers

Sub-problem 3

Sub-problem 1 Sub-problem 2

Figure 1

Behavioral Aspects of Hierarchy

Structural Aspects of Hierarchy

• Question: Horizontal coordination: Sequential or parallel work? • Result: Sequential work is faster and produces higher quality.

• 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.

• Question: Vertical coordination: How far to delegate decisions? • Result: Delegate decisions all the way to front-line management.

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

Article submitted to

Mihm et al.: Hierarchical Structure and Search Management Science; manuscript no. MS-01261-2007-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 2000, Winter et al. 2007), adapting to industry life cycles (Doty et al. 1993, Levinthal 1997, Nelson and Winter 1982), building organizational capabilities (Bruderer and Singh 1996, Gavetti 2005), and new product development (Fleming and Sorenson 2001, Mihm et al. 2003, Yassine et al. 2003). Search may contain elements of local optimization by intelligent local agents (e.g., Gavetti 2005, Gavetti and Levinthal 2000, Knudsen and Levinthal 2007, Sorenson 2002, Winter et al. 2007), 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 2007, Simon 1969, Van Zandt 1999). 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 1963) 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 manageable tasks performed by specialized groups while

dividing

the overall search problem into

coordinating those groups to obtain cohesive

action (March and Simon 1958). Search has been a topic in organization theory since its infancy (Cyert and March 1963, March and Simon 1958, Simon 1969), but interest has surged with the introduction of formal models to organization theorynotably the with

K

NK

model (e.g., Levinthal 1997), which models actors as

N

nodes

interdependencies; it was developed in physics as a spin glass model and then applied

to evolutionary biology (Kauman 1993). Simulation work on the

NK

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 2004, Huberman and Wilkinson 2005, Mihm et al. 2003): ongoing choices in some groups make the requirements for other groups inherently unstable (Thomke 1997, Van Zandt 1999). 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.

1

We take

1

There are other coordination mechanismssuch as liaisons, cross-unit groups, and

vertical hierarchy to mean that multiple individuals, or multiple groups, report to the same manager.

Mihm et al.:

Hierarchical Structure and Search Management Science; manuscript no. MS-01261-2007-R2

5

Article submitted to

informal networksbut a vertical hierarchy is present in virtually every organization (Nadler and Tushman 1997). Groundbreaking work on the role of hierarchies in search (Rivkin and Siggelkow 2003, Siggelkow and Rivkin 2005) 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 converge to a solution at all (see

NK

simulations, but also the ability to

Mihm et al. 2003, Yassine et al. 2003). 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 1994, Child 1984, Keren and Levhari 1983, Moldoveanu and Bauer 2004, Nadler and Tushman 1997, Radner 1993, Williamson 1991, Van Zandt 1999).

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: (1) 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. 2003). We then show how the presence of a hierarchy inuences decision making and interactions among the decision makers.

6

Article submitted to

3.1.

Mihm et al.: Hierarchical Structure and Search Management Science; manuscript no. MS-01261-2007-R2

A Basic Search Model without Hierarchy

An organization of

N

employees collectively produces an output of value

a function of all the employees' decisions,

P (h1 , . . . , hN ).

P.

The output value is

In order to allow decentralized problem

solving, the organization divides the overall problem into subproblems

i of value Pi , where problem i

is delegated to one specialist decision maker i. For simplicity, suppose that each subproblem thus each employee) has one decision variable,

Pi

(and

hi , and that Pi 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

P=

N X

αi :3

αi Pi .

(1)

i=1

If the sub-problems were independent, that is, each

Pi (hi ) 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:

Pi = Pi (h1 , . . . , hN ). We know from

N K -model

(2)

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. 2003). Then we show how a hierarchy changes the search. The organization starts with a previous solutionfor example, the result obtained in the last search,

(h01 , ..., h0N ). 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

P,

i

makes a decision aimed at optimizing the overall performance function

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

[0, 1]. The ideal 2

PN

j=1 bj,i

αj Pj (h1 , ..., hN ),

where

would be that all employees act fully holistically, bj,i

bi,i = 1

and the other

bj,i ∈

= 1. The opposite (pessimistic)

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

a more general function merely complicates the exposition.

P = f (P1 , ..., PN );

Mihm et al.:

Hierarchical Structure and Search Management Science; manuscript no. MS-01261-2007-R2

7

Article submitted to

assumption would be that

bj,i = 0

when

i 6= 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

hj , j 6= i.

But information overload may prevent her from

seeking by-the-minute updates from the other subproblems. Thus the status of the other subproblems at the last time

hj , j 6= i,

reect the decision

i obtained an update from j . The employee maintains

this assumption until she has another opportunity to be updated. Then the decision by employee can be written as

h∗i =

argmax

N X

bj,i αj Pj (hi , {hj }).

i

(3)

j=1

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 1977). 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

hi ,

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. 2003a,b, Lounamaa and March 1987). 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 (1993) have shown that modeling a naturally asynchronous system with a synchronous

mathematical process may introduce distorting artifacts, and Sorenson (2002) stresses the importance of this result for organizations.

8

Article submitted to

Mihm et al.: Hierarchical Structure and Search Management Science; manuscript no. MS-01261-2007-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 through the space of decisions

h

(the vector of all decision variables

hi ).

search

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 2007). In this stable state, decision makers are correctly updated about the overall solution status (hj

= hj )

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. 3.2.

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

bj,i = 1).

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

Mihm et al.:

Hierarchical Structure and Search Management Science; manuscript no. MS-01261-2007-R2

9

Article submitted to

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)), where

g(h(t))

(4)

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

gi (h(t)) = h∗i (h(t)) and for other agents gk6=i (h(t)) = hk (t). In this way, we are able to examine

incremental

changes in the problem-solving status for any point in time,

g(h(t)) − h(t). The system has reached a xed point i

makes a move over

∆t:

if

t: h(t + ∆t) − h(t) =

g(h(t)) − h(t) = 0 independent of which player

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

h0

and a perturbation



with random elements

i

centered about

0

with some nite variance. Since

we are interested in what happens for an arbitrarily small perturbation, solving systems can be described by a linearization: Dene the

[Ji ]ij = ∂gi (h)/∂hj |h=h0 ∂gi (h)/∂hj

for

most relevant problem-

partial Jacobian matrix Ji ,

i 6= j , [Ji ]ii = ∂gi (h)/∂hi |h=h0 , [Ji ]jj = 1,

and

[Ji ]ji = 0.

i, m, . . . , v

containing

n

hn

be the system state after

with

The derivative

expresses the interdependency, which is the eect that a small change in

has on the optimal decision of i. Let the set

,

j 's

decision

n players have moved. Then, with

elements, the problem-solving dynamics can be described by

hn − h0 ≈ Ji Jm . . . Jv  The problem-solving system becomes

where

unstable

i, m, . . . , v ∈ {1 . . . N } .

(5)

if mutual interactions among subproblems enlarge

a small perturbation from the xed point. Here is where our rst condition becomes relevant: if all

bj,i = 1,

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

∂gi (h)/∂hj

become zero and

10

Article submitted to

thus, for all i, the

Ji

Mihm et al.: Hierarchical Structure and Search Management Science; manuscript no. MS-01261-2007-R2

become zero matrices. Hence, no instability arises. However, if

bj,i < 1 for i 6= 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. 1999). Thus, interdependencies

∂gi (h)/∂hj

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 1: 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 (bj,i

0, 0 < C  < 1 there is a nite size of the matrix N such that for all N > N the Q  i probability P J  > κ > C for any nite number of steps k .  i∈Cl Ck

Note that the proposition implies that given an appropriate system size, the reaction to the small shock drives the rst player in his or her rst step arbitrarily far from his or her original position. Furthermore, for any nite number of steps the resulting state vector of those elements that have made a move will venture arbitrarily far away from their original position. It is relatively simple to substantially generalize the proposition  though at the cost of extended technical exposition. First, not all matrix elements need be non-zero. It is sucient that the probability of a new entry being non-zero does not decay to 0 as N grows. Second, not all Ji,j have to from the same distribution. It is sucient that the variance of their distributions has a lower positive bound. Third, even correlations between elements can be accommodated, as long as the Ji,j can be ordered such that the correlations between them taper o with distance. Proposition 1 generalizes previous results (Mihm et al. 2003, Yassine et al. 2003, Huberman and Wilkinson 2005) in that it models asynchronous (not synchronous) changes and does not require Gaussian distributions of interaction parameters.

30

Article submitted to

Mihm et al.: Hierarchical Structure and Search Management Science; manuscript no. MS-01261-2007-R2

Proof of Proposition 1:

prove a statement equivalent to the one in the proposition: There is N such    We will  Q i that for all N > N , P  ≤ κ < C for any 0 < C . In a fully random asynchronous system, a i∈Cl J Ck

step Jl may be repeated. In order to reduce the complexity of the proof, we rst show that for large system sizes, cases with steps carried out more than once. For notational simplicity dene P (6=) as   we can disregard  Q Ji  ≤ κ conditional on no step being repeated, Ji 6= Jj ∀i 6= j , and P (=) for all other cases. P i∈Cl Ck    Q i ≤ κ = P (Ji 6= Jj ∀i 6= j)P (6=) + P (Ji 6= Jj ∀i 6= j)P (=) ≥ P (Ji 6= Jj ∀i 6= j)P (6=). As J  Then P i∈Cl the Poisson arrival process for steps assures that all sequences of moves are equally likely, P (Ji 6= Jj ∀i 6= j) = N! (N −k)! Nk

. As limN →∞

N! (N −k)! Nk

= 1, it is sucient to show in the following that limN →∞ P (6=) = 1.

The proof is straightforward. Note that in case there is no repeated move we can reorder the columns and Q Qk rows in the matrices without loss of generality such that Ck = 1...k and Ck Ji becomes i=1 Ji . Clearly, as for i > 1, the rst row of matrices J i has J i1,1 = 1 and J i1,j = 0 for j > 1, the rst element of the state vector Q  P N = Ji  is dened in the rst move and then left unchanged. Then |N |1...k ≥ |N J  |. 1 |=| i∈Cl j 1,j j Since j and J1,j are iid, so will be J1,j j . Since both j and J1,j have nite expectation and nite but positive variance, so does J1,j j . To see this note that because of independence µJ = E[J1,j j ] = E[J1,j ]E[j ]. Second, from nite variance of the variables it follows that E[J12,j ] and E[2j ] exist and thus so does its variance

s2J = E[J12,j 2j ] − E[J1,j j ]2 = E[J12,j ]E[2j ] − E[J1,j ]2 E[j ]2 . P P P Now note that | j J1,j j | ≤ κ implies that both J  ≤ κ and − j J1,j j ≤ κ. Then, j 1,j j  Q    P   P J1,j j −N µJ µJ µJ √ √ P Ji  ≤ κ ≤ P | j J1,j j | ≤ κ = P −κ−N ≤ j √N s ≤ κ−N . Now as of the i∈Cl N sJ N s J J Ck

central limit theorem,

P

j

J1,j j −N µJ √ N sJ



converges in distribution against the standard normal distri   P J1,j j −N µJ κ−N µJ µJ −κ−N µJ √ √ √ = lim Φ( ≤ j √N s ≤ κ−N ) − Φ( ) . N →∞ Ns Ns Ns

bution Φ. Then, limn→∞ P J J J J √ κ−N µJ −κ−N µJ √ √ N sJ grows in N without bound, limN →∞ Φ( N s ) = limN →∞ Φ( N s ). But then But since J J   κ−N µJ µJ √ limN →∞ Φ( √N s ) − Φ(− κ−N ) = 0 and thus Ns J

−κ−N µJ √ N sJ

J

 ! Y i J  lim P  N →∞ i∈Cl

 ≤ κ = 0 Ck

Finally note that this is true for arbitrary k . Proposition 2.

(8)



Let Pi , i = 1..N , be the performance functions for N agents and P be the system overall

performance as dened in section 3.1. Let hk be the system state at time k . Let the manager only accept performance improvements. Then there is nite k such that P (hk ) ≥ P (hi ) −  for any i ∈ N and  > 0. Proof of Proposition 2.



The proof is very simple. First note that P is bounded: there is P such that for all

hk , P (hk ) ≤ P . Furthermore, the manager only accepting improvements implies P (hk+i ) ≥ P (hk ) for i ≥ k . Finally, P (h0 ) has an initial value of P > −∞. As a result there must exist limit performance K such that

limk→∞ P (hk ) = K < ∞. But then any performance K − ,  > 0 must be reached with nite k or K cannot be the limit performance.



Mihm et al.:

Hierarchical Structure and Search Management Science; manuscript no. MS-01261-2007-R2

Article submitted to

31

Proposition 3.

Assume a system of players with performance function of Equation (6). Assume the system

is instable for bj,i = 1 ∀j, i and for the inuence of hk not reduced. Group the agents in z non-overlapping departments, D1 . . . Dz . Assume that (a) bj,i = 1 if j, i are in the same department, and bk,i ∈ [0 . . . 1] if k, i in dierent departments and (b) the inuence of hk is potentially buered. Then, there is a choice of z and

bk,i , k, i in dierent departments, and a choice of buer for hk such that the system is stable. Proof of Proposition 3.



The proof is simple. Assume to the extreme that z = N and Di = {i} (each person

is a department). Choose bk,i = 0, for k and i in dierent departments and choose the buer such that

δPi δhk

=0

for k and i in dierent departments. Then, Pi is not a function of any hj with j 6= i and any optimization can be carried out once and for all in the rst step component maker i carries out.



Note that Pi is continuous in bk,i . Now if buering can be carried out in such a ne grained way that Pi is also continuous in the level of buering, it can be shown that there exist department partitions with positive interaction strength that are stable. Thus stability can be expected for intermediate levels of interaction.