# Emergence of Coordination in Growing Decision-Making Organizations: The Role of Complexity, Search Strategy, and Cost of Effort

Authors: Friederike Wall
Journal: Complexity
 (2019)

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## Abstract

Coordination among decision-makers of an organization, each responsible for a certain partition of an overall decision-problem, is of crucial relevance with respect to the overall performance obtained. Among the challenges of coordination in distributed decision-making systems (DDMS) is to understand how environmental conditions like, for example, the complexity of the decision-problem to be solved, the problem’s predictability and its dynamics shape the adaptation of coordination mechanisms. These challenges apply to DDMS resided by human decision-makers like firms as well as to systems of artificial agents as studied in the domain of multiagent systems (MAS). It is well known that coordination for increasing decision-problems and, accordingly, growing organizations is in a particular tension between shaping the search for new solutions and setting appropriate constraints to deal with increasing size and intraorganizational complexity. Against this background, the paper studies the adaptation of coordination in the course of growing decision-making organizations. For this, an agent-based simulation model based on the framework of NK fitness landscapes is employed. The study controls for different levels of complexity of the overall decision-problem, different strategies of search for new solutions, and different levels of cost of effort to implement new solutions. The results suggest that, with respect to the emerging coordination mode, complexity subtly interferes with the search strategy employed and cost of effort. In particular, results support the conjecture that increasing complexity leads to more hierarchical coordination. However, the search strategy shapes the predominance of hierarchy in favor of granting more autonomy to decentralized decision-makers. Moreover, the study reveals that the cost of effort for implementing new solutions in conjunction with the search strategy may remarkably affect the emerging form of coordination. This could explain differences in prevailing coordination modes across different branches or technologies or could explain the emergence of contextually inferior modes of coordination.


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## Body

## 1. Introduction

The coordination of decision-making among a set of agents, with each being responsible for a certain partition of an overall decision-problem, is a fundamental issue in the design of distributed decision-making systems (DDMS)—may they be, for example, organizations like firms or teams of robots (e.g., [1–6]). Coordination within DDMS is studied in various domains of organizational thinking. For example, a predominant issue in organization theory is mechanisms to manage interdependencies between activities within an organization [7–9]; in the domain of management control, the so-called Management Control Systems (MCS) are intended to ensure that decision-making is consistent with objectives and strategies of a firm by employing a multitude of mechanisms and techniques to coordinate managers’ choices [10–12]; in the domain of multiagent systems, for example, “plan merging” in terms of integrating partial plans into an overall plan is one of the issues discussed in the context of coordination (e.g., [5, 13]).A common, though mostly implicit, idea in these schools of organizational thinking is, however, that there is a well-informed designer knowing the “true” nature of the overall decision-problem and, with this, the coordination need as affected by the problem’s complexity; hence, the designer can organize the DDMS accordingly. However, this assumption does not necessarily universally apply, since the system may be newly set up or it may operate in an environment which has undergone an external shock. In a similar vein, the underlying decision-problem does not need to keep its structure over time; it may, for example, grow in size: A firm may produce and sell additional products or an increased geographical area may be provided with services by a (growing) fleet of unmanned aerial vehicles.In this line of thought, for coordination theory in multiagent systems (MAS), Lesser and Corkill [14] recently argue that among the issues so far underrepresented in MAS research is to explicitly take into account environmental conditions like, for example, the predictability of task characteristics and the dynamic adaptation of coordination when the environment or the problem to solve is evolving dynamically.Against this background, the research objective of this paper is to studywhich types of coordination emerge in DDMS when the underlying decision-problem is not known in advance and is subject to growth. The paper seeks to provide some answers to this research question while adopting the perspective of contingency theory, i.e., assuming that the performance of a system is shaped by the fit between its situational context and its internal arrangements, taking the particular interrelation among complexity theory and contingency theory ([15], pp. 411) into account. Based on this understanding, the research endeavor presented takes three contingent factors into account.


### 1.1. Complexity of the Decision-Problem

Corresponding to the seminal paper of Simon [16], the complexity of an overall decision-problem and, hence, the need for coordination is shaped by the interactions among its components: In case the overall decision-problem is (nearly) decomposable, it can be separated into (nearly) disjoint sub-problems such that intra-sub-problem linkages are stronger than the inter-sub-problem interactions. In consequence, sub-problems can be solved independent from each other without taking positive or negative interactions with respect to the overall problem’s solution into account. In contrast, if an overall decision-problem is nondecomposable, no decomposition into sub-problems can be found that (nearly) diminishes inter-sub-problem interactions (e.g., [17, 18]). Hence, if complexity in terms of interactions is high, also the need for coordination across the sub-problems is high when superior solutions to the overall decision-problem are pursued.


### 1.2. Search Strategy

The strategies for finding new solutions for decision-problems have been studied in various domains. For DDMS “resided” by human decision-makers a cornerstone was set by Simon [19] who argues that in “situations of any complexity” (pp. 104) decision-makers are unable to survey the entire search space and, hence, cannot identify the optimal solution of their decision-problem “at once”; rather, they search stepwise for superior solutions following a satisfying approach. In the domains of organizational search and innovation, the idea of stepwise search often is characterized by the extent of changes in terms of the distance of the newly found options compared to a status quo, i.e., explorative, exploitative, or ambidextrous strategies of search (e.g., [20, 21]). Constraints regarding the allowed or enforced extent of changes induced by alternative solutions are part of the boundary system in MCS: The boundary system sets behavioral constraints to decision-makers and is regarded as a prerequisite for the delegation of decision-making [22–24]. In growing organizations, the boundary system is in a particular tension between shaping the search for new opportunities and innovation on the one hand and on the other, setting behavioral constraints to deal with increasing size and intraorganizational complexity [25, 26].


### 1.3. Cost of Effort

The search strategy shapes the potential extent of changes made compared to a status quo and changes may come along with notable extra costs often termed “switching cost” or “cost of effort,” e.g., for cognitive effort or additional consumption of time or resources. For example, organizational changes may cause costs for reorganization and for handling resistance of certain stakeholders [27], choosing another supplier could result in costs for technical conversions [28, 29], changing the direction of a robot’s movement in a multirobot system could cost some extra time [30–32], and altering the task assignment in the course of job scheduling may rise some extra costs of, for example, learning—in case of humans as well as “artificial” agents [33, 34]. The level of cost of effort, therefore, affects the propensity to implement an alternative to the status quo as identified according to the search strategy.The paper employs an agent-based simulation modelto “grow” organizations from scratch and to observe which modes of coordination predominantly emerge for different settings of the aforementioned contingent factors. Simulation appears an appropriate method since it allows to capture long-term and processual phenomena that—depending on the subject—would require rather challenging longitudinal studies in empirical research (e.g., [35–37]). Simulation further helps to analyze “borderline” cases which could be hardly studied in a sufficient number empirically in order to systematically explore the effects of contingency the effects of contingency factors in interaction with each other [15]. To capture and be able to control the complexity of a DDMS’s overall decision problem, the study relies on the framework of NK fitness landscapes which was originally introduced in the domain of evolutionary biology [38, 39] and, since then, broadly employed in managerial science (for overviews, see [21, 37]). In particular, an agent-based simulation is employed in which growing organizations—with an increasing number of decentralized decision-makers—search for superior solutions to their task (specified according to the NK framework) and where the organizations may employ different search styles. From time to time, the organizations adapt their coordination mode based on reinforcement learning.The remainder of this paper is organized as follows: Section2 relates this paper to streams of prior research. Section 3 introduces a theoretical model of growing DDMS before in Section 4 the simulation experiments including the parameter settings are described. Section 5 reports and discusses the results obtained from the simulations, and Section 6 provides concluding remarks.


## 1.1. Complexity of the Decision-Problem

Corresponding to the seminal paper of Simon [16], the complexity of an overall decision-problem and, hence, the need for coordination is shaped by the interactions among its components: In case the overall decision-problem is (nearly) decomposable, it can be separated into (nearly) disjoint sub-problems such that intra-sub-problem linkages are stronger than the inter-sub-problem interactions. In consequence, sub-problems can be solved independent from each other without taking positive or negative interactions with respect to the overall problem’s solution into account. In contrast, if an overall decision-problem is nondecomposable, no decomposition into sub-problems can be found that (nearly) diminishes inter-sub-problem interactions (e.g., [17, 18]). Hence, if complexity in terms of interactions is high, also the need for coordination across the sub-problems is high when superior solutions to the overall decision-problem are pursued.


## 1.2. Search Strategy

The strategies for finding new solutions for decision-problems have been studied in various domains. For DDMS “resided” by human decision-makers a cornerstone was set by Simon [19] who argues that in “situations of any complexity” (pp. 104) decision-makers are unable to survey the entire search space and, hence, cannot identify the optimal solution of their decision-problem “at once”; rather, they search stepwise for superior solutions following a satisfying approach. In the domains of organizational search and innovation, the idea of stepwise search often is characterized by the extent of changes in terms of the distance of the newly found options compared to a status quo, i.e., explorative, exploitative, or ambidextrous strategies of search (e.g., [20, 21]). Constraints regarding the allowed or enforced extent of changes induced by alternative solutions are part of the boundary system in MCS: The boundary system sets behavioral constraints to decision-makers and is regarded as a prerequisite for the delegation of decision-making [22–24]. In growing organizations, the boundary system is in a particular tension between shaping the search for new opportunities and innovation on the one hand and on the other, setting behavioral constraints to deal with increasing size and intraorganizational complexity [25, 26].


## 1.3. Cost of Effort

The search strategy shapes the potential extent of changes made compared to a status quo and changes may come along with notable extra costs often termed “switching cost” or “cost of effort,” e.g., for cognitive effort or additional consumption of time or resources. For example, organizational changes may cause costs for reorganization and for handling resistance of certain stakeholders [27], choosing another supplier could result in costs for technical conversions [28, 29], changing the direction of a robot’s movement in a multirobot system could cost some extra time [30–32], and altering the task assignment in the course of job scheduling may rise some extra costs of, for example, learning—in case of humans as well as “artificial” agents [33, 34]. The level of cost of effort, therefore, affects the propensity to implement an alternative to the status quo as identified according to the search strategy.The paper employs an agent-based simulation modelto “grow” organizations from scratch and to observe which modes of coordination predominantly emerge for different settings of the aforementioned contingent factors. Simulation appears an appropriate method since it allows to capture long-term and processual phenomena that—depending on the subject—would require rather challenging longitudinal studies in empirical research (e.g., [35–37]). Simulation further helps to analyze “borderline” cases which could be hardly studied in a sufficient number empirically in order to systematically explore the effects of contingency the effects of contingency factors in interaction with each other [15]. To capture and be able to control the complexity of a DDMS’s overall decision problem, the study relies on the framework of NK fitness landscapes which was originally introduced in the domain of evolutionary biology [38, 39] and, since then, broadly employed in managerial science (for overviews, see [21, 37]). In particular, an agent-based simulation is employed in which growing organizations—with an increasing number of decentralized decision-makers—search for superior solutions to their task (specified according to the NK framework) and where the organizations may employ different search styles. From time to time, the organizations adapt their coordination mode based on reinforcement learning.The remainder of this paper is organized as follows: Section2 relates this paper to streams of prior research. Section 3 introduces a theoretical model of growing DDMS before in Section 4 the simulation experiments including the parameter settings are described. Section 5 reports and discusses the results obtained from the simulations, and Section 6 provides concluding remarks.


## 2. Related Work

The emergence of coordination is a key question in various domains and, accordingly, the related approaches in research are rather manifold. For example, complex systems science studies the emergence of “collective intelligence” (for an overview, see [40]) as apparent in linguistic conventions [41]; in economics, it is examined how individuals learn to coordinate in respect to collective objectives like in public good games (e.g., [42, 43]); prominent issues of multiagent systems are the self-adaptation of consensus mechanisms in variable networks or the convergence of consensus algorithms (for an overview, see [44]).While these streams of research clearly are of general relevance for this research endeavor, with respect to the research question posed in the Introduction, the paper particularly focuses on the emergence of coordination in DDMS which face decision-problems of increasing size by means of an agent-based simulation.Herewith, the paper also relates to the two domains of “agent-based computing” as Niazi and Hussain [45] put it: The research question is predominantly directed to the understanding of emergent phenomena in the context of complex decision-problems and, thus, it is related to the stream of research employing agent-based simulation as a research method—often termed as “social simulation” [46]; however, the research question also relates to the domain of multiagent systems (MAS) with its primary design focus [45, 47].With this, the paper particularly builds on three lines of research which are outlined subsequently: (1) coordination and, in particular, constraint-setting mechanisms via MCS in growing organizations; (2) agent-based simulation in the domain of organizations and management control; and (3) coordination of decision-making in MAS.


### 2.1. Management Control Systems in Growing Organizations

In this line of research, the empirical study of Davila [48] on the emergence of management control systems (MCS) in small growing organizations provides a cornerstone according to which organizational size has a positive impact on the overall level of use of MCS. A key argument is that when organizations grow, informal controls become too costly and/or ineffective and, hence, more formal controls for purposes of motivation and monitoring are employed as they are captured in MCS. As mentioned in the Introduction, the research endeavor of this paper addresses complexity in terms of interactions among sub-problems in growing decision-problems. However, in this respect, research findings on MCS are somewhat ambiguous. While low levels of interactions have been found to be linked to budgets, operating procedures, and statistical reports, the latter combined with informal coordination were found to be employed when complexity is high [49]. Some findings indicate that with higher levels of complexity, more emphasis is put on communication between subordinate and superior decision-makers and that aggregated and integrative information becomes more important [50]. Moreover, it was argued that with higher levels of inter-sub-problem interactions the uncertainty derived from a lack of control over the supplying decision-makers increases and this, in turn, may result in more formal controls, while, at the same time, increasing the need for flexibility resulting in an emphasis on informal controls [51]. In a recent study [24], a high level of interdependencies was found to, in tendency, be related to the establishment of vertical information flows for solving coordination problems. With respect to the search strategy (see Introduction), first of all, it is worth mentioning that MCS are intended to provide focus for the search for novel solutions, i.e., aligning them with the organization-wide strategic orientation [25]. Constraining the decision-making scope via boundary systems (as subsystems of MCS) was found to be positively associated with performance of exploitative innovations with their typically tightly coupled activities—particularly, because risk is reduced that subordinate decision-makers pursue activities which are not in line with established processes [26, 52]. In contrast, in the long run, boundary systems are argued to reduce the propensity of exploration in terms of experimentation [53].


### 2.2. Agent-Based Simulation in the Domain of Organizations and Management Control Systems

Several studies employ agent-based modeling—following the tradition of “social simulation”—in the domain of organizations [54], and, as in this paper, employ the NK-framework—with its particular capabilities of capturing the level of complexity of the decision-problem and of studying the effects of different search strategies for superior solutions at the level of the entire organization (for overviews, see [21, 37, 55, 56]) For example, the performance of different coordination mechanisms in turbulent and complex task environments is analyzed by Siggelkow and Rivkin [57]: the authors identify—for different levels of task complexity—organizational configurations which appear appropriate to cope with environments undergoing some external shock; e.g., when complexity and turbulence are at high levels, ample coordinative power and strategies fostering broad search processes turn out to perform best; however, the results also suggest that subtle interferences with distribution of decision-making authority in the organizations may affect overall performance. These results relate to prior research which indicates on the subtle interaction between complexity and the appropriate organizational design, namely, hierarchical coordination (e.g., [55, 58, 59]). Regarding the search strategy (see Introduction), not only experiential search (i.e., based on processes of local search) has been studied but also compared to, for example, forward-looking search (capturing decision-makers’ potentially imprecise beliefs about the actions-outcome-relations) [60] or “search” via imitation [61]: imitation turns out to be effective for the imitator particularly at intermediate levels of complexity while with high complexity of the decision-problem small errors in imitation can lead to severe deviations from the intended solutions and performance losses [62]. With respect to cost of effort for implementing new solutions (see Introduction), prior research suggests on subtle interactions with problem complexity: On the one hand, higher cost of effort makes changes less attractive and, hence, increase the peril of inertia, which is a particular relevant aspect in complex decision-problems; on the other hand, they may stabilize search and prevent frequent mutual (“hyperactive”) adjustments within an organization [63, 64].


### 2.3. Coordination of Decision-Making in Multiagent Systems

Coordination among agents is a key issue of multiagent systems (MAS) and, accordingly, can build on a vast body of prior research (for reviews, see [3, 6, 65–67]). With respect to the issues focused in this paper, the unpredictability of the environment, imperfect information about the environment as well as the fellow agents behavior, and imperfect communication are of particular interest: For multiagent systems, a general theoretical specification of this kind of problem is given by the framework of decentralized partially observable Markov decision processes (dec-POMDPs) [67, 68] and overviews of the multitude of techniques for coordinating distributed (or local) plans are given by [5, 13, 69]. According to the recent survey of Torreño et al. [69], the multitude of multiagent planning can be classified according to six key features: (1) agent distribution (involvement of multiple agents in formation and/or execution of plans), (2) computational processes (centralized or split among several processing units), (3) plan synthesis schemes (i.e., how and when the coordination takes place), (4) communication mechanisms (which is highly related to aspects 2 and 3), (5) heuristic search processes, and (6) privacy preservation (i.e., coordination of plans without that sensitive information becomes publicly available).Within this framework, the research question studied in this paper is mainly related to aspect 3, plan synthesis (“plan merging”): Theformation of plans is, at least, partially distributed across agents, and once each agent has formed a plan regarding its sub-problem, the emerging modes of whether, and, if so, how the local plans are synthesized into an overall plan [2, 13] are studied for different levels of the planning problem’s complexity and search strategies (aspect 5 as mentioned above).In MAS, two principle ways for plan synthesis are distinguished:unthreaded planning and coordination in terms of sequential activities [69] vs. interleaved coordination which implies an immediate integration of planning and coordination [70]. This paper focuses on unthreaded planning. A general model of coordination of plans according to unthreaded planning was introduced by Martial [2] including negotiations among agents for resolution of conflicts. In this paper, synthesizing schemes like concatenation of distributed plans according to certain rules [71] or iterative response planning (i.e., sequential information and adjustment) [72] are employed (for details, see Section 3.4).For coordination theory in MAS, recently Lesser and Corkill [14] raised four challenges which they argue based on empirical observations, so far, are reflected in research to a lesser extent. Two of these challenges reflect key aspects of the research endeavor of this paper (see Introduction). In particular, according to Lesser and Corkill [14] (in a similar vein [69])(1)
environmental conditions like, for example, predictability of task characteristics shape which coordination strategies are appropriate(2)
the dynamic adaptation of coordination seems crucial when the environment or the problem to solve is evolving dynamicallyBoth aspects correspond to key issues of this research effort (see Introduction): the first since this paper adopts a contingency perspective on coordination taking different levels of complexity, different types of search behavior, and different levels of cost of effort into account; the second since the research question posed in this paper boils down to which type of coordination emerges for evolving, in terms of growing decision-problems.


## 2.1. Management Control Systems in Growing Organizations

In this line of research, the empirical study of Davila [48] on the emergence of management control systems (MCS) in small growing organizations provides a cornerstone according to which organizational size has a positive impact on the overall level of use of MCS. A key argument is that when organizations grow, informal controls become too costly and/or ineffective and, hence, more formal controls for purposes of motivation and monitoring are employed as they are captured in MCS. As mentioned in the Introduction, the research endeavor of this paper addresses complexity in terms of interactions among sub-problems in growing decision-problems. However, in this respect, research findings on MCS are somewhat ambiguous. While low levels of interactions have been found to be linked to budgets, operating procedures, and statistical reports, the latter combined with informal coordination were found to be employed when complexity is high [49]. Some findings indicate that with higher levels of complexity, more emphasis is put on communication between subordinate and superior decision-makers and that aggregated and integrative information becomes more important [50]. Moreover, it was argued that with higher levels of inter-sub-problem interactions the uncertainty derived from a lack of control over the supplying decision-makers increases and this, in turn, may result in more formal controls, while, at the same time, increasing the need for flexibility resulting in an emphasis on informal controls [51]. In a recent study [24], a high level of interdependencies was found to, in tendency, be related to the establishment of vertical information flows for solving coordination problems. With respect to the search strategy (see Introduction), first of all, it is worth mentioning that MCS are intended to provide focus for the search for novel solutions, i.e., aligning them with the organization-wide strategic orientation [25]. Constraining the decision-making scope via boundary systems (as subsystems of MCS) was found to be positively associated with performance of exploitative innovations with their typically tightly coupled activities—particularly, because risk is reduced that subordinate decision-makers pursue activities which are not in line with established processes [26, 52]. In contrast, in the long run, boundary systems are argued to reduce the propensity of exploration in terms of experimentation [53].


## 2.2. Agent-Based Simulation in the Domain of Organizations and Management Control Systems

Several studies employ agent-based modeling—following the tradition of “social simulation”—in the domain of organizations [54], and, as in this paper, employ the NK-framework—with its particular capabilities of capturing the level of complexity of the decision-problem and of studying the effects of different search strategies for superior solutions at the level of the entire organization (for overviews, see [21, 37, 55, 56]) For example, the performance of different coordination mechanisms in turbulent and complex task environments is analyzed by Siggelkow and Rivkin [57]: the authors identify—for different levels of task complexity—organizational configurations which appear appropriate to cope with environments undergoing some external shock; e.g., when complexity and turbulence are at high levels, ample coordinative power and strategies fostering broad search processes turn out to perform best; however, the results also suggest that subtle interferences with distribution of decision-making authority in the organizations may affect overall performance. These results relate to prior research which indicates on the subtle interaction between complexity and the appropriate organizational design, namely, hierarchical coordination (e.g., [55, 58, 59]). Regarding the search strategy (see Introduction), not only experiential search (i.e., based on processes of local search) has been studied but also compared to, for example, forward-looking search (capturing decision-makers’ potentially imprecise beliefs about the actions-outcome-relations) [60] or “search” via imitation [61]: imitation turns out to be effective for the imitator particularly at intermediate levels of complexity while with high complexity of the decision-problem small errors in imitation can lead to severe deviations from the intended solutions and performance losses [62]. With respect to cost of effort for implementing new solutions (see Introduction), prior research suggests on subtle interactions with problem complexity: On the one hand, higher cost of effort makes changes less attractive and, hence, increase the peril of inertia, which is a particular relevant aspect in complex decision-problems; on the other hand, they may stabilize search and prevent frequent mutual (“hyperactive”) adjustments within an organization [63, 64].


## 2.3. Coordination of Decision-Making in Multiagent Systems

Coordination among agents is a key issue of multiagent systems (MAS) and, accordingly, can build on a vast body of prior research (for reviews, see [3, 6, 65–67]). With respect to the issues focused in this paper, the unpredictability of the environment, imperfect information about the environment as well as the fellow agents behavior, and imperfect communication are of particular interest: For multiagent systems, a general theoretical specification of this kind of problem is given by the framework of decentralized partially observable Markov decision processes (dec-POMDPs) [67, 68] and overviews of the multitude of techniques for coordinating distributed (or local) plans are given by [5, 13, 69]. According to the recent survey of Torreño et al. [69], the multitude of multiagent planning can be classified according to six key features: (1) agent distribution (involvement of multiple agents in formation and/or execution of plans), (2) computational processes (centralized or split among several processing units), (3) plan synthesis schemes (i.e., how and when the coordination takes place), (4) communication mechanisms (which is highly related to aspects 2 and 3), (5) heuristic search processes, and (6) privacy preservation (i.e., coordination of plans without that sensitive information becomes publicly available).Within this framework, the research question studied in this paper is mainly related to aspect 3, plan synthesis (“plan merging”): Theformation of plans is, at least, partially distributed across agents, and once each agent has formed a plan regarding its sub-problem, the emerging modes of whether, and, if so, how the local plans are synthesized into an overall plan [2, 13] are studied for different levels of the planning problem’s complexity and search strategies (aspect 5 as mentioned above).In MAS, two principle ways for plan synthesis are distinguished:unthreaded planning and coordination in terms of sequential activities [69] vs. interleaved coordination which implies an immediate integration of planning and coordination [70]. This paper focuses on unthreaded planning. A general model of coordination of plans according to unthreaded planning was introduced by Martial [2] including negotiations among agents for resolution of conflicts. In this paper, synthesizing schemes like concatenation of distributed plans according to certain rules [71] or iterative response planning (i.e., sequential information and adjustment) [72] are employed (for details, see Section 3.4).For coordination theory in MAS, recently Lesser and Corkill [14] raised four challenges which they argue based on empirical observations, so far, are reflected in research to a lesser extent. Two of these challenges reflect key aspects of the research endeavor of this paper (see Introduction). In particular, according to Lesser and Corkill [14] (in a similar vein [69])(1)
environmental conditions like, for example, predictability of task characteristics shape which coordination strategies are appropriate(2)
the dynamic adaptation of coordination seems crucial when the environment or the problem to solve is evolving dynamicallyBoth aspects correspond to key issues of this research effort (see Introduction): the first since this paper adopts a contingency perspective on coordination taking different levels of complexity, different types of search behavior, and different levels of cost of effort into account; the second since the research question posed in this paper boils down to which type of coordination emerges for evolving, in terms of growing decision-problems.


## 3. A Theoretical Model of Distributed Decision-Making in Growing Organizations

This part introduces a model of distributed decision-making in organizations facing a decision-problem which grows over time reflected in organizational growth accordingly. The organizations learn about appropriate modes of coordination in terms of plan synthesis (Section2.3) in the course of growth. The description of the model is structured into the following steps: first, the model of growing decision-problems is described (Section 3.1); next, Section 3.2 introduces the two types of agents “residing” in the organizations before Section 3.3, in particular, goes into details of the distributed (or local) decision-making agents including their respective sub-problems, their search strategies, and their formation of preferences. The different schemes of plan synthesis for coordination among distributed decision-makers captured in the model are described in Section 3.4 before the organizations’ learning about the appropriate mode of coordination is modeled (Section 3.5).


### 3.1. Decision-Problem Based on Growing NK Fitness Landscapes

The decision-problems to be solved by the artificial organizations are modeled according to the framework of NK-fitness landscapes. A major feature of NK fitness landscapes is that they allow to easily control for the complexity of anN-dimensional decision-problem captured by parameter K; from a “technical” perspective, NK landscapes are stochastically generated pseudo-Boolean functions with N bits, i.e., F:0,1N⟶ℝ+ [73, 74], for a brief description of NK landscapes as used in managerial science (see [75]). In line with the NK-framework, at time step t the organizations face an N-dimensional binary decision problem, i.e., dt→=d1t,…,dNt with dit∈0,1, i=1,…,N, out of 2N different binary vectors possible. Each of the two states dit∈0,1 provides a contribution Cit to the overall performance Vdt→, where the Cit are randomly drawn from a uniform distribution with 0≤Cit≤1. The parameter K (with 0≤K≤N−1) reflects the number of those choices djt, j≠i which also affect the performance contribution Cit of choice dit and, thus, captures the complexity of the decision problem to be solved in terms of the interactions among the single decisions. Hence, contribution Cit may not only depend on the single choice dit but also on K other choices:(1)Cit=fidit;di1t,…,diKt,with i1,…,iK⊂1,…,i−1,i+1,…,N. In case of no interactions among choices, K equals 0, and K is N−1 for the maximum level of complexity where each single choice i affects the performance contribution of each other binary choice j≠i. The overall performance Vt achieved in period t results as normalized sum of contributions Cit from(2)Vt=Vdt→=1N∑i=1NCit.While this, so far, captures the key elements of the “standard” NK framework, a distinguishing feature of the model presented in this study, is that—due to growth—the number of single choices to be made by an organization may increase over time, i.e.,Nt, and, with this, also the level of complexity may rise, i.e., Kt. Figure 1(b) gives an example reflecting, in the first growth stage, a rather small decision-problem with N1=6 and K1=3, in the second stage, a medium-sized decision-problem with N2=9 and K2=4 growing up to a size N3=12 and K3=5 in the third growth stage. More formally, we have(3)Nt=N1,for0≤t≤t1,⋮Ns,forts−1<t≤ts,⋮NS,fortS−1<t≤T,where s=1,…,S captures the growth stage and T denotes the entire observation period. Correspondingly, Ks gives the level of complexity in the respective periods of time. The overall performance (see equation (2)), herewith, modifies to(4)Vt=Vdt→=1Ns∑i=1NsCit.Figure 1
Interaction structures and growth stages in the simulation experiments: (a) decomposable and (b) nondecomposable.
(a)
(b)With this, the overall performance is “dynamically” normalized to the problem size Ns; moreover, in the analysis of the simulation results, the final performance VT 750 is given in relation to the global maxima of the respective performance landscapes: otherwise the results could not be compared over time and across different performance landscapes.


### 3.2. Agents and Their Capabilities

Two types of agents reside in the organizations:(1)
The distributed (or local) “decision-making agents” (type 1) which are in primary responsibility of decision-making where each of these agents is in charge for a distinct sub-problem of the overall (growing) decision-problem. These agents could, for example, reflect managers each being the head of a department in an organization.(2)
Each organization has one central agent (type 2)—capturing, for example, the headquarter of a firm—being responsible for(a)
learning-based selection of a mode of coordination in everyTL-th period(b)
eventually—i.e., subject to the coordination mode selected—actively intervening in plan synthesis(c)
registering the performance achieved by each distributed decision-making agent (see above) and eventually rewarding it accordinglyAs familiar in agent-based computational economics (e.g., [76, 77]), the agents captured in the model show some form of bounded rationality [19]. In particular, both types of agents, i.e., distributed decision-making agents and headquarter, are assumed to decide on basis of bounded information in terms of not knowing the entire search space and having limited computational power [78–80]. Hence, although distributed decision-makers are self-interested pursuing their particular goals and, in a similar vein, the central agent seeks to maximize overall performance (equation (4)), the agents are not optimizers but conduct stepwise search processes in terms of neighborhood search. Moreover, the agents are not able to perfectly evaluate newly found options to their respective decision-problems.The properties of the distributed decision-making agents (type 1) are described more precisely in Section3.3 while the central agent (type 2) is depicted more in detail in Sections 3.4 and 3.5.


### 3.3. Distributed Decision-Making Agents’ Search and Formation of Preferences

#### 3.3.1. Decomposition and Delegation

TheNs-dimensional decision problem is partitioned into Ms disjoint partial problems of, for the sake of simplicity, equal size Nsr. Each of these sub-problems is delegated to one distributed decision-maker r (i.e., one agent of type 1)—with particular competencies of decision-maker r related to the respective sub-problem being subject to the mode of coordination implemented at that time. However, the distributed decision-makers are, at least, preparing choices regarding their partition of the overall Ns-dimensional decision-problem in each period t (i.e., depicting “unthreaded planning” from MAS; see Section 2.3).The growth processes in the problem space are reflected in the number of distributed decision-makers (type 1), accordingly. More formally, we have the number of distributed decision-makers being a function of time, i.e.,Mst and, at each given growth stage s, the scope of competencies of the distributed decision-makers are of equal size in terms of number of single choices Nsr assigned to them (see, for example, the shaded areas in Figure 1).


#### 3.3.2. Search Strategies

In the model, as mentioned above, distributed decision-makers are not able to survey the entire search space of their decision-problem and, hence, they are not able to “locate” the optimal solution of their partial decision problem “at once”; rather, they search stepwise for superior solutions. In each time stept distributed decision-maker r considers two alternative solutions dtr,a1→ and dtr,a2→ for its partial decision-problem compared to the status quo dt−1r∗→. With this, in each of the search strategies modeled and in every time step, decision-maker r has three options to choose from—including staying with the status quo.However, the model captures that—as a part of the boundary system in the MCS—some boundaries related to thedistances of the two alternatives dtr,a1→ and dtr,a2→ compared to the status quo are set and enforced by the central unit (type 2 agent). In particular, the boundary system enforces search strategies being of an exploitative, explorative, or ambidextrous nature [20, 58, 81] and, accordingly, the model comprises the following three search strategies:(i)
“exploitation only”: both alternatives dtr,a1→ and dtr,a2→ discovered by distributed decision-maker r differ from the status quo dt−1r∗→ by one digit, respectively. Hence, the Hamming distance hdr,a1→=∑i=1Nrdt−1r∗→−dtr,a1→ of the first alternative to the status quo equals 1 as well as is the case for the second alternative (i.e., hdr,a2→=1).(ii)
“exploitation and exploration”: the local decision-makers are allowed to alternatively consider an option with one bit flipped and another where two digits are altered, i.e., the Hamming distances are hdr,a1→=1 and hdr,a2→=2.(iii)
“exploration only”: in each of both alternatives dtr,a1→ and dtr,a2→, two bits are flipped compared to the status quo dt−1r∗→, i.e., we have hdr,a1→=2andhdr,a2→=2.


#### 3.3.3. Distributed Decision-Makers’ Objectives and Cost of Effort

Each local decision-makerr, r=1,…,Ms pursues its “own” objective (i.e., related to its respective sub-problem). In particular, each distributed decision-maker seeks to identify that option dtr→ out of dt−1r∗→, dtr,a1→, and dtr,a2→ which promises the highest net performance Ar, i.e., the highest difference between partial performance Ptr—resulting from the contributions Cit of those particular single choices assigned to decision-maker r—and the related cost of efforts for implementing dtr→:(5)Atrdtr→=Ptrdtr→−Ztrdtr→,where the partial performance Ptrdtr→ of decision-maker r’s contribution to overall performance Vt (see equation (4)) is given by(6)Ptrdtr→=1N∑i=1+wNsrCit,with w=∑p=1r−1Nsp for r>1 and w=0 for r=1.In case of interactions across the sub-problems, captured byKsex>0, choices of decision-maker r might affect the contribution of the other decision-makers’ q≠p choices on q’s parochial performance and vice versa.In the model, the number of single choices altered in terms of the Hamming distancehdt−1r∗→,dtr→ to the status quo (i.e., the number of bits flipped) is regarded as the effort to be taken by decision-maker r in order to implement an option. With this, the possible range of effort is shaped by the search strategy as introduced in Section 3.3.2: the lower bound for the effort in all three search strategies equals 0 for the case that the status quo dt−1r∗→ is chosen to be kept; the upper bound is 1 in the “exploitation only” and equals 2 in the “exploration only” as well as in the ambidextrous strategy.For modeling the cost of effort, as customary in economics (e.g., [82, 83]), it is assumed that higher levels of effort are increasingly costly. Hence, for distributed decision-maker r’s cost of effort Zr we have Zrh′>0 and Zrh″>0. In particular, the cost of effort Zrdtr→ of decision-maker r is modeled to be quadratically increasing with the Hamming distance h to the status quo, i.e.,(7)Ztrdtr→=zr⋅hdt−1r∗→,dtr→2,where zr is a cost coefficient. For the sake of simplicity, in the simulation experiments the cost coefficient is the same for every decision-makers r and does not change over time. For this reason, subsequently, the index r indicating on a decision-maker r is skipped when addressing the cost coefficient (considering decision-makers which are heterogeneous in respect to their cost functions (as it could be captured by different cost coefficients zr) would be a natural extension of the research effort presented here).


#### 3.3.4. Evaluation of Options

When evaluating their options, the decentralized decision-makers suffer from two types of limited information: First, without eventual further communication prescribed by the coordination mechanism (see Section3.4), decision-maker r is not able to anticipate the other decision-makers’ q≠r choices and, thus, assumes that they will stay with the status quo, i.e., that they will opt for dt−1q∗→ (for this see also [84]). This is particularly relevant in case of inter-sub-problem interdependencies when other local decision-makers’ actions may affect performance of decision-maker r.Second, decentralized decision-makers are not able to perfectly ex-ante evaluate their newly discovered options’dtr,a1→ and dtr,a2→ effects on their partial performance Ptrdtr→ (see equation (6)). Rather the ex-ante evaluation is afflicted with some noise. In particular, for the sake of simplicity, a decision-maker r’s perceived partial performance Pr˜ is distorted by a relative error imputed to the true performance. With this, our local decision-makers search on noisy partial performance landscapes (for further types of errors, see [85]). The error terms follow a Gaussian distribution N0;σ with expected value 0 and standard deviations σr for every decision-maker r; errors are assumed to be independent from each other. Hence, the perceived performance Ptr˜dt→ of distributed decision-maker r is given by(8)Ptr˜dtr→=Ptrdtr→+erdtr→,and the objective function in equation (5) modifies to(9)Atr˜dtr→=Ptr˜dtr→−Ztrdtr→.With equation (8), each distributed decision-maker r has a distinct partial and distorted “view” of the true fitness landscape: each decision-maker r is exclusively in charge of an “own” part of the entire N-dimensional decision-problem as it is shaped in the current growth stage and imperfectly estimates the performance contributions of newly discovered options. Thus, at a given growth stage, the model captures Ms+1 heterogeneous (distorted) views of the true landscape, i.e., one view per local decision-maker r=1,…,Ms plus a headquarters’ perspective (for the latter, see Section 3.4.3).However, for the status quo option, we assume that each decision-makerr remembers the performance obtained from the last period and, with this, knows the actual performance Pt−1r of the status quo should it be implemented in period t again.Based on the evaluation of options, each decision-makerr compiles a list Ltr=dtr,p1→,dtr,p2→,dtr,p3→ of preferences where dtr,p1→ indicates the most preferred option out of dt−1r∗→,dtr,a1→, and dtr,a2→. Correspondingly, dtr,p2→ denotes the second-most and dtr,p3→ the third-most preferred option.


### 3.4. Coordination Modes for Synthesizing Plans

As a result of the search and (imperfect) evaluation of options, so far each distributed decision-maker has an ordered listLtr of preferences which, in terms of multiagent systems, reflects the individual [2] or local plan [69]. The next step within each period t is to come to a decision on the organization’s overall problem dt→, i.e., to merge the decentralized decision-makers’ preferences on dtr→ captured in the Ms lists Ltr into the overall configuration dt→. This requires to employ a mode of coordination for synthesizing the local plans—and the very core of this paper is to study which mode emerges under which conditions of search strategy and cost of effort.As it was outlined in Section2, a multitude of modes for synthesizing local plans has been proposed in various domains. Out of the numerous feasible mechanisms, the model comprises three modes which could represent particular pronounced (for not to say “extreme”) forms of coordination regarding(i)
(de-)centralization in terms of (locus of) authority(ii)
direction of alignment, i.e., lateral vs. vertical(iii)
parallelization vs. sequencing of final decision regarding the local plansIn particular, the three types of coordination captured in the model (see Table1) differ in the communication channels (the model assumes that communication in the course of coordination works perfectly; for an investigation of the effects of unintended communications errors, see [86]), the information employed in decision-making, the locus of final decision-making, and, in the end, the tightness of coordination provided (for an overview and further modes, see, e.g., [2, 57, 69, 87, 88]).Table 1
Overview of the modes of coordination in the model.
Type of coordination mode
Decentralized
Sequential
Proposal
Lateral communication
No
Yes
No
Vertical communication
No
No
Yes
Headquarter intervening
No
No
Yes
Re-evaluation of local plans
No
Yes
Yes
Information employed for final decisions
Expected parochial performance
Expected parochial performance taking predecessor’s choices into account
Expected organization-wide aggregated performance
Locus of final decisions on local plans
Decentralized
Decentralized
Centralized
Temporal order of final decisions on local plans
In parallel
Asynchronously (in sequence)
Synchronously (“at once”)
Final configurations for distributed decision-makers’ sub-problems
First preferencesdtr,p1→∀r∈1,…,Ms
First preferences subject to preceding decision-makers’ choices, i.e.,dtr,p1→dtr−1,p1→ for r>1; dtr,p1→ for r=1
First (or second) preferencesdtr,p1→ [or dtr,p2→] if V˜dC→≥Vdt−1→


#### 3.4.1. Decentralized Mode

The highest level of autonomy is granted to theMs distributed decision-makers if each of them is allowed to choose its most preferred option. Then, the overall organizational configuration dt→ results from(10)dt→=dt1,p1→,…,dtr,p1→,…,dtMs,p1→.Hence, for decision-makerr’s partial decision problem Nsr the option according to(11)dtr∗→=dtr,p1→∀r∈1,…Ms,is implemented. Obviously, this type of coordination does not require any form of communication between the decentralized decision-makers or with the headquarter (central agent). The headquarter does not intervene in decision-making directly and its role is limited to registering the achieved performances Vtdt→ and Ptrdtr→ in the end of each period t and, eventually, to reward the decentralized decision-makers accordingly.


#### 3.4.2. Sequential Mode

This mode captures the idea of “sequential planning”: The distributed decision-makers make their final choices sequentially with, for the sake of simplicity, the sequence being given by the indexr of the decision-makers. In particular, in time step t decision-maker r−1 informs decision-maker r with 1<r≤Ms about the choices made so far, i.e., made by the “preceding” decision-makers <r. Decision-maker r>1 re-evaluates its “own” options dt−1r∗→, dtr,a1→, and dtr,a2→ taking these “prior” choices into account—potentially resulting in an adjusted list of preferences Ltr=dtr,p1→;dtr,p2→;dtr,p3→ and chooses dtr,p1→ which here depends on dtr−1,p1→, i.e., is a function of the choices of the preceding decision-makers. Hence, only decision-maker 1 does not have to consider previous choices and the choice of decision-maker r=1,…,Ms is made according to(12)dtr∗→=dtr,p1→,ifr=1,dtr,p1→dtr−1,p1→,if1<r≤Μ,and with equation (12) the overall configuration is given by dt→=dtM∗→. The headquarter (type 2 agent) does not intervene in decision-making and—similar to the decentralized mode—is confined to observing the performances achieved and, eventually, rewarding the decentralized decision-makers accordingly.


#### 3.4.3. Proposal Mode

Each local decision-maker transfers its listLtr of preferences to the headquarter which compiles the first preferences to a composite vector dC→=dt1,p1→,…,dtr,p1→,…,dtMs,p1→, and then evaluates the overall performance VdC→ (see equation (4)) that this solution promises.However, as with decentralized decision-makers, in the model it is assumed that the headquarter is not capable to perfectly ex-ante evaluate new options, i.e., other solutions than the status quo (see Section3.2). Rather, similar to the local decision-makers, the headquarter suffers from some relative noise following a Gaussian distribution with expected value 0 and standard deviations σcent resulting in a perceived overall performance V˜dC→. The headquarter decides in favor of the composite vector, i.e., dt→=dC→, if dC→ promises the same or a higher performance as the status quo dt−1→, i.e., if(13)V˜dC→≥Vdt−1∗→.If the composite vectordC→ assembled from the local decision-makers’ first preferences does not meet the condition in equation (13), the headquarter evaluates a vector composed from the decentralized decision-makers’ second preferences according to equation (13). If this also does not, at least, promise the performance of the status quo, then the organization stays with the status quo, i.e., then dt→=dt−1→.


### 3.5. Learning-Based Adaptation of Coordination

The very core of this research endeavor is to study which modes of coordination emerge within growth processes of the artificial organizations. In order to capture some kind of self-adaptation of coordination, the model employs a simple mode of reinforcement learning (for overviews, see [89, 90]) based on statistical learning, i.e., a generalized form of the Bush–Mosteller model [91, 92].This mode of learning is chosen for the following reasons: In the vast multitude of forms of learning studied in various domains (e.g., psychology, economics, and computer science), reinforcement learning is regarded to be among the most basic forms of learning [92]. It represents a fundamental possible form of humans’ behavior but also provides some basis for learning of artificial agents [90]—both regarded as DDMS in this paper. Moreover, in this model, as a form of experiential learning the mid-termed reinforcement learning on the coordination mode corresponds to the experiential type of short-termed adaptation [60] when the decision-making systems, in every single time-step, search for a superior solution compared to the status quo. However, it would be a natural extension of the model to let the DDMS employ other and, in particular, more advanced forms of learning about which mode of coordination is appropriate within the stage of growth and for the search strategy employed and the cost of effort.In the model, reinforcement learning on coordination for synthesizing local plans is represented in the following way: In the end of each time stept, the central agent (type 2 agent; see Section 3.2) receives information about the overall performance Vt according to equation (4). This allows the central agent, in every TLth period, to compute the performance enhancements achieved within the last TL periods. Moreover, in every TLth period, the organizations can alter the type of coordination mode as introduced in Section 3.4. Hence, our DDMS face the mid-termed decision-problem which type of coordination mode act∈Ac (with Ac=3; see Section 3.4) to implement in the next TL periods. Let pac,t denote the probability of an alternative act to be chosen at time t (with 0≤pac,t≤1 and ∑aac∈Acpac,t=1).The key idea of reinforcement learning is that the probabilities of options are updated according to the positive or negative stimuli resulting from these options. In our context, whether the performance enhancementΔVt obtained under the regime of a certain mode of coordination act in the previous TL periods is regarded positive or negative, depends on whether, or not, it at least equals an aspiration level v. ΔVt of type act of coordination is defined as the relative performance enhancement achieved within the last TL periods of the adaptive walk, i.e.,(14)ΔVt=Vt−Vt−TLVt−TL.Hence, the stimulusτt is(15)τt=1,ifΔVt≥v,−1,ifΔVt<v.The probabilities of optionsac∈Ac are updated according to the following rule, with λ (where 0≤λ≤1) giving the reinforcement strength [92]:(16)pac,t+1=pac,t+λ⋅1−pac,t,ifac=act∧τt=1,−pac,t,ifac=act∧τt=−1,−pac,t,ifac≠act∧τt=1,pac,t⋅pact,t1−pact,t,ifac≠act∧τt=−1.After the probabilities are updated as given in equation (16) the “next” mode of coordination to be employed from t+1 to t+TL is determined at random—according to the updated probabilities.


## 3.1. Decision-Problem Based on Growing NK Fitness Landscapes

The decision-problems to be solved by the artificial organizations are modeled according to the framework of NK-fitness landscapes. A major feature of NK fitness landscapes is that they allow to easily control for the complexity of anN-dimensional decision-problem captured by parameter K; from a “technical” perspective, NK landscapes are stochastically generated pseudo-Boolean functions with N bits, i.e., F:0,1N⟶ℝ+ [73, 74], for a brief description of NK landscapes as used in managerial science (see [75]). In line with the NK-framework, at time step t the organizations face an N-dimensional binary decision problem, i.e., dt→=d1t,…,dNt with dit∈0,1, i=1,…,N, out of 2N different binary vectors possible. Each of the two states dit∈0,1 provides a contribution Cit to the overall performance Vdt→, where the Cit are randomly drawn from a uniform distribution with 0≤Cit≤1. The parameter K (with 0≤K≤N−1) reflects the number of those choices djt, j≠i which also affect the performance contribution Cit of choice dit and, thus, captures the complexity of the decision problem to be solved in terms of the interactions among the single decisions. Hence, contribution Cit may not only depend on the single choice dit but also on K other choices:(1)Cit=fidit;di1t,…,diKt,with i1,…,iK⊂1,…,i−1,i+1,…,N. In case of no interactions among choices, K equals 0, and K is N−1 for the maximum level of complexity where each single choice i affects the performance contribution of each other binary choice j≠i. The overall performance Vt achieved in period t results as normalized sum of contributions Cit from(2)Vt=Vdt→=1N∑i=1NCit.While this, so far, captures the key elements of the “standard” NK framework, a distinguishing feature of the model presented in this study, is that—due to growth—the number of single choices to be made by an organization may increase over time, i.e.,Nt, and, with this, also the level of complexity may rise, i.e., Kt. Figure 1(b) gives an example reflecting, in the first growth stage, a rather small decision-problem with N1=6 and K1=3, in the second stage, a medium-sized decision-problem with N2=9 and K2=4 growing up to a size N3=12 and K3=5 in the third growth stage. More formally, we have(3)Nt=N1,for0≤t≤t1,⋮Ns,forts−1<t≤ts,⋮NS,fortS−1<t≤T,where s=1,…,S captures the growth stage and T denotes the entire observation period. Correspondingly, Ks gives the level of complexity in the respective periods of time. The overall performance (see equation (2)), herewith, modifies to(4)Vt=Vdt→=1Ns∑i=1NsCit.Figure 1
Interaction structures and growth stages in the simulation experiments: (a) decomposable and (b) nondecomposable.
(a)
(b)With this, the overall performance is “dynamically” normalized to the problem size Ns; moreover, in the analysis of the simulation results, the final performance VT 750 is given in relation to the global maxima of the respective performance landscapes: otherwise the results could not be compared over time and across different performance landscapes.


## 3.2. Agents and Their Capabilities

Two types of agents reside in the organizations:(1)
The distributed (or local) “decision-making agents” (type 1) which are in primary responsibility of decision-making where each of these agents is in charge for a distinct sub-problem of the overall (growing) decision-problem. These agents could, for example, reflect managers each being the head of a department in an organization.(2)
Each organization has one central agent (type 2)—capturing, for example, the headquarter of a firm—being responsible for(a)
learning-based selection of a mode of coordination in everyTL-th period(b)
eventually—i.e., subject to the coordination mode selected—actively intervening in plan synthesis(c)
registering the performance achieved by each distributed decision-making agent (see above) and eventually rewarding it accordinglyAs familiar in agent-based computational economics (e.g., [76, 77]), the agents captured in the model show some form of bounded rationality [19]. In particular, both types of agents, i.e., distributed decision-making agents and headquarter, are assumed to decide on basis of bounded information in terms of not knowing the entire search space and having limited computational power [78–80]. Hence, although distributed decision-makers are self-interested pursuing their particular goals and, in a similar vein, the central agent seeks to maximize overall performance (equation (4)), the agents are not optimizers but conduct stepwise search processes in terms of neighborhood search. Moreover, the agents are not able to perfectly evaluate newly found options to their respective decision-problems.The properties of the distributed decision-making agents (type 1) are described more precisely in Section3.3 while the central agent (type 2) is depicted more in detail in Sections 3.4 and 3.5.


## 3.3. Distributed Decision-Making Agents’ Search and Formation of Preferences

### 3.3.1. Decomposition and Delegation

TheNs-dimensional decision problem is partitioned into Ms disjoint partial problems of, for the sake of simplicity, equal size Nsr. Each of these sub-problems is delegated to one distributed decision-maker r (i.e., one agent of type 1)—with particular competencies of decision-maker r related to the respective sub-problem being subject to the mode of coordination implemented at that time. However, the distributed decision-makers are, at least, preparing choices regarding their partition of the overall Ns-dimensional decision-problem in each period t (i.e., depicting “unthreaded planning” from MAS; see Section 2.3).The growth processes in the problem space are reflected in the number of distributed decision-makers (type 1), accordingly. More formally, we have the number of distributed decision-makers being a function of time, i.e.,Mst and, at each given growth stage s, the scope of competencies of the distributed decision-makers are of equal size in terms of number of single choices Nsr assigned to them (see, for example, the shaded areas in Figure 1).


### 3.3.2. Search Strategies

In the model, as mentioned above, distributed decision-makers are not able to survey the entire search space of their decision-problem and, hence, they are not able to “locate” the optimal solution of their partial decision problem “at once”; rather, they search stepwise for superior solutions. In each time stept distributed decision-maker r considers two alternative solutions dtr,a1→ and dtr,a2→ for its partial decision-problem compared to the status quo dt−1r∗→. With this, in each of the search strategies modeled and in every time step, decision-maker r has three options to choose from—including staying with the status quo.However, the model captures that—as a part of the boundary system in the MCS—some boundaries related to thedistances of the two alternatives dtr,a1→ and dtr,a2→ compared to the status quo are set and enforced by the central unit (type 2 agent). In particular, the boundary system enforces search strategies being of an exploitative, explorative, or ambidextrous nature [20, 58, 81] and, accordingly, the model comprises the following three search strategies:(i)
“exploitation only”: both alternatives dtr,a1→ and dtr,a2→ discovered by distributed decision-maker r differ from the status quo dt−1r∗→ by one digit, respectively. Hence, the Hamming distance hdr,a1→=∑i=1Nrdt−1r∗→−dtr,a1→ of the first alternative to the status quo equals 1 as well as is the case for the second alternative (i.e., hdr,a2→=1).(ii)
“exploitation and exploration”: the local decision-makers are allowed to alternatively consider an option with one bit flipped and another where two digits are altered, i.e., the Hamming distances are hdr,a1→=1 and hdr,a2→=2.(iii)
“exploration only”: in each of both alternatives dtr,a1→ and dtr,a2→, two bits are flipped compared to the status quo dt−1r∗→, i.e., we have hdr,a1→=2andhdr,a2→=2.


### 3.3.3. Distributed Decision-Makers’ Objectives and Cost of Effort

Each local decision-makerr, r=1,…,Ms pursues its “own” objective (i.e., related to its respective sub-problem). In particular, each distributed decision-maker seeks to identify that option dtr→ out of dt−1r∗→, dtr,a1→, and dtr,a2→ which promises the highest net performance Ar, i.e., the highest difference between partial performance Ptr—resulting from the contributions Cit of those particular single choices assigned to decision-maker r—and the related cost of efforts for implementing dtr→:(5)Atrdtr→=Ptrdtr→−Ztrdtr→,where the partial performance Ptrdtr→ of decision-maker r’s contribution to overall performance Vt (see equation (4)) is given by(6)Ptrdtr→=1N∑i=1+wNsrCit,with w=∑p=1r−1Nsp for r>1 and w=0 for r=1.In case of interactions across the sub-problems, captured byKsex>0, choices of decision-maker r might affect the contribution of the other decision-makers’ q≠p choices on q’s parochial performance and vice versa.In the model, the number of single choices altered in terms of the Hamming distancehdt−1r∗→,dtr→ to the status quo (i.e., the number of bits flipped) is regarded as the effort to be taken by decision-maker r in order to implement an option. With this, the possible range of effort is shaped by the search strategy as introduced in Section 3.3.2: the lower bound for the effort in all three search strategies equals 0 for the case that the status quo dt−1r∗→ is chosen to be kept; the upper bound is 1 in the “exploitation only” and equals 2 in the “exploration only” as well as in the ambidextrous strategy.For modeling the cost of effort, as customary in economics (e.g., [82, 83]), it is assumed that higher levels of effort are increasingly costly. Hence, for distributed decision-maker r’s cost of effort Zr we have Zrh′>0 and Zrh″>0. In particular, the cost of effort Zrdtr→ of decision-maker r is modeled to be quadratically increasing with the Hamming distance h to the status quo, i.e.,(7)Ztrdtr→=zr⋅hdt−1r∗→,dtr→2,where zr is a cost coefficient. For the sake of simplicity, in the simulation experiments the cost coefficient is the same for every decision-makers r and does not change over time. For this reason, subsequently, the index r indicating on a decision-maker r is skipped when addressing the cost coefficient (considering decision-makers which are heterogeneous in respect to their cost functions (as it could be captured by different cost coefficients zr) would be a natural extension of the research effort presented here).


### 3.3.4. Evaluation of Options

When evaluating their options, the decentralized decision-makers suffer from two types of limited information: First, without eventual further communication prescribed by the coordination mechanism (see Section3.4), decision-maker r is not able to anticipate the other decision-makers’ q≠r choices and, thus, assumes that they will stay with the status quo, i.e., that they will opt for dt−1q∗→ (for this see also [84]). This is particularly relevant in case of inter-sub-problem interdependencies when other local decision-makers’ actions may affect performance of decision-maker r.Second, decentralized decision-makers are not able to perfectly ex-ante evaluate their newly discovered options’dtr,a1→ and dtr,a2→ effects on their partial performance Ptrdtr→ (see equation (6)). Rather the ex-ante evaluation is afflicted with some noise. In particular, for the sake of simplicity, a decision-maker r’s perceived partial performance Pr˜ is distorted by a relative error imputed to the true performance. With this, our local decision-makers search on noisy partial performance landscapes (for further types of errors, see [85]). The error terms follow a Gaussian distribution N0;σ with expected value 0 and standard deviations σr for every decision-maker r; errors are assumed to be independent from each other. Hence, the perceived performance Ptr˜dt→ of distributed decision-maker r is given by(8)Ptr˜dtr→=Ptrdtr→+erdtr→,and the objective function in equation (5) modifies to(9)Atr˜dtr→=Ptr˜dtr→−Ztrdtr→.With equation (8), each distributed decision-maker r has a distinct partial and distorted “view” of the true fitness landscape: each decision-maker r is exclusively in charge of an “own” part of the entire N-dimensional decision-problem as it is shaped in the current growth stage and imperfectly estimates the performance contributions of newly discovered options. Thus, at a given growth stage, the model captures Ms+1 heterogeneous (distorted) views of the true landscape, i.e., one view per local decision-maker r=1,…,Ms plus a headquarters’ perspective (for the latter, see Section 3.4.3).However, for the status quo option, we assume that each decision-makerr remembers the performance obtained from the last period and, with this, knows the actual performance Pt−1r of the status quo should it be implemented in period t again.Based on the evaluation of options, each decision-makerr compiles a list Ltr=dtr,p1→,dtr,p2→,dtr,p3→ of preferences where dtr,p1→ indicates the most preferred option out of dt−1r∗→,dtr,a1→, and dtr,a2→. Correspondingly, dtr,p2→ denotes the second-most and dtr,p3→ the third-most preferred option.


## 3.3.1. Decomposition and Delegation

TheNs-dimensional decision problem is partitioned into Ms disjoint partial problems of, for the sake of simplicity, equal size Nsr. Each of these sub-problems is delegated to one distributed decision-maker r (i.e., one agent of type 1)—with particular competencies of decision-maker r related to the respective sub-problem being subject to the mode of coordination implemented at that time. However, the distributed decision-makers are, at least, preparing choices regarding their partition of the overall Ns-dimensional decision-problem in each period t (i.e., depicting “unthreaded planning” from MAS; see Section 2.3).The growth processes in the problem space are reflected in the number of distributed decision-makers (type 1), accordingly. More formally, we have the number of distributed decision-makers being a function of time, i.e.,Mst and, at each given growth stage s, the scope of competencies of the distributed decision-makers are of equal size in terms of number of single choices Nsr assigned to them (see, for example, the shaded areas in Figure 1).


## 3.3.2. Search Strategies

In the model, as mentioned above, distributed decision-makers are not able to survey the entire search space of their decision-problem and, hence, they are not able to “locate” the optimal solution of their partial decision problem “at once”; rather, they search stepwise for superior solutions. In each time stept distributed decision-maker r considers two alternative solutions dtr,a1→ and dtr,a2→ for its partial decision-problem compared to the status quo dt−1r∗→. With this, in each of the search strategies modeled and in every time step, decision-maker r has three options to choose from—including staying with the status quo.However, the model captures that—as a part of the boundary system in the MCS—some boundaries related to thedistances of the two alternatives dtr,a1→ and dtr,a2→ compared to the status quo are set and enforced by the central unit (type 2 agent). In particular, the boundary system enforces search strategies being of an exploitative, explorative, or ambidextrous nature [20, 58, 81] and, accordingly, the model comprises the following three search strategies:(i)
“exploitation only”: both alternatives dtr,a1→ and dtr,a2→ discovered by distributed decision-maker r differ from the status quo dt−1r∗→ by one digit, respectively. Hence, the Hamming distance hdr,a1→=∑i=1Nrdt−1r∗→−dtr,a1→ of the first alternative to the status quo equals 1 as well as is the case for the second alternative (i.e., hdr,a2→=1).(ii)
“exploitation and exploration”: the local decision-makers are allowed to alternatively consider an option with one bit flipped and another where two digits are altered, i.e., the Hamming distances are hdr,a1→=1 and hdr,a2→=2.(iii)
“exploration only”: in each of both alternatives dtr,a1→ and dtr,a2→, two bits are flipped compared to the status quo dt−1r∗→, i.e., we have hdr,a1→=2andhdr,a2→=2.


## 3.3.3. Distributed Decision-Makers’ Objectives and Cost of Effort

Each local decision-makerr, r=1,…,Ms pursues its “own” objective (i.e., related to its respective sub-problem). In particular, each distributed decision-maker seeks to identify that option dtr→ out of dt−1r∗→, dtr,a1→, and dtr,a2→ which promises the highest net performance Ar, i.e., the highest difference between partial performance Ptr—resulting from the contributions Cit of those particular single choices assigned to decision-maker r—and the related cost of efforts for implementing dtr→:(5)Atrdtr→=Ptrdtr→−Ztrdtr→,where the partial performance Ptrdtr→ of decision-maker r’s contribution to overall performance Vt (see equation (4)) is given by(6)Ptrdtr→=1N∑i=1+wNsrCit,with w=∑p=1r−1Nsp for r>1 and w=0 for r=1.In case of interactions across the sub-problems, captured byKsex>0, choices of decision-maker r might affect the contribution of the other decision-makers’ q≠p choices on q’s parochial performance and vice versa.In the model, the number of single choices altered in terms of the Hamming distancehdt−1r∗→,dtr→ to the status quo (i.e., the number of bits flipped) is regarded as the effort to be taken by decision-maker r in order to implement an option. With this, the possible range of effort is shaped by the search strategy as introduced in Section 3.3.2: the lower bound for the effort in all three search strategies equals 0 for the case that the status quo dt−1r∗→ is chosen to be kept; the upper bound is 1 in the “exploitation only” and equals 2 in the “exploration only” as well as in the ambidextrous strategy.For modeling the cost of effort, as customary in economics (e.g., [82, 83]), it is assumed that higher levels of effort are increasingly costly. Hence, for distributed decision-maker r’s cost of effort Zr we have Zrh′>0 and Zrh″>0. In particular, the cost of effort Zrdtr→ of decision-maker r is modeled to be quadratically increasing with the Hamming distance h to the status quo, i.e.,(7)Ztrdtr→=zr⋅hdt−1r∗→,dtr→2,where zr is a cost coefficient. For the sake of simplicity, in the simulation experiments the cost coefficient is the same for every decision-makers r and does not change over time. For this reason, subsequently, the index r indicating on a decision-maker r is skipped when addressing the cost coefficient (considering decision-makers which are heterogeneous in respect to their cost functions (as it could be captured by different cost coefficients zr) would be a natural extension of the research effort presented here).


## 3.3.4. Evaluation of Options

When evaluating their options, the decentralized decision-makers suffer from two types of limited information: First, without eventual further communication prescribed by the coordination mechanism (see Section3.4), decision-maker r is not able to anticipate the other decision-makers’ q≠r choices and, thus, assumes that they will stay with the status quo, i.e., that they will opt for dt−1q∗→ (for this see also [84]). This is particularly relevant in case of inter-sub-problem interdependencies when other local decision-makers’ actions may affect performance of decision-maker r.Second, decentralized decision-makers are not able to perfectly ex-ante evaluate their newly discovered options’dtr,a1→ and dtr,a2→ effects on their partial performance Ptrdtr→ (see equation (6)). Rather the ex-ante evaluation is afflicted with some noise. In particular, for the sake of simplicity, a decision-maker r’s perceived partial performance Pr˜ is distorted by a relative error imputed to the true performance. With this, our local decision-makers search on noisy partial performance landscapes (for further types of errors, see [85]). The error terms follow a Gaussian distribution N0;σ with expected value 0 and standard deviations σr for every decision-maker r; errors are assumed to be independent from each other. Hence, the perceived performance Ptr˜dt→ of distributed decision-maker r is given by(8)Ptr˜dtr→=Ptrdtr→+erdtr→,and the objective function in equation (5) modifies to(9)Atr˜dtr→=Ptr˜dtr→−Ztrdtr→.With equation (8), each distributed decision-maker r has a distinct partial and distorted “view” of the true fitness landscape: each decision-maker r is exclusively in charge of an “own” part of the entire N-dimensional decision-problem as it is shaped in the current growth stage and imperfectly estimates the performance contributions of newly discovered options. Thus, at a given growth stage, the model captures Ms+1 heterogeneous (distorted) views of the true landscape, i.e., one view per local decision-maker r=1,…,Ms plus a headquarters’ perspective (for the latter, see Section 3.4.3).However, for the status quo option, we assume that each decision-makerr remembers the performance obtained from the last period and, with this, knows the actual performance Pt−1r of the status quo should it be implemented in period t again.Based on the evaluation of options, each decision-makerr compiles a list Ltr=dtr,p1→,dtr,p2→,dtr,p3→ of preferences where dtr,p1→ indicates the most preferred option out of dt−1r∗→,dtr,a1→, and dtr,a2→. Correspondingly, dtr,p2→ denotes the second-most and dtr,p3→ the third-most preferred option.


## 3.4. Coordination Modes for Synthesizing Plans

As a result of the search and (imperfect) evaluation of options, so far each distributed decision-maker has an ordered listLtr of preferences which, in terms of multiagent systems, reflects the individual [2] or local plan [69]. The next step within each period t is to come to a decision on the organization’s overall problem dt→, i.e., to merge the decentralized decision-makers’ preferences on dtr→ captured in the Ms lists Ltr into the overall configuration dt→. This requires to employ a mode of coordination for synthesizing the local plans—and the very core of this paper is to study which mode emerges under which conditions of search strategy and cost of effort.As it was outlined in Section2, a multitude of modes for synthesizing local plans has been proposed in various domains. Out of the numerous feasible mechanisms, the model comprises three modes which could represent particular pronounced (for not to say “extreme”) forms of coordination regarding(i)
(de-)centralization in terms of (locus of) authority(ii)
direction of alignment, i.e., lateral vs. vertical(iii)
parallelization vs. sequencing of final decision regarding the local plansIn particular, the three types of coordination captured in the model (see Table1) differ in the communication channels (the model assumes that communication in the course of coordination works perfectly; for an investigation of the effects of unintended communications errors, see [86]), the information employed in decision-making, the locus of final decision-making, and, in the end, the tightness of coordination provided (for an overview and further modes, see, e.g., [2, 57, 69, 87, 88]).Table 1
Overview of the modes of coordination in the model.
Type of coordination mode
Decentralized
Sequential
Proposal
Lateral communication
No
Yes
No
Vertical communication
No
No
Yes
Headquarter intervening
No
No
Yes
Re-evaluation of local plans
No
Yes
Yes
Information employed for final decisions
Expected parochial performance
Expected parochial performance taking predecessor’s choices into account
Expected organization-wide aggregated performance
Locus of final decisions on local plans
Decentralized
Decentralized
Centralized
Temporal order of final decisions on local plans
In parallel
Asynchronously (in sequence)
Synchronously (“at once”)
Final configurations for distributed decision-makers’ sub-problems
First preferencesdtr,p1→∀r∈1,…,Ms
First preferences subject to preceding decision-makers’ choices, i.e.,dtr,p1→dtr−1,p1→ for r>1; dtr,p1→ for r=1
First (or second) preferencesdtr,p1→ [or dtr,p2→] if V˜dC→≥Vdt−1→


### 3.4.1. Decentralized Mode

The highest level of autonomy is granted to theMs distributed decision-makers if each of them is allowed to choose its most preferred option. Then, the overall organizational configuration dt→ results from(10)dt→=dt1,p1→,…,dtr,p1→,…,dtMs,p1→.Hence, for decision-makerr’s partial decision problem Nsr the option according to(11)dtr∗→=dtr,p1→∀r∈1,…Ms,is implemented. Obviously, this type of coordination does not require any form of communication between the decentralized decision-makers or with the headquarter (central agent). The headquarter does not intervene in decision-making directly and its role is limited to registering the achieved performances Vtdt→ and Ptrdtr→ in the end of each period t and, eventually, to reward the decentralized decision-makers accordingly.


### 3.4.2. Sequential Mode

This mode captures the idea of “sequential planning”: The distributed decision-makers make their final choices sequentially with, for the sake of simplicity, the sequence being given by the indexr of the decision-makers. In particular, in time step t decision-maker r−1 informs decision-maker r with 1<r≤Ms about the choices made so far, i.e., made by the “preceding” decision-makers <r. Decision-maker r>1 re-evaluates its “own” options dt−1r∗→, dtr,a1→, and dtr,a2→ taking these “prior” choices into account—potentially resulting in an adjusted list of preferences Ltr=dtr,p1→;dtr,p2→;dtr,p3→ and chooses dtr,p1→ which here depends on dtr−1,p1→, i.e., is a function of the choices of the preceding decision-makers. Hence, only decision-maker 1 does not have to consider previous choices and the choice of decision-maker r=1,…,Ms is made according to(12)dtr∗→=dtr,p1→,ifr=1,dtr,p1→dtr−1,p1→,if1<r≤Μ,and with equation (12) the overall configuration is given by dt→=dtM∗→. The headquarter (type 2 agent) does not intervene in decision-making and—similar to the decentralized mode—is confined to observing the performances achieved and, eventually, rewarding the decentralized decision-makers accordingly.


### 3.4.3. Proposal Mode

Each local decision-maker transfers its listLtr of preferences to the headquarter which compiles the first preferences to a composite vector dC→=dt1,p1→,…,dtr,p1→,…,dtMs,p1→, and then evaluates the overall performance VdC→ (see equation (4)) that this solution promises.However, as with decentralized decision-makers, in the model it is assumed that the headquarter is not capable to perfectly ex-ante evaluate new options, i.e., other solutions than the status quo (see Section3.2). Rather, similar to the local decision-makers, the headquarter suffers from some relative noise following a Gaussian distribution with expected value 0 and standard deviations σcent resulting in a perceived overall performance V˜dC→. The headquarter decides in favor of the composite vector, i.e., dt→=dC→, if dC→ promises the same or a higher performance as the status quo dt−1→, i.e., if(13)V˜dC→≥Vdt−1∗→.If the composite vectordC→ assembled from the local decision-makers’ first preferences does not meet the condition in equation (13), the headquarter evaluates a vector composed from the decentralized decision-makers’ second preferences according to equation (13). If this also does not, at least, promise the performance of the status quo, then the organization stays with the status quo, i.e., then dt→=dt−1→.


## 3.4.1. Decentralized Mode

The highest level of autonomy is granted to theMs distributed decision-makers if each of them is allowed to choose its most preferred option. Then, the overall organizational configuration dt→ results from(10)dt→=dt1,p1→,…,dtr,p1→,…,dtMs,p1→.Hence, for decision-makerr’s partial decision problem Nsr the option according to(11)dtr∗→=dtr,p1→∀r∈1,…Ms,is implemented. Obviously, this type of coordination does not require any form of communication between the decentralized decision-makers or with the headquarter (central agent). The headquarter does not intervene in decision-making directly and its role is limited to registering the achieved performances Vtdt→ and Ptrdtr→ in the end of each period t and, eventually, to reward the decentralized decision-makers accordingly.


## 3.4.2. Sequential Mode

This mode captures the idea of “sequential planning”: The distributed decision-makers make their final choices sequentially with, for the sake of simplicity, the sequence being given by the indexr of the decision-makers. In particular, in time step t decision-maker r−1 informs decision-maker r with 1<r≤Ms about the choices made so far, i.e., made by the “preceding” decision-makers <r. Decision-maker r>1 re-evaluates its “own” options dt−1r∗→, dtr,a1→, and dtr,a2→ taking these “prior” choices into account—potentially resulting in an adjusted list of preferences Ltr=dtr,p1→;dtr,p2→;dtr,p3→ and chooses dtr,p1→ which here depends on dtr−1,p1→, i.e., is a function of the choices of the preceding decision-makers. Hence, only decision-maker 1 does not have to consider previous choices and the choice of decision-maker r=1,…,Ms is made according to(12)dtr∗→=dtr,p1→,ifr=1,dtr,p1→dtr−1,p1→,if1<r≤Μ,and with equation (12) the overall configuration is given by dt→=dtM∗→. The headquarter (type 2 agent) does not intervene in decision-making and—similar to the decentralized mode—is confined to observing the performances achieved and, eventually, rewarding the decentralized decision-makers accordingly.


## 3.4.3. Proposal Mode

Each local decision-maker transfers its listLtr of preferences to the headquarter which compiles the first preferences to a composite vector dC→=dt1,p1→,…,dtr,p1→,…,dtMs,p1→, and then evaluates the overall performance VdC→ (see equation (4)) that this solution promises.However, as with decentralized decision-makers, in the model it is assumed that the headquarter is not capable to perfectly ex-ante evaluate new options, i.e., other solutions than the status quo (see Section3.2). Rather, similar to the local decision-makers, the headquarter suffers from some relative noise following a Gaussian distribution with expected value 0 and standard deviations σcent resulting in a perceived overall performance V˜dC→. The headquarter decides in favor of the composite vector, i.e., dt→=dC→, if dC→ promises the same or a higher performance as the status quo dt−1→, i.e., if(13)V˜dC→≥Vdt−1∗→.If the composite vectordC→ assembled from the local decision-makers’ first preferences does not meet the condition in equation (13), the headquarter evaluates a vector composed from the decentralized decision-makers’ second preferences according to equation (13). If this also does not, at least, promise the performance of the status quo, then the organization stays with the status quo, i.e., then dt→=dt−1→.


## 3.5. Learning-Based Adaptation of Coordination

The very core of this research endeavor is to study which modes of coordination emerge within growth processes of the artificial organizations. In order to capture some kind of self-adaptation of coordination, the model employs a simple mode of reinforcement learning (for overviews, see [89, 90]) based on statistical learning, i.e., a generalized form of the Bush–Mosteller model [91, 92].This mode of learning is chosen for the following reasons: In the vast multitude of forms of learning studied in various domains (e.g., psychology, economics, and computer science), reinforcement learning is regarded to be among the most basic forms of learning [92]. It represents a fundamental possible form of humans’ behavior but also provides some basis for learning of artificial agents [90]—both regarded as DDMS in this paper. Moreover, in this model, as a form of experiential learning the mid-termed reinforcement learning on the coordination mode corresponds to the experiential type of short-termed adaptation [60] when the decision-making systems, in every single time-step, search for a superior solution compared to the status quo. However, it would be a natural extension of the model to let the DDMS employ other and, in particular, more advanced forms of learning about which mode of coordination is appropriate within the stage of growth and for the search strategy employed and the cost of effort.In the model, reinforcement learning on coordination for synthesizing local plans is represented in the following way: In the end of each time stept, the central agent (type 2 agent; see Section 3.2) receives information about the overall performance Vt according to equation (4). This allows the central agent, in every TLth period, to compute the performance enhancements achieved within the last TL periods. Moreover, in every TLth period, the organizations can alter the type of coordination mode as introduced in Section 3.4. Hence, our DDMS face the mid-termed decision-problem which type of coordination mode act∈Ac (with Ac=3; see Section 3.4) to implement in the next TL periods. Let pac,t denote the probability of an alternative act to be chosen at time t (with 0≤pac,t≤1 and ∑aac∈Acpac,t=1).The key idea of reinforcement learning is that the probabilities of options are updated according to the positive or negative stimuli resulting from these options. In our context, whether the performance enhancementΔVt obtained under the regime of a certain mode of coordination act in the previous TL periods is regarded positive or negative, depends on whether, or not, it at least equals an aspiration level v. ΔVt of type act of coordination is defined as the relative performance enhancement achieved within the last TL periods of the adaptive walk, i.e.,(14)ΔVt=Vt−Vt−TLVt−TL.Hence, the stimulusτt is(15)τt=1,ifΔVt≥v,−1,ifΔVt<v.The probabilities of optionsac∈Ac are updated according to the following rule, with λ (where 0≤λ≤1) giving the reinforcement strength [92]:(16)pac,t+1=pac,t+λ⋅1−pac,t,ifac=act∧τt=1,−pac,t,ifac=act∧τt=−1,−pac,t,ifac≠act∧τt=1,pac,t⋅pact,t1−pact,t,ifac≠act∧τt=−1.After the probabilities are updated as given in equation (16) the “next” mode of coordination to be employed from t+1 to t+TL is determined at random—according to the updated probabilities.


## 4. Simulation Experiments: Processual Structure, Parameters, and Analysis

This section is intended to introduce the principle structure of the simulation experiments based on the theoretical model as presented in the previous Section3. For this, first, Section 4.1 gives an overview of the principle processual structure of the simulations before the parameters settings are motivated (Section 4.2) and the metrics employed for analysis of experiments are introduced (Section 4.3).


### 4.1. Process Overview of the Simulation Model

Figure2 depicts the principle processual structure of the simulation model based on the theoretical model as introduced in the previous Section 3. In particular, the simulation in its core is characterized by three loops capturing three temporal horizons.Figure 2
Principle processual structure of the simulation model.In the short term, in each time stept, the artificial organizations search for superior solutions of their Ns-dimensional decision-problem where the overall problem is segmented into Ms sub-problems with each delegated to Ms local decision-making agents accordingly (type 1 agents in Section 3.2). In the mid-term, i.e., in each TLth time step, the central unit (type 2 agent) evaluates the current mode of coordination, learns from this evaluation via reinforcement, and, eventually, chooses another mode of coordination for the next TL periods. In the long term, in every TGth time step, the decision-problem grows by a fixed number Nsr of additional single choices to be made and the number Ms of distributed decision-makers increases by 1 (for details, see Section 3.1).


### 4.2. Parameter Settings

#### 4.2.1. Parameters Fixed for all Experiments

In the simulation experiments—parameter settings are listed in Table2—organizations are observed over T=750 periods where in every TL=10th period, eventually the mode of coordination is switched (the observation period T and the learning interval TL=10 were fixed based on pretests which indicate that the results do not principally change when the organizations are observed for a longer time; the similar holds for an extension of the learning interval (e.g., to TL=20); however, shortening the learning period notably below 10 periods does not leave the different coordination modes “enough time” to unfold their particular potential with respect to the aspiration level).Table 2
Parameter settings.
Parameter
Values/types
Parameters fixed for all experiments (see Section4.2.1)
Observation period
T
=
750
Interval of learning
T
L
=
10
Interval of growth stages
T
G
=
250
Growth stages
s
=
1,2,3
Number of choices
In growth stages=1: Ns=1=6 In growth stage s=2: Ns=2=9 In growth stage s=3: Ns=3=12
NumberMs of distributed decision-makers
In growth stages=1: M1=2 distributed decision-makers with sub-problems d1→=d1,d2,d3 and d2→=d4,d5,d6In growth stage s=2: M2=3, i.e., one additional distributed decision-maker 3 with sub-problem d3→=d7,d8,d9In growth stage s=3: M3=4, i.e., one further additional decision-maker 4 with sub-problem d4→=d10,d11,d12
Precision of ex-ante evaluation
Distributed decision-makers:σr=0.05∀rCentral agent (headquarter): σcent=0.1
Modes of coordination
a
c
∈
A
c
=
“decentralized
,
” “sequential
,
” “proposal”
Aspiration level
v
=
0
Learning strength
λ
=
0.5
Parameters subject to variation across experiments (see Section4.2.2)
Complexity of decision problem
“Decomposable”; “nondecomposable”, for details of interaction structures, see Figure 1
Search strategy
“Exploitation only”:hdr,a1→=1 and hdr,a2→=1“Exploitation and exploration”: hdr,a1→=1 and hdr,a2→=2“Exploration only”: hdr,a1→=2 and hdr,a2→=2
Cost of effort (cost coefficient)
Baseline scenarios: z=0.001Sensitivity analysis: z∈0,0.003,0.005,0.01,0.02In everyTG=250th period, the organizations undergo a growth in problem space as well as in the number of distributed decision-makers:(i)
In their first growth stage, the organizations face anNs=1=6-dimensional overall decision-problem decomposed into two sub-problems of equal size (i.e., Ns=1r=3∀r∈1,2) assigned to two decision-makers (Ms=1=2) accordingly(ii)
In the second growth stage (i.e., fromt=251to500), three additional binary choices are to be made by the organizations for which one additional local decision-maker is responsible—hence, the organizations comprise 3 decentralized decision-makers, i.e., Ms=2=3(iii)
In the third stage, the decision-problem grows by three further binary choices and; hence, finally the organizations deal with anNs=3=12-dimensional problem and a fourth decentralized decision-maker comes into play (Ms=3=4)The simulations are run for a moderate level of noise captured by parametersσr and σcent relevant for the ex-ante evaluations of options by the local decision-makers and, in case of the proposal mode, the central agent, respectively (see Sections 3.2 and 3.4.3). It is assumed that the information of the local decision-makers—only related to their respective Nr-dimensional sub-problems—are more precise than the information of the central unit which is related to the entire Ns-dimensional decision problem (i.e., σr<σcent). This is intended to capture differentiation and specialization [8, 93] due to division of labor. Some empirical evidence suggests that noise of about 10% of the true value in the domain of management control is at a reasonable range [94].The modes of coordination which the organizations choose of are motivated and introduced in Section3.4. The organizations employ the same initial mode of coordination, namely, the “decentralized” mode. This “setup” procedure is chosen to make sure that the adaptive processes start from a “defined” initial configuration and without having the learning processes overlaid by the strong performance enhancements that are typically made in the very first periods of the adaptive walks (notwithstanding, further simulations have shown that the overlaying effects appear to be rather negligible). However, an intuitive “story” behind this setup procedure is that the organizations start, in fact, without any particular mode of coordination, i.e., the decentralized mode (for example, because they were newly founded), and keep this until time t=10; in this period, they “discover” the two alternative modes and choose randomly one option ac∈Ac out of the three options (i.e., Ac=3) where each mode has the same initial probability 1/Ac. Then, in every TL=10th period, probabilities are adjusted according to the performance enhancements. The parameters for learning correspond to a moderate strength of learning [95]. On basis of the updated probabilities, eventually, a new coordination mode is established.


#### 4.2.2. Contingency Factors: Parameters Subject to Variation across Experiments

As argued in the Introduction, this research effort takes a contingency perspective and, in particular, intends to study the interacting effects of (1) complexity of decision-problem, (2) search strategy, and (3) cost of effort. Accordingly, these three contingent factors are subject to variation across simulation experiments (see also the lower part in Table2).(1) Complexity of Decision-Problem. In line with the idea of factorial design of simulation experiments [96], the experiments distinguish between two different levels of complexity of the decision problem to be solved by the organizations which capture two rather pronounced cases.Based on the seminal work of Simon [16] on the architecture of complexity (near) decomposability as compared to nondecomposability is the key aspect in the understanding of complexity (see also Introduction). Against this background, the simulations are run for a growing decomposable and a growing nondecomposable interaction structure, where the principle “type” of complexity is kept over the growth stages. Hence, the growth processes simulated may capture some kind of organic growth [97–99]. Figure 1 provides a graphical representation of interaction matrices of the growing decision-problems which could capture the following situations (in the simulation experiments, the number of decision-makers mirrors the growth of the decision-problem for avoiding interference with effects of varying size of decision-makers’ scope of competency (for further references, see [55, 100, 101])):(i)
“decomposable” (Figure 1(a)): This type of organization and growth relates to the idea of an organization consisting of self-contained “sub-problems” or “units” [7, 8] which have intense intra‐unit interactions, but no cross-unit interactions. In the beginning, the organization’s overall decision-problem is decomposable into 2 distinct sub-problems [17], which, for example, might be related to different products without any interrelations between them. For the 2 products, 2 business units are responsible. In the course of the growth stages, the organization “adds” further products and business units, correspondingly, without interrelations among the “old” and “new” products and, accordingly, the units.(ii)
“nondecomposable” (example given in Figure 1(b)): This case of interactions may capture what—according to the prominent classification of Thompson [7]—is called reciprocal interdependencies. In particular, this structure could represent an organization with functional specialization showing the typical high level of interrelations between sub-problems—and departments accordingly. In the course of growth, it may be that the vertical integration is increased (e.g., establishing an inhouse production of certain intermediate products or sales logistics).(2) Search Strategy. As described in Section 3.3.2, the decentralized decision-makers employ one of three fundamental search strategies which are intended to capture exploitation, exploration, and an ambidextrous strategy [20] as enforced by the boundary system—each characterized by the number of options and the Hamming distances allowed. For example, if an exploration strategy is pursued, with M3=4 local decision-makers and N3=12 in the final growth stage, at maximum, 8 digits of the overall decision-problem could be flipped, i.e., two-thirds of the configuration dt→ could be altered in one time step.(3) Cost of Effort. While the search strategy shapes the maximum of binary choices that could be flipped, the actual alterations are affected by the local decision-makers’ preferences together with the coordination mode employed. According to equations (7) and (9) in Section 3, the distributed decision-makers’ preferences are affected by the cost of effort and, in particular, by cost coefficient z.However, in order to be clear and concise, the simulation experiments are conducted in two steps. First, to gain a basic understanding of the emergence of coordination, in the baseline scenarios the cost of effort is at a moderate levelz=0.001 for all local decision-makers. Second, a sensitivity analysis is conducted in whose course the cost of effort is varied from costless effort (i.e., z=0) to higher levels of cost of effort (i.e., z>0.001) in order to analyze the effects of cost of effort on the emergence of coordination.


### 4.3. Metrics Employed for Analysis of Simulations

In the baseline scenarios, with 2 interaction structures and 3 search strategies under investigation at a given level of cost of effort, 6 different scenarios of parameters are simulated. For the sensitivity analysis, 5 additional levels of cost of effort are simulated which results in 30 further scenarios. For each of, in sum, the 36 scenarios, 2,500 simulations are run with 10 runs on 250 performance landscapes.The research question of this paper boils down to the question which coordination modes emerge predominantly for which contingencies in terms of complexity, search strategy, and cost of effort. Hence, for answering this question, the key metric is therelative frequencies of the coordination modes in the end of the observation period t=750.Moreover, the relative frequencies of coordination modes over observation time are depicted. This allows to gain an understanding of how their relative shares evolve in the course of growth and learning.As described in Section3.5 the propensities to opt for the one or the other mode of coordination is shaped by the performance gains achieved under the current mode in the last TL periods. Hence, the level of performance according to equations (2) and (4), respectively, in Section 3.1, i.e., the final performance achieved at the end of the observation period for the different coordination modes is of interest. For this, the 2,500 simulations for each scenario were grouped according to that mode of coordination which was “active” (i.e., has emerged) in the last period t=750 of the observation time.These subgroups were analyzed individually and, in particular, themean of the final performance Vt=750 achieved for each coordination mode selected at t=750 is computed as well as the respective confidence interval at a 99.9% level of confidence.Moreover, another metric informing about the effectiveness of the search processes is therelative frequency of the global maximum found in t=750 of the respective performance landscape which is computed for each subgroup (based on the coordination mode in t=750) separately.In order to gain some deeper understanding of the search processes conducted by the DDMS, two further metrics were observed and analyzed for each subgroup: theratio of periods in which the status quo is altered and the ratio of periods with false-positive alterations, i.e., alterations in favor of a false-positive option (i.e., reducing Vt) to the T=750 periods of observation. These metrics put some focus on the efficiency of search.


## 4.1. Process Overview of the Simulation Model

Figure2 depicts the principle processual structure of the simulation model based on the theoretical model as introduced in the previous Section 3. In particular, the simulation in its core is characterized by three loops capturing three temporal horizons.Figure 2
Principle processual structure of the simulation model.In the short term, in each time stept, the artificial organizations search for superior solutions of their Ns-dimensional decision-problem where the overall problem is segmented into Ms sub-problems with each delegated to Ms local decision-making agents accordingly (type 1 agents in Section 3.2). In the mid-term, i.e., in each TLth time step, the central unit (type 2 agent) evaluates the current mode of coordination, learns from this evaluation via reinforcement, and, eventually, chooses another mode of coordination for the next TL periods. In the long term, in every TGth time step, the decision-problem grows by a fixed number Nsr of additional single choices to be made and the number Ms of distributed decision-makers increases by 1 (for details, see Section 3.1).


## 4.2. Parameter Settings

### 4.2.1. Parameters Fixed for all Experiments

In the simulation experiments—parameter settings are listed in Table2—organizations are observed over T=750 periods where in every TL=10th period, eventually the mode of coordination is switched (the observation period T and the learning interval TL=10 were fixed based on pretests which indicate that the results do not principally change when the organizations are observed for a longer time; the similar holds for an extension of the learning interval (e.g., to TL=20); however, shortening the learning period notably below 10 periods does not leave the different coordination modes “enough time” to unfold their particular potential with respect to the aspiration level).Table 2
Parameter settings.
Parameter
Values/types
Parameters fixed for all experiments (see Section4.2.1)
Observation period
T
=
750
Interval of learning
T
L
=
10
Interval of growth stages
T
G
=
250
Growth stages
s
=
1,2,3
Number of choices
In growth stages=1: Ns=1=6 In growth stage s=2: Ns=2=9 In growth stage s=3: Ns=3=12
NumberMs of distributed decision-makers
In growth stages=1: M1=2 distributed decision-makers with sub-problems d1→=d1,d2,d3 and d2→=d4,d5,d6In growth stage s=2: M2=3, i.e., one additional distributed decision-maker 3 with sub-problem d3→=d7,d8,d9In growth stage s=3: M3=4, i.e., one further additional decision-maker 4 with sub-problem d4→=d10,d11,d12
Precision of ex-ante evaluation
Distributed decision-makers:σr=0.05∀rCentral agent (headquarter): σcent=0.1
Modes of coordination
a
c
∈
A
c
=
“decentralized
,
” “sequential
,
” “proposal”
Aspiration level
v
=
0
Learning strength
λ
=
0.5
Parameters subject to variation across experiments (see Section4.2.2)
Complexity of decision problem
“Decomposable”; “nondecomposable”, for details of interaction structures, see Figure 1
Search strategy
“Exploitation only”:hdr,a1→=1 and hdr,a2→=1“Exploitation and exploration”: hdr,a1→=1 and hdr,a2→=2“Exploration only”: hdr,a1→=2 and hdr,a2→=2
Cost of effort (cost coefficient)
Baseline scenarios: z=0.001Sensitivity analysis: z∈0,0.003,0.005,0.01,0.02In everyTG=250th period, the organizations undergo a growth in problem space as well as in the number of distributed decision-makers:(i)
In their first growth stage, the organizations face anNs=1=6-dimensional overall decision-problem decomposed into two sub-problems of equal size (i.e., Ns=1r=3∀r∈1,2) assigned to two decision-makers (Ms=1=2) accordingly(ii)
In the second growth stage (i.e., fromt=251to500), three additional binary choices are to be made by the organizations for which one additional local decision-maker is responsible—hence, the organizations comprise 3 decentralized decision-makers, i.e., Ms=2=3(iii)
In the third stage, the decision-problem grows by three further binary choices and; hence, finally the organizations deal with anNs=3=12-dimensional problem and a fourth decentralized decision-maker comes into play (Ms=3=4)The simulations are run for a moderate level of noise captured by parametersσr and σcent relevant for the ex-ante evaluations of options by the local decision-makers and, in case of the proposal mode, the central agent, respectively (see Sections 3.2 and 3.4.3). It is assumed that the information of the local decision-makers—only related to their respective Nr-dimensional sub-problems—are more precise than the information of the central unit which is related to the entire Ns-dimensional decision problem (i.e., σr<σcent). This is intended to capture differentiation and specialization [8, 93] due to division of labor. Some empirical evidence suggests that noise of about 10% of the true value in the domain of management control is at a reasonable range [94].The modes of coordination which the organizations choose of are motivated and introduced in Section3.4. The organizations employ the same initial mode of coordination, namely, the “decentralized” mode. This “setup” procedure is chosen to make sure that the adaptive processes start from a “defined” initial configuration and without having the learning processes overlaid by the strong performance enhancements that are typically made in the very first periods of the adaptive walks (notwithstanding, further simulations have shown that the overlaying effects appear to be rather negligible). However, an intuitive “story” behind this setup procedure is that the organizations start, in fact, without any particular mode of coordination, i.e., the decentralized mode (for example, because they were newly founded), and keep this until time t=10; in this period, they “discover” the two alternative modes and choose randomly one option ac∈Ac out of the three options (i.e., Ac=3) where each mode has the same initial probability 1/Ac. Then, in every TL=10th period, probabilities are adjusted according to the performance enhancements. The parameters for learning correspond to a moderate strength of learning [95]. On basis of the updated probabilities, eventually, a new coordination mode is established.


### 4.2.2. Contingency Factors: Parameters Subject to Variation across Experiments

As argued in the Introduction, this research effort takes a contingency perspective and, in particular, intends to study the interacting effects of (1) complexity of decision-problem, (2) search strategy, and (3) cost of effort. Accordingly, these three contingent factors are subject to variation across simulation experiments (see also the lower part in Table2).(1) Complexity of Decision-Problem. In line with the idea of factorial design of simulation experiments [96], the experiments distinguish between two different levels of complexity of the decision problem to be solved by the organizations which capture two rather pronounced cases.Based on the seminal work of Simon [16] on the architecture of complexity (near) decomposability as compared to nondecomposability is the key aspect in the understanding of complexity (see also Introduction). Against this background, the simulations are run for a growing decomposable and a growing nondecomposable interaction structure, where the principle “type” of complexity is kept over the growth stages. Hence, the growth processes simulated may capture some kind of organic growth [97–99]. Figure 1 provides a graphical representation of interaction matrices of the growing decision-problems which could capture the following situations (in the simulation experiments, the number of decision-makers mirrors the growth of the decision-problem for avoiding interference with effects of varying size of decision-makers’ scope of competency (for further references, see [55, 100, 101])):(i)
“decomposable” (Figure 1(a)): This type of organization and growth relates to the idea of an organization consisting of self-contained “sub-problems” or “units” [7, 8] which have intense intra‐unit interactions, but no cross-unit interactions. In the beginning, the organization’s overall decision-problem is decomposable into 2 distinct sub-problems [17], which, for example, might be related to different products without any interrelations between them. For the 2 products, 2 business units are responsible. In the course of the growth stages, the organization “adds” further products and business units, correspondingly, without interrelations among the “old” and “new” products and, accordingly, the units.(ii)
“nondecomposable” (example given in Figure 1(b)): This case of interactions may capture what—according to the prominent classification of Thompson [7]—is called reciprocal interdependencies. In particular, this structure could represent an organization with functional specialization showing the typical high level of interrelations between sub-problems—and departments accordingly. In the course of growth, it may be that the vertical integration is increased (e.g., establishing an inhouse production of certain intermediate products or sales logistics).(2) Search Strategy. As described in Section 3.3.2, the decentralized decision-makers employ one of three fundamental search strategies which are intended to capture exploitation, exploration, and an ambidextrous strategy [20] as enforced by the boundary system—each characterized by the number of options and the Hamming distances allowed. For example, if an exploration strategy is pursued, with M3=4 local decision-makers and N3=12 in the final growth stage, at maximum, 8 digits of the overall decision-problem could be flipped, i.e., two-thirds of the configuration dt→ could be altered in one time step.(3) Cost of Effort. While the search strategy shapes the maximum of binary choices that could be flipped, the actual alterations are affected by the local decision-makers’ preferences together with the coordination mode employed. According to equations (7) and (9) in Section 3, the distributed decision-makers’ preferences are affected by the cost of effort and, in particular, by cost coefficient z.However, in order to be clear and concise, the simulation experiments are conducted in two steps. First, to gain a basic understanding of the emergence of coordination, in the baseline scenarios the cost of effort is at a moderate levelz=0.001 for all local decision-makers. Second, a sensitivity analysis is conducted in whose course the cost of effort is varied from costless effort (i.e., z=0) to higher levels of cost of effort (i.e., z>0.001) in order to analyze the effects of cost of effort on the emergence of coordination.


## 4.2.1. Parameters Fixed for all Experiments

In the simulation experiments—parameter settings are listed in Table2—organizations are observed over T=750 periods where in every TL=10th period, eventually the mode of coordination is switched (the observation period T and the learning interval TL=10 were fixed based on pretests which indicate that the results do not principally change when the organizations are observed for a longer time; the similar holds for an extension of the learning interval (e.g., to TL=20); however, shortening the learning period notably below 10 periods does not leave the different coordination modes “enough time” to unfold their particular potential with respect to the aspiration level).Table 2
Parameter settings.
Parameter
Values/types
Parameters fixed for all experiments (see Section4.2.1)
Observation period
T
=
750
Interval of learning
T
L
=
10
Interval of growth stages
T
G
=
250
Growth stages
s
=
1,2,3
Number of choices
In growth stages=1: Ns=1=6 In growth stage s=2: Ns=2=9 In growth stage s=3: Ns=3=12
NumberMs of distributed decision-makers
In growth stages=1: M1=2 distributed decision-makers with sub-problems d1→=d1,d2,d3 and d2→=d4,d5,d6In growth stage s=2: M2=3, i.e., one additional distributed decision-maker 3 with sub-problem d3→=d7,d8,d9In growth stage s=3: M3=4, i.e., one further additional decision-maker 4 with sub-problem d4→=d10,d11,d12
Precision of ex-ante evaluation
Distributed decision-makers:σr=0.05∀rCentral agent (headquarter): σcent=0.1
Modes of coordination
a
c
∈
A
c
=
“decentralized
,
” “sequential
,
” “proposal”
Aspiration level
v
=
0
Learning strength
λ
=
0.5
Parameters subject to variation across experiments (see Section4.2.2)
Complexity of decision problem
“Decomposable”; “nondecomposable”, for details of interaction structures, see Figure 1
Search strategy
“Exploitation only”:hdr,a1→=1 and hdr,a2→=1“Exploitation and exploration”: hdr,a1→=1 and hdr,a2→=2“Exploration only”: hdr,a1→=2 and hdr,a2→=2
Cost of effort (cost coefficient)
Baseline scenarios: z=0.001Sensitivity analysis: z∈0,0.003,0.005,0.01,0.02In everyTG=250th period, the organizations undergo a growth in problem space as well as in the number of distributed decision-makers:(i)
In their first growth stage, the organizations face anNs=1=6-dimensional overall decision-problem decomposed into two sub-problems of equal size (i.e., Ns=1r=3∀r∈1,2) assigned to two decision-makers (Ms=1=2) accordingly(ii)
In the second growth stage (i.e., fromt=251to500), three additional binary choices are to be made by the organizations for which one additional local decision-maker is responsible—hence, the organizations comprise 3 decentralized decision-makers, i.e., Ms=2=3(iii)
In the third stage, the decision-problem grows by three further binary choices and; hence, finally the organizations deal with anNs=3=12-dimensional problem and a fourth decentralized decision-maker comes into play (Ms=3=4)The simulations are run for a moderate level of noise captured by parametersσr and σcent relevant for the ex-ante evaluations of options by the local decision-makers and, in case of the proposal mode, the central agent, respectively (see Sections 3.2 and 3.4.3). It is assumed that the information of the local decision-makers—only related to their respective Nr-dimensional sub-problems—are more precise than the information of the central unit which is related to the entire Ns-dimensional decision problem (i.e., σr<σcent). This is intended to capture differentiation and specialization [8, 93] due to division of labor. Some empirical evidence suggests that noise of about 10% of the true value in the domain of management control is at a reasonable range [94].The modes of coordination which the organizations choose of are motivated and introduced in Section3.4. The organizations employ the same initial mode of coordination, namely, the “decentralized” mode. This “setup” procedure is chosen to make sure that the adaptive processes start from a “defined” initial configuration and without having the learning processes overlaid by the strong performance enhancements that are typically made in the very first periods of the adaptive walks (notwithstanding, further simulations have shown that the overlaying effects appear to be rather negligible). However, an intuitive “story” behind this setup procedure is that the organizations start, in fact, without any particular mode of coordination, i.e., the decentralized mode (for example, because they were newly founded), and keep this until time t=10; in this period, they “discover” the two alternative modes and choose randomly one option ac∈Ac out of the three options (i.e., Ac=3) where each mode has the same initial probability 1/Ac. Then, in every TL=10th period, probabilities are adjusted according to the performance enhancements. The parameters for learning correspond to a moderate strength of learning [95]. On basis of the updated probabilities, eventually, a new coordination mode is established.


## 4.2.2. Contingency Factors: Parameters Subject to Variation across Experiments

As argued in the Introduction, this research effort takes a contingency perspective and, in particular, intends to study the interacting effects of (1) complexity of decision-problem, (2) search strategy, and (3) cost of effort. Accordingly, these three contingent factors are subject to variation across simulation experiments (see also the lower part in Table2).(1) Complexity of Decision-Problem. In line with the idea of factorial design of simulation experiments [96], the experiments distinguish between two different levels of complexity of the decision problem to be solved by the organizations which capture two rather pronounced cases.Based on the seminal work of Simon [16] on the architecture of complexity (near) decomposability as compared to nondecomposability is the key aspect in the understanding of complexity (see also Introduction). Against this background, the simulations are run for a growing decomposable and a growing nondecomposable interaction structure, where the principle “type” of complexity is kept over the growth stages. Hence, the growth processes simulated may capture some kind of organic growth [97–99]. Figure 1 provides a graphical representation of interaction matrices of the growing decision-problems which could capture the following situations (in the simulation experiments, the number of decision-makers mirrors the growth of the decision-problem for avoiding interference with effects of varying size of decision-makers’ scope of competency (for further references, see [55, 100, 101])):(i)
“decomposable” (Figure 1(a)): This type of organization and growth relates to the idea of an organization consisting of self-contained “sub-problems” or “units” [7, 8] which have intense intra‐unit interactions, but no cross-unit interactions. In the beginning, the organization’s overall decision-problem is decomposable into 2 distinct sub-problems [17], which, for example, might be related to different products without any interrelations between them. For the 2 products, 2 business units are responsible. In the course of the growth stages, the organization “adds” further products and business units, correspondingly, without interrelations among the “old” and “new” products and, accordingly, the units.(ii)
“nondecomposable” (example given in Figure 1(b)): This case of interactions may capture what—according to the prominent classification of Thompson [7]—is called reciprocal interdependencies. In particular, this structure could represent an organization with functional specialization showing the typical high level of interrelations between sub-problems—and departments accordingly. In the course of growth, it may be that the vertical integration is increased (e.g., establishing an inhouse production of certain intermediate products or sales logistics).(2) Search Strategy. As described in Section 3.3.2, the decentralized decision-makers employ one of three fundamental search strategies which are intended to capture exploitation, exploration, and an ambidextrous strategy [20] as enforced by the boundary system—each characterized by the number of options and the Hamming distances allowed. For example, if an exploration strategy is pursued, with M3=4 local decision-makers and N3=12 in the final growth stage, at maximum, 8 digits of the overall decision-problem could be flipped, i.e., two-thirds of the configuration dt→ could be altered in one time step.(3) Cost of Effort. While the search strategy shapes the maximum of binary choices that could be flipped, the actual alterations are affected by the local decision-makers’ preferences together with the coordination mode employed. According to equations (7) and (9) in Section 3, the distributed decision-makers’ preferences are affected by the cost of effort and, in particular, by cost coefficient z.However, in order to be clear and concise, the simulation experiments are conducted in two steps. First, to gain a basic understanding of the emergence of coordination, in the baseline scenarios the cost of effort is at a moderate levelz=0.001 for all local decision-makers. Second, a sensitivity analysis is conducted in whose course the cost of effort is varied from costless effort (i.e., z=0) to higher levels of cost of effort (i.e., z>0.001) in order to analyze the effects of cost of effort on the emergence of coordination.


## 4.3. Metrics Employed for Analysis of Simulations

In the baseline scenarios, with 2 interaction structures and 3 search strategies under investigation at a given level of cost of effort, 6 different scenarios of parameters are simulated. For the sensitivity analysis, 5 additional levels of cost of effort are simulated which results in 30 further scenarios. For each of, in sum, the 36 scenarios, 2,500 simulations are run with 10 runs on 250 performance landscapes.The research question of this paper boils down to the question which coordination modes emerge predominantly for which contingencies in terms of complexity, search strategy, and cost of effort. Hence, for answering this question, the key metric is therelative frequencies of the coordination modes in the end of the observation period t=750.Moreover, the relative frequencies of coordination modes over observation time are depicted. This allows to gain an understanding of how their relative shares evolve in the course of growth and learning.As described in Section3.5 the propensities to opt for the one or the other mode of coordination is shaped by the performance gains achieved under the current mode in the last TL periods. Hence, the level of performance according to equations (2) and (4), respectively, in Section 3.1, i.e., the final performance achieved at the end of the observation period for the different coordination modes is of interest. For this, the 2,500 simulations for each scenario were grouped according to that mode of coordination which was “active” (i.e., has emerged) in the last period t=750 of the observation time.These subgroups were analyzed individually and, in particular, themean of the final performance Vt=750 achieved for each coordination mode selected at t=750 is computed as well as the respective confidence interval at a 99.9% level of confidence.Moreover, another metric informing about the effectiveness of the search processes is therelative frequency of the global maximum found in t=750 of the respective performance landscape which is computed for each subgroup (based on the coordination mode in t=750) separately.In order to gain some deeper understanding of the search processes conducted by the DDMS, two further metrics were observed and analyzed for each subgroup: theratio of periods in which the status quo is altered and the ratio of periods with false-positive alterations, i.e., alterations in favor of a false-positive option (i.e., reducing Vt) to the T=750 periods of observation. These metrics put some focus on the efficiency of search.


## 5. Results and Discussion

The results are presented in two steps. First, for gaining a basic understanding of the effects of intraorganizational complexity and search strategy on the emergence of coordination, the baseline scenarios are presented and analyzed (Section5.1). While in the baseline scenarios the distributed decision-makers operate at a rather moderate, though nonzero level of cost of effort, the sensitivity analysis (Section 5.2) illustrates the effect of cost of effort at the local decision-makers’ side on the coordination mode emerging at the system’s level.


### 5.1. Complexity and Search Strategy: Baseline Scenarios

#### 5.1.1. Overview

For the baseline scenarios, Table3 displays condensed results obtained from the simulation experiments according to the metrics introduced in Section 4.3. In addition, Table 4 reports on the significances of mean differences of the final performances Vt=750 achieved on average for the simulation runs grouped by the modes of coordination active in the last observation period employing Welch’s method [102, 103]. The plots in Figure 3 display—for the two interaction structures and the three search strategies under investigation in the baseline scenarios—the relative frequencies of the three modes of coordination within the observation time.Table 3
Condensed results of the baseline scenarios (moderate cost coefficient of effortz=0.001).
Search strategy
Frequency of coordination modeac in t=750
Final performanceVt=750 (conf. interval∗)
Frequency of global maximum found int=750
Frequency of alterations (false positives) ofdt→
(a)
(b)
(c)
(d)
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decomposable structure
Exploitation only
33.9%
35.0%
31.1%
0.953
±
0.0052
0.950
±
0.0055
0.952
±
0.0055
18.8%
17.5%
15.8%
4.3% (2.3%)
4.7% (2.0%)
(2.1%)
Exploitation and exploration
41.4%
38.8%
19.8%
0.992
±
0.0017
0.992
±
0.0018
0.988
±
0.0038
52.5%
57.2%
47.7%
4.3% (1.9%)
4.5% (2.0%)
(2.2%)
Exploration only
35.2%
36.3%
28.4%
0.913
±
0.0070
0.915
±
0.0066
0.915
±
0.0075
5.7%
5.3%
5.0%
3.0% (1.4%)
3.1% (1.4%)
3.0% (1.3%)
Nondecomposable structure
Exploitation only
27.9%
35.5%
36.6%
0.896
±
0.0083
0.894
±
0.0070
0.911
±
0.0064
7.6%
6.4%
7.5%
4.3% (2.1%)
4.3% (2.0%)
4.4% (2.1%)
Exploitation and exploration
22.1%
33.0%
45.0%
0.891
±
0.0142
0.910
±
0.0092
0.933
±
0.0064
10.3%
9.3%
18.3%
12.4% (6.2%)
12.4% (6.1%)
14.2% (7.0%)
Exploration only
17.5%
27.1%
55.4%
0.847
±
0.0133
0.848
±
0.0100
0.872
±
0.0057
3.0%
2.1%
2.6%
8.0% (4.0%)
9.0% (4.5%)
9.7% (4.8%)
∗Confidence intervals for Vt=750 are given at a level of 99.9%. Each row represents averaged/aggregated results of 2,500 simulation runs: 250 landscapes with 10 runs on each.Table 4
Mean differences of final performancesVt=750 and half-lengths of individual 99.9% confidence intervals for pairwise comparisons in the baseline scenarios (moderate cost coefficient of effort z=0.001).
Search strategy
Mode of coordination
Decentralized
Sequential
Decomposable structure
Exploitation only
Proposal
0.0015 ± 0.0059
−0.002 ± 0.0061
Decentralized
−0.0034 ± 0.0059
Exploitation and exploration
Proposal
0.004 ± 0.0033∗
0.004 ± 0.0033∗
Decentralized
0.0000 ± 0.0019
Exploration only
Proposal
−0.0017 ± 0.0079
−0.0001 ± 0.0078
Decentralized
0.0017 ± 0.0074
Nondecomposable structure
Exploitation only
Proposal
−0.015 ± 0.0082∗
−0.0169 ± 0.0074∗
Decentralized
−0.0019 ± 0.0085
Exploitation and exploration
Proposal
−0.0418 ± 0.0122∗
−0.0234 ± 0.0088∗
Decentralized
0.0184 ± 0.0133∗
Exploration only
Proposal
−0.0256 ± 0.0113∗
−0.0239 ± 0.009∗
Decentralized
0.0017 ± 0.013
∗Significant difference. For parameter settings, see Table 2.Figure 3
Relative frequencies of coordination modes in the course of growth in the baseline scenarios (i.e., moderate cost coefficient of effortz=0.001). Each plot represents results of 2,500 simulation runs: 250 landscapes with 10 runs on each. For parameter settings, see Table 2. (a) Decomposable-exploitation only, (b) decomposable-exploitation and exploration, (c) decomposable-exploration only, (d) nondecomposable-exploitation only, (e) nondecomposable-exploitation and exploration, and (f) nondecomposable-exploration only.
(a)
(b)
(c)
(d)
(e)
(f)According to the results, the coordination modes emerging in the course of growth and adaptation differ remarkably across the two interaction structures and for the three search strategies.Broadly speaking, for the growing decomposable structure, the results suggest that with the exploitative and the explorative search strategy no particular coordination mode emerges predominantly; in contrast, when the organizations allow more flexibility in terms of an ambidextrous search strategy, coordination modes prevail that leave the decision-making authority at the side of the decentralized decision-makers.For growing nondecomposable structures, hierarchical coordination (proposal mode) increasingly predominates in the course of growth. Moreover, the level of predominance varies with the search strategy employed. A purely explorative strategy is most likely to emphasize hierarchical elements for coordination of decentralized decision-making.These results are discussed for the two types of interaction structures more into detail subsequently.


#### 5.1.2. Decomposable Interaction Structure

For a closer analysis, a starting point is the coordination need, and in the decomposable structure there is, in fact, no need for coordination across the sub-problems which could result from the nature of the “growing” task: The sub-problems do not show any interactions among each other (see Figure1(a)). With this, intuition suggests that—for a given search strategy and as far as no cost of coordination are taken into account like in this study—the three mechanisms of coordination under investigation should not show remarkable differences in terms of performance Vt provided and, in consequence, the learning-based frequencies of occurrence.For the “exploitation only” and the “exploration only” search strategies this conjecture is broadly supported by the results: frequencies of occurrences throughout the observation period (Figures3(a) and 3(c)) and final performances (columns (a) and (b) in Table 3) are at a similar level for the three modes of coordination. However, for the ambidextrous search strategy, results suggest that the proposal mode (i.e., employing hierarchy), in the course of growth, is increasingly predominated by the other modes and the final performances achieved with the decentralized and the sequential mode go beyond the level obtained when the proposal mode is “active” at t=750 (employing Welch’s method [102, 103], the performance differences against the proposal mode are significant at a confidence level of 99.9% as can be seen in Table 4.). Moreover, it is worth mentioning that in the ambidextrous strategy the final performances exceed the levels obtained with the “exploitation only” strategy by around 3.5 and with the “exploration only” strategy by around 8 points of percentage.Hence, from these observations, two interrelated questions arise: (1) What may cause the performance excess of the ambidextrous strategy compared to the other strategies and (2) what drives the imbalance in the coordination modes in the ambidextrous strategy?For suggesting answers to these questions, it appears helpful to consider the sources of coordination need which are, broadly speaking,(1)
interactions across sub-problems and distributed decision-makers accordingly which result from decomposition into sub-problems (e.g., [16, 93])(2)
distributed decision-makers pursuing not the DDMS’s overall, but their parochial objectives—as extensively elaborated in contract theory with its applications in management control (e.g., [83, 104])(3)
decision-makers having different and imperfect information of the problem to be solved—may it be in the tradition of information economics as elaborated in the seminal paper of Sah and Stiglitz [105], in contract theory (e.g., [83]) or following the tradition of bounded rationality according to Simon [19]Since in the decomposable structure, there are no interactions across sub-problems and distributed decision-makers, no inter-sub-problem coordination need (1) exists, and for this reason even the merely parochial objectives (2) of the local decision-makers should not affect the overall performance achieved which here results—across all growth stages—as a sum of the decentralized decision-makers’ performances without complementarities or substitutes to be taken into account. However, apparently aspect (3) is of relevance: in the model, none of the decision-makers—neither the distributed nor the central type (see Section3.2)—disposes of perfect information and, in the proposal mode, the rather imprecise information of the central agent enters in decision-making without that its broad perspective makes a relevant contribution since there is no need for coordination resulting from aspects (1) or (2).Apparently, this is particularly crucial in the ambidextrous strategy: This strategy gives the distributed decision-makers the highest flexibility in terms of shaping the novelty of solutions to their partial problems, and, combined with their rather precise information, allows them to adjust rather fast to the local maxima of their particular sub-problems. These beneficial effects of flexibility of search captured in the ambidextrous strategy show up in the frequency of the global maximum found (column (c) in Table3) which is at a considerably higher level than in the other strategies. However, involving the central agent—with its imprecise information whose broadness does not contribute in case of decomposable structures—reduces the effectiveness of search and, hence, the proposal mode is less often selected.More broadly speaking, the results, so far, suggest that the search strategy subtly interferes with the coordination mode emerging even if coordination need is at a low level.


#### 5.1.3. Nondecomposable Interaction Structure

In contrast to the decomposable structure, for noncomposable decision-problems, the emergence of coordination modes differs remarkably across search strategies. For a start, it is worth emphasizing that now—due to cross-problem interactions—superior configurations or even the global optimum in the performance landscape cannot be found by just locating superior (or optimal) solutions to the sub-problems. Moreover, stepwise search processes, particularly, when conducted in a decentralized manner as in our model, are likely to end up in local maxima causing inertia of the search (with further references [21, 37, 73]).Each of the aforementioned three sources of coordination need is relevant now: (1) cross-departmental interactions with the complexity increasing in the course of growth, (2) distributed decision-makers focusing on parochial performance which in the nondecomposable structure is not necessarily in line with overall performance, and (3) decision-makers operating with imperfect information in various senses.As can be seen in Table3, in the “exploitation only” strategy, the three modes of coordination emerge with rather similar relative frequencies. In contrast, in the ambidextrous strategy, the proposal mode’s share reaches a level of 45%, and in the “exploitation only” strategy, it is established in more than 55% of the runs in the end of the observation period. Figures 3(d)–3(f) illustrate these differences in the emergence over time.As reported in Table3, the final performances achieved with the different settings of search strategy and coordination mode differ remarkably. With “exploitation only”, in all coordination modes, a medium level of around 90% of the maximal performance of 1 is achieved—though the proposal mode significantly provides the best results (according to the Welch’s test (see Table 4), the performance excess is significant at a confidence level of 99.9%.). In contrast, with “pure” exploration, final performance, at maximum, is around 87%, and in the decentralized and the sequential mode around 2.5 points of percentage less. In the ambidextrous strategy, the final performance achieved in the proposal mode is significantly higher than that obtained with the decentralized (+4.18 points of percentage) and the sequential mode (+2.34 points of percentage), and with 93.3%, it is even the highest obtained for this structure across all search strategies. The frequency of the global maximum found directs in a similar direction.Apparently, in growing task environments with high complexity throughout growth, the central agent is increasingly involved in coordination in terms of using upward communication and employing aggregate, organization-wide information for decision-making. This corresponds to results of empirical studies examining the effect of intraorganizational interdependencies as a contingent factor on management controls as reported in Section2 indicating that higher levels of interdependencies are associated by more vertical information flows and use of aggregated information [24, 49, 50]. In a similar vein, though not explicitly controlling for internal complexity, Davila [48] argues that the use of formal controls increasing in organizational size may be driven by increasing complexity which is supported by a positive relation between size and emphasis on action controls (corresponding to the boundary system) in his empirical study.While the aforementioned effect shows up for all search strategies, the different search strategies appear to be differently sensitive to the coordination mode emerging: As mentioned above, the purely explorative strategy performs worst in this interaction structure, with hierarchical coordination (“proposal mode”) leading to the relatively best results and emerging, by far, most often. This indicates on the tension incorporated in the boundary system in general and as also captured in the model: As argued by Simons et al. [53, 106], the boundary system could facilitate renewal enforcing decision-makers to search for largely new ways—which our model seeks to capture in the “exploration only” strategy; at the same time, boundaries as established in the “proposal mode” help to align parochial choices to the overall objectives of the organization. However, rigid limits on the scope of search as established in the “exploitation only” strategy reduce the diversity of search and, in this sense, more rigid coordination (as with the proposal mode) may be beneficial (as is supported by the empirical study of Bedford [26]) though less than that in the “exploration only” strategy. This is broadly reflected in the simulation results since in the “exploitation only” strategy the final performances obtained in the three modes of coordination reach a similar level and the predominance of the proposal mode is not as clear as for the other search strategies.The ambidextrous search strategy appears to be particularly sensitive to the coordination mode in respect of the (spread of) performance levels obtained. As mentioned before, this strategy grants the highest flexibility to the distributed decision-makers in terms of allowing for varying levels of novelty of their solutions; apparently, the combination of flexibility on the side of decentralized decision-makers and centralized final choices provides the best results when complexity is (increasingly) high. This relates to the tension captured in ambidextrous strategies which, in face of empirical results, lets Bedford [26] argue that in organizations pursuing this search strategy setting boundaries may be an alternative or substitute to other components of the MCS rather than being necessarily balanced with the other components.


### 5.2. Sensitivity to Cost of Effort

#### 5.2.1. Overview

So far, the emergence of the coordination mode was studied for organizations with the distributed decision-makers operating at a moderate level of cost of effort (i.e., cost coefficientz=0.001). In the next step of this simulation study, the effect of cost of effort is analyzed. In particular, for the two interaction structures and the three search strategies under investigation, simulations with different levels of cost of effort are run. Apart from simulations for costless effort (i.e., z=0), also experiments for cost coefficients z∈0.003,0.005,0.01,0.02 were conducted. Figures 4 and 5 present results obtained for a medium cost level z=0.005 and a high cost level of z=0.02, respectively.Figure 4
Relative frequencies of coordination modes in the course of growth with medium cost of effort (cost coefficientz=0.005). Each plot represents results of 2,500 simulation runs: 250 landscapes with 10 runs on each. For parameter settings, see Table 2. (a) Decomposable-exploitation only, (b) decomposable-exploitation and exploration, (c) decomposable-exploration only, (d) nondecomposable-exploitation only, (e) nondecomposable-exploitation and exploration, and (f) nondecomposable-exploration only.
(a)
(b)
(c)
(d)
(e)
(f)Figure 5
Relative frequencies of coordination modes in the course of growth with high cost of effort (cost coefficientz=0.02). Each plot represents results of 2,500 simulation runs: 250 landscapes with 10 runs on each. For parameter settings, see Table 2. (a) Decomposable-exploitation only, (b) decomposable-exploitation and exploration, (c) decomposable-exploration only, (d) nondecomposable-exploitation only, (e) nondecomposable-exploitation and exploration, and (f) nondecomposable-exploration only.
(a)
(b)
(c)
(d)
(e)
(f)In order to be clear and concise, only the key findings and selected aspects of the sensitivity analysis are addressed explicitly. For this reason, the case of zero cost of effort is not discussed more in detail, since the experiments suggest that results obtained in the baseline scenarios (i.e., moderate cost level) do not change substantially when the decentralized decision-makers operate withcostless effort. However, with increasing cost of effort the emergence of coordination modes (given by their relative frequencies) notably changes compared to the baseline scenarios. The most interesting results show up for the nondecomposable structure which is why this will be discussed more extensively—particularly in conjunction with the “exploitation and exploration” strategy.With respect to thedecomposable structure, comparing results obtained for the moderate cost level (Figure 3) against those for medium and high cost (Figures 4 and 5) reveals the most obvious difference that—in each search strategy—hierarchical coordination (proposal mode) is the less often implemented, the higher the cost level. This effect is more pronounced for the ambidextrous and the explorative strategies. Analyzing the results more in detail reveals an interesting dichotomy between the relative frequency of emergence and the final performance Vt=750 of the coordination modes; this effect is even stronger in the nondecomposable structures and, therefore, will be discussed in the context below.In contrast, in thenondecomposable structure, the emerging coordination mode appears to be remarkably sensitive to an increase in the cost of effort. In particular, for the ambidextrous search strategy and the purely explorative strategy the “order” of coordination mechanisms changes compared to the baseline scenarios: recall that at the moderate cost level (baseline), the proposal mode clearly predominates (with a share up to more than 55%); now, with higher cost, this is the least often emerging coordination mode (partially with only about 12%).Moreover, this low frequency of occurrence of hierarchical coordination goes along with thehighest final performances compared to the other coordination modes. This clearly runs against intuition since one would expect that the better performing coordination mode is selected more often as it is the case for lower cost levels as shown in the baseline scenarios (see Section 5.1).To illustrate this effect, Figure6 plots—for the different cost levels simulated—the final performances Vt=750 and the relative frequencies in the end of the observation time for the case of the ambidextrous search strategy in the nondecomposable structure, while Table 5 reports on the respective details. As it becomes apparent, with a cost coefficient level of z=0.003 and higher, final performance and frequency of the hierarchical coordination run apart—or put the other way round: the worse performing coordination modes emerge more often (a similar effect shows up in the “explorative only” search strategy, and even for the “explorative only,” it is worth mentioning that the coordination modes show similar frequencies of occurrence while the final performance obtained with hierarchical coordination (“proposal”) goes remarkably beyond the level obtained with the other coordination modes).Figure 6
Sensitivity of final performance and relative frequencies of coordination modes to the cost of effort with the “exploitation and exploration” search strategy for the nondecomposable interaction structure. Each plot represents results of 2,500 simulation runs: 250 landscapes with 10 runs on each. For parameter settings, see Table2. (a) z = 0, (b) z = 0.001(baseline), (c) z = 0.003, (d) z = 0.005, (e) z = 0.01, and (f) z = 0.02.
(a)
(b)
(c)
(d)
(e)
(f)Table 5
Condensed results obtained for the nondecomposable interaction structure in the “exploitation and exploration” search strategy for different levels of cost of effort.
Cost of effort (cost coefficient)
Frequency of coordination modeac in t=750
Final performanceVt=750 (conf. interval∗)
Frequency of global maximum found int=750
Frequency of alterations (false positives) ofdt→
(a)
(b)
(c)
(d)
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
z
=
0
17.9%
28.4%
53.8%
0.878
±
0.0185
0.899
±
0.0120
0.932
±
0.0058
13.0%
12.4%
18.5%
14.4% (7.1%)
15.4% (7.6%)
16.3% (8.0%)
z
=
0.001
22.1%
33.0%
45.0%
0.891
±
0.0142
0.910
±
0.0092
0.933
±
0.0064
10.3%
9.3%
18.3%
12.4% (6.2%)
12.4% (6.1%)
14.2% (7.0%)
z
=
0.003
27.6%
37.2%
35.2%
0.903
±
0.0102
0.915
±
0.0079
0.934
±
0.0068
11.1%
14.1%
19.2%
8.5% (4.2%)
8.3% (4.1%)
11.3% (5.5%)
z
=
0.005
29.8%
43.7%
26.4%
0.914
±
0.0080
0.916
±
0.0063
0.935
±
0.0080
11.7%
11.9%
17.7%
5.8% (2.8%)
6.0% (2.9%)
9.0% (4.4%)
z
=
0.01
38.8%
45.6%
15.6%
0.901
±
0.0071
0
.
905
±
0.0062
0.940
±
0.0100
7.0%
9.1%
20.7%
3.2% (1.5%)
3.1% (1.4%)
5.0% (2.4%)
z
=
0.02
43.5%
44.2%
12.3%
0.877
±
0.0070
0.880
±
0.0070
0.940
±
0.0100
3.0%
4.2%
18.9%
1.9% (0.8%)
1.8% (0.8%)
3.0% (1.3%)
∗Confidence intervals for Vt=750 are given at a level of 99.9%. Each row represents averaged/aggregated results of 2,500 simulation runs: 250 landscapes with 10 runs on each.


#### 5.2.2. On the Dichotomy of Coordination Modes in Performance and Frequency

Hence, an interesting question is what may cause this dichotomy in final performance and frequency of emergence for higher cost of effort in the “exploitation and exploration” strategy as shown in Figure6 and Table 5.A starting point for an explanation is that higher cost of effort at the distributed decision-makers’ side increase the propensity that they prefer to stay with the status quo. This, in turn, is best overcome by hierarchical coordination though with the risk of false positive alterations.In particular, with an increase in cost of effort, from a local decision-maker’s perspective, leaving the status quo in favor of an alternative configuration becomes less rewarding, or in other words, the higher the cost of effort the more promising an alternative to the status quo—inducing effort—has to be for being selected, i.e., only alterations that promise more than the cost of effort are preferable from a decentralized decision-maker’s perspective. The search strategies enforce different levels of effort to be taken for leaving the status quo; for example, the purely explorative strategy requires making long jumps (i.e., switching two single choices) which—according to the quadratic cost function in equation (7)—induces rather high cost of effort.Hence, with an increasing cost coefficientz, in tendency, in all three modes of coordination, keeping the status quo becomes more attractive for the local decision-makers and this effect is the more pronounced the higher the effort enforced by the search strategy. However, the coordination modes are differently prone to this kind of “inertia”: In the decentralized mode the local decision-makers’ first preferences are implemented without any further revision. In the sequential mode, distributed decision-makers’ first preferences may be revised, though from a parochial perspective.In contrast, in the proposal mode it is more likely that the decentralized decision-makers’ preferences are overrided by the central agent with respect to the overall performance and, hence, alterations, though causing cost on the decentralized side, may be induced. This, in turn, makes it more likely that the status quo is abandoned and a new configurationdt→ is implemented. However, since the central agent disposes of rather noisy information (σcent), these alterations could also be in favor of a false positive alteration. This explanation is broadly confirmed by the ratios of periods with alterations and with false positive alterations (for the metrics, see Section 4.3): These ratios are particularly higher in case of hierarchical coordination (proposal mode) as reported in Table 5 for the example of the ambidextrous search strategy (with increasing levels z of the cost coefficient the alterations decrease for all coordination modes under investigation but the relative differences among the modes increase; the confidence intervals of the final performance achieved shows in a similar direction: with levels of cost coefficient z=0.005 and higher the confidence intervals for the proposal mode are higher than those of the other coordination modes).With respect to the emergence of coordination modes, the stability in terms of, at least, keeping a once achieved performance level provided by a particular mode of coordination drives the propensity of being implemented in the future. Hence, since the decentralized and the sequential mode in combination with higher cost coefficients of effort and a search strategy enforcing exploration induce more inertia and, on average, lower performance than hierarchical coordination, according to the simulation results, these coordination mechanisms predominate.In this sense, the somewhat counter-intuitive results for higher levels of cost of effort may provide an explanation for the ambiguous results obtained in empirical studies on the relation between tightness of coordination on the one hand and organizational size as well as intraorganizational complexity on the other hand as reported in Section2.For a further interpretation of results, it appears worth mentioning that the level of cost of effort in the model (captured by a simple cost coefficientz) may represent rather different aspects and—depending on its particular context—the results of this study show in quite different directions. For example, the cost of effort could capture “switching costs” for altering a status quo in favor of a new solution which may be specific to a certain branch or industry due to technological aspects [107, 108]. In this sense, the results may be regarded as an indication that, for example, in different branches different coordination modes might be predominant.In another interpretation, cost of effort could capture the costs for dealing with resistance of certain stakeholders or (emotional) costs for loosing old and building new intra- or extraorganizational relations when new solutions are to be implemented [109–111]. Regarded in this sense, the results suggest that organizations where, for example, high resistance or high relational costs for implementing the “new” are to be expected, in tendency, might stick to decentralized coordination modes—though hierarchical coordination could lead to higher levels of performance.


## 5.1. Complexity and Search Strategy: Baseline Scenarios

### 5.1.1. Overview

For the baseline scenarios, Table3 displays condensed results obtained from the simulation experiments according to the metrics introduced in Section 4.3. In addition, Table 4 reports on the significances of mean differences of the final performances Vt=750 achieved on average for the simulation runs grouped by the modes of coordination active in the last observation period employing Welch’s method [102, 103]. The plots in Figure 3 display—for the two interaction structures and the three search strategies under investigation in the baseline scenarios—the relative frequencies of the three modes of coordination within the observation time.Table 3
Condensed results of the baseline scenarios (moderate cost coefficient of effortz=0.001).
Search strategy
Frequency of coordination modeac in t=750
Final performanceVt=750 (conf. interval∗)
Frequency of global maximum found int=750
Frequency of alterations (false positives) ofdt→
(a)
(b)
(c)
(d)
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decomposable structure
Exploitation only
33.9%
35.0%
31.1%
0.953
±
0.0052
0.950
±
0.0055
0.952
±
0.0055
18.8%
17.5%
15.8%
4.3% (2.3%)
4.7% (2.0%)
(2.1%)
Exploitation and exploration
41.4%
38.8%
19.8%
0.992
±
0.0017
0.992
±
0.0018
0.988
±
0.0038
52.5%
57.2%
47.7%
4.3% (1.9%)
4.5% (2.0%)
(2.2%)
Exploration only
35.2%
36.3%
28.4%
0.913
±
0.0070
0.915
±
0.0066
0.915
±
0.0075
5.7%
5.3%
5.0%
3.0% (1.4%)
3.1% (1.4%)
3.0% (1.3%)
Nondecomposable structure
Exploitation only
27.9%
35.5%
36.6%
0.896
±
0.0083
0.894
±
0.0070
0.911
±
0.0064
7.6%
6.4%
7.5%
4.3% (2.1%)
4.3% (2.0%)
4.4% (2.1%)
Exploitation and exploration
22.1%
33.0%
45.0%
0.891
±
0.0142
0.910
±
0.0092
0.933
±
0.0064
10.3%
9.3%
18.3%
12.4% (6.2%)
12.4% (6.1%)
14.2% (7.0%)
Exploration only
17.5%
27.1%
55.4%
0.847
±
0.0133
0.848
±
0.0100
0.872
±
0.0057
3.0%
2.1%
2.6%
8.0% (4.0%)
9.0% (4.5%)
9.7% (4.8%)
∗Confidence intervals for Vt=750 are given at a level of 99.9%. Each row represents averaged/aggregated results of 2,500 simulation runs: 250 landscapes with 10 runs on each.Table 4
Mean differences of final performancesVt=750 and half-lengths of individual 99.9% confidence intervals for pairwise comparisons in the baseline scenarios (moderate cost coefficient of effort z=0.001).
Search strategy
Mode of coordination
Decentralized
Sequential
Decomposable structure
Exploitation only
Proposal
0.0015 ± 0.0059
−0.002 ± 0.0061
Decentralized
−0.0034 ± 0.0059
Exploitation and exploration
Proposal
0.004 ± 0.0033∗
0.004 ± 0.0033∗
Decentralized
0.0000 ± 0.0019
Exploration only
Proposal
−0.0017 ± 0.0079
−0.0001 ± 0.0078
Decentralized
0.0017 ± 0.0074
Nondecomposable structure
Exploitation only
Proposal
−0.015 ± 0.0082∗
−0.0169 ± 0.0074∗
Decentralized
−0.0019 ± 0.0085
Exploitation and exploration
Proposal
−0.0418 ± 0.0122∗
−0.0234 ± 0.0088∗
Decentralized
0.0184 ± 0.0133∗
Exploration only
Proposal
−0.0256 ± 0.0113∗
−0.0239 ± 0.009∗
Decentralized
0.0017 ± 0.013
∗Significant difference. For parameter settings, see Table 2.Figure 3
Relative frequencies of coordination modes in the course of growth in the baseline scenarios (i.e., moderate cost coefficient of effortz=0.001). Each plot represents results of 2,500 simulation runs: 250 landscapes with 10 runs on each. For parameter settings, see Table 2. (a) Decomposable-exploitation only, (b) decomposable-exploitation and exploration, (c) decomposable-exploration only, (d) nondecomposable-exploitation only, (e) nondecomposable-exploitation and exploration, and (f) nondecomposable-exploration only.
(a)
(b)
(c)
(d)
(e)
(f)According to the results, the coordination modes emerging in the course of growth and adaptation differ remarkably across the two interaction structures and for the three search strategies.Broadly speaking, for the growing decomposable structure, the results suggest that with the exploitative and the explorative search strategy no particular coordination mode emerges predominantly; in contrast, when the organizations allow more flexibility in terms of an ambidextrous search strategy, coordination modes prevail that leave the decision-making authority at the side of the decentralized decision-makers.For growing nondecomposable structures, hierarchical coordination (proposal mode) increasingly predominates in the course of growth. Moreover, the level of predominance varies with the search strategy employed. A purely explorative strategy is most likely to emphasize hierarchical elements for coordination of decentralized decision-making.These results are discussed for the two types of interaction structures more into detail subsequently.


### 5.1.2. Decomposable Interaction Structure

For a closer analysis, a starting point is the coordination need, and in the decomposable structure there is, in fact, no need for coordination across the sub-problems which could result from the nature of the “growing” task: The sub-problems do not show any interactions among each other (see Figure1(a)). With this, intuition suggests that—for a given search strategy and as far as no cost of coordination are taken into account like in this study—the three mechanisms of coordination under investigation should not show remarkable differences in terms of performance Vt provided and, in consequence, the learning-based frequencies of occurrence.For the “exploitation only” and the “exploration only” search strategies this conjecture is broadly supported by the results: frequencies of occurrences throughout the observation period (Figures3(a) and 3(c)) and final performances (columns (a) and (b) in Table 3) are at a similar level for the three modes of coordination. However, for the ambidextrous search strategy, results suggest that the proposal mode (i.e., employing hierarchy), in the course of growth, is increasingly predominated by the other modes and the final performances achieved with the decentralized and the sequential mode go beyond the level obtained when the proposal mode is “active” at t=750 (employing Welch’s method [102, 103], the performance differences against the proposal mode are significant at a confidence level of 99.9% as can be seen in Table 4.). Moreover, it is worth mentioning that in the ambidextrous strategy the final performances exceed the levels obtained with the “exploitation only” strategy by around 3.5 and with the “exploration only” strategy by around 8 points of percentage.Hence, from these observations, two interrelated questions arise: (1) What may cause the performance excess of the ambidextrous strategy compared to the other strategies and (2) what drives the imbalance in the coordination modes in the ambidextrous strategy?For suggesting answers to these questions, it appears helpful to consider the sources of coordination need which are, broadly speaking,(1)
interactions across sub-problems and distributed decision-makers accordingly which result from decomposition into sub-problems (e.g., [16, 93])(2)
distributed decision-makers pursuing not the DDMS’s overall, but their parochial objectives—as extensively elaborated in contract theory with its applications in management control (e.g., [83, 104])(3)
decision-makers having different and imperfect information of the problem to be solved—may it be in the tradition of information economics as elaborated in the seminal paper of Sah and Stiglitz [105], in contract theory (e.g., [83]) or following the tradition of bounded rationality according to Simon [19]Since in the decomposable structure, there are no interactions across sub-problems and distributed decision-makers, no inter-sub-problem coordination need (1) exists, and for this reason even the merely parochial objectives (2) of the local decision-makers should not affect the overall performance achieved which here results—across all growth stages—as a sum of the decentralized decision-makers’ performances without complementarities or substitutes to be taken into account. However, apparently aspect (3) is of relevance: in the model, none of the decision-makers—neither the distributed nor the central type (see Section3.2)—disposes of perfect information and, in the proposal mode, the rather imprecise information of the central agent enters in decision-making without that its broad perspective makes a relevant contribution since there is no need for coordination resulting from aspects (1) or (2).Apparently, this is particularly crucial in the ambidextrous strategy: This strategy gives the distributed decision-makers the highest flexibility in terms of shaping the novelty of solutions to their partial problems, and, combined with their rather precise information, allows them to adjust rather fast to the local maxima of their particular sub-problems. These beneficial effects of flexibility of search captured in the ambidextrous strategy show up in the frequency of the global maximum found (column (c) in Table3) which is at a considerably higher level than in the other strategies. However, involving the central agent—with its imprecise information whose broadness does not contribute in case of decomposable structures—reduces the effectiveness of search and, hence, the proposal mode is less often selected.More broadly speaking, the results, so far, suggest that the search strategy subtly interferes with the coordination mode emerging even if coordination need is at a low level.


### 5.1.3. Nondecomposable Interaction Structure

In contrast to the decomposable structure, for noncomposable decision-problems, the emergence of coordination modes differs remarkably across search strategies. For a start, it is worth emphasizing that now—due to cross-problem interactions—superior configurations or even the global optimum in the performance landscape cannot be found by just locating superior (or optimal) solutions to the sub-problems. Moreover, stepwise search processes, particularly, when conducted in a decentralized manner as in our model, are likely to end up in local maxima causing inertia of the search (with further references [21, 37, 73]).Each of the aforementioned three sources of coordination need is relevant now: (1) cross-departmental interactions with the complexity increasing in the course of growth, (2) distributed decision-makers focusing on parochial performance which in the nondecomposable structure is not necessarily in line with overall performance, and (3) decision-makers operating with imperfect information in various senses.As can be seen in Table3, in the “exploitation only” strategy, the three modes of coordination emerge with rather similar relative frequencies. In contrast, in the ambidextrous strategy, the proposal mode’s share reaches a level of 45%, and in the “exploitation only” strategy, it is established in more than 55% of the runs in the end of the observation period. Figures 3(d)–3(f) illustrate these differences in the emergence over time.As reported in Table3, the final performances achieved with the different settings of search strategy and coordination mode differ remarkably. With “exploitation only”, in all coordination modes, a medium level of around 90% of the maximal performance of 1 is achieved—though the proposal mode significantly provides the best results (according to the Welch’s test (see Table 4), the performance excess is significant at a confidence level of 99.9%.). In contrast, with “pure” exploration, final performance, at maximum, is around 87%, and in the decentralized and the sequential mode around 2.5 points of percentage less. In the ambidextrous strategy, the final performance achieved in the proposal mode is significantly higher than that obtained with the decentralized (+4.18 points of percentage) and the sequential mode (+2.34 points of percentage), and with 93.3%, it is even the highest obtained for this structure across all search strategies. The frequency of the global maximum found directs in a similar direction.Apparently, in growing task environments with high complexity throughout growth, the central agent is increasingly involved in coordination in terms of using upward communication and employing aggregate, organization-wide information for decision-making. This corresponds to results of empirical studies examining the effect of intraorganizational interdependencies as a contingent factor on management controls as reported in Section2 indicating that higher levels of interdependencies are associated by more vertical information flows and use of aggregated information [24, 49, 50]. In a similar vein, though not explicitly controlling for internal complexity, Davila [48] argues that the use of formal controls increasing in organizational size may be driven by increasing complexity which is supported by a positive relation between size and emphasis on action controls (corresponding to the boundary system) in his empirical study.While the aforementioned effect shows up for all search strategies, the different search strategies appear to be differently sensitive to the coordination mode emerging: As mentioned above, the purely explorative strategy performs worst in this interaction structure, with hierarchical coordination (“proposal mode”) leading to the relatively best results and emerging, by far, most often. This indicates on the tension incorporated in the boundary system in general and as also captured in the model: As argued by Simons et al. [53, 106], the boundary system could facilitate renewal enforcing decision-makers to search for largely new ways—which our model seeks to capture in the “exploration only” strategy; at the same time, boundaries as established in the “proposal mode” help to align parochial choices to the overall objectives of the organization. However, rigid limits on the scope of search as established in the “exploitation only” strategy reduce the diversity of search and, in this sense, more rigid coordination (as with the proposal mode) may be beneficial (as is supported by the empirical study of Bedford [26]) though less than that in the “exploration only” strategy. This is broadly reflected in the simulation results since in the “exploitation only” strategy the final performances obtained in the three modes of coordination reach a similar level and the predominance of the proposal mode is not as clear as for the other search strategies.The ambidextrous search strategy appears to be particularly sensitive to the coordination mode in respect of the (spread of) performance levels obtained. As mentioned before, this strategy grants the highest flexibility to the distributed decision-makers in terms of allowing for varying levels of novelty of their solutions; apparently, the combination of flexibility on the side of decentralized decision-makers and centralized final choices provides the best results when complexity is (increasingly) high. This relates to the tension captured in ambidextrous strategies which, in face of empirical results, lets Bedford [26] argue that in organizations pursuing this search strategy setting boundaries may be an alternative or substitute to other components of the MCS rather than being necessarily balanced with the other components.


## 5.1.1. Overview

For the baseline scenarios, Table3 displays condensed results obtained from the simulation experiments according to the metrics introduced in Section 4.3. In addition, Table 4 reports on the significances of mean differences of the final performances Vt=750 achieved on average for the simulation runs grouped by the modes of coordination active in the last observation period employing Welch’s method [102, 103]. The plots in Figure 3 display—for the two interaction structures and the three search strategies under investigation in the baseline scenarios—the relative frequencies of the three modes of coordination within the observation time.Table 3
Condensed results of the baseline scenarios (moderate cost coefficient of effortz=0.001).
Search strategy
Frequency of coordination modeac in t=750
Final performanceVt=750 (conf. interval∗)
Frequency of global maximum found int=750
Frequency of alterations (false positives) ofdt→
(a)
(b)
(c)
(d)
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decomposable structure
Exploitation only
33.9%
35.0%
31.1%
0.953
±
0.0052
0.950
±
0.0055
0.952
±
0.0055
18.8%
17.5%
15.8%
4.3% (2.3%)
4.7% (2.0%)
(2.1%)
Exploitation and exploration
41.4%
38.8%
19.8%
0.992
±
0.0017
0.992
±
0.0018
0.988
±
0.0038
52.5%
57.2%
47.7%
4.3% (1.9%)
4.5% (2.0%)
(2.2%)
Exploration only
35.2%
36.3%
28.4%
0.913
±
0.0070
0.915
±
0.0066
0.915
±
0.0075
5.7%
5.3%
5.0%
3.0% (1.4%)
3.1% (1.4%)
3.0% (1.3%)
Nondecomposable structure
Exploitation only
27.9%
35.5%
36.6%
0.896
±
0.0083
0.894
±
0.0070
0.911
±
0.0064
7.6%
6.4%
7.5%
4.3% (2.1%)
4.3% (2.0%)
4.4% (2.1%)
Exploitation and exploration
22.1%
33.0%
45.0%
0.891
±
0.0142
0.910
±
0.0092
0.933
±
0.0064
10.3%
9.3%
18.3%
12.4% (6.2%)
12.4% (6.1%)
14.2% (7.0%)
Exploration only
17.5%
27.1%
55.4%
0.847
±
0.0133
0.848
±
0.0100
0.872
±
0.0057
3.0%
2.1%
2.6%
8.0% (4.0%)
9.0% (4.5%)
9.7% (4.8%)
∗Confidence intervals for Vt=750 are given at a level of 99.9%. Each row represents averaged/aggregated results of 2,500 simulation runs: 250 landscapes with 10 runs on each.Table 4
Mean differences of final performancesVt=750 and half-lengths of individual 99.9% confidence intervals for pairwise comparisons in the baseline scenarios (moderate cost coefficient of effort z=0.001).
Search strategy
Mode of coordination
Decentralized
Sequential
Decomposable structure
Exploitation only
Proposal
0.0015 ± 0.0059
−0.002 ± 0.0061
Decentralized
−0.0034 ± 0.0059
Exploitation and exploration
Proposal
0.004 ± 0.0033∗
0.004 ± 0.0033∗
Decentralized
0.0000 ± 0.0019
Exploration only
Proposal
−0.0017 ± 0.0079
−0.0001 ± 0.0078
Decentralized
0.0017 ± 0.0074
Nondecomposable structure
Exploitation only
Proposal
−0.015 ± 0.0082∗
−0.0169 ± 0.0074∗
Decentralized
−0.0019 ± 0.0085
Exploitation and exploration
Proposal
−0.0418 ± 0.0122∗
−0.0234 ± 0.0088∗
Decentralized
0.0184 ± 0.0133∗
Exploration only
Proposal
−0.0256 ± 0.0113∗
−0.0239 ± 0.009∗
Decentralized
0.0017 ± 0.013
∗Significant difference. For parameter settings, see Table 2.Figure 3
Relative frequencies of coordination modes in the course of growth in the baseline scenarios (i.e., moderate cost coefficient of effortz=0.001). Each plot represents results of 2,500 simulation runs: 250 landscapes with 10 runs on each. For parameter settings, see Table 2. (a) Decomposable-exploitation only, (b) decomposable-exploitation and exploration, (c) decomposable-exploration only, (d) nondecomposable-exploitation only, (e) nondecomposable-exploitation and exploration, and (f) nondecomposable-exploration only.
(a)
(b)
(c)
(d)
(e)
(f)According to the results, the coordination modes emerging in the course of growth and adaptation differ remarkably across the two interaction structures and for the three search strategies.Broadly speaking, for the growing decomposable structure, the results suggest that with the exploitative and the explorative search strategy no particular coordination mode emerges predominantly; in contrast, when the organizations allow more flexibility in terms of an ambidextrous search strategy, coordination modes prevail that leave the decision-making authority at the side of the decentralized decision-makers.For growing nondecomposable structures, hierarchical coordination (proposal mode) increasingly predominates in the course of growth. Moreover, the level of predominance varies with the search strategy employed. A purely explorative strategy is most likely to emphasize hierarchical elements for coordination of decentralized decision-making.These results are discussed for the two types of interaction structures more into detail subsequently.


## 5.1.2. Decomposable Interaction Structure

For a closer analysis, a starting point is the coordination need, and in the decomposable structure there is, in fact, no need for coordination across the sub-problems which could result from the nature of the “growing” task: The sub-problems do not show any interactions among each other (see Figure1(a)). With this, intuition suggests that—for a given search strategy and as far as no cost of coordination are taken into account like in this study—the three mechanisms of coordination under investigation should not show remarkable differences in terms of performance Vt provided and, in consequence, the learning-based frequencies of occurrence.For the “exploitation only” and the “exploration only” search strategies this conjecture is broadly supported by the results: frequencies of occurrences throughout the observation period (Figures3(a) and 3(c)) and final performances (columns (a) and (b) in Table 3) are at a similar level for the three modes of coordination. However, for the ambidextrous search strategy, results suggest that the proposal mode (i.e., employing hierarchy), in the course of growth, is increasingly predominated by the other modes and the final performances achieved with the decentralized and the sequential mode go beyond the level obtained when the proposal mode is “active” at t=750 (employing Welch’s method [102, 103], the performance differences against the proposal mode are significant at a confidence level of 99.9% as can be seen in Table 4.). Moreover, it is worth mentioning that in the ambidextrous strategy the final performances exceed the levels obtained with the “exploitation only” strategy by around 3.5 and with the “exploration only” strategy by around 8 points of percentage.Hence, from these observations, two interrelated questions arise: (1) What may cause the performance excess of the ambidextrous strategy compared to the other strategies and (2) what drives the imbalance in the coordination modes in the ambidextrous strategy?For suggesting answers to these questions, it appears helpful to consider the sources of coordination need which are, broadly speaking,(1)
interactions across sub-problems and distributed decision-makers accordingly which result from decomposition into sub-problems (e.g., [16, 93])(2)
distributed decision-makers pursuing not the DDMS’s overall, but their parochial objectives—as extensively elaborated in contract theory with its applications in management control (e.g., [83, 104])(3)
decision-makers having different and imperfect information of the problem to be solved—may it be in the tradition of information economics as elaborated in the seminal paper of Sah and Stiglitz [105], in contract theory (e.g., [83]) or following the tradition of bounded rationality according to Simon [19]Since in the decomposable structure, there are no interactions across sub-problems and distributed decision-makers, no inter-sub-problem coordination need (1) exists, and for this reason even the merely parochial objectives (2) of the local decision-makers should not affect the overall performance achieved which here results—across all growth stages—as a sum of the decentralized decision-makers’ performances without complementarities or substitutes to be taken into account. However, apparently aspect (3) is of relevance: in the model, none of the decision-makers—neither the distributed nor the central type (see Section3.2)—disposes of perfect information and, in the proposal mode, the rather imprecise information of the central agent enters in decision-making without that its broad perspective makes a relevant contribution since there is no need for coordination resulting from aspects (1) or (2).Apparently, this is particularly crucial in the ambidextrous strategy: This strategy gives the distributed decision-makers the highest flexibility in terms of shaping the novelty of solutions to their partial problems, and, combined with their rather precise information, allows them to adjust rather fast to the local maxima of their particular sub-problems. These beneficial effects of flexibility of search captured in the ambidextrous strategy show up in the frequency of the global maximum found (column (c) in Table3) which is at a considerably higher level than in the other strategies. However, involving the central agent—with its imprecise information whose broadness does not contribute in case of decomposable structures—reduces the effectiveness of search and, hence, the proposal mode is less often selected.More broadly speaking, the results, so far, suggest that the search strategy subtly interferes with the coordination mode emerging even if coordination need is at a low level.


## 5.1.3. Nondecomposable Interaction Structure

In contrast to the decomposable structure, for noncomposable decision-problems, the emergence of coordination modes differs remarkably across search strategies. For a start, it is worth emphasizing that now—due to cross-problem interactions—superior configurations or even the global optimum in the performance landscape cannot be found by just locating superior (or optimal) solutions to the sub-problems. Moreover, stepwise search processes, particularly, when conducted in a decentralized manner as in our model, are likely to end up in local maxima causing inertia of the search (with further references [21, 37, 73]).Each of the aforementioned three sources of coordination need is relevant now: (1) cross-departmental interactions with the complexity increasing in the course of growth, (2) distributed decision-makers focusing on parochial performance which in the nondecomposable structure is not necessarily in line with overall performance, and (3) decision-makers operating with imperfect information in various senses.As can be seen in Table3, in the “exploitation only” strategy, the three modes of coordination emerge with rather similar relative frequencies. In contrast, in the ambidextrous strategy, the proposal mode’s share reaches a level of 45%, and in the “exploitation only” strategy, it is established in more than 55% of the runs in the end of the observation period. Figures 3(d)–3(f) illustrate these differences in the emergence over time.As reported in Table3, the final performances achieved with the different settings of search strategy and coordination mode differ remarkably. With “exploitation only”, in all coordination modes, a medium level of around 90% of the maximal performance of 1 is achieved—though the proposal mode significantly provides the best results (according to the Welch’s test (see Table 4), the performance excess is significant at a confidence level of 99.9%.). In contrast, with “pure” exploration, final performance, at maximum, is around 87%, and in the decentralized and the sequential mode around 2.5 points of percentage less. In the ambidextrous strategy, the final performance achieved in the proposal mode is significantly higher than that obtained with the decentralized (+4.18 points of percentage) and the sequential mode (+2.34 points of percentage), and with 93.3%, it is even the highest obtained for this structure across all search strategies. The frequency of the global maximum found directs in a similar direction.Apparently, in growing task environments with high complexity throughout growth, the central agent is increasingly involved in coordination in terms of using upward communication and employing aggregate, organization-wide information for decision-making. This corresponds to results of empirical studies examining the effect of intraorganizational interdependencies as a contingent factor on management controls as reported in Section2 indicating that higher levels of interdependencies are associated by more vertical information flows and use of aggregated information [24, 49, 50]. In a similar vein, though not explicitly controlling for internal complexity, Davila [48] argues that the use of formal controls increasing in organizational size may be driven by increasing complexity which is supported by a positive relation between size and emphasis on action controls (corresponding to the boundary system) in his empirical study.While the aforementioned effect shows up for all search strategies, the different search strategies appear to be differently sensitive to the coordination mode emerging: As mentioned above, the purely explorative strategy performs worst in this interaction structure, with hierarchical coordination (“proposal mode”) leading to the relatively best results and emerging, by far, most often. This indicates on the tension incorporated in the boundary system in general and as also captured in the model: As argued by Simons et al. [53, 106], the boundary system could facilitate renewal enforcing decision-makers to search for largely new ways—which our model seeks to capture in the “exploration only” strategy; at the same time, boundaries as established in the “proposal mode” help to align parochial choices to the overall objectives of the organization. However, rigid limits on the scope of search as established in the “exploitation only” strategy reduce the diversity of search and, in this sense, more rigid coordination (as with the proposal mode) may be beneficial (as is supported by the empirical study of Bedford [26]) though less than that in the “exploration only” strategy. This is broadly reflected in the simulation results since in the “exploitation only” strategy the final performances obtained in the three modes of coordination reach a similar level and the predominance of the proposal mode is not as clear as for the other search strategies.The ambidextrous search strategy appears to be particularly sensitive to the coordination mode in respect of the (spread of) performance levels obtained. As mentioned before, this strategy grants the highest flexibility to the distributed decision-makers in terms of allowing for varying levels of novelty of their solutions; apparently, the combination of flexibility on the side of decentralized decision-makers and centralized final choices provides the best results when complexity is (increasingly) high. This relates to the tension captured in ambidextrous strategies which, in face of empirical results, lets Bedford [26] argue that in organizations pursuing this search strategy setting boundaries may be an alternative or substitute to other components of the MCS rather than being necessarily balanced with the other components.


## 5.2. Sensitivity to Cost of Effort

### 5.2.1. Overview

So far, the emergence of the coordination mode was studied for organizations with the distributed decision-makers operating at a moderate level of cost of effort (i.e., cost coefficientz=0.001). In the next step of this simulation study, the effect of cost of effort is analyzed. In particular, for the two interaction structures and the three search strategies under investigation, simulations with different levels of cost of effort are run. Apart from simulations for costless effort (i.e., z=0), also experiments for cost coefficients z∈0.003,0.005,0.01,0.02 were conducted. Figures 4 and 5 present results obtained for a medium cost level z=0.005 and a high cost level of z=0.02, respectively.Figure 4
Relative frequencies of coordination modes in the course of growth with medium cost of effort (cost coefficientz=0.005). Each plot represents results of 2,500 simulation runs: 250 landscapes with 10 runs on each. For parameter settings, see Table 2. (a) Decomposable-exploitation only, (b) decomposable-exploitation and exploration, (c) decomposable-exploration only, (d) nondecomposable-exploitation only, (e) nondecomposable-exploitation and exploration, and (f) nondecomposable-exploration only.
(a)
(b)
(c)
(d)
(e)
(f)Figure 5
Relative frequencies of coordination modes in the course of growth with high cost of effort (cost coefficientz=0.02). Each plot represents results of 2,500 simulation runs: 250 landscapes with 10 runs on each. For parameter settings, see Table 2. (a) Decomposable-exploitation only, (b) decomposable-exploitation and exploration, (c) decomposable-exploration only, (d) nondecomposable-exploitation only, (e) nondecomposable-exploitation and exploration, and (f) nondecomposable-exploration only.
(a)
(b)
(c)
(d)
(e)
(f)In order to be clear and concise, only the key findings and selected aspects of the sensitivity analysis are addressed explicitly. For this reason, the case of zero cost of effort is not discussed more in detail, since the experiments suggest that results obtained in the baseline scenarios (i.e., moderate cost level) do not change substantially when the decentralized decision-makers operate withcostless effort. However, with increasing cost of effort the emergence of coordination modes (given by their relative frequencies) notably changes compared to the baseline scenarios. The most interesting results show up for the nondecomposable structure which is why this will be discussed more extensively—particularly in conjunction with the “exploitation and exploration” strategy.With respect to thedecomposable structure, comparing results obtained for the moderate cost level (Figure 3) against those for medium and high cost (Figures 4 and 5) reveals the most obvious difference that—in each search strategy—hierarchical coordination (proposal mode) is the less often implemented, the higher the cost level. This effect is more pronounced for the ambidextrous and the explorative strategies. Analyzing the results more in detail reveals an interesting dichotomy between the relative frequency of emergence and the final performance Vt=750 of the coordination modes; this effect is even stronger in the nondecomposable structures and, therefore, will be discussed in the context below.In contrast, in thenondecomposable structure, the emerging coordination mode appears to be remarkably sensitive to an increase in the cost of effort. In particular, for the ambidextrous search strategy and the purely explorative strategy the “order” of coordination mechanisms changes compared to the baseline scenarios: recall that at the moderate cost level (baseline), the proposal mode clearly predominates (with a share up to more than 55%); now, with higher cost, this is the least often emerging coordination mode (partially with only about 12%).Moreover, this low frequency of occurrence of hierarchical coordination goes along with thehighest final performances compared to the other coordination modes. This clearly runs against intuition since one would expect that the better performing coordination mode is selected more often as it is the case for lower cost levels as shown in the baseline scenarios (see Section 5.1).To illustrate this effect, Figure6 plots—for the different cost levels simulated—the final performances Vt=750 and the relative frequencies in the end of the observation time for the case of the ambidextrous search strategy in the nondecomposable structure, while Table 5 reports on the respective details. As it becomes apparent, with a cost coefficient level of z=0.003 and higher, final performance and frequency of the hierarchical coordination run apart—or put the other way round: the worse performing coordination modes emerge more often (a similar effect shows up in the “explorative only” search strategy, and even for the “explorative only,” it is worth mentioning that the coordination modes show similar frequencies of occurrence while the final performance obtained with hierarchical coordination (“proposal”) goes remarkably beyond the level obtained with the other coordination modes).Figure 6
Sensitivity of final performance and relative frequencies of coordination modes to the cost of effort with the “exploitation and exploration” search strategy for the nondecomposable interaction structure. Each plot represents results of 2,500 simulation runs: 250 landscapes with 10 runs on each. For parameter settings, see Table2. (a) z = 0, (b) z = 0.001(baseline), (c) z = 0.003, (d) z = 0.005, (e) z = 0.01, and (f) z = 0.02.
(a)
(b)
(c)
(d)
(e)
(f)Table 5
Condensed results obtained for the nondecomposable interaction structure in the “exploitation and exploration” search strategy for different levels of cost of effort.
Cost of effort (cost coefficient)
Frequency of coordination modeac in t=750
Final performanceVt=750 (conf. interval∗)
Frequency of global maximum found int=750
Frequency of alterations (false positives) ofdt→
(a)
(b)
(c)
(d)
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
z
=
0
17.9%
28.4%
53.8%
0.878
±
0.0185
0.899
±
0.0120
0.932
±
0.0058
13.0%
12.4%
18.5%
14.4% (7.1%)
15.4% (7.6%)
16.3% (8.0%)
z
=
0.001
22.1%
33.0%
45.0%
0.891
±
0.0142
0.910
±
0.0092
0.933
±
0.0064
10.3%
9.3%
18.3%
12.4% (6.2%)
12.4% (6.1%)
14.2% (7.0%)
z
=
0.003
27.6%
37.2%
35.2%
0.903
±
0.0102
0.915
±
0.0079
0.934
±
0.0068
11.1%
14.1%
19.2%
8.5% (4.2%)
8.3% (4.1%)
11.3% (5.5%)
z
=
0.005
29.8%
43.7%
26.4%
0.914
±
0.0080
0.916
±
0.0063
0.935
±
0.0080
11.7%
11.9%
17.7%
5.8% (2.8%)
6.0% (2.9%)
9.0% (4.4%)
z
=
0.01
38.8%
45.6%
15.6%
0.901
±
0.0071
0
.
905
±
0.0062
0.940
±
0.0100
7.0%
9.1%
20.7%
3.2% (1.5%)
3.1% (1.4%)
5.0% (2.4%)
z
=
0.02
43.5%
44.2%
12.3%
0.877
±
0.0070
0.880
±
0.0070
0.940
±
0.0100
3.0%
4.2%
18.9%
1.9% (0.8%)
1.8% (0.8%)
3.0% (1.3%)
∗Confidence intervals for Vt=750 are given at a level of 99.9%. Each row represents averaged/aggregated results of 2,500 simulation runs: 250 landscapes with 10 runs on each.


### 5.2.2. On the Dichotomy of Coordination Modes in Performance and Frequency

Hence, an interesting question is what may cause this dichotomy in final performance and frequency of emergence for higher cost of effort in the “exploitation and exploration” strategy as shown in Figure6 and Table 5.A starting point for an explanation is that higher cost of effort at the distributed decision-makers’ side increase the propensity that they prefer to stay with the status quo. This, in turn, is best overcome by hierarchical coordination though with the risk of false positive alterations.In particular, with an increase in cost of effort, from a local decision-maker’s perspective, leaving the status quo in favor of an alternative configuration becomes less rewarding, or in other words, the higher the cost of effort the more promising an alternative to the status quo—inducing effort—has to be for being selected, i.e., only alterations that promise more than the cost of effort are preferable from a decentralized decision-maker’s perspective. The search strategies enforce different levels of effort to be taken for leaving the status quo; for example, the purely explorative strategy requires making long jumps (i.e., switching two single choices) which—according to the quadratic cost function in equation (7)—induces rather high cost of effort.Hence, with an increasing cost coefficientz, in tendency, in all three modes of coordination, keeping the status quo becomes more attractive for the local decision-makers and this effect is the more pronounced the higher the effort enforced by the search strategy. However, the coordination modes are differently prone to this kind of “inertia”: In the decentralized mode the local decision-makers’ first preferences are implemented without any further revision. In the sequential mode, distributed decision-makers’ first preferences may be revised, though from a parochial perspective.In contrast, in the proposal mode it is more likely that the decentralized decision-makers’ preferences are overrided by the central agent with respect to the overall performance and, hence, alterations, though causing cost on the decentralized side, may be induced. This, in turn, makes it more likely that the status quo is abandoned and a new configurationdt→ is implemented. However, since the central agent disposes of rather noisy information (σcent), these alterations could also be in favor of a false positive alteration. This explanation is broadly confirmed by the ratios of periods with alterations and with false positive alterations (for the metrics, see Section 4.3): These ratios are particularly higher in case of hierarchical coordination (proposal mode) as reported in Table 5 for the example of the ambidextrous search strategy (with increasing levels z of the cost coefficient the alterations decrease for all coordination modes under investigation but the relative differences among the modes increase; the confidence intervals of the final performance achieved shows in a similar direction: with levels of cost coefficient z=0.005 and higher the confidence intervals for the proposal mode are higher than those of the other coordination modes).With respect to the emergence of coordination modes, the stability in terms of, at least, keeping a once achieved performance level provided by a particular mode of coordination drives the propensity of being implemented in the future. Hence, since the decentralized and the sequential mode in combination with higher cost coefficients of effort and a search strategy enforcing exploration induce more inertia and, on average, lower performance than hierarchical coordination, according to the simulation results, these coordination mechanisms predominate.In this sense, the somewhat counter-intuitive results for higher levels of cost of effort may provide an explanation for the ambiguous results obtained in empirical studies on the relation between tightness of coordination on the one hand and organizational size as well as intraorganizational complexity on the other hand as reported in Section2.For a further interpretation of results, it appears worth mentioning that the level of cost of effort in the model (captured by a simple cost coefficientz) may represent rather different aspects and—depending on its particular context—the results of this study show in quite different directions. For example, the cost of effort could capture “switching costs” for altering a status quo in favor of a new solution which may be specific to a certain branch or industry due to technological aspects [107, 108]. In this sense, the results may be regarded as an indication that, for example, in different branches different coordination modes might be predominant.In another interpretation, cost of effort could capture the costs for dealing with resistance of certain stakeholders or (emotional) costs for loosing old and building new intra- or extraorganizational relations when new solutions are to be implemented [109–111]. Regarded in this sense, the results suggest that organizations where, for example, high resistance or high relational costs for implementing the “new” are to be expected, in tendency, might stick to decentralized coordination modes—though hierarchical coordination could lead to higher levels of performance.


## 5.2.1. Overview

So far, the emergence of the coordination mode was studied for organizations with the distributed decision-makers operating at a moderate level of cost of effort (i.e., cost coefficientz=0.001). In the next step of this simulation study, the effect of cost of effort is analyzed. In particular, for the two interaction structures and the three search strategies under investigation, simulations with different levels of cost of effort are run. Apart from simulations for costless effort (i.e., z=0), also experiments for cost coefficients z∈0.003,0.005,0.01,0.02 were conducted. Figures 4 and 5 present results obtained for a medium cost level z=0.005 and a high cost level of z=0.02, respectively.Figure 4
Relative frequencies of coordination modes in the course of growth with medium cost of effort (cost coefficientz=0.005). Each plot represents results of 2,500 simulation runs: 250 landscapes with 10 runs on each. For parameter settings, see Table 2. (a) Decomposable-exploitation only, (b) decomposable-exploitation and exploration, (c) decomposable-exploration only, (d) nondecomposable-exploitation only, (e) nondecomposable-exploitation and exploration, and (f) nondecomposable-exploration only.
(a)
(b)
(c)
(d)
(e)
(f)Figure 5
Relative frequencies of coordination modes in the course of growth with high cost of effort (cost coefficientz=0.02). Each plot represents results of 2,500 simulation runs: 250 landscapes with 10 runs on each. For parameter settings, see Table 2. (a) Decomposable-exploitation only, (b) decomposable-exploitation and exploration, (c) decomposable-exploration only, (d) nondecomposable-exploitation only, (e) nondecomposable-exploitation and exploration, and (f) nondecomposable-exploration only.
(a)
(b)
(c)
(d)
(e)
(f)In order to be clear and concise, only the key findings and selected aspects of the sensitivity analysis are addressed explicitly. For this reason, the case of zero cost of effort is not discussed more in detail, since the experiments suggest that results obtained in the baseline scenarios (i.e., moderate cost level) do not change substantially when the decentralized decision-makers operate withcostless effort. However, with increasing cost of effort the emergence of coordination modes (given by their relative frequencies) notably changes compared to the baseline scenarios. The most interesting results show up for the nondecomposable structure which is why this will be discussed more extensively—particularly in conjunction with the “exploitation and exploration” strategy.With respect to thedecomposable structure, comparing results obtained for the moderate cost level (Figure 3) against those for medium and high cost (Figures 4 and 5) reveals the most obvious difference that—in each search strategy—hierarchical coordination (proposal mode) is the less often implemented, the higher the cost level. This effect is more pronounced for the ambidextrous and the explorative strategies. Analyzing the results more in detail reveals an interesting dichotomy between the relative frequency of emergence and the final performance Vt=750 of the coordination modes; this effect is even stronger in the nondecomposable structures and, therefore, will be discussed in the context below.In contrast, in thenondecomposable structure, the emerging coordination mode appears to be remarkably sensitive to an increase in the cost of effort. In particular, for the ambidextrous search strategy and the purely explorative strategy the “order” of coordination mechanisms changes compared to the baseline scenarios: recall that at the moderate cost level (baseline), the proposal mode clearly predominates (with a share up to more than 55%); now, with higher cost, this is the least often emerging coordination mode (partially with only about 12%).Moreover, this low frequency of occurrence of hierarchical coordination goes along with thehighest final performances compared to the other coordination modes. This clearly runs against intuition since one would expect that the better performing coordination mode is selected more often as it is the case for lower cost levels as shown in the baseline scenarios (see Section 5.1).To illustrate this effect, Figure6 plots—for the different cost levels simulated—the final performances Vt=750 and the relative frequencies in the end of the observation time for the case of the ambidextrous search strategy in the nondecomposable structure, while Table 5 reports on the respective details. As it becomes apparent, with a cost coefficient level of z=0.003 and higher, final performance and frequency of the hierarchical coordination run apart—or put the other way round: the worse performing coordination modes emerge more often (a similar effect shows up in the “explorative only” search strategy, and even for the “explorative only,” it is worth mentioning that the coordination modes show similar frequencies of occurrence while the final performance obtained with hierarchical coordination (“proposal”) goes remarkably beyond the level obtained with the other coordination modes).Figure 6
Sensitivity of final performance and relative frequencies of coordination modes to the cost of effort with the “exploitation and exploration” search strategy for the nondecomposable interaction structure. Each plot represents results of 2,500 simulation runs: 250 landscapes with 10 runs on each. For parameter settings, see Table2. (a) z = 0, (b) z = 0.001(baseline), (c) z = 0.003, (d) z = 0.005, (e) z = 0.01, and (f) z = 0.02.
(a)
(b)
(c)
(d)
(e)
(f)Table 5
Condensed results obtained for the nondecomposable interaction structure in the “exploitation and exploration” search strategy for different levels of cost of effort.
Cost of effort (cost coefficient)
Frequency of coordination modeac in t=750
Final performanceVt=750 (conf. interval∗)
Frequency of global maximum found int=750
Frequency of alterations (false positives) ofdt→
(a)
(b)
(c)
(d)
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
Decent.
Sequ.
Prop.
z
=
0
17.9%
28.4%
53.8%
0.878
±
0.0185
0.899
±
0.0120
0.932
±
0.0058
13.0%
12.4%
18.5%
14.4% (7.1%)
15.4% (7.6%)
16.3% (8.0%)
z
=
0.001
22.1%
33.0%
45.0%
0.891
±
0.0142
0.910
±
0.0092
0.933
±
0.0064
10.3%
9.3%
18.3%
12.4% (6.2%)
12.4% (6.1%)
14.2% (7.0%)
z
=
0.003
27.6%
37.2%
35.2%
0.903
±
0.0102
0.915
±
0.0079
0.934
±
0.0068
11.1%
14.1%
19.2%
8.5% (4.2%)
8.3% (4.1%)
11.3% (5.5%)
z
=
0.005
29.8%
43.7%
26.4%
0.914
±
0.0080
0.916
±
0.0063
0.935
±
0.0080
11.7%
11.9%
17.7%
5.8% (2.8%)
6.0% (2.9%)
9.0% (4.4%)
z
=
0.01
38.8%
45.6%
15.6%
0.901
±
0.0071
0
.
905
±
0.0062
0.940
±
0.0100
7.0%
9.1%
20.7%
3.2% (1.5%)
3.1% (1.4%)
5.0% (2.4%)
z
=
0.02
43.5%
44.2%
12.3%
0.877
±
0.0070
0.880
±
0.0070
0.940
±
0.0100
3.0%
4.2%
18.9%
1.9% (0.8%)
1.8% (0.8%)
3.0% (1.3%)
∗Confidence intervals for Vt=750 are given at a level of 99.9%. Each row represents averaged/aggregated results of 2,500 simulation runs: 250 landscapes with 10 runs on each.


## 5.2.2. On the Dichotomy of Coordination Modes in Performance and Frequency

Hence, an interesting question is what may cause this dichotomy in final performance and frequency of emergence for higher cost of effort in the “exploitation and exploration” strategy as shown in Figure6 and Table 5.A starting point for an explanation is that higher cost of effort at the distributed decision-makers’ side increase the propensity that they prefer to stay with the status quo. This, in turn, is best overcome by hierarchical coordination though with the risk of false positive alterations.In particular, with an increase in cost of effort, from a local decision-maker’s perspective, leaving the status quo in favor of an alternative configuration becomes less rewarding, or in other words, the higher the cost of effort the more promising an alternative to the status quo—inducing effort—has to be for being selected, i.e., only alterations that promise more than the cost of effort are preferable from a decentralized decision-maker’s perspective. The search strategies enforce different levels of effort to be taken for leaving the status quo; for example, the purely explorative strategy requires making long jumps (i.e., switching two single choices) which—according to the quadratic cost function in equation (7)—induces rather high cost of effort.Hence, with an increasing cost coefficientz, in tendency, in all three modes of coordination, keeping the status quo becomes more attractive for the local decision-makers and this effect is the more pronounced the higher the effort enforced by the search strategy. However, the coordination modes are differently prone to this kind of “inertia”: In the decentralized mode the local decision-makers’ first preferences are implemented without any further revision. In the sequential mode, distributed decision-makers’ first preferences may be revised, though from a parochial perspective.In contrast, in the proposal mode it is more likely that the decentralized decision-makers’ preferences are overrided by the central agent with respect to the overall performance and, hence, alterations, though causing cost on the decentralized side, may be induced. This, in turn, makes it more likely that the status quo is abandoned and a new configurationdt→ is implemented. However, since the central agent disposes of rather noisy information (σcent), these alterations could also be in favor of a false positive alteration. This explanation is broadly confirmed by the ratios of periods with alterations and with false positive alterations (for the metrics, see Section 4.3): These ratios are particularly higher in case of hierarchical coordination (proposal mode) as reported in Table 5 for the example of the ambidextrous search strategy (with increasing levels z of the cost coefficient the alterations decrease for all coordination modes under investigation but the relative differences among the modes increase; the confidence intervals of the final performance achieved shows in a similar direction: with levels of cost coefficient z=0.005 and higher the confidence intervals for the proposal mode are higher than those of the other coordination modes).With respect to the emergence of coordination modes, the stability in terms of, at least, keeping a once achieved performance level provided by a particular mode of coordination drives the propensity of being implemented in the future. Hence, since the decentralized and the sequential mode in combination with higher cost coefficients of effort and a search strategy enforcing exploration induce more inertia and, on average, lower performance than hierarchical coordination, according to the simulation results, these coordination mechanisms predominate.In this sense, the somewhat counter-intuitive results for higher levels of cost of effort may provide an explanation for the ambiguous results obtained in empirical studies on the relation between tightness of coordination on the one hand and organizational size as well as intraorganizational complexity on the other hand as reported in Section2.For a further interpretation of results, it appears worth mentioning that the level of cost of effort in the model (captured by a simple cost coefficientz) may represent rather different aspects and—depending on its particular context—the results of this study show in quite different directions. For example, the cost of effort could capture “switching costs” for altering a status quo in favor of a new solution which may be specific to a certain branch or industry due to technological aspects [107, 108]. In this sense, the results may be regarded as an indication that, for example, in different branches different coordination modes might be predominant.In another interpretation, cost of effort could capture the costs for dealing with resistance of certain stakeholders or (emotional) costs for loosing old and building new intra- or extraorganizational relations when new solutions are to be implemented [109–111]. Regarded in this sense, the results suggest that organizations where, for example, high resistance or high relational costs for implementing the “new” are to be expected, in tendency, might stick to decentralized coordination modes—though hierarchical coordination could lead to higher levels of performance.


## 6. Conclusion

This paper presents a computational study on the emergence of the mode of coordination in the course of an increasing decision-problem and, in consequence, a growing number of decentralized decision-makers in DDMS. In the theoretical model and the simulation model accordingly, the mode of coordination employed is subject to learning-based emergence, while the search strategy—being exploitative, explorative or of an ambidextrous nature—is regarded as given (e.g., by a central authority). The same holds for the level of cost of effort that the distributed decision-makers face for implementing new solutions.For moderate levels of cost of effort, the results suggest that DDMS facing a growing nondecomposable decision-problem would increasingly employ vertical information flows, broad information, and decision-making via hierarchy rather than granting high autonomy to decentralized decision-makers. In contrast, when growth means that additional self-contained decision-problems are to be solved, according to the simulation results, with the exploitative and the explorative search strategy, no predominance in the coordination mode appears; in an ambidextrous search strategy, coordination modes tend to prevail that leave the decision-making authority at the side of decentralized decision-makers. For the levels of complexity studied, results suggest that the ambidextrous strategy bears the highest potential of superior performance while enforcing high levels of novelty by a “purely” explorative strategy leads to inferior performance.However, the results suggest that the higher the cost of effort on the side of decentralized decision-makers are, the more likely it may become that those coordination mechanisms emerge which provide high autonomy to decentralized decision-makers though employing hierarchical coordination could lead to higher performance obtained by the DDMS.These results could, for example, be regarded as an explanation for some ambiguous empirical findings since different industries may differ in the “switching” costs in favor of new solutions due to technological aspects. This study may also provide an explanation for organizational inertia when resistance against change is of relevance or change comes along with high relational costs to decision-makers.Moreover, a particular contribution of this study may also lie in its method: It appears worth mentioning that with the computational study introduced here some empirical findings related to (the emergence of) the coordination mode as a part of the boundary system obtained in prior empirical research [26, 48–51, 106] were “replicated.” In particular, via growing DDMS “from scratch,” the coordination mode at the system’s level emerged from some rather simple (for not to say simplistic) components and behavioral rules at the individual agents’ level. In this sense, computational studies directed to growth processes could contribute to the research in the various domains of organizational thinking since they allow to capture growth processes without having to deal, for example, with methodological challenges of longitudinal empirical research as well as hardly controllable contingent factors [35].This study calls for further research activities: First, it is worth mentioning that this study, by far, does not represent the width of modes of coordination. Hence, a natural extension of the research effort presented here would be to integrate further coordination modes, like, for example, negotiations among distributed decision-makers.Second, the learning mechanism employed in this study, i.e., reinforcement learning, is rather simple, and more sophisticated learning modes should be studied. For example, in contexts representing organizations resided by human decision-makers, belief-based learning [92] could be a reasonable alternative.Third, it is to be mentioned that the model presented here builds on rather simple cost functions: As such, the model does not capture any cost for coordination which, of course, vary across coordination modes (e.g., [6]). Furthermore, the cost of effort could be modeled in a more sophisticated way by, for example, distinguishing between industry-specific technological “switching cost,” cost of effort related to the capabilities of agents, as well as costs due to intraorganizational resistance or costs due to specific investments to name but a few.Fourth, a natural extension would be to study the emergence of coordination in the context of further types of controls: For example, in the context of DDMS with human decision-makers, according to the “Levers of Control” framework [22, 106], the emergence of coordination in the context of commonly shared values as part of the beliefs system or of incentive systems being part of the diagnostic control system is of interest. These ideas relate to the growing body of research emphasizing the internal fit and the balance between the various types of control mechanisms. Hence, further computational studies may build on more comprehensive models of mechanisms to control decision-making and examine the interrelations among these mechanisms for different contingent factors like, for example, the complexity of the decision-problem.



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*Source: 4398620-2019-12-06.xml*