Introduction

Complex systems are systems consisting of interacting agents. Such systems are ubiquitous, arising in the biological, social and physical sciences. Examples include human society with interacting people; an ecosystem of interacting species; the solar system in which the planets and the sun interact through gravity; a bee colony; the human brain consisting of interacting neurons; a business organization consisting of workers and managers; a financial market of buyers and sellers; the human body consisting of interacting cells; the military; and the internet with interacting computers.

While in recent times researchers have borrowed ideas, methodologies, and mathematical models from the complex systems literature to study business problems (discussed in more detail below), the thesis here is that complex-systems researchers can benefit significantly from some of the ideas, methodologies, and mathematical models used in business research. Specifically, much of the study of complex systems has been based on the fascinating property of self-organization, in which individual agents in a complex system organize themselves, with no external influence, in such a way as to produce interesting and useful emergent system behaviors. Self-organization has been used to explain behavior in many socioeconomic systems including market economies (Ruzavin, 1994; Markose, 2005), politics (Rhee, 2000), entrepreneurship (Nicholls-Nixon, 2005), innovation projects (Harkema, 2003), and task-specialization (Solow and Szmerekovsky, 2005). Self-organization has also been observed in both biological systems (Bonabeau and Meyer, 2001; Capra, 2005) and physical systems (Richardson, 2005). In business organizations, however, central organization, with leadership in particular, plays a significant role in the organization’s performance (see later).

While many complex systems may initially arise from self-organization (e.g., the planets, the World Wide Web, and perhaps even biological organisms), as many of these systems evolve it is the emergence of some sort of centralized organization that allows for high levels of performance that might not have been achieved otherwise. An obvious example is human society, which may have started out more in a self-organizing mode but has since evolved hierarchical governmental structures involving central control that account, to a large degree, for social successes over the years. Other examples of complex systems whose performance is due in large part to central organization include the human body, controlled by the brain and the nervous system, and the solar system, in which leadership is embodied in the role of the sun.

To the extent, then, that the behavior of certain complex systems is affected greatly by central organization, our understanding of the behavior of these systems should include the study of how central organization and leadership affect system performance. For instance, it is commonly accepted that one role of central organization is to exert control over the agents of a complex system. But how much control should be exercised to achieve optimal system performance; or, in other words, under what conditions do systems benefit from different amounts of central control? There are systems that function well with little or no central control (such as the internet), while other systems require high levels of central control to achieve good performance (such as the military or the solar system as controlled by the gravitational force of the sun), and still other systems that function best with intermediate levels of control (the human body, for example). The answers to these questions have practical implications. For example, how much control should the Federal Reserve board exercise in the form of interest rates? To what degree should the federal government control the price of heating oil and natural gas? To be sure, these are very difficult, maybe even intractable questions. However, mathematical models are presented later in this paper and in the appendix to indicate that indeed there are mathematical conditions under which optimal performance of a complex system can be achieved with no central control, other conditions under which intermediate levels of control are best, and still other conditions under which the system requires full control for optimal performance.

The impact of complex-systems research on business research

Because the behavior of a complex system generally results from interaction among the agents (and not from any single agent), a complex-systems approach is to study the system as a whole, attempting to understand how agent interaction affects the system (rather than to take a “reductionist” approach of studying the individual agents). This kind of “holistic” approach also arises in the study of business organizations, a prominent example being the supply chain, consisting of suppliers, manufacturers, retailers, and customers. Rather than studying each individual player in the supply chain in isolation, supply chain management requires studying how the constituents interact with order quantities and sales through demand uncertainties and prices (see, for example, Mantrala and Raman, 1999; Munson and Rosenblatt, 2001; and Kolay et al., 2004).

Another example of holistic complex-systems thinking in business research is in the realization that the actions and decisions in one functional area affect the performance of other functional areas. Thus, rather than using the reductionist approach of studying each functional area separately — as was often done in the past — much recent research in business includes the interaction of two or more functional areas. Kaeter (1993) describes the business world’s transition from specialized “silo” structures to more general cross-functional team structures. Current theory suggests the use of cross-functional teams in a variety of areas to improve organizational performance. These areas are as diverse as in-store IT development (Rowen, 2006), business partnerships (Boedeker and Hughes, 2005), safety enhancement (Smith, 2005), new product development (Fredericks, 2005; Mosey, 2005), and global-sourcing (Gopal et al., 2004). However, these improvements are not automatic. For example, Webber (2002) indicates the importance of trust in improving cross-functional team performance, and Rowe (2004) identifies the role of accounting procedures and team structure in eliminating free-riders in cross-functional teams. In addition, Mohamed et al. (2004) recommend combining cross-functionality and knowledge management to achieve superior results.

Other examples where complex-systems approaches have influenced research in business include the NK model, which was initially developed by Kauffman and Levin (1987) and Kauffman (1993) to study the impact of interactions among genes on chromosome evolution by way of the notion of a fitness landscape and has been used in the study of business organizations since the mid-1990s (Levinthal, 1997; Levinthal and Warglien, 1999; and McKelvey, 1999). Rivkin (2000) suggests that firms can use a fitness landscape to prevent competitors from copying successful business strategies. Another application of the NK model arises in the study of an organizational team consisting of interacting workers (Rivkin and Siggelkow, 2002). Solow et al. (2002) use the NK model to provide managerial insights on how worker interaction affects the performance of various worker-replacement policies.

How business research can influence complex-systems research

While much of the foregoing business literature borrows from the complex-systems approach, business research can provide ideas that have an equally valuable impact on the study of complex systems. One such example, as mentioned in the introduction, is the impact of leadership on organizations, for which there is a large and rich literature that we briefly summarize into four categories (Luthans et al., 1988):

  • Traditional management activities, such as planning (Kotter, 1982; Bennis and Townsend, 1995; Chemers, 1997), decision making (Mintzberg, 1973; Kotter, 1982), and controlling (Kotter, 1982; Bennis and Townsend, 1995; Chemers, 1997);

  • Human resource management activities, such as motivating/reinforcing (Fiedler, 1967; Mintzberg, 1973; Kotter, 1982; Manz and Sims, 1989; Cole, 1996; Chemers, 1997), disciplining/punishing (Mintzberg, 1973), seeking cooperation and managing conflict (Mintzberg, 1973; Kotter, 1982), staffing and training/development (Mintzberg, 1973);

  • Communication activities, such as exchanging information (Mintzberg, 1973; Kotter, 1982; Chemers, 1997) and handling paperwork (Mintzberg, 1973);

  • Networking activities, such as interacting with outsiders (Mintzberg, 1973) and socializing/politicking (Kotter, 1982).

Some business researchers who have borrowed ideas from the complex-systems literature have realized the need to include some of the foregoing leadership roles in their work. For example, when studying teams, Solow and Leenawong (2003) and Solow et al. (2005) modify the NK model to include a leader and show how the effects of cooperational and motivational leadership, in addition to worker interactions, affect team performance. In a different direction, Rivkin and Siggelkow (2003) extend the NK model to capture management’s decision-making role. Their paper considers two department managers and a CEO, where decisions made in each department affect other decisions throughout the firm. Insights are provided into how managerial ability, incentives, interaction, and the way decisions are assigned have impacts on firm performance.

The thesis presented here is that the study of complex systems would benefit significantly by considering how leadership affects system performance. Thus, a careful analysis of the foregoing leadership roles is needed to identify which ones apply to other complex systems; or better yet, to broad collections of complex systems. For example, consider the role of a leader as one who exerts control on the individuals in the organization. By “control” is meant any of the direct and indirect means by which a central authority seeks to achieve enhanced performance by changing the behavior of the individuals. This change in behavior can be realized, for example, in an authoritarian manner by issuing rules, orders, and regulations (as in bureaucratic control); in a motivating manner by providing directives and incentives (Eisenhardt, 1989; Fama, 1980; Jensen and Meckling, 1976); in a cooperative manner by achieving coordination among the agents (Lee, 2001); or in a passive-responsive manner by exerting influence only when the system is not behaving within specified limits.

The concept of control applies to many complex systems. For example, governments control their populations through laws and regulations; the brain and central nervous system control the human body through bio-feedback loops; the sun’s gravitational field controls the orbits of the planets in our solar system. Thus, as mentioned in the introduction, an interesting and important question is how much central control is good for complex systems; or, more accurately, under what conditions do complex systems benefit from low, intermediate, and high levels of central control? While no specific answers are provided here, a model is presented to show that there are mathematical conditions under which optimal performance of a complex system is achieved with no central control, other conditions under which partial levels of control are best, and still other conditions under which the system requires full control for optimal performance.

A mathematical framework for studying central control

To study the effects of different amounts of central control, a complex system is represented as a group of n agents together with a leader. The effort of each agent i under the influence of the leader is assumed to result in a real number, xi. These efforts are collectively denoted by x = (x1, …, xn) and are called the agents’ outputs. System performance depends on the agents’ outputs and is represented by the real number p(x), with larger values of p(x) denoting better system performance.

The key issue is how the leader’s control affects system performance. To that end, control is assumed to change the agents’ behaviors, which, in this model, are represented by the agents’ outputs. Thus, control by the leader changes the agents’ outputs, which in turn change system performance.

When the leader has direct and immediate influence on the individual agents, it is possible and reasonable for the leader to exert a different amount of control on each agent. This would be the case, for example, with the manager of a department in a business organization or with a platoon leader in the army. However, when the leader has less immediate contact with the agents (for example, the CEO of a large corporation or the Secretary of Defense for the military), it is impractical (or unfair) for the leader to set different amounts of control for each agent. In this case, the leader’s control, in theory, affects all agents equally. This latter situation is modeled here by the assumption that the control by the leader on each agent is the same. As such, the amount of control is denoted by a real number λ > 0, in which λ = 0 corresponds to no control and λ = 1 to full control, with increasing values of λ corresponding to increasing amounts of control. A small value for λ might reflect a system in which the agents require little supervision or oversight, such as the internet or the faculty at a university. In contrast, a value for λ close to 1 might correspond to detailed regulations that are aggres-sively enforced through constant supervision of the agents, such as in the military.

When left on their own with no control from the leader (λ = 0), each agent i has no restrictions and is therefore allowed to choose its own output, which is represented by the real number xi(0). The specific way in which the agent chooses this output value is not important in the model proposed here. Thus, x(0) = ( x1(0),…,xn(0) ) represents the collective outputs of the agents when there is no control and the associated system performance is p(x(0)).

As the leader exerts more control by increasing the value of λ, the effect is to change the agents’ behaviors and hence their outputs from x(0) to new values denoted by x(λ) = (x1(λ), …, xn(λ)). An important question is how the values of x(λ) are determined in that regard. It is assumed that the leader has an a priori estimate of the agents’ output values, say, x(1) = (x1(1),…,xn(1)), that would, in the leader’s opinion, provide maximum system performance. Note that one measure of the skill of the leader is how close the leader’s values of x(1) are to the actual agent outputs that result in optimal system performance. In any event, an obvious question to ask is: If the leader believes that the output values x(1) are best for system performance, then why should the leader not exert full control? One answer is because the leader does not know the performance function explicitly, for example in a nascent business. Therefore, by exerting full control, the leader may find that the outputs x(1) do not result in the best system performance. Thus, the leader needs to determine the optimal amount of control λ to exert so that the agents’ outputs, x(λ), maximize the system performance p(x(λ)). In order to analyze this optimization problem, it is necessary to know how the agents’ outputs, x(λ), vary as the amount of control λ varies and also the specific form of p(x); that is, how the performance of the system varies as the agents’ outputs vary.

Analytical results from a specific form of the general model

Specific forms are now given for how the agents’ outputs, x(λ), change as a function of the amount of control and for how system performance, p(x), changes as the agents’ outputs change. Keep in mind that the forms proposed here are not meant to be realistic or to reflect any specific complex system. Rather, the proposed forms of x(λ) and p(x) are used to show that there are mathematical conditions under which optimal system performance is achieved with no control, other conditions under which the system performs best with an intermediate amount of control, and still other conditions for which it is best to exert full control. To simplify the model, the system is assumed here to consist of a leader and a single agent (and is extended in the appendix to include n agents).

Turning first to how the amount of control λ affects the agent’s output, as the leader increases control from 0 to 1, the agent’s output is assumed to change in a linear way from x(0) (the outputs chosen by the agent under no control) to x(1) (the agent’s output deemed best by the leader). That is, for any amount of control 0 ≤ λ ≤ 1, the output of the agent is given mathematically as x(λ) = (1 – λ)x(0) + λx(1).

Turning to the performance function, a quadratic form is assumed; that is, when the output of the agent is x, the system achieves the performance p(x) = qx2, where q is a given negative number. Observe that the maximum possible system performance is p(x) = 0 and is achieved when the agent’s output is x = 0. The value of q represents how fast system performance deteriorates as the agent’s output deviates from its optimal value of 0.

Having specified both x(λ) and p(x), the goal of the leader is to determine the optimal amount of control – that is, the value of λ – that solves the following problem:

Max p(x(λ)) = qx(λ)2 (1)

Subject to x(λ) = (1 – λ)x(0) +λx(1) 0 ≤ λ ≤ 1

Because of the specific form chosen for p(x) and the fact that q is negative, it is possible to solve the optimization problem in (1) by setting the derivative of p(x(λ)) with respect to λ equal to 0 and solving for λ. Details of doing so are given in the appendix, but the result is that the solution, λ*, to the optimization problem is given in the following cases.

Case 1: No control (λ*= 0).

The optimal solution to (1) is λ* = 0 if and only if x(0)x(1) ≥ x(0)2.

Case 2: Full control (λ* = 1).

The optimal solution is λ* = 1 if and only if, x(1)2x(0)x(1).

Case 3: Partial control (0 < λ* < 1).

Finally, 0 < λ* < 1 if and only if x(0)2 > x(0)x(1) and x(1)2 > x(0)x(1).

While the foregoing mathematical model is not realistic, it is possible to conjecture conditions under which real-world systems benefit from differing amounts of control. For instance, it is likely that a system will benefit from low levels of control when the agents’ choices of outputs under low control are closely aligned with those outputs that result in good system performance. One such example is the faculty at a university who, when left alone, generally choose actions that result in good performance for the university. Another circumstance in which low levels of control are likely to be best is when the survival of the individual agents depends on the survival of the system, as in early human civilizations where self-organizing behavior, more than leadership, resulted in the survival of the species. A final circumstance under which systems might benefit from low levels of control is when enforcing the leader’s will on the agents would result in outputs that degrade rather than improve system performance. Though much speculation is often made as to when this situation occurs, no general scheme is proposed here.

A condition on real-world systems under which exerting a large amount of control is likely to be beneficial occurs when the agents’ self-chosen outputs under no control conflict with the performance and survival of the system. One such example is a prison. The prisoners would like to escape the prison, whereas the goal of the prison system is their continued incarceration. Other conditions under which high levels of central control might be beneficial are when the leader has accurate knowledge of the agents’ outputs that indeed are good for the system, and small deviations from those outputs result in poor system performance. One example of such a system is the classic assembly line, where the leader knows what each worker should do to get the job done and so exerts a lot of control by specifying what, when, and how tasks should be done. Further, should a single member of the line fail in his or her duties, the final product will prove defective and/or the assembly line may be forced to shut down. Hence, it is necessary and beneficial for central control to be enforced on assembly lines. Even some more modern manufacturing methods such as just in time (JIT) and statistical process control (SPC) require regimented control of the system agents. Employing these methods requires a clear target for each agent. Determining such targets requires experience of which behaviors result in good system performance. Hence, “full control” is likely to be beneficial in very stable environments where system needs do not change significantly over time.

As we have seen, low and high levels of control are beneficial if the system is volatile or stable, is in danger of extinction unless the agents choose appropriate outputs, or if the leader’s knowledge of the system is extremely poor or accurate. Moderate levels of control might be appropriate under less severe conditions. For example, if the system is facing rapid change then it is not always clear what agent outputs will be optimal for system performance and if the system is not on the edge of collapse, agents will likely pursue their own interests along with the system’s interests. Under these circumstances “full control” is not appropriate because the leader’s knowledge of the outputs that are best for the system is not complete. Instead, the leader must depend in part on the agents to determine what is best for the system. However, if “no control” is used, the agents could easily abuse their freedom at the expense of the system or oth-erwise choose outputs that do not result in good system performance. Hence, some level of “partial control” will prove best. This situation occurs in such areas as entrepreneurship, venture capitalism, new product development, and emerging markets. A similar scenario arises when the success of a system requires inputs from agents with diverse areas of expertise. The leader is not likely to be an expert in all areas and hence cannot exercise “full control,” but must depend partially on the agents to determine their own inputs. Of course, “no control” is again an unsatisfactory option. This situation occurs with many contemporary management structures, including cross-functional teams, matrix structures, project management, and organic structures.

A model has been presented for which there are conditions under which it is optimal for the leader to exert no control (λ* = 0), partial control (0 < λ* < 1), and full control (λ* = 1) over the agents. However, this model has many deficiencies. For example, no mention has been made of how the agents determine their preferred input levels. A model that overcomes these deficiencies and allows the agents to choose their outputs based on their own individual utility functions is presented in Solow and Szmerekovsky (2006).

Conclusion

While the study of complex systems and complex-systems thinking has influ-enced research and approaches to research in business, the reverse appears not to have taken place to any significant degree. Recognizing that central organization and leadership have a substantial impact on business organizations, it has been argued that there are potential benefits to be gained by considering the role played by central organization and leadership in general complex systems. Consideration of numerous examples has shown that there are many complex systems whose extraordinary performance is attributable, in large part, to central control exerted by a leader. A simple (but non-realistic) model has been presented here to illustrate that there are mathematical conditions under which optimal performance of a complex system is achieved with no central control, other conditions under which intermediate levels of control are best, and still other conditions under which the system requires full control for optimal performance

In the authors’ opinion, this model is the tip of an iceberg that represents the wealth of results and insights that can be realized from research in complex systems that takes into account the role, value, and importance of central organization and leadership. For example, one interesting question is to understand under what conditions complex systems benefit from each different type of central control (authoritarian, motivational, cooperational, and passive-responsive). Another direction is to investigate the extent to which leadership roles other than control, as identified in this paper, affect the performance of complex systems in general. No central organization is needed to conduct this research, but the authors are grateful to the central organization of the editors that led to this special issue of E:CO.

Mathematical appendix

The model given in (1) above is now generalized to include n agents. To that end, when the leader increases control from 0 to 1, the output of each agent i is assumed to change in a linear way from xi(0) (the output chosen by that agent under no control) to xi(1) (the agent output deemed best by the leader). That is, for any amount of control 0 ≤ λ ≤ 1:

xi(λ) = (1 – λ)xi(0) + λx(1),

or equivalently, using vectors,

x(λ) = (1- λ)x(0) + λ x(1). (2)

Turning to the performance function, a quadratic form is assumed; that is, when the outputs of the agent are x, the system achieves the following level of performance:

p(x) = xTQx, (3)

where Q is a given (n×n) negative semidefinite symmetric matrix. Observe that the maximum value of p(x) in (3) is 0 and is achieved when the agents’ outputs are all 0; that is, when x = (0, ., 0). The values of Q represent the way – that is, how fast – system performance deteriorates as the agents’ outputs deviate from their optimal values of 0.

Having specified both x(λ) and p(x), the goal of the leader is to determine the optimal amount of control – that is, the value of λ – that solves the following problem:

Max p(x (λ)) = x)TQ x(λ) (4)

Subject to x(λ) = (1 – λ)x(0) + λx(1) 0 ≤ λ ≤ 1

Because of the specific form chosen for p(x) in (3) and the fact that Q is negative semidefinite (and hence p(x (λ)) is concave in λ), it is possible to solve the optimization problem in (4) by setting the derivative of p(x(λ)) with respect to λ equal to 0 and solving for λ. To that end, notationally letting y = x(0) and z= x(1), for x(λ) = (1 – λ)y + λz, it follows from (3) that

p(x(λ)) = [(1 – λ)y + λz]TQ[(1 – λ)y + λ z] = (1 – λ)2yTQy + 2λ(1 – λ)yTQz + λ 2zTzQz = (yTQy – 2 yTQz + zTQz2 + (2 yTQz – 2 yTQy)λ + yTQy.

Letting,

a = yTQy – 2 yTQz + zTQz = y zTQ (y z), b = 2 yTQz – 2 yTQy = 2(yTQz yTQy), c = yTQy, f(λ) = p(x (λ)),

pa]it follows that

f(λ) = aλ2 + bλ + c and (5)

f’(λ) = 2aλ + b (6)

The solution, λ*, to the optimization problem in (4) for the foregoing special form of f’(λ) in (5) and f’(λ) in (6) is given in the following cases.

Case 1: No control (λ* = 0). It is easy to prove that the optimal solution to (4) is λ* = 0 if and only if ’(0) < 0; that is, if and only if f’(0) = b = 2(yTQz yTQy) ≤ 0; that is, yTQzyTQy.

Case 2: Full control (λ*= 1). Likewise, the optimal solution to (4) is λ* = 1 if and only if f’(1) ≥ 0, that is, if and only if f’(1) = 2a + b = 2(zTQz yTQz) ≥ 0; that is, zTQz yTQz.

Case 3: Partial control (0 < λ* < 1). Finally, if f’(0) < 0 and f’(1) < 0, the optimal solution to (4) is the value of λ* for which f’(λ*) = 0, that is,

λ* = –b / 2a = (yTQyyTQz) / (yTQy – 2yTQz + zTQz),

with 0 < λ* < 1 if and only if 0 < b < -2a; that is, if and only if yTQz > yTQy and zTQz < yTQz.