On the Nonlinear Nature of Competition
Some of the fundamental studies on competitive dynamics are based on an equilibrium relation between a firm’s performance, its competitive activity, and the levels of cooperation and competition throughout an industry. Nevertheless, research on the theory of the firm and evolutionary economics indicates that other industry participants also influence how companies take “make-or-buy” decisions that can indirectly affect the cooperative and competitive landscape at both firm and industry levels. In this essay, competition is analyzed through the lens of nonlinear dynamics, a tool that has been successfully used in the social and natural sciences because it can accurately describe complex and chaotic behaviors emerging in nature. From this perspective, all industry participants are considered independent agents, each one with sets of strategies and possible actions to take given a particular state of the world; the aggregate of those actions defines the final state of the industry. This approach is based on the understanding of the collective behavior of multiple independent agents and how feedback loops emerge from the inevitable inefficiencies inherent in complex dynamical systems.
Carnegie Mellon Professor Steven Keppler’s very influential work on firm survival and the evolution of oligopoly developed a model where firms differentiate by their research and development (R&D) productivity and total output is impacted by new entrants. Under Keppler’s model, firms must make three major decisions: when to enter and exit the industry, how much to invest in R&D, and how much to spend on growth and meeting new demand. Under this model, long-term performance and survival expectations are sensitive to these initial conditions (Keppler, 2002) and help shape the number of competing organizations in industrial populations (Carroll & Hannan, 1989).
Sensitive dependence on initial conditions applies to a broad range of dynamical systems (Glasner and Weiss, 1993), which possess a high degree of adaptability to their ever-changing environment. Such complex adaptive systems are often modeled as networks of independent interacting agents that “evolve in time to produce a complicated interplay between deterministic and stochastic outputs” (Manrique et al., 2018) that generate feedback loops directing the whole system toward a specific final state. These systems can be centralized — with all agents sharing information through a central node — or decentralized — with control agents having “nonidentical information on the system structures, states, parameters, and so on”. This definition of decentralization is common in economic literature (Aoki, 1971) and is consistent in competitive markets with different types of firms acting as system components.
The decentralized nature of competing firms is accentuated by a dynamic conception of competitivity that has its origins in Joseph Schumpeter’s ideas of “creative destruction,” the Austrian School of Economics’ emphasis on market processes, and the structure-conduct-performance paradigm of industrial organization. All of these are based on the idea that competition is a dynamic market process that moves toward equilibrium without ever reaching it because “the profit forces for action will disrupt any stable state or status quo” (Smith, Ferrier and Ndofor, 2001). In general, firms take “externally directed, specific, and observable competitive moves to enhance [their] relative competitive position” (Young, Smith and Grimm, 1996) and reach their individual optimization goals.
Firms’ competitive actions create positive reinforcement loops where those that are ahead get further ahead and those losing advantage lose further advantage consistently with the law of increasing returns (Arthur, 1988). Economists have long known that increasing returns can cause multiple equilibria and that this kind of local positive feedback loops create unstable states and stable critical points known as attractors (Arthur, 1988). These dynamic systems are nonlinear and present in other subfields of economics that allow self-reinforcement mechanisms, such as spatial economics and industrial organization (Arthur, 1988), that not only share similar dynamical behaviors to competitive dynamics but are also interrelated with the competitive nature of firms. Nevertheless, existing research on the field does not provide clear insights about types of positive reinforcement loops, unstable states, or attractors of systems of competitive actions and responses that conform to competitive dynamics. Therefore, a deeper analysis of the field’s nonlinear characteristics is required to develop a new framework that is not linear or centralized.
In order to define such a framework, the following assumptions are made: (1) Firms need to cooperate and compete. (2) The distribution of firm costs or quality can be mapped to firm and industry profits. (3) The levels of differentiation among competitors are positively related to the magnitude and scope of competitive actions. Therefore, any set of strategies should consider a firm’s self-interest and how its aggregated actions can affect the competitive landscape. In a competitive market, when a company opts to integrate vertically instead of outsource (or vice versa), such a decision may affect the supply of raw materials, labor markets, production levels, demand for professional services, and research activities at universities, among others, even when it acts independently and consider no other information beyond what is required to pursue its own benefits. Also, since firms look to create and defend competitive advantages over other firms, no central authority can coordinate the actions between all agents in the market; thus, they are decentralized. Therefore, when the level of coordination between actors increases — either by increased vertical integration or by other noncompetitive practices — the level of centralization increases and the marketing decision power is limited to a reduced number of participants. This introduces increasing returns to scale — where larger companies produce more efficiently than small companies — thus creating multiple local feedback loops that amplify deviations from certain states, making them unstable equilibrium points.[1]
As in many subfields of economics, competitive dynamics’ critical points — or “equilibria” — lie within the boundaries of some smooth manifold,[2] implying the existence of other stable fixed points or attractors (Arthur, 1988). A standard example of complex competitive interactions is when a company that is experienced and highly integrated vertically announces a new product with improved performance that competes with another company’s bottom line. If one company decides to accept reduced profits and cedes their market share to the other, it will create unstable equilibria. If both decide to either engage in aggressive competitive actions or accept reduced returns at the same time, it will create local minima, allowing the previous market structure to be replaced by a monopoly/duopoly or by an open market, with new participants enticed by the increased demand — a result of aggressive competition. In all cases, outputs and returns will stabilize at different levels, and they will depend on the initial conditions and the characteristics (e.g., radicality, scope, magnitude, and irreversibility) of the firms’ competitive and collaborative actions (Smith, Ferrier and Ndofor, 2001)
The nonlinear nature of the competitive landscape highlights two fundamental themes of the self-reinforcing mechanism in economics: (1) Since there is no accepted dynamic about how agents should compete or cooperate, there is the question of how a particular equilibrium is selected from various candidates, what Arthur (1) calls the “selection problem,” which remains mostly unresolved. (2) Self-reinforcement mechanisms accumulate into an economic advantage that forms barriers separating the minima. So once a market structure occupies a local minimum, it will be locked into that position until it receives the sufficient influx of energy to overcome the barrier.[3]
From the perspective of complexity economics, the selection problem is approached adaptively. Initial conditions combined with early random events or fluctuations push the dynamics into the domain of some emergent structures, thus selecting the market configuration that the system eventually locks into.[4] These structures emerge through initial market fluctuations, so agents compete to occupy a local minimum and get ahead by exercising competitive exclusion on their rivals; the prevailing agents — those with better market positions — have a “selectional advantage” (Arthur, 1988). As we can see, the nonlinear nature of these systems makes them indeterminate (i.e., outputs cannot be predeterminate) and with nonergodicity — path dependence where small events cumulate, making the systems gravitate toward an outcome rather than others (Arthur, 1994). Therefore, it is only by studying how random fluctuations build up that it is possible to observe how outcomes are selected.
One potential “solution” to the selection problem can also be found in nature, where brainless (i.e., decentralized) larvae display goal-oriented behaviors, realized through biased random walks where paths are extended in the direction of the stimulus through an error-correction mechanism of temporal sampling known as weathervaning or klinotaxis (Manrique et al. 2018). This model has a system with a number of agents. Based on the set of strategies that each agent holds, each of which is a look-up table of actions to take given a particular history of recent global outcomes, the aggregate action of these agents will dictate the system’s trajectory during the next and subsequent timesteps (Manrique et al. 2018). This can be considered under a market environment where agents represent a firms, each one with some optimal level of returns. Then, the global outcome produced at each timestep and the information about previous outcomes, will inform whether the aggregate of strategies that firms use to decide their actions was good or bad in terms of moving toward or away from the target returns.
The initial state of the system is arbitrary and influenced by an initial set of endogenous variables, such as the level of industry experience, the time of entry into the market, and the levels of vertical integration.[5] If the eigenvalues and eigenvectors describing that initial state are calculated at point p0 and timestep t0, the system reaches subsequent points p0, p1,…, pn at timesteps t0, t1,…, tn. If it is also assumed that the target is located at a fixed point pn, the firm, at any point, may take (among many others) a set of pricing, marketing, product, capacity- and scale-related services and operations, and signaling actions to perform a system-target alignment over time[6] so the eigenvector will “point” in the target’s general direction. To determine whether an individual action is good, we look at the change it produces at the direction of the system at time tm compared to time tm-1; a good action would improve the alignment toward pn.
It is important to understand that since the model is following a random walk, there is a chance that the market gets stuck in a local minimum that is not efficient (i.e., does not give optimal returns) before it reaches the point of maximum returns; if it reaches that point (as discussed above, it could when one company decides to accept reduced profits and cedes market share to the other), it will be an unstable equilibrium. This is not only consistent with the idea of bounded rationality[7]; but it also addresses the nonlinear nature of the mechanisms through which particular equilibria are selected. Also, it can shed light on why monopolies and oligopolies are so hard to break or the emergence of arbitrage opportunities in highly competitive markets (Gebarowski, Watorek and Drozdz, 2019). It can even help measure the level of competitiveness since a market with many buyers and sellers, undifferentiated products, low or no transaction costs, no barriers to entry and exit, and perfect information about the price of goods will not allow positive feedback loops. Also, the system will not move away from the equilibrium point because any deviation from the current state will be small. Therefore, as markets become less competitive, it will become more unstable, and the chances of being locked into a local minimum will increase.
From the beginning, neoclassical economics was based on the idea of the existence of equilibria in the markets, from Adam Smith’s “invisible hand” to Alfred Marshall’s supply-and-demand curve to homo economicus, a world of rational choices where positive feedback loops were rare and used equilibria to create representations of the economy to maximize utility functions. This approach has been dominant in economic analysis for many reasons, but its empirical failings have attracted a wide range of alternative models (Levin and Milgrom, 2004). Competitive dynamics, which has its origins in Schumpeter’s theory of creative destruction and evolutionary economics, presents itself as one of these models but is still reliant on some of the fundamentals of economics theory (Baum And Korn, 1996). Nevertheless, the dynamic nature of competition allows for a nonlinear analysis approach where multiple agents with independent strategies and actions create nonergodic systems that are unpredictable and potentially inefficient. Therefore, even if agents take rational decisions, it is impossible to know if the long-term outcome was the best possible one. These, cumulated inefficiencies combined with random fluctuations can create positive feedback loops — which could be increasing returns to scale in competitive dynamics — and provide the conditions for the emergence of nonlinearities, minima and maxima, that in the long term will define which outcomes are eventually “selected.”
This nonlinearity nature of the dynamics of competition provides a framework that addresses some major issues with neoclassical economics where rationally unbounded agents have perfect knowledge and behave selfishly; that is, components are statically interconnected and are very well defined. More formally, “general equilibrium models do not have many elements with many degrees of freedom; they are too interconnected.” This results in models that are “very rigid and ignore the possibility of evolution, creation, transition, and adaptation in an economy” (Sharma, 2019) Conversely, complex systems more accurately describe the interactions of agents with transient chaotic environments by not limiting their scope to static input/output relations.
In sum, economic models based on linear determinism are unable to present accurate representations of the dynamics that rule competition between independent complex agents, and nonlinear approaches can provide a broader picture by adapting to the environment. Nevertheless, the unstable and unpredictable nature of these systems makes them hard to grasp for traditional computational paradigms. This issue is being addressed by a new generation of computers, including — but not limited to — quantum computers. As these state-of-the-art technologies evolve and are made available, it is reasonable to think that new areas of inquiry beyond neoclassical economics will open, uncovering stronger and deeper connections between nonlinear dynamic systems and the behavior of economic competitive agents.
Notes
[1] This phenomenon can be observed in monopolistic firms that utilize increased demand for its products to eliminate competition, reducing the consumer’s choice and increasing even more the demand for its products and its pricing power. This, in turn, amplifies the deviations of market equilibrium under regular competitive market conditions.
[2] Since functions that describe profit maximization and market returns are usually continuous, they are differentiable over their domain, and the vector field of the system is considered to be a smooth manifold (Lee, 2019).
[3] A particular equilibrium “is locked in to a degree measurable by the minimum cost to effect change over to an alternative equilibrium” (Arthur, 1988).
[4] W. Brian Arthur, like other authors, sees analogies with systems with local feedbacks in physics, chemical kinetics, and theoretical biology that tend to possess a multiplicity of asymptotic states that are stable within the system (Arthur, 1994).
[5] We assume that competitive actions taken by agents can be described as a linear system of equations associated with some eigenvalues. This is a different approach from that of Manrique et al., where the magnitude of the individual contributions of two agents are cancelled out if they select opposite actions and added up if they select the same action; the net change in the direction of the system is the sum of the agents’ individual contributions.
[6] This point also differs from Manrique et al., a discrete time-step approach compared to this case, which is a continuous time system.
[7] In a world of unbounded rationality, the dynamics described by a deterministic path would never fall into a local minimum.
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