In Volume 3 Karen Vaughn wrote about Hayek's theory of market order, which he sometimes characterized as a complex, adaptive system and other times as a spontaneous order.
In some of his writing, he described an economy, or market, as a complex phenomena. The complexity of a system depends on a sufficient number of elements showing a pattern of characteristic attributes. The more the number of elements, the more complex it is. Complex systems demonstrate emergent properties, characteristics that cannot be simply reduced to an account of the individual elements.
Hayek took evolutionary biology as a prime example of a complex system to illustrate that the exact outcome of evolution depended on relationships between an overwhelming number of variables, the exact relationships among which could never be fully specified.
Hayek used Walrasian general equilibrium (#1, #2) as an example of a complexity system in social science. General equilibrium describes a pattern of price relationships more or less observed in the real world but the theory itself can never be predictive of actual prices because the initial conditions can never be fully specified. The primary value of general equilibrium theory, he argued, was not to predict the future course of events, but only to provide a general description of a particular kind of order.
Vaughn says that Hayek focused on the number of variables and the complexity of relationships among the elements, but he did not identify the defining features of modern complex adaptive systems: non-linearity of the elements and the path-dependent interactions among them. On the other hand, his emphasis elsewhere on the process of learning in a market suggests non-linearity and path dependency.
He described a spontaneous order as a social pattern that that emerges as "the result of human action but not of human design," that is, "from the bottom up" through the actions of individuals directed to their own purposes. The order is unplanned in the sense that it was not designed by some higher intelligence or a central planner. The reason that order can emerge from the self-directed actions of individuals is that their actions are not only purposeful but the actors are also rule followers. These rules consist of laws, customs, and habits that emerge as an unintended consequence of individual action -- "emergent properties" in the parlance of complexity theory.
Piecing together Hayek's description of a market economy from his various writing, Vaughn thinks that his understanding of markets is much closer to a complex, adaptive system than it is an example of Walrasian general equilibrium, in terms of a system of simultaneous linear equations. She gives several reasons for this.
"While many may remain skeptical, I am cautiously optimistic that the science of complexity, by clarifying the characteristics of non-linear, adaptive systems may prove to be an aid in articulating a more Hayekian understanding of market order. For instance, the mature theory of complex adaptive systems can help support Hayek's central argument against central planning.
Consider the "mathematical solution" which held that all one needed was to set up a system of simultaneous [linear] equations to generate equilibrium prices that could guide planners. At the time of the economic calculation debate, computers were not available to attempt to set up and solve those equations, so even the socialist sympathizers agreed that such a scheme was "practically" impossible, at least for the immediate future. Oskar Lange's "trial and error" solution to the problem of factor pricing was a fall-back to achieve the same goal without computers. However, Lange never gave up on the possibility that one day, computers could be used to direct a centrally planned economy, making his clumsy trial and error solution obsolete. The science of complexity can finally put that fantasy to rest, and at the same time, vindicate Hayek's insight" (p. 344).