> 5@ αbjbj22 "XXHHH8$<7"$$$$$$/7171717171717$8R:U7Q+$$++U7$$7444+$$/74+/744w6w6$.<H1Jw6/7707w6;-4^;w6;w6$Tx 4F$R'$$$U7U7H4HEPISTEMOLOGICAL IMPLICATIONS OF ECONOMIC COMPLEXITY
J. Barkley Rosser, Jr.
Professor of Economics
James Madison University
Harrisonburg, VA 22807 USA
HYPERLINK "mailto:rosserjb@jmu.edu" rosserjb@jmu.edu
January, 2004
EPISTEMOLOGICAL IMPLICATIONS OF ECONOMIC COMPLEXITY
I. INTRODUCTION
Over the last several decades the view that economic reality is somehow fundamentally complex has increasingly taken hold among economists, not only those focused on abstract theory but even policymakers as well (Greenspan, 2004). Consequently such economists have been forced to wrestle with the problem not only of how to forecast, which has always been difficult, but even how to understand the apparently most simple economic phenomena in principle as well. More to the point, even how to think about how to understand such phenomena has become a serious challenge. In short, economists increasingly must grapple with fundamental problems of epistemology, how to know what they know in a complex economic reality.
This paper will consider three foundational aspects of this problem. The first involves to what extent the source of the problem is endogenous to nonlinear dynamics. The second involves to what extent it is logical and computational. The third will consider whether or not the epistemological problem is really an ontological problem.
The problem of nonlinear dynamics and epistemology is most clearly seen in relation to chaotic dynamics, particularly the problem of sensitive dependence on initial conditions, also known popularly as the butterfly effect. If minute changes in initial conditions, either of parameter values controlling a system or of initial starting values, can lead to very large changes in ultimate outcomes of a system, then it may essentially require an infinite exactness of knowledge to completely understand the system. Likewise such problems as fractality of basin boundaries in systems with multiple basins of dynamic attraction can lead to similar problems when there is even the slightest amount of stochastic noise in the dynamical system (Rosser, 2000a).
The problem of logic or computation arises in complex systems of multiple interacting heterogeneous agents thinking about each others thinking. Although game theoretic solutions such as Nash equilibria may present themselves, these may involve a certain element of ignorance, a refusal to fully know the system. Efforts to fully know the system may prove to be impossible due to problems of infinite regress or self referencing that lead to non-computability (Binmore, 1987; Albin with Foley, 1998; Koppl and Rosser, 2002).
The final issue is one that has been raised in the context of economic complexity by Paul Davidson (1996), who argues that complexity is not ontologically a source of uncertainty in economic analysis but merely an epistemological one in that if one had infinite computing power and knowledge the problem would be resolved. Rosser (1998) has contested his arguments, but only in a partial way, arguing that complexity implies an essentially ontological foundation for fundamental uncertainty in economic analysis. However, here I wish to pursue this argument further, to say that indeed the fundamental source of the difficult epistemology of complexity is ultimately ontological. This project can also be seen as asking for the basis of Herbert Simons (1957) assertion of the boundedness of rationality for economic agents.
The next section of the paper will note the competing definitions of complexity. The following three will pursue the points noted above in order. In addition there will be a concluding section.
II. WHAT IS ECONOMIC COMPLEXITY?
In The End of Science (1996, p. 303) John Horgan reports on 45 definitions of complexity that have been gathered by the physicist Seth Lloyd. Some of the more widely used conceptualizations include informational entropy (Shannon, 1948), algorithmic complexity (Chaitin, 1987), stochastic complexity (Rissanen, 1989), and hierarchical complexity (Simon, 1962). Curiously, at least three other definitions have been more frequently used in economics that do not appear on Lloyds list, namely simple complicatedness in the sense of many different sectors with many different interconnections (Pryor, 1995; Stodder, 1995), dynamic complexity (Day, 1994), and computational complexity (Lewis, 1985; Velupillai, 2000). We shall not consider the problems associated with mere complicatedness, as the epistemological issues involved with this form of complexity are essentially trivial from a philosophical perspective.
Following Richard Day (1994), dynamic complexity can be defined as arising from dynamical systems that endogenously fail to converge to either a point, a limit cycle, or a smooth explosion or implosion. Nonlinearity is a necessary but not sufficient condition for such complexity. Rosser (1999) identifies this definition with a big tent view of complexity that can be subdivided into four sub-categories: cybernetics, catastrophe theory, chaos theory, and small tent complexity. The latter does not possess a definite definition, however Arthur, Durlauf, and Lane (1997) argue that such complexity exhibits six characteristics: 1) dispersed interaction among locally interacting heterogeneous agents in some space, 2) no global controller that can exploit opportunities arising from these dispersed interactions, 3) cross-cutting hierarchical organization with many tangled interactions, 4) continual learning and adaptation by agents, 5) perpetual novelty in the system as mutations lead it to evolve new ecological niches, and 6) out-of-equilibrium dynamics with either no or many equilibria and little likelihood of a global optimum state emerging. Certainly such systems offer considerable scope for problems of how to know what is going on in them.
Computational complexity essentially amounts to a system being non-computable. Ultimately this is depends on a logical foundation, that of non-recursiveness due to incompleteness in the Gdel sense. In actual computer problems this problem manifests itself most clearly in the form of the halting problem (Blum, Cucker, Shub, and Smale, 1998), that the halting time of a program is infinite. Ultimately this form of complexity has deep links with several of the others listed above, such as Chaitins algorithmic complexity. These latter two approaches are the ones we shall consider in more detail in the next two sections.
III. EPISTEMOLOGY OF DYNAMIC COMPLEXITY
To the extent that we live in a dynamically complex system, the problem of epistemology in such systems becomes the epistemological problem more generally. However, to address this issue we need to pin it down a bit more precisely. Thus, we shall consider the more specific problem of being able to know the consequences of an action that we take in such a system. Let G(xt) be the dynamical system in an n-dimensional space. Let an agent possess an action set A. Let a given action by the agent at a particular time be given by ait. For the moment let us not specify any actions by any other agents, each of whom also possesses his or her own action set. We can identify a relation whereby xt = f(ait). In effect the epistemological problem for the agent in question becomes, can the agent know the reduced system G(f(ait) when this system possesses complex dynamics due to nonlinearity?
First of all, it may be possible for the agent to be able to understand the system and to know that he or she understands it, at least to some extent. One reason why this can happen is that many complex nonlinear dynamical systems do not always behave in erratic or discontinuous ways. For many fundamentally chaotic systems there is a pattern of transiency (Lorenz, 1992). In effect, the system can move in and out of behaving chaotically, with long periods passing during which the system will effectively behave in a non-complex manner, either tracking a simple equilibrium or following an easily predictable limit cycle. While the system remains in this pattern, actions by the agent may have easily predicted outcomes, and the agent may even be able to become confident regarding his or her ability to manipulate system in a systematic manner. However, this is essentially avoiding the question.
Let us consider three forms of complexity: chaotic dynamics, fractal basin boundaries, and discontinuous phase transitions in heterogeneous agent situations. For the first of these there is a clear problem for the agent, the existence of sensitive dependence on initial conditions. If an agent moves from action ait to action ajt, where lait ajtl < < 1, then no matter how small is, there exists an m such that lG(f(ait+t ) G(f(ajt+t )l > m for some t for each . As approaches zero, m/ will approach infinity. It will be very hard for the agent to be confident in predicting the outcome of changing his or her action. This is the problem of the butterfly effect or sensitive dependence on initial conditions. More particularly, if the agent has an imperfectly precise awareness of his or her actions, with the zone of fuzziness exceeding , clearly the agent will be facing a potentially large range of uncertainty regarding the outcome of his or her actions. It is worth remembering the outcome of Edward Lorenz s original experience in this matter when he discovered chaos (Lorenz, 1963). When he restarted his simulation of a three-equation system of fluid dynamics partway through, the roundoff error that triggered a subsequent dramatic divergence was too small for his computer to perceive (at the four decimal place).
However, there are two offsetting elements for the situation of chaotic dynamics. Although an exact knowledge is effectively impossible, requiring essentially infinitely precise knowledge (and knowledge of that knowledge), a broader approximate knowledge over time may be possible. Thus, chaotic systems are generally bounded and often ergodic. While in the short run relative trajectories for two slightly different actions may sharply diverge, the trajectories will at some later time return toward each other, becoming arbitrarily close to each other before once again diverging. Not only may the bounds of the system be knowable, but the long run average of the system may be knowable. There are still limits as one can never be sure that one is not dealing with a long transient of the system, with it possibly moving into a substantially different mode of behavior later. But the possibility of a substantial degree of knowledge, with even some degree of confidence regarding that knowledge is not out of the question for chaotically dynamic systems.
Now let us consider the problem of fractal basin boundaries, first identified for economic models by Hans-Walter Lorenz (1992) in the same paper in which he discussed the problem of chaotic transience. Whereas in a chaotic system there may be only one basin of attraction, albeit with the attractor being fractal and strange and thus generating erratic fluctuations, the fractal basin boundary case involves multiple basins of attraction, whose boundaries with each other take fractal shapes. The attractor for each basin may well be as simple as being a single point. However, the boundaries between the basins may lie arbitrarily close to each other in certain zones.
In such a case, although it may be difficult to be certain, for the purely deterministic case once one is able to determine which basin of attraction one is in, a substantial degree of predictability may ensue, although again there may be the problem of transient dynamics, with the system taking a long and circuitous route before it begins to get anywhere close to the attractor, even if the attractor is merely a point in the end. The problem arises if the system is not strictly deterministic, if G includes a stochastic element, however small. In this case one may be easily pushed across a basin boundary, especially if one is in a zone where the boundaries lie very close to one another. Thus there may be a sudden and very difficult to predict discontinuous change in the dynamic path as the system begins to move toward a very different attractor in a different basin. The effect is very similar to that of the sensitive dependence on initial conditions in epistemological terms, even if the two cases mathematically quite distinct.
Nevertheless, in this case as well there may be something similar to the kind of dispensation over the longer run we noted for the case of chaotic dynamics. Even if exact prediction in the chaotic case is all but impossible, it may be possible to discern broader patterns, bounds and averages. Likewise in the case of fractal basin boundaries with a stochastic element, over time one should observe a jumping from one basin to another. Somewhat like the pattern of long run evolutionary game dynamics studied by Binmore and Samuelson (1999), one can imagine an observer keeping track of how long the system remains in each basin and eventually developing a probability profile of the pattern, with the percent of time the system spends in each basin gradually approaching some fixed asymptotic value. However, this is contingent on the nature of the stochastic process itself, as well as the degree of complexity of the fractal pattern of the basin boundaries. A non-ergodic stochastic process may render it very difficult, even impossible to observe convergence on a stable set of probabilities for being in the respective basins, even if those are themselves few in number with simple attractors.
Finally we consider the case of phase transitions in systems of heterogeneous locally interacting agents, the world of the so-called small tent complexity. Brock and Hommes (1997) have developed a useful model for understanding such phase transitions, based on statistical mechanics. This is a stochastic system and is driven fundamentally by two key parameters, a strength of interactions or relationships between neighboring agents and a degree of willingness to switch behavioral patterns by the agents. For their model the product of these two parameters is crucial, with a bifurcation occurring for their product. If the product is below a certain critical value, then there will be a single equilibrium state. However, once this product exceeds a particular critical value two distinct equilibria will emerge. Effectively the agents will jump back and forth between these equilibria in a herding pattern. For financial market models (Brock and Hommes, 1998) this can resemble oscillations between optimistic bull markets and pessimistic bear markets, whereas below the critical value the market will have much less volatility as it tracks something that may be a rational expectations equilibrium.
For this kind of a setup there are essentially two serious problems. One is determining the value of the critical threshold. The other is understanding how the agents jump from one equilibrium to the other in the multiple equilibrium zone. Certainly the second problem resembles somewhat the discussion from the previous case, if not involving as dramatic a set of possible discontinuous shifts.
Of course once a threshold of discontinuity is passed it may be recognizable when it is approached again. But prior to doing so it may be essentially impossible to determine its location. The problem of determining a discontinuity threshold is a much broader one that vexes policymakers in many situations, such as attempting to avoid catastrophic thresholds that can bring about the collapse of a species population or of an entire ecosystem. One does not want to cross the threshold, but without doing so, one does not know where it is. However, for less dangerous situations involving irreversibilities, it may be possible to determine the location of the threshold as one moves back and forth across it.
On the other hand in such systems it is quite likely that the location of such thresholds may not remain fixed. Often such systems exhibit an evolutionary self-organizing pattern in which the parameters of the system themselves become subject to evolutionary change as the system moves from zone to zone. Such non-ergodicity becomes consistent not only with Keynesian style uncertainty, but may also come to resemble the complexity identified by Hayek (1948, 1967) in his discussions of self-organization within complex systems. Of course for market economies Hayek evinced an optimism regarding the outcomes of such processes. Even if market participants may not be able to predict outcomes of such processes, the pattern of self-organization will ultimately be largely beneficial if left on its own. Although Keynesians and Hayekian Austrians are often seen as in deep disagreement, some observers have noted the similarities of viewpoint regarding these underpinnings of uncertainty (Shackle, 1972; Loasby, 1976; Rosser, 2001). Furthermore, this approach leads to the idea of the openness of systems that becomes consistent with the critical realist approach to economic epistemology (Lawson, 1997).
IV. EPISTEMOLOGY OF COMPUTATIONAL COMPLEXITY
We shall ultimately argue that there are links between dynamic complexity and computational complexity, but for initially we must consider the problem of computational complexity on its own. Velupillai (2000) provides definitions and general discussion and Koppl and Rosser (2002) provide a more precise formulation of the problem, drawing on arguments of Kleene (1967), Binmore (1987), Lipman (1991), and Canning (1992). Velupillai defines computational complexity straightforwardly as intractability or insolvability. Halting problems such as studied by Blume, Cucker, Shub, and Smale (1998) provide excellent examples of how such complexity can arise.
In particular, Koppl and Rosser reexamined the famous Holmes-Moriarty problem of game theory, in which two players who behave as Turing machines contemplate a game between each other involving an infinite regress of thinking about what the other one is thinking about. Such a situation has a Nash equilibrium solution, but the problem is whether or not hyper-rational Turing machines can arrive at knowing that solution or not. The conclusion is that they cannot, ultimately due to the halting problem. This arises from the self-referencing involved that involves problems fundamentally similar to those underlying the Gdel Incompleteness Theorem (Kleene, 1967, p. 246). More precisely, the best reply functions of these agents are not computable.
Binmores (1987, pp. 209-212) response to this sort of failure of undecidability in self-referencing systems was to invoke a sophisticated form of Bayesian updating. This he hoped could bring about convergence in a computational sense on the Nash equilibrium solution for such players, without encountering the logical paradoxes arising from hyper-rationality. In short, epistemologically, the agents might be able to function if they are a bit more ignorant. More broadly Koppl and Rosser agree, the only way that agents can indeed operate in such an environment is not to be so knowledgeable, to accept limits on knowledge and operate accordingly, perhaps on the basis of intuition or Keynesian animal spirits (Keynes, 1936). The earlier discussion in effect says that no matter how hyper-rational the agents are, they cannot have complete knowledge, essentially for the same reason that Gdel showed that no logical system can be complete within itself.
However, even for Binmores proposed solution there are also limits. Thus, Diaconis and Freedman (1986) have shown that Bayes Theorem fails to hold in an infinite dimensional space. There may be a failure to converge on the correct solution through Bayesian updating, notably when the basis is discontinuous. The kind of thing that can happen instead is a convergence on a cycle in which agents are jumping back and forth from one probability to another, neither of which is correct. In the simple example of coin tossing, they might be jumping back and forth between assuming priors of 1/3 and 2/3 without ever being able to converge on the correct probability of 1/2. Nyarko (1991) has studied such kinds of cyclical dynamics in learning situations in generalized economic models.
Koppl and Rosser compare this issue to that of the Keyness problem (1936, chap. 12) of the beauty contest in which the participants are supposed to win if they most accurately guess the guesses of the other participants. Again, there is potentially an infinite regress problem with the participants trying to guess how the other participants are going to be guessing about their guessing and so forth. There is not necessarily a convergence to a solution in this case for the fully rational agent. A solution only comes by in effect choosing to be somewhat ignorant or boundedly rational and operating at a particular level of analysis. However, as there is no way to determine rationally the degree of boundedness, which itself involves an infinite regress problem (Lipman, 1991), this decision also ultimately involves an arbitrary act, based on animal spirits or whatever, an decision ultimately made without knowledge.
A curiously related point here is the newer literature (Gode and Sunder, 1993; Mirowski, 2002) on the behavior of zero intelligence traders. Gode and Sunder have shown that in many artificial market setups zero intelligence traders following very simple rules can converge on market equilibria that may even be efficient. Not only may it be necessary to limit ones knowledge in order to behave in a rational manner, but one may be able to be rational in some sense while being completely without knowledge whatsoever.
Fundamentally, the epistemological problem for computationally complex economic systems is essentially the same as that for logical systems more broadly. Incompleteness implies both an ultimate non-decidability that translates as a non-computatability. This incompleteness suggests an incompleteness of knowledge, at least of knowledge obtained by the use of logic or calculation, although other methods of obtaining knowledge may remain available.
V. IS THE COMPLEXITY EPISTEMOLOGICAL PROBLEM ONTOLOGICAL?
Finally we wish to consider briefly the question regarding the ultimate source of the epistemological problem in complex economic systems. Is it simply a matter of difficulties in discerning relationships or precise values of parameters or gathering of data or limits on the computing power of the human mind, or is it more fundamentally derived from ontological foundations. The implication of the latter is that the epistemological problem cannot ultimately be overcome; it is inherent in the nature of complex systems. If the former is the case, then there is the possibility that somehow the problem can be resolved by improvements in theory construction, data gathering, or computer power.
We note again particularly the argument of Paul Davidson (1996) regarding the source of fundamental uncertainty of the Keynesian variety in economic analysis. Whereas Rosser (1998, 1999) argues that the existence of systemic complexity in economics provides an explanation for such uncertainty, Davidson dismisses this argument. He argues that this is merely epistemological, due to the sorts of limits on data gathering or simple computational power alluded to above. However, he argues instead that such uncertainty must be ontologically founded on an axiomatic assertion regarding the non-ergodicity of economic systems. Rosser has responded that such non-ergodicity is in fact an empirical issue rather than one of axiomatics and that uncertainty may occur in complex systems even when they are ergodic sometimes. He has also suggested that the epistemological problem of complexity may be effectively ontological, although that in effect grants Davidsons ultimate argument to some extent.
We are now in a position to further evaluate this discussion. Most of this earlier discussion was in terms of what we are labeling here dynamic complexity. As already noted, in effect, to have exact knowledge of a chaotically dynamic system would require an arbirtrarily high information cost, effectively approaching infinity. This is the basis of the argument regarding the epistemological problem being effectively ontological. No matter how precise one gets in ones measurements, say to the sixth decimal place, an error at an even smaller order of magnitude, the seventh decimal place, can still encounter the problem of sensitive dependence on initial conditions and lead one to make a wildly incorrect forecast or prediction. One simply cannot guarantee exact prediction, or even very close prediction with any certainty, as long as one is expending a finite effort to obtain information regarding the system, its internal relations, its initial conditions, its parameter values, and so forth. One may be able to learn some things with certainty, such as the outer bounds of the dynamical path of the system, but one will not be able to know its exact path, or even be certain that one is going to be very near its exact path at all, other than being within these very broad bounds.
Regarding the question of computational complexity there would seem to be a somewhat different problem. We have emphasized that ultimately the problem is one of logic, especially of the Gdel problem of incompleteness. This problem becomes especially acute in systems that attempt to understand themselves, that are self-referencing. However, this may not be as ineluctable as it may seem. The issue is that these problems only apply to the logical analysis carried out by a Turing machine. A human being, who may use other approaches to gaining knowledge, including intuition, may well be able to obtain sufficient information to understand a computationally complex system, even if a Turing machine cannot. There would not seem to be an ultimate ontological problem here.
Further complications suggest themselves. A serious one has to do with the systems of analysis being used themselves, their evolution and their connection with the fundamental reality. Lawson has argued that the true reality is a deep structure, even in an non-dynamically or computationally complex universe. Such deep structures are not readily amenable to empirical or analytical discovery. The modes of analysis themselves may by their very nature involve artificial boundaries or categories that do not correspond to the underlying reality. So it is that Foucault (1972, Introduction) argues that epistemological acts or thresholds arise in the historical development of ideas, discontinuities that may not correspond with the underlying reality. If one uses dialectical modes of analysis, categories may be artificially opposed to one another, even as dialectical processes can be seen as a foundation for nonlinear dynamical systems (Rosser, 2000b). The search for a common language in mathematics itself may also have led economists astray. Communication and clarity may have been achieved as abstraction has proceeded, even as understanding of actual ontology may have decreased (Mirowski, 2002; Weintraub, 2002).
Ultimately this question cannot be resolved for epistemological reasons. We cannot know if the source of the epistemological problem for complex economic systems is ultimately ontological or not because we cannot know for certain the ontology itself.
VI. CONCLUSIONS
We have reviewed issues related to the epistemological problem as it relates specifically to complex economic systems. While noting that there are many competing definitions of complexity, we have identified two that have been most frequently used in economics, dynamic complexity and computational complexity. Each has its own sort of epistemological problem. Dynamic complexity is subject to such issues as the sensitive dependence on initial conditions of chaos theory, or the uncertainty due to fractal basin boundaries in stochastic nonlinear systems, or the pattern of phase transitions and self-organizing transformations that can occur in systems with interacting heterogeneous agents. Such problems imply that in effect only an infinite degree of precision of knowledge will allow one to fully understand the system, which is impossible.
In computationally complex systems the problem is more related to logic, the problems of infinite regress and undecidability associated with self-referencing in systems of Turing machines. This can manifest itself as the halting problem, something that can arise even for a computer attempting to precisely calculate even a dynamically complex system as for example the exact shape of the Mandelbrot set (Blum, Cucker, Shub, and Smale, 1998,). A Turing machine cannot understand fully a system in which its own decisionmaking is too crucially a part. However, knowledge of such systems may be gained by other means.
Regarding the ontological foundations of the epistemological problem for complex systems, this ultimately runs into the fundamental epistemological problem of all, how do we know that we understand true ontology? This leaves open-ended a resolution of the debate over whether or not the special problems we have been discussing are ultimately ontological in nature or not.
In the end, the serious epistemological problems associated with complex economic systems do imply that there exist serious bounds on the rationality of economic agents. These bounds take many forms, inability to understand the internal relations of a system, inability to fully know crucial parameter values, inability to identify critical thresholds or bifurcation points, inability to understand the interactions of agents, especially when these agents are thinking about how each other are thinking about each others thinking. Infinite regress problems imply non-decidability and non-computability for hyper-rational Turing machine agents. Thus, economic agents must ultimately rely on arbitrary acts and decisions, even if those simply involve deciding what will be the bounds beyond which the agent will no longer attempt to solve the epistemological problem.
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