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Thursday, November 1, 1990

The Two-Edged Sword of Chaos

Mr. McGath is a software consultant in Hollis, New Hampshire.

In any debate, it’s a great temptation to refute the other person on his own terms. Sometimes this is the right approach; if you and your opponent agree on a basic premise and disagree only on how to implement it, then it makes sense to show that your proposals are better by his standards. But if the disagreement is one of basic principles, and if beating your opponent on his own ground entails adopting his frame of reference, your victory may well be Pyrrhic. Even if you adopt his premise just for the purpose of showing his inconsistency, you’ve allowed the debate to be conducted on his terms, and you can easily lose sight of what you should really be proving: that he’s wrong not just in particulars, but in his basic approach.

For instance, an advocate of the free market and a socialist might agree that prosperity is desirable. if the socialist claims that his system will create prosperity, it’s proper to answer the claim on its own terms, showing that socialism in fact wouldn’t achieve the goal for which its advocate hopes. On the other hand, suppose the socialist claims that his system will provide a more just distribution of wealth. Anyone who knows a little history may see a chance for an easy comeback here and point to all the cases in which socialism has created its own privileged classes, thus showing that socialism fails to distribute wealth justly. But the person who follows this approach is granting a key element of the socialist’s argument: that there is some ethically proper mount of wealth that each person should have. This line of argument may indeed force him to admit that existing socialist societies haven’t measured up to the ideal, but he can respond that the remedy lies in a more consistent application of economic justice. Moreover, he can now take the offensive, pointing out that socialism at least aims at a “just distribution,” whereas capitalism leaves no room for the government to correct the alleged inequities created by the market.

The right way to approach a mistaken claim is to answer it at the most basic level at which it% mistaken, not to pick at its details. Neglecting this principle is sure to lead to fruitless debating of side issues and failure to recognize errors at their source.

A case in point is the implications that some advocates of the free market have seen in so-called “chaos theory.” This area of study has captured the interest of the educated lay public in the past few years, particularly as a result of James Gleick’s 1987 book, Chaos: Making a New Science. Aside from the mathematical fascination of the theory, it may have wide-ranging implications for deciding what kinds of problems are tractable. The theory suggests that certain types of systems, although they can be described mathematically, behave in such outrageous ways that predicting their future behavior mathematically is all but useless. However, it tells us that even such seemingly “chaotic” systems can be analyzed and described by applying new mathematical methods.

Traditionally, builders of mathematical models try to construct systems that are, so to speak, “well behaved.” A falling rock, for example, can be modeled in a well-behaved way. A fairly simple equation, taking into account gravity and air resistance, will predict how the rock will fall. Minor air currents or tiny errors will result in only slight deviations between the predicted and the actual result.

In contrast with well-behaved systems, some types of systems are “chaotic.” A tiny change in the system may result in large changes in its later behavior. The smoke rising from a cigarette is an example of a chaotic system. A little difference in the temperature of the smoke or the conditions of the air can result in a completely different pattern of smoke. A mistake in the ninth decimal place when calculating the behavior of a chaotic system may result in a 200 percent error in the outcome. Building a model that will accurately predict the behavior of this type of system is a virtually impossible task.

Long-range weather predictions also face the problems of chaotic systems. According to chaos theory, a sneeze in Minnesota may affect whether it will rain in Virginia three months later. Because of the practical impossibility of measuring the current situation with sufficient accuracy, and of carrying out the calculations to enough decimal places, accurate long-range weather forecasts may be beyond human reach.

Chaos and Economics

Do similar considerations apply to economics? Some advocates of the free market have suggested that they do. Chaos theory, they have suggested, rebuts those who want to model and centrally direct the economy on a mathematical basis. Tom G. Palmer, editor of the Humane Studies Review, says that chaos theory shows that “Technocratic prediction of the future—as would be necessary for a ‘planned’ society—is impossible on mathematical grounds. It turns out that certain systems are ‘initial condition sensitive,’ meaning that a tiny change in the initial conditions can produce enormous changes in the results.”

Noted libertarian economist Murray Rothbard writes in a similar vein: “The neo-classicals have for a long while employed their knowledge of math and their use of advanced mathematical techniques as a bludgeon to discredit Austrian [economists]; now come the most advanced mathematical theorists to replicate, unwittingly, some of the searching Austrian critiques of the unreality and distortions of orthodox neo-classical economics. In the current mathematical pecking order, fractals, non-linear thermodynamics, the Feigenbaum number, and all the rest rank far higher than the old-fashioned techniques of the neo-classicals.”

Much of the philosophy underlying chaos theory is attractive to anyone who distrusts the mathematization of economic systems. Gleick tells us that the theory may be “turning back a trend in science toward reductionism.” He echoes F. A. Hayek when he writes: “Yet order arises spontaneously out of these [chaotic] systems—chaos and order together. Only a new kind of science could begin to cross the great gulf between knowledge of what one thing does—one water molecule, one cell of heart tissue, one neuron—and what millions of them do.”

Rhetoric such as this is attractive, and can add to the temptation to challenge mathematical economists on their own ground by throwing still more complex mathematical systems against them. However, this attempt to enlist chaos theory is mistaken and potentially harmful to a proper defense of the free market. To see why, it’s necessary to understand just what chaos theory says and doesn’t say.

The contribution of chaos theory isn’t that it tells us that there are unpredictable systems. We’ve always known that. Rather, the theory applies to a certain type of system: one that can be described by a set of equations or a computer program. Set up the starting conditions, run the program or solve the equations, and you can see the system unfold itself.

Chaos theory’s point is that for some systems, the results of running the program won’t even be close to the real-life behavior of the system. The equations are perfectly legitimate, but the system is extremely sensitive to tiny perturbations. Unless every tiny input to the system is measured with impossible accuracy, and unless the calculations are performed with outrageous precision, the results will be completely wrong. How-ever—and this is crucial—such systems may still be susceptible to analysis by tools which belong neither to traditional deterministic mathematics nor to statistics.

As an example, consider the Japanese game called “pachinko.” In this game, a ball is launched with a spring, then falls through a field of pins. The player’s goal is to make it fall toward certain targets. Pachinko exhibits “sensitive dependence on initial conditions”; a tiny change in the force of launching the ball will make it fall a completely different way. Using traditional modeling methods, pachinko would be just a game of chance. There are, however, patterns in the relationship between the launching and the course the ball follows, and expert players can direct it toward the high-scoring targets. These players might be considered intuitive chaos theorists. They are able to find order where there seems to be none.

The mathematical systems that Keynesians and their allies have used to describe the economy, on the other hand, aren’t like a pachinko game. Their equations are completely well-behaved. Pump a nickel more or less into the economy, and nothing drastic will happen. If you decide not to go shopping on Tuesday, you don’t throw world trade into turmoil. Displayed on a chart, the Keynesian equations show a nice smooth relationship between input and output. The chart of a genuine chaotic system looks like something from a psychedelic art museum.

The mathematical economists’ equations should be challenged on a more basic level. The proper question to ask is whether they, or any set of equations, actually model the course of the economy. Before asking whether a mathematical model of the economy fails in its predictions because it is chaotic, we should ask whether any such model is valid in the first place.

An economic system is, in fact, the sum of a vast number of ongoing human choices. These choices aren’t totally arbitrary, but reflect people’s perceived needs and desires, so certain statistical generalizations can be made about them. Supply and demand curves can be drawn for particular commodities. These curves are, however, simply empirical generalizations reflecting the aggregate of people’s choices at a given time. Nothing says that their preferences will be the same next year. People’s desires may change, or new discoveries may lead to new options.

Ludwig von Mises wouldn’t accept the premise of mathematical economic models, and didn’t quibble over their accuracy. Rather, he challenged the premise at its root: “There is no such thing as quantitative economics. All economic quantifies we know about are data of economic history. No reasonable man can contend that the relation between price and supply is in general, or in respect of certain commodities, constant. We know, on the contrary, that external phenomena affect different people in different ways, that the reactions of the same people to the same external events vary, and that it is not possible to assign individuals to classes of men reacting in the same way.”

Human economic activity is “chaotic” in the sense that it is unpredictable and not subject to mathematical analysis, but this has nothing to do with chaos theory. Pachinko-like systems are completely determined by simple mechanical laws; their behavior depends entirely on the way they are set in motion, and chaos theory provides tools for analyzing them in spite of their apparent unpredictability. Human behavior, in contrast, is unpredictable not because its flow displays “sensitive dependence on initial conditions,” but because it is not dependent on initial conditions at all. The future may change due to events that no amount of precise calculation can predict.

Chaos as a Planner’s Tool

Rather than being an argument against centralized planning based on economic models, chaos theory may offer the planners new weapons. For instance, Gleick discusses scientist Benoit Mandelbrot’s exploration in 1960 of cotton-price fluctuations. “Economists,” Gleick states, “generally assumed that the price of a commodity like cotton danced to two different beats, one orderly and one random. Over the long term, prices would be driven steadily by real forces in the economy—the rise and fall of the New England textile industry, or the opening of international trade routes. Over the short term, prices would bounce around more or less randomly.”

But Mandelbrot took a different view. “Instead of separating tiny changes from grand ones, his picture bound them together. He was looking for patterns not at one scale or another, but across every scale.” He found that “Each particular price change was random and unpredictable. But the sequence of changes was independent of scale: curves for daily price changes and monthly price changes matched perfectly.”

Independence of scale is a key idea in chaos theory. By using “fractals,” a kind of curve which contains miniature replicas of itself, which in turn contain yet smaller replicas of the same pattern, and so on, modelers can create pictures that are rich in detail out of relatively simple formulas. This permits great depth of detail without information overload. Fractal economics could create the ultimate nightmare for opponents of central planning: a model that claims to unite both the macro-eco-nomic and micro-economic realms, that describes not only the broad course of the economy but the detailed activities of each individual With such a model, planners could claim to know enough to delve into each person’s life, without having to know anything about that particular individual. Gleick cites mathematician Ralph Abraham’s dream of using mathematical models to “educate children to be better members of the board of directors of the planet.”

Advocates of government intervention have talked about “priming the pump” of the economy, only to be refuted by the impossibility of figuring out which pump should be primed and to what extent. But the ideas of chaos theory could encourage new variants on this model, replacing the old idea of priming the pump with one of flipping the handle of the pachinko machine. With these new tools, they might imagine, it will be possible to decide exactly how far to pull the handle, even though traditional analysis can’t offer a due. “Of all the possible pathways of disorder,” Gleick tells us, “nature favors just a few.” Economic planners, turning this idea into pop science, could easily conclude that the unpredictability of human activity is no obstacle to classifying and controlling human behavior.

Murray Rothbard regards the discovery of patterns in seemingly random events as a challenge to the idea that market expectations can accommodate “perfect knowledge” of the future apart from random, unpredictable fluctuations. Yet this seems odd. Chaos theory, far from challenging the predictability of future market behavior, claims to find patterns even in the variations that have previously been regarded as random. Whether these patterns really exist is a matter for study; if they do exist, then the investor who learns to predict them will have an advantage over people who use traditional analysis.

In any event, statistical patterns are meaningful only for large numbers of stocks (or other commodities) over long periods of time. The rise or fall of a particular commodity% price is the result of particular events and people’s response to them, not of abstract mathematical forces. People who anticipate these events and responses will do better than the average investor; those who follow statistical averages will simply obtain average results. Again, the key to refuting the claim that statistics provide perfect knowledge lies in recognizing that they are simply mathematical descriptions, not in trying to one-up existing descriptions with new ones.

In fact, though, chaos has no more to offer to the planners than it offers to the free market. In analyzing a particular phenomenon under steady conditions, Mandelbrot’s methods may well produce a description that closely matches the observed phenomena. But these methods can provide no information about the phenomena of human action which affect an economy in vital ways. A new invention, the emergence of a new political movement, changing economic habits resulting from changing philosophies of life, and similar phenomena are not simply random fluctuations, or even new inputs to a mechanistic system; the analysis of unconscious systems has only limited applicability to the realm of human choice. The modeler can only devise formulas after the fact to fit the data, with no guarantees that these formulas will describe the future.

Chaos theory is a fascinating area to study, and it is very likely to have applications in the analysis of current trends. However, it offers neither support nor refutation to the idea that an economy can be mathematically planned. It doesn’t refute classical methods of mathematical planning, because it simply doesn’t apply to them. Nor does it provide new methods of plotting an economy’s long-term course, because it is as helpless as any other mathematical method to anticipate the consequences of changing choices and emerging knowledge. The best strategy for advocates of the free market is to stick with the basic principles that have shown its moral and economic superiority, and to avoid trying to undercut the champions of mathematical planning on their ground. 

1.   Tom G. Palmer, review of Chaos: Making a New Science in the August 1988 Laissez Faire Books catalogue.

2.   Murray N. Rothbard, “Chaos Theory: Destroying Mathematical Economics from Within?” The Free Market, March 1988.

3.   James Gleick, Chaos: Making a New Science (New York: Penguin Books, 1987), p. 8.

4.   Ludwig von Mises, Human Action, Third Revised Edition (Chicago: Henry Regnery Company, 1966), p. 351.

5.   Gleick, pp. 83-86.

6.   Ibid., p. 279.

7.   Ibid., p. 267.

8.   Rothbard, op. cit.

  • Gary McGath is a freelance software engineer living in Nashua, New Hampshire.