Chaos -- Mathematical and Financial

Ask an ace pool player to place a ball at any spot on the table. Then have him (or her) aim toward the obstacles and predict the exact trajectory of the shot.

Forecasting the first three or four bounces and caroms of the ball will usually be easy. Even if he's off by the merest fraction of a degree in his reckoning, however, his mis-estimation will be greatly magnified by successive hits of the round obstacles.

Soon the pool ball will run up against an obstacle that the player didn't intend to hit or miss one that he did intend to hit, and then all bets about where the ball will go are off.

The sensitivity of the ball's path to tiny variations in its initial angle is suggestive of the disproportionate effect of seemingly inconsequential events and actions. The amplification of these slight deviations is just one of the factors explaining why nonlinear dynamical systems -- and the economy is one -- are so resistant to dependable and accurate long-range forecast.

To push the policy maker/pool player analogy a bit further, we can interpret the effect of the opacity and minimal regulation of mortgage securities as perhaps akin to the effect of hand tremors on the pool player. Both lead to less control and even greater unpredictability of the ball/policy's trajectory.

Doubt, Certitude, and Competence

Dynamical systems are generally more complicated than the pool table example, but even a vague, intuitive understanding of their behavior and of the effect of so many nonlinearly interacting variables, sensitively dependent systems, feedback and so on, should be sufficient to arouse a certain wariness of simplistic pronouncements delivered with overweening confidence.

Our standard economic statistics are notoriously imprecise and unreliable, and this imprecision and unreliability (like the pool ball above) quickly propagate through the system.

Interestingly, chaos theory also hints at some constructive, albeit vague ideas for partially managing the economy and even the present bailout.

One is that real change in a system often requires a reorganization of its structure.

Another is that to effect such change we must search for points of maximum leverage, points that are often not obvious and are sometimes many steps removed from their intended effects.

A third idea is that there is evidence indicating some chaos (in the mathematical sense) is necessary for the stability and resilience of systems.

The bottom line is that when dealing with complex systems we should act with a certain humble dubiousness, not with an unblinking rashness.

A bit of psychology backs up this counsel. A study a few years ago by Cornell psychologist David Dunning found that incompetent people are generally not aware of their own incompetence. Part of the explanation, Dunning wrote, is that the skills that constitute competence are often the very ones needed to recognize incompetence.

This is perhaps why the most clueless politicians are the most self-confident.

Too often in politics, economics, religion, and everyday life, knowledge and tentativeness battle ignorance and certitude.

John Allen Paulos, a professor of mathematics at Temple University, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as of the just-released "Irreligion: A Mathematician Explains Why The Arguments for God Just Don't Add Up " His "Who's Counting?" column on ABCNEWS.com appears the first weekend of every month.

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