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Many of the present econophysicists are familiar with the statistical mechanics of phase transitions. There, some classic problems like the Ising model have become widely accepted as basic cases on which new theoretical ideas should be tested and with which experiment should be compared..

The crucial quantity is the distribution of relative price changes for some given time interval, (e.g. of the Dow Jones index from day to day). These distributions are roughly symmetric if positive and negative changes are compared, have a round peak at small changes, follow a power law for intermediate magnitudes of the changes, and may be truncated for large changes. What is the exponent for the power law? Is it universal? When Mandelbrot wrote his classic papers a third of a century ago [4], a universal exponent near -1.4 seemed to hold, but more recently [5] deviations where seen, leading to a higher exponent near -3, or to an exponential truncation. If there is such a truncation, is it a simple exponential, a Gaussian [3], or a stretched exponential [6] ? There seems to a consensus that there is no correlation (except for very short times) in the sign of two consecutive price changes, while their absolute values (the market volatility) are strongly correlated: Big changes on one day often produce big changes the next day, but one cannot predict reliably the sign of this following change.

Econophysics Complex Multi-Agent Models

If these questions are answered by observing real markets carefully, then we are at the level analogous to scaling laws for phase transitions. What is the analog of the Ising model or other microscopic descriptions? Here, individuals decide to buy or to sell according to some criteria, usually also involving randomness, and the balance of supply and demand then determines the price.
There are numerous microscopic models of this type [1] and some of them give the above shape for the distribution of price fluctuations, including a crossover to a Gaussian for long time intervals. It is not clear which one of them is best, and this lack of clarity is understandable if the above questions about the behavior in real markets are partially still open.

How to predict (and thus prevent) stock market crashes?
How to take into account properly the risk of huge losses of investments?
Since reality gave us only a few big crashes this century, their computer simulation or analytic description could give better information to clarify e.g. the controversial log-periodic oscillations [8] before such a crash. If twenty years from now market crashes are prevented by insights gained now from our econophysics attempts, then later economics textbooks may decribe this as the major contribution of physicists around the year 2000.

Finally, economics Nobel laureate Markowitz commented on the econophysics chapter of our book: ``I believe that microscopic market simulations have an important role to play in economics and finance. If it takes people from outside economics and finance -- perhaps physicists -- to demonstrate this role, it won't be for the first time that outsiders have made substational contributions to these fields.

Next: Spatially distributed social and economic simulations
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