Conclusions

Abstract

These notes are intended to contribute to the growing field of research dealing with the use of AI and CI techniques in economics. We have shortly summarized some of the most important contributions of the recent years and have afterwards concentrated on a thorough analysis of a special CI technique, namely genetic algorithms. We have argued that GAs, due to their decentralized structure which very naturally resembles to a group of economic agents and their interactions, are especially well-suited to simulate the behavior of an economic system. Further we have shown how to interpret the single operators contained in a genetic algorithm in an economic sense, but have also pointed out the problems we have with the economic interpretation of certain aspects of the algorithm. The fact that in economic setups the fitness of a single string depends on the current state of the whole population has led us to the conclusion that the analytical models which describe the behavior of a genetic algorithm used to solve an optimization problem can not be applied in an economic system. Perhaps the main results are several propositions describing the limit behavior of a genetic algorithm in systems where the fitness function is state dependent. These theoretical results, but of course also the learning ability of a GA as such, have been illustrated afterwards with several examples from the field of game theory and economics. The simulations showed that our mathematical theory enables us not only to understand but also to predict the behavior of GAs in economic systems. Ilowever these examples made also clear that a local analysis like the one done in chapter 4 will in many cases not be able to explain the complete behavior of the system. In several cases we had to give heuristic explanations for a certain kind of behavior instead of a rigorous mathematical analysis. We have to face the fact that the current insight to high-dimensional non-linear difference equations does not enable us to describe the global behavior of such a system. Nevertheless, we are confident that the application of special theories like the graph theoretical approach taken here in proposition 4.2.1 or the theory of nonlinear dynamical systems will permit further mathematical insights into the behavior of GAs in SDF systems.