Adaptive Learning by Genetic Algorithms: Analytical Results and Applications to Economical Models

172 Pages·1996·8.129 MB·English

by  Dr. Herbert Dawid (auth.)

Lecture Notes in Economics
and Mathematical Systems 441
Founding Editors:
M. Beckmann
H. P. Kiinzi
Editorial Board:
H. Albach, M. Beckmann, G. Feichtinger, W. Giith, W. Hildenbrand,
W. Krelle, H. P. Kiinzi, K. Ritter, U. Schittko, P. Schonfeld, R. Selten
Managing Editors:
Prof. Dr. G. Fandel
Fachbereich Wirtschaftswissenschaften
Fernuniversitiit Hagen
Feithstr. 140lAVZ II, D-58097 Hagen, Germany
Prof. Dr. W. Trockel
Institut flir Mathematische Wirtschaftsforschung (IMW)
Universitlit Bielefeld
Universitlitsstr. 25, D-33615 Bielefeld, Germany
Springer
Berlin
Heidelberg
New York
Barcelona
Budapest
Hong Kong
London
Milan
Paris
Santa Clara
Singapore
Tokyo
Herbert Dawid
Adaptive Learning
by Genetic Algorithms
Analytical Results and Applications
to Economical Models
Springer
Author
Dr. Herbert Dawid
University of Vienna
Institute of Management Science
BrunnersstraBe 72
A-121O Vienna, Austria
Library of Congress Cataloging-in-Publication Data
Dawid, Herbert.
Adaptive learning by genetic algorithms: analytical results and
applications to economic models / Herbert Dawid.
p. cm. -- (Lecture notes in economics and mathematical
systems; 441)
ISBN 978-3-540-61513-2 ISBN 978-3-662-00211-7 (eBook)
DOI 10.1007/978-3-662-00211-7
1. Economics--Mathematical models. 2. Genetic algorithms.
I. Title. II. Series.
HB135.D387 1996
330' .01'5118--dc20
ISSN 0075-8442
ISBN 978-3-540-61513-2
This work is subject to copyright. All rights are reserved, whether the whole or part
ofthe material is concerned, specifically the rights of translation, reprinting, re-use
of illustrations, recitation, broadcasting, reproduction on microfilms or in any other
way, and storage in data banks. Duplication of this publication or parts thereof is
permitted only under the provisions ofthe German Copyright Law of September 9,
1965, in its current version, and permission for use must always be obtained from
Springer-Verlag. Violations are liable for prosecution under the German Copyright
Law.
The use of general descriptive names, registered names, trademarks, etc. in this
publication does not imply, even in the absence of a specific statement, that such
names are exempt from the relevant protective laws and regulations and therefore
free for general use.
© Springer-Verlag Berlin Heidelberg 1996
Typesetting: Camera ready by author
SPIN: 10544056 42/3142-543210 -Printed on acid-free paper
Preface
I started to deal with genetic algorithms in 1993 when I was working on
a project on learning and rational behavior in economic systems. Initially
I carried out simulations in an overlapping generations model but soon got
dissatisfied with the complete lack of theoretical foundation for the observed
behavior. Thus, I started to work on a mathematical representation of the
behavior of a simple genetic algorithm in the special setup of an interacting
population of economic agents and step by step arrived at the results collected
here. However, I believe that much more can and has to be done in this field.
I would like to thank Gustav Feichtinger who not only supervised my
doctoral thesis but always supported and encouraged me throughout the
last few years. Special thanks are also due to K. Hornik, A. Mehlmann and
M. Kopel who contributed largely to the work. During the preparation of
the monograph I also benefited from helpful comments of A. Geyer-Schulz,
G. Rote, G. Tragler and A. Rahman. Special thanks to W. A. Muller from
Springer-Verlag for his support. Financial support from the Austrian Science
Foundation under contract number P9112-S0Z is gratefully acknowledged.
Vienna, May 1996
Table of Contents
Preface....................................................... V
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Bounded Rationality and Artificial Intelligence . . . . . . . . . . . 7
2.1 Bounded Rationality in Economics......... ......... ... . . . 7
2.2 Artificially Intelligent Agents in Economic Systems ... . . . . .. 11
2.3 Learning Techniques of Artificially Intelligent Agents. . . . . . .. 13
2.3.1 Genetic Algorithms and Related Techniques.. .... . .. 13
2.3.2 Classifier Systems ................................ 13
2.3.3 Neural Networks ................................. 17
2.3.4 Cellular Automata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21
2.4 Some Applications of CI Methods in Economic Systems .. . .. 25
2.4.1 Bidding Strategies in Auctions .................... , 25
2.4.2 The Iterated Prisoner's Dilemma. . . . . . . . . . . . . . . . . .. 27
2.4.3 Market Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31
2.4.4 Further Simple Economic Models. . . . . . . . . . . . . . . . . .. 33
2.5 Potentiality and Problems of CI Techniques in Economics ... 34
3. Genetic Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37
3.1 What are Genetic Algorithms? ......... " ... , . . .. .... . . .. 37
3.2 The Structure of Genetic Algorithms.. . . . . . . . . . . . . . . . . . . .. 38
3.3 Genetic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39
3.3.1 Selection........................................ 39
3.3.2 Crossover ....................................... 41
3.3.3 Mutation........................................ 42
3.3.4 Other Operators ................................. 42
3.3.5 An Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43
3.4 Genetic Algorithms with a Non-Standard Structure.. ... . . .. 44
3.5 Economic Interpretation of Genetic Learning. . . . . . . . . . . . . .. 45
3.6 Some Analytical Approaches to Model Genetic Algorithms. .. 49
3.6.1 The Schema Theorem ............................. 49
3.6.2 The Quantitative Genetics Approach ............... 52
3.6.3 Markov Chain Models ............................ 54
VIII Table of Contents
4. .G enetic Algorithms with a State Dependent Fitness
:Function.. .. .. ........ ...... .... .. .......... .. .... .. .. .... 61
4.1 State Dependency in Economic Systems ................... 61
4.2 A Markov Model for Systems with a State Dependent Fitness
Function....... ........ ... . ... . . .. .. .. . . ...... . ....... 62
4.3 The Difference Equations Describing the GA . . . . . . . . . . . . . .. 67
4.4 Deviation from the Markov Process. . . . . . . . . . . . . . . . . . . . . .. 68
4.5 A Numerical Example ................................... 72
4.6 Stability ofthe Uniform States... .. ... ... .... . ... ........ 73
4.7 Two-Population Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81
5. Genetic Learning in Evolutionary Games .......... . . . . . .. 87
5.1 Equilibria and Evolutionary Stability ..................... 87
5.2 Learning in Evolutionary Games... ... . ...... . ... ........ 88
5.3 Learning by a Simple Genetic Algorithm.. .... . . .. . ....... 91
5.3.1 Rock-Scissors-Paper Games.... . . ... . ... ....... . ... 92
5.3.2 A GA Deceptive Game. . . . . . . . . . . . . . . . . . . . . . . . . . .. 97
5.3.3 Learning in Non Deceptive Games .................. 101
5.4 Two-Population Contests ................................ 104
6. Simulations with Genetic Algorithms in Economic Systems111
6.1 A Model of a Competitive Market ........................ 111
6.1.1 Pure Quantity Decision ........................... 114
6.1.2 Exit and Entry Decisions .......................... 117
6.2 An Overlapping Generations Model with Fiat Money ....... 122
6.2.1 Learning of Cyclical Equilibria ..................... 126
6.2.2 Learning of Sunspot Equilibria ..................... 130
7. Stability and Encoding ................................... 133
7.1 The Cobweb Example Revisited .......................... 133
7.2 Impact of a Change in Encoding and Scaling ............... 137
7.3 A Method for Finding Economic Equilibria ................ 139
8. Conclusions ............................................... 141
A. Basic Definitions and Results Used ....................... 143
A.1 Time Homogeneous Markov Chains ....................... 143
A.2 Nonlinear Difference Equations and Stability ............... 145
B. Calculation of the Equilibria of the Evolutionary Games in
Chapter 5 ................................................ 147
B.1 Rock-Scissor-Paper Games ............................... 147
B.2 The GA Deceptive Game GAD ........................... 149
B.3 The Games Gl and G2 .................................. 150
References .................................................... 151
Table of Contents IX
List of Figures ................................................. 159
List of Tables .................................................. 163
Index ......................................................... 165
1. Introduction
The analysis of mathematical models describing the learning behavior of ra
tional agents has been one of the major topics of economic research for the
last few years or even decades. Many different models have been proposed and
their analysis has given the researchers some insights into the phenomenon of
formation of equilibria in economic systems. On the other hand, the modern
development of computer technology has caused the rise of a new dynamic
field of research, which deals entirely with the understanding and imitation of
human behavior, namely the "artificial intelligence" research. Although there
is considerable overlap between these two fields they have long developed in
isolation from one another. Only recently the interest of some economists in
certain techniques mainly from the AI related field of "computational intelli
gence" (el) has increased and has led to the applications of these techniques
to economic models.
The main reason for the weak interaction between economists and CI re
searchers is that the tools and goals of these two groups are quite different.
The traditional approach of economists is to analyze economic systems with
the help of mathematical theory. Relying on plausible behavioral assump
tions about economic agents an adaptive learning model is constructed. The
economists use mathematical representations of the model and try to derive
analytical results for these models. In order to keep these models analyti
cally tractable most of these models use rather simple behavioral assump
tions. Nevertheless, a large number of conclusions could be derived with this
approach. Often the mathematical analysis allows structural insights and ex
plains similarities and differences in the behavior of different models. On the
other hand, the majority of models allow only a few weak or local results.
In these cases a mathematical analysis draws only a coarse picture of the
learning behavior.
Quite the opposite holds true for the approach taken by computer sci
entists. They like to deal with learning models which may be efficiently im
plemented on a computer. Usually mathematical considerations are of minor
importance, and the algorithms rely on heuristic arguments and similarities
to nature. Normally the analysis of these algorithms is done by testing them
with a large number of actual problems. The -obtained numerical results are
used to build up conjectures regarding the performance of the algorithm in
2 1. Introduction
different set-ups. On one hand this approach allows to use more complex
learning models than the analytical approach described above but on the
other hand simulations can never prove a certain feature of the model but
only suggest it. In general the structural insights which may be obtained by
numerical simulations are not comparable to general mathematical results
but simulations allow some ideas about the behavior even if a mathematical
analysis is completely impossible.
In this monograph we connect the two approaches and analyze a special
learning algorithm developed in the CI literature with analytical methods
and simulations. We concentrate on genetic algorithms (GAs) and show how
these learning algorithms may be interpreted as models of the learning be
havior of a population of adaptive agents. We further translate this algorithm
into a framework of traditional learning models in economics, namely math
ematics. In doing so we get the opportunity to take both the analytical and
numerical approach to this algorithm. In the analytical part we do a more
or less standard analysis of an economic learning rule and try to describe
the long run behavior of a population which acts according to this learning
model. We derive by means of Markov theory a result which characterizes the
possible long run states of the population. In analyzing a system of difference
equations, we furthermore give conditions for local stability or instability of
an arbitrary state. Based on these results we show that the calibration of pro
cesses like genetic algorithms is very important. Different coding mechanisms
and parameter values may lead to a completely different long run behavior
of the system. These results will be first shown by mathematical proofs and
also illustrated by examples from the fields of game theory and economics.
However, the part of these notes in which we present simulations is not
intended as a pure illustration of the mathematical results, but should be
regarded as an important part of the analysis of GAs in economic systems.
The simulations show several properties of the process we could not prove
analytically and we also present simulations in models where the assump
tions of our analytical result.s are not fulfilled. Thus we believe that in these
notes both the analytical approach and the simulation approach are of great
importance. Both approaches together will give quite a fine picture of the
behavior of GAs in economic systems.
Besides the technical aspects a major goal of these notes is also to give an
economic interpretation of the GA as a learning rule. We will try to use this
interpretation to understand simulation results in different economic models
from an economic point of view.
Some words of motivation are in order here. Why do we consider GAs
as a model of adaptive learning and why do we think that this analysis is
of interest. First of all, as already mentioned and as elaborated in chapter 2
various researchers have used GAs for simulating the behavior of a population
of interacting agents. Although this per se should not be seen as a striking
argument that GAs are a useful model oflearning our analysis can in any case