Case-based decision theory and financial markets [Elektronische Ressource] / vorgelegt von Ani Vladimirova Guerdjikova

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Case-Based Decision Theory andFinancial MarketsDissertation zur Erlangung des Grades eines Doktors derWirtschaftswissenschaften der Sozial- undWirtschaftswissenschaftlichen Fakultät der UniversitätHeidelbergvorgelegt vonAni Vladimirova GuerdjikovaJuli 2004, HeidelbergAcknowledgments .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . . 61. Introduction . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . . 91.1 Developments in the Theory of Decision-Making underUncertainty . . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 111.2 The Case-Based Decision Theory .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 151.2.1 Axiomatic Representation .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 171.2.2 Applications of the Case-Based Decision Theory.. . .. . .. .. . .. . .. . .. . 211.3 Decision Theory and Financial Markets . . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 241.4 Case-Based Reasoning in Models of Financial Markets . .. . .. . .. . .. . 291.4.1 Anchoring. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 311.4.2 Reference Levels . . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 331.4.3 Representativeness Heuristic .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 341.4.

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Case-Based Decision Theory and
Financial Markets
Dissertation zur Erlangung des Grades eines Doktors der
Wirtschaftswissenschaften der Sozial- und
Wirtschaftswissenschaftlichen Fakultät der Universität
Heidelberg
vorgelegt von
Ani Vladimirova Guerdjikova
Juli 2004, HeidelbergAcknowledgments .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . . 6
1. Introduction . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . . 9
1.1 Developments in the Theory of Decision-Making under
Uncertainty . . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 11
1.2 The Case-Based Decision Theory .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 15
1.2.1 Axiomatic Representation .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 17
1.2.2 Applications of the Case-Based Decision Theory.. . .. . .. .. . .. . .. . .. . 21
1.3 Decision Theory and Financial Markets . . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 24
1.4 Case-Based Reasoning in Models of Financial Markets . .. . .. . .. . .. . 29
1.4.1 Anchoring. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 31
1.4.2 Reference Levels . . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 33
1.4.3 Representativeness Heuristic .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 34
1.4.4 Belief Perseverance and Confirmatory Bias .. . .. . .. . .. . .. .. . .. . .. . .. . 35
1.4.5 Learning . . .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 36
1.4.6 Learning through Induction. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 37
1.5 Overview of the Thesis .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 39
2. Conceptual Issues .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. 47
2.1 Problems and Acts .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 48
2.2 Memory .. . .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 50
2.3 Aspiration Level .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 54
2.3.1 The Concept of Aspiration Level in Psychology . .. . .. . .. .. . .. . .. . .. . 54
2.3.2 Aspiration Levels in the Economic Literature . .. . .. . .. . .. .. . .. . .. . .. . 56
2.3.3 The Operationalization of the Concept of Aspiration Level in
the Applications of the Case-Based Decision Theory. . .. .. . .. . .. . .. . 59
2.4 Similarity . .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 61
2.4.1 Similarity — A Philosophical View. .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 61
2.4.2 Similarity in Economics . . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 62
2.4.3 Similarity in Financial Markets .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 64
2.5 Conclusion .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 67
3. Portfolio Choice and Case-Based Reasoning .. . .. 69
3.1 Case-Based Decision-Making . . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 71
13.2 The Portfolio-Choice Problem . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 74
3.3 Case-based Decision Theory and Expected-Utility Maximization . . 76
3.4 The Case of Constant Aspiration Level .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 79
3.5 Max-Min Updating Rules. .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 81
3.6 Interpretation of the Results . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 83
3.6.1 Choosing an Efficient Portfolio .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 84
3.6.2 Arbitrage Restrictions . . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 85
3.6.3 Diversification and Familiarity Effects .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 86
3.6.4 Excessive Trading . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 88
3.7 Collecting Additional Information . . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 89
3.8 Portfolio Choice with Similarity Considerations . .. . .. . .. .. . .. . .. . .. . 94
3.8.1 The Case of Constant Aspiration Level .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 95
3.8.2 Learning with Similarity. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . 99
3.9 Conclusion .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 103
Appendix . .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 105
4. On the Definition and Existence of an
Equilibrium in an OLG Economy with
Case-Based Decisions . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 135
4.1 The Economy.. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 138
4.2 Investment Decision of the Young Investors.. . .. . .. . .. . .. .. . .. . .. . . 139
4.3 Individual Demand for Assets . . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 141
4.4 Walras’ Law for an Economy with Case-Based Decision-Makers . 144
4.5 Definition of a Temporary Equilibrium .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 146
4.6 Existence of a Temporary Equilibrium. .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 148
4.7 The Case of Two Assets.. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 157
4.8 Conditions for a Non-Degenerate Equilibrium .. . .. . .. . .. .. . .. . .. . . 161
4.9 Discussion of the Results . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 166
4.10 Conclusion .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 168
Appendix . .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 170
5. Asset Price Dynamics with Case-Based
Decisions .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 173
5.1 The Economy.. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 179
25.2 Temporary Equilibrium .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 181
5.3 Price-Dynamics with One-Period Memory . .. . .. . .. . .. . .. .. . .. . .. . . 183
5.3.1 The Case of Low Aspiration Levels .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 185
5.3.2 The Case of Intermediate Aspiration Levels . . .. . .. . .. . .. .. . .. . .. . . 186
5.3.2.1 Computation of the Equilibrium.. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 187
5.3.2.2 Discussion of the Results . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 187
5.3.3 The Case of High Aspiration Levels .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 192
5.3.3.1 Computation of the Equilibrium.. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 192
5.3.3.2 Discussion of the Results . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 193
5.4 Price-Dynamics with Long Memory .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 194
5.4.1 Investor Types . . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 195
5.4.2 Price Dynamics — An Example . .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 196
5.4.3 Discussion of the Results . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 197
5.4.4 Price Dynamics for General Probability Distributions . .. .. . .. . .. . . 202
5.5 Hypothetical Cases .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 204
5.5.1 Individual Portfolio Choice . . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 205
5.5.2 Limit Behavior with Long Memory. .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 206
5.6 Similarity in Asset Markets. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 209
5.6.1 OLG-Model with Two Types of Investors .. .. . .. . .. . .. . .. .. . .. . .. . . 210
5.6.2 Equilibrium Paths . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . ..211
5.6.3 Price Dynamics with One-Period Memory . .. . .. . .. . .. . .. .. . .. . .. . . 212
5.6.4 Price Dynamics with Long Memory .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 217
5.7 Conclusion .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 220
Appendix . .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 223
6. Fitness and Survival of Case-Based Decisions
— An Evolutionary Approach. .. .. . .. . .. . .. . .. .. . .. . .. . . 259
6.1 Survey of the Literature .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 261
6.2 The Economy.. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 263
6.2.1 Information and Individual Decisions . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 265
6.2.1.1 Case-Based Decision-Makers . . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 265
6.2.1.2 Expected Utility Maximizers .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 269
6.2.2 Temporary Equilibrium .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 270
36.3 The Evolution of Investor Types . .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 271
6.3.1 Replicator Dynamic . .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 273
6.3.2 Temporary Equilibrium with Replicator Dynamic . . .. . .. .. . .. . .. . . 276
6.4 Analysis of the Dynamic. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 277
6.4.1 Stationary States .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 277
6.4.2 Stability ofe= 1 . . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 279
6.4.2.1 The Case of High Aspiration Levels .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 281
6.4.2.2 The Case of Low Aspiration Levels .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 285
6.5 Asset Prices in the Presence of Case-Based Decision-Makers .. . . 286
6.6 CRRA Utility .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 290
6.7 CRRA Utility with Diversification. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 293
6.8 Conclusion .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 297
Appendix . .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 300
7. Conclusion .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 325
7.1 Main Results . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 325
7.1.1 Portfolio Choice with Case-Based Decisions. . .. . .. . .. . .. .. . .. . .. . . 325
7.1.2 Asset Prices in an Economy with Case-Based Decision-Makers. . 326
7.1.3 Fitness of Case-Based Decisions . .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 327
7.2 Outlook .. . .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 328
References . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . . 331
4 Acknowledgments
This thesis would not have been possible without the help and support of many people. First of
all, I am deeply indebted to my advisor, Jürgen Eichberger, who introduced me to the exciting
world of decision-making under uncertainty and financial markets. He guided me throughout
the whole duration of my PhD project. He gave me the freedom to develop my ideas and en-
couraged me to search for answers. I would like to thank my second advisor, Hans Gersbach.
His critical questions and comments helped to clarify many of the issues addressed in this the-
sis. I also benefited a lot from discussions with Malte Faber. The Faculty of Economics and
Social Sciences at the University of Heidelberg, as well as the Sonderforschungsbereich 504
at the University of Mannheim provided through their weekly seminars and invited lecturers
a unique possibility to communicate with international scientists. I have benefited a lot from
the comments and suggestions of Hans Haller, Uzi Seagal, Paolo Ghirardato, Clemens Puppe
and Atsushi Kajii. The German Research Foundation and the German Economic Association
provided financial support which allowed me to participate at many international conferences.
This gave me the opportunity to meet Itzhak Gilboa, David Schmeidler, Klaus Nehring, David
Easley, Lawrence Blume, Ho Mou Wu, Thorsten Hens and discuss my work with them. I would
like to thank them for illuminating discussions and for many helpful comments. Thanks are also
th th thdue to the participants of the 57 ESEM in V enice, the 8 SMYE in Leuven, the 58 ESEM in
thStockholm, the 18 annual meeting of EEA in Stockholm, the annual meeting of the German
Economic Association and the NCCR workshop in Rigi and especially to Florian Engelmeier,
Markus Reisinger, Burckhard Schipper, David Kelsey, Klaus Ritzberger, Alexander Koch and
Laurens Swinkles.
I would like to thank to all my colleagues at the University of Heidelberg and, especially to Ute
Schumacher, for the nice working atmosphere during these four years. The participants at the
PhD seminar provided many useful comments which enriched my work. Switgard Feuerstein,
V erena Liessem, Ralph Winkler and Alexander Zimper all read parts of this work and made
helpful comments and suggestions. I would like to thank them for their help. Special thanks
are due to my colleague Johannes Becker and to my brother Stefan Gerdjikov, who were always
ready to share their mathematical knowledge with me. Their comments helped to make some of
the proofs more comprehensive. I am thankful to my colleagues Mi-Yong Lee-Peuker, Michael
6Rimmler, Damian Damianov and Dmitri Vinogradov, for long hours of discussion and for their
moral support.
I am indebted to my parents, Svetla Drenska and Vladimir Gerdjikov, my brother, to Miroljuba
Drenska and Viktor Badulin and to my boyfriend, Markus Hero, for the moral support I received
during these four years and for their patience and readiness to discuss topics of this thesis with
me. Without their belief in me, this work would not have been possible.
7Chapter 1. Introduction
The case-based decision theory has been recently proposed by Gilboa and Schmeidler (1995) as
an alternative theory for decision-making under uncertainty. Differently from the expectedutility
theory, the case-based decision theory models decisions in situations of structural ignorance,
in which neither states of the world, nor their probabilities can be naturally derived from the
description of the problem. It is assumed that a decision-maker can only learn from experience,
by evaluating an act based on its past performance in similar circumstances. An aspiration level
is used as a bench-mark in the evaluation process. It distinguishes results considered satisfactory
(those exceeding the aspiration level), which make the alternative more attractive, from the
unsatisfactory ones, which influence negatively the evaluation of the alternative.
Similarity considerations play an important role in case-based reasoning. The evaluation of an
alternative depends not only on its own performance, but also on the utility realizations achieved
from similar alternatives in similar circumstances.
1Although the case-based decision theory has been applied in several economic contexts , it has
not been used to model decision-making in financial markets up to now. Still, a model of fi-
nancial markets, in which expected utility maximization is replaced by case-based reasoning is
of interest for several reasons. First, it allows to gain a deeper understanding of the case-based
decision theory itself. In such a model the operationalization of theoretical concepts such as
the aspiration level, the past experience of the decision-maker, as well as his similarity percep-
tions becomes necessary. It is therefore possible to examine the inf luence of these concepts on
individual behavior. The results achieved further allow to interpret these concepts in an eco-
nomically meaningful way.
Second, the application of the case-based decision theory to financial markets contributes to
the literature on behavioral finance, by describing the dynamics of portfolio holdings and asset
prices in a market with case-based investors. The analysis of the behavior implied by case-based
reasoning allows for comparisons to the predictions of the standard financial theory, as well as
2to the results obtained from alternative decision theories .
1 This literature is reviewed in section2 of the introduction.
2 The implications of an abstract decision theory can only be understood by applying it to a specific economic
9