Algebraic decomposition of individual choice behavior
mpib_escidoc_33561
Materialien aus der Bildungsforschung Nr. 63
Martin Lages
ALGEBRAIC DECOMPOSITION
OF INDIVIDUAL CHOICE BEHAVIOR
MaxPlanckInstitut für Bildungsforschung
Berlin 1999
GW ISSN 01733842
ISBN 3879850704
.. *
2
*
.
4
Bestellungen werden erbeten an die Verwaltung des Instituts bei gleichzeitiger
Überweisung von DM 25, (einschließlich 7% Mehrwertsteuer) zuzüglich
Versandpauschale pro Band DM 3, auf das Konto Nr. 417 12 11 der Deutschen Bank,
Bankleitzahl 100 700 00
Nachdruck, auch auszugsweise, ist nur mit Zustimmung des Instituts gestattet.
61999 MaxPlanckInstitut für Bildungsforschung, Lentzeallee 94, D14195 Berlin.
GW ISSN 01733842
D16
ISBN 3879850704
2
Contents
List of Figures
List of Tables
Acknowledgements
Abstract
1 Introduction
1.1
Rationality and Probability of Choice
Deterministic Models.
1.2
* * * * * * * * * * * * * * * *
1.2.1 Order Structures
* * * * * * * * * * * * * * * * * * * * * * * * * * * * .
1.2.2 Expected Utility Models
1.2.3 NonExpected Utility Models
* * * * * * * * * * * * * * * * * * *
1.2.4 Miscellaneous Models
Probabilistic Models
1.3
* * * * * * * * * * * * * * * * * * * *
1.3.1 Constant Utility Models
* * * * *
1.3.2 Random Utility Models
* * * * * * * * * * * *
Multiattribute Models
1.4
* * * * * * * * * * *
. .
. .
1.4.1 Deterministic Multiattribute Models
1.4.2 Probabilistic Multiattribute Models
1.5
Algebraic Decomposition
. .
1.6 Summary
Theory
2.1
Directed Cycles and Tournaments
2.2 Decomposition into Strong Components
2.2.1 Polynomials
2.2.2 Critical Arcs in Tournaments
2.2.3 Examples of Tournaments
Ear Decomposition
2.3
2.3.1 Cycle Space and Dicycle Basis
2.3.2 Ear Decomposition by Sequence
2.3.3 Example of Ear Decomposition by Sequence.
2.4
Completion by Cuts
2.4.1 Families of Intransitive Subchains
2.4.2 Closure and Lattice ..
* * * * * * * * * * * *
* * * *
2.4.3 Example of Completion by Cuts
vii
ix
xiv
XV
1
2
9
11
15
16
17
17
20
22
22
24
26
29
31
34
36
39
44
46
53
54
56
57
60
61
63
66
CONTENTS
vi
68
2.5 Summary
3 Experiments
71
3.1 General Method ....
72
* * * * *
76
3.2 Hypotheses and Statistical Testing ..
* * * * * * * *
3.3
80
Experiment 1A: Riskless Choice
80
3.3.1 Method
82
3.3.2 Results
Experiment 1B: Riskless Choice
87
3.4
* * * *
*. .
3.4.1 Method.
87
* * *
88
3.4.2 Results
Discussion
3.5
91
* * * * * * * * * * * * * * * * * * * * * * * * * * * *
3.6
Experiment 2A: Risky Choice
93
* * * * * * * * * * * * * * * * * * * * *
3.6.1 Method
93
* * * * * * * *
95
3.6.2 Results
Experiment 2B:
Risky Choice
101
3.7
3.7.1 Method
101
* * *
3.7.2 Results
102
* * * *
Discussion..
3.8
107
. .
Experiment 3A:
Discrimination ..
109
3.9
3.9.1 Method
109
* * * * *
.
* * * * * * * * * *
3.9.2 Results
112
* * * * * * * *
* * *
* * * *
* * .
* * * * * * * *
. * .
117
3.10
Experiment 3B: Discrimination ....
117
3.10.1 Method
* .
* .
118
3.10.2 Results.
122
3.11 Discussion..
* * * * * *
124
3.12 General Discussion
4 Conclusions
127
127
4.1 Theoretical and Empirical Implications
4.2 Individual vs Group Data
130
* * * * *
131
4.3 Toward a Qualitative Theory of Error
A
133
Mathematical Background
133
A.1 Notations and Basics ..
* * * *
134
A.2 Order Theory..
136
A.3 Graph Theory
137
A.4 Algebra
* * * * *
137
A.4.1 Matrix Theory...
* * *
.
139
A.4.2 Polynomial Rings
* * * * * * *
141
B Supplements
141
B.1
Experiment 1A and 1B ...
141
B.1.1 Instructions .....
142
B.1.2 Stimuli
* * * * * *
143
B.1.3 Results of Experiment 1
*
147
B.1.4 Results of Experiment 1B
.
* * * * *
* * * * * * *
*
151
B.2 Experiment 2A and 2B
CONTENTS
B.2.1 Instructions
Stimuli
B.2.2
. .
* * *
* * * * *
* * * * * * * * * * *
B.2.3
Results of Experiment 24
* * * * * * * * * *
B.2.4
Results of Experiment 2B...
* * * * *
B.3
Experiment 3A and 3B ..
* * * * * * * * *
. .
B.3.1
Instructions
* * * * * * * * * * * *
Stimuli
B.3.2
* * * * * * * * * * * * * * * * * * * * * * * * * * *
. .
B.3.3 Results of Experiment 3A
* * * * * * * * *
B.3.4 Results of Experiment 3B
C ProgramListings
*
C.1 Prolog
* .
* * * * *
C.2 Mathematica.
* * * * * * *
C.3 CProgram
Bibliography
Author and Subject Index
vii
151
152
153
159
163
163
164
166
172
177
177
184
185
187
197
List of Figures
1.1 Examples of inconsistent choices: A. Preference reversal; B. Intransitive triple;
C. Intransitive preferences between three and four objects.
5
* * *
Tournament A differs from B by a single arc.
2.1
35
2.2
45
Tournament A differs from B by a single arc.
* * * * * * *
)
2.3
Hasselike diagram of Subject 1 in Session 1 (Exp 2B)
49
2.4 Diagram of Subject 1 in Session 2 (Exp 2B)
50
2.5 Hasselike diagram of Subject 1 in Session 3 (Exp 2B)
51
2.6
53
Illustration of 'ears' in an undirected graph.
2.7
58
Ear decomposition of tournament for Subject 1 in Session 1 (Exp 2B).
2.8
59
Continued ear decomposition of tournament.
67
Hasse diagram of T and C.
2.9
3.1 Complete graph of order 12
72
3.2
Graph of resolution block Bo
73
3.3 Graph of repetition block B
74
* * * * * *
3.4 Kendall's ( for all groups of Experiment 1A and 1B
91
3.5 Display of two lotteries in a single choicetrial
94
3.6 Lotteries plotted by payoff and chance of win on a loglinear scale.
95
107
3.7 Kendall's ( for all groups of Experiment 2A and 2B.
110
3.8 Luminances of center and surround.
3.9 Display of stimuli on screen.
111
. * * * * * * *
3.10 Kendall's ( for all groups of Experiment 3A and 3B
123
*
B.1 Illustration of stimuli.
165
ix
List of Tables
1.1 The 'Ellsberg Paradox'
3.1 Resolution and Repetition Block Designs
* * * * * * * * * * * *
3.2 Mean Response Times of Choices (Exp 1A)
3.3 ANOVA on Mean Response Times (Exp 14)
3.4
Intransitive Triples of Preference (Exp 1A)
ANOVA on Kendall's ( (Exp 14)
3.5
3.6
Stepwise Discriminant Analysis on Dicycles (Exp 1A)
3.7
14
(Exp
Stepwise Discriminant Analysis on Ear Dicycles
Reversals of Preference (Exp 1A)
3.8
Mean Response Times of Choice Trials (Exp 1B)
3.9
3.10
ANOVA for Mean Response Times (Exp 1B)
3.11 Intransitive Triples of Preference (Exp 1B)
3.12 ANOVA for Kendall's ( (Exp 1B)
3.13 Reversals of Preference (Exp 1B)
3.14 Mean Response Times of Choices (Exp 24)
3.15 ANOVA for Mean Response Times (Exp 24)
3.16 Intransitive Triples of Preference (Exp 24)
3.17 ANOVA for Kendall's ( (Exp 24)
3.18 Stepwise Discriminant Analysis on Ear Dicycles (Exp 24)
3.19 Average Size of Strong Components (Exp 24,
3.20 ANOVA for Size of Strong Components (Exp 24)
3.21 Reversals of Preference (Exp 24)
3.22 ANOVA for Preference Reversals (Exp 24)
* *
3.23 Mean Response Times of Choices (Exp 2B)
* * *
3.24 ANOVA for Mean Response Times (Exp 2B)
3.25 Intransitive Triples of Preference (Exp 2B)
ANOVA for Kendall's ( (Exp 2B)
3.26
3.27
Stepwise Discriminant Analysis on all Dicycles (Exp 2B)
3.28 Stepwise Discriminant Analysis on Ear Dicycles (Exp 2B)
3.29 Average Size of Strong Components (Exp 2B)
3.30 ANOVA for Size of Strong Components (Exp 2B)
3.31 Reversals of Preference (Exp 2B)
. .
3.32 ANOVA for Preference Reversals (Exp 2B)
3.33 Mean Response Times for Discrimination (Exp 34)
3.34 ANOVA for Mean Response Times (Exp 34)
3.35 Intransitive Triples of Discrimination
* * *
* .
3.36 ANOVA for Kendall's ( (Exp 34)
* * * * * * * * * * *
Xi
14
75
82
83
83
84
84
85
85
88
89
89
90
90
96
96
97
97
98
98
99
99
100
102
102
103
103
104
104
105
105
106
106
112
113
113
114
xii
3.37 Stepwise Discriminant Analysis on Dicycles (Exp 34)
3.38 Stepwise Discriminant Analysis on Ear Dicycles
34)
(Exp
3.39 Reversals of Discrimination (Exp 34)
3.40 ANOVA on Discrimination Reversals (Exp 34)
3.41 Mean Response Times for Discrimination (Exp
3B)
3.42 ANOVA for Mean Response Times (Exp 3B)
3.43 Intransitive Triples of Discrimination (Exp 3B)
3.44 ANOVA for Kendall's ( (Exp 3B)
3.45 Stepwise Discriminant Analysis on EarDicycles (Exp 3B)
3.46 Reversals of Discrimination (Exp 3B)
3.47 ANOVA on Preference Reversals (Exp 3B)
B.1 Chocolate Bars
B.2 Coefficients of y for Each Subject and Session (Exp 14)
B.3 Coefficients of % for Each Subject and Session (Exp 14)
B.4 Number of Ear Dicycles for Each Subject and Session (Exp 14)
B.5 Number of Ear Dicycles for Each Subject and Session (Exp 14)
B.6 Coefficients ofy for Each Subject and Session (Exp 1B)
B.7 Coefficients of y for Each Subject and Session (Exp 1B)
B.8 Number of Ear Dicycles for Each Subject and Session (Exp 1B)
B.9 Number of Ear Dicycles for Each Subject and Session (Exp 1B)
B.10 Description of Lotteries
B.11 Coefficients of % for Each Subject and Session (Exp
24)
24,
B.12 Coefficients of y for Each Subject and Session (Exp
B.13 Coefficients of% for Each Subject and Session (Exp 24)
B.14 Numbers of Ear Dicycles for Each Subject and Session (Exp 24)
B.15 Number of Ear Dicycles for Each Subject and Session
(Exp 24)
B.16 Number of Ear Dicycles for Each Subject and Session
(Exp 24)
2B)
B.17 Coefficients of % for Each Subject and Session (Exp
B.18 Coefficients of % for Each Subject and Session (Exp
2B)
(Exp 2B)
B.19 Number of Ear Dicycles for Each Subject and Session
B.20 Number of Ear Dicycles for Each Subject and Session (Exp 2B)
B.21 Description of Disks
B.22 Coefficients of % for Each Subject and Session (Exp 34)
B.23 Coefficients of y for Each Subject and Session (Exp 34,
B.24 Coefficients of % for Each Subject and Session (Exp 34)
B.25 Number of Ear Dicycles for Each Subject and Session (Exp 34)
B.26 Number of Ear Dicycles for Each Subject and Session (Exp 34)
B.27 Number of Ear Dicycles for Each Subject and Session (Exp 34)
B.28 Coefficients of % for Each Subject and Session (Exp 3B)
B.29 Coefficients of % for Each Subject and Session (Exp 3B)
B.30 Number of Ear Dicycles for Each Subject and Session (Exp 3B)
B.31 Number of Ear Dicycles for Each Subject and Session (Exp 3B)
LIST OF TABLES
114
115
115
116
118
118
119
119
120
120
121
*.
142
143
. * .
144
*
145
146
147
148
..
149
150
152
153
154
155
156
157
158
159
160
161
162
164
166
167
168
169
170
171
172
173
174
175
Meinen Eltern gewidmet
Acknowledgements
1 would like to thank all the people and institutions who made this dissertation possible.
First, 1 am grateful to my supervisor Prof. Dr. Dietrich Albert for his steady support and
valuable advice.
started work on the dissertation at the Psychological Institute, University of Heidelberg
and I finished it during my stay at the Department of Experimental Psychology, University
of Oxford. Therefore, I am especially grateful to Dr. Michel Treisman who supported con¬
tinuation of work at Oxford.
Finally, I would like to thank Rhona Johnston and Martin Schrepp for proofreading,
Michael Brady for advice on programming in Open Prolog and all subjects for participating
in experiments.
This dissertation was supported by a stipend of the Land BadenWürttemberg (§10
LGFG), supplemented by a grant of the Deutscher Akademischer Austausch Dienst (DAAD).
and extended by a doctoral studentship of the Studienstiftung des deutschen Volkes.
Abstract
In this study a general framework for algebraic decomposition models of individual decision
behavior is proposed. A specific decomposition that is based on the sequence of intransitive
choices is investigated in classical domains of binary decision tasks.
In the theoretical part a general algebraic decomposition is applied to individual choice
behavior. It is based on directed cycles of length k, the graphtheoretical analogue of in¬
transitive choice. Two major results are obtained by applying techniques from graph theory
and algebra to pair comparisons.
First, internal consistency of choice behavior can be characterized by a polynomial ex¬
pression that provides detailed information about the number and size of strong components
together with the number of kdicycles in each component. The attempt, however, to iden¬
tify a minimal set of critical choices which can explain all intransitive choices leads to a well
known optimization problem.
Second, a specific decomposition model, the ear decomposition by sequence, is suggested
which is based on the sequence of intransitive choices in a pair comparison. The ear de¬
composition determines a unique and minimal collection of dicycles that can generate all
intransitive choices. In an additional technique, that makes slightly stronger assumptions,
intransitive subchains are defined on the sequence of choices. It is shown that under an
appropriate closure operation these subchains form a lattice structure.
In the experimental part it is investigated whether or not the algebraic decomposition
of choice behavior provides a suitable alternative to traditional algebraic and probabilistic
models. In three domains inconsistent choice behavior is studied in block designs with
differently optimized arrangements of choicetrials. Experiments in the domain of riskless
and risky choice show that subjects interact with the sequence of choicetrials in different
block designs. This supports the ear decomposition by sequence rather than the general
assumption of independent choices that underlies classical decision models. Experiments
on the visual discrimination of contrast do not provide such evidence and inconsistency of
choice varies unsystematically.
Algebraic decompositions of choice behavior are considered as an important step to¬
ward a qualitative theory of error on ordinal scale and further experimental and theoretical
developments are encouraged.
Zusammenfassung
In der vorliegenden Arbeit wird ein Rahmenkonzept für die algebraische Zerlegung individu¬
ellen Entscheidungsverhaltens vorgestellt. Eine spezifische Zerlegung, die auf der Reihenfolge
von intransitiven Wahlen beruht, wird in klassischen Alternativenbereichen binärer Entschei¬
dungsaufgaben untersucht.
Im theoretischen Teil wird eine generelle algebraische Zerlegung auf individuelles Wahl¬
verhalten angewandt. Sie beruht auf gerichteten Zyklen der Länge k, dem graphentheoretis¬
chen Analog intransitiven Wählens. Die Anwendung von Techniken aus der Graphentheorie
und Algebra auf Paarvergleiche hat zwei grundlegende Ergebnisse.
Zum einen kann die interene Konsistenz des Wahlverhaltens durch einen polynomischen
Ausdruck charakterisiert werden, der detailierte Information über die Anzahl und Größe
streng zusammenhängender Komponenten, sowie die Anzahl gerichteter Zyklen in jeder
Komponente bereithält. Allerdings führt der Versuch, eine minimale Menge von kritis¬
chen Wahlen zu identifizieren, die sämtliche intransitiven Wahlen erklären kann, zu einem
bekannten Optimierungsproblem.
Zum anderen wird ein spezifisches Zerlegungsmodell, die EarZerlegung nach Reihen¬
folge, vorgestellt, die auf der Reihenfolge von intransitiven Wahlen im Paarvergleich beruht.
Mit Hilfe der EarZerlegung kann eine eindeutig bestimmte minimale Menge gerichteter
Zyklen ermittelt werden, die sämtliche intransitiven Wahlen generieren. In einer weiteren
Technik, die auf etwas strengeren Annahmen beruht, werden intransitive Teilketten auf der
Reihenfolge der Wahlen definiert. Es wird gezeigt, daß diese Teilketten unter geeigneter
Abschlußoperation eine Verbandsstruktur bilden.
Im experimentellen Teil wird überprüft, ob die algebraische Zerlegung von Entschei¬
dungsverhalten eine Alternative zu klassischen algebraischen und probabilistischen Model¬
len darstellt. In drei Alternativenbereichen wird anhand von optimierten Anordnungen der
Wahldurchgänge in unterschiedlichen Blockdesigns inkonsistentes Entscheidungsverhalten
untersucht. Experimente zu Entscheidungen mit und ohne Risiko zeigen, daß die Ver¬
suchspersonen mit der Anordnung der Wahldurchgänge in unterschiedlichen Blockdesigns
interagieren. Ein Befund, der für die EarZerlegung nach Reihenfolge spricht und die für klas¬
sische Entscheidungsmodelle grundlegende Annahme von unabhängigen Wahldurchgängen
widerlegt. Experimente mit Diskriminierungsaufgaben zur visuellen Kontrastwahrnehmung
zeigen keinen vergleichbaren Befund und inkonsistentes Entscheidungsverhalten variiert un¬
systematisch.
Die algebraische Zerlegung von Entscheidungsverhalten wird als ein wichtiger Schritt in
Richtung einer qualitativen Fehlertheorie auf ordinalem Meßniveau betrachtet und mögliche
experimentelle und theoretische Entwicklungen werden angeregt.
Chapter 1
Introduction
In this study a general framework for algebraic decomposition models of
individual choice behavior is developed. It is argued that algebraic decom¬
positions provide an alternative to classical deterministic and probabilistic
models of choice, especially when the sample is small and choicetrials are
not repeated.
The thesis consists of four chapters and three appendices which are orga¬
nized as follows. In the first chapter rationality and probability of choice are
discussed. Both concepts have been essential to models of human decision¬
making and provide a normative framework. Classical models of choice be¬
havior are reviewed that are exposed to violations of rational choice principles.
No attempt is made to provide a complete survey of decisionmaking theories.
Only a few models from different approaches are highlighted with a prefer
ence for axiomatized models. Finally, algebraic decomposition techniques are
outlined which have been concerned with substructures in binary data. In
the second chapter theoretical considerations lead to a characterization of
choice behavior in terms of intransitive choices, that is directed cycles in a
directed graph. This characterization includes a general algebraic decompo¬
sition of a preference matrix into irreducible components. In the following
a more specific model is established. The ear decomposition by sequence
employs the sequence of choicetrials in a pair comparison which leads to a
unique algebraic decomposition of choice behavior. In the third chapter it is
experimentally investigated whether choice behavior in pair comparisons can
be modeled by algebraic decompositions such as the ear decomposition by
sequence. In six experiments inconsistency of choice behavior is tested un¬
der different block designs and in three different domains: decisionmaking
under certainty, decisionmaking under uncertainty or risk, and decisions in
psychophysical discrimination tasks. In the fourth chapter conclusions are
2
CHAPTER 1. INTRODUCTION
drawn from the theoretical and experimental results and possibilities for fur¬
ther research is discussed.
At this point a remark on the presentation of the first two chapters appears
necessary. The survey of decision models in the first chapter is far from being
complete. It was attempted to present concepts and models which have been
influential on decision making research or which can be linked to algebraic
decompositions. In both chapters it was tried to provide a comprehensible
account of theoretical ideas whilst avoiding a mixture of intuitive ideas with
mathematical terms. Therefore, both forms of presentation were only given
whenever this seemed advisable and straightforward. Progressing in this way
has the disadvantage that formal definitions and expressions may disturb the
flow of reading and that Appendix A needs to be addressed which provides
short accounts of the mathematical background.
On the other hand, all theoretical ideas are given precisely. In addition,
redundancy is avoided because each theoretical idea is defined only once
and can be referred to throughout the text. A subject and author index
is provided at the end. To some extent such a presentation of ideas might
even improve the comprehensibility of this dissertation, especially when more
technical sections are skipped at first reading.
1.1 Rationality and Probability of Choice
Describing human decision behavior in terms of deterministic models has a
longer tradition than psychology as an empirical science and dates back to
the eighteenth century. For example, the mathematician Daniel Bernoulli
(1738) laid down the foundations of expected utility, the majority rule traces
back to Condorcet (1785), who applied this rule in the context of social
choice, and the weighted sets of differences rule was already used by Benjamin
Franklin (1772) in the context of individual choice. Since then researchers in
disciplines such as mathematics, economics, philosophy, and psychology have
been involved in the development of decision models.
In general, models of individual choice can be divided into deterministic
and probabilistic models depending on their particular use of mathematical
concepts. A further classification which is based on the alternatives was
proposed by Edwards (1954c). He distinguished between risky choice in which
lA deterministic model may be regarded as a probabilistic one by assigning a probability of O to non¬
preference, 4 to indifference, and 1 to preference but this does not change the qualitative nature of the
deterministic model.
3
1.1. RATIONALITY AND PROBABILITY OF CHOICE
decisions are made under risk or uncertainty about the outcomes, and riskless
choice in which decisions are made under certainty.? Additionally, one may
distinguish between models for single and multiattribute alternatives and
between models for singlestage and multistage decision processes. Models
of the latter type are omitted here (see however Keeney & Raiffa, 1976; 1993).
Before a brief survey of models from some of these categories is given,
two important aspects of decision theory are introduced: rationality and
probability. A rather simplistic view of rationality and probability is adopted
here for there is no universal definition of rationality or probability, especially
not in the context of human decisionmaking. Nevertheless, both concepts
play an important role in deterministic and probabilistic models of choice
behavior.
In general, choice may be described as the specification of a nonempty
subset X of objects from a set S, with finite or infinite cardinality, where
every possible subset of the set denotes a possible choice. Consequently, the
complete choice behavior of an individual decisionmaker is only captured
if all chosen subsets of alternatives from all possible subsets of S are col¬
lected. This is clearly impracticable and a number of suitable restrictions are
normally imposed. Then of course, choice behavior is reduced to a subcol¬
lection of choices. Choosing a single object out of a set of objects is more
common than selecting two or more alternatives which might be considered
as repeated choice of single alternatives from the same set.? In fact, binary
choice or choosing a single object out of two in a forced choice pair com¬
parison is considered as a valid way of assessing individual choice behavior.
In the usual incomplete forced choice pair comparison procedure a person is
forced to make a choice between unordered pairs of objects. Clearly, under
this paradigm choice behavior satisfies connectedness and asymmetry, for ar¬
bitrary alternatives a,b e S satisfy either a » b or b a, where » denotes
the preference relation (see Definition A.2.1). The preference relation is of¬
ten said to be a subset of the cartesian product of the set of alternatives,
written as »C S x S. Hence, in the case of a complete pair comparison the
preference relation is a connected subset of the cartesian product whereas
in the case of an incomplete forced choice pair comparison binary prefer¬
ences are restricted to an asymmetric and connected subset of the cartesian
2In the following risky choice means choice with known probabilities whereas choice under uncertainty
also includes ambiguous or unknown probabilities.
»From this point of view ranking of alternatives is another special case because it is equivalent to choosing
single objects from a given set of objects without replacement.
CHAPTER 1. INTRODUCTION
4
product. This is easily overlooked because it is convenient to assume that
choice behavior is at least asymmetric and therefore unaffected by this stan¬
dard paradigm. The restriction to asymmetric and connected relations in
an incomplete forced choice pair comparison gives a trivial example of how
the experimental paradigm affects choice behavior since preference reversals
and missing preferences are simply excluded. A nontrivial interaction be¬
tween the pair comparison paradigm and choice behavior is investigated in
Chapter 3.
Despite their presupposed properties, incomplete forced choice pair com¬
parisons' have been used in numerous empirical investigations and are closely
related to the axiomatization of preference by binary relations. By imposing
reasonable constraints or axioms on a preference relation an ordered structure
can be defined as described in the next section and Appendix A.2. When
defining an ordered structure it is important whether the set of alternatives S
is considered as a set of finite or infinite cardinality, and whether the objects
in the set are represented as single or as multiattribute alternatives.
Rational choice behavior has been interpreted as optimal choice, empha¬
sizing the economic origin of decision research. Optimal choice maximizes
the outcome of one or more choices. This interpretation requires an interval
scale where different outcomes can be compared quantitatively. Such a scale
is the utility scale that was originally defined in terms of money but became a
more abstract construct since the days of Bernoulli (1738). The interval scale
requires stronger assumptions than the more fundamental ordinal scale which
only preserves the order of objects (Stevens, 1946; 1951). Consequently, the
assumptions for the representation on ordinal scale are of primary interest
and rational choice behavior is discussed in Chapter 2 and 3 exclusively in
terms of ordinal preference relations.
Transitivity of choice is a necessary condition for the existence of an ordi¬
nal utility scale. Together with connectedness, transitivity is also a sufficient
condition for the existence of such a scale, provided the number of alterna¬
tives is finite or countable. Since transitivity is so fundamental for the ordinal
representation of choice behavior it is considered as a cornerstone of norma¬
tive and descriptive decision theories (Edwards, 1954c; Luce, 1990). Indeed,
persons behave irrationally if they violate transitivity in their choice behav¬
ior and the following example is commonly cited: Assume a person chooses
intransitively by preferring alternative a over alternative b, b over c, and c
over a. Then, it can be argued that this person is willing to pay a certain
4In the following pair comparison always refers to an incomplete forced choice comparison.
1.1. RATIONALITY AND PROBABILITY OF CHOICE
5
amount of money to exchange a for b, another amount to exchange c for a,
and a third amount to exchange b for c. Thereby, the person loses money but
ends up with the same alternative. Obviously, the circular argument a » b,
6 » c, and c » a can be used recursively and may be extended to more than
three alternatives.
B.
A.
C.
4
—
Figure 1.1: Examples of inconsistent choices: A. Preference reversal; B.
Intransitive triple; C. Intransitive preferences between three and four ob¬
jects.
On the other hand, if we think of choices in terms of preference and indif
ference, written as a £ b, b £ c, and c 2 a, the phenomenon of intransitive
choices may be interpreted as indifference between alternatives. This can be
modeled by an equivalence relation between indifferent alternatives, written
as ab, b  c, and c  a (Definition A.2.3). However, it has been shown
that indifference is not necessarily transitive, and therefore not an equivalence
relation. The economist Armstrong (1939) was one of the first to argue that
indifference is not necessarily transitive. He discussed the following example:
Suppose a boy is indifferent about receiving as a gift a pony or a bicycle. He
will undoubtedly prefer the bicycle with a bell attached to the bicycle with
out a bell. But he is still likely to be indifferent about the bicycle with bell
and the pony. Intransitive indifference in the form of limited discriminatory
ability of the decisionmaker has been incorporated into deterministic and
probabilistic models.
Another axiom that has been associated with rational choice and that has
implications for the representation on ordinal scale is asymmetry. If some¬
body prefers a over b, he or she should not prefer b over a. Again, the phe¬
nomenon of preference reversals might be considered as transitive or intran¬
sitive indifference, but only a few preference reversals between alternatives
6
CHAPTER 1. INTRODUCTION
are believed to be nonsystematic caused by indifference or limited discrim¬
inatory ability between multiattribute alternatives (e.g., Tversky, 1969). As
mentioned before, in an incomplete forced choice pair comparison preference
reversals are ruled out by the experimental design unless the pair comparison
itself is repeated.
Preference reversals and intransitivity of preference may be considered as
violations of rational choice behavior especially when indifference between al¬
ternatives can be excluded as a possible explanation. The phenomena are re¬
lated and have a common feature in the language of graph theory. Preference
reversals and intransitivity can be described as directed cycles connecting two,
three, or more objects in a digraph (see Chapter 2 and Appendix A.2). It
is suggested here that irrational choice is conveniently described as directed
kcycles, including directed 2cycles, rather than a violation of stronger prin
ciples that can be derived from axiom systems of expected utility theory and
related theories. The graphtheoretical definition of intransitive preference
is exploited in the next two chapters. Intransitive preferences which form
directed cycles between more than two alternatives are investigated theoreti
cally in Chapter 2 and empirically in Chapter 3. In the following the violation
of asymmetry and transitivity in terms of preference reversals and intransi
tive preferences is considered as irrational choice behavior. To acknowledge
this simplistic view of rationality the general term inconsistent choice or in¬
consistency instead of irrationality is used throughout the text and refers
to preference cycles of any length. Figure 1.1 gives a graphical representa¬
tion of inconsistencies. Graph A on the lefthand side shows a preference
reversal (2dicycle), an intransitive triple (3dicycle) is depicted in Graph B
and a preference cycle of length 4 (4dicycle) appears in Graph C on the
righthand side. It is argued that longer preference cycles are less convinc¬
ingly interpreted in terms of intransitive indifference because all alternatives
that belong to a directed cycle have to be indifferent. This means as soon
as a single preference connects the least with the most preferred alternative
all previously ordered alternatives become indifferent. To explain d £ a in
Graph C, for example, would require that abc d. On the contrary,
it seems more plausible to assume that the preference order between a, b, c,
and d is valid and that the choice d a marks a new decision strategy. The
problem of critical arcs is addressed Section 2.2.2.
The concept of probability plays a crucial role in modeling riskless as well
as risky or uncertain choice behavior. It offers a possibility to model vi¬
olations of rational principles such as asymmetry and transitivity. It also
7
1.1. RATIONALITY AND PROBABILITY OF CHOICE
provides theoretical tools for modeling risky or uncertain outcomes. When
stating a probabilistic choice model, it is essential how and where the concept
of probability is incorporated. In models of riskless choice, for example, the
relative frequency of repeated choices between the same pair of alternatives
is modeled by functions of constant or random utilities, whereas in models of
risky choice the (subjective) probabilities of the outcomes are taken into ac¬
count to model a single choice between alternatives with uncertain outcomes.
Both types of models point to different underlying concepts of probability
which have been labeled as the frequentist and Bayesian approach. According
to the frequentist approach probability of an event means that if an exper¬
iment is repeated over and over again the Laplace probability" tends to be
the probability. According to the Bayesian approach, probability of an event
is a quantification of the speaker’s uncertainty about the outcome of the ex¬
periment and thus has a personal or subjective notion; the probability for
a certain event may be different for different speakers, depending on their
experience and knowledge of the situation. Ramsey (1931) and de Finetti
(1937) laid the groundwork for a subjective theory of probability. There has
been vigorous debates among proponents of various versions of these points
of view (for a collection of papers on this important topic see Kyburg &
Smokler, 1980) Without entering this controversy we give the following gen¬
eral definition of a probability measure on a countably additive probability
space which is widely accepted. It was first correctly stated by Kolmogorow
(1933).
Definition 1.1.1 (Probability Measure) A probability measure on the
probability space 9 % 0 is a function P from a galgebra A of subsets of
Q into 0,1 that satisfies the following axioms:
(i) P(O) =1.
(ii) IfACA, then P(A) 2 0.
(iii) If Aj, A2,... E A are mutually disjoint, then
P
04
£P(A.).
1
11
The first two axioms are quite obvious. Since the sample space 12 consists
of all possible outcomes, its probability equals P(9) = 1. The second axiom
The Laplace probability of event A is defined as the number of possible ways A can occur divided by
the number of all possible outcomes in a suitable sample space 82.
8
CHAPTER 1. INTRODUCTION
simply states that a probability is nonnegative. In the finite case, the third
axiom requires that if A and B are disjoint then P(AU B) = P(A) + P(B),
and in the infinite case it is assumed that this property extends to limits.6
The importance of the definition for the infinite case lies in the fact that it
holds for discrete and continuous probability measures and allows to distin¬
guish elementary probabilities from any probabilistic or stochastic mechanism
operating upon this structure.
The probability measure on the sample space determines the probabilities
of a random variable X; in the discrete case by a frequency function p, and
in the continuous case by a density function f. The expectancy E(X) of a
discrete or continuous random variable X will be of importance in the context
of expected utility and is defined as follows.
Definition 1.1.2 (Expectancy) Let X be a discrete random variable with
frequency function p(x), then
(1.1)
E(X)= Lap(x.)
is the expectancy provided that Lxp(x:) « o. If X is a continuous vari¬
able with density f(x), then
E(X)
(1.2)
xf(x) da
Lo"
provided that ff(x) da «00.
The definition explicitly states that the expectancy is undefined if the sum
or integral does not converge.
Probabilistic mechanisms and their distributional assumptions rather than
the probability measure itself are usually at the center of interest when study
ing choice behavior. But the qualitative aspects of probability have also been
employed to model choice behavior (e.g., Luce & Suppes, 1965; Narens, 1991).
A few traditional probabilistic choice models are discussed in Section 1.3 but
it is already pointed out here that properties of these models are derived
from limit theorems in which underlying mechanisms are studied in the infi¬
nite case; a powerful assumption which demands reasonably large samples of
individual choice behavior levelling out any systematic changes.
Having specified rationality and probability as the main concepts for de¬
terministic and probabilistic investigations of choice behavior a brief survey
of deterministic and probabilistic decision models is provided in the next
sections.
6In the more unusual case of a finitely additive probability space, A is reduced to an algebra of subsets
of Q.
1.2. DETERMINISTIC MODELS
1.2 Deterministic Models
The following order structures have been suggested to model preference as
well as indifference in binary choice without using probabilities. They are
defined by a set of axioms or conditions which are imposed on a collection of
binary choices. They all require or imply transitivity or negative transitivity
(see Definition A.2.1) but some of them are too weak for a representation into
the numbers, thus not even an ordinal utility function is available. The defini¬
tions combine preference and transitive indifference if the ordered structures
are reflexive but indifference is intransitive or excluded if they are irreflex¬
ive. In the following the reflexive relation £ always denotes preference and
indifference and the irreflexive relation denotes strict preference.
1.2.1 Order Structures
The weak order allows for a unique ordinal representation of alternatives into
the real numbers R if the set S of alternatives is finite. In the infinite case
the underlying set S has to be orderdense.
Definition 1.2.1 (Weak Order) Let S be a nonempty set and 2 a binary
relation on S, with 2C S X S. £ is called a weak order on S, if and only if
for all a, b,cE S holds:
(transitive)
(i) if a bandbc, then a c
(ii) aborbra
(connected)
The axiomatization of the weak order has been generalized in two ways to
accommodate infinite and multiattribute sets of alternatives as described
in Section 1.4.1. Only the simple case of a weak order on a finite set of
singleattribute alternatives is considered here. The axioms of the weak order
guarantee the existence of an ordinal utility function u : S — R so that the
representation
abeu(a) 2u(b)
(1.3)
exists where u is unique up to monotone increasing transformations, thus
defining an ordinal scale.
The corresponding representation theorem is due to Cantor (1895) and
has been extended to infinite sets by Milgram (1939), Birkhoff (1967), and
Krantz, Luce, Suppes and Tversky (1971). As mentioned before, empirical
results suggest that the indifference relation is not necessarily transitive, and
therefore not necessarily an equivalence relation.
10
CHAPTER 1. INTRODUCTION
Definition 1.2.2 (Semiorder) Let S be a nonempty set and » a binary
relation on S, with »C S X S. » is called a semiorder on S, if and only if
for all a,b, c,d E S holds:
(i) not a a
(irreflexive)
(ii) if ab and cd, then a d orc»b
(iii) if ab andbc, then a dord»c
The semiorder allows for intransitive indifference whilst the preference rela¬
tion remains transitive. It has a representation into the real numbers
axbeu(a)u(b) +6
(1.4)
where u is unique up to monotone transformations and ò is a positive number.
It denotes a certain threshold or justnoticeable difference which must be
exceeded in order to establish a preference relation (Luce, 1956). Therefore,
the threshold ô, which may be set to 1 by scaling, represents the limited
ability to discriminate between alternatives. If only axiom (i) and (ii) of
Definition 1.2.2 are required the structure is called an interval order with a
representation given by ordered intervals rather than real numbers (Fishburn,
1970).
(1.5)
abeu(a) u(b) +9(6)
This representation generalizes Eq. 1.4. Clearly, every interval order is a
semiorder.
Definition 1.2.3 (Strict Partial Order) Let S be a nonempty set and »
a binary relation on S, with »C S X S. » is called a strict partial order on
S, if and only if for all a, b,cE S holds:
(irreflexive)
(i) not a a
(transitive)
(ii) if abandbc, then a c
Every semiorder is a strict partial order because transitivity follows from (i)
by taking d = c in (iii) of Definition 1.2.2. The strict partial order as well as
the biorder are too weak for a representation into the real numbers but they
do appear in algebraic decomposition models.
Definition 1.2.4 (Biorder) Let S, and D be nonempty sets and » a bi¬
nary relation with »C S X D. » is called a biorder, if and only if for all
a,bE S, and for all d,e E D
11
1.2. DETERMINISTIC MODELS
(i) ax dandb e imply a e or b » d.
If the underlying sets S and D are the same then an irreflexive biorder on S
is connected and transitive and equivalent to an interval order. A reflexive
biorder on a set S is asymmetric and negatively transitive (Definition A.2.1)
and corresponds to a partial order (Doignon, Ducamp & Falmagne, 1984).
For a thorough introduction into measurement theory and the role of order in
measurement theory consult Krantz, Luce, et al. (1971) and Narens (1985)."
The weak order and the semiorder form the backbone of deterministic
choice models. Together with the strict partial order and the biorder, which
are both beyond a representation on ordinal scale, they share a common
property: They obey transitivity or, in the case of a reflexive biorder, negative
transitivity. The next section deals with classical deterministic models in
which alternatives are described by the probabilities of outcomes.
1.2.2 Expected Utility Models
Gabriel Cramer (1728) and Daniel Bernoulli (1738) stated hypotheses which
marked a change in the understanding of risky choices. Until then, it was
assumed that individual decisionmaking is best modeled by the evaluation
of alternative monetary gambles on the basis of their expected values, so
that a lottery offering the payoffs (xi,...,n) with respective probabilities
(P1.....Pa) would yield as much satisfaction as a sure payment equal to its
expected value Ex;p;. Such an approach was justified by appealing to the
law of large numbers, which states that if a gamble is indefinitely and inde
pendently repeated, its longrun average payoff will necessarily converge to
its expected value. Instead of applying plain expectancies the psychologi¬
cally significant concept of utility was introduced which may be expressed as
a squareroot or logarithmic function.
With this utility function it was tried to explain the so called 'St. Pe¬
tersburg Paradox’, first presented in 1728 by Nicholas Bernoulli, a cousin of
Daniel Bernoulli: Why does nearly everybody prefer the option of receiving
about 825 for sure instead of winning 82" where n is the number of tails'
until the first toss of head' when flipping a fair coin?? If the two options are
compared in terms of their expectancy values to reach an optimal decision
Further developments in measurement theory are summarized in Suppes, Krantz, Luce, and Tversky
(1989) and Luce, Krantz, Suppes, and Tversky (1990).
Note that the problem is stated here in dollars rather than in ducats as in the original problem. It has
been shown that different currencies can have an effect on the utility function (Roskam, 1987).
CHAPTER 1. INTRODUCTION
12
the following problem arises. According to Definition 1.1.2, the expectancy
of the gamble is simply not defined when the sum or integral is infinite.
Since the probability that k tails' followed by one head' is 2+1), which is
P(X = 2*) = 2, it follows that E(X) =
LRp(n)  20224/41 = 00.
Consequently and contrary to the original claim, there is no paradox to be
solved for the infinite or unlimited gamble. However, if the number of throws
is finite, people still tend to choose in favor of a lower expectancy value based
on certainty rather than a higher expectancy value based on risky or uncer
tain outcomes. Bernoulli suggested a logarithmic utility function u for the
subjective value of money which gives reasonable small expected utilities in
this gamble.
There are several arguments which cast doubt on the solution of this prob¬
lem in terms of logarithmic utility." For instance, Savage (1954) suggested a
modified version of this gamble where the first toss of head' leads to a win
of 822".
Even if a logarithmic utility function is assumed the expected utility
quickly becomes very large but most people are still willing to sell the option
for a considerably lower but certain amount of money.
More than two centuries passed by until von Neumann and Morgenstern
(1944) formally axiomatized the idea of expected utility in the second an
third edition of their classic book Theory of Games and Economic Behavior
(1947; 1953). They established a system of axioms which is based on a
binary preference relation on a set of alternatives which includes probability
mixtures. A probability mixture denoted by (x,p, y) can be understood as
a gamble where outcome x occurs with probability p and outcome y with
probability 1  p.
Definition 1.2.5 (Expected Utility) Let be a relation on a set of out¬
comes S which includes all probability mixtures (x,p,y). For all æ, y,z e S
and p,q %0,1
(i) ù is a weak order on S, strict preference, and  indifference.
(ii) (x,p.y), 4, 9 (x,pq,9).
(iii) Ifay, then (x,p,2) (y,p, 2).
(iv) Ifxy, then x (x,p,y) y.
For example, it can be questioned: What is the probability of winning more than about 825 in this
m
gamble? This turns out to be P(X » 2") = 1 P(X £ 2m) = 1. For k = 5 the probability
P(X » 25) « 0.03 is already quite low; see Vlek and Wagenaar (1979), Lopes (1981), and Treisman (1983)
for different heuristics.
1.2. DETERMINISTIC MODELS
13
(v) Ifxyz, then there exists a probability p such that y (x,p, 2).
This structure is knoun as the expected utility model.
The axioms permit a representation of the alternatives into the reals. The
use of a utility function implies a stronger notion of rationality than ordinal
preference because probability mixtures are represented by their expectancy
and choice behavior is therefore required to maximize utility.
(1.6)
a2yu(x) 2u(y)
u(x,p, y) = pu(x) + (1  p)u(y)
Although this axiomatization of expected utility is very important because it
laid down the foundation of game theory and provided a framework to test
conditions underlying expected utility, its empirical limitations for modeling
choice behavior were soon discovered (for a review on expected utility theory
and its empirical violations see Camerer, 1989).
Modified conditions for the representation of linear expected utility have
been suggested. Jensen (1967), for example, formulated the following set of
axioms. For all x, y,2 e S and for all e (0,1)
(1)
Order: » is asymmetric; and are transitive
(ii) Independence: If x » y and 0 « ) « 1 then Ax + (1  A)2 »
)y + (1  ))2)
(iii) Continuity: Ifay zthen ax+(1a)2 y and y Br+(18)2
for some a,B e (0,1)
Fishburn (1982) explored a system of axioms which yields a representation
on ratio scale by replacing the assumption of independence and transitivity
by a symmetry condition.
The expected utility model uses utility as a source of subjectivity but
entertains an undesirable numerical representation of probability in form of
probability mixtures, thereby representing numerical objects into numbers.
The first theory with a subjective concept of probability was sketched by
Ramsey (1931) in his treatise on the philosophy of beliefs. Savage (1954)
proposed a system of axioms which employs subjective probability and util¬
ity. In his work on the foundations of statistical inference Savage stated
conditions on preference relations which are sufficient for the derivation of
both a subjective probability measure and a utility function.° By means of
10Scott (1964) formulated necessary and sufficient conditions for the finite case.
14
CHAPTER 1. INTRODUCTION
several axioms, Savage proved the existence of a subjective probability func¬
tion s, which obeys the axioms of a probability measure (Definition 1.1.1),
and a utility function u on interval scale such that
2yu(x) u(y)
(1.7)
u(x, A,y) = s(A)u(x) + [1 s(A))u(y)
where (x, A, y) denotes the gamble where outcome x is obtained if event A
occurs and outcome y, otherwise.
Generalized versions of utility and subjective probability are frequently
used in economics and psychology, but systematic violations of some of the
axioms are well documented (Schoemaker, 1982; Lopes, 1990; Birnbaum,
1992). For example Ellsberg (1961) created a problem" which can be de¬
scribed as follows: Imagine an urn that contains 90 balls. Thirty of the
balls are red; the remaining 60 are black and yellow in unknown proportion.
One ball is to be drawn at random from the urn. Consider the two choice
situations in A and B and their payoffs in Table 1.1.
Ellsberg Paradox'
Table 1.1: The
30
Number of Balls
60
Yellow
Red
Black
Situation A
30
50
3100
1. Bet on red
80
80
8100
2. Bet on black
Situation B
Red
Yellow
Black
50
3. Bet on red or yellow
5100
5100
8100
8100
4. Bet on black or yellow
80
If a person bets on red balls in Situation A he or she will win §100 if a
red ball is drawn and nothing if a black or yellow ball is drawn. If the person
bets on black or yellow balls in Situation B he or she will win 8100 if a black
or yellow ball is drawn and nothing if a red ball is drawn.
In this problem, the extended surething principle which immediately fol¬
lows from the independence axiom as well as the axioms of subjective ex¬
pected utility is violated. It implies that a decision maker who maximizes
utility must choose either Option 1 and 3 or Options 2 and 4 in the two situ
11The earlier Allais Paradox' (Allais, 1953) is similar but has the disadvantage that it incorporates a
probability of 1 and a payoff of 0 as well as large sums of money. The problem was reformulated and tested
by Slovic and Tversky (1974) but still includes extreme values which make the interpretation of the empirical
result less convincing.
1.2. DETERMINISTIC MODELS
15
ations. The extended surething principle which is also related to the domi¬
nance principle asserts that if two alternatives have a common outcome under
a particular state of nature, then the ordering of the alternatives should be
independent of the value of that common outcome. This is illustrated by the
fact that the options of the Ellsberg Paradox' become equivalent if the last
column in Table 1.1 is ignored. However, most people select Option 1 and 4,
thus violating the principle. Presumably, they prefer to bet on payoffs whose
probabilities are known precisely rather than on payoffs with ambiguous or
uncertain probabilities (Grether & Plott, 1979). This phenomenon is impor¬
tant because it does not only violate expected utility but also generalized
versions such as Fishburn's nontransitive expected utility, and nonexpected
utility models which are discussed next.
1.2.3 NonExpected Utility Models
The systematic violations of expected utility led to generalizations of this
model introducing nonlinear functions for expected values in form of non¬
expected utility models. Many of these models are flexible enough to maintain
desirable properties of expected utility in a nonexpected utility framework
and have proven to be both theoretically and empirically useful. As for
expected utility, preferences can be empirically assessed and used to predict
individual choice behavior in other situations.
Luce and Narens (1985) investigated how general the representation of util¬
ity could be and still retain the property of interval scalability. These models
are centered around evidence that utility of risky or uncertain outcomes and
probability are not independent (Birnbaum & Stegner, 1979; Lopes, 1984;
1987; Birnbaum, Coffey, Mellers & Weiss, 1992).
The rankdependent utility model was developed for uncertain choice al¬
ternatives and for the restricted case of twooutcome gambles (Luce, 1992).
The rankdependent utility of a gamble (x,A,y), where outcome x occurs
when event A happens and outcome y occurs otherwise, depends on whether
ay or y and is defined as:
(1.8)
a2yu(x) u(y)
u(x, A,y) = s(A)u(x) + [1 s(A)u(y) + r(A)u(x)  u(y)
In comparison to Equation 1.7 the expression on the righthand side is ex¬
tended by the weighted difference of utilities where r is the rankdependent
weighting function which can have different forms (Weber, 1994; Luce &
16
CHAPTER 1. INTRODUCTION
von Winterfeldt, 1994). The rankdependent model was generalized by Luce
(1988; 1991) and by Luce and Fishburn (1991) to a rank and signdependent
linear utility model resulting in a representation for multiattribute alterna¬
tives.
Although these models permit modeling and analysis of preferences which
are more general than those allowed by expected utility, they have two limita¬
tions. First, each model requires a different set of conditions on its component
functions to model desirable properties of choice behavior so that expected
utility theorems linking properties of utility to such aspects of choice behavior
will typically not extend to the corresponding properties of the component
functions. Second, nonexpected utility models replace the independence ax¬
iom by some other more general restriction on preferences, possibly subject
to similar types of systematic empirical violations already observed.
An alternative approach to the study of nonexpected utility considers
nonlinear functions in general and uses calculus to extend results from ex¬
pected utility in the same manner as it is used to extend results involving
linear functions (Machina, 1983). More specifically, linear approximations in
form of the first order Taylor expansion of a smooth differentiable preference
function are investigated. In the case of a preference function over cumula¬
tive distribution functions, this expansion includes the classical multivariate
derivative of the function plus an remainder in terms of the standard L'
norm.
1.2.4 Miscellaneous Models
The failures of expected utility as revealed by the Ellsberg Paradox' and
related phenomena inspired researchers to find alternative explanations of
choice behavior which are not directly linked to expected utility theory. Ed¬
wards (1953, 1954a, 1954b) was the first to study probability and variance
preferences. Coombs (1969; 1975) proposed portfolio theory which highlights
risk preference as a determinant of choices among gambles. One important
characteristic of portfolio theory is the single peakedness of the curves of pref¬
erence indifference which has been investigated theoretically and empirically
by Coombs and Avrunin (1977a, 1977b) and Aschenbrenner (1981; 1984).
Kahneman and Tversky (1979) developed an deterministic framework,
called prospect theory, to overcome some of the failures of expected util¬
ity theory. Three phenomena, the certainty effect (overestimating certain
outcomes), the reflection effect (risk aversion for positive expectancies vs risk
1.3. PROBABILISTIC MODELS
17
proneness for negative expectancies), and the isolation effect (simplifying
gambles) were identified and incorporated in their model. In a later article
Tversky and Kahneman (1992) extended the model to cumulative prospect
theory which corresponds to a rank and signdependent expected utility
model for multiattribute alternatives.
Bell (1982), Loomes and Sugden, (1982) and Sage and White (1983) de¬
veloped sophisticated alternative theories of decisionmaking under risk and
uncertainty. In their models, the preference a » b means that choosing a
and rejecting b is preferable to choosing b and rejecting a. Since every choice
depends on the regret involved in the particular pair of options being consid¬
ered, transitivity of choice is not ensured in models of regret theory.
In many cases these models originated from a list of side conditions which
were derived from experimental results and have been extended to a full
theory at a later stage. Some of the models received a mathematical ax¬
iomatization which revealed their multiattribute nature. For example, an
axiomatization was developed which casts portfolio theory in the framework
of conjoint measurement with weak orderings in respect to expected value,
risk, and preference (van Santen, 1978; Suck & Getta, 1994).
1.3 Probabilistic Models
Traditional probabilistic choice models are outlined. Some of their properties
have an deterministic counterpart. To admit preference reversals and intran¬
sitive preferences in individual decision behavior axioms of rational choice
are replaced by corresponding probabilistic assumptions. So transitivity is
replaced by weak, strong and other probabilistic versions of transitivity. Al¬
though the distinction between constant and random utility models is not
entirely adequate, probabilistic models are introduced under these categories
to keep this account brief and simple.
1.3.1 Constant Utility Models
Constant utility models are models of riskless choice. They are closely related
to probabilistic versions of transitivity. Their interrelationship is summarized
at the end of this section. The description here is due to Roberts (1979,
Chapter 6) and Falmagne (1985, Chapter 5) but can be found elsewhere
(e.g., Luce & Suppes, 1965).
In the following a system (S,p) is called a system of pair comparison
18
CHAPTER I. INTRODUCTION
probabilities if for all distinct alternatives a % b in a set S holds: Pa+P = 1
with
Pab  P(a » b)
(1.9)
the empirical probability that alternative a is preferred to alternative b. For
convenience assume that the probability of choosing between two identical
alternatives equals pa.a
= 3.
First, the representation for the weak utility model is introduced. If for
all a,beS
(1.10)
PaPa(a)u(b)
then this model is equivalent to a probabilistic version of transitivity, called
weak (stochastic) transitivity2 which is defined as follows:
(1.11)
Pab 2 and pc 2  Pa,c 2
Two stronger probabilistic versions of transitivity, moderate and strong (stoch
astic) transitivity, are defined below:
(1.12)
Pa,b 2 and p,c 2 Pa,c 2 min (Pab,P,c)
(1.13)
Pa 2 and p 2 Pa2 max (Pa,,P.c)
A collection of different probabilistic versions of transitivity can be found in
Fishburn (1979). The strong utility model fulfills
(1.14)
Pap Pedu(a)  u(b) u(c)  u(d)
and implies the strong version of probabilistic transitivity.
A general variant of the strong utility model is the Fechnerian utility
model. A forced choice pair comparison system (S,p) satisfies this model if
there is a realvalued function u on S and a (strictly) monotone increasing
function % : R— R so that for all a,b E S
(1.15)
Pa,b  % u(a)  u(b)
This model also appears in psychophysics where it is used to relate a phys¬
—
ical intensity to a psychological sensation. The model implies that pa»
Pod = u(a)  u(b) = u(c)
u(d). Thus, pairs of stimuli which are equally
often confused are equally far apart. This property was a critical assumption
in Fechner's derivation of the psychophysical function as a logarithmic one
(Fechner, 1860).
12 Probabilistic forms of transitivity are usually called stochastic transitivity but as Luce and Suppes (1965)
have pointed out the term stochastic should only refer to some process over time.
1.3. PROBABILISTIC MODELS
19
Luce's choice axiom (1959), reformulated by Luce and Suppes (1965),
states that choices from any subset are independent of what else may have
been available.
Definition 1.3.1 (Luce's Choice Axiom) A set of preference probabili¬
ties defined for all subsets of a finite set S satisfy Luce's choice axiom provided
that for all x, Y and X such that x eY CXCS whenever the conditional
probability exists,
(1) py(x) = px(x Y).
More specifically, the axiom requires that the probability of choosing a from
Y when only Y was presented, written as py(x), equals the probability of
those occasions where x is chosen from Y even though X may have been
presented, written as px(x Y). For binary choice the axiom leads to a simple
requirement for the utility function u also known as the BradleyTerryLuce
model (BTL model).
u(a)
(1.16)
Pa,b
u(a) + u(5)
However, for all a,b e Sthe additional assumption that pa %0, 1 has to be
fulfilled. This model was also used by Zermelo (1929) to measure the playing
power of a chess player. Adams and Messick (1957) pointed out that if (S,p)
meets this assumption, the BTL model implies the Fechnerian model as it is
shown here. Define u' = In u for all u(a) » 0, a e S. Let ø be the logistic
distribution function o(A) =
Te5. Then
u(a)
Pa,b
u(a) + u(6)
1+ub)/u(a)
1+ exp( u'(a)  u'(5))
= %u (a)  u'(6))
so u' and % satisfy the Fechnerian utility scale (see Equation 1.15). Conse¬
quently, if the set S is finite, the BTL model implies the Fechnerian utility
model. The Fechnerian implies the strong utility model for
a(d))
Pap ped u(a) u(b) »u(c)
u(a)  u(b)» u(c)  u(d).
20
CHAPTER 1. INTRODUCTION
Moreover, when S is finite, they are equivalent (cf., Roberts, 1979, Chap. 6).
The strong utility model implies the three probabilistic transitivity condi¬
tions, where weak transitivity is equivalent to the weak utility model. Note
that if S is infinite neither of the probabilistic versions of transitivity implies
the weak utility model. Hence, weak (stochastic) transitivity is as fundamen
tal to constant utility models as transitivity to deterministic models.
The most thoroughly examined ideas are the three probabilistic versions
of transitivity. Experimental findings are not too conclusive because these as¬
sumptions are difficult to test (e.g., Davidson & Marschak, 1959). Neverthe¬
less, Tversky (1969) demonstrated systematic violations of weak transitivity
under a carefully constructed situation where lotteries varied in probability
and payoff. Although evidence against transitivity is less convincing in the
context of money, other domains with multiattribute outcomes appear to be
good candidates for intransitive choices (May, 1954).
1.3.2 Random Utility Models
In a related approach, one can assign a random variable Ua to each aE S so
that
(1.17)
Pa,b = P (Ua 2 Ub)
This model is known as the random utility model (Block & Marschak, 1960).
The interpretation is that the utilities are no longer assumed to stay fixed,
but are determined by some probabilistic process. Indeed, the constant utility
models may be considered as special cases of the random utility model with
random variables remaining constant.
The following models originated from applications within psychophysics
but have also been considered as utility models. They can be looked at as ran¬
dom utility models or as special versions of the Fechnerian model introduced
in the previous section.
In variation of the Fechnerian model, if the function o is required to be a
cumulative distribution function, then % u(a)  u(b) is the probability that
u(a) is larger than u(b). This idea goes back to Thurstone (1927a, 1927b)
and is known as Thurstone's Law of Comparative Judgment'. Scale values
u(a) are sought so that for all a,be S
u(a)u(b)
(1.18)
N(x) da
Pa,b
0
where N(x) denotes the standard normal distribution. Obviously, this is a
special case of the random utility model introduced above. The cases most
1.3. PROBABILISTIC MODELS
21
frequently used arise when the random variables are pairwise independent
(Thurstone's Case III) and have equal variances (Case V).
As mentioned earlier, another variation of the Fechnerian model uses ø as
the logistic distribution o(x) 
L— and seeks scale values u(a) so that
(1.19)
Pa,b
1 + elu(a)u(6)
The logistic model is due to Guilford (1954) and Luce (1959). If the random
variables are double exponential distributions then their difference is the lo¬
gistic distribution and therefore equivalent to Luce’s Choice Axiom and the
BTL model (Yellott, 1977).
Models of random utility with different distributional assumptions have
been used and tested against each other extensively. Underlying assump
tions or axioms of random utility, however, have been scarcely tested, simply
because no set of necessary and sufficient conditions for the representation
of binary choice probabilities have been found. Despite some progress in
this area this problem remains a major theoretical and empirical obstacle
(Koppen, 1995).
It seems straightforward to combine expected utility with probabilistic
models leading to further generalizations. However, most of these models are
not very different in their predictions from subjective expected utility models
and similar violations have been reported (Becker, DeGroot, & Marschak,
1963a, 1963b, 1963c). The development of random utility models on the basis
of nonexpected utility is in some respect similar to rank and signdependent
utility models but addresses more abstract issues within measure theory and
probability theory (Gilboa, 1987; Schmeidler, 1989; Wakker, 1989). This
generalization of expected utility can accommodate preference reversals as
observed in the Ellsberg Paradox because individuals' probabilistic beliefs
are represented by nonadditive probability measures which do not satisfy
Definition 1.1.1. However, some axioms of these models are impossible to
test. Therefore, they have limited psychological interpretation and relevance.
As with nonexpected utility models the success of these models depends on
the extent to which they can usefully address issues in the theory of individual
and group choice under uncertainty.
22
CHAPTER 1. INTRODUCTION
1.4 Multiattribute Models
Another generalization is of interest here. It originates from economics and
the representation of choice alternatives as commodity bundles. A multi¬
attribute alternative a can be expressed as a vector'3 of attributes a =
(a1,a2,..., a») where the attributes belong to sets in a cartesian product
Sjx ... x S.. Such product structures are mostly represented by additive
conjoint measurement.
Most of the deterministic and probabilistic models in the previous sec¬
tions can be generalized to multiattribute alternatives. However, a number
of additional assumptions are needed to establish a representation into the
numbers (Krantz, Luce, Suppes, & Tversky 1971).
1.4.1 Deterministic Multiattribute Models
A simple conjoint structure of multiple attributes is given by this represen¬
tation in conjunctive form
(1.20)
a Ebe(a) 2 u(b) A...Aun(an) 2 un (bn)
where a preference holds if and only if the utilities of all attributes in one
alternative exceed the utilities of all attributes in another alternative. In
other words reversed utilities on a single dimension would prevent a preference
relation from being established. The righthand side of Equation 1.20 is also
known as the dominance principle. The representation in disjunctive form is
defined dually
(1.21)
a Ebeu(a) 2u(b) V... Vun(an) 2 un (bn)
where a difference on a single dimension suffices to create a preference.
Coombs (1964) and Dawes (1964) defined a conjunctive and disjunctive deci¬
sion rule in terms of thresholds. The conjunctive rule states that the attribute
values for the chosen alternative must exceed a threshold specific on each at
tribute and that for at least one attribute, the other alternative falls below
this threshold. The disjunctive rule states that the evaluation on at least one
attribute for the chosen alternative must exceed a threshold specific to that
attribute and that all aspects of the other alternative fall below a critical
value specific to each attribute.
13To keep the notation simple no distinction is made between singleattribute and multiattribute alterna¬
tives which are sometimes written as vectors in boldfaced letters.
1.4. MULTIATTRIBUTE MODELS
23
The weak order (Definition 1.2.1) has to fulfill the dominance and a conti¬
nuity condition to yield the following representation of multiattribute alter
natives with a single utility function.
(1.22)
a Ebeu(ai,...,an) 2u(b...., bn)
In contrast to conjoint structures the multiattribute alternatives are repre¬
sented by a single utility function.
The most prominent multiattribute representation is additive conjoint
measurement where different attributes are being measured conjointly (De¬
breu, 1959; Luce & Tukey, 1964).
(1.23)
axbeLu(a)2Lu.(b.)
i=1
1
Several axioms are needed to formulate a representation for the finite as well
as the infinite case (Luce & Tukey, 1964; Krantz, Luce, et al. 1971). Polyno¬
mial conjoint measurement is an example for a nonadditive representation
and has also been studied by Krantz, Luce, et al. (1971). In general, multiat¬
tribute utility models decompose the utility function u into singleattribute
u;, and then aggregate according to the appropriate model.
It has been argued that gambles or lotteries as introduced in Section 1.2.2
are in fact multiattribute alternatives which can be characterized by their
probability of winning, amount to win, probability of losing, and amount
to lose (Slovic & Lichtenstein, 1968). In the same line of argument rank
and signdependent utility models have been extended to multiple outcomes
and can accommodate empirical findings such as dependence between out
comes and probability as well as asymmetrical loss and gain functions (Luce
& Fishburn, 1991; Tversky & Kahneman, 1992). The axioms of utility and
probability are generalized while preserving some desirable properties of ex¬
pected utility, namely the dominance principle.
Additional examples of deterministic multiattribute models are the ma¬
jority and lexicographic rule. The majority rule
n
(1.24)
axbeLsu(a)
v su;(b))
2=
1
where v is defined as
1:u, (a.) » u. (b;)
y =
0: otherwise
CHAPTER 1. INTRODUCTION
24
can produce intransitive choice behavior as demonstrated by May (1954).
The rule counts the number of dominating attributes for each multiattribute
alternative in order to establish a preference relation between the two alter
natives. If, for example, certain attributes are missing then intransitivity can
occur.
The lexicographic rule selects the alternative which dominates on the most
important dimension. If alternatives have equal utility on this attribute the
next important is considered and so forth. Hence,
abfor some (1 jSn)u(a;)» u(b;)
(1.25)
and for all (i « j) u(a;) = u(b;)
As long as the importance of the dimensions remains unchanged and u is
monotonically increasing the lexicographic rule is transitive. Again, if certain
attributes are indifferent or missing then intransitive choice can result (Luce,
1956).
Since the information processing approach was launched in cognitive psy
chology it has been at the center of interest in psychological decision research.
Mainly initiated by Simon (1959) and his notion of bounded rationality the
cognitive limitations of human information processing were investigated in
decisionmaking. A number of simple decision rules, heuristics and strategies
which are based on multiattribute alternatives emerged from this approach
Another example that has also been
(e.g., Gigerenzer & Goldstein, 1996).
applied in statistical decision theory is the maximin and the maximax rule
(Dahlstrand & Montgomery, 1984). According to the first rule, the chosen
alternative has the highest lowest attribute value. The second rule implies
that the chosen alternative has the highest highest evaluation.
It is not possible to review all the models here as it is difficult to eval¬
uate and categorize the different algorithms and heuristics, especially when
they are lacking a precise mathematical description. For various attempts to
provide a framework for different choice heuristics and strategies consult for
example Payne (1976; 1982), Beach and Mitchell (1978), Billings and Marcus
(1983), and Ford, Schmitt, Schlechtman, Hults, and Doherty (1989).
1.4.2 Probabilistic Multiattribute Models
Restle (1961) proposed a specific BTL model where each alternative is un¬
derstood as a set A of aspects A = a1,a2,..., an. He assumed that one ex¬
amines only certain relevant aspects of each stimulus before making a choice.
1.4. MULTIATTRIBUTE MODELS
25
He also assumed that any characteristic contained by both alternatives be¬
comes irrelevant to the decision process, and that the same is true for aspects
contained in neither alternative. In a comparison of alternative A and B, the
utility for alternative A depends on the utilities of all the aspects contained
in A minus those contained in the intersection, written as AO B. Hence,
the utility of A paired with B could be represented by u(A) u(AD B) and
similarly for B by u(B) u(AOB). In terms of the BTL model the following
representation results
u(A)u(AOB)
(1.26)
PA,B
u(A) + u(B) 2u(AOB)
If the sets A and B are disjoint, then Restle's model and the BTL model
make equivalent predictions about choice probabilities. An example of the
superiority of Restle's choice model over the BTL model has been demon¬
strated by Edgell, Geisler, and Zinnes (1973), who based their analysis on
data collected by Rumelhart and Greeno (1971).
Tversky's elimination by aspects rule (Tversky 1972a, 1972b) views each
alternative as a collection of measurable attributes called aspects and de¬
scribes choice as a covert process of successive elimination. At each stage
in the process, an attribute of the alternatives is selected with a probability
proportional to its value. Any alternative that does not include the chosen
attribute is eliminated and the process continues until a single alternative
remains. In the case of binary choice, elimination by aspects coincides with
Restle’s model. Moreover, elimination by aspects can be considered as a
probabilistic version of the lexicographic rule.
In addition to the empirical violations in the context of expected utility,
there are a number of empirical phenomena which are known to occur among
multiattribute alternatives and which have a potential to create inconsistent
choice. For instance, choice behavior is affected if an alternative is added to
the set of multiattribute alternatives. It was recognized that the frequency
of choosing alternatives can be either reduced or increased (Kahneman &
Tversky, 1979; Huber, Payne, & Puto, 1982; Wedell, 1991) depending on cer¬
tain attributes alternatives share with the added alternative. The systematic
selection of a set of multiattribute alternatives due to similarity (Rumelhart
& Greeno, 1971) or difference in attractiveness can also have a considerable
effect on individual decision behavior (Albert, Aschenbrenner & Schmalhofer,
1989).
As the number of empirically verified effects increase, the requirements
26
CHAPTER 1. INTRODUCTION
for decision models become more and more demanding. It seems reason¬
able to assume that most alternatives are multiattribute and that decision
making processes vary even when the experimental context of choices does
not. So far choice heuristics are the only candidates which try to explain
inconsistent preferences in terms of adaptive individual choice behavior (e.g.,
Payne, 1976; Payne, Bettman, & Johnson, 1988; 1990; Schmalhofer, Albert,
Aschenbrenner & Gertzen, 1986). Unfortunately, these models and in fact
all multiattribute models, require a lot of domainspecific information before
they can be applied.
In the next chapter a nonprobabilistic approach for modeling adaptive
choice is developed which has been neglected in decision research. In this ap¬
proach it is tried to assess and explain inconsistent choices in a decomposition
model without imposing strong assumptions on domainspecific attributes or
underlying information processing. Inconsistencies are viewed as signals of
adapting individual choice behavior. This view suggests the decomposition
of a preference structure into substructures which is based on inconsistencies.
In the next section models and techniques are discussed which can be seen
as predecessors of an algebraic decomposition approach.
1.5 Algebraic Decomposition
Some nonprobabilistic multidimensional scaling techniques can be related
to the decomposition of a preference structure into substructures. Usually
substructures refer to attributes or dimensions of alternatives underlying the
decision process. In general, substructures meet stronger assumptions than
the preference structure itself. Therefore, these models are related to deter¬
ministic multiattribute models.
In a theoretical paper Doignon, Ducamp, and Falmagne (1984) showed
that an arbitrary relation R, defined on the cartesian product of two finite
sets RC SxD, can be expressed by the intersection of R biorders B;.14 Kop¬
pen (1987) investigated reductions and algorithms that compute the minimal
number of biorders for data matrices of moderate size using tools of graph
theory. He used the fact that the minimal number of biorders or bidimen¬
sion of a relation is identical with the chromatic number of an associated
hypergraph. This idea also applies to square preference matrices resulting in
specific partial orders instead of biorders. In a more general approach Chubb
14 R denotes the cardinal number of the complement of relation R.
1.5. ALGEBRAIC DECOMPOSITION
27
(1986) reported an algebraic technique that can reduce preference matrices
to a 0core of relations to find the minimal number of dimensions.
A more explicit idea for the decomposition of a pair comparison was pro¬
posed by Barthélemy (1990). He suggested that substructures of the decom¬
position should refer to different points of view of the decision maker. The
decomposition follows from a lexicographic decision rule that includes in its
generalized form both the conjunctive and disjunctive rule as well as decision
rules with thresholds. The main goal is a minimization of the number of sub¬
structures. Thereby, the lexicographic sum of the minimal number of partial
orders can explain all kinds of preferences, including intransitive choices.
In order to give an example of this decomposition the preference relation
R, defined on the cartesian product of a set of four elements S = a,b,c, d),
is mapped onto a (0,1)matrix A =a of size 4 x 4 by the indicator function
1:SXS (0,1).
1:(a,b)ER
((a;;) 
1
0: (a,b) #R
Let R = ((a,b), (a,c), (a,d), (b,c), (c,d), (d,b)) be a set of preferences on S
then the (0,1)matrix is written as
abcd
0111
a
0 0 1 0
A = b
0 0 0 1
c
d0100
Note that the pairs ((b,c), (c,d), (d,b)) C R describe an intransitive triple.
Due to the generalized lexicographic rule, matrix A may be decomposed into
two transitive partial orders Oj, and O2. At this point it should be empha¬
sized that Barthélemy did not suggest an explicit decomposition technique
which can determine the underlying partial orders. He derived upper bounds
for the minimal number of partial orders. Nevertheless, the lexicographic
decomposition is illustrated in the example below to give an idea of how such
an algebraic decomposition might work.
0111
0 1 111
0 1 1 11
0 0 1 0
0 0 1 0
000 0
—
0
0 0 0
0 0 0 1
0 0 0 1
0 10 0
00 0 0
o 1 0 0
—
A
0
02
01
15The indices of a;; run over i, j — (1,..., 4); l’s denote preference of alternative in row i over alternative
in column j, O's denote no preference.
28
CHAPTER 1. INTRODUCTION
The operation ' denotes the lexicographic sum which for the case of a
combination of only two partial orders is defined as
(a,b)eO2 (a,b)eOor ((a,b) eO2 and (b,a) £ 0i) (1.27)
The two partial orders Oj and O2 can be interpreted as different points of
view held by the same person."° The lexicographic decomposition can ex¬
plain intransitive choices in A. According to the lexicographic rule, it is not
possible to compensate the preference on a more important dimension by less
important ones.
In a theoretical note Lages (1989) suggested a simple decomposition which
is based on the embedding of a partial order into the cartesian product of two
linear orders. Suck (1994) established solvability conditions for the general
case of an embedding into product structures. In another approach Doignon
(1995) and Doignon and Falmagne (1997) investigated the family of all pos¬
sible semiorders on a given set. If a family of semiorders or similar order
relations is well graded it can be described by a permutahydron where the
addition or removal of a single relation generates semiorders. These com¬
binatorial results can be used to model a stochastic learning process which
operates on this lattice structure.
The following critical remarks apply to the decompositions mentioned so
far. None of the decompositions has been tested empirically as a model
of choice behavior. The reasons for the lack of empirical data are quickly
explained. These models are new to decision research and their application to
choice behavior was not always considered. Moreover, no explicit techniques
for assessing substructures have been proposed. This problem arises because
algorithms for decompositions are inefficient and solutions are not necessarily
unique. Depending on the observed preference structure, there can be several
sets of substructures, each of which is computationally hard to determine
but explains a given preference structure. As a consequence, if techniques
for the previously described decompositions were available, they would be
difficult to test. This is true because such techniques have at least one trivial
but adequate solution and observe preferences as the only variable. So far
no other variables have been considered or incorporated in a decomposition
model.
For the reason that decomposition models are difficult to falsify they
1e For example, when choosing between motorbikes, O, may reflect preference due to price, and O» pref
erence due to fuel consumption with the two points of view decreasing in importance, respectively.
1.6. SUMMARY
29
should be based on weak and plausible assumptions. A variable which is
easy to observe is the sequence of choicetrials. In decision research the se¬
quence of choices has been treated as a nuisance variable of the experimental
procedure or has been ignored altogether. It is proposed here that the se¬
quence of choicetrials can be used as a variable of choice behavior which
offers weak and plausible assumptions for an algebraic decomposition.
In the next chapter an explicit decomposition model, the ear decompo¬
sition by sequence, is proposed which is efficient and solves the uniqueness
problem by incorporating the sequence of inconsistent choices.
It follows from the survey and discussion of decision models in this chap¬
ter that the violation of asymmetry and transitivity has been crucial for
the development of deterministic and probabilistic decision models. What
is missing is a general approach in which individual choice behavior can be
characterized in terms of inconsistencies and which provides a psychologically
plausible algebraic decomposition. This decomposition should have a mini¬
mally and uniquely determined solution. In Chapter 2, we will pursue this
goal by employing methods of graph theory and algebra. A graphtheoretical
definition of intransitivity is suggested which leads to the algebraic decom¬
position of preference matrices into irreducible components, first established
by Frobenius (1912). In contrast to other decompositions it naturally follows
from intransitive choice behavior. This leads on to a more specific decompo¬
sition, the ear decomposition of socalled irreducible components, which has
a unique solution if it is based on the sequence of intransitive choices. The
solution is a directed ear basis, a minimal collection of choices which covers all
intransitivities within an irreducible component. In Chapter 3 quantitative
measures of inconsistency are tested in order to determine if choice behavior
is adaptive so that the sequence of choicetrials affects inconsistency as pos¬
tulated by the ear decomposition. In six experiments the ear decomposition
by sequence is applied to three different domains of alternatives and various
quantitative measures of inconsistency are compared.
1.6 Summary
This chapter has been concerned with basic assumptions of rational choice
behavior. Asymmetry and transitivity were identified as fundamental as
sumptions for the ordinal representation of rational choice. Both assumptions
appear in almost all deterministic choice models such as order structures, sub¬
jective expected and expected utility and related models. The violation of
30
CHAPTER 1. INTRODUCTION
transitivity and related rational principles such as the extended surething
principle might favor a representation in form of probabilistic choice mod¬
els satisfying probabilistic versions of transitivity. But probabilistic models
such as constant utility and random utility require a lot more empirical data.
These models have been successful when random noise enters the decision pro¬
cess as in psychophysical decision tasks, but violations of weak probabilistic
versions of transitivity have been demonstrated in a number of preference
data. It was argued that all traditional single and multiattribute determin¬
istic and probabilistic models face the problem of explaining systematic or
adapting inconsistencies.
In Chapter 3 we will provide evidence that different arrangements of
choicetrials in a pair comparison can affect the consistency of choice be¬
havior; a finding that cannot be explained by classical deterministic or prob¬
abilistic models and makes a strong case for adaptive choice behavior and the
application of decomposition models as introduced in Chapter 2.
Chapter 2
Theory
A decomposition model is developed which assumes that inconsistent choice
originates from adaptive choice behavior. Consequently, inconsistent choices
do not occur randomly but in response to changing choice situations. Intran¬
sitivities are therefore no longer regarded as random errors; they are consid¬
ered as indicators of adapting choice behavior signaling a shift of strategy or a
change in the information process of the decision maker. In a slightly stronger
model it is assumed that subjects maintain their strategy when choosing in
a pair comparison until they are confronted with a critical choice pair that
suggests a different way to reach a decision. This is most likely due to an
attribute which was ignored or forgotten in preceding choices but becomes im¬
portant in current choices. The reasons for inconsistencies caused by altered
information processing such as selective information processing are believed
to be highly subjective and domainspecific eluding efforts to be described in
the framework of a normative model such as expected utility, prospect theory
or related models.
Consider, for example, a person who chooses between magazines in a pair
comparison. In the first few choices he or she might develop a strong prefer¬
ence for magazines with a large section on cooking recipes. When confronted
with a magazine which has a detailed TV preview he or she realizes that this
is another attribute of importance. Taking this attribute into account would
have reversed an earlier choice and leads now to inconsistencies in terms of
intransitive preferences.
The difficulty is to identify critical choices that caused a shift in perspective
without assessing domainspecific information by costly methods as in verbal
protocols (Ericsson & Simon, 1984), without introducing strong assumptions
as in multiattribute decision models, and without running into computational
problems as in techniques of nonprobabilistic multidimensional scaling and
31
32
CHAPTER 2. THEORY
related decompositions.
In the previous chapter we have encountered some decision rules which
can create intransitive choices. The majority rule and the lexicographic rule,
for example, are heuristics that can cause preference cycles. However, it can
be argued that such heuristics are not adaptive and should produce invariant
amounts of inconsistency unless selective or altered information processing is
assumed.
As outlined in Chapter 1 the assumption of transitivity is predominant
in both algebraic and probabilistic choice theories. Perhaps due to the fo¬
cus on transitivity, elaborate measures of inconsistency have not yet been
developed. The methods almost invariably use Kendall's consistency index (
and r (Kendall, 1970). Slater (1961) proposed a measure of consistency by
computing the minimal number of comparisons in which the choice should
be reversed to obtain a linear order. A linear order resulting from reversing a
suitable set of choices is called the nearest adjoining order and the number is
known as Slater's i. Again, determining such a minimal set leads to an opti
mization problem which is related to the minimal acyclic subgraph problem'
mentioned in Section 2.2.1 and 2.2.2. Bezembinder (1981) suggested a more
sophisticated measure of inconsistency. This measure is related to the size of
strong components, a characteristic of pair comparisons which is discussed in
the following sections.
In the following preferences from a pair comparison are represented by
sets, graphs, and matrices. First, intransitive choice in a pair comparison
is defined in terms of directed cycles in a tournament. A theorem which
goes back to Frobenius (1912) is applied and leads to a decomposition of
Related to the decomposition is
any tournament into strong components.
the partition of a polynomial expression which can be used as a quantitative
measure to characterize intransitive choice behavior in a tournament (Lages,
1995). Simple matrix operations on the adjacency matrices of tournaments
are discussed that were designed to identify critical arcs of intransitive choices.
As for the characterization of intransitive choice a fundamental optimization
problem is encountered.
Second, a more specific decomposition is outlined which is based on the
sequence of intransitive choices in a pair comparison. The ear decomposi¬
tion by sequence has the advantage that it determines a small and uniquely
defined subset of directed cycles that generate all intransitivities within a
feasible algebraic structure. Moreover, the underlying algorithm is known to
be efficient.
33
Finally, in a digression on the sequence of choices the completion by cuts
is applied to a family of intransitive choices. This technique does not result
in a unique set of critical choices but it offers a refined interpretation for
solutions of algebraic decompositions.
As mentioned before, theoretical ideas in this chapter are presented in
mathematical terms. They are summarized in an example at the end of each
section. Although Appendix A provides some of the mathematical back
ground, it is beyond the scope of this thesis to give a selfcontained account
of the mathematics involved. Standard textbooks on order theory, graph
theory and algebra are cited in Appendix A.
34
CHAPTER 2. THEORY
2.1 Directed Cycles and Tournaments
There are three equivalent ways to represent a collection of relations between
distinct objects: sets, graphs, and matrices. Each representation refers to a
different theoretical approach: Sets of ordered pairs are mainly used in order
theory, diagrams in graph theory, and matrices in algebra. The basic concepts
of each approach which are of interest here are provided in Appendix A2,
A3, and A4, respectively. In the following, diagrams are used to exemplify
structures, whereas sets and matrices are mainly used to achieve theoretical
results.
A definition of intransitive relations is given by directed cycles, abbreviated
to dicycles (see also Appendix A.3).
Definition 2.1.1 (Directed Cycle) Let D be a digraph. A closed directed
walk of length k is of the form
(o, ), (a1,a2),, (a1, a4)
where ap = ao are identical vertices. If ax, and ao are the only identical
vertices then the closed directed walk is a directed cycle of length k or k
dicycle.
A directed walk may also be denoted by
do — aj — — —1 — 4.
Note that preference reversals correspond to directed cycles of length 2 or
2dicycles and intransitive triples are equivalent to 3dicycles.
Two vertices a and b are called strongly connected provided there are di¬
rected walks from a to b and from b to a. A single vertex is regarded as
strongly connected to itself. Strong connectivity between vertices is reflexive,
symmetric, and transitive. Hence, strong connectivity defines an equivalence
relation on the vertices of D and yields a partition
VIUVU...UV
of the vertices V. The subdigraphs Di,D2,..., D. formed by taking the
vertices in an equivalence class and the arcs incident to them are called the
strong components of D. The digraph D is strong if it has exactly one strong
component.
Directed cycles' are also called circuits' in the context of networks. Directed cycles should not be
confused with undirected cycles that are called 'cycles' when dealing with undirected graphs.
2.1. DIRECTED CYCLES AND TOURNAMENTS
35
A tournament is a digraph where every pair of vertices is connected by
a single arc. Typically, such a digraph is the result of a pair comparison or
roundrobin tournament, in which each contestant battles every other con
testant and with wins or losses as the only outcomes. There are 2(2) possible
tournaments of order n.
Two tournaments of order 5 are depicted in Figure 2.1. Tournament A
differs from Tournament B by a single arc (1 — 2 and 2 — 1, respectively).
Tournament A is strong and has three 3dicycles (3 — 4 — 5 — 3; 1
2 — 5 — 1; 1 — 4 — 5 — 1), two 4dicycles (1 — 2 — 4 — 5 — 1;
1 —3 — 4 — 5 — 1), and one 5dicycle (1 — 2 — 3 — 4 — 5 — 1) whereas
Tournament B has two 3dicycles (1 — 4 — 5 — 1; 3 — 4 — 5 — 3), one
4dicycle (1 — 3 — 4 — 5 — 1) and no 5dicycle. This simple example
illustrates that a single arc quickly changes the number of dicycles.
B.
A.
Figure 2.1: Tournament A differs from B by a single arc.
According to Reid and Beineke (1978) very little is known about dicycles
of length greater than 4. However, Bermond and Thomassen (1981) provide a
survey on dicycles in digraphs. It is concluded that dicycles of length greater
than 3 do describe inconsistencies in a pair comparison and should be taken
into account if a measure of inconsistency is established.
The following lemma was obtained by Moon (1968) and characterizes tour
naments which are strongly connected abbreviated to strong tournaments. It
shows that for the special case of a strong tournament each vertex in a strong
component is part of some directed cycles of length greater than 3.
Lemma 2.1.2 Each vertex of a strong tournament of order n is contained
in a kdicycle, for (3 £ k £ n).
36
CHAPTER 2. THEORY
Proof. This is proven by induction over k. Let D be a tournament of
order n, and let ap be a vertex in V. First, ao must be on a 3dicycle, since
there must be a vertex 6 so that (ao,b) E D, and a vertex c so that (c,ap)
and (b,c) e D because D is strong. Now assume that ao lies on a kdicycle
C = (ao, a1), (ai, a2),...,(a1, a0) with k « n. We define the sets of vertices
X, Y and Z as X = (xEV (x,a.) E D), Y = (yeV (a,y) e D), and
Z = zeV 24C,X,Y). If Z +0, then since D is strong, X and Y cannot
be empty, and for some vertex be X and ce Y it must hold (b,c) E D.
In this case ao lies on a k + 1dicycle (ao, b), (b,c), (c, a2),..., (ax1, ao). On
the other hand, if there exists a vertex z £ Z, then there must be vertices
a; and a1 in C such that (a;,2) and (z,a+1) e D; then again ao lies on a
□
k + 1dicycle. The result then follows by induction.
However, not every arc in a strong tournament belongs to a dicycle of length
k with k e (1,2,...,n).
Let A be the adjacency matrix of a tournament, J the all l’s matrix, and
I the identity matrix. Then A is a (0,1)matrix satisfying the equation
A+AT = JI
(2.1)
A more detailed description of tournaments and their graphtheoretical prop
erties can be found in Moon (1968) and Reid and Beineke (1978).
2.2 Decomposition into Strong Components
Let M, be the set of all square (0,1)matrices of size n x n and A E Mp.
Let B denote a certain class of matrices. A decomposition is an expression
of the form
(2.2)
A = Bi+B2+.. + B + X
where Bj, B2,..., B. are in the class B. Usually, X is required to be the zero
matrix O. The purpose of such a decomposition is to minimize or maximize
t or another quantity associated with the decomposition.
Two simple decompositions of tournaments are presented as examples. A
trivial decomposition results if the adjacency matrix A of a tournament is
decomposed into acyclic matrices Bj,..., B. (t £ i x j) where each matrix
contains a single entry a;; % 0 of A and O's otherwise. Another simple
decomposition of an adjacency matrix A into matrices without directed cycles
2Three goals are usually achieved in a decomposition theorem: (1) The decomposition is well defined, (2)
the existence of the decomposition, and (3) the uniqueness of the decomposition is shown.
2.2. DECOMPOSITION INTO STRONG COMPONENTS
37
is given by A = Bi + B2 where Bi contains the nonzero entries of the upper
triangular matrix of A and O's elsewhere, and B2 contains the nonzero entries
of the lower triangular matrix of A and O's elsewhere. The decomposition
holds for any simultaneous permutation of A, written as PAP where P
is a permutation matrix. This minor result is summarized in the following
lemma.
Lemma 2.2.1 Let A EM.. Then there exists a decomposition
A = Bi + B2
(2.3)
where Bj and Ba are acyclic matrices (without directed cycles). Bj and B2
are unique up to simultaneous permutations of A.
Proof. A matrix with O's in the lower (upper) triangular matrix and the
main diagonal implies that the vertices of A can be ordered aj,a2,...an
so that each arc of Bj (B2) is of the form (a;, a;) for some i and j with
1Kixjsn (12ij2n). It follows that neither Bi nor Bz contain a
D
directed cycle for this would require an arc (a;,a;).
The next lemma shows the equivalence between irreducible adjacency ma¬
trices and strong components of a digraph. This result is well known (e.g.,
Reid & Beineke, 1978) but is presented here with proof because it will be
needed later.
Lemma 2.2.2 Let A E M, be an adjacency matrix of the digraph D. Then
A is irreducible if and only if D is strongly connected.
Proof. Assume that A is reducible. Then the vertex set V of D can be
partitioned into two nonempty sets Vj and Va in such a way that there is no
arc from a vertex in Vj to a vertex in V. If a e Vj and b e Va there is no
directed walk from a to b. Hence D is not strongly connected. Now assume
that D is not strongly connected. Then there are distinct vertices a and b in
Vfor which there is no directed walk from a to b. Let Wi consist of b and all
vertices of V from which there is a directed walk to b, and let Wa consist of
a and all vertices to which there is a directed walk from a. The sets Wi and
Wa are nonempty and disjoint. Let Wz be the set consisting of those vertices
which belong to neither Wi and W2. By simultaneously permuting the lines
of A so that the lines corresponding to the vertices in Wa come first followed
by those corresponding to the vertices in Wz we obtain:
Aii A12 A13
PAPT
A21 A22 A23
A31 A32 A33
38
CHAPTER 2. THEORY
Since there is no directed walk from a to b there is no arc from a vertex in
Wa to a vertex in Wj. Also there is no arc from a vertex c in Wa to a vertex
in Wi, because such an arc implies that c belongs to Wj. Hence Aj3 = O and
□
A23 = O, and A is reducible.
The decomposition into irreducible components is introduced next. It
simplifies the study of any adjacency matrix A by breaking it down into
submatrices which contain all directed cycles. The decomposition theorem
can be derived from the specific arrangement of a matrix also known as the
Frobenius normal form (Frobenius, 1912). In the case of tournaments it is
shown that this form is unique up to simultaneous permutations within the
irreducible submatrices. The following theorem and proof is due to Brualdi
and Ryser (1991).
Theorem 2.2.3 Let A be a matrix of size nx n. Then there exists a permu¬
tation matrix P of order n and an integer t 2 1 such that
Ai A12 Alt
O A2... A2
PAPT =
:
:
O.. 4
where Aj, A»,..., Az are square irreducible submatrices. The irreducible sub¬
matrices are uniquely determined up to simultaneous permutation of their
rows and columns, but their ordering as diagonal components is not neces¬
sarily unique.
Proof. see Brualdi and Ryser (1991), Chapter 3, pp. 5758
Note that Theorem 2.2.3 guarantees that any (0,1)matrix can take on the
Frobenius normal form, and that a permutation matrix P exists. The prob¬
lem, however, of how to obtain such a permutation matrix is not answered
by the theorem. The following decomposition of tournaments is a special
case of Theorem 2.2.3 and the proof provides a simple way to determine a
permutation matrix P for the adjacency matrix of a tournament.
Corollary 2.2.4 Let A be an adjacency matrix of a tournament of order n.
Then there exists a permutation matrix P of order n and an integer t 2 1
3Tarjan (1972) developed a depthfirst search algorithm to determine strong components and their order¬
ing in a digraph.
2.2. DECOMPOSITION INTO STRONG COMPONENTS
39
such that
Ai J12 Jut
O A2... Jat
PAPT =
:
::
O... 4.
where Aj, A2,..., A are square irreducible submatrices. The irreducible sub¬
matrices are uniquely determined up to simultaneous permutation of their
rows and columns, and their ordering as diagonal components is unique.
Proof. The Frobenius normal form PAPT of a tournament A has a de¬
scending order of row sums si £ s2 £ ... £ s». Therefore, permuting the
rows so that the row sums of A decrease monotonically determines a permu¬
tation matrix P.
Only the special form of the Frobenius normal form has to be shown; the
uniqueness of the order of the diagonal components follows immediately. If
A is a tournament and PAP the Frobenius normal form with Aj,...,A.
strong subtournaments then the matrices denoted by O contain O's only. It
follows from the asymmetry and connectedness of the tournament A that the
D
matrices A;; = J for i je (1,...,t).
It is easy to see that the decomposition has a minimal number of irre¬
ducible subdigraphs. This decomposition of adjacency matrices can reduce
the computational effort for determining polynomial expressions. A polyno¬
mial expression which has directly interpretable coefficients is defined in the
following section.
2.2.1 Polynomials
Associated with the adjacency matrix A of an arbitrary digraph D is the
characteristic polynomial (x) as defined in Equation A.5. The characteristic
polynomial of a tournament or a digraph without loops is a monic polyno¬
mial or monom. Research on the characteristic polynomial of digraphs has
been concerned with eigenvalues of A or their nearest bounds in the complex
plane (Wilkinson, 1988). But the coefficients of the characteristic polynomial
also have an interesting interpretation in terms of directed cycles: There is a
correspondence between collections of disjoint dicycles whose lengths add up
to k and the kth coefficient of the characteristic polynomial. The explicit
relationship was shown by Mowshowitz (1972) in the context of cospectral
graphs. For an arbitrary digraph D of order k, let fp (i,12,...,i.)) de¬
note the number of collections of disjoint directed cycles in D of lengths
40
CHAPTER 2. THEORY
i1,12,...,i., where i; 2 1 (1 Sj Sr) and ij + 12 +... + i, = k. Using the
formula for the determinant of the adjacency matrix of a digraph (see Theo¬
rem A.4.6) Mowshowitz obtained the following result.
Theorem 2.2.5 Let D be a digraph of order n and A its adjacency matrix.
Then for 1 k S n, the kth coefficient zy of the characteristic polynomial
(x) of A is given by
42 I131
fo (Li1, 12 i.))
L=1
where the summation is taken over all rank r partitions i,12,...,ir) of k
with (1 KrSk), and zo = 1.
Proof. Only a sketch of the proof is presented. It is known that the kth
coefficient z (1 £ k £ n) is equal to the sum of all principal minors of order
k. Since each k order principal submatrix of A is the adjacency matrix of a
subdigraph of D containing k vertices, it is clear that any principal minor of
A is the determinant of the adjacency matrix of a subdigraph of D. Thus, the
coefficients of the characteristic polynomial of A can be expressed in terms
of determinants of matrices belonging to subdigraphs of D.
In the following a modification of the characteristic polynomial is sug¬
gested. It is shown that the kth coefficient of this modified polynomial
equals the number of kdicycles in any digraph.
To establish this polynomial the characteristic polynomial is first written as
the signed sum of products
(2.4)
9(a) = L(sign r) I(2I  A):()
1
Each fixedpoint i of 7 will contribute either x or a; to the product, while
each nonfixed point i will contribute a). Reordering the sum and orga¬
gives
nizing it by S
n
—
k
(2.5)
o(x) =
sign (r)(1) I)
L
L
ies
k0
S =k TEP(S)
where P(S) denotes the set of permutations on S. If the sign function is
dropped and the permutations TE P(S) are replaced by permutations re
P(S) with a single permutation cycle, the expression becomes
n
L
(r)  Lark
(2.6)
Ir
k0
Sk rep9ies
2.2. DECOMPOSITION INTO STRONG COMPONENTS
41
The so defined polynomial counts only kdicycles as shown in the following
proposition.
Proposition 2.2.6 Let D be a digraph of order n and A its adjacency matrix.
Then for (1 S k S n), the kth coefficient zi of the polynomial y(x) of A is
given by
24 = fD (Li))
where fp is the collection of all directed cycles of length k and zo = 1.
Proof. Follows immediately from the definition ofy(x), Theorem 2.2.5,
and the onetoone correspondence between directed cycles and permutations
D
with a single permutation cycle as shown in Lemma A.4.2.
Although the polynomial(x) describes any digraph in terms of dicycles,
it does not factor into irreducible polynomials.* In the following, polynomials
are discussed which factor into irreducible polynomials, each corresponding
to a strong component.
Definition 2.2.7 Let A be the adjacency matrix of a directed graph, X the
diagonal matrix
Xj0.. 0
0 x2... 0
X =
::
0 0 ... En
and f a combinatorial matrix function. Then
% = O(x,an) =f(A+X)
is a polynomial in the polynomial ring Rsæi,.,x.
The factorization can be established by showing that a strong component
corresponds to an irreducible polynomial. The following result holds for
polynomials where the combinatorial matrix function f is the determinant
(Frobenius, 1912; Schneider, 1977), and for polynomials where f takes on a
form related to the permanent (König, 1936; Ryser, 1973).
Theorem 2.2.8 Let D be a directed graph. Then A is the adjacency matrix
of a strong directed graph if and only if o is an irreducible polynomial in the
polynomial ring Rsai,n.
An unsolved question in this context is: Which type of digraph has strong components that correspond
to irreducible polynomials of (x) in Zsæl, the characteristic polynomial in the polynomial ring over the
integers. It follows from Proposition 2.2.6 that a digraph whose strong components have no vertexdisjoint
partition into dicycles is a sufficient condition.
42
CHAPTER 2. THEORY
Proof. see Schneider (1977).
D
If the polynomial ring is defined over the integers then the uniqueness of the
result follows from the fact that Zx..., is a unique factorization domain
(Herstein, 1975, Chapter 3, and Corollary A.4.11).
Although the polynomial % has a factorization which corresponds to the
strong components its coefficients neither equal nor determine the number
of kdicycles. To overcome this limitation a different approach is suggested
next. In the following, a polynomial in the polynomial ring over the integers is
defined. This polynomial can be partitioned in its indeterminates describing
the strong components of a digraph in terms of kdicycles.
Definition 2.2.9 Let A be the adjacency matrix of a digraph. The combina¬
torial matrix function q: M— Z is given by the formula
n.
g(A) =
(2.7)
aa
L
k=0 S=k TEP(S)ieS
where the third summation extends over all permutations T in S with a single
permutation cycle.
Notice that in contrast to the determinant as defined in Equation 2.4 or
the permanent only permutations with a single permutation cycle appear in
Definition 2.2.9 and that the factor a; in the product determines the support
of each dicycle.
Definition 2.2.10 Let D be a digraph of order n and A the associated adja¬
cency matrix. The polynomial expression y(xj,...,xn) of A is given by
(2.8)
y = V(xi..,an) = 9 (4+X)
where the combinatorial matrix function q and matrix X are defined as before.
Proposition 2.2.11 (Partition) Let A be the adjacency matrix of a digraph
of order n, and y the nonzero polynomial expression associated with the
digraph. Then A is irreducible if and only if y cannot be partitioned into
nonconstant polynomial expressions in disjoint sets of indeterminates.
Proof. Assume that A is reducible. Then there exist positive integers r
and n  r and a permutation matrix P such that
Ar + X.
P(A+X)PT =
*
Anr + Xnr
2.2. DECOMPOSITION INTO STRONG COMPONENTS
43
—
where A, + X, and An + X» are square matrices of order r and n
r,
respectively. The union of their diagonal elements is sæ1,x2,...,xn).
It
follows thaty can be written as a partition of two nonconstant polynomials
in Zsæj xn).
Now assume that the matrix A = a; is irreducible. Suppose that y has
a partition
V = p+9
(2.9)
into two nonconstant polynomials in Zaj,...,. Without loss of generality
let p be a polynomial in the indeterminates x1,...,x. and q a polynomial
in the indeterminates (.+1...,n).
Consider the digraph D associated with the adjacency matrix A. By ap¬
plying Lemma 2.2.2 we know that A is irreducible if and only if the associated
digraph D is strongly connected. It follows that there is at least one dicy¬
cle whose support has at least one vertex in x1,...,x. and at least one
in ,1...,. Consequently, y cannot be expressed as the sum of two
polynomials in distinct sets of indeterminates, which renders a contradiction.
O
Hence, y has no partition in its indeterminates.
The partition ofy in its indeterminates is uniquely determined. The de¬
velopment of an algorithm for the computation of the polynomial and its
partition is limited by the complexity of the digraph. Explicit considerations
on the computational complexity are beyond the scope of this thesis and
have been omitted. However, counting all dicycles in a polynomial is not an
efficient technique although complexity can be reduced by the partition of
the polynomial expression. A slightly better way to assess all directed cycles
in a digraph was implemented in the Prolog program listed in Appendix C.1
which employs a depthfirst strategy. Directed cycles in digraphs have been
subject to extensive research in connection with the linear ordering problem
and the corresponding acyclic subgraph problem' (see for example Fishburn,
1992; Jünger, 1985; Reinelt, 1985). These problems among many others are
known to be NPcomplete which means that if there exists an algorithm
which solves one of them in polynomial time, there would exist a polynomial
time algorithm for solving them all. Garey and Johnson (1979) provide a
comprehensive treatment of the theory of NPcompleteness and a compila¬
tion of all contemporary NPcomplete problems.?
In a straightforward application of results the adjacency matrix A of a
tournament is first brought into Frobenius normal form. Then the polyno¬
There is an occasional survey on recent developments in the field by Johnson in Journal of Algorithms.
44
CHAPTER 2. THEORY
mial y is computed for each irreducible submatrix Aj,..., A, using single
indeterminates. According to Proposition 2.2.11 the resulting polynomials
can be written in a sum which contains equivalent information to the poly¬
nomial y(x....,.). Compared to an implementation of the polynomial
% in n indeterminates this technique reduces the computational complexity
and size of the suggested polynomial expression and determines the number
of kdicycles.
2.2.2 Critical Arcs in Tournaments
So far we have established a general algebraic decomposition which breaks
down a digraph (tournament) into subdigraphs (subtournaments) on the ba¬
sis of its dicycles. To take this approach a step further, it appears promising
to explore which arcs are particularly responsible or critical for dicycles. This
question leads to a fundamental optimization problem. The following discus¬
sion is mostly restricted to tournaments.
The following proposition suggests how to perform simple matrix opera¬
tions on an adjacency matrix to find the number of 3dicycles intersecting in
each arc.
Proposition 2.2.12 Let A E M be the adjacency matrix of a directed graph
be the matrix
without loops. Let R = r
RJ = A?AT
where * denotes the Hadamard product. Then each entry in Ry records the
number of times arc a;; belongs to a directed cycle of length 3.
Proof. The Hadamard product is defined as A*B = a,b for (1 i,j
n). A2 records all directed walks of length 2 from a; to a;. The transposed
matrix of the Hadamard product between A2 and A', has entry r; » 0 if
there is an arc a; — a; completing r:; directed 3cycles. Consequently, arc
D
a; belongs to r;; different 3dicycles.
Note that this result does not hold for multigraphs where 2dicycles and loops
are allowed. However, the matrix
RT =A1 AT
(2.10)
for 3 % k § 5 identifies kdicycles in a tournament and R+ = A*A can be
used to determine preference reversals between pair comparisons.
2.2. DECOMPOSITION INTO STRONG COMPONENTS
45
Lemma 2.2.13 Let A be the adjacency matrix of a tournament. Then the
following two statements are equivalent.
A is acyclic
(1)
(ii) (42) +A=0
Note that in this context an acyclic tournament is equivalent to a transitive
tournament without directed cycles.
Proof. That (i) implies (ii) is clear. Now assume that A has no 3dicycles
but contains an ndicycle. If a; — a; — a is a directed walk within the n
dicycle then there is an arc a;— — a because A represents a tournament,
hence there is a directed cycle of length n  1. By induction a 3dicycle
remains and a contradiction is reached. Consequently, (ii) implies (i).
This result does not ensure that a tournament is acyclic or without in¬
consistent choices if arcs from 3dicycles in a tournament are successively
removed until no 3dicycle remains. This is illustrated in Figure 2.2. If the 3
dicycles in Tournament A are eliminated by removing arcs 1 — 3 and 2 — 4
from Tournament A then the subdigraph is acyclic. If the same arcs are re¬
moved from Tournament B then the 4dicycle 1 — 4 — 3 — 2 — 1 remains.
In this case the removal of arc 3 — 2 which belongs to both 3dicycles and
B.
A.
25
24
Figure 2.2: Tournament A differs from B by a single arc.
the 4dicycle in Tournament B results in acyclic digraphs.
It is conjectured that an iterative algorithm which takes into account the
number of intersecting kdicycles for each arc of a digraph eventually leads to
an acyclic digraph with a minimal set of removed arcs. For example, consider
46
CHAPTER 2. THEORY
the following iterative lexicographic heuristic: First select the highest entry
rij e Rj, delete the corresponding entry a;; e A then recompute Ry from
the modified matrix A and continue this process until all dicycles in A are
resolved. If the entries of R4 are employed whenever the entries in Ry are
tied and the entries of Rj whenever the entries of R and RT are tied and
so on should lead to a minimal solution.
Example. Consider the adjacency matrix A representing a tournament of
size 8 and Rj from the first step in this process.
0 3 0
0 100000 0
0
00000 1 1 1
00000 3 3
11000001
0 2 0 0 0 0 0
1110010 0
0 2 100 10 0
A =
111101 o (42)744 
0 2 1 100 1 0
1010100 0
10 20 300 0
0 1 10 10
0
1021010
100 1111 0
10 0 2 2 1 1 0
Four 3dicycles are removed if entry a38 = ed. This entry
corresponds to the highest entry r38 =
in Rj. In the next step and
the following steps there is no unique highest entry in Rj. If we delete in
subsequent steps a12, 665, a32,442, and a52 the highest entries of Rø in this
order then all directed cycles are resolved but if a26, a27, a28, aes the highest
O
entries in Rj and R are selected no sixth arc has to be removed.
Thomassen (1989) showed that strong tournaments with identical 4di¬
cycles are isomorphic or antiisomorphic (Definition A.1.5) solving a problem
raised by Goldberg and Moon (1971). It is not possible to conclude from this
result that an iterative heuristic which takes into account all 4dicycles in a
strong tournament is indeed an algorithm which finds a minimal number of
arcs because the first removal renders a digraph and not a strong tournament.
In general this problem is linked to the 'acyclic subgraph problem' which is
known to be NPcomplete. This problem arises in any digraph where the
removal of a number of arcs may be minimal for 3dicycles or 4dicycles but
does not necessarily guarantee an optimal solution for all dicycles. However,
heuristic methods and approximate algorithms have been developed which
can find solutions in polynomial time (e.g. Jünger, 1985).
2.2.3 Examples of Tournaments
The theoretical concepts are explained and summarized in the following ex¬
amples. The correspondence between strong components of a tournament
2.2. DECOMPOSITION INTO STRONG COMPONENTS
47
and the partition of the associated polynomial expression is illustrated. This
section also gives an idea how quantitative and qualitative information can
be extracted from adjacency matrices if they are decomposed into irreducible
submatrices.
Note that the following (0,1)matrix A does not contain artificial data
but describes the preferences of Subject 1 in Session 1 of Experiment 2B (see
Chapter 3).
0 110 1001000 1
0000000 00 0 0 1
O10111111101
110010010001
0 100000000 0 1
110110111001
A =
110110010001
0 10 0 1000000 0
110110110001
110111111001
11111111110
0 0 0 0 0 0 0 1 0 0 0 0
The index of the rows and columns refers to the number of the twelve lotteries
as listed in Table B.10. The subject made choices between all possible pairs
of lotteries in a forced choice pair comparison resulting in a tournament. The
entry a;; in matrix A is 1 if the subject preferred the Lottery i in row i over
the Lottery j in columnj and 0 otherwise. By simultaneous permutations of
the rows and columns according to their row sums, the following Frobenius
normal form of the tournament can be found.
1 1
111111
1
10
1 1 1 1
1 0
0 1111
0
1 11 1
0 011111
1 1 1 1
0001111
1
0 0 0 0 1 1 1
1 1 1
0 0 0 0 0 1 1
1 1 11
PAPT =
0 00000 0
1111
10000 10
1 1 11
0 1 0 11
0 0 000 0 0
0
0 0 1 1
0 0 0 0 0 0 0
00 000 0 0
100 0
00 00000000 10
The irreducible submatrices along the diagonal, which correspond to the
strong components of the tournament, are indicated by square brackets.
48
CHAPTER 2. THEORY
The characteristic polynomial is computed by
%(x) = det(xI  A) = 2x" + 2121 +... + 2n12 + 2n
(2.11)
where x is indeterminate in Z and I is the identity matrix. The coefficients zi
of the characteristic polynomial correspond to certain collections of dicycles
within the matrix. The characteristic polynomial o(x) of matrix A has the
form
9(x) = g1272° 112° 10x' + 5a“ + 242' + 30r' + 20a' +722+r (2.12)
The factorization of the polynomial (x) results in three irreducible poly¬
nomials." The disadvantages of the characteristic polynomial are twofold:
First, the coefficients of the characteristic polynomial detect dicycles of any
given length together with collections of disjoint dicycles which add up to the
same given length. Consequently, if there are disjoint dicycles within strong
components then the coefficients (before and after factorization) do not equal
the number of dicycles. Second, the factorization of %(x) does not necessarily
correspond to the strong components of a tournament.
To overcome both problems the polynomial y(x1,...,x») is used instead.
It counts only directed cycles and has a partition in its indeterminates. The
partitioned polynomial corresponds to the strong components of any digraph
including tournaments. For computational reasons the polynomials y in sin¬
gle indeterminates are computed for the irreducible submatrices of the Frobe¬
nius normal form. According to Theorem 2.2.11 the resulting polynomials
can be written as a sum.
The Hasselike diagram in Figure 2.3 illustrates the preferences of matrix A
in a special way. The alternatives are arranged in descending order according
to their preference scores. As in a Hasse diagram preference decreases from
top to bottom along the solid lines without arrowheads. In the example of
Figure 2.3 Lottery 11 is preferred over any other lottery, followed by Lotter¬
ies 3, 10, 6, 9, 7, 4 and 1. The arc pointing from Lottery 1 to 3 indicates a
preference which violates the topdown order and creates many intransitivi¬
ties among the seven lotteries in this strong component. However, all these
lotteries are preferred to Lottery 8. Finally, Lottery 8 is preferred over Lot¬
tery 5, followed by Lottery 2, and 12. Here the arc pointing from Lottery 12
In this special case the three factors correspond to the decomposition of the tournament A into three
strong components Aj, A», and Ag. Moreover, the coefficients of the three irreducible polynomials equal the
number of directed cycles within the strong components. But we know from Theorem 2.2.5 and Proposi¬
tion 2.2.6 that both results do not hold for arbitrary tournaments.
2.2. DECOMPOSITION INTO STRONG COMPONENTS
A(11e
)
3
A2
10
6
9
7
4
8 —
A3
5
2
12
Figure 2.3: Hasselike diagram of Subject 1 in Session 1 (Exp 2B)
to 8 introduces intransitivities among the four lotteries in this strong com¬
ponent. The three strong components are indicated by surrounding boxes.
They correspond to the partition of the polynomial y.
By applyingy to the irreducible matrices in the Frobenius normal form
and by writing the resulting polynomials in a sum the following polynomial
expression results:
v = (x' + 526 + 1025 + 1024 + 523) + (y+ + 2y3)
(2.13)
This expression is interpreted as follows: The tournament of order 12 has
three strong components, one of order 1 (Aj), another of order 7 (A2), and
the third of order 4 (A3). Note that the strong components are the small¬
est subdigraphs, which are linearly ordered within the tournament. The
strong component of order 7 has a single directed 7cycle (x'), five directed
6cycles (52°), ten directed 5cycles (102°), ten directed 4cycles (102*), and
five directed 3cycles (52*), whereas the strong component of order 4 has one
directed 4cycle (y*), and two directed 3cycles (2y’). No directed 2cycles
or loops (directed 1cycles) were detected because the subject chose in a pair
comparison.
The strong components of the pair comparisons of the same subject in
49
CHAPTER 2. THEORY
Ar
Ar9
7
6
11
10
4
14
8
43.8
5
24
1
Figure 2.4: Diagram of Subject 1 in Session 2 (Exp 2B)
Session 2 and 3 are illustrated in Figures 2.4 and 2.5, respectively. The
preference structures are more consistent in the second and third session
than in the first. This improvement is reflected in the decreasing number
of strong components with size greater than 1 as well as the total number
of kdicycles. The partition of% results in the following expression for the
tournament of Session 2:
y = 2° + 22' + 323
(2.14)
The tournament has seven strong components of order 1 (Aj,A3,..., Ag) and
one strong component of order 5 (42). This strong component has a single
directed 5cycle (2*), two directed 4cycles (22*), and three directed 3cycles
(32°). This example shows how consistency of choices improved over Session 1
and 2, and were even perfect in the last session.
With the above decomposition technique at hand tournaments can be
characterized in great detail on ordinal level. The number of directed cy¬
cles of different length as well as the number and size of strong components
can be assessed at once. With these measures inconsistency of individual
choice behavior can be studied quantitatively in an exhaustive way. On the
other hand, the enumeration of all directed cycles in a digraph can be very
2.2. DECOMPOSITION INTO STRONG COMPONENTS
37A12
69
74
114
104
94
4
11
8
5
2
124
Figure 2.5: Hasselike diagram of Subject 1 in Session 3 (Exp 2B)
time consuming and almost impossible for strong digraphs of order n 2 30.
Furthermore, these measures do not reflect the adaptive nature of individual
choice behavior.
It was suggested that matrix operations may help to identify critical arcs
which can explain all inconsistencies in a tournament. If the matrix opera¬
tions are applied to A of the previous example then we observe
0 0 5 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 10 110 1100
10 0 000000000
0 0 0 0 0 0 0 0 0 0 0 1
10 0 0 0 00 0 00 0 0
R3 = (42) +A 
10 0 00 00 00 0 0 0
0 1 0 0 1000000 0
10 00 0 0 0 000 0 0
10 0 00 000 00 0 0
00 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 2 o 0 0 0
The two entries ri,3 = 5
and 7128 = 2 of matrix R indicate that the
removal of arcs 1 — 3 and 12 — 8 in A will break up all directed 3cycles.
However, resolving all 3dicycles in a tournament does not necessarily lead to
a digraph without dicycles of greater length. For a graphical representation
51
52
CHAPTER 2. THEORY
of the tournament A we refer to Figure 2.3: The arcs belonging to directed
cycles which do not fit into the topdown arrangement of the Hasse diagram
are drawn with an arrow. In this example two arcs are easily identified which
are responsible for all directed cycles in this tournament. They correspond to
the arrows pointing from 1 — 3, and from 12 — 8. The same applies to the
example in Figure 2.4. The arcs 6 — 9 and 10 — 9 resolve all inconsistencies
within this tournament. As described in Section 2.2.2 matrix operations may
help to determine these critical arcs'. In general, however, solutions are hard
to determine and not necessarily unique. There can be several minimal sets
of arcs explaining all directed cycles or inconsistencies within a tournament.
A different approach which can overcome some of the problems is suggested
in the next two sections.
2.3. EAR DECOMPOSITION
53
2.3 Ear Decomposition
In the following a technique is described which decomposes the strong compo¬
nents of a digraph. This technique is known as ear decomposition. Folklore in
mathematics tells that the name was derived from a graphical illustration of
a simple ear decomposition of an undirected graph as pictured in Figure 2.6.
The idea of an ear structure for undirected graphs was first suggested by
Whitney (1932) and has been useful in proving many graph theoretical re¬
sults (Lovász & Plummer, 1986).
Figure 2.6: Illustration of ears' in an undirected graph.
The sequence of choices in a forced choice pair comparison is employed
to ensure an ear decomposition with uniquely determined ear bases. Each
ear basis describes a minimal collection of dicycles in a strong component
generating all dicycles together with all cycles of the associated undirected
graph which are not dicycles. If all possible combinations (modulo 2 sums)
of these dicycles are generated then a lattice results which covers the strong
component and has a minimal number of dicycles as a basis.
The psychological interpretation of ear bases in a pair comparison is straight¬
forward. Each ear basis describes a minimal set of intransitive preferences
of a strong subtournament that can account for all intransitive preferences
and all cyclic comparisons which are not intransitive. This means if all possi¬
ble combinations of the minimal set of intransitive preferences are generated,
that is by combining preferences that belong to either of two preference cycles
but not both, then a structure results covering all preferences in a strong sub
tournament. The ear decomposition is unique if the minimal set of dicycles
is determined by the sequence of intransitive choices.
54
CHAPTER 2. THEORY
2.3.1 Cycle Space and Dicycle Basis
The cycle space of an undirected graph G'is the subspace GF2E generated
by the incidence vectors of the cycles of G. The cycle space of a directed
graph D is the cycle space of the underlying undirected graph. A cycle basis
is a basis of the cycle space of G, equivalently a minimal set of elements of
the cycle space such that any cycle of G is a modulo 2 sum of these elements.
A set D of directed cycles of D which is a cycle basis is called a directed
cycle basis, i.e. any incidence vector of a cycle of the underlying graph of D
is a modulo 2 sum of incidence vectors of some directed cycles of D. The
modulo 2 sum of incidence vectors may be interpreted as the combination of
edges in the undirected graph.
The lattice generated by a set A of vectors is the set of all integer linear
combinations of vectors of A. It is a wellknown fact (e.g. Schrijver, 1986)
that each lattice generated by a finite set of rational vectors has a basis,
i.e. a set of linearly independent vectors (over rationals) such that any other
element of the lattice is an integer linear combination. Since the elements of
a lattice are integer linear combinations of elements of each lattice basis, each
element of the lattice is also an integer linear combination mod q of vectors
of each lattice basis, for any q » 1. For short, an integer linear combination
mod q is sometimes called a qcombination.
The directed cycle bases which are naturally defined from an ear decom
position of a digraph are bases of the lattice generated by the directed cycles
as well. Galluccio and Loebl (1996) showed that the lattices generated by
the cycles of undirected graphs have bases consisting of directed cycles. This
result can be applied to a lattice whose elements are cycles with arcs instead
of edges.
A digraph is strongly connected if and only if it may be built up from a
vertex by sequentially adding arcs and by subdividing arcs. This property
leads to the concept of an ear decomposition.
Definition 2.3.1 An car decomposition of a digraph D is a successive ex¬
tension Do,..., D = D of subdigraphs of D such that Do consists of a single
vertex and no arc, and D; arises from D; by adding a directed path P; whose
endvertices (which are not necessarily distinct) belong to D; while the arcs
and intermediate vertices of P; do not.
The paths P; are called ears and the endvertices of P; are called initial ver¬
tices of the ear. A digraph is strongly connected if and only if it has an ear
2.3. EAR DECOMPOSITION
55
decomposition (Whitney, 1932). It is well known that from each ear decom
position of a strongly connected digraph it is possible to obtain a directed
cycle basis by simply completing each new ear to a directed cycle using a
directed path in the already built subdigraph. Such directed cycle bases are
called directed earbases.
The following result concerning the lattice generated by the directed cycles
of a digraph D is due to Galluccio and Loebl (1996).
Theorem 2.3.2 Let D be a strongly connected digraph. Any directed ear
basis of D is a basis of the lattice generated by the directed cycles of D.
Hence for any q » 0, the incidence vector of each directed cycle of D is a
qcombination of incidence vectors of a directed ear basis of D.
Proof. Let B = xg...,Xq, denote a directed cycle basis of digraph D,
i.e. the set of incidence vectors of the directed cycles C; obtained from an ear
decomposition Do, Di,..., D = D by completing the directed path P; into
a directed cycle of D;. Let B; = x,,...,Xq, be the set of incidence vectors
from the ear decomposition of D;. In order to prove the theorem it has to
be shown that the vectors of B are linearly independent over the rationals
and that the characteristic vectors of directed cycles of D are integer linear
combinations of them.
The linear independence follows from the construction of the directed ear
basis: for each j «i k the directed cycle C; contains no arc of the path P:
while C; contains P;.
Let &a be the set of eulerian vectors of D whose components are integers.
Define 2; to be the set of vectors of £ having nonzero components only on
arcs of D;. Hence, £m = 84. It is proved by induction on i that each element of
8; is an integer linear combination of elements of B;. This finishes the proof of
the theorem since the incidence vector of each directed cycle of D is eulerian.
Let ( be any vector in 8; 81. Since ( is eulerian, the components of (
corresponding to the arcs of P; are equal, say p. Hence, the vector ( pxg.,
O
belongs to 8;1, and the result follows from the induction hypothesis.
In the following this result is applied to the special case of strong tourna¬
ments and integer linear combinations with q = 2 only.
Each element in such a lattice is a collection of arcs whose underlying
undirected graph G forms a cycle. Therefore, such an element is either a
dicycle or an acyclic subset of arcs. Consequently, determining a basis in this
lattice is the minimal set of directed cycles generating all the elements which
56
CHAPTER 2. THEORY
form directed cycles or not. Such a basis offers a characterization of a strong
tournament in terms of directed cycles and acyclic subsets.
2.3.2 Ear Decomposition by Sequence
In a straightforward application an ear decomposition is established which
determines a directed ear basis for a given sequence of arcs in a strong digraph.
The directed ear basis of such a labeled digraph is unique.
Thereto, we define a sequence on the arcs or preference relations. When
considering the trials of a pair comparison their sequence may be expressed
as an ntuple. Hence, a sequence of trials in a pair comparison describes
a single outcome in the sample space of all possible sequences.
Together
with the observed choices a simple pair comparison leads to a (3)tuple of
preferences ordered in time.
((a,b), (c,d),.., (v, w)
If a neighborhood relation CS2XS2 is introduced to describe the sequence
in time, then the tuple can be expressed as a chain Ko of pairs:
(2.15)
Ko:= (a,b)b (c,d)b... (v,w) for all (x,y)ESxS
The chain Ko can be understood as a set of ordered pairs ordered in time.
Definition 2.3.3 (Ear Decomposition by Sequence) Let Ko be a sequ¬
ence defined on the arcs of a strong digraph. An ear decomposition by se¬
quence of digraph D is a successive extension Di,...,D4 = D of subdigraphs
of D such that Di consists of the first arcs in Ko which form a directed cycle,
and D; arises from D; by adding the first directed path P; in Ko whose
endvertices (which are not necessarily distinct) belong to D; while the arcs
and intermediate vertices of P; do not.
The definition above results in a unique collection of directed ears each of
which can be completed to an ear dicycle by adding arcs, according to their
sequence, from the already built subdigraph. From this collection of ear
dicycles the dicycle space can be generated. Because these directed ears sub¬
divide arcs which already belong to the subdigraph but appeared earlier in
the sequence than any other directed ears they are believed to reflect adap¬
tiveness in individual decision behavior. The ear decomposition by sequence
is adaptive because it can model the process in which the strong components
and their dicycles evolved across trials in a pair comparison.
2.3. EAR DECOMPOSITION
57
Corollary 2.3.4 The ear decomposition by sequence of a strongly connected
digraph D is uniquely determined.
Proof. It follows from Theorem 2.3.2 that the ear decomposition by se¬
quence exists and is well defined. Since every ear has at least one distinctive
labeled arc their sequence is uniquely determined. They can be uniquely
completed to dicycles using the first directed path in the already built sub¬
digraph. Hence, the resulting ear dicycles form a directed ear basis and are
D
uniquely determined.
In the next section the ear decomposition by sequence is applied to real
data from a forced choice pair comparison. The data corresponds to a tour¬
nament with more than one strong component. The underlying graph of a
tournament is a complete graph of order n where each arc belongs to a cycle of
length k with (3 k£ n). Because the strong components of a tournament
are strong subtournaments the directed cycle basis of a strong component
generates all cycles and thus all dicycles within this subtournament. Collect¬
ing the directed ear bases of all strong components in a tournament leads
to disjoint dicycle spaces for each strong subtournament. The number of
subtournaments and their sizes can be derived from the directed ear bases.
2.3.3 Example of Ear Decomposition by Sequence
If the preferences of Subject 1 in Session 1 (Exp 2B) are ordered by their
sequence the following ear decomposition results. In the first fifteen steps
an ear decomposition by sequence of the strong component A2 is obtained.
Because Aj has a single vertex it cannot be decomposed (see Figure 2.3).
By finding the first dicycle in the sequence of preferences Ko the first and
second step of Definition 2.3.3 are executed in one step. The resulting fifteen
directed ears and the corresponding completed ear dicycles from each step
are listed below.
In the first step the dicycle of intransitive preferences with the lowest
trial numbers is found. In the following step the arcs of a directed path are
collected that start and end in a vertex of the subdigraph Di and that have
the lowest trial numbers. This ear is then attached to the subdigraph Di
extending it to D2. In the next step a directed path is searched that starts
and ends in a vertex of the extended subdigaph D2 and has the lowest trial
number. The search is terminated when no further ear is found that can be
attached to the subdigraph. Figure 2.7 illustrates the first four steps of this
58
CHAPTER 2. THEORY
Step Ear
Step Ear Ear Dicycle
Ear Dicycle
7 — 4
10 — 1 — 3 — 10
9.
— 1 — 3 — 10 — 7
1.
10 — 7 — 1 — 3 — 10
10. 9— 4
— 1 — 3 — 10 — 9
2.
3 — 7 — 1 — 3
3.
11. 6 —4 — 1 — 3 — 10 — 6
4. 10 — 9 — 1 — 3 — 10
12. 10 — 4 — 1 — 3 — 10
5.
3 — 9 — 1 — 3
13. 6 —1 — 3 — 10 — 6
9— 7 — 1 — 3 — 10 — 9
14. 3 — 4 — 1 — 3
6.
10 —6 — 7 — 1 — 3 — 10
15. 3 — 6 — 7 — 1 — 3
7.
6 — 9 — 1 — 3 — 10 — 6
8.
process. D displays the first ear which is a dicycle in Ko, D2 corresponds
to D extended by the second ear, Dz corresponds to D» extended by the
third ear, and so forth. The decomposition proceeds until Dis (not shown)
is found which corresponds to the strong component A» in Figure 2.3.
D,
D,
D,
D.
1
+
3
3
3
4
10
10
10
10
Figure 2.7: Ear decomposition of tournament for Subject 1 in Session 1
(Exp 2B). Only the first four steps are illustrated. See text for explana¬
tion.
The ear decomposition is continued if another disjoint dicycle is found as
in this example. The disjoint dicycle initiates the ear decomposition of the
strong component A3. The three ear dicycles which are found in successive
steps are listed below.
The associated subdigraphs are illustrated in Figure 2.8. Ej corresponds to
the first ear or dicycle in Ko which is disjoint from Dis, E2 corresponds to Ei
extended by the second ear, and Ez finally corresponds to E2 extended by the
third and last ear. The decomposition terminates because no more ears can
be found and no more dicycles or strong components are left. Ez in Figure 2.8
2.3. EAR DECOMPOSITION
Step Ear Ear Dicycle
8 — 5 — 12 — 8
1.
2. 8 — 2 — 12 — 8
3. 5 — 2 — 12 — 8 — 5
is equivalent to the strong component Az in Figure 2.3. Note that the ear
dicycles are a subset of all dicycles leading to a smaller number ex of kdicycles
in comparison with the coefficients zi of y. A Prolog program is listed in
Appendix C.1 that performs a depth and breadthsearch on a given sequence
of preferences. It determines the ear dicycles of the ear decomposition by
sequence. This unique collection of ear dicycles corresponds to the directed
ear basis of each strong component.
E,
E,
E,
5
12
12
Figure 2.8: Continued ear decomposition of tournament for Subject 1 in
Session 1 (Exp 2B). See text for explanation.
The ear decomposition by sequence solves the problems encountered ear¬
lier. Unlike the polynomialy and other techniques it works efficiently, that
is in polynomial time. It provides a minimal solution in the sense that the
ear dicycles form a basis of the cycle space. If the sequence of choices in a
pair comparison are taken into account then the solution is also unique and
can be studied experimentally.
59
60
CHAPTER 2. THEORY
2.4 Completion by Cuts
In this digression a slightly stronger assumption about the sequence of choice¬
trials is made. The following technique does not follow from the algebraic
decompositions studied in the previous sections but it offers a more detailed
explanation of inconsistent choice behavior. The completion by cuts is ap¬
plied extending the idea of a sequence of intransitive choices. A technique
is suggested which is based on latent preference orders and the embedding
of intransitive subchains into a lattice structure. However, a computerized
implementation needs to be developed before it can be applied to empirical
data. Therefore, this technique was not applied to the experimental data
presented in the next chapter.
The technique is motivated by the following simple observation. If the
choices in a pair comparison are brought into a sequence then intransitive
choices may appear anywhere in this sequence. This means, they are often
separated by a number of choices which might or might not belong to other
intransitivities. Hence, a change in information processing or shift of view
point yielding a different preference order may have occurred not just in the
intransitive choices but in any choicetrial in between. It is assumed that
a consistent preference order persists until the next shift of viewpoint in a
given sequence of choices occurs.
As in the previous section, suppose the chain Ko of pairs represents the
outcome of a pair comparison of a single subject. Every subchain of Ko can
be understood as a set of choices with the usual set operations, union and
intersection, working upon them. The intersection of subchains is again a
subchain, but the union of subchains is not necessarily a subchain. Struc¬
tures which are closed under intersection are well known in computer science
(domain theory) and sometimes denoted as algebraic Qstructures. The prop¬
erties of such an algebraic structure are employed in a simple model which
suggests that a shift of perspective occurs in the intersection of subchains
enclosed by intransitive choices.
Intransitive and transitive choices may belong to different numbers of la¬
tent order structures (see Section 1.2.1). Within a pair comparison intran
sitivities may occur if pairs belong to different orders associated with the
alternatives. It is emphasized that the use of a linear order is not essential
for the development of ideas and may be replaced by weak orders, semiorders
or partial orders.
As mentioned before, consistent choice in a pair comparison should lead
2.4. COMPLETION BY CUTS
61
to a transitive order of its alternatives. The order of the alternatives can be
understood as a permutation of its elements. The number of permutations for
k of n objects (k £ n) isn (n1):(nk+1). The number of permutations
of length n on n objects is given by (n!).
Example. The three elements (a,b,c,...) C S can be brought into 3! = 6
different linear orders. The orders Rj,..., Re are presented here as sets of
pairs.
Ri = ((a,b), (b,c), (a,c),...) R4 = ((b, a), (b,c), (a,c),...)
R2 = ((a,b), (c,b), (a,c)....) Rz = ((b, a), (b,c), (c,a),...)
Rz = ((a,b), (c,b), (c,a),...) Re = ((b,a), (c,b), (c,a),...)
O
2.4.1 Families of Intransitive Subchains
In the following, it is postulated that in a pair comparison a change in infor¬
mation processing may occur from one choicetrial to the next. Consequently,
intransitivities can be the result of different latent preference orders where a
single intransitivity in a pair comparison indicates a change of the preference
order.
As in the previous section the sequence of trials in a pair comparison can
be expressed as an ntuple describing a single outcome in the sample space
of all possible sequences. The neighborhood relation bC S2X S2 describes
the sequence of choicetrials in a chain Ko of pairs:
(2.16)
Ko:= (a,b)b (c,d)b... (v,w) for all (x,y)eSxS
The chain can be understood as a set of ordered pairs ordered in time. Instead
of the decomposition of a preference matrix A into submatrices the partition
of such a chain into subchains with acyclic preferences is considered. For
Ko C S2x S2 we write Ko = Uer K;. Thereby K; denotes a subchain which
is a sequence of choices of any length in Ko. For the chain Ko on n alternatives
in a pair comparison there are
(n)
/n)
21 12121 411
(2.17)
2
1
possible subchains which are collected in the set of all subchains K with i e I
as the index set for the elements of K. All subchains K; e K are sets for which
62
CHAPTER 2. THEORY
intersections and subsets can be defined in the usual way. This is legitimate
because the sequence of pairs is the same for all subchains of a given chain
Ko.
In order to establish weak assumptions about the sequence of choices in
a pair comparison a shift of perspective or shift for short, and an unaltered
perspective or link for short, are defined. A shift of perspective simply des¬
ignates any change of viewpoint or information processing which causes an
alteration of the underlying preference order R;.
Definition 2.4.1 Let (a, b)b (c,d) C Ko be a subchain where (a,b) E R; and
(c,d) E R; are elements of orders on S with i,j E (1,...,nl). A subchain
(a, b) b (c,d) is called shift if and only if (a,b) E R., (c,d) E R; and R, + R;
and link otherwise.
It is stressed here that the linear orders R; are not directly observable.
Consequently, shifts and links are not directly observable and their locations
have to be inferred from intransitive choices.
Every ordered pair (a,b) e S x S in a pair comparison can be thought of
belonging to an order R; on S. The definition states that subsequent pairs
are links and therefore belong to the same order unless a shift has occurred.
Therefore, chains of subsequent pairs belong to the most recent order as long
as no shift has occurred.
In the previous section we have encountered the basis of an ear decom¬
position as a family of dicycles. These dicycles naturally form a family of
intransitive subchains I =
(K;jey. The set of all dicycles can also be trans¬
lated into the family of all intransitive subchains. Both families are proper
subsets of K. They are eligible to the following operations because each sub¬
chain is just enclosed by intransitive choices and therefore has at least a single
shift. The intransitive subchain K; is defined as
(2.18)
K;:=KkCek.)
where C; denotes the choices of a kdicycle. Some straightforward conse
quences of the definitions are collected in the following corollary
Corollary 2.4.2 Let (K;) jey C T be intransitive subchains in K. Then the
following statements are true:
(i) Every intransitive subchain contains at least one shift.
2.4. COMPLETION BY CUTS
63
(ii)
Let 1 withK intransitive subchains in T. Then
exactly one shift in Oe K; is sufficient for the intransitive subchains
K;,jeJ.
Proof. (i) is obvious, It follows from (i) that a shift occurs in every K;.
Because a single shift in Ojey K; splits every K; into two subchains without
directed cycles a single shift is sufficient.
O
Statement (ii) motivates the investigation of the algebraic intersection struc
ture of subchains which is discussed in the next section. Thereto, the follow
ing lemma is already provided.
Lemma 2.4.3 Every family of subchains A C K ordered by C is a partial
order.
Proof. It is easily established that C is reflexive, transitive, and antisym¬
D
metric.
This result is needed for the family of intransitive subchains T.
If exactly t  1 shifts occurred in a chain Ko then there are exactly t
disjoint successive subchains KjU KU...UK, = Ko whose elements belong
to different orders; it follows that there are at least 2 alternating and at most
t different orders involved. If the elements in a subchain belong to t orders
then at least t — 1 shifts must have occurred.
Example. The intransitive subchain (a,b) b... b (b,c) p... » (c,a)
may be explained by a single shift with (a,b) p... p (b,c) » ... e
Rj and ... b (c,a) £ R4. Correspondingly, (a,b) D... E R5,... b (b,c) »
... E Rz and ... b (c,a) e Re is an example for an intransitive subchain
which contains two shifts of perspective.
2.4.2 Closure and Lattice
In this section it is shown that the DedekindMacNeille completion or comple¬
tion by cuts on a family of intransitive subchains defines a closure operator.
This closure on a set of intransitive subchains corresponds to a lattice struc¬
ture which can be used to investigate subchains in more detail.
Before we can derive some results the following notation is introduced:
The set of all upper bounds and the set of all lower bounds of a set A (see
64
CHAPTER 2. THEORY
also Definition A.2.4) is written as A" and A', respectively. In the case of
intransitive subchains they are defined by
4" := (Ke (vekcK
A : (Ke (ekck)
First we establish the closure operator on the partially ordered set of sub¬
chains.
Lemma 2.4.4 Let A = K;er be a family of subchains in T the partially
ordered set of intransitive subchains including Ko. Then the mapping C :
27 — 27 with
C(A): A
defines a closure operator on T.
Proof. We have to show (i) to (iii) of Definition A.2.6, the properties of a
closure operator.
(i) We have K; C K; for all K; e A and all K; e A" therefore
AC (A")=Au. Dually, AC A.
(ii) If AC B, then B" C A", hence every lower bound of A" is also a
ul
lower bound of B", thus belongs to B“
ul
(iii) It follows from (i) that A" C (A")", clearly, (Au)
Au because
(i) applied to A“ gives A" C (A") and every lower bound of Auu
is also a lower bound of A" hence belongs to A".
D
The following theorem is well known (e.g., Davey & Priestley, 1990) and
stresses the relationship between various definitions of lattices.
Theorem 2.4.5 Let C be a nonempty, partially ordered set. Then the fol¬
lowing statements are equivalent.
(i) L is a lattice.
inf A) in C for every subset A of C.
(ii)
(iii) C has a maximum and infA exists in E for every nonempty
subset A of C.
2.4. COMPLETION BY CUTS
65
Proof. (i) implies (ii) and (ii) implies (iii) because the infimum of the
empty subset of C exists only if C has a maximum. Finally, (iii) implies (i)
by the use of Lemma A.2.5.
The following result holds for any partial order and is equivalent to the
completion by cuts. It is shown here for a family of intransitive subchains on
the chain of choicetrials.
Proposition 2.4.6 Let I = K;ey be the family of intransitive subchains
of Ko including Ko itself. Then
C:=(ACT A=A)
is a lattice in which, when ordered by C, infimum and supremum is defined
as
infKiel QK;
iel
ul
sup (K. i1) K.
Liel
Proof. By Theorem 2.4.5 it suffices to show that C has a maximum and
that the infimum of every nonempty subset of £ exists in C.
Clearly, Ko is the maximal element in C. Let ;er be a nonempty
subset of C and assume that OK;ier is not in L. For OKier C T and
ul
ul
OKK
K.er
 Ke
a contradiction is reached. Hence, OKiler E C. Since Oer K; C K; for
all j e I it follows that Oer K; is a lower bound of (Kier. IfKeCisa
lower bound of (K:)er then K C K; for all i e I and therefore K COier K;.
Hence, Oer K; is the greatest lower bound of (K;)er in C, which can be
expressed as
inf(K; iel=OK.
iel
Thus (C,C) is a lattice. Since Ko e C is an upper bound of (K;er in L, it
follows from Lemma A.2.5 that
supK iel = infKi
 Kec
(Vi EIKCK)
 eukck)
iel
□
66
CHAPTER 2. THEORY
The closure system on a set of intransitive subchains with Ky as the maxi¬
mum, £ := C(T), forms a lattice in regard of C.
The minimal elements of the lattice can be investigated further. The min¬
imal elements are subchains containing sets of arcs ordered in time. It is
assumed that the smallest sets of subchains are responsible for the inconsis
tent choice behavior.
2.4.3 Example of Completion by Cuts
In the following example the hypothetical outcome of a pair comparison is
studied. The example illustrates the theoretical concepts discussed in the
previous two sections.
Example. Let S = a,b,c,d,e, f,g) be a set of alternatives and
R= ((c,f), (b, d), (a,g), (d,e), (b, f), (a,b), (c,d),
(d,g), (b,c), (e,c), (b, g), (d,a), (b,e), (a,c),
(a,f), (e,f), (c,g), (a,e), (f,g), (b,f), (g,e))
the result of a pair comparison written as a tuple of (3 =21 preference rela¬
2
tions. First we define the chain Ko := (c,f) b ... b (g,e) and identify a
family of intransitive subchains T = Kjey as:
Ki = (b, d)b (a,g)b (d,e)v (b,f)D (a,b (c,d)D (d,g)D (b,c)»
(e,c) v (b, 9) b (d,a),
K2 = (d,e)b (b,f)b (a,b)b (c,d)b (d,g)b (b,c) (e,c),
Kz = (c,d)b (d,g)b (b,c)b(e,c)b (b,g)b (d,a)b (b,e) (a,c),
K4 = (e,f)b (c,g)b (a,e)b (f,g)b (b,f)(g,e))
In each subchain Kj,..., K4 the pairs which belong to the defining intran¬
sitive triples are underlined. On the lefthand side of Figure 2.9 the Hasse
diagram of intransitive subchains is shown. The subchains are partially or¬
dered as subsets of Ko. In the next step the closure C is applied to T and
the lattice structure £ is obtained. The lattice then contains all possible
subchains closed under intersections including two new subchains
K5 = (c,d)b (d,g)b (b,c)b (e,c) b (b,g) b (d, a),
Ke = (c,d) b (d,g)b (b,c)» (e,c)
Note that unlike the ear decomposition by sequence an overlap between intransitive subchains repre¬
senting dicycles from different strong components suggests a common source of intransitivity.
2.4. COMPLETION BY CUTS
K,
K,
K
K3
1
—
k.
K,
K2
45
9
Figure 2.9: Hasse diagram of T and C. See text for explanation.
together with the empty set 0. In Figure 2.9 the lattice £ is represented in
a Hasse diagram on the righthand side. The newly obtained subchains are
denoted by open circles. The minimal elements in £ are the subchains Ke
and K4. It follows from Corollary 2.4.2 that a single shift in both subchains
□
would explain all intransitive triples within the pair comparison.
In this example the minimal elements are uniquely determined. In general the
minimal elements in a lattice are not necessarily unique and subchains may
have to be selected, according to their sequence in Ko, until all intransitive
triples are covered. The lattice structure may be investigated further by
exploiting properties of lattices (Birkhoff, 1967). The completion by cuts
can be applied to any family of intransitive subchains such as the family of
subchains defined by ear dicycles or dicycles.
It is emphasized that the completion by cuts is not unique in terms of
critical choicetrials and therefore predictions of the model may not be very
precise. However, the completion by cuts provides a general model and more
specific assumptions can be incorporated. The model is based on the testable
assumption of a sequential dependency between trials as stated in Defini
tion 2.4.1.
67
CHAPTER 2. THEORY
68
2.5 Summary
In this chapter we defined intransitive choice as kdicycles and discussed the
general algebraic decomposition of digraphs into strong components which is
related to the identification of all kdicycles. The decomposition into strong
components can be achieved by matrix operations and corresponds to the
factorization or partition of associated polynomials. The partition has the
advantage that it completely characterizes the intransitivities within a di
graph. The coefficients of the partitioned polynomial y equal the number of
dicycles in each strong component. Associated with the problem of identi¬
fying a minimal set of critical arcs that are responsible for all dicycles in a
digraph is the so called acyclic subgraph problem. For tournaments a simple
matrix technique was suggested and examples of decompositions into strong
components were presented.
In a more specific model the ear decomposition was introduced. This
technique is known to be efficient, and has a minimal solution. The ear
decomposition by sequence finds a unique set of dicycles that constitutes
a directed ear basis in a suitable space of incidence vectors. By using the
sequence of intransitive choices in a pair comparison a unique basis can be
identified.
In a digression related to the ear decomposition by sequence the comple¬
tion by cuts on the sequence of choicetrials was suggested. This is a simple
technique which can be performed on subchains associated with any family of
intransitive dicycles. It leads to subsets of choicetrials which may be respon¬
sible for most intransitivities. Some aspects of the completion by cuts were
discussed which may have implications on the detection of critical choices.
One might presume that we are now wellequipped with empirically test
able assumptions which can be derived from the theoretical models. At this
point, however, a few cautious words may be appropriate. The algebraic de¬
composition as outlined in Section 2.2 is not a model of choice but an exhaus¬
tive descriptive characterization of individual choice behavior. It therefore
offers limited opportunity for testing except for comparative purposes. As
mentioned before, the completion by cuts introduced in Section 2.4 has no
computerbased implementation and was not applied. The ear decomposition
is based on the sequence of intransitive choicetrials. Consequently, it should
be sensitive to systematic changes of the sequence of choicetrials in a pair
comparison. This specific assumption was tested in the next chapter. The
completion by cuts makes a slightly stronger assumption about the sequence
2.5. SUMMARY
69
of choicetrials in Definition 2.4.1. It states that subsequent choicetrials be¬
long to the same point of view unless a shift of perspective has occurred. It
is stressed here that in contrast to assumptions about the sequence of choice¬
trials the independence of choicetrials is an implicit or explicit assumption
of all classical algebraic and probabilistic models as discussed in Chapter 1.
In the next chapter an experimental design is proposed which tests the
independence of choices in regard of intransitive choice behavior. If we can
show that inconsistent choice behavior systematically adapts to different ar¬
rangements of choicetrials then this would favor the ear decomposition by
sequence and reject classical algebraic or probabilistic models. Algebraic de¬
composition models are designed to study individual decision behavior. How¬
ever, experimental testing is usually based on group data. Because individual
choice varies additional qualitative information provided by the decomposi¬
tions had to be excluded. As a consequence, the experimental testing in the
next chapter focuses on a comparison of quantitative measures of inconsis¬
tency, that is Kendall's ( and r, with new measures derived from the algebraic
decompositions: the number of kdicycles, ear dicycles, and the size of strong
components. A qualitative validation of the decomposition techniques is left
to future studies.
Chapter 3
Experiments
The main objective of this chapter is to produce empirical evidence for the
algebraic decompositions introduced in the previous chapter. More specifi¬
cally, it is tested if inconsistent choice behavior is sensitive to changes in the
design of pair comparisons.
If intransitive choices are not random errors but are related to changes
in the decision process, then it should be possible to induce systematic dif¬
ferences experimentally. It was claimed that the Ellsberg Paradox is an ex¬
ample of a persistent change in the information process induced by carefully
constructed alternatives. A more subtle factor which has theoretical and
empirical implications is tested by changing the sequence of choicetrials in
a pair comparison. If no differences in intransitive choice between specially
designed pair comparisons are observed, then the assumption of independent
information processing in successive choicetrials cannot be replaced by weak
assumptions about adapting choice behavior. If, however, inconsistency of
choice behavior is systematically affected by the sequence of choicetrials and
if there is an effect over sessions then an alternative approach such as the ear
decomposition by sequence is recommended.
The ear decomposition by sequence offers an explanation of adaptive in¬
dividual choice behavior because it operates on the sequence of intransitive
choices. The finding that intransitive choices are not random but are related
to the sequence of choicetrials would be difficult, if not impossible, to explain
within the framework of traditional algebraic or probabilistic models.
Six experiments on individual decision behavior in three different domains
are reported in this chapter to explore whether or not this new approach can
be applied to different domains. The first and second experiment investi¬
gate riskless choice between named alternatives in the domain of chocolate
bars. The third and fourth experiment study risky choice between described
71
CHAPTER 3. EXPERIMENTS
Figure 3.1: Complete graph of order 12
alternatives in the domain of lotteries, and the fifth and sixth experiment
investigate psychophysical discrimination between disks varying in bright¬
ness contrast. The aim of the experiments is to decide in which of the three
domains algebraic decomposition models offer an alternative to classical alge¬
braic and probabilistic decision models. In particular, the ear decomposition
by sequence is expected to trace adapting intransitive decision behavior pro¬
viding a better measure of inconsistency. The completion by cuts as outlined
in Section 2.4 was not applied to the data sets because a computerized im¬
plementation has yet to be developed.
3.1 General Method
In the following experiments subjects chose in two or three sessions between
the same set of alternatives. In a single session subjects chose in an incomplete
pair comparison between 12 alternatives comprising 66 choicetrials. The
sequence of trials was arranged in different block designs which are explained
below. In Figure 3.1 a complete graph of order 12 is depicted that also
describes a pair comparison. Each of the 66 lines connecting two vertices
refer to a single pair comparison or binary choice. If an orientation is assigned
to each connecting line a tournament of order 12 results, which completely
describes the preference relations in a pair comparison.
The block designs were constructed so that in each block of six choicetrials
3.1. GENERAL METHOD
11
Figure 3.2: Graph of resolution block Bo
an alternative of the preceding trial was either repeated in the successive trial
or appeared only once in each block. In the following the resulting two pair
comparisons are called repetition block design and resolution block design,
respectively. In combinatorial theory the latter is also known as the resolution
of a balanced incomplete block design (Street & Street, 1987, Chapter 2).
Figure 3.2 and Figure 3.3 illustrate the difference between the two block
designs and Table 3.1 contains the arrangement of choicetrials used in the
experiments. In the resolution block design all alternatives appear in each
block whereas the repetition block design covers only half of the alternatives
in a single block. The construction of the resolution block design can be
expressed in mathematical terms as follows: Let V = w,0,1,...,2n 2) be
the set of alternatives in a pair comparison (resulting in a complete graph of
order 2n). Define addition on the set V to be carried out modulo 2n  1,
except that w + i = w for all i EV.
Definition 3.1.1 Let V = w,0,1,...,2n — 2 be a set where addition is
defined as before. Then for i = 0,1,2,...,2n 2
B;: (w,i), (1 + i,2n2 +i), (2 +i,2n 3+ i),...,(ni+1,n+ i)
is called resolution block.
Collecting the blocks (Bo, B2,..., B2n2) constitutes a resolution block de¬
sign.
73
CHAPTER 3. EXPERIMENTS
10
8
Figure 3.3: Graph of repetition block B,
It is emphasized here that both block designs are incomplete pair compar
isons. They differ only by their arrangement of choicetrials. The repetition
blocks B; are constructed by replacing the trials of every second column in the
resolution block B; by trials of every second column in Bi+1. Consequently,
half of the trials appear at the same position or trial number in both designs
(see Table 3.1). The important feature of the two block designs is that they
are optimal in the sense that the number of repetitions between successive
trials is maximal under the repetition block design and minimal under the
resolution block design.
In the third block design subjects chose in an individually randomized se
quence of trials leading to repetition and resolution blocks of random length
between one and six choicetrials. Randomization is a standard and widely
used technique in the social sciences because it is convenient to assume that a
randomized sequence of trials rules out unwanted sequential effects. Putting
aside the problems associated with generating (pseudo) random sequences,
this assumption is challenged here for small samples. It is argued that ran¬
domized choicetrials are subject to similar adaptive choice behavior as the
repetition block design. A design with randomized trials appeared in Exper¬
iment 2A and 2B, and Experiment 3A and 3B and is called random block
design. Note that the number of repetitions of alternatives in successive tri¬
als under the random block design reaches on average an intermediate level
compared to the repetition and resolution block design.
3.1. GENERAL METHOD
Table 3.1: Resolution and Repetition Block Designs
Resolution Block Design
Block
(12,11)
13,8)
(1,10)
Bo:
(4,7
(2,9)
(5,6)
Bi:
(1,12)
(8,5)
(11,2)
(10,3)
(9,4)
(7,6)
(4,11)
B2:
(12,2)
5,10)
(6,9)
(7,8)
(3,1)
(10,7)
(11,6)
(2,4
B3:
(3,12)
(9,8)
(1,5)
(6,2)
(12,4)
(5,3)
B4:
(8,11)
(9,10)
(7,1)
B5:
(4,6)
(3,7)
(1,9)
(5,12)
(11,10)
(2,8)
(10,2)
(8,4)
(11,1)
(12,6)
(9,3)
Be:
(7,5)
(5,9)
(6,8)
(3,11)
(7,12)
(4,10)
B;:
(2,1)
(10,6)
(1,4)
(11,5)
(12,8)
(9,7)
Bg:
(2,3)
Bo:
(8,10)
(5,2)
(6,1)
19,12)
(4,3)
(7,11)
(11,9)
(2,7)
14,5)
11,8)
(3,6)
(12,10)
Bio:
Repetition
Block Design
Block
17,6)
Bö:
19,4)
(11,2)
(12,11)
(4,7
(2,9)
(10,3)
(5,10)
B:
(1,12)
(3,1)
18,5)
(7,8)
(11,6)
(4,11)
19,8)
(6,9)
(12,2)
(2,4)
B;:
(9,10)
B;:
(5,3)
(1,5)
(7,1)
(3,12)
(10,7)
(12,4)
B;:
(4,6)
(8,11)
(6,2)
(2,8)
(11,10)
(9,3)
B;:
(1,9)
(11,1)
(5,12)
(3,7)
(7,5)
(10,2)
(2,1)
(8,4)
(12,6)
(6,8)
Bg:
(4,10)
(5,9)
B;:
(2,3)
(3,11)
(11,5)
(7,12)
(9,7)
Bg:
(10,6)
(4,3)
(8,10)
16,1)
(1,4)
(12,8)
(5,2)
(11,9)
(4,5)
Bg:
(9,12)
(7,11)
(2,7)
15,6)
13,8)
13,6)
(12,10)
Bio:
(1,10)
(1,8)
Note: The numbers 1 to 12 refer to alternatives. Their position in each
pair indicates the actual presentation (left and right, up and down) on
screen.
Two experiments were conducted in each domain of alternatives. In the
first experiment block designs were tested between subjects. In the second
experiment block designs were tested within subjects. It was hoped that
the test of block designs between and within subjects would help to vali¬
date results and increase the understanding of adapting choice behavior. In
Experiment 1A the block designs varied between subjects of Group S and
P, and in Experiment 2A, and 3A between subjects of Group S, P, and N.
Each group label corresponds to the third letter of the block design, that
is Group P stands for rePetition block design, Group S for reSolution block
design and Group N for raNdomized block design. In these experiments pair
comparisons under the same block designs were repeated over two or three
sessions. Note that the repetition of a whole pair comparison with the same
75
76
CHAPTER 3. EXPERIMENTS
block design can be understood as an extension of the same block design.
The block designs in Experiment 1B, 2B, and 3B varied within subjects over
successive sessions. Experiment 1B investigates the effect of exchanging ses
sions for the resolution and repetition block design in Group SP and PS. In
Experiment 2B and 3B it was tested if an exchange of the resolution and
repetition block design in the first and last session of Group SNP and PNS
has an effect.
3.2 Hypotheses and Statistical Testing
In the following the hypotheses for the experiments are summarized. In gen
eral it was predicted that individuals interact with block designs by adapting
to the sequence of choicetrials. How subjects interact with the block design
is believed to be highly domainspecific and hard to predict. Therefore, the
alternative hypothesis remained unspecified in the first experiment of each
domain and was tested against the null hypothesis of no effect of block design.
This leads to twotailed testing on a reduced significance level of 2.5 percent
(Winer, Brown & Michels, 1991). The following hypotheses were tested in
the first experiment of each domain.
(1) The resolution and repetition block design should lead to differences
in inconsistent choice behavior. Random blocks are a mixture of repetition
and resolution blocks. Therefore, inconsistency should reach an intermediate
level under the random block design.
(2) It is expected that over sessions increasing familiarity with task and
alternatives reduces inconsistency and shortens mean response times.
Conducting two experiments in each domain also had the purpose of con¬
firming results from the first experiment in the second. In particular, the
effect of block designs in the first experiment was used to specify the alter¬
native hypothesis in the first session of the second experiment. Therefore
planned (onetailed) ttests were conducted on a 5 percent significance level
to detect differences between block designs in the first session of the second
experiment.
From the construction of the block designs the following more specific pre
dictions about inconsistent choice behavior were made. It can be predicted
that under the resolution block design familiarity with initially unfamiliar
In a factorial design all possible sequences of block designs in sessions would have required nine groups.
In order to cut down the number of subjects, the random block design was only presented in the second
session of Experiment 2B and 3B.
77
3.2. HYPOTHESES AND STATISTICAL TESTING
alternatives is achieved earlier because all alternatives appear in each block.
Therefore, resolution blocks should facilitate exhaustive or independent in¬
formation processing over successive trials leading to more consistent choice
behavior. Accordingly, one would predict that under the repetition block de¬
sign familiarity with initially unfamiliar alternatives is achieved later because
all the alternatives appear only in every two blocks. Moreover, repetition of
multiattribute alternatives is likely to encourage selective information pro¬
cessing because subjects can reduce the cognitive effort when the same al¬
ternative appears in successive choicetrials. Thereby an important attribute
of the alternatives may have been disregarded which comes into play in a
later choicetrial. Consequently, a repetition block design should cause more
inconsistent choice behavior.
Furthermore, if selective information processing occurs in successive trials
then this should produce more and longer dicycles whereas complete informa¬
tion processing between trials should lead to less and shorter dicycles. More
specifically, the ear decomposition by sequence should produce directed ear
bases with more and longer ear dicycles under the repetition block design
than under the resolution block design.
The three block designs were compared to investigate the effect of re
peated alternatives in successive trials on inconsistent choice behavior. Ac¬
cordingly, measures of intransitive choice behavior should discriminate be¬
tween the three block designs. The experiments reported in this chapter em¬
ploy pair comparisons and statistical analyses focus on quantitative measures
of intransitive choice such as the number of 3dicycles, number of kdicycles
and number of ear dicycles of length k, (3 £ k £ n). Analyses of variance
were conducted on univariate measures of inconsistency (Kendall’s (, and
size of strong components) whereas stepwise discriminant analyses were ap
plied to find the best discriminatory variables among multivariate measures
of inconsistency (kdicycles and ear dicycles of length k). Further evidence
was expected from analyses of variance on preference reversals between pair
comparisons and mean response times.
At this point some remarks about the statistical methods appear neces¬
sary. Statistical testing of the number of kdicycles and ear dicycles creates
its own problems because the distributions for these variables are unknown.
The use of group data for statistical analysis has the advantage that the
mean number of kdicycles and ear dicycles is approximately normal dis¬
tributed regardless of their underlying distribution (Central Limit Theorem,
cf. Rice, 1988). The same argument applies to the analyses of mean response
78
CHAPTER 3. EXPERIMENTS
times. Considering the small sample sizes (N=10) in the experiments, it is
likely that this assumption was violated in some of the analyses. However,
the analysis of variance is known to be robust even if the assumption of a
normally distributed variable is not fulfilled (Winer, Brown & Michels, 1991).
Therefore, this assumption remained untested.
As the analysis of variance the discriminant analysis is related to lin¬
ear multiple regression and hypothesis testing requires normally distributed
variables and homogeneous covariances. The stepwise discriminant analy¬
sis selects a subset of quantitative variables to produce a good discrimination
model using stepwise selection (Klecka, 1980). The set of variables that make
up each class is assumed to be multivariate normal with a common covari¬
ance matrix. Variables are chosen to enter or leave the model according to
the following criterion: The significance level of an F test from an analysis
of covariance, where the variables already chosen act as covariates and the
variable under consideration is the dependent variable. Because the partial
Fstatistics in successive steps are not independent it is not advisable to use
critical values of the F distribution. In most applications, all variables con¬
sidered have some discriminatory power, however small. To choose the model
that provides the best discrimination using the sample estimates one has to
guard against estimating more parameters than can be reliably estimated
with the given sample size; hence, a moderate significance level of 15 percent
is appropriate.
Stepwise selection started like forward selection with no variables in the
model. At each step during the analysis the model was examined. If the
variable in the model that contributed least to the discriminatory power of
the model as measured by Wilk's A failed to meet the criterion to stay,
then that variable was removed. Otherwise, the variable not in the model
that contributed most to the discriminatory power of the model was entered.
When all variables in the model met the criterion to stay and none of the other
variables met the criterion to enter, the stepwise selection process stopped.
It is important to realize that in the selection of variables for entry, only
one variable can be entered into the model at each step. The selection process
does not take into account relationships between variables that have not yet
been selected. Thus, some important variables could have been excluded in
2Costanza and Afifi (1979) used Monte Carlo studies to compare alternative stopping rules for the forward
selection method in a twogroup multivariate normal classification problem. They conclude that the use of
a moderate significance level, in the range of 10 percent to 25 percent, often performs better than the use of
a much larger or a much smaller significance level. Hence, a significance level of 15 percent was the criterion
for entering or removing variables in stepwise discriminant analyses.
3.2. HYPOTHESES AND STATISTICAL TESTING
79
the process.
The stepwise discriminant analyses on the number of all dicycles and the
number of ear dicycles can be compared in terms of their discriminatory
power. It is hypothesized that the ear decomposition by sequence discrimi¬
nates better between block designs because the mean number of ear dicycles
have smaller variances and are based on the sequence of intransitive choices.
All statistical analyses were conducted using procedures of the statistical
package SAS/STAT (Version 6.0) on a VAX computer.
80
CHAPTER 3. EXPERIMENTS
3.3 Experiment 1A: Riskless Choice
In the first experiment it was investigated if individual choice behavior is af¬
fected by the resolution and repetition block design. More specifically, it was
asked if forced choice pair comparisons with successive binary choices display
different inconsistencies when a labeled or named alternative is repeated from
trial to trial (repetition block design) or when no alternative is repeated over
blocks of six choices (resolution block design).
3.3.1 Method
By using forced choice pair comparisons, subjects chose in two separate sessions between the
same set of named alternatives under different fixed block designs. The block designs were
constructed so that in each block either an alternative of the preceding trial was repeated in
the successive trial (repetition block design) or each alternative appeared only once in each
block (resolution block design).
Design
In a 2 x 2 design (block design by session) with repeated measurement on the second factor
two groups of subjects chose in forced choice pair comparisons with different block designs
in two consecutive sessions. The block designs were varied between subjects.
Group S: In the first and second session subjects chose under the reSolution block
design.
Group P: In the first and second session subjects chose under the rePetition block
design.
Dependent variables were inconsistencies and mean response times. Measures of inconsis¬
tency were derived from the choices in terms of preference reversals (Kendall's 7), intransitive
triples (Kendall's (), directed kcycles (coefficients z4 of polynomial y), and ear dicycles of
length k (of a directed ear basis). Kendall's r is a standardized measure for the number of
reversed preferences in a set of S = n alternatives. It is defined as
(3.1)
7 = 1 (4R/R.)
where R, = n(n  1), and R is the number of reversed relations (Kendall, 1970). Values
of Kendall's 7 may vary between 1.0 and +1.0, for the maximal and minimal number of
reversals, respectively.
Depending on S = n, the number of intransitive triples in a pair comparison can range
from T = 0 up to a maximal number Th of
n(n?1)
Ta =
for n odd and
24
n(n?—4)
for n'even.
(3.2)
Tn 
24
3.3. EXPERIMENT 1A: RISKLESS CHOICE
81
Kendall's ( is then defined as
(3.3)
(=1T/T.
where T is the observed number of intransitive triples or 3dicycles (Kendall, 1970; Kendall &
BabingtonSmith, 1940). The standardized measure of intransitive triples ( can range from
0 to 1.0, for maximal and minimal number of 3dicycles, respectively. Listings of programs
which recorded the number of reversals between pair comparisons, dicycles (expressed as
coefficients z4 for each pair comparison), and ear dicycles are presented in Appendix C.
Response times of 66 choicetrials were averaged for each subject and session and are
referred to as mean response times.
Subjects
A total of 20 subjects participated in this experiment. Half of them were students at the
University of Heidelberg and the other half had various occupations. They were assigned to
two groups: Group S (average 26.4 years of age, range 1833), Group P (average 25.5 years
of age, range 1732). Subjects in each group were balanced in gender. They received 12.
Deutsche mark per hour for their participation in an unrelated experiment which took place
between the two sessions.
Materials and Apparatus
Twelve chocolate bars and their names (see Table B.1) were presented to the subjects. If
the subject was unfamiliar with one of the chocolate bars then the experimenter offered a
slice of the chocolate bar to the subject who tasted it. During choicetrials only the names
for a pair of chocolate bars were displayed on a computer screen in 9 x 16 dot characters
with a refresh rate of 70 Hz. The screen intensity was adjusted to an easy reading level and
was maintained at this level throughout the experiment. The experiment was programmed
in MEL and run on an Tandon 386 (IBMAT compatible) computer allowing response time
measures from the keyboard with an accuracy of up to 68 milliseconds (Schneider, 1988;
1990).
Procedure
The subject had two sessions each with
= 66 choicetrials. The two sessions were
separated by an unrelated experiment in the field of social psychology which lasted between
half an hour and an hour. Before the first session all twelve chocolate bars were presented to
the subject. The subject was asked to taste any of the chocolate bars they were unfamiliar
with. The experimenter recorded which bars were tasted (see Table B.1). Subsequently,
the subject was seated in front of a computer screen and keyboard. First, the subject read
the general instruction explaining the experiment (Appendix B.1.1). On each choicetrial
the following question was displayed on screen: Which chocolate bar tastes better? Two
seconds later the names of two chocolate bars were displayed on the left and righthand
side of the screen and response timing was initiated. Then the subject chose either the
left or right alternative by pressing the 'F' or 'J' key with their left and right index finger,
respectively. Each session started with three training trials so that the subject knew how
to express their preference by pressing the appropriate key. Each session lasted between six
and twelve minutes.
82
CHAPTER 3. EXPERIMENTS
3.3.2 Results
This section is divided according to the three dependent variables response times, preference
cycles and preference reversals.
The hypotheses are summarized as follows: The resolution and repetition block design
should show different inconsistent choice behavior as measured by Kendall's ( and 7, number
of dicycles, and ear dicycles. In comparison the number of ear dicycles should discriminate
better between block designs than the total number of dicycles. Moreover, a decrease in
response times and inconsistency was expected over sessions.
Response Times
Mean response times were analyzed to compare groups and to test effects of learning and
familiarity over sessions. In Table 3.2 the mean and standard deviations of mean response
times are tabulated for each group and session.
Table 3.2: Mean Response Times of Choices (Exp 14)
Session 1
Session 2
Mean“
SD
SD
Mean
76
2.40
Group S (N=10)
1.45
43
2.50
Group P (N10)
1.83
82
55
Note: Statistical significant effect between Session 1 and 2,
F(1, 18) = 25.86, p = 0.0001).
ein seconds
For the two block designs mean response times are significantly longer in the first session
than in the second. On average the mean response times are reduced by 0.81 secs from 2.45
secs in the first session to 1.64 secs in the second.
Mean response times were entered into a 2 by 2 analysis of variance (ANOVA) with
repeated measurement on the second factor. Block design (resolution, repetition) served
as fixed factor between subjects, session (first, second) as fixed factor within subjects and
subjects acted as random factor (see Table 3.3 for detailed results).3 The analysis revealed
no statistically significant effect between block designs (FI1,18 « 1, ns). However, there
was a highly significant effect of session and the hypothesis of no effect across sessions is
rejected (F[1, 18 = 25.86, p = 0.0001). No other effects approached statistical reliability.
Preference Cycles
Table 3.4 lists the mean and standard deviation of Kendall's ( for each group and session.
Kendall's ( is a standardized measure of the number of intransitive triples (3dicycles) within
each preference matrix. As mentioned before, if there are no intransitive triples in a pair
3For the reason that only two levels of the withinsubject variable were present no test of homogeneity is
required. The normality assumption remained untested because the Fstatistic is considered as sufficiently
robust against violations and response times were averaged across 66 trials for each subject (Winer, Brown,
& Michels, 1991).
3.3. EXPERIMENT 1A: RISKLESS CHOICE
Table 3.3: ANOVA on Mean Resvonse Times (Exp 14)
df
S5
Source
MS
F

Between subjects:
581.2
1
581.2
Block Design (BD)
0.94
11120.3
Error
617.8
18
Within subjects:
6495.8 25.86***
1
6495.8
Session
1
BD by Session
0.78
195.7
195.7
Session by Error
4520.8
18
251.2
Note: ***p « 0.0001.
comparison then (
should equal 1, and 0 if there is a maximal number of intransitivities
(Kendall, 1970).
Table 3.4: Intransitive Triples of Preference (Exp 1A)
Kendall's (
Session 1
Session 2
SD
Mean
Mean
SD
89
90
Group S
15
09
06
97
96
Group P
04
Note: ( = 1 T/T, with T, = n(n2 4)/24 = 70 for n = 12, and
T number of 3dicycles.
In their first and second session subjects chose under the resolution block design in
Group S and under the repetition block design in Group P. In summary, the (values for the
first sessions suggest that the repetition block design leads to more consistent choices. On
the other hand, subjects displayed more intransitive triples if they chose under the resolution
block design. The same type of analysis as for mean response times was applied to Kendall's
( (Table 3.5).
Although the effect of block design nearly reached the significance level of 2.5 percent
the null hypothesis of no effect between block designs could not be rejected (F[1,18 = 5.86,
p = 0.026). No effect across sessions (FI1, 18 « 1, ns) was detected and no other effect
approached statistical significance.
Stepwise discriminant analyses were performed on the number of kdicycles, i.e. the
coefficients zy of the polynomial %, as well as the number of ear dicycles from the ear
decomposition by sequence." The number of kdicycles were entered stepwise as dependent
variables starting with dicycles who discriminated best between the two groups. Table 3.6
and 3.7 gives the results of the analyses. Variables were entered and excluded at a significance
level of a = 0.15.
Two Prolog programs are listed in Appendix C.1. The first detects all dicycles in a coded digraph and
the second the ear dicycles of an ear bases.
83
84
CHAPTER 3. EXPERIMENTS
Table 3.5: ANOVA on Kendall's ( (Exp 14)
85
Source
MS
4
E
Between subjects:
0.046
0.046
1
Block Design (BD)
5.86'
0.141
0.008
18
Error
Within subjects:
0.0003
0.03
Session
1
0.0003
BD by Session
0.00001
0.00001
0.00
1
Session by Error
0.0098
0.177
18
Note: *p « 0.05.
Table 3.6: Stepwise Discriminant Analysis on Dicycles (Exp 14)
Number
Partial F
Step Entered Removed
Partial R
P2F
Session 1
0.049
I
3cycles
0.199
4.477
1.—
2.650
0.135
0.122
2
2. 4cycles
Session 2
No variables entered
In Session 1 the two groups can be discriminated by the number of 3dicycles, and in
addition by the number of 4dicycles. No variable reached the significance level in Session 2.
The number of kdicycles for each subject and session are listed in Table B.2 and B.3 to¬
gether with their mean and standard deviation. A comparison of Subject 39 in Session 1
and Subject 45 in Session 2 illustrates that 3dicycles alone are not sufficient to describe
inconsistency in a pair comparison.
The two groups can be discriminated by the number of ear dicycles of length 3 and 4.
The considerably higher Fvalues in this analysis show that the use of ear dicycles instead
of dicycles adds discriminatory power to the model. Accordingly, the ear dicycles of length 3
also reached the significance level in Session 2. The number of ear dicycles for each subject
and session are listed in Table B.4 and B.5 together with mean and standard deviation.
Preference Reversals
A (twotailed) ttest was performed on the number of reversed preferences between the two
sessions using Kendall's 7 as dependent variable.° Kendall's r is a standardized measure for
the number of reversed preferences and may vary between values of —1 and +1.
In Table 3.8 the mean and standard deviation of Kendall's r are shown for both groups.
There was no statistical significant effect between the two groups (t = 1.62, df=18, p = 0.12).
In general, mean response times are significantly shorter for both groups in the second
session but there was no significant difference between block designs. The analyses on mean
response times and Kendall's ( suggest that no improvement of consistency took place over
5The number of preference reversals were detected by a program written in C listed in Appendix C.3.
3.3. EXPERIMENT 1A: RISKLESS CHOICE
Table 3.7: Stepwise Discriminant Analysis on Ear Dicycles (Exp 14)
1
Number
Removed
Step Entered
Partial R7
Partial
PE
Session 1
3dicycles
1.
9.90
0.355
0.006
2
0.374
4dicycles
2.
10.17
0.005
Session 2
0.140
2.94
0.104
1. 3dicycles
Table 3.8: Reversals of Preference (Exp 14)
Kendall's 7
Mean
SD
Group S
77
11
09
84
Group P
—
Note: 7 = 1  (4R/R») with R» = n(n1) = 132 for
n = 12, and R number of reversed preferences.
the two sessions. In their second session subjects performed quicker but not more consistent
than in their first session.
A nearly significant difference between the two block designs was detected. Subjects
who chose under the resolution block design (Group S) displayed a more inconsistent choice
behavior in terms of Kendall's ( than subjects under the repetition block design (Group P).
This was also reflected in the discriminant analyses on the number of dicycles and the number
of ear dicycles. It was shown that 3dicycles and 4dicycles together with ear dicycles of
length 3 and 4 distinguished between groups. The number of ear dicycles discriminated
considerably better between groups than the total number of dicycles. It is believed that
high familiarity with the alternatives may have contributed to the reversed intransitivities
under the two block designs:
As mentioned earlier, subjects who chose under the resolution block design encountered
all alternatives in a single block whereas under the repetition block design they made deci¬
sions involving all alternatives every two blocks. As a consequence, they should have been
able to establish a preference structure at an earlier stage under the resolution block design.
It was hypothesized that this results in more consistent choice behavior under the resolution
block design than under the repetition block design. However, the presentation and tasting
of all chocolate bars before the actual pair comparisons and the fact that alternatives were
nonartificial, highly familiar objects probably reversed the hypothesized effect.
It was hypothesized that the repetition of alternatives from one trial to the next would
facilitate selective information processing thereby increasing inconsistent choice behavior.
Although the alternatives in this domain may be considered as multiattribute as the taste of
chocolate bars is closely related to the ingredients of each chocolate bar (e.g. plain or milk
chocolate, peanuts, hazelnuts, caramel, raisins, waffle), it is argued that subjects employed
named alternatives in a simpler decision process. Encouraged by the presentation and tasting
85
86
CHAPTER 3. EXPERIMENTS
of chocolate bars before the choices, subjects might have used a preference value of the
alternatives rather than comparing the two alternatives in an information process. Hence,
inconsistency is not decreased under the resolution block design but consistency is increased
under the repetition block design because it is likely that the same preference values are
employed in a decision process if highly familiar alternatives are repeated in successive trial.
It is assumed that these arguments together with the fact that alternatives covered a wide
range of attractiveness can account for the highly consistent choice behavior under both block
designs and the significant difference between the number of ear dicycles of length 3 and 4.
This effect is a consequence of the advantage of the repetition block design over the resolution
block design in this domain of named and highly familiar alternatives.
3.4. EXPERIMENT IB: RISKLESS CHOICE
87
3.4 Experiment 1B: Riskless Choice
In Experiment 1A it was tested if inconsistency of individual choice is af¬
fected by the resolution and repetition block design. The results indicated
an advantage for Group P where subjects chose under the repetition block
design. This experiment investigates if this effect can be replicated when the
block designs are varied within rather between subjects. It was expected that
in the first session similar (values as in the first session of Experiment 1A
would occur and that reversed values would appear in the second session due
to the interchanged block designs.
3.4.1 Method
By using pair comparisons, subjects chose in two separate sessions between the same set of
named alternatives under different fixed block designs. The block designs were constructed
in the same way as in Experiment 1A.
Design
In a 2 x 2 design (sequence by session) with repeated measurement on the second factor, the
same subject chose under different pair comparisons in two consecutive sessions. The block
designs were varied within subjects and the sequence of the block designs between subjects.

Group SP: In their first session subjects chose under the reSolution block design and
in their second session under the rePetition block design.
Group PS: In their first session subjects chose under the rePetition block design and
in their second session under the reSolution block design.
Dependent variables were the binary choices and their response times. The same measures
of inconsistency as in Experiment 1A were employed.
Subjects A total of 22 subjects participated in this experiment.“ Half of the subjects were
students of the University of Heidelberg and the other half had various occupations. 10
subjects were assigned to each group: Group SP (average 29.0 years of age, range 2243
years), and Group PS (average 26.5 years of age, range 2138 years). The subjects for
each group were balanced in gender. They received 12. Deutsche mark per hour for their
participation in an unrelated experiment which took place between the two sessions.
Materials and Apparatus
The same twelve chocolate bars as in Experiment 1A (see Table B.1) were presented to the
subjects. Before the first pair comparison subjects were allowed to taste any of the chocolate
bars if they were unfamiliar with them. The same setup and equipment as in Experiment 1A
was employed. During choicetrials only the names of the chocolate bars were displayed on
screen.
6Two subjects were excluded and replaced because they were unfamiliar with more than five alternatives.
88
CHAPTER 3. EXPERIMENTS
Procedure
The same procedure as in Experiment 1A was applied. Each session lasted between six and
twelve minutes.
3.4.2 Results
This section is divided according to the dependent variables response times, preference cycles
and preference reversals. The hypotheses were the same as for Experiment 1A with the ex¬
ception that the alternative hypothesis for the effect of block design was specified and tested
against the null hypothesis. According to the findings in Experiment 1A choice behavior
should be more consistent under the repetition block design than under the resolution block
design and should appear in the first session.
Response Times
In Table 3.9 mean and standard deviations of mean response times are tabulated for both
groups and sessions.
Times of Choice Trials (Exp
1B)
Table 3.9: Mean Response
1
Session 2
Session
Mean
Mean"
SD
SD
—
89
2.94
2.16
51
Group SP (N=10)
1.19
2.94
Group PS (N=10)
2.05
61
Note: Statistical significant effect between Session 1 and 2,
F1, 18 = 28.3, p « 0.0001).
in seconds
For every group mean response times were significantly longer in the first session than in
the second. On average the mean response times were reduced by 0.84 secs from 2.94 secs
in the first session to an average of 2.1 secs in the second session.
Mean response times were entered into a 2 by 2 analysis of variance with repeated mea¬
sures on the second factor. Sequence of block designs (resolutionrepetition, repetition¬
resolution) served as fixed factor between subjects, session (first, second) as fixed factor
within subjects and subjects acted as random factor (see Table 3.10 for detailed results).
The analysis revealed no statistically significant effect between groups (F1,18 £ 1, ns).
However, the hypothesis of no effect across sessions was rejected (F 1,18 = 25.31, p =
0.0001). No other effects approached statistical reliability.
Preference Cycles
Table 3.11 shows the mean and standard deviation of Kendall's ( for each group and session
as a standardized measure of the number of intransitive triples in each pair comparison.
In the first session Group SP chose under the resolution block design whereas Group PS
made choices under the repetition block design. As in Experiment 1A the (values in the first
7For the reason that only two levels of the withinsubject variable were present no test of homogeneity is
required.
3.4. EXPERIMENT IB: RISKLESS CHOICE
Table 3.10: ANOVA for Mean Response Times (Exp 1B)
F
4
Source
55
MS
Between subjects:
29.9
Sequence (S)
1
29.9 0.02
Error
18
21550.7
1197.3
Within subjects:
28.31***
1
Session
7055.6
7055.6
S by Session
35.4
35.4
1
0.14
4485.5
Session by Error
249.2
18
Note: ***p = 0.0001.
Table 3.11: Intransitive Triples of Preference (Exp 1B)
Kendall's (
Session 1
Session 2
Mean
Mean
SD
SD
Group SP
94
98
08
02
Group PS
97
03
97
02
Note: ( = 1 T/T. with T, = n(n2 4)/24 = 70 for n = 12, and
T number of 3dicycles.
session suggest that the repetition block design enhanced consistent choice. On the contrary,
subjects displayed more inconsistent choice behavior under the resolution block design. In
the second session Group SP chose under the repetition block design and Group PS under
the resolution block design. There was no difference between the two block designs and
subjects displayed approximately the same (values.
The same analysis was applied to the number of intransitive triples for each preference
matrix using Kendall's ( as a standardized measure. Table 3.12 gives the results of the
analysis in full detail.
As can be expected from the results of Experiment 1A and the design of this experiment
there was no significant main effect between groups (F[1,18 « 1, ns) and no significant
effect between sessions (FI1, 18 = 2.11,p = 0.16). A planned ttest between the first session
of Group SP and PS did not reveal a significant effect of block design (minimal significant
difference = 0.056).
As in Experiment 1A stepwise discriminant analyses were performed on the number of
kdicycles, i.e. the coefficients z4 of the polynomial y, and the number of ear dicycles from
the ear decomposition by sequence. None of the variables reached a significance level of
p = 0.15 in Session 1 or Session 2. The number of kdicycles for each subject and session are
listed in Table B.6 and B.7. When comparing the number of kdicycles in Table B.2 with
B.3 only a few subjects had dicycles with length greater than 3. The number ear dicycles
together with means and standard deviation are presented in Table B.8 and B.9
89
90
CHAPTER 3. EXPERIMENTS
Table 3.12:
ANOVA for Kendall's ( (Exp 1B)
F
S5
Source
MS
4
Between subjects:
0.011
1
Sequence (S)
0.011
0.47
0.044
Error
18
0.002
Within subjects:
0.0037
Session
0.0037
1
2.11
1
0.0023
S by Session
1.28
0.0023
0.0317
18
0.0018
Session by Error
Preference Reversals
A (onetailed) ttest was performed for the number of reversed preferences between the two
sessions using Kendall's 7 as a standardized measure.
Table 3.13: Reversals of Preference
(Exp 1B)
7
Kendall's
Mean
SD
Group SP
08
82
81
08
Group PS
Note: r = 1  (4R/Rn) where R, = n(n  1) = 132
for n = 12, and R number of reversed preferences.
In Table 3.13 the mean and standard deviation of Kendall's r is shown for each group.
There was no statistical significant effect between groups (t = 0.25, df 18, p = 0.80).
As in Experiment lA the analysis of variance on mean response times suggests that
subjects performed quicker in the first session regardless of the block designs. From the
already high (values with small standard deviations in the first session it was expected
that only Group SP may improve consistency of their choice behavior in the second session.
Accordingly, the average (values increased from .94 to .98 in Group SP but remained at a
value of .97 in Group PS. If choice behavior were not adaptive then the block designs should
have an independent effect in each session and the (value for Group PS in the second session
should have dropped to a value around .94. The unchanged values point to the conclusion
that a more consistent preference structure persists and is hardly affected by the change of
block designs.
In the first session subjects in Group PS established a more consistent preference structure
while choosing under a repetition block design. The same explanation as in Experiment 1A
is offered for the differences between block designs. It is very plausible that the preference
structure from the first session was maintained in the second session leading to similar (¬
values and suppressing effects of the block design. This explanation is supported by the
comparable rvalues in Group SP and PS.
3.5. DISCUSSION
91
3.5 Discussion
The results for the first two experiments may indicate that contrary to the
original hypotheses no significant effect between block designs occurred for
Kendall's ( and that no improvement in consistency took place over the two
sessions. Subjects performed quicker but not more consistent in the second
session regardless of the block designs.
However, from the already high (values with small standard deviations
in the first session of Experiment 1B it can be expected that only Group SP
would improve in consistency due to a ceiling effect. Accordingly, the average
(values increased from .94 to .98 but remained at .97 for Group PS.
Session 1
Session 2
1
0.8
0.6
0.4
0.2
9
SP
PS
Group
Figure 3.4: Kendall's ( for all groups of Experiment 1A and 1B
In line with our hypothesis about the effect of the block designs, there
were systematic lower average (values for the resolution blocks of Group S
and SP compared to the repetition blocks of Group P and PS in the first
session. This effect also appeared for 4dicycles in Experiment 1A as shown
by the stepwise discriminant analysis on dicycles and ear dicycles. The same
difference was present in the second session of Experiment 1A but did not re¬
verse when the block designs were interchanged between groups in the second
session of Experiment 1B. Therefore, improvement in consistency is regarded
as irreversible unless more drastic alterations are introduced.
92
CHAPTER 3. EXPERIMENTS
From these results the following question arises: Why did subjects exhibit
less intransitivities under the repetition block design when a named alterna
tive is repeated from trial to trial? As mentioned earlier, the answer probably
lies in the way these highly familiar alternatives were perceived and compared.
It was claimed that under the resolution block design early familiarity with
multiattribute alternatives leads to more exhaustive and therefore more in¬
dependent decision processes facilitating consistent choice whereas under the
repetition block design information processing is more likely to be selective
when the same alternative is repeated in successive choicetrials. Selective
information processing would lead to more inconsistent choices.
In the domain of familiar and named alternatives, however, the repetition
of one alternative from trial to trial might facilitate consistent choice behavior
because the subject simplifies the decision process by recalling preference val¬
ues only. It is emphasized that identification of alternatives is a preliminary
and necessary part of the decision process in the domain of named alterna¬
tives. The identification is likely to be accompanied by a recall of preference
values if the alternatives are sufficiently familiar.
More specifically, it is believed that in the domain of chocolate bars iden¬
tification led to reduced inconsistencies in a repetition block design because
names of chocolate bars were familiar and easily recognizable. In fact, in
each choicetrial the alternatives had to be identified by their names before a
choice could be made. Their preference value" could have been recalled from
preceding trials simplifying the decision process. It is also likely that sub¬
jects already evaluated the chocolate bars when they were introduced to all
alternatives and tasted unfamiliar bars at the beginning of the first session.
The main objective of our investigation was to identify inconsistency as a
discriminatory variable of individual choice behavior. The systematic differ¬
ences between block designs for 3 and 4dicycles in Experiment 1A illustrate
that at least some subjects interacted with the arrangement of the choice
trials. This contradicts the assumption of independent choicetrials which
is essential to traditional probabilistic or nonadaptive algebraic models. Its
violation supports the algebraic decomposition approach in this domain, even
though results were not convincingly confirmed in Experiment 1B.
3Note that the term preference value may be replaced by holistic evaluation', utility', information
integration', or 'accumulative value' depending on the terminology of the theoretical framework it refers to.
3.6. EXPERIMENT 2A: RISKY CHOICE
93
3.6 Experiment 24: Risky Choice
In the following experiment it was tested if the inconsistency of choice behav¬
ior is affected by different block designs in the domain of risky alternatives.
The study of decision making under risk or uncertainty, mostly by using
gambles or lotteries, has a long tradition and still is a popular reserach topic.
The design of the previous experiments was extended. In addition to
the fixed resolution and repetition block design a random block design was
introduced which should have an intermediate effect on inconsistency. Each
subject had three sessions and the alternatives were generated so that they
were almost indifferent in terms of expected utility.
3.6.1 Method
By using pair comparisons subjects chose in three sessions between the same set of lotteries"
varying in probability and amount of winning. The fixed block designs, repetition and
resolution block design, were constructed as in Experiment 1A and 1B. The random block
design had a sequence of choicetrials which was randomized for each subject and each pair
comparison.
Design
The experiment had a 3 x3 design with repeated measurement on the second factor. In three
consecutive sessions subjects chose between risky alternatives in different block designs. The
three different block designs were varied between subjects:
Group S: In all three sessions subjects chose in a reSolution block design.
Group P: In all three sessions subjects chose in a rePetition block design.
Group N: In all three sessions subjects chose in a raNdom block design.
Dependent variables were the choices and their response times as well as different measures
of inconsistency in terms of preference cycles and preference reversals.
Subjects
A total of 30 undergraduate and graduate students were recruited from the subject panel
of the Department of Experimental Psychology, Oxford University. There were 10 subjects
in each group (Group S: average age 24.5 years, SD=5.21; Group P: average age 27.3 years,
SD=6.43; Group N: average age 21.9 years, SD 3.21). The subjects for each group were
balanced in gender. Each subject received £3 per hour plus traveling expenses for their
participation in this experiment and in Experiment 3A or 3B. In each experiment subjects
were assigned to different groups.
"Gambles are usually called lotteries if the decision maker is provided with probabilities rather than
uncertain events and if the consequences are stated in terms of money.
94
CHAPTER 3. EXPERIMENTS
Materials and Apparatus
Twelve lotteries with equivalent expectancy values were constructed by varying probability
of winning (in percentages) and payoff (in pounds). Any possible loss x2 was fixed to fl
with probability 1  p whereas the probability of winning p and the win itself x, was varied
to gain an approximate constant expectancy value of £9.1 (see Table B.10). The expectancy
values of the lotteries varied around £9.1 with standard deviation SD=0.28.
E(X) = pri + (1p)(1) = 9.1
(3.4)
The probabilities for each lottery were displayed in a pie chart with percentages written out.
A legend above the pie chart gave information about possible win and loss in pounds. During
choicetrials two lotteries differing in probability and payoff were displayed on a 21 inch Two¬
Page Monochrome Macintosh computer screen as illustrated in Figure 3.5. In Figure 3.6 the
255.50
629.90
E1
8:
Figure 3.5: Display of two lotteries in a single choicetrial
payoff of the twelve lotteries is plotted on a logarithmic scale (in log £) against the chance of
winning (in percent). If the probability scale were logarithmic, too, then the fitted function
would be linear. The experiment was programmed in SuperLab and run on a Macintosh Ilcx
(enhanced by a floating point unit) allowing time measures on the keyboard with accuracy
of up to 4 millisecs.
Procedure
/12)
= 66 choicetrials plus 3 training trials with
Each subject had three sessions with
2
different alternatives. Sessions were separated by a session of Experiment 3A or 3B which
lasted between six and twelve minutes.
The subject was seated in front of the computer screen and the keyboard. First, they
read a general instruction about the experiment, followed by more detailed instructions (see
Appendix B.2.1). In three training trials they learned how to give a response by pressing key
1°The 12 lotteries were selected from a pool of 24 lotteries. In a pretest four additional subjects computed
roughly the expectancy value for each of the 24 lotteries following an instruction. Only lotteries which had
on average approximately the same estimated expectancy value were selected.
3.6. EXPERIMENT 24: RISKY CHOICE
a
100
80
3
60
5
5
40
à
20
10
100
Payoff in £ (log scale)
Figure 3.6: Lotteries plotted by payoff and chance of win on a
loglinear scale. The fitted function is linear when plotted on
loglog scales.
’D' or 'K' on the keyboard. In each choicetrial the following question was displayed first:
Which gamble do you prefer to play?". After 3.0 seconds two lotteries were displayed on
the left and righthand side of the screen and response timing was initiated. The two lotteries
stayed visible until the subject chose the left or right lottery by pressing the appropriate key.
Each session lasted between ten and fifteen minutes.
3.6.2 Results
This section is divided according to the dependent variables response times, intransitive pref¬
erences, size of strong components, and preference reversals. The size of strong components
is an example of an additional consistency measure which can be derived from the algebraic
decompositions.
The hypotheses can be summarized as follows: The resolution and repetition block design
should show different consistent choice behavior as measured by Kendall's ( and 7, number
of dicycles and ear dicycles. The random block design should lead to intermediate consistent
choices. In comparison the number of ear dicycles should discriminate better between block
designs than the total number of dicycles. Moreover, a decrease in response times and
inconsistency can be expected over sessions.
11The question remained on the screen for 1.7 seconds.
95
96
CHAPTER 3. EXPERIMENTS
Response Times
As in the previous experiments mean response times and standard deviations are listed in
Table 3.14 for each group and session.
Mean Response Times of Choices (Exp
Table 3.14:
24)
Session 1
Session 2
Session
3
Mean"
Mean
SD
SD
Mean
SD
2.69
3.33
Group S (N10)
2.16
2.20
2.17
1.48
3.28
4.15
Group P (N=10)
2.27
2.32
2.87
2.12
4.13
2.07
Group N (N10)
2.79
1.01
1.37
87
Note: Statistical significant effect between sessions,
= 36.79, p
Fs2,541
0.0001) adjusted for heterogeneity.
in seconds
The mean response times were significantly shorter in the second and third session than
in the first session. On average mean response times were reduced by 0.95 secs from Session 1
to 2 and again by 0.54 secs from Session 2 to 3.
Mean response times were entered into a 3 by 3 analysis of variance (ANOVA) with
repeated measurement on the second factor. Block design (resolution, repetition, random)
served as fixed factor between subjects, session (first, second) as fixed factor within subjects
and subjects acted as random factor (see Table 3.15 for complete results).
Table 3.15: ANOVA for Mean Response
Times (Exp 24)
Source
MS
F
85
4
Between subjects:
737.7
368.9
Block Design (BD)
0.40
2
915.6
Error
24720.2
27
Within subjects:
1711.5 36.79***
3422.9
2
Session
271.0
BD by Session
1.46
67.8
4
46.5
2511.8
Session by Error
54
Note: ***p = 0.0001.
A sphericity test revealed a significant violation of homogeneity (approximate y’2 =
16.76,p « 0.0002) so that a conservative test statistic is recommended. Accordingly, degrees
of freedom were adjusted by e = 0.754 as suggested by Huynh and Feldt (1976) and Huynh
(1978).
The analysis showed no statistical significant effect between block designs (Fs2,27 « 1,
ns). However, the hypothesis of no effect across sessions was rejected (Fs2,54 = 36.79, with
p « 0.0001 adjusted for heterogeneity). No other effects approached statistical reliability.
3.6. EXPERIMENT 2A: RISKY CHOICE
97
Preference Cycles
The mean (values are listed in Table 3.16 for each block design and session. The three
groups show distinctive average (values in the first session. During the first session subjects
in Group S made choices under a resolution block design reaching average (values of .64,
whereas subjects in Group P had a value of .48. As expected, the average (value of 56 for
subjects choosing in the random block design lies in between.
Table 3.16: Intransitive Triples of Preference (Exp 24)
Kendall's (
Session
3
Session 1
Session
2
Mean
SD
SD
SD
Mean
Mean
70
25
20
64
16
77
Group S
28
64
65
22
Group P
48
24
Group N
56
71
73
26
29
27
Note: ( = 1  T/T, with T, = n(n2  4)/24 = 70 for n = 12, and T
number of 3dicycles.
The same analysis as for mean response times was applied to the number of intransitive
triples for each preference matrix using Kendall's ( as a standardized measure. A sphericity
test revealed a significant violation of homogeneity for the repeated measurements over
sessions (approximate y’2 = 15.66,p « 0.001) and degrees of freedom were adjusted by
é = 0.767.
Table 3.17: ANOVA for Kendall's ( (Exp 24)
55
F
MS
Source
4
Between subjects:
0.191
0.095
2
Block Design (BD)
0.85
27
0.112
3.015
Error
Within subjects:
6.88**
0.196
0.393
2
Session
0.028
BD by Session
0.007
0.25
4
0.029
54
1.542
Session by Error
Note: **p « 0.01.
There was no significant effect between block designs (F2,27 « 1, ns) but the analysis
revealed a highly significant effect across sessions (Fs2,54 = 6.88, p = 0.005, adjusted for
heterogeneity). Table 3.17 shows the detailed results of the analysis.
For each session stepwise discriminant analyses were performed on the total number of
dicycles, i.e. the coefficients z of the polynomial y, and number of ear dicycles from the ear
decomposition by sequence. The number of kdicycles were selected stepwise as dependent
variables starting with dicycles whose number discriminated best between groups. For the
98
CHAPTER 3. EXPERIMENTS
analysis of all dicycles no variables could be entered in any of the sessions. The number of
kdicycles for each subject and session are listed in Table B.11 for Group S, in Table B.12
for Group P and in Table B.13 for Group N characterizing all intransitivities in the pair
comparisons. A comparison of polynomials with the same z3 coefficients, i.e. number of
3dicycles, illustrates that in some cases the coefficients z4 with k » 3 can be considerably
different.
The stepwise discriminant analysis on the number of ear dicycles revealed differences
between groups. The results are summarized in Table 3.18. The ear dicycles of length 10
24)
Table 3.18: Stepwise Discriminant Analysis on Ear Dicycles (Exp
Removed
Number
Partial R
Partial F
P2F
Step Entered
Session 1
1
0.286
0.011
5.40
1. 10dicycles
Session 2
2.21
0.141
0.129
1. 10dicycles
Session 3
0.159
2.54
0.097
1. 3dicycles
—
had considerable discriminatory power in the first session but less so in the second session.
Ear dicycles of length 3 did also discriminate in the last session. The number of ear dicycles
from the ear decomposition by sequence for each subject and session are listed in Table B.14
for Group S, in Table B.15 for Group P, and in Table B.16 for Group N.
Strong Components
The average size of strong components was determined for each pair comparison (see Ta¬
ble 3.19) as an additional consistency measure. The size of strong components can be derived
from the ear decomposition and a larger size indicates inconsistent choices between more al¬
ternatives.
Table 3.19: Average Size of Strong Components (Exp 24)
3
Session 2
Session
Session 1
Mean
SD
SD
Mean
Mean
SD
3.77
6.77
7.43
9.77
4.78
Group 5
4.92
3.25
8.87
9.22
4.67
Group P
4.23
10.97
5.66
8.04
5.61
Group N
4.59
4.62
4.26
An analysis of variance was conducted on the average size of strong components for each
pair comparison per session. Subjects from the three different groups were compared over
three sessions leading to a 3 by 3 design with repeated measurement on the second factor as
in the previous analyses.
12A sphericity test revealed no significant violation of homogeneity with approximate x?21 = 1.79,p 
0.41.
3.6. EXPERIMENT 2A: RISKY CHOICE
Table 3.20: ANOVA
for Size of Strong Components (Exp 24)
SS
F
Source
af
MS
Between subjects:
Block Design (BD)
158.6
79.3
2.28
2
940.9
27
Error
34.8
Within subjects:
4.91*
2
110.7
Session
55.4
BD by Session
0.7
.06
2.8
4
11.3
Session by Error
54
608.9
Note: **p = 0.01.
As for the analysis of the (values, there was no significant effect between block designs
(F2,27 = 2.28, p = 0.12) but the analysis revealed a signifiçant effect across sessions
(F2,54 = 4.91, p = 0.01). Table 3.20 provides the results of the analysis in detail.
Preference Reversals
The number of preference reversals which occurred between each of the three sessions are
listed in Table 3.21 using Kendall's r as a standardized measure. Table 3.21 shows the
pattern of rvalues for the three comparisons indicating that more reversals occurred between
Session 2 and 3 than Session 1 and 2 or Session 1 and 3.
Table 3.21: Reversals
of Preference (Exp 24)
Kendall's 7
Session 2 vs 3
Session 1 vs 3
Session
1 vs 2
SD
SD
SD
Mean
Mean
Mean
58
51
23
25
Group S
21
47
27
34
34
52
32
Group P
34
68
43
29
22
48—
1
20
Group N
n(n 1) = 132 for n = 12, and R
Note: r = 1  (4R/Rn) where Rn =
number of reversed relations.
An analysis of variance with repeated measurement was performed on the number of
preference reversals which occurred between each of the three sessions. Kendall's r served as
dependent variable. A sphericity test revealed a significant violation of homogeneity for the
repeated measurements (approximate y° 2 = 13.40, p « 0.001) and the degrees of freedom
were adjusted by e = 0.797.
Again, there was no statistical significant effect between block designs (Fs2, 27 =1.04,
p0.37) but there was a highly significant effect for the three pairwise comparisons of sessions
(Fs2, 54 =12.23, p=0.0002, adjusted for heterogeneity).
Although not significant the different rvalues between block designs correspond to the (¬
values and number of ear dicycles. Especially in Session 1 the pattern of results indicates that
99
100
CHAPTER 3. EXPERIMENTS
Table 3.22: ANOVA for Preference Reversals (Exp 24)
S8
MS
Source
4
Between subjects:
0.155
Block Design (BD)
0.310
1.04
Error
27
4.042
0.150
Within subjects:
9***
12.23
2
0.551
Combination (C)
0.276
BD by C
0.072
0.018
.79
4
C by Error
0.023
1.217
54
Note: ***p = 0.0002.
subjects chose more consistent under the resolution block design (Group S), less consistent
under the random block design (Group N), and worst under the repetition block design
(Group P).
In general, mean response times are significantly shorter for all three groups in the second
and third session. As for Experiment 1A there was no significant difference between block
designs. The analyses of mean response times and Kendall's ( suggest that some form of
improvement in consistency took place over the three sessions. Subjects performed quicker
and more consistent in the second and third session. This dynamic was also reflected in
the significantly smaller size of strong components in later sessions. Surprisingly, the steady
improvement over sessions was accompanied by a different number of reversals between ses¬
sions. This means that despite the increasing (values over Session 1, 2, and 3 the adjacency
matrices in Session 3 were more similar to the matrices in Session 1 than to the matrices in
Session 2 as shown by the values of 7.
No significant difference between the block designs was detected except for the number of
ear dicycles. There is a tendency in the first two sessions that subjects who chose under the
resolution block design in Group S displayed a more consistent choice behavior especially in
terms of longer ear dicycles than subjects under the random block design in Group N and
the repetition block design in Group P.
As mentioned earlier, subjects who chose under the resolution block design encountered
all alternatives in every single block of six choicetrials whereas under the repetition block
design they encountered all alternatives in every two blocks. As a consequence subjects may
have established a preference structure earlier when choosing under the resolution blocks.
Hence, it can be expected that subjects are choosing more consistent under a resolution
block design.
The alternatives in this domain are explicitly defined by their chance of winning and
their payoff. After a sufficient number of trials it is possible that lotteries were identified
by their pie charts thereby simplifying the choice process and facilitating consistency in the
repetition blocks as in Experiment 1A and 1B.
3.7. EXPERIMENT 2B: RISKY CHOICE
101
3.7 Experiment 2B: Risky Choice
The experimental design was changed similar to the design in Experiment 1B.
The three block designs were varied within subjects and two different se
quences of block designs were investigated. From the results of Experi¬
ment 2A a similar effect of the block designs on the measures of inconsistency
and a comparable learning effect over sessions was expected.
3.7.1 Method
The same method and equipment as in Experiment 2A was employed.
Design
The experiment had a 2 x 3 design (sequence by session) with repeated measurement on the
second factor. In three consecutive sessions each subject chose between risky alternatives
under different block designs. Two different sequences of block designs in three sessions were
tested in Group SNP and PNS:
Group SNP: In the first session subjects chose under a reSolution block design, in the
second under a raNdom block design, and in the third under a rePetition block design.
Group PNS: In the first session subjects chose under a rePetition block design, in the
second under a raNdom block design, and in the third under a reSolution block design.
Dependent variables were the choices and their response times as well as measures of incon¬
sistency derived from preference cycles and preference reversals.
Subjects
A total of 20 undergraduate and graduate students were recruited from the subject panel of
the Department of Experimental Psychology, Oxford University. There were 10 subjects in
each group (Group SNP: average age 24.9 years, SD 4.09; Group PNS: average age 21.5 years,
SD 3.5). Subjects for both groups were balanced in gender. Each subject received £3 per
hour plus traveling expenses for their participation in this experiment and in Experiment 3A
or 3B in a different group.
Materials and Apparatus
The same stimuli (lotteries) and the same equipment as in Experiment 2A was used.
Procedure
The same procedure as in Experiment 2A was employed. Each session lasted between ten
and fifteen minutes.
CHAPTER 3. EXPERIMENTS
102
3.7.2 Results
This section is divided according to the dependent variables response times, preference cycles,
size of strong components, and preference reversals. The hypotheses were the same as for
Experiment 2A with the exception that the alternative hypothesis for the effect of block
design was specified and tested against the null hypothesis. According to the findings in
Experiment 2A choice behavior in the first session should be more consistent under the
resolution block design than under the random block design and more consistent under the
random block design than under the resolution block design.
Response Times
Table 3.23 shows mean response times and standard deviations for each group and session.
As in Experiment 2A mean response times were significantly shorter in the second and third
session than in the first session. On average mean response times are reduced by 1.7 secs
from Session 1 to 2 and again by 1.23 secs from Session 2 to 3.
Response Times of Choices
(Exp
2B)
Table 3.23: Mean
Session 1
Session 2
Session 3
SD
SD
Mean SD
Mean"
Mean
—
2.43
1.86
Group SNP (N10)
4.54 4.41
5.69
4.99
Group PNS (N10)
3.23 1.74
2.88 1.82
5.48
4.52
Note: Statistical significant effect between sessions, Fs2,36  8.40, p « 0.005
adjusted for heterogeneity.
ein seconds
Mean response times were entered into a 2 by 3 analysis of variance (ANOVA) with
repeated measurement on the second factor. Sequence of block designs (resolutionrandom¬
repetition, repetitionrandomresolution) served as fixed factor between subjects, session
(first, second, third) as fixed factor within subjects and subjects acted as random factor (see
Table 3.24).
Times (Exp 2B)
Table 3.24: ANOVA for Mean Response
F
MS
88
4
Source
Between subjects:
303.7
0.28
303.7
1
Sequence (S)
1082.9
19491.5
18
Error
Within subjects:
24.97**
1182.7
2365.5
2
Session
2
98.1
1.04
49.1
S by Session
1705.4 36
47.4
Session by Error
Note: ***p = 0.0001.
3.7. EXPERIMENT 2B: RISKY CHOICE
103
A sphericity test revealed a significant violation of homogeneity (approximate y22 =
15.29,p « 0.001) and the degrees of freedom were adjusted by e = 0.69. The analysis
showed no statistical significant effect between groups (F[1, 18 « 1, ns) but the hypothe¬
sis of no effect across sessions was rejected (Fs2,36 = 8.40, with p = 0.004 adjusted for
heterogeneity).
Intransitive Preferences
Mean (values for each group and session are displayed in Table 3.25. During the first session
subjects in Group SNP made choices in a resolution block design resulting in an average (¬
value of .65 whereas subjects in Group PNS reached a value of .41. A planned ttest revealed
a significant difference between the two values (minimal significant difference = 0.199).
Table 3.25: Intransitive Triples of Preference (Exp 2B)
Kendall's
Session 1
Session 2
Session 3
Mean
Mean
Mean
SD
SD
SD
71
28
Group SNP
65
23
29
82
64
26
Group PNS
24
I1
55—
41
—
—
Note: ( = 1 T/T, with T, = n(n2  4)/24 = 70 for n = 12, and T
number of intransitivities.
The same type of analysis as for mean response times was applied to number of in¬
transitive triples in each preference matrix using Kendall's (as a standardized measure (see
Table 3.26). A sphericity test revealed a significant violation of homogeneity for the repeated
measurements of factor session (approximate y22 =6.50, p0.039) and degrees of freedom
were adjusted by e = 0.767.
Table 3.26:
ANOVA for Kendall's ( (Exp 2B)
F
4
S5
Source
MS
Between subjects:
1
0.536
0.536
4.12
Sequence (S)
2.341
0.130
Error
18
Within subjects:
9.06***
0.199
0.397
2
Session
0.36
0.016
0.007
S by Session
2
0.022
36
0.789
Session by Error
Note: ***p « 0.001.
As before, there was a significant effect across sessions (F2, 36 =9.06, p=0.001, adjusted
for heterogeneity) but no significant effect between groups (F1, 18 =4.12, p0.057)
For each session stepwise discriminant analyses were performed on the number of dicycles
i.e. the coefficients zy of the polynomialy, and the number of ear dicycles derived from the eai
CHAPTER 3. EXPERIMENTS
104
decompositions by sequence. The number of kdicycles were entered stepwise as dependent
variables starting with dicycles whose number discriminated best between groups. Table 3.27
and Table 3.28 show the results of the analyses. Variables were entered and excluded at a
significance level of p « 0.15.
Table 3.27: Stepwise Discriminant Analysis on all Dicycles (Exp 2B)
Number
Partial R' Partial F
Step Entered Removed
P2E
Session
1
0.254
6.121
1 3cycles
0.024
—
Session 2
No variables
entered
Session
3
2.538
0.129
0.124
1 3cycles
In Session 1 and 3 the two groups can be discriminated by the number of 3dicycles
whereas in Session 2 no variable could be entered. The number of kdicycles for each subject
and each session are listed in Table B.17 for Group SNP and in Table B.18 for Group PNS
summarizing all inconsistencies within pair comparisons. A comparison of polynomials with
equal coefficients z3, i.e. number of 3dicycles, shows that in some cases the coefficients z4
for k » 3 are considerably different.
A stepwise discriminant analysis on the number of ear dicycles also showed differences
between groups. The results are summarized in Table 3.28.
(Exp 2B)
Table 3.28: Stepwise Discriminant Analysis on Ear Dicycles
Number
F
Removed
Partial R Partial
P2F
Step Entered
Session 1
0.002
0.412
1.—
12.62
8dicycles
Session 2
6.60
0.268
1. 7dicycles
0.019
Session 3
5.76
0.243
0.027
1. 6dicycles
The ear dicycles of length 8 discriminated best between groups in the first session. Ear
dicycles of length 7 discriminated in the second session and finally ear dicycles of length 6
discriminated in the last session. The number of ear dicycles from the ear decomposition
by sequence for each subject and session are listed in Table B.19 for Group SNP, in and in
Table B.20 for Group PNS.
Strong Components
The average size of strong components was determined for each pair comparison as an
additional consistency measure which can be derived from the algebraic decompositions
(Table 3.29).
An analysis of variance was conducted on the average size of strong components for each
pair comparison per session. Subjects from the two different groups were compared over
3.7. EXPERIMENT 2B: RISKY CHOICE
105
Table 3.29: Average Size of Strong Components (Exp 2B)
3
Session 1
Session 2
Session
SD
SD
Mean
Mean
SD
Mean
3.76
6.64
6.17
Group SNP
4.39
4.69
5.20
6.90
8.70 3.59
Group PNS 11.40 1.90
4.58
three sessions leading to a 2 by 3 design with repeated measurement on the second factor as
in the previous analyses.!s
Table 3.30: ANOVA for Size of Strong Components
(Erp 2B)
MS
Source
38
4
Between subjects:
181.2
181.2
Sequence (S)
4.93*
1
660.9
36.7
Error
18
Within subjects:
8.48**
Session
2
68.6
137.1
6.6
2
S by Session
0.8
13.2
8.1
36
291.1
Session by Error
Note: *p = 0.04,***p = 0.001.
In contrast to the analysis on Kendall's ( a significant effect of sequence (F[1, 18 = 4.93,
p = 0.04) and a highly significant effect of session (Fs2,36 = 8.48, p = 0.001) was detected.
Table 3.30 gives detailed results of this analysis.
Preference Reversals
The pattern of rvalues, as shown in Table 3.31, indicates that more preference reversals
occurred between Session 2 and 3 than Session 1 and 2 or Session 1 and 3. The significant
difference of rvalues for reversals between Session 1 to 2 corresponds with the (values and
supports the finding that subjects chose more consistent under the resolution block design
in Group SNP and less consistent under the repetition block design in Group PNS.
An analysis of variance with repeated measurement was performed on the number of
preference reversals which occurred between each pair of the three sessions. Kendall's r
is a standardized measure for the number of reversals and served as dependent variable. A
sphericity test showed no significant violation of homogeneity for the repeated measurements
(approximate x22 = 4.24,p = 0.12).
Despite the nonsignificant overall test betweensubjects (F]1,18 = 4.18, p = 0.056) a
Scheffe test showed a significant difference (minimal significant difference = 0.207) between
the rvalues 54 and 30 for Group SNP and PNS, respectively. As for Experiment 2A there
was a highly significant effect for the three combinations of sessions (F2,54 = 10.56,p =
0.0003).
13A sphericity test showed no significant violation of homogeneity with approximate x?21  3.79, p  0.15.
106
CHAPTER 3. EXPERIMENTS
Table 3.31: Reversals of Preference (Exp 2B)
Kendall's 7
Session 1 vs 2
Session 2 vs 3
Session 1 vs 3
SD
Mean
SD
Mean
Mean
SD
27
—
52
92
28
54
Group SNP
65
—
17
32
50
30— 13
Group PNS
22
Note: 7 = 1  (4R/R,) where R, = n(n 1) = 132 for n = 12, and R
number of reversed relations.
Table 3.32: ANOVA for Preference Reversals (Exp 2B)
F
df
85
Source
MS
Between subjects:
1
0.624
0.624
Sequence (S)
4.18
Error
0.149
18
2.687
Within subjects:
10.56***
2
0.270
0.135
Combination (C)
S by C
0.008
0.004
.32
2
0.013
0.461
36
C by Error
***p = 0.0003.
Note:
As in Experiment 2A mean response times were significantly shorter in the second and
third session but displayed no significant difference between groups. The analyses on mean
response times and Kendall's ( suggest that some form of learning or improvement in consis¬
tency took place over the three sessions. Regardless of the block designs, subjects performed
quicker and also more consistently in the second and especially in the third session.
An analysis of the size of strong components showed a significant effect for the sequence
of block designs. The effect is most likely due to the large difference between the block
designs in the first session and corresponds to the decreasing length of discriminating ear
dicycles over sessions.
Confirming the results of Experiment 2A similar differences between block designs ap¬
peared in the first session of both experiments. Subjects who chose under the resolution
block design in Group SNP displayed a more consistent choice bchavior in terms of intransi¬
tive triples and ear dicycles than subjects under the repetition block design in Group PNS.
Accordingly, groups differ by the number of intransitive triples in the first session and more
significantly by the number of ear dicycles. In Session 3 (values for Group SNP are still
higher than for Group PNS, suggesting that as in Experiment 1B improvement in consistency
is irreversible.
If consistent choice persists and cannot be reversed by block designs then the significant
difference of preference reversals between Session 2 and 3 also supports the finding that
subjects chose more consistently under resolution blocks in the first session.
3.8. DISCUSSION
107
3.8 Discussion
In both experiments mean response times decreased considerably over ses¬
sions. Subjects in each group learned to solve the decision task quicker due
to increasing familiarity with the task and the alternatives. At the same time
the number of intransitivities decreased across sessions. This result suggests
that some form of learning or improvement took place so that from session to
session subjects were able to choose more consistently in later sessions. It is
assumed that the strong effect across sessions covers up differences between
block designs, especially in the second and third session of Experiment 2A.
Nevertheless, in both experiments the measures of inconsistency in terms of
(values, size of strong components, number of dicycles, and number of ear
dicycles show marked differences between block designs in the first session.
Session 1
Session 2
Session 3
0.8
6
0.4
0.2
PNS
SNP
Group
Figure 3.7: Kendall's ( for all groups of Experiment 2A and 2B.
The differences between block designs in the first sessions were strongest
for ear dicycles confirming the predictions for the effect of block designs on
individual choice behavior. These effects are interpreted as follows: Under
the resolution block design the subject encountered all twelve unfamiliar al¬
ternatives in each block of six choices. This enabled the subject to establish a
more consistent set of preferences compared with the repetition block design
where only half of the alternatives are introduced in each block.
108
CHAPTER 3. EXPERIMENTS
Furthermore, the repetition of multiattribute alternatives possibly favored
selective information processing which leads to more inconsistent choices.
The resolution blocks on the other hand should have enhanced more complete
and therefore independent information processing resulting in more consistent
choices.
Although pointing in the opposite direction, the differences between block
designs in the first session strengthen the conclusion from Experiment 1A
and 1B that subjects interact with the arrangement of choices. This was
reflected in all measures of inconsistency. As in Experiment 1A and 1B
this finding is incompatible with classical algebraic or probabilistic models.
The highly significant effect of session is also incompatible with algebraic or
probabilistic decision models and supports adaptive behavior as modeled by
algebraic decomposition. Probabilistic and nonadaptive algebraic decision
models do not permit any systematic changes of choice behavior over time.
No serious attempt is made to offer an explanation for the finding that
in both experiments significantly more preference reversals occurred between
the second and third session although intransitivities decreased over all three
sessions. It is possible, however, that subjects decided to adopt a simpler and
therefore more consistent strategy in the last session. A strategy they had
used occasionally in the first session. The steadily decreasing response times
across sessions seem to support this explanation.
3.9. EXPERIMENT 3A: DISCRIMINATION
109
3.9 Experiment 3A: Discrimination
By using discrimination tasks in the domain of visual contrast perception
this experiment investigates if different block designs can have an effect on
inconsistency of discrimination.
If two center fields have the same luminance't but surrounding fields of
different luminances then the brightness of the center field with a bright sur
round is perceived darker than the center field with a dark surround. This
effect is known as brightness contrast and was quantified in classical exper¬
iments by Heinemann (1955). In the following experiment the brightness
contrast was used to create indifferent stimuli which were indifferent in per
ceived brightness near the discrimination threshold although they were quite
different in absolute luminance. If two center fields with different luminance
are surrounded by fields of different luminance then the brightness contrast
can create the impression of equivalent brightness between the center fields.
Because each stimulus consisted of an indifferent center field and a different
surround it was assumed that repetition of stimuli with the same surround
over subsequent trials would influence individual discrimination performance
similar to the choice tasks in Experiment 2A. The same unspecified hypothe
ses were investigated in this first experiment.
3.9.1 Method
As in Experiment 2A all stimuli remained constant. Subjects chose in pair comparisons (two¬
alternative forced choice) in three separate sessions between the same set of visual stimuli
which differed in luminance but were nearthreshold in perceived brightness. The fixed block
designs (repetition and resolution block design) and the random block design were used as
before. The sequence of trials in the random block design was randomized for each subject
and each pair comparison.
Design
The same design as in Experiment 2A was employed. In a 3 x 3 design (block design
by session) with repeated measurement on the second factor each subject discriminated
between square disks in three sessions under different block designs. The block designs in
three sessions were varied between subjects leading to three groups labeled S, P and N.
Group S: In all three sessions subjects chose under the reSolution block design.

Group P: In all three sessions subjects chose under the rePetition block design.
14In the following luminance always refers to physical brightness measured in candela per square meter
(cd/m2).
110
CHAPTER 3. EXPERIMENTS
Group N: In all three sessions subjects chose under the raNdom block design.
Dependent variables were the choices and response times as well as measures of inconsistency
(reversals, intransitive triples, dicycles, and ear dicycles). As in the previous experiments
it was hypothesized that the different block designs would affect the consistency of the
discrimination performance.
Subjects
A total of 30 undergraduate and graduate students who also took part in Experiment 2A
or 2B were recruited from the subject panel of the Department of Experimental Psychology,
Oxford University. Each group had 10 subjects and was balanced in gender. (Group S:
average age 24.9 years SD 4.09; Group P: average age 21.5 years SD 3.5; Group N: average
age 21.9 years, SD 3.21). All subjects had normal or correctedtonormal vision. Each subject
received «3 per hour plus traveling expenses for their participation in Experiment 2A or 2B
and this experiment.
Materials and Apparatus
Twelve visual stimuli with different luminance of center and surround were generated. The
center field was a square disk within a square surround. The luminances of center and
surround were adjusted in luminance so that all centers appeared to have approximately the
same perceived brightness (see Appendix B.3.2). All stimuli appeared monochromatic. The
—
70
— y =21.34 + 1.92x
60
50
40
5
30
20
10
10 20 30 40 50 60 70
Luminance of Center in cdlm’
Figure 3.8: Luminances of center and surround. A linear func¬
tion is fitted to the two luminances of the stimuli.
3.9. EXPERIMENT 3A: DISCRIMINATION
111
luminances of center and surround of the stimuli were varied along a linear axis with stimuli
close to indifference (see Figure 3.8). It is well documented that small linear differences in
contrast do not produce a linear simultaneous contrast effect (Takasaki, 1966; Semmelroth,
1970). Therefore, only stimuli were selected which had a sufficient difference in luminance
between center and surround. The fitted regression line has a slope of 1.92 and an intercept
of 21.34 (R2 = 0.997).
3°3
3°3
101
Figure 3.9: Display of stimuli on screen.
During choicetrials two stimuli with centers of different luminance were displayed on
a 21 inch Twopage Macintosh monochrome monitor with a frame rate of 66.7 Hz. The
center field subtended 1°1' degree visual angle and the surround had a visual angle of 3°3'.
The two stimuli were displayed vertically 3°3' apart on a black background as illustrated in
Figure 3.9.15 No fixation point was displayed and the screen was observed binocularly from a
distance of 114 cm while the head was supported by a chinrest. The images were drawn on
screen in an interval of 3.0 secs between last response and next display. During this interval
15The square disks were displayed vertically rather than horizontally to reduce the effect of voltage droop
(Pelli & Zhang, 1991). This bias on luminance was revealed during calibration of the monitor. The effect
was stronger for a horizontal than a vertical display.
112
CHAPTER 3. EXPERIMENTS
the screen remained black. The program waited until the image was completed which took
16
between 300 and 500 msec.
Each stimulus pair was displayed on screen by changing the
color lookup table (CLUT) in frame rate so that the whole image became instantly visible.
The experiment was programmed in SuperLab and run on a Macintosh Ilcx (enhanced
by a floatingpoint unit) allowing time measures on the keyboard with an accuracy of up to
4 millisecs.
Procedure
Each subject had three sessions with (½) = 66 choicetrials plus 3 training trials. Each
2/
session was preceded by a session of Experiment 2A or 2B which lasted between ten and
fifteen minutes. The background for the instructions and stimuli of Experiment 2A and
2B had approximately the same average luminance of 12 cd/m2 as the stimuli so that the
subjects were sufficiently dark adapted before each session.
Subject were seated in front of the computer screen and keyboard in a completely dark¬
ened cubicle. They were asked to place their head on a chinrest to keep viewing angle and
distance constant.
First, they read a general instruction about the experiment, which was followed by a more
detailed instruction (see Appendix B.3.1). In three training trials they learned how to give
a response by pressing the 'U' and 'N' key on the keyboard. In each trial they were asked to
decide which disk in the center of the two squares appeared brighter. A black background
was shown for three seconds before response timing was initiated and the two stimuli were
displayed in the upper and lower half of the screen. The two stimuli stayed visible until the
subject chose the upper or lower alternative by pressing the appropriate key. Each session
lasted between six and twelve minutes.
Table 3.33: Mean
for Discrimination
Response Times
(Exp 34)
Session
Session
3
Session
2
1
Mean"
SD
Mean
Mean
SD
SD
87
1.10
2.03
1.65
55
1.29
Group S (N10)
86
2.02
72
1.74
1.39
53
Group P (N10)
—
60
132
146
55
1.94
Group N (N10)
42
Note: Statistical significant effect between sessions, Fs2,54 = 34.41,p «
0.0001).
in seconds
3.9.2 Results
In this section the results of analyses on the dependent variables response times, discrimi¬
nation cycles, and discrimination reversals are reported.
The hypotheses were the same as before and are summarized as follows: The resolution
block design should result in different discrimination performance (measured by Kendall's
1eIn each trial the varying delay was subtracted from the time interval between trials.
3.9. EXPERIMENT 3A: DISCRIMINATION
Table 3.34: ANOVA for Mean Response Times (Exp 34)
df
F
Source
S5
MS
Between subjects:
2
Block Design (BD)
146.6
293.2
0.11
Error
27
34682.7
1284.5
Within subjects:
24.16***
6578.8
Session
2
3289.4
50.2
BD by Session
200.7
4
0.37
7352.2
Session by Error
54
136.1
Note: ***p = 0.0001.
(, 7, number of dicycles and ear dicycles) than the repetition block design. The random
block design should lead to an intermediate level of consistent discrimination performance.
In comparison the number of ear dicycles should discriminate better between block designs
than the total number of dicycles. Moreover, a decrease in response times and inconsistency
can be expected over sessions.
Response Times
Mean response times and standard deviations are listed in Table 3.33. In every group mean
response times decreased over sessions. On average response times were reduced by 0.49 secs
from Session 1 to 2 and again by 0.17 secs from Session 2 to 3.
Mean response times were entered into a 3 by 3 analysis of variance (ANOVA) with
repeated measurement on the second factor. Block design (resolution, repetition, random)
served as fixed factor between subjects, session (first, second, third) as fixed factor within
subjects and subjects acted as random factor (see Table 3.34 for results). A sphericity test
showed no significant violation of homogeneity (approximate y’2 = 2.03,p = 0.36).
The analysis revealed no statistical significant effect between block designs (Fs2, 27 « 1,
ns). However, the null hypothesis of no effect over sessions was rejected (Fs2,54 = 24.16,
p « 0.0001). No other effects approached statistical reliability.
Table 3.35: Intransitive Triples of Discrimination
Kendall's
Session 3
Session 2
Session
1
SD
SD
Mean
Mean
SD
Mean
59
66
73
Group S
24
28
22
28
69
68
66
22
20
Group P
17
21
.79
82
Group N
18
17
Note: ( = 1  T/T. with T, = n(n2  4)/24 = 70 for n = 12, and T
number of 3dicycles.
113
114
CHAPTER 3. EXPERIMENTS
Table 3.36:
ANOVA for Kendall's ( (Exp 34)
Source
85
MS
E
4
Between subjects:
0.255
Block Design (BD)
0.128
2
0.99
3.479
Error
27
0.129
Within subjects:
0.088
Session
0.044
2
2.11
0.028
BD by Session
0.007
0.25
4
1.542
Session by Error
0.029
54
Discrimination Cycles
In Table 3.35 the average (values with standard deviations for each group and session are
listed. All groups had comparable average (values in the first and more or less in the
following sessions. In the first session subjects in Group S made choices under a resolution
block design reaching an average (value of .73, whereas subjects in Group P had an average
value of .66 under the repetition block design. The subjects of Group N made choices under
the random block design and reached a higher average (value of .77.
Table 3.37: Stepwise Discriminant Analysis on Dicycles (Exp 34)
Removed
Step Entered
Number Partial R° Partial F pF
Session 1
No variables entered
Session 2
1
0.143
1 3dicycles
0.125
2.250
Session 3
No variables entered
The same type of analysis was applied to the number of intransitive triples using Kendall's
(as a standardized measure (see Table 3.36).17 There was no significant effect of block design
(F[2,27 « 1, ns) and no effect of session (Fs2,54 = 2.11, p = 0.131).
For each session stepwise discriminant analyses were performed on the number of dicycles,
i.e. the coefficients z of the polynomial %, and the number of ear dicycles from the ear
decompositions by sequence. The numbers of kdicycles were entered stepwise as dependent
variables starting with dicycles which discriminated best between groups. The results of the
analysis are given in Table 3.37. Variables were entered and excluded at a significance level
of p « 0.15.
Only in Session 2 the three block designs could be weakly discriminated by the number of
3dicycles whereas in Session 1 and 3 no variables were entered. The stepwise discriminant
analysis on the number of ear dicycles revealed some differences between groups in all three
sessions. The results are summarized in Table 3.38.
17A sphericity test revealed no significant violation of homogeneity for the repeated measurements over
sessions (approximate x22 = 5.25,p = 0.07).
3.9. EXPERIMENT 3A: DISCRIMINATION
Table 3.38: Stepwise Discriminant Analysis on Ear Dicycles (Exp 34)
Step Entered
Number
Removed
Partial R? Partial F
P2F
Session 1
1. 5dicycles
0.150
0.111
2.39
Session 2
1. 7dicycles
5.06
0.273
0.014
1
0.143
2. 12dicycles
0.134
2.18
Session 3
0.159
1. 9dicycles
2.54
0.097
0.141
2. 10dicycles
0.129
2.21
The ear dicycles of length 7 had considerable discriminatory power in the second session.
Ear dicycles of length 5 discriminated in the first session and dicycles of length 9 and 10 in
the third session. The number of ear dicycles together with mean and standard deviation
are listed for each subject and session in Table B.25 for Group S, in Table B.26 for Group P
and in Table B.27 for Group N.
Table 3.39: Reversals of Discrimination (Exp 34)
Kendall's 7
Session 1 vs 2
Session 2 vs 3
Session 1 vs 3
SD
Mean
SD
Mean
SD
Mean
30
Group S
43
27
33
393
—
38
53
18
Group P
20
22
43
48
17
13
22
52
65 —
Group N 57—
Note: r = 1  (4R/R.) where R» = n(n1) = 132 for n = 12, and R
number of reversed relations.
Discrimination Reversals
The average rvalues and standard deviations are listed for each group and session in Ta
ble 3.39. The pattern of rvalues for the three comparisons indicates that slightly more
reversals occurred between Session 2 and 3 than Session 1 and 2 or Session 1 and 3. Al¬
though not significant the rvalues for Group N, the random block design, were higher than
for Group S or P which corresponds to the higher (values in Group N.
An analysis of variance with repeated measurement was performed on the number of pref
erence reversals between each combination of the three sessions (see Table 3.40). Kendall's
r was used as standardized dependent variable. A sphericity test showed no significant vio¬
lation of homogeneity for the repeated measurements (approximate y’ 2 = 4.24,p = 0.12).
There was no statistical significant effect of block design (Fs2,27 = 2.19,p = 0.13) but a
significant effect for the three combinations of sessions (Fs2,54 = 5.03,p = 0.01).
No systematic effects between groups were detected suggesting that the block designs
had no significant effect on consistency of discrimination. Beside the session effect for mean
115
116
CHAPTER 3. EXPERIMENTS
Table 3.40: ANOVA on Discrimination Reversals (Exp 34)
F
4
Source
55
MS
Between subjects:
0.564
Block Design (BD)
0.282
2
2.19
27
Error
0.129
3.475
Within subjects:
2
0.086
5.03*
0.172
Combination (C)
BD by C
0.001
0.004
.06
4
0.017
C by Error
0.925
54
Note: *p « 0.01.
response times there was a similar effect for reversals between sessions as in Experiment 2A.
Subjects reversed more discriminations between Session 2 to 3 than between Session 1 and
2 and Session 1 and 3. No explanation for this effect is offered. Although not significant, (
and rvalues in this experiment appeared to be higher for the random block design. The ear
dicycles of length 5 in the first session and some combination of ear dicycles in Session 2 and
3 discriminated between block designs but these results are inconclusive because numbers
are pointing in different directions (see Table B.25 B.26 and B.27).
3.10. EXPERIMENT 3B: DISCRIMINATION
117
3.10 Experiment 3B: Discrimination
The results from Experiment 3A suggest that there is no effect of block design
on consistency of discrimination. Hence, the alternative hypothesis could not
be specified and no systematic effects were expected from the variation of
block designs across sessions.
3.10.1 Method
The same method and stimuli as in Experiment 3A was applied.
Design
The same design as in Experiment 2B was applied. In a 2 x 3 design (sequence by session)
with repeated measurement on the second factor each subject chose between stimuli in three
sessions under different block designs. Two different sequences of block designs over three
sessions were tested in Group SNP and PNS.

Group SNP: In the first session subjects chose under a reSolution block design, in the
second under a raNdom block design, and in the third under a rePetition block design.
Group PNS: In the first session subjects chose under a rePetition block design, in the
second under a raNdom block design, and in the third under a reSolution block design.
Dependent variables were the choices and response times as well as measures of inconsistency
(reversals, intransitive triples, dicycles, and ear dicycles). As in the previous experiments
it was hypothesized that the different block designs would affect the consistency of the
discrimination performance.
Subjects
A total of 20 undergraduate and graduate students who also took part in Experiment 2A
or 2B were recruited from the subject panel of the Department of Experimental Psychology,
Oxford University. There were 10 subjects in each group (Group SNP: average age 24.5
years, SD 5.21; Group PNS: average age 27.3 years, SD 6.43). Both groups were balanced
in gender. All subjects had normal or correctedtonormal vision. Each subject received £3
per hour plus traveling expenses.
Materials and Apparatus
The same equipment, stimuli and setup as in Experiment 3A was used.
Procedure
The procedure was the same as in Experiment 3A. Each session was preceded by a session
of Experiment 24 or 2B and lasted between six and twelve minutes.
CHAPTER 3. EXPERIMENTS
118
3.10.2 Results
In the following the results from analyses on the dependent variables response times, discrim¬
ination cycles and reversals are presented. The hypothesis are the same as for Experiment 3A
with the exception that differences between block designs should appear over sessions.
Response Times
Mean response times and standard deviations are listed in Table 3.41. In both groups mean
response times decreased over sessions. On average response times were reduced by 0.29 secs
from Session 1 to 2 and again by 0.20 secs from Session 2 to 3.
for Discrimination (Exp 3B)
Table 3.41: Mean Response Times
Session
3
2
Session 1
Session
Mean"
SD
SD
Mean
Mean
SD
1.18
1.47
1.75
61
Group SNP (N10)
55
48
1.44
1.54
72
184
76
59
Group PNS (N10)
—
= 35.6,p «
Note: Statistical significant effect between sessions, F(2,90)
0.0001).
ein seconds
Mean response times were entered into a 2 by 3 analysis of variance (ANOVA) with
repeated measurement on the second factor. Sequence of block designs (resolutionrandom¬
repetition, repetitionrandomresolution) served as fixed factor between subjects, session
(first, second, third) as fixed factor within subjects and subjects acted as random factor (see
Table 3.42). A sphericity test showed no significant violation of homogeneity (approximate
x22 = 2.32,p = 0.31).
Table 3.42: ANOVA for Mean Response Times (Exp 3B)
Source
88
MS
4
F
Between subjects:
303.7
0.28
303.7
1
Sequence (S)
18
19491.5
Error
1082.9
Within subjects:
24.97**
2365.5
2
Session
1182.7
49.1
2
1.04
98.1
S by Session
1705.4
47.4
36
Session by Error
Note: ***p = 0.0001.
There was no statistical significant effect between groups (FI1, 18 «1, ns) but the null
hypothesis of no effect of session was rejected (Fs2,36 = 25.0, p £ 0.0001).
3.10. EXPERIMENT 3B: DISCRIMINATION
119
Discrimination Cycles
In Table 3.43 the average (values are given for each group and session. As shown in Ta
ble 3.43 subjects in Group SNP first made discriminations under a resolution block design
reaching an average (value of .64, whereas subjects in Group PNS discriminated under the
repetition block design and displayed a value of .68. The values increase slightly in the last
session.
Table 3.43: Intransitive Triples of Discrimination (Exp 3B)
Kendall's (
Session 2
Session 1
Session 3
Mean
SD
Mean
Mean
SD
SD
Group SNP
76
64
25
26
63
16
Group PNS
68
73
26
64
27
33
Note: ( = 1 T/T, with T, = n(n2  4)/24 = 70 for n = 12, and T
number of intransitivities.
The same type of analysis was applied to the number of intransitive triples using Kendall's
( as a standardized measure (see Table 3.44). A sphericity test detected no significant
= 0.95,p =
violation of homogeneity for the repeated measurements (approximate x2s21
0.15). Again, there was no significant effect of sequence of block designs (Fsl, 18 « 1, ns)
and no effect of session (F2,36 « 1, ns).
Table 3.44:
ANOVA for Kendall's ( (Exp 3B)
F
4
85
MS
Source
Between subjects:
0.005
Sequence (S)
0.005
1
0.04
Error
2.123
0.118
18
Within subjects:
0.65
0.015
2
0.031
Session
0.070
S by Session
2
0.035
1.49
36
0.851
0.024
Session by Error
For each session stepwise discriminant analyses were performed on the number of dicycles,
i.e. the coefficients z4 of the polynomial %, as well as the number of ear dicycles from the ear
decomposition by sequence. The numbers of kdicycles were entered stepwise as dependent
variables starting with dicycles which discriminated best between the two groups. For the
number of dicycles none of the variables reached a significance level of p + 0.15. The number
of kdicycles together with mean and standard deviation are listed for each subject and each
session in Table B.28 for Group SNP and in Table B.29 for Group PNS. The results of the
stepwise discriminant analysis on the number of ear dicycles are shown in Table 3.45.
Only in the first session ear dicycles of length 3 discriminated weakly between the block
designs. The number of ear dicycles from the ear decomposition by sequence are listed for
120
CHAPTER 3. EXPERIMENTS
Table 3.45: Stepwise Discriminant Analysis on EarDicycles (Exp 3B)
Removed
Number
Step Entered
Partial R? Partial F
P2F
Session 1
0.126 —
2.60—
0.124
1. 3dicycles
Session 2
No variables entered
Session 3
No variables entered
each subject and session in Table B.30 for Group SNP, and in Table B.30 for Group PNS.
In the next section the number of dicrimination reversals between sessions are discussed.
Discrimination Reversals
In Table 3.46 mean Kendall's r are listed for each group and session as a standardized
measure of reversals between pair comparisons. The rvalues do not exhibit any systematic
variations. Especially, the average rvalues in Group PNS remained almost constant.
Table 3.46: Reversals of Discrimination (Exp 3B)
Kendall's 7
Session 1 vs 2
Session 2 vs 3
Session 1 vs 3
SD
SD
Mean
Mean
Mean
SD
36
39
29
22
46
Group SNP
32
50
33
33
Group PNS
50
50
32
Note: r = 1  (4R/R.) where R, = n(n  1) = 132 for n = 12, and R
number of reversed relations.
An analysis of variance with repeated measurement was performed on the number of
reversals between each combination of the three sessions (see Table 3.47). Kendall's r was
used as standardized dependent variable.1s
There is no statistical significant effect of sequence of block designs (FI1, 18 « 1, ns) and
no significant effect of combination of sessions (Fs2,36 = 1.51,p = 0.23).
The results are quickly summarized. In Experiment 3B no effects other than the session
effect for mean response times were present. In general the block designs had no systematic
effect on intransitive triples or dicycles. The number of ear dicycles of length 3 discriminated
weakly between block designs in the first session but their means do not correspond to the
findings in Experiment 3A.
The (values for discrimination under the random block design in Session 2 of Group SNP
and PNS were lower than or equal to the values of Session 1 and 3. Hence, the tendency
for more consistent discrimination under the random block design in the results of Experi¬
ment 3A was not confirmed. In Experiment 3A the rvalues for reversals between Session 2
13 A sphericity test showed no significant violation of homogeneity for the repeated measurements (approx¬
imate x22 = 1.10,p = 0.58).
3.10. EXPERIMENT 3B: DISCRIMINATION
Table 3.47: ANOVA on Preference Reversals (Exp 3B)
F
Source
88
MS
4
Between subjects:
0.220
Sequence (S)
0.220 0.92
4.325
18
Error
0.240
Within subjects:
0.030
0.060
2
1.51
Combination (C)
2
S by C
0.055
0.028
1.38
36
0.020
0.721
C by Error
and 3 were slightly higher but no corresponding effect appeared in Experiment 3B. We con¬
clude that no systematic changes of consistency occurred, neither over sessions nor between
block designs.
121
CHAPTER 3. EXPERIMENTS
122
3.11 Discussion
In general mean response times for Experiment 3A and 3B are shorter in com¬
parison to Experiment 1A and 1B as well as Experiment 2A and 2B but they
also decrease significantly across sessions. Although subjects in each group
solve the discrimination tasks quicker probably due to increasing familiarity
with the task but not the stimuli, increasing consistency over sessions and
varying consistency between groups was not observed: Intransitivities seem
to vary unsystematically across groups and sessions (see for example Fig¬
ure 3.10). In contrast to Experiment 1A and 1B as well as Experiment 2A
and 2B the results suggest that subjects did not interact with the arrange
ment of choicetrials and did not improve in consistency over sessions. Hence,
subjects were not able to choose more consistently under a particular block
design or in a later session.
This result suggests that in this domain subjects perform discrimination
tasks independently of block designs. Therefore, the results do not support
the application of algebraic decompositions and are compatible with prob¬
abilistic decision models as favored in psychophysics. However, much more
data needs to be accumulated before probabilities can be reliably estimated
and before any conclusions about probabilistic choice behavior can be drawn.
Although Experiment 3A and 3B was conducted under the same design
as Experiment 2A and 2B there are two important differences between these
experiments. First, the alternatives in Experiment 3A and 3B were physi
cal stimuli, more precisely, visual stimuli which were matched in perceived
brightness leading to indifferent alternatives or stimuli near a discrimina
Although the stimuli can
tion threshold (or point of subjective equality).
be described as twodimensional just as the lotteries in Experiment 2A and
2B, they were identifiable only by the perceived brightness of their surround
which is a different and more difficult task than remembering a pie chart with
numbers. Second, the choice task was a discrimination task which does not
involve preferences, i.e. an evaluation of the stimuli.
Mean (values varied unsystematically between sessions and groups as
shown in Figure 3.10 and the same holds for the total number of dicycles
and ear dicycles. It is assumed that the indifferent or nearthreshold stimuli
did not allow subjects to improve in consistency. Subjects could not benefit
from previous trials nor simplify the discrimination task over sessions as in
Experiment 2A and 2B. This is confirmed by the absence of any systematic
differences between block designs and sessions. The subjects were unable
3.11. DISCUSSION
Session 1
Session 2
Session 3
0.8
0.6
0.4
0.2
2
SNP
PNS
Group
Figure 3.10: Kendall's ( for all groups of Experiment 3A and 3B
to take advantage of the repetition of a stimulus in successive trials simply
because the stimulus was not recognized and repetitions went by unnoticed.
The subject certainly noticed that the stimuli had surrounds which differed in
brightness but from the present results it seems unlikely that they were able
to identify stimuli by their surround. However, effects similar to the results in
Experiment 2A and 2B may occur in a different domain where visual stimuli
are easily distinguishable by attributes irrelevant to the discrimination task.
123
CHAPTER 3. EXPERIMENTS
124
3.12 General Discussion
Six experiments in three domains were conducted leading to different results.
Named and familiar alternatives under riskless choice showed more consis
tent choice behavior under the repetition block design and an effect of block
design on 3dicycles (intransitive triples) and 4dicycles was observed. An ex¬
planation was offered which takes into account that the named alternatives
were highly familiar so that existing preference values were simply recalled
if alternatives were repeated from one trial to another thereby increasing
consistency. Although these results were not in line with the original predic
tions about information processing in multiattribute alternatives they reject
the null hypothesis which is linked to the assumption that choice behavior
should be independent of arrangements of the choicetrials.
The effect of block design was not very prominent for indifferent alterna¬
tives under risky choice but appeared consistently in the first session of both
experiments. It was argued that the effect of block designs on consistency
was overshadowed by a stronger effect across sessions. The consistency effect
over sessions proved to be irreversible and increased regardless of the initial
block design. This improvement is not seen as a pure learning effect because
subjects did not receive feedback on their choice behavior. It is suggested
that subjects eventually adopted a simpler decision strategy which shortened
response times and improved consistency over sessions. The strong session
effect also casts doubt on the assumption of nonadaptive choice behavior,
particularly in domains with multiattribute alternatives.
Finally, no significant effects of block design or session on consistency were
found for visual nearthreshold stimuli in a psychophysical discrimination
task. Discrimination of stimuli according to their induced brightness contrast
showed unsystematic variations of inconsistency between groups and sessions.
The polynomialy which assessed all inconsistencies in a pair compari
son occasionally had the same number of 3cycles in one pair comparison
or session but distinctive numbers of kdicycles for k » 3 in another. Be¬
cause the number of kdicycles with k » 4 increased exponentially in cases of
very inconsistent choice behavior causing huge standard deviations (see for
example Subject 39 and Subject 4 in Experiment 3A and 3B, respectively)
statistical tests on the number of dicycles are not reliable. On the other
hand, it was not possible to establish an adequate test statistic because the
distributional properties of the kdicycles are unknown. Nevertheless, the
polynomials provide an exhaustive way to characterize individual choice be¬
3.12. GENERAL DISCUSSION
125
havior and may prove fruitful in domains with few indifferent alternatives
such as named alternatives. In general, the number of ear dicycles proved to
be a better measure of inconsistency. Their numbers increased linearly with
the size of the strong components and discriminated better between block de¬
signs than dicycles. It is concluded that the sequence of intransitive choices
as employed by the ear decomposition offers a valid way to capture adaptive
choice behavior in a quantitative way.
In summary, there is evidence that adaptive choice behavior is reflected
in inconsistent choice behavior. Inconsistency may be induced by systematic
arrangements of choicetrials but different results were obtained for familiar
named alternatives and unfamiliar multiattribute alternatives. It is concluded
that inconsistency is efficiently characterized in terms of ear dicycles.
Chapter 4
Conclusions
In this chapter the theoretical and experimental results are briefly discussed
and their implications on decision making theory are evaluated. Recommen¬
dations for experimental studies and the analyses of empirical data are given
and further theoretical developments are encouraged. The chapter concludes
with remarks on the concept of error in human decision behavior.
It is emphasized that algebraic decomposition models can serve as a uni¬
versal tool to analyze adaptiveness in binary data. In the case of choice
behavior it is not necessary to specify a particular decision rule that is sup¬
posed to apply to all decisions in successive choicetrials. Often this is not
even desirable because the specification of a decision rule is only possible if
all attributes of the alternatives are known and remain unchanged over time.
This describes a rather domainspecific and artificial choice situation. In
this context it is important to realize that the specification of an underlying
knowledge structure of attributes usually favors a certain type of decision
rule. In essence this means that declarative knowledge implies procedural
knowledge and vice versa; a simple fact that has not been fully appreciated
and hampers progress in psychological decision research.
4.1 Theoretical and Empirical Implications
Further developments on the algebraic decomposition of individual choice be¬
havior are necessary to find appropriate representations. The identification of
a directed ear basis as suggested in Chapter 2 is one of several possible rami¬
fications. For example, the decomposition of strong components into acyclic
substructures such as spanning trees are investigated in problems of linear
optimization and opens up promising paths for future research. In addition,
the rich theory of algebra and developments on oriented matroids may offer
127
CHAPTER 4. CONCLUSIONS
128
different and exciting ways to decompose digraphs and strong components
(e.g., Björner, Las Vergnas, Sturmfels, White, & Ziegler, 1993).
More generally, algebraic decompositions can be linked to research issues
in theories of knowledge representation. The theory of knowledge structures
as proposed by Doignon and Falmagne (1985) shares some interesting features
with algebraic decomposition models. For an introduction to the theory of
knowledge structures consult Falmagne, Koppen, Villano, Doignon, and Jo¬
hannesen (1990). The assessment of knowledge states in a knowledge space
can be translated into the assessment of preference orders in a family of possi¬
ble preference orders. A theory which may bring the two approaches together
is currently under investigation (e.g., Doignon & Falmagne, 1997). At present
the theory of knowledge structures is based on group data because a knowl¬
edge space has to be assessed and verified by experts before it can be applied.
Moreover, such a knowledge space is believed to be universal and therefore
it should be applicable to any subject and knowledge state. Algebraic de¬
compositions, on the other hand, have the advantage that they are based on
individual preference data with no universal structure superimposed.
In this work a basic assumption of an algebraic decomposition model has
been investigated which contradicts independence of choice trials. The ear
decomposition by sequence makes weak assumptions about the sequence of
intransitive choices. A stronger dependency between choicetrials was sug¬
gested in Definition 2.4.1 for the completion by cuts. This dependency may be
regarded as a deterministic counterpart of the Markov property in stochastic
processes. These or similar assumptions might offer a way to extend algebraic
decompositions to an adaptive probabilistic model.
In the domain of named alternatives the experimental study of block de¬
signs revealed differences between choice behavior in terms of consistency.
The data and results of experiments in the domain of risky alternatives
showed a steady improvement of consistent choice behavior over three consec¬
utive sessions as well as differences between block designs in the first session.
It is believed that these results hint at the intricate domainspecificity of indi¬
vidual decision behavior which is imminent for any study of choices between
single or multiattribute alternatives. The results demonstrate that subjects
do not express their preferences independently trial after trial. It is assumed
that they interact with the alternatives, experimental task and paradigm in
stead. If slight experimental manipulations such as a different sequence of
choicetrials and the repetition of a pair comparison show significant effects on
inconsistent choice behavior then it seems likely that other alterations bring
4.1. THEORETICAL AND EMPIRICAL IMPLICATIONS
129
about more drastic changes. Consequently, an adaptive analysis is needed to
understand and explain individual choice behavior.
Three categories of possible experimental manipulations with several di¬
chotomous subcategories are listed below already leading to 212 possible com¬
binations. Each of these combinations may have a different impact on choice
behavior. More categories are easily obtained extending the number of possi¬
ble combinations. Obviously, only some of the combinations have been stud¬
ied whereas others have been ignored. Experimental studies which use the
suggested decompositions might not elucidate governing principles of adapt
ing choice behavior. On the contrary, it is believed that decision research car
ried out with different alternatives, choice tasks or choice paradigms would
illustrate the limitations of normative or nonadaptive decision models. If
only some of the categories listed below interact with each other then this
would also limit efforts to derive general principles from the observed choice
behavior. Any experimental indication of higher order effects between these
(a) choice alternatives:
real or artificial
i.
.*
certain or uncertain outcomes
11.
*.
single or multiattribute
111.
similar or dissimilar
iv.
choice task:
(b)
forced or nonforced choice
i.
ii.
binary or nary choice
iii.
with or without timelimit
with or without coverstory
iv.
choice paradigm:
(c)
small or big set size
i.
ii. complete or incomplete pair comparison
...
iii. fixed or random sequence of choicetrials
iv. same or different sessions
categories would emphasize the difficulty to explain choice behavior in an
nonadaptive choice model.
On the basis of weak assumptions algebraic decomposition techniques are
able to detect where individual choices have adapted. No costly tracing
methods (e.g., Ericsson & Simon, 1984) are necessary to single out choices or
subsets of choices which are at the center of inconsistent choice behavior. As
mentioned before, the question of how the choice process has adapted cannot
130
CHAPTER 4. CONCLUSIONS
be answered without detailed knowledge about the particular domain and
the underlying information process. This is not only difficult to assess but
also difficult to generalize because the changes in information processing are
known to be context and domainspecific. It has been shown that at least
in some domains the hierarchical organisation of semantic memory which is
recalled in binary decision tasks (Lages, 1991; Albert & Lages, in preparation)
makes the study of human decision making even more demanding.
4.2 Individual vs Group Data
The algebraic decomposition of choice behavior into critical sets of choices is
an excellent tool to study individual choice behavior in greater detail without
imposing strong assumptions upon the choice process. In Chapter 2 some
techniques were developed that might help to extract not only quantitative
but also qualitative information from a set of choices.
In Chapter 3 we have employed standard and new measures of inconsis¬
tency for the statistical analyses of experimental data. The average size of
strong components together with the number of ear dicycles was analyzed in
Experiment 2A and 2B as an example of another measure of inconsistency.
The size of strong components is related to the longest dicycles and can be
derived from the decomposition into strong components as well as the ear
decomposition by sequence (see Table 3.19 and 3.29). The analyses on size of
strong components revealed effects between groups which were not discovered
by standard measures of inconsistency as provided by Kendall's (.
As mentioned before, the stepwise discriminant analysis is a crude sta¬
tistical method for investigating the number of dicycles and ear dicycles.
Unlike the analysis of variance it only allowed testing between subjects but
not within subjects. If testing across sessions were possible it is assumed that
the strong learning effect for 3dicycles across sessions as found in Experi¬
ment 2A and 2B would have been confirmed by a corresponding analysis of
dicycles and ear dicycles.
Determining all kdicycles in a polynomial expression provides limited
additional information about individual choice behavior. For group data it
might be sufficient to assess only 3dicycles in a pair comparison. For the com¬
parison of individual choice behavior, however, assessment of all kdicycles
is recommended. The tabulated coefficients of every subject and session in
Appendix B show that intransitive triples alone are often not sufficient to
capture the extent of inconsistent choice behavior. The assessment of ear
4.3. TOWARD A QUALITATIVE THEORY OF ERROR
131
dicycles, however, is efficient and showed more discriminatory power than di¬
cycles. It is concluded that ear dicycles give a clearer picture of inconsistent
choice because they do not increase exponentially and because they take into
account how inconsistencies evolved in a pair comparison.
As discussed in the introduction an untested assumption of any pair com¬
parison is asymmetry. It is believed that an incomplete pair comparison does
not sufficiently describe individual choice behavior unless all possible ordered
pairs are presented, that is all pairs in the cartesian product of a set except for
the diagonal elements in the adjacency matrix. Under such a paradigm the
adjacency matrix can either remain a (0,1)matrix if preferences are counted
only once, or the matrix may be extended to a multigraph with weighted arcs.
Allowing 2cycles together with kdicycles provides a more detailed picture of
inconsistent choices and choices can be analyzed in an ear decomposition by
sequence in the same way as the choices in an incomplete pair comparison.
In a straightforward application the preference matrices of different sub¬
jects or sessions may be added together creating a multigraph. A difficulty lies
in the fact that such a multigraph may have only one strong component due
to individual differences in preference. This would considerably increase the
complexity of the multigraph and a characterization in terms of all dicycles
would be no longer suitable. Again, an ear decomposition is not restricted
to adjacency matrices of moderate complexity and size and the same type of
analysis can be performed. In general, the ear decomposition seems to be a
robust technique which also works for preferences in a nonforced choice task
or pair comparisons with missing data. It is highly recommended for small
samples such as a pair comparison.
4.3 Toward a Qualitative Theory of Error
One of the major challenges in the field of social sciences is the understand
ing of error in human choice behavior (e.g., Luce, 1996). The probabilistic
viewpoint is unsatisfying because it identifies error with randomness.
A probabilistic choice model, for example, describes preferences between
lotteries or gambles by a probabilistic preference relation between alterna¬
tives. Thereby, the alternatives themselves are represented in terms of prob¬
abilities, thus subjecting a single preference to randomness in choice and
randomness in utility. In this case and for most decisions of everyday's life a
probabilistic model of individual decision making does not appeal.
On the other hand, if sensitivity to a physical intensity varies randomly
CHAPTER 4. CONCLUSIONS
132
and affects the decision process as in psychophysical tasks then probabilis¬
tic models appear justified. In Experiment 3A and 3B, for example, the
energy modulation of luminances may have had probabilistic properties so
that perceived contrast varies randomly due to the light source, the optical
characteristics of the eye and some lowlevel visual processing.
The information processing approach including conjoint measurement, al¬
though successful under some circumstances, requires vectorlike represen¬
tations of alternatives which always results in artificial and domainspecific
applications.
In general, deterministic and probabilistic theories address the problem of
structure and variability in behavioral data. The dilemma is that classical al¬
gebraic approaches are inflexible and cannot account for variability whereas
the probabilistic approach often does not provide a desirable algebraic or
qualitative representation. An algebraic decomposition model, however, in¬
corporates variability without loosing its qualitative appeal.
Appendix A
Mathematical Background
A.1 Notations and Basics
The following mathematical notations and abbreviations are used throughout the text with¬
out further comment.
sets
X,Y,(...)
empty set
9
cartesian product of the sets Xj,..,Xn
XIX...XXn
ordered ntuple
(x1Xn)
element of, proper subset of, subset of
c
logical and, or, negation
A, V, —
S
implies
V, 2
for all, exists
equivalent
greater than, greater or equal than
2
,
matrix
A
AT
transposed matrix
identity matrix (of order n)
I (I.)
all l’s matrix (of order n)
J
(J.)
zero matrix (of order n)
O (On)
set of all (0, 1)matrices (of order n)
Mn
set of natural, real, and complex numbers
Z,R,C
polynomial ring on Z in one indeterminate a
Zsæ)
The symbol □ indicates the end of an example or proof.
133
134
APPENDIX A. MATHEMATICAL BACKGROUND
Some elementary mathematical definitions are introduced next.
Definition A.1.1 Let Xj,..., Xn + 0 be sets. A subset R of the cartesian product Xi X
... X Xn is called nary relation on Xj,...,Xn. A 2ary relation on X,Y is called binary
relation on X, Y, and if X = Y binary relation on X.
Definition A.1.2 Let RC XX Y be a binary relation on X,Y. The domain and image of
R is defined as:
(A.1)
D(R) = (xeXyeY(x,y)eR)
I(R) = lyeYxeX(x,y)eR)
(A.2)
Instead of (x,y) E R we can also write xRy.
If (x,y) E R then (y,x) e R defines the complement R of a binary relation R.
Definition A.1.3 A binary relation F on X,Y is called a mapping from X to Y (F: X
Y), if Vx E X there is exactly one y £ Y so that xFy. In this case we write F(x) = y
instead of xFy.
Definition A.1.4 A mapping F: X — Y is called
(i) injective : Va,2 e X(ifxi 4 x2 then F(xi) 4 F(x2))
(ii) surjective : Vy e Yd e X(F(x) = y)
(iii) bijective : Fis injective and surjective
Definition A.1.5 Let X,Y %0 be sets, Ri,..., R, relations on X and Si,..., S» relations
on Y. A mapping F: X  Y is called homomorphism of (X, Ri,..., R.) into (Y, Si,..., S.)
if and only if Vær,2 e X and vi = (1,...,n) ((zi,a2) e R.  (F(xi), F(x2)) E S.).
A bijective homomorphism is called isomorphism and an antiisomorphic mapping is a bi¬
jective homomorphism of the complement R.
A.2 Order Theory
In this section some basic definitions of order theory are presented. For a complete intro¬
duction to order theory and its application consult Davey and Priestley (1990).
In the following X 0 is a set.
Definition A.2.1 A binary relation on X is called
(i) reflexive : Va e X(x2x)
(ii) irreflexive: Vr eX(xEx)
(iii) transitive : Va,y,zeX((yy z)
(iv) negatively transitive : Va, y,zeX(xy (xzVzy))
A.2. ORDER THEORY
135
(v) antisymmetric: Va, y eX((xyAya)x=y)
(vi) symmetric : Va,yeX(yyx)
(vii) asymmetric : Va, y e X(ise 2yx)
(viii) connected : Va, y eX(x2yVyex)
Definition A.2.2 A reflexive and transitive binary relation on X is called partial quasiorder
on X. An antisymmetric partial quasiorder on X is called partial order on X. A transitive
and connected binary relation on X is called a weak order on X. An antisymmetric weak
order is called a linear order on X.
Definition A.2.3 (Equivalence) A reflexive, symmetric and transitive binary relation on
X is called equivalence relation on
X. Ifa denotes an equivalence relation on X we define
for anyx e X the equivalence class sa of x by:
[x:= syeXasy)
For x, y e X holds either sx =
yl or Oy = 0, which means that two equivalence classes
are either identical or disjoint.
We write  for the symmetric complement of relation », so that x  y iff neither x » y
nor y x, and 2 is the union of and .
Definition A.2.4 Let X be an ordered set and let A C X. Then
(i) a e A is called minimal element of A if a 2xe A implies a = x, and minimum
of A ifa Sx for every xE A.
(ii)
x e X is called lower bound of A, if x £a for all a E A.
(iii)
Let x be a lower bound of A, so that y Sx for all lower bounds y of A, then x is
the greatest lower bound or infimum of A.
A maximal element, maximum, upper bound, and the least upper bound or supremum is
defined dually.
The following notation for the infimum and supremum of a set A is used in the text:
inf (A) and sup (4).
Lemma A.2.5 Let X be a partially ordered set such that inf (A) E X for every nonempty
subset A of X. Then sup A exists in X for every subset A of X which has an upper bound
in X; in fact sup A = inf x eX asxfor allae A) the infimum of the set of all upper
bounds of A.
D
Proof. see David and Priestley (1990), p.33.
Definition A.2.6 (Closure) Let X be a set. A mapping C:2* — 2* is a closure operator
on X if, for all A,BCX,
(i) ACC(A),
APPENDIX A. MATHEMATICAL BACKGROUND
136
(ii) if ACB, then C(A) CC(B),
(iii) C(C(A)) = C(A).
A subset A of X is called closed (with respect to C) if C(A) = A.
Definition A.2.7 (Lattice) Let X be a finite and nonempty partially ordered set. If
sup(A) e X and inf (A) e X for all A CX, then X is called a lattice.
A.3 Graph Theory
Ore (1962) and Harary (1969) are classic textbooks on graph theory. A more extensive
account on graph theory is contained in Bergé (1976). More specifically, Moon (1968) wrote
a small book on tournaments and Reid and Beineke (1978) included a chapter on tournaments
in their book.
A graph G consists of a set V = (aj,a2,..., an) of elements called vertices together with
a prescribed set E of unordered pairs of distinct vertices of V. The number n of vertices is
called the order of the graph G. Every unordered pair Ja,b) of vertices in E is called an
edge of the graph G. Two vertices on the same edge or two distinct edges with a common
vertex are adjacent. Also, an edge and a vertex are incident with one another if the vertex
is contained in the edge. A complete graph is one in which all possible pairs of vertices are
edges. A digraph (directed graph) D consists of a set V of vertices together with a prescribed
set E of ordered pairs of not necessarily distinct vertices of V. Every ordered pair (a, b) of
vertices in E is called an arc (or directed edge) of the digraph D.
A directed walk is of the form
(ao,a), (a1, a2) (d.1, a4).
If az = ao this is a closed directed walk. Moreover, if az and ao are the only identical vertices
it is said to be a directed cycle of length k or kdicycle for short. Thereby, the length of
the walk k» 0 is the length of the directed cycle.1 Note that intransitive pairs are directed
cycles of length 3 or 3dicycles.
Two vertices a and b are called strongly connected provided there are directed walks
from a to b and from b to a. A single vertex is regarded as strongly connected to itself.
Strong connectivity between vertices is reflexive, symmetric, and transitive. Hence, strong
connectivity defines an equivalence relation on the vertices of D and yields a partition
VUVU...UV
of the vertices V. The subdigraphs D(VI), D(V2),..., D(V.) formed by taking the vertices
in an equivalence class and the arcs incident to them are called the strong components of D.
The digraph D is strong if it has exactly one strong component.
Directed cycles are sometimes called circuits.
A.4. ALGEBRA
137
A.4 Algebra
Only two topics from algebra are touched in the following subsections: (combinatorial)
matrix theory and polynomial rings. For a proper introduction to algebra consult for example
Allerby (1991) or Herstein (1975). The main focus here is on (ir)reducibility of matrices in
Mn, the set of square (0,1)matrices, and the factorization of polynomials in Zsæl, the
polynomial ring of the integers.
A.4.1 Matrix Theory
The presentation here is due to Brualdi and Ryser (1991), Biggs (1993) and Cameron (1994).
First some algebraic aspects of permutations are discussed.
There are two ways of regarding a permutation.
Definition A.4.1 Let X = (1,2,...,n) be a finite set. A permutation r : X — X is a
onetoone mapping from X onto itself.
For the second representation, we assume that there is a natural ordering of the elements
of X, say (1,2,...,n). Then the permutation r can be represented as as ordered ntuple
(x(1), r (2),...,(n)). If i e X does not change position in the ntuple, r(i) = i, then it is
called a fixpoint. The set of all permutations of (1,2,...,n), equipped with the operation
of composition, is a group. It is known as the symmetric group of degree n, denoted by Sn.
A permutation T can be represented in socalled twoline notation as
n
1 2 ...
1r 2r... n
The top row can be in any order, as long as ær is directly under x for all a e X.
There is another representation of a permutation, called the cycle form. A permutation
cycle, or cyclic permutation, is a permutation of a set X which maps
Xj — 2 — — — X1
(A.3)
where xj,...,x, are all the elements of X in some order. If the cyclic permutation has n—k
fixpoints then the permutation cycle has length k. A permutation cycle (of any length) is
not unique and can start at any point.
Lemma A.4.2 Any permutation can be written as the composition of permutation cycles on
pairwise disjoint subsets. The representation is unique, apart from the order of the factors,
and the startingpoints of the cycles.
D
Proof. for example Cameron (1994), p.30.
The onetoone correspondence of permutation cycles and directed cycles follows imme¬
diately.
A simple indicator function is defined which is needed to establish a homomorphism
between digraphs and adjacency matrices. Let D be a digraph of order n with a, b e V the
set of vertices and whose set of arcs is ECVxV. The indicator function i: VXV — 10,1)
is given by
1: (a,b)EE
((a;;) =
0: (a,b) 4 E
Next, the adjacency matrix of a digraph is defined by the indicator function t.
APPENDIX A. MATHEMATICAL BACKGROUND
138
Definition A.4.3 Let D be a digraph of ordern whose set of vertices is V  a, a,..., an).
The adjacency matrix is a (0, 1)matrix of size n x n A = a,(i,j = 1,2,...,n) whose
entries a;; are given by the indicator function i: V XV — (0,1),
Let D be a digraph and let C be a collection of directed cycles C; C E. Define an
indicator function k by
1: (a,b)EC.
«(cij) =
0: (a,b) 4 C.
An incidence matrix for the dicycles of a digraph can now be defined by the indicator function
K.
Definition A.4.4 Let D be a digraph whose set of arcs is E CVXV and C the family of
directed cycles C; CE. The incidence matrix of dicycles is a (0,1)matrix C = sc, (i,j =
1,2,..., n) whose entries c;; are given by the indicator function k : CX E — 10,1),
Definition A.4.5 Let x be an integer vector indexed by the arcs of a digraph D. The
indegree of a vertex of D in x is the sum of the entries of x corresponding to the arcs
entering that vertex. The outdegree of a vertex of D in x is the sum of the entries ofx
corresponding to the arcs leaving that vertex. A vector x is eulerian if each vertex of D has
equal indegree and outdegree.
The n! permutation matrices of order n are obtained from the identity matrix I, by
arbitrary permutations of rows and columns of I,. A permutation matrix P of order n
satisfies the matrix equations
PPT = PTP  In
(A.4)
A matrix A of order n is called reducible if by simultaneous permutations of its lines a
matrix can be obtained of the form
O
A1
PAPT =
A21 A2
where Aj and A» are square matrices of order at least one. If A is not reducible, then A is
called irreducible. Notice that a matrix of order 1 is irreducible.
The determinant of a matrix A is a matrix function defined by the formula
(A.5)
det (A)  sign )(1)d2(2) Gn(n)
7
where the summation extends over all permutations r of 1,2,...,n). Suppose that the
permutation r consists of k permutation cycles of sizes l1,l»,..., l, respectively, where li +
12 +... + 1 = n. Then signr can be computed by
sign 7 = (1)414½14.4½1  (—1)»  (1)"(1)'.
(A.6)
The following formula can be applied to compute the determinant of the adjacency matrix
of a digraph.
Theorem A.4.6 If D is a digraph whose linear subgraphs are D., i = 1,...,n and D; has
e; even cycles, then
7
det(A) =Y(1)“
(A.7)
1
O
Proof. see Harary (1969), p.151.
A.4. ALGEBRA
139
A.4.2 Polynomial Rings
It is impossible to give a selfcontained and short account on polynomials and polynomial
rings. Therefore, only a definition, and two theorems are presented here. The omitted proofs
can be found in most textbooks on (abstract) algebra (e.g., Herstein, 1975, chap. 3). Let
R be a commutative ring with unit element. By the polynomial ring in x over R, denoted
as Rx we shall mean an expression of the form
o(2) or" + +2 ++
where co,..., ca are coefficients in R, 1,...,n are positive integers, and x is indeterminate.
The degree of a polynomial o(x) e Rx is the highest exponent n with a nonzero coefficient.
A polynomial is called monic, if co equals unity and primitive if the greatest common divisor
of the coefficients equals one. The polynomial o(x) = LLcx' of degree n is called reducible
if and only if it can be expressed by the product of nonzero polynomials of lower degree,
and irreducible otherwise. Whether or not a polynomial is irreducible depends on which
polynomial ring it is considered as belonging to.?
Definition A.4.7 Let R be an integral domain. R is a unique factorization domain if and
only if
(i) every nonzero nonunit element p of R can be written as p = 9192...qm, the q:
being irreducibles, and
(ii)
if p=q192...qm =q9...q then m =n and q = uq, pair off with u some unit
of R.
Theorem A.4.8 If R is a unique factorization domain and if p(x) is a primitive polynomial
in Rsxl , then it can be factored in a unique way as the product of irreducible elements in
Rsæ).
Proof. see Herstein (1975), chap. 3, p.164.
The next theorem offers a very helpful tool to decide if a polynomial ring is a unique
factorization domain with the desired factorization properties.
Theorem A.4.9 If R is a unique factorization domain, then so is Rla.
□
Proof. see Herstein (1975), chap. 3, p.165.
We conclude this section with two corollaries. The first states that Zx the polynomial
ring in x over the integers Z is a unique factorization domain.
Corollary A.4.10 The polynomial ring Zixl over the integers Z is a unique factorization
domain and every monic polynomial can be factored in a unique way as the product of irre¬
ducible elements in Zsxl.
2For example every polynomial in Clal the polynomial ring over the complex numbers C has a unique
representation as linear factors. This is known as the fundamental theorem of algebra.
APPENDIX A. MATHEMATICAL BACKGROUND
140
Proof. Only a sketch of a proof is given. From the fundamental theorem of arithmetic
Z are a unique factorization domain. It follows from Theo¬
it is known that the integers
rem A.4.9 that the polynomial ring Zsx is a unique factorization domain. Clearly every
monic polynomial %(x) in Zsæ) is primitive. Hence, according to Theorem A.4.8 %(x) can be
D
uniquely factored into irreducible polynomials.
It follows that every %(x) in Zx has a unique representation in terms of irreducible
polynomials in Zsæ).
Corollary A.4.11 The polynomial ring Zsæ,...,x in the indeterminates xi,...,xn over
the integers Z is a unique factorization domain and every monic polynomial can be factored
in a unique way as the product of irreducible elements in Zsæi....,x.l.
The characteristic polynomial of an adjacency matrix is given by
9(x) = det (xIn  A)
(A.8)
where x is indeterminate and I, is the identity matrix.
If the characteristic polynomial is set equal to zero the solutions for the indeterminate
are called eigenvalues of the matrix (e.g., Wilkinson, 1988).
Appendix B
Supplements
PI
B.1 Experiment 1A and 1B
B.1.1 Instructions
Instruction
This is a simple study on preferences between chocolate bars. In the following you are
asked: Which chocolate bar tastes better?" Shortly afterwards the name of a chocolate bar
is displayed on the left and right side of the screen. By pressing key 'F' you choose the left
chocolate bar, and by pressing key J' you choose the right bar. There are no correct or
incorrect choices. After reading this instruction, please press key F' or J' to exercise the
choice tasks in three examples.
After three training trials a similar instruction was displayed.
Instructions of Experiment 1 were translated from German.
141
APPENDIX B. SUPPLEMENTS
142
B.1.2 Stimuli
Stimuli were names of twelve chocolate bars which are listed in Table B.1. The code for each
chocolate bar is given in the left column, the name of the chocolate bar in the middle column.
and in the rightmost column it is listed how many of the 40 subjects in Experiment 1A and 1B
tasted a chocolate bar prior to the forced choice comparison. Each subject was encouraged
to taste any of the chocolate bars they were unfamiliar with before they chose between names
of chocolate bars in two pair comparisons.
Table B.1: Chocolate Bars
Code
Name
Tasted
1
9
Balisto
Banjo
2
3
Mars
1
4
Milky Way
5
11
Lion
6
Snickers
1
7
Bounty
1
8
Duplo
0
9
Nussini
19
4
10
Nuts
11
Twix
12
Kitkat
8
—
B.1. EXPERIMENT IA AND IB
143
B.1.3 Results of Experiment 1A
The following tables list the number of all dicycles and ear dicycles for each subject and
session as determined by the prolog programs in Appendix C.1. The number of all dicycles
can also be described by the coefficients 24 of polynomial y. The number of ear dicycles ex
which constitute directed ear bases are listed in separate tables.
Table B.2: Coefficients of y for Each Subject and Session (Exp 14)
Croup 8 (Resolution Block Designs
Subj
27
25
28
29
211
210
26
23
24
212
X
5
1
3
4
0
35
0
0
5
3
0
4
0
1
0
0
0
0
39
79
482
2986
206
2725
1032
24
1031
3504
1936
3
0
0
1
2
3
0
0
0
73
2
8
10
12
8
41
3
2
0
0
0
2
6
9
2
9
0
43
0
3
0
0
0
0
2
0
0
0
4
1
45
36
2291
628
267
105
1747
1239
1958
621
10
8
8
47
4
1
0
2
4
3
0
0
0
0
49
2
0
1
0
4
4
0
37
0
0
0
6
0
0
1
3
2
0
0
0
7
16
0
15
11
53
3
D
0
9
26
10
31
24
17
0
0
5
1
4
54
0
0
0
0
0
2
103.1
51.9
298.6
194.1
272.5
105.2
7.0
Mean
25.3
350.4
12.7
65.3
126.3
62.8
174.7
30.3
14.3
196.7
229.1
7.2
944.3
108
861.7
63.8
SD
326.0
6.4
6120
23.7
325.7
151.2
198.6
552.4
83.5
724.5
618.9
391.0
32.2
195.5
10.4
Note: Subject numbers in the leftmost column were assigned to each subject in an un¬
related experiment between sessions. The first and second line for each subject refers to
Session 1 and 2, respectively.
144
B. SUPPLEMENTS
Table B.3: Coefficients of % for Each Subject and Session (Exp 14)
Group
P Repetition Bock Designs
Subject
29
27
25
23
26
28
210
211
24
212
2
3
0
4
0
1
0
34
0
0
4
0
0
0
2
0
0
36
0
0
0
0
0
0
38
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
D
40
0
0
0
0
0
0
0
0
0
7
0
7
0
0
0
42
0
0
0
0
0
1
0
0
2
14
0
0
0
4
10
0
44
10
0
0
0
0
0
0
0
oa
0
46
0
0
0
0
0
0
48
0
0
0
0
0
0
0
0
0
50
0
0
1
79
14
44
66
12
38
26
67
0
)
6
8
8
4
0
1.
0
0
52
6
1
4
4
0
0
0
2.3
Mean
1.8
0.8
2.2
0.1
2.4
6.8
4.8
2.6
7.9
6.6
3.8
1.2
3.5
1.
2.8
0.3
4.8
0
3.6
3.7
SD
0
21.2
8.1
4.3
13.8
12.0
25.0
20.9
3.8
0
Note: Subject numbers in the leftmost column were assigned to each subject in an un¬
related experiment between sessions. The first and second line for each subject refers to
Session 1 and 2, respectively.
B.1. EXPERIMENT 1A AND 1B
Table B.4: Number of Ear Dicycles for Each Subject and Session (Exp 14)
Group
Dieeiciution Block Designs)
e7
63
e5
Subj
ee
e8
e4
810
89
e11
0
3
4
0
35
2
0
0
0
0
0
3
1
3
0
5
—
7
14
2
6
5
39
6
3
7
4
2
0
0
0
5
5
6
41
3
2
0
6
2
43
2
0
0
—
0
0
4
45
0
6
6
2
9
6
0
13
8
6
47
0
0
0
0
2
2
49
0
0
3
0
0
1
2
51
0
0
0
0
0
0
0
0
6
4
2
53
—
6
6
2
5
5
5
0
54
2
0
0
0
0
1
0
0.7
0.3
0.6
1.5
4.9
0.9
1.5
2.3
Mean
3.0
0.6
0
0.9
0.2
2.2
3.6
2.7
1.3
0.1
0.9
4.4
1.5
28
2.2
2.1
1.9
28
SD
2.2
4.1
0.6
1.9
1.4
0.3
1.9
2.6
1.3
0
0
0
0
612
0
145
146
B. SUPPLEMENTS
Table B.5: Number of Ear Dicycles for Each Subject and Session (Exp 1A)
D(Repetion Block Designs
Group
e7
69
e11
65
e8
e6
610
Subj
e4
612
63
0
5
5
5
0
2
3
4
34
0
0
4
0
2
0
0
0
0
0
36
0
0
0
0
0
0
0
0
0
0
0
0
38
0
0
0
0
0
0
0
0
0
40
0
0
1
0
0
0
0
0
5
42
2
0
0
4
3
2
44
0
0
0
)
0
2
46
0
0
0
0
1
1
0
6
48
0
0
0
0
0
0
0
0
1
0
50
0
0
8
8
3
6
5
0
0
2
6
52
0
0
4
4
2
Mean
1.9
1.0
0.6
1.6
1.0
0
1.5
0.6
0.8
2.0
0.5
0.3
)
SD
2.1
1.8
0
2.5
1.1
0.3
2.6
2.6
1.9
1.6
1.9
0.9
1
—
0
0
0
5
0
0
0
0
B.I. EXPERIMENI IA AND IB
B.1.4 Results of Experiment 1B
Table B.6: Coefficients of% for Each Subject and Session (Exp 1B)
(ResolutionRepetition Block Design
Sroup sp
29
28
25
210
27
26
24
211
Subj
23
5
0
0
0
0
0
2
5
0
0
0
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
1
13
0
0
0
0
0
0
0
0
90
45
405
178
430
317
19
125
295
15
3
0
0
1
0
0
0
2
0
170
6
8
7
0
2
0
0
0
0
0
0
0
4
0
10
10
9
0
21
1
0
0
2
0
0
0
9
4
4
0
23
0
0
0
0
0
0
0
0
0
2
0
0
0
25
0
0
0
0
0
0
0
0
0
0
0
0
0
0
27
0
4
0
3
0
0
4
0
0
0
2
0
29
1
0
0
0
0
0
0
0
0
0
33
0
0
0
0
0
0
2
0
1
0
18.7
40.5
31.
43.0
Mean
3.8
5.9
12.5
29.9
10.2
0
0
0
0.2
0
0
0.8
1.5
0
SD
0
128.1
100.2
5.7
136.0
13.4
56.0
93.2
39.5
28.2
1.0
1.5
0.4
Note: Subject numbers in the leftmost column were assigned to each subject in an unre¬
lated experiment between sessions (see Experiment 1A). Therefore some subject numbers
are missing. The first and second line for each subject refers to Session 1 and 2, respec¬
tively.
Subject 17 was excluded because he was unfamiliar with more than 5 chocolate bars.
212
0
0
0
0
D
0
147
148
APPENDIX B. SUPPLEMENTS
Table B.7: Coefficients of y for Each Subject and Session (Exp 1B)
RepetitionResolution
P5
Group
Bock Designs)
Subject
25
27
28
23
29
24
26
210
211
212
3
0
3
0
0
4
0
0
0
0
1
0
0
2
60
0
0
0
0
0
2
12
0
2
0
0
2
0
14
0
0
0
0
0
0
0
0
0
16
0
0
0
0
5
1
3
0
18
3
0
1
0
0
0
3
0
0
4
0
24
0
0
0
0
0
0
0
0
26
0
0
0
0
3
4
28
3
1
0
C
0
2
0
0
0
0
0
0
30
1
3
0
0
0
5
3
0
0
0
2
0
32
7
6
0
0
3
6
0
1
0.6
2.1
1.4
Mean
0
1.0
1.1
1.8
0.3
0
9
5
0
0
1.0
1.4
1.4
SD
0.9
2.3
2.1
2.0
0
0
L
—
Note: Subject numbers in the leftmost column were assigned to each subject in an un¬
related experiment between sessions (see Experiment 1A). Therefore some numbers are
missing. The first and second line for each subject refers to Session 1 and 2, respectively.
«Subject 6 was excluded because he was unfamiliar with more than 5 chocolate bars.
B.1. EXPERIMENT IA AND IB
Table B.8: Number of Ear Dicycles for Each Subject and Session (Exp 1B)
gns)
Desi
E (ResolutionRepetition Block
Group
e7
29
e3
e5
Subj
e8
810
e6
e11
24
5
0
2
5
0
0
0
0
0
0
0
0
0
0
0
0
—
9
0
0
0
13
C
0
0
0
2
6
6
15
0
0
0
2
1
4
170
3
2
0
1
0
9
3
4
21
2
2
0
23
3
1
U
2
0
25
0
0
0
0
2
0
0
0
27
1
0
2
0
4
0
3
2
0
29
0
1
0
1
0
0
0
0
1
33
0
0
0
0
0
1
2
0.6
0.6
0.1
0.5
1.0
1.1
1.6
Mean
2.3
0
0
0.7
0
0
1.5
0
0.2
0.3
1.6
SD
2.3
2.3
1.9
1.9
2.5
1.9
0
0
0.4
0.8
1.5
L
4Subject 17 was excluded because he was unfamiliar with more than 5 chocolate bars.
612
18
0
0
0
7
149
UPPLEMENTS
Table B.9: Number of Ear Dicycles for Each Subject and Session (Exp 1B)
Croup PS (RepetitionResolution Block Designs)
e8
e7
e9
810
ee
611
e5
812
e4
63
Subj
5
5
0
2
0
2
0
2
4
0
0
0
0
0
0
1
0
0
0
0
0
60
0
0
2
0
0
0
0
0
0
0
2
1
0
12
0
0
0
0
0
2
0
0
2
0
1
6
0
14
0
0
0
0
0
5
0
0
16
D
0
0
0
0
0
—
3
2
0
18
1
0
0
0
0
3
1
24
0
0
0
0
26
0
0
0
0
0
0
0
0
0
—
2
0
2
0
2
28
0
2
0
0
0
0
0
0
30
1
0
3
3
0
0
0
0
0
32
2
0
0
0
4
2
2
2
0
0
Mean
0.5
1.1
1.8
1.4
0
0.5
0.5
0.2
(
0
SD
1.1
1.1
0.7
1.3
0.6
1.1
0.7
0
0
Subject 6 was excluded because he was unfamiliar with more than 5 chocolate bars.
8
0
0
0
B.2. EXPERIMENT 2A AND 2B
151
B.2 Experiment 2A and 2B
B.2.1 Instructions
Instruction
This is a simple study on preferences between playing gambles. You are asked to choose
between two gambles which are described by their possible win and loss (in pounds) and
by their chance of winning and losing (in percentages). For your convenience the chance of
winning and losing is also displayed as a pie chart for each gamble.
After reading the instructions there will be 3 training trials followed by 66 trials. In each
trial you are asked:" Which gamble do you prefer to play?" Shortly afterwards two gambles
are displayed on the left and right hand side and you are asked to make your decision as
quickly and as accurately as possible.
On the next page you will see an example and you will learn how to give a response.
Press 'SPACEBAR' to continue¬
As soon as you know which gamble you prefer to play press key D' if you prefer to play
the gamble on the left, and press key "K' if you prefer to play the gamble on the right.
• Please always place your left index finger on key D' and your right index finger on key
K' to ensure an undelayed response.
• Please respond as quickly and as accurately as possible.
APPENDIX B. SUPPLEMENTS
152
B.2.2 Stimuli
In a pretest four additional subjects computed roughly the expectancy value for each of 24
lotteries with approximately the same expectancy. From this pool 12 lotteries were selected
which had equivalent expectancy values in spite of possible rounding errors by the subjects
and which covered the widest range of probabilities and payoffs. The lotteries are listed in
Table B.10 together with their expectancy values.
Table B.10: Description of Lotteries
Chance of Win
Payoff
Expectancy
Lottery
Value
(in %)
Code
(in 2)
3
31
29.90
9.21
9.49
12
79.10
2
3
89
8.72
9.80
45
9.18
20.40
4
5
9
9.51
105.70
8.77
10.70
82
6
2
8.81
11.90
74
40.70
8
9.36
23
51
9.08
9
17.80
9.01
15.80
57
10
8.97
65
13.80
11
55.50
9.44
17
12
B.2. EXPERIMENT 24 AND 2B
B.2.3 Results of Experiment 2A
Table B.11: Coefficients ofy for Each Subject and Session (Exp 24)
Croup 8 (Resolution Block
Design
27
25
26
Subj
23
24
28
210
29
211
44
3
8729
1060
4783
7462
136
2452
398
6785
24
57
275
228
363
0
122
329
106
0
19
125
149
46
114
75
0
46
255
495
7
28
104
987
881
791
520
181
800
617
559
378
79
27
798
235
191
12
0
14
4
12
0
0
0
0
47
486
17
366
103
14
510
348
203
157
88
877
435
766
1129
1227
23
219
362
53
414
447
597
119
19
643
146
236
3
0
0
0
0
0
1
0
0
19
0
7
0
0
2
4
6
4
1
0
0
56
38
68
4
24
69
22
50
22
14
6
10
0
0
10
0
9
2
45
673
205
375
362
20
620
100
549
16
13
0
21
28
0
0
13
22
58
43
15
0
65
15
61
39
25
96
195
258
394
96
39
320
16
409
171
66
26
374
882
1429
33
1374
710
1123
1615
1143
34
533
1820
92
1987
1030
239
548
87
32
2881
2049
1843
1157
2853
232
753
6264
40
64
27019
36631
2299
38369
238
14419
0
0
0
0
0
0
0
0
0
3
3
0
0
0
1
0
0
0
43
368
408
95
184
20
438
46
210
304
5447
40
138
4356
3673
904
363
6037
1997
3
0
1
0
2
0
0
0
0
78
456
269
22
58
433
282
144
512
47
96
36
625
3096
2247
1296
2117
3159
264
589
743
255
1820
1809
100
1143
2210
34
4914
1138
2279
73.3
5002
4493
197.9
3825
25.1
Mean
1287
610.2
988.8
58.6
316.7
21.0
821.3
1257
145.1
354.5
681.9
554.9
601.5
89.5
39.5
16.2
320.3
189.2
4494
11484
70.6
12022
8452
1939
228.6
705.4
SD
17.4
1422
47.5
1973
1791
673.7
14.1
312.1
128.8
1221
449.1
977.8
586.3
220.6
93.0
765.4
1022
34.5
11.1
212
2633
0
28
43
0
30
65
0
0
0
107
0
0
15
280
388
559
17589
0
0
52
1511
0
710
0
2061
271.7
68.1
5517
495.0
175.7
153
154
APPENDIX B. SUPPLEMENTS
Table B.12: Coefficients of y for Each Subject and Session (Exp 24)
Group P (Repetition
Block Design
29
25
26
27
Subj
28
24
23
210
211
212
44
1072
400
4
4923
138
2491
7293
7784
9248
2834
30
468
425
1486
84
917
203
1929
1945
1276
555
91
32
1140
1962
2724
232
2075
2893
719
765
60
26
129
9
652
262
452
669
98
397
79
51
35
120
167
154
19
189
80
21
8
14
52
56
42
28
62
32
21
1
0
9
5
0
6
0
0
15
1
9
0
0
0
0
0
0
0
0
0
4
6
4
1
0
0
0
67
329
280
40
21
48
20
194
119
285
166
41
78
79
124
163
21
177
163
26
129
69
869
563
1048
497
919
26
150
304
118
30
92
21
1857
235
1101
2424
545
2261
1331
369
0
0
0
0
0
0
0
0
0
0
3
6
0
4
0
0
0
0
61
34
5761
13142
710
207
2150
34272
24351
32426
15474
45
147
8028
417
1114
2604
6929
5113
2212
9323
45
422
9009
5375
10767
8711
2679
3816
140
1137
93
127
41
4495
1004
385
2579
6482
8282
7039
41
2322
69
1665
391
1528
170
26
786
862
205
1299
44
15
98
65
0
33
9
111
141
144
4430
55
184
9604
8916
16872
1735
48
19943
597
22405
27
365
0
928
647
583
144
914
70
169
8
260
145
49
19
322
197
109
50
289
50
1420
164
7053
3430
507
12158
14572
5058
11684
50
721
296
38
1541
3870
1014
4112
105
2903
2727
435
1167
47
159
2743
6948
2444
9441
5323
8261
28
367
680
71
51
1088
635
1224
172
1024
172
145
45
2674
419
1128
8682
10622
3585
8734
5346
34
604
2790
98
256
2081
2818
547
1225
1782
867.5
9308
8083
Mean
36.2
4303
7247
1092
320.7
2086
3555
439.0
2838
2547
1723
945.2
748.8
2101
69.6
180.4
25.2
411.9
2038
1614
883.9
2703
67.1
170.8
2397
765.2
24.2
4459
10756
11467
730.0
8213
5088
1960
65.2
236.5
17.0
SD
3980
156.0
2035
3180
52.5
3290
1016
425.9
15.8
1220
4069
1060
2115
3260
54.3
444.2
1313
3386
161.5
15.2
Note: The first, second and third line for each subject refers to Session 1, 2 and 3,
respectively.
B.2. EXPERIMENT 24 AND 2E
Table B.13: Coefficients of y for Each Subject and Session (Exp 24)
Group
(Random
block
Design)
Subj
29
23
25
27
26
28
24
211
210
212
85
453
5
1209
6
31
271
832
211
869
1284
91
552
31
243
836
1054
1981
1661
1648
183
23
51
273
198
112
127
260
0
0
0
56
10
626
183
4736
24606
22140
9993
1828
10352
18327
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
15
82
69
34
13
54
112
110
22
0
0
777
30
86
422
775
1171
1108
193
228
0
37
64
18
165
109
239
89
219
18
191
356
7237
914
38
128
2067
3910
5618
18
519
2160
98
30
60
63
127
116
13
0
15
0
73
984
178
672
373
659
216
28
937
56
58
8
28
8
70
2
28
24
35
35
1
21
21
5
5
0
3
0
1
0
0
722
35
110
2670
300
3909
943
27
1519
3742
2706
47
15
27
58
20
0
52
0
25
0
8
0
19
11
18
23
0
25
35
36
48
55
13
32
22
0
6
0
2
0
0
0
7
9
11
10
4
12
11
8
0
0
470
5767
48
3225
6742
14873
1311
37
152
11463
13053
779.
64
713
14011
2145
6048
18183
26202
37(32
36822
247
709
34758
64
5996
16886
35985
2039
267
13946
25420
50
4386
1464
531
7095
11501
3517
43
13941
173
11157
38
863
343
3900
119
5197
5625
1263
3542
1922
475
85
30
963
1988
203
2099
1180
319
1602
85
10
45
9
26
22
21
7
0
0
0
0
0
3
0
0
0
0
0
0
201
6546
1614
5261
5495
30.6
692.0
Mean
3224
266.2
2325
93.2
4179
4581
3462
2045
1000
417.3
20.6
163.6
1963
62.7
3624
2887
19.1
808.6
3884
1698
1722
323.2
54.8
130.6
6470
7617
666.0
SD
3703
3397
1714
8569
65.0
18.0
224.9
8158
5713
1886
11534
4359
11780
226.5
675.1
77.2
20.1
4336
7946
11297
5329
1852
10945
80.5
626.2
216.6
18.9
Note: The first, second and third line for each subject refers to Session 1, 2 and 3,
respectively.
0
3
155
156
7.8
5.6
6.1
3.3
3.5
2.9
Table B.14: Numbers
63
Subj
e4
7
3
11
8
4
7
7
7
10
11
8
6
9
14
8
12
6
.
1
19
2
2
7
22
4
7
10
5
28
9
7
6
6
4
10
12
33
6
7
10
12
40
0
46
3
6
47
7
4
Mean
7.6
5.9
4.8
SD
2.6
2.7
1.9
7
APPENDIX B. SUPPLEMENTS
of Ear Dicycles for Each Subject and Session (Exp 24)
Designs
Block
5 (Resolution
Group
e7
e9
e6
eg
e5
810
e11
612
6
8
3
15
10
1
6
5
4
4
0
4
3
5
2
0
)
)
7
9
I1
5
1
9
6
11
8
2
0
4
2
8
9
9
10
0
7
4
11
13
9
5
5
7
7
2
0
0
0
2
0
0
9
3
5
4
4
0
0
12
6
10
0
0
4
0
0
0
3
2
5
9
6
9
9
2
5
2
11
11
0
4
10
6
3
10
2
8
2
4
13
10
0
0
0
)
0
0
2
8
8
10
4
1
7
8
7
9
5
11
1
0
0
0
0
9
5
10
0
0
1
12
5
3
8
6
1
0.7
1.6
7.3
4.1
6.3
0.1
8.1
5.0
0.9
6.4
4.2
6.3
0.2
2.3
3.8
4.7
4.8
3.8
0.7
2.2
1.4
4.1
1.4
0.3
0.7
3.5
2.9
3.4
3.9
3.3
4.1
4.7
1.1
0.4
2.2
3.6
3.4
4.2
3.9
2.7
0.9
1.9
0
0
B.2. EXPERIMENT 2A AND 2B
Table B.15: Number of Ear Dicycles for Each Subject and Session (Exp 24)
Block Desiens
Group
P (Repetition
Subj
e5
e7
89
ee
e4
83
e8
810
e11
612
9
6
5
8
e
3
10
4
1
10
6
10
0
10
0
0
10
6
8
9
10
0
8
6
5
9
4
10
10
1
5
6
8
6
4
4
9
11
0
7
8
5
7
4
2
0
0
5
2
4
0
15
5
6
5
0
1
0
4
—
8
6
20
2
9
9
10
10
8
8
11
2
6
8
11
12
10
21
0
0
0
0
0
0
0
6
0
4
0
0
0
f
6
6
8
6
Z
1
34
A
4
11
1
7
4
6
10
—
9
11
10
0
41
1
9
6
8
1
9
0
10
0
12
7
0
2
8
4
48
8
6
—
4
6
11
10
7
0
0
7
9
11
5
10
12
8
5
8
50
2
2
9
8
11
10
0
12
2
5
9
8
9
4
1
13
9
9
4
6
11
10
4
51
0
6
4
6
6
10
6
2
3
9
10
7
4
10
1
9
6.0
4.1
6.6
1.3
7.3
8.9
2.7
Mean
7.0
6.0
5.3
1.0
2.3
6.8
4.9
7.4
2.9
6.2
6.3
1.3
3.6
5.6
0.5
6.4
2.8
7.3
7.9
8.1
2.2
2.5
2.1
2.9
2.4
1.2
2.0
2.7
SD
2.9
3.5
4.0
2.8
4.0
1.9
2
2.7
3.6
3.9
1.7
4.0
4.1
1.0
2.3
3.2
2.9
2.4
2.8
0
0
4
0
0
0
0
0.6
0
1.3
0
157
158
APPENDIX B. SUPPLEMENTS
Table B.16: Number of Ear Dicycles for Each Subject and Session (Exp 24)
Designs
Croup N (Kandom Block
e9
e8
e7
e5
ee
e11
810
84
Subj
63
e12
3
8
5
7
9
5
12
1
5
6
15
12
8
6
4
1
7
0
1
X
9
6
8
8
3
6
3
10
0
0
0
0
0
0
3
0
4
2
8
2
10
0
13
—
6
8
0
6
6
0
0
6
4
3
6
10
6
8
3
6
5
10
7
1
3
14
18
0
3
6
0
6
6
5
8
7
10
9
3
6
4
0
4
24
4
13
0
0
7
0
4
2
5
0
5
4
6
9
13
10
27
4
1
4
4
0
0
0
3
1
4
0
3
—
3
5
4
10
7
0
32
0
0
2
0
0
4
6
0
0
0
8
0
7
1
0.
8
7
12
8
2
37
4
3
0
9
—
5
5
6
8
4
11
5
9
4
6
10
2
19
4
4
15
10
43
11
2
1
8
10
6
6
10
1
2
2
15
11
12
6
4
4
45
0
2
0
0
0
0
0
0
0
0
1
0
0
3
0
0
0
7.0
8.6
4.6
1.0
2.7
Mean
5.4
7.0
8.2
0.4
6.4
5.3
5.5
3.4
3.0
3.8
0.4
2.1
0.9
5.9
3.4
5.3
4.7
6.3
2.4
0.6
1.1
1.6
SD
2.9
2.1
2.9
2.9
3.
2.3
1.2
1.8
1.0
4.5
4.6
3.0
3.2
3.5
2.9
3.2
1.7
0.7
4.3
3.0
2.5
3.6
3.9
3.3
2.8
2.2
1.1
0.1
0.3
0
0
0
B.2. EXPERIMENT 24 AND 2B
B.2.4 Results of Experiment 2B
Table
B.17: Coefficients of % for Each Subject and Session (Exp 2B)
Group
Snt (ResclutionKandom Repetition Block Desien)
27
Subj
25
28
29
23
26
24
210
211
7
0
5
0
11
10
2
3
1
0
0
0
0
0
0
0
0
785
6
877
30
355
188
73
667
0
387
89
793
30
1056
562
993
470
221
127
51
157
70
193
21
0
163
99
0
49
12559
473
15376
3347
1333
7154
12
16834
153
89
32
1120
1995
225
3203
2896
534
2407
0
0
0
0
2
0
0
43
2114
948
6661
133
4857
4001
16
5997
371
165
923
23
778
626
345
363
1022
69
43
11
34
0
31
11
22
0
0
58
0
5
49
46
23
32
22
14
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
90
30
34
1834
1058
528
225
1919
2691
2549
705
2131
61
212
5677
23634
33013
12846
31045
4474
39
801
1731
4839
3114
109
3512
323
5
10
9
13
35
12
14
1
11
0
9
5
11
0
10
0
11
0
0
3
0
0
2
0
1
0
0
0
0
6
4
1
36
0
2
0
1
0
0
0
0
0
0
0
0
53
570
178
14745
13995
3991
17615
8346
1613
39
795
37
4393
318
4715
3063
111
3363
1703
410
2507
4900
6510
42
1083
8945
141
7712
3
49
0
0
0
0
0
0
0
1
2
7
0
0
0
8
8
0
12
12
4416
3690
3590
2224
1124
190.2
24.6
488.7
68.8
Mean
1988
428.5
4227
3294
3731
58.8
992.4
164.8
20.0
446.4
1002
12.6
1226
818.8
1378
33.8
87.9
209.2
5455
7074
6191
19.5
1511
66.8
3188
607.5
214.3
SD
3956
9672
7301
10245
69.5
20.1
1750
663.5
224.4
2228
3063
2677
1733
902.3
16.1
394.9
151.4
51.2
212
3
0
0
0
7179
940
0
1732
79
0
0
0
691
14772
1288
0
0
6151
1192
2102
0
0
1575
1698
339.0
2749
4615
740.0
)
159
160
INDIX B. SUPPLEMENTS
Table B.18: Coefficients of% for Each Subject and Session (Exp 2B)
—
Broup Pps (RepetitionKandomResclution Block Design
27
28
Subj
26
210
29
25
23
24
212
211
437
150
5313
1166
8341
47
9745
7502
2
2702
2826
66
22
432
890
359
183
2156
1547
2189
1344
4
9
11
16
14
0
0
0
37
1880
2317
1930
621
761
268
106
1208
8
0
41
4942
379
2126
5788
3909
6500
1907
972
130
469
1486
31
208
1349
878
88
1649
260
892
4597
436
13091
10295
2935
46
11
6113
145
11066
1206
11855
4897
169
12048
520
14663
1445
51
7182
3493
60
12166
30205
27911
21985
5456
2086
696
13011
207
58
9245
857
97
2361
547
2982
1175
17
2082
230
14398
52
12933
7613
16669
6226
525
1487
172
3633
4144
3061
1056
747
1601
4397
107
2894
301
38
50
7244
13509
5831
1411
15715
3466
159
12251
502
25
0
0
48
87
69
30
73
13
0
25
0
0
0
26
11
35
20
13
29
5874
899
5301
4315
10
3595
1561
1959
125
353
26
1253
419
60
1037
639
0
333
1019
151
21
0
0
0
0
0
0
0
12005
6179
47
2989
153
10469
13596
5430
439
1215
31
55
9544
4211
9248
16595
22535
568
1659
187
20630
3521
10056
4983
394
8404
1047
45
8128
133
2484
406
591
2749
251
2823
2089
1633
38
91
1220
33
0
105
20
55
241
21
119
218
309
329
0
0
46
94
115
51
156
10
143
18
476
4403
10772
42
13058
48
6421
3162
10532
1313
152
75
405
214
228
331
368
51
113
23
43
38
146
95
250
0
252
171
292
20
775
418
69
122
872
164
350
962
612
44
47
9
86
55
136
106
30
120
15
0
47
42
14
83
0
28
70
72
15
7973
2603
6434
6627
4184
2143
931.9
41.0
124.7
3556
Mean
2293
5389
6397
3146
691.9
5133
1584
94.5
266.1
31.4
3625
4631
1785
4031
477.9
2188
1094
68.6
188.0
24.9
227
5674
5110
4193
2299
2323
1027
SD
377.9
31.9
7.7
118.8
7558
8430
3416
3576
1633
633.6
6322
63.1
208.9
16.5
4096
6976
9548
3877
8803
666.5
1745
221.4
65.0
18.3
Note: The first, second and third line for each subject refers to Session 1, 2 and 3,
respectively.
B.2. EXPERIMENT 24 AND 2B
Table B.19: Number of Ear Dicycles for Each Subject and Session (Exp 2B)
Group SNP (ResolutionKandom Repetition Block Design
e5
Subj
e7
e9
e8
63
e10
e4
e6
611
612
6
6
6
0
0
0
10
2
3
0
0
0
0
0
0
0
0
5
—
—
4
4
4
5
8
6
0
0
0
—
9
2
4
10
2
0
7
3
4
4
8
2
0
2
8
6
8
4
5
12
2
6
9
11
11
11
1
3
0
0
0
0
0
9
4
1
10
11
4
16
3
10
10
11
4
2
6
4
3
5
0
—
5
4
3
0
23
0
0
0
1
0
0
0
0
0
0
0
—7
9
8
6
30

7
10
6
4
4
6
10
6
9
2
6
6
2
35
1
0
0
0
0
1
0
0
0
0
0
7
4
0
36
.
3
9
0
0
0
20
0
0
0
0
4
4
9
6
39
2
2
6
8
10
13
6
0
4
10
6
3
8
6
12
0
0
1
0
49
0
0
0
0
0
0
5
0
5
3
0
0
6
0
1
0
6.1
4.8
15
4.4
2.9
0.5
0.1
Mean
53
6.4
2.7
4.3
0.3
3.3
5.6
4.8
0.9
0.3
6.3
1.9
2.2
2.6
0.1
1.8
2.0
0
0.7
1.8
2.4
3.7
3.3
2.3
3.6
3.7
2.8
2.9
0.8
2.1
0.3
2.6
2.2
SD
4.6
4.2
3.1
1.4
4.7
3.5
0.9
3.0
0.9
2.4
1.6
4.0
3.4
3.2
3.0
0.3
2.9
2.9
3.3
Note: The first, second and third une for each subject refers to Session 1, 2 and 3,
respectively.
0
161
162
APPENDIX B. SUPPLEMENTS
Table B.20: Number of Ear Dicycles for Each Subject and Session (Exp 2B)
Design
RepetitionKandom Resclution Block
Group PNS
e9
88
e10
e5
e11
ee
e7
612
83
e4
Subj
5
15
3
6
2
3
5
10
i
2
5
9
5
5
9
9
6
10
10
0
5
3
4
2
1
3
0
4
5
5
6
5
10
8
5
8
1
3
0
1
11
6
10
10
2
6
6
10
10
9
8
2
6
6
8
12
6
6
8
11
e
5
10
J
1
8
6
6
5
8
10
4
0
X
3
5
6
3
6
6
17
4.
6
4
11
6
9
77
4
2
6
8
11
10
7
2
5
6
8
3
10
25
5
0
4
4
4
0
4
4
4
2
0
—
8
6
3
6
6
6
6
5
8
26
5
4
6
4
8
6
0
0
6
4
4
3
10
I1
31
2
8
5
6
4
11
9
2
5
5
7
8
2
9
12
9
6
9
6
6
0
38
4
1
6
7
2
13
11
0
0
3
9
8
8
6
0
0
0
—7
1
10
42
12
0
9
2
12
5
8
6
6
10
—
9
5
44
11
3
10
11
5
1
7
6
6
8
9
9
1
3
4
2
4
6
8
5
2
0
Mean
7.5
8.2
7.1
6.6
7.9
6.4
4.8
2.9
1.9
7.7
6.7
8.0
8.0
3.9
6.6
3.7
1.1
1.8
5.6
6.0
5.9
5.9
3.1
5.1
3.9
1.4
0.2
SD
1.7
2.6
2.2
2.5
2.6
1.5
2.0
2.0
1.8
2.1
2.4
3.2
2.1
2.6
1.9
2.1
3.2
1.3
2.8
3.0
2.8
3.8
3.3
3.3
2.5
0.4
3.1
0
0.7
0.2
0.3
0.9
0.4
0.7
0
1
0
163
B.3. EXPERIMENT 3A AND 3B
B.3 Experiment 3A and 3B
B.3.1 Instructions
Instruction
This is a simple study on the perception of brightness. Therefore, the light in your
room should be switched off. In this study you are asked to compare two squares which are
surrounded by grey frames.
Äfter reading the instructions there will be 3 training trials followed by 66 trials. In
each trial you are asked:"Which square does look brighter to you?" Shortly afterwards two
framed squares are displayed on the upper and lower part of the screen and you are asked
to make your decision as quickly and as accurately as possible. On the next page you will
see an example and you will learn how to give a response.
Press ’SPACEBAR' to continue¬
Every square is surrounded by a frame of different grey, but you are asked to compare
only the brightness of the square in the center. As soon as you know which square in the
center looks brighter to you, press key 'U' if the square within the frame on the upper part
of the screen looks brighter, and press key 'N' if the square within the frame on the lower
part of the screen looks brighter.
• Please place your head on the chinrest during all trials. Don’t try to adjust the seat
or the chinrest during the trials.
• Please always place your left index finger on key 'U' and your right index finger on key
D’to ensure an undelayed response.
• Please respond as quickly and as accurately as possible.
Press 'U' or 'N' to start first training trial¬
• The following 66 trials are run without a break. (The question "Which square does
look brighter to you?" does not appear on the screen any more.)
• The light has to be switched off.
• Please place your head on the chinrest during all trials. Don’t try to adjust the seat
or the chinrest during the session.
• Please place your left index finger on key 'U' and your right index finger on key 'N' to
ensure an undelayed response.
• Please respond as quickly an as accurately as possible.
Press 'U' or 'N' to start¬
APPENDIX B. SUPPLEMENTS
164
B.3.2 Stimuli
The stimuli used in Experiment 3 were square disks in the center of a square surround.
The luminance of the center and surround of each stimulus was chosen in such a way that
the center was almost indiscriminate in brightness among stimuli. Figure B.1 gives only a
impression of the stimuli and the levels of gray because the resolution of dots per inch (dpi)
is varied rather than the luminance of pixels on screen (see Table B.21). The unattenuated
monitor which appeared monochromatic was calibrated using a Minolta LS110 photometer.
Each pixel on the monitor was controlled by three 8 bit digital to analog converters (DAC)
and a color lookup table (CLUT) allowing for 256 luminances (or levels of gray) on screen.
Consequently, there were 256 by 256 possible combinations of gray levels for center and
surround. The stimuli were generated by using the method of adjustment.
Table B.21: Description of Disks
Luminance of
Luminance of
Disk
Center (in cd/m'
Surround (in ed/m
Code
18.08
20.80
1
33.29
41.67
2
3
14.70
7.54
4
60.86
42.52
5
35.71
29.99
11.77
6
17.92
7
31.88
28.79
15.23
8
18.74
38.88
54.07
9
4.00
12.76
10
1.46
11.61
11
47.31
34.87
12
B.3. EXPERIMENT 3A AND 3B
.
r
1
Figure B.1: Illustration of stimuli: The square disks in the center appear
indifferent in brightness due to brightness contrast of center and surround
165
166
APPENDIX B. SUPPLEMENTS
B.3.3 Results of Experiment 3A
Table B.22: Coefficients of y for Each Subject and Session (Exp 34
Group s (Resolution Block
Design
28
25
Subj
27
210
26
24
29
212
23
211
57
443
254
126
665
792
24
90
1
674
361
79
407
30
184
1431
446
1210
778
1131
0
100
719
4910
284
37
5845
4685
3130
1627
1891
92
31
2358
240
3805
3483
1273
884
588
2695
6
166
50
1471
7414
523
3593
5000
15230
12487
12279
512
3500
162
12051
51
1423
15129
7226
5237
12642
0
0
2
0
0
1
0
0
0
12
0
43
71
77
26
33
101
15
113
120
4
0
2
0
3
0
0
0
0
29
17
85
185
202
52
16
161
83
133
20
549
86
3164
3338
29
2203
2177
227
656
1187
76
41
48
83
67
21
18
10
3
5
5
6
0
0
1
0
23
0
49
0
36
12
18
0
50
21
37
4725
719
1680
4371
1592
5582
107
3025
285
34
146
761
810
557
60
132
918
30
312
24
427
1113
78
826
422
1371
1768
189
30
1808
335
31
61
110
21
157
87
180
158
15
7
4
0
0
0
1
0
0
35
7
0
0
0
0
2
0
0
0
0
0
2
3
36
4
0
0
0
190
438
75
28
891
2120
1532
416
2100
1376
0
0
41
24
35
9
13
0
29
0
22874
10007
4520
1757
56
594
10607
39
24827
18104
197
500
48
12174
6995
1372
5075
167
3372
14565
11632
36
2216
0
1740
97
1191
1869
677
263
607
197
46
421
398
65
102
321
49
234
20
0
43
4719
970
4060
9051
364
124
2305
7949
10282
5059
4071
37
3372
116
1795
5623
806
1416
319
2390
19.0
1173
49.8
2651
724.8
3046
319.9
1445
127.1
Mean
4852
4049
82.3
3888
1555
2557
1310
225.7
574.9
28.7
2647
3402
1022
180.2
1889
997.5
449.2
2917
23.9
66.2
SD
60.0
539.3
7740
1391
3326
182.3
7154
3094
5622
16.4
6096
4773
2206
5197
529.8
2813
1330
184.7
15.7
57.3
4053
4866
1679
2366
480.4
56.5
1162
3903
175.4
17.0
B.3. EXPERIMENT 3A AND 3B
Table B.23: Coefficients of y for Each Subject and Session (Exp 34)
Design
Group P (Repetition
Block
29
25
Subj
28
210
27
211
212
24
26
23
7401
14488
49
12691
16526
159
499
6381
3498
1412
2
7391
16761
14651
4008
19128
8509
562
1594
53
172
8559
426
8148
46
10268
3149
2688
5346
145
1157
8
49
4104
504
1382
161
6524
10441
10137
12582
3274
4
4
0
0
0
0
1
0
0
0
38
70
186
237
288
83
11
18
301
152
64
1303
140
592
970
284
1297
864
26
307
11
40
41
0
49
15
0
0
26
14
0
425
9370
5664
2791
4346
46
11786
144
10028
1147
5
70
40
16
0
32
55
17
63
14
—
7
11
2
13
8
0
0
12
0
0
3
2
0
0
0
4
0
0
277
21
47
335
0
282
97
25
184
122
50
6955
161
486
3323
11808
15156
1360
5601
13004
999
79
30
681
1716
2406
490
2579
210
1870
3
8
100
10
46
0
143
150
26
12
117
2
7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
82
50
41
31
10
128
145
212
19
242
207
49
354
395
162
37
91
167
326
257
21
48
570
506
483
97
99
194
21
308
333
287
444
461
538
0
205
148
65
38
24
8
20
27
3
17
21
11
12
26
0
445
50
356
463
263
112
219
68
19
15
42
9
17
15
20
6
1
13
21
93
96
58
28
51
12
18
124
123
6
15
24
21
11
0
24
17
78
57
25
114
44
38
16
64
102
24
102
44
9859
4508
423
5884
2844
1173
12504
141
10593
486
647
24
142
61
385
627
108
287
0
861.3
397.8
63.8
256
Mean
1645
3102
1080
1632
23.9
2562
4054
447.3
1063
21.9
4717
3682
61.0
1754
169.4
2186
2597
791.4
58.4
1464
2060
151.8
367.0
22.1
2227
828.6
4790
14.2
1343
6119
182.3
534.9
53.2
SD
2831
2260
5247
68.7
7671
1632
650.2
5924
3460
19.5
6649
225.5
2897
3773
1069
439.6
4522
3625
158.2
51.7
2189
15.2
1578
Note: The first, second and third line for each subject refers to Session 1, 2 and 3, respectively.
0
167
168
APPENDIX B. SUPPLEMENTS
Table B.24: Coefficients ofy for Each Subject and Session (Exp 34)
Random
Group
Block Desien
27
Subj
23
25
28
26
29
24
211
210
212
44
5
383
141
4905
3811
978
5243
2122
2351
0
3
8
0
10
0
0
2
11
15
15
0
12
15
76
58
9
48
28
18
10
5
27
43
52
63
68
24
13
5
6
0
0
4
0
0
1
0
186
68
13
904
2196
22
1576
438
2225
393
1398
8
4
6
8
0
0
0
1
0
72
65
47
0
83
13
0
25
23
3
7
0
7
0
0
9
0
0
18
0
0
9
4
10
966
850
70
152
749
27
320
575
300
57
129
1295
335
605
309
139
92
909
24
24
2707
90
582
562
1944
2665
1135
1799
34
242
245
21
34
157
202
0
114
43
92
222
0
8
1
5
16
14
15
11
11
0
27
0
0
0
0
0
0
0
0
9
0
3
0
0
0
1
0
0
93
140
62
34
0
16
102
32
32
139
0
5
1
0
2
0
0
0
8
8
4
0
0
3
0
37
0
0
15
0
13
10
12
0
0
5
61
22
67
131
34
124
110
18
95
6
14
0
12
10
8
43
0
0
0
1
0
0
0
O
4
0
0
793
705
1309
401
165
45
1625
23
71
1377
180
2556
44
8521
4660
1050
402
6636
2409
134
7269
7980
4301
40
9259
330
120
3398
882
2094
7228
984.4
788.6
1048
89.3
42.7
536.3
233.9
104.3
466.1
16.3
Mean
1123
665.6
362.1
995.8
73.0
843.5
29.1
313.8
170.3
12.9
833.6
339.8
1019
853.0
559.7
311.0
154.6
14.4
71.1
32.3
43.0
811.6
154.3
1595
121.0
680.5
311.8
SD
1225
1689
12.4
2735
1530
45.6
808.8
801.7
137.1
2358
2112
354.7
14.5
1075
2513
2905
37.6
2260
275.0
1340
651.5
103.4
12.1
Note: The first, second and third line for each subject refers to Session 1, 2 and 3,
respectively.
0
0
2
0.5
0.6
0.5
1.3
1.0
0.8
B.3. EXPERIMENT 3A AND 3B
of Ear Dicycles for Each Subject and Session (Exp 34)
Table B.25: Number
DiResclution Block Design
Group
e7
ee
e5
e8
810
Subj
e4
e9
83
e11
612
O
8
1
8
15
6
0
%
5
10
4
6
2
0
8
10
)
—
9
6
4
4
3
6
0
6
1
6
9
10
D
15
0

2
8
8
1
6
9
2
0
12
0
5
0
7
4
8
4
10
10
8
7
2
0
O
0
4
0
9
0
0
16
1
9

c
6
10
12
1
4
9
5
8
9
3
0
4
4
3
23
9
12
0
6
8
5
2
2
8
17
0
30
4
8
11
6
7
6
6
8
10
12
0
9
8
6
7
0
6
3
5
4
6
7
0
35
0
2
0
0
0
0
2
0
4
0
0
0
0
7
4
2
36
0
0
0
0
0
9
1
8
6
5
8
6
4
5
0
0
0
6.
39
8
8
9
4
—
2
6

7
8
6
10
3
1
8
2
9
o
8
3
12
49
1
9
8
6
10
7
9
4
4
12
9
10
5.1
1.0
4.3
6.5
1.5
Mean
5.3
2.6
6.2
7.8
5.1
6.0
6.9
6.9
1.8
7.9
3.9
4.4
6.6
6.1
5.4
3.3
1.0
6.2
5.0
SD
4.3
3.1
5.1
3.4
3.2
2.7
2.3
1.9
3.7
2.0
3.8
3.1
1.8
3.3
2.9
3.5
3.8
2.9
2.7
2.8
3.6
3.0
3.8
0.9
0
8
0
0
0.1
0
0.3
0
0
9
0
169
170
APPENDIX B. SUPPLEMENTS
Table B.26: Number of Ear Dicycles for Each Subject and Session (Exp 34)
Group P (Repetition
Block Design)
e8
e9
e7
ee
es
Subj
e3
610
e4
e11
812
8
3
10
4
5
2
7
7
3
5
4
9
9
0
10
8
1
11
10
11
10
0
2
0
8
2
2
6
8
11
13
8
3
3
0
0
0
0
5
9
11
8
3
7
9
6
10
7
11
0
5
9
2
0
0
.
3
2
11
11
11
9
7
A
4
9
6
17
6
e
0
0
1
0
4
0
0
0
0
—
—
4
2
6
8
0
25
6
9
10
4
11
3
9
0
6
12
0
11
0
26
0
0
0
0
0
0
1
0
2
0
—
5
5
9
6
6
31
9
2
9
4
8
11
10
11
5
7
8
8
8
6
12
3
5
4
4
6
2
6
38
6
9
9
4
0
8
6
0
0
5
—
8
11
0
8
42
7
9
1
11
7
11
2
2
5
4
0
4
0
2
8
8
9
2
6
3
44
6
e
7
5
5
5
3
10
4
6
0
3
0
11
8
11
7.5
4.7
6.6
0.7
1.6
2.6
8.3
Mean
8.0
6.2
4.0
0.6
5.3
4.3
2.1
1.7
6.6
5.5
5.9
4.0
5.7
2.4
0.9
5.8
7.0
0.5
7.3
6.6
2.9
1.3
2.0
1.8
1.9
1.0
SD
2.0
2.5
2.1
3.4
2.4
4.1
4.1
1.6
2.0
4.1
4.2
3.2
3.6
1.0
2.1
1.0
2.8
2.9
4.1
3.5
4.6
0
0.2
0.2
0
0.4
0.4
0
0
0
B.3. EXPERIMENT 3A AND 3B
Table B.27: Number of Ear Dicycles for Each Subject and Session (Exp 34)
Group N (Random Block Design
e7
e9
e5
e8
63
e6
Subj
e4
810
e11
812
9
5
12
5
10
4
2
4
1
5
3
7
4
0
0
4
0
4
4
6
6
3
0
+
8
12
0
10
0
7
7
6
10
4
0
0
0
0
3
8
11
8
3
6
13
1
5
4
6
0
0
6
8
6
0
0
2
0
4
0
4
18
O
0
7
5
4
5
6
4
6
6
10
10
9
24
4
8
10
2
4
10
10
7
10
2
5
5
5
4
7
6
6
27
2
0
0
0
0
0
0
0
3
0
0
0
5
1
2
6
6
10
32
0
9
0
0
0
0
1
0
4
0
5
0
0
5
0
0
0
0
37
0
0
0
8
0
0
0
0
1
0
6
4
8
1
11
1
4
0
4
43
0
0
0
4
2
0
0
0
2
5
5
5
6
4
11
45
*
6
5
12
2
5
7
10
6
6
2
9
8
4.7
2.6
0.8
0.9
3.4
7.1
3.0
5.9
5.5
Mean
5.4
0.4
0.7
1.1
4.8
1.3
1.9
2.9
3.8
5.9
5.0
5.4
0.9
4.5
2.1
0.1
2.9
1.2
1.2
50
3.5
2.8
3.1
3.2
3.2
1.3
2.8
SD
3.5
4.0
3.1
0.8
1.5
3.5
2.8
1.9
2.3
4.4
0.3
1.6
3.2
2.3
3.1
3.2
2.5
1.4
0
0.1
0
0.1
0
0
0.3
0
—
0
0
1
171
172
APPENDIX B. SUPPLEMENTS
B.3.4 Results of Experiment 3B
Table B.28: Coefficients of % for Each Subject and Session (Exp 3B)
Grour
S (ResolutionKandom Repetition Bock Design
Subj
27
29
24
28
23
26
25
210
211
212
3
3680
1459
49
166
513
17645
13513
7836
15397
6718
842
1085
33
1160
466
90
214
0
512
)
7
1
12
0
10
0
0
0
0
O
80
178
0
49
20
133
43
186
7
133
17
46
37
169
118
178
130
67
0
489
470
701
60
272
660
134
25
163
0
40
4649
359
9169
976
3046
2312
7514
121
14
7454
0
0
0
0
1
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
750
403
403
186
1276
31
81
19
6
992
1122
4653
57
18186
605
22578
200
10221
10409
1773
24691
1585
233
91
376
521
1443
33
1122
967
9
4
11
13
9
13
0
0
0
0
22
206
464
83
895
31
598
1737
1423
1443
0
44
159
31
121
103
153
70
9
12
0
7770
52
14322
19046
28
168
17066
528
8192
1510
3811
516
3370
6825
51
11049
1407
176
13190
10184
3736
85
449
787
450
211
32
959
1062
12
8
15
20
16
21
1
33
6
0
0
0
17
8
12
0
15
9
19
0
14
116
16
58
68
35
114
155
0
152
73
414
29
1611
40
1268
791
1496
884
224
186
47
55
74
15
30
76
13
0
0
0
49
736
418
364
231
642
625
112
97
21
11
6
0
0
0
2
10
0
0
46
10
5
38
84
37
119
234
159
258
212
16
3
0
0
0
3
1
47
2
0
1398
31
1010
501
238
890
360
1283
93
35
45
166
246
234
90
127
281
19
493.4
3838
1776
2327
1155
4840
4120
68.8
Mean
189.2
25.1
4101
3376
2140
1415
3381
1114
498.3
75.9
200.2
26.1
443.4
96.1
40.9
92.5
437.9
319.9
9.7
293.2
187.5
17.1
6791
3042
5785
7653
3314
1538
207.9
603.6
64.3
SD
17.8
6189
7454
3371
3502
1598
8292
67.3
208.5
611.8
17.9
152.6
30.7
183.8
535.2
376.3
499.5
339.2
31.9
82.6
11.2
)
0
B.3. EXPERIMENT 3
Table B.29: Coefficients of y for Each Subject and Session (Exp 3B)
Group
Pns (RepetitionKandomResclution Block Design)
27
25
Subj
26
28
210
23
29
211
24
212
55
4326
183
587
1698
4
8630
9363
21701
16389
19313
65
761
40455
6414
14911
2326
240
19114
28132
39192
30372
678
60
5358
2029
13472
21885
28421
12014
207
25
480
119
415
2
268
139
353
60
3
9
27
581
1077
957
911
161
184
339
613
67
551
941
27
66
345
920
163
161
1079
624
71
78
124
15
181
34
157
19
201
15
0
15
0
3
21
12
0
22
5
0
10
10
11
10
0
1
54
564
190
4238
22743
9317
1645
16788
20592
20
9332
59
678
21346
5314
12476
195
26914
29213
11818
2012
43
6649
4241
4352
397
6378
1025
147
1191
2321
8
58
22
13
27
21
53
0
38
22
5
0
13
0
14
0
10
0
12
2
0
0
6
6
4
0
0
0
22
344
30
486
542
272
47
104
203
463
34
46
416
147
9764
1146
11148
5759
2787
12636
5030
50
3
42
68
87
24
18
105
94
13
5
5
3
0
0
0
4
41
1
0
50
4
57
20
13
42
54
34
23
0
2
6
6
6
0
0
0
0
27
45
79
0
222
199
140
18
127
48
78
9
47
66
30
66
0
35
15
66
43
66
80
44
17
25
80
15
3
0
0
0
50
0
0
0
0
0
3
9
10
0
0
0
0
0
0
14
56
42
0
56
41
51
16
24
0
3
4
0
0
0
1
0
2
0
19
8
9
14
2
17
17
0
0
0
3441
4515
2017
417.4
4020
981.7
162.4
Mean
1803
60.8
22.3
1528
7784
3352
25.3
6040
8329
3680
74.2
598.0
214.5
2950
1749
1483
3806
3325
18.6
50.6
136.2
850.4
356.9
3867
3783
8403
6932
666.5
68.9
SD
9337
1745
18.3
222.0
14060
5614
6757
14746
10429
294.7
901.7
23.0
87.3
2453
4229
1741
8916
9561
668.5
70.5
3852
6943
226.7
19.1
Note: The first, second and third line for each subject refers to Session 1, 2 and 3,
respectively.
0
0
173
174
APPENDIX B. SUPPLEMENTS
Table B.30: Number of Ear Dicycles for Each Subject and Session (Exp 3B)
ResolutionKkandom Repetition Block Design
Group SNI
Subi
89
e5
e6
e8
e7
e4
e10
e11
63
e12
3
8
5
5
9
3
5
6
2
2
2
4
6
4
0
0
0
0
—
8
6
8
3
7
0
8
6
6
10
0
1
8
3
8
6
0
6
1
8
8
9
6
7
8
8
0
0
1
14
0
0
0
2
0
0
0
0
0
U
0
0
0
0
0
9
5
6
8
5
10
19
0
11
6
1
6
—.
12
10
2
2
10
0
7
2
8
2
22
0
7
0
8
0
—
6
8
3
6
7
5
3
6
3
4
12
28
6
8
70
6
6
0
8
6
6
4
4
7
9
6
3
0
6
0
33
6
6
4
1
0
8
5
5
10
6
7
0
0
0
6
8
8
6
9
0
40
2
6
4
6
0
5
8
9
3
7
2
10
10
1
8
6
0
0
2
46
4
0
11
13
14
.
0
0
3
2
0
0
47
0
6
4
6
2
2
4
10
11
0
8
0
0
6
9
4
0
11
4.5
5.8
0.6
3.9
6.4
2.9
Mean
5.9
1.1
1.1
5.4
7.8
6.6
7.4
3.1
2.1
0.7
5.1
0.3
5.7
6.8
5.2
5.9
1.9
3.5
3.3
0.3
1.0
21
3.4
1.1
3.0
3.4
2.3
2.0
2.9
3.3
SD
3.1
3.9
2.1
3.1
1.1
2.3
2.3
0.9
3.3
4.0
2.6
2.8
1.5
2.6
3.1
4.0
0.5
3.8
0.4
0.1
0
0.8
0.3
7
0
B.3. EXPERIMENT 3A AND 3B
Table B.31: Number of Ear Dicycles for Each Subject and Session (Exp 3B)
Block Design
RepetitionKandomResolution
Group PNS
810
e8
e5
Subj
e7
ee
e9
84
23
e11
612
7
5
6
7
4
7
7
0
5
4
6
6
4
9
8
4
12
6
7
8
4
6
3
0
11
9
4
4
0
0
0
9
9
8
10
2
4
7
4
10
12
8
9
4
2
3
1
5
7
15
0
5
0
0
0
0
5
5
0
3
0
2
6
—
3
4
6
5
2
4
11
20
4
7
7
5
10
2
12
—
9
1,
4
4
4
10
11
N
0
4
10
21
1
4
0
4
0
1
0
2
0
0
5
8
4
2
6
7
34
9
0
10
1
13
12
5
8
1
0
15
11
12
O
4
0
0
5
0
41
8
6
3
8
7
1
0
4
0
0
0
3
2
0
3
8
9
4
48
7
8
4
6
0
0
10
2
5
6
6
?
0
0
50
0
4
0
0
1
0
0
9
8
3
6
8
6
51
0
3
2
4
0
0
0
8
0
8
8
4
0
0
0
0
4.6
2.0
3.0
5.8
6.1
5.1
Mean
D.1
1.2
6.2
1.5
0.1
6.7
3.5
5.0
7.0
6.7
3.3
1.0
0.3
2.2
0.9
0.5
2.3
2.0
0.2
5.9
5.6
3.1
6.3
7.4
0.3
1.9
3.1
3.0
SD
2.8
1.5
2.4
3.2
2.1
3.7
3.5
2.6
3.7
0.7
1.5
2.7
3.0
0.3
3.5
2.9
4.6
4.0
2.9
1.0
2.4
3.8
2.3
0.4
1.3
2.9
Note: The first, second and third line for each subject refers to Session 1, 2 and 3,
respectively.
0
0
0
0
0
175
Appendix C
ProgramListings
C.1 Prolog
The following two listings are written in Open Prolog (Version 1.0.2). The first program
performs a depthfirst search (Bratko, 1990) for directed cycles of any length in a directed
graph. The program reads in the arcs (binary relations) of a coded digraph and lists all
dicycles within the digraph. The algorithm is slow and does not operate in polynomial time.
Consequently, depending on the complexity of the digraph, the application can be space and
time consuming. The second program executes an ear decomposition by sequence. It finds
directed ears of the coded digraph in a depthbreadth search by using a given sequence of
the arcs. The ears are completed to ear dicycles using the same search procedure. The ear
dicycles of each strong component form a directed ear basis as described in Section 2.3.
********** OPEN PROLOG (Version 1.0.2d0) ***
last change 29/8/95
7 by M. Lages
7 program finds dicycles in coded digraph
7 running the program:
by typing "start. «ENTER»"
start program
% inputfile
digraph1/2/3 (coded digraphs for Exp1/2/3)
(temporary clauses)
7 helpfile
helpfile
outputfile
circuit1/2/3 (list of dicycles)
start :
see (digraph3),
read (graph (Inf,Arcs)),
process (graph (Inf,Arcs)).
process(end of file) :
!.
process (graph(L,1)) :
search.
177
178
APPENDIX C. PROGRAMLISTINGS
process(graph(Inf,Arcs)) :
tell(helpfile),
writelist (Inf),
writelist (Arcs),nl,
told,
start.
******* HELPROUTINES **
writelist ([).
Z write list
:
writelist (IXILJ)
write(X), write(**),nl,
writelist (L).
writelist2 (D).
writelist2 (IXIL]) :
write (X),nl,
writelist2 (L).
conc (L,L,L).
concatenate two lists
conc (IXILIJ,L2, LXIL3J) :
conc (Li,L2,L3).
firstm(X, IXILJ).
% first member in list
maxlist (IX Y,Z) : maxOfList (Y,X,Z). 7 maximum in list
maxOfList (Ll,X,X)
maxOfList (IH T),M,X) : H»M,!,maxOfList (T,H,X).
maxOfList (L. Tl,X,Y) : maxOfList (T,X,Y).
member (X, IXILJ).
member of list
:
member (X, IVILJ)
member (X,L).
reverse (X,Y) : reverse (X, U,Y).
reverse (I,X,X).
reverse (IX YJ,Z,A) : reverse (Y, IXIZJ,A).
tellall(Query,Template) :
call(Query)
write (Template),
write(* '),
nl,fail.
tellall(,.).
*.****** MAIN PROGRAM *******
search :
reconsult (helpfile),
plain (Inf),
C.1. PROLOG
tell(circuit3),
write('graph(I'),
write (Inf),write('1,'),
tellall(solve (Nodel,Resolution), Resolution),
tell(circuit3),
writelist2 (List),write (*).*), n1,
7 List is a list of lists
write('graph(L, L).'),nl,nl,
start.
solve (Nodel,Resolution) :
depth(,Nodel,Node2,Solution),
reverse (Solution, Resolution).
plain(i(Vp,Sess,Exp)) :
i (Vp,Sess,Exp).
depth (Path, Nodel,Node2, Nodel Pathl) :
reverse (Path,Repath),
firstm (Node2,Repath),
condition for cycle
maxlist (Path,M),
Node2 »= M,
unique cycle
Node2 »= Nodel.
:
depth (Path, Nodel,Node2,Sol)
p(Nodel,Node2,Triali),
p(Node2,Node3,Trial2),
not member (Node2,Path),
depth(Nodei Pathl,Node2,Node3,Sol)
************ OPEN PROLOG (Version 1.0.2d0) *************
7 Last Change 7/2/97
7 by M. Lages
program performs ear decomposition by sequence
Finds directed ears of coded digraph in depthbreadth
% search by using sequence of arcs. Ears are completed
% to directed ear cycles using the same search procedure
% start program by typing "start."«ENTER»
4 inputfile
digraph1/2/3 (coded digraphs)
(temporary clauses)
Z helpfile
helpfile
dears1/2/3
outputfile
(list of ear dicycles)
*.********.******.***********************************
start :
see (digraph3),
read (graph (Inf,Arcs)),
process (graph (Inf,Arcs)).
179
APPENDIX C. PROGRAMLISTINGS
180
process(endoffile) :
!.
process(graph(l,D)) :
search.
process(graph(Inf,Arcs)) :
tell(helpfile),
writelist (Inf),
writelist (Arcs),nl,
told,
start.
********* HELPROUTINES
writelist ([).
writelist (IXIL]) :
write(X), write ('.'),nl,
writelist (L).
concatenate two lists
conc(U,L,L).
conc (IXILIl,L2, IXIL3J) : conc (Li,L2,L3).
maxlist (IX YJ,2) : maxOfList (Y,X,Z). % maximum in list
maxOfList (Ll,X,X).
maxOfList (IH T,M,X) : H»M,!,maxOfList (T,H,X).
maxOfList (L Tl,X,Y) : maxOfList (T,X,Y).
% member of list
member (X, IXILJ).
:
member (X, IYILJ)
member (X,L).
add(X, L, LxJ).
add (X,L,L) : member (X,L),!.
add (X,L, IXILJ).
del (LXILJ,X,L).
firstm(X,L) : !,fail.
7 first member in list
firstm(X, IXILJ).
last (X,L) : !,fail.
% last member in list
last(X,L) :
conc(, LXJ,L).
7
C.1. PROLOG
delete (L,R) :
remove directed ear
s(S),
delete (L,L,Ni,S.R),
retract(s(S)),!.
7 ! prevents backtracking
delete(Il,L,Ni,S,S).
delete (L,L2,Ni,S,Z) :
firstm (Ni,L2),
firstm (N2,L),
remove pclause
retract (p(Ni,N2,Trial)),
assume qclause
assert(q(Ni,N2,Trial)),
write (Ni),write (N2),write(**),
del (L,First,Li),
add (First,S,Si),
delete (Li, [First L21,First,Si,Z).
:
test if arcs in Sub
delete (L,L2,Ni,S,Z)
firstm (Ni,L2),
firstm (N2,L),
(Ni,N2,Trial),
write(Ni), write (N2),write(***)
del (L,First,Li),
add (First,S,Si),
delete (Li, [First L21,First,S,Z).
:
% add first vertex
delete (L,L2,Ni,S,Z)
del (L,First,Li),
add (First,S,Si),
delete (Li, [First L21,First,Si,Z).
retractq :
retract(q(.)).
retractq.
Zreverse (Il,Y).
reverse (X,Y) : reverse (X,L,Y).
reverse (L,X,X).
reverse (IXIY,Z,A) : reverse (Y, IXZ1,A).
shell(0,T) :
iterate3 (Ni,S,Sol),
reverse (Sol,Resol),
remove ear (pclauses)
delete (Resol,NS),nl
assertz(s(NS))
assert new Sub
tellall(0,T),
retract(s(X)),
start new component
assertz(s([)),
retract
remove all qclauses
181
182
APPENDIX C. PROGRAMLISTINGS
fail.
shell(,.).
tellall(Query,Template) :
call (Query),
2
write (Template),
write(* *),
7
nl,fail.
tellall(,).
******* MAIN PROGRAM *******
search :
reconsult (helpfile),
plain(Inf),
tell (dears3).
write ('graph('),
write (Inf), write(','),nl,
assertz(s(U)),
shell(solve (Sub,Resolution),Resolution),
write ('graph (L,L).'),nl,nl,
retract(s (Sub)),
start.
solve (NSub,Resolution) :
7 do Query
iterate (Sub,Solution),
reverse (Solution, Resolution).
delete (Resolution, NSub),
remove arcs
assertz(s (NSub)).
% assert new Sub
7 write (NSub),nl.
plain(i (Vp,Sess,Exp)) :
i (Vp,Sess,Exp)
:
iterate (Sub,Solution)
trydepth (Sub, Sol,1),
7 write (Sol),nl,
firstm (Ni,Sol),
depth2(L,Sol,N1,Solution).
trydepth (Sub,Sol,Trial) :
depthi(L,Sub, Nodel,Sol,Trial)
Trial«66,
NTrial is Triali,
trydepth (Sub,Sol,NTrial).
depthi (Path, Sub, Nodel, Nodel Pathl,Trial) :
N is last vertex in Path
firstm (N,Path),
% test if arc exists
P(N, Node1,T),
C.1. PROLOG
last (Nod,Path),
s (Sub),
% Nod first vertex is in Sub
member (Nod, Sub)
Nodel last vertex is in Sub
member (Nodel,Sub).
Z write('e').
depthi (Path,Sub,Nodel, Nodel Pathl,Trial) :
last (Nodel,Path),
s (Sub)
member (Nodel,Sub).
4 first and last vertex is in Sub
write('x').
7
depthi (Path,Sub,Nodel,Sol,Trial) :
search for Path in pclauses
p (Nodel,Node2,Triali),
Trial=Triali,
not member (Node2,Path),
depthi([Nodel Path,Sub,Node2,Sol,Trial).
depth2 (Path,Sol,Ni, INi Soll) :
reverse (Sol,Resol),
firstm (Ni,Resol),
directed cycle
firstm(N,Path),
last vertex
q (N,Ni,T),
del (Resol,Ni,Res),
firstm (N2,Res),
P(Ni,N2,).
Z write ('b').
depth2 (Path,Sol,Ni,Solution) :
7 search for Path in qclauses
(Ni,N2,Triali),
not member (N2,Path),
depth2 (NiPath, INiSol,N2,Solution).
iterate3 (Nodel,Sub,Sol) :
trydepth3 (Nodel,Sub,Sol,1).
trydepth3 (Nodel,Sub,Sol,Trial) :
depth3(L,Sub,Nodel,Node2,Sol,Trial)
*
Trial«66,
NTrial is Trialti,
trydepth3 (Nodel,Sub,Sol,NTrial).
depth3(Path,Sub,Nodel,Node2, Node2,Nodel Pathl,Trial) :
last (Node2,Path).
directed cycle
write('c').
depth3 (Path,Sub,Nodel,Node2,Sol,Trial) :
183
184
APPENDIX C. PROGRAMLISTINGS
search for Path in pclauses
7
p(Nodel,Node2,Triali),
Trial»=Triall,
p(Node2,Node3,Trial2),
Trial)=Trial2,
not member (Node2,Path),
depth3([Nodel Pathl,Sub,Node2, Node3,Sol,Trial).
C.2 Mathematica
The following program was written for Mathematica 2.0. The listing of the function Read¬
Graphl is given that reads in the adjacency matrix of an arbitrary digraph of order 12
and prints out the factorization of its characteristic polynomial. Useful functions for doing
discrete mathematics are provided by Skiena (1990) and Wolfram (1991).
(* writen by ML based on ReadGraphL, last change 9.5.95 *)
ReadGraphl (fileName0 String, fileNamel Stringl :
Modulelifileo, expr, mi, poly, coef, facs),
fileo  OpenRead fileNameol;
filel = OpenWrite(fileNameil;
Ifl fileo == 3Failed, Returnl 1;
Whilel True,
expr = Readl fileo, (Number, Number, Number) 1;
If expr ===
EndOfFile, Breakll 1;
mi = Readl fileo, Table(Number, Number, Number, Number,
Number, Number, Number, Number,
Number, Number, Number, Number),
(12)) 1;
poly = DetL mi x IdentityMatrix 121 1;
coef  CoefficientListspoly,x;
facs  Factorpolyl;
Writelfilel, expr
coef
Writelfilel,
Print ["Subject"
expr
Printfacsl
(*Printspolyl*)
*)
(* PrintMi
*)
(*Print(facs.
1;
Closelfileo
Closelfileil
185
C.3. CPROGRAM
C.3 CProgram
The following Cprogram was written for Think C 6.0 and counts preference reversals be¬
tween two adjacency matrices using standard ANSI C language (Kernighan & Ritchie, 1988).
Another program was developed in Mathematica utilizing the builtin function for the
Hadamard product and the function ReadGraphl].
/*********************************
* Last revision 5.2.94, M. Lages
*
reads adjacency.o
* writes reverse.o
**********************************/
tinclude stdio.h)
/tinclude math.h *
FILE filein;
FILE fileout;
main
()
/row and column indices
int i,j,m,n;
/ preference matrix */
int al 121 (121, a2 121 [121;
int subj, sess, exp, deci, coun;
/Read data
filein  fopen ("adjacency.o", "r");
if (filein == NULL)
printf ("Couldnt open file adjacency.on");
fileout  fopen ("reverse.o", "w");
if (fileout = NULL)
printf ("Couldnt open file reverse.oln"); / fp); */
else
while (!feof(filein))
/ feof() returns nonzero at end of file
fscanf (filein,"Yd 7d %d", asubj, asess, dexp);
printf("Zd 7d ZdAn", subj, sess, exp);
if (sess = 1) (
for (m=1;m(13;++m)
for (n=1;n(13;+n)
(
fscanf (filein,"72d ", aimnl);
/ aimlin  getc (filein); */
printf ("72d ", almnl);
);
printf("In");
);
);
fscanf (filein,"Yd %d 7d", asubj, åsess, aexp);
186
APPENDIX C. PROGRAMLISTINGS
printf ("72d 7d %dIn", subj, sess, exp);
if (sess == 2) (
for (m=1;m(13;++m) (
for (n=1;n(13;++n)
1
fscanf (filein,"72d ", ta2m n));
1*
*/
a2mn  getc(filein);
printf ("72d ", a2m nl);
);
).
printf("Vn"),
);
);
/comparing matrices *
coun = O;
for (m=1;m(13;++)
for (n=1;n(13;+n)
1
if ((m:=n) &ap (ainm==0) aap (a2mn=0)) (
oun++;
fprintf (fileout,"72d %d 72d 72d 72d 72d 72dn",
subj, exp, coun, m, n, allmnl, a2nl ml);
);
):
);
/ fprintf (fileout, "72d %d 72dn", subj, exp, coun); */
);
J:
fclose (filein);
fclose (fileout);
return;
Bibliography
Adams, E., & Messick, S. (1957). An axiomatization of Thurstone's successive intervals and
paired comparisons scaling methods. Technical report 12, Stanford University, Applied
Mathematics and Statistics Laboratory.
Albert, D., Aschenbrenner, K., & Schmalhofer, F. (1989). Cognitive choice processes and the
attitudebehavior relation. In Upmeyer, U. (Ed.), Attitudes and behavioral decisions,
pp. 6199. Springer, New York.
Albert, D., & Lages, M. Semantic relations and binary choice. in preparation.
Allais, M. (1953). Le comportement de l'homme rationnel devant le risque: critique des
postulats et axiomes de l'école americaine. Econometrica, 21, 269290.
Allerby, R. (1991). Rings, fields and groups. An introduction to abstract algebra. Edward
Arnold, London.
Arend, L., & Spehar, B. (1993). Lightness, brightness, and brightness contrast: 1. Illumi¬
nance variation. Perception & Psychophysics, 54 (4), 446456.
Armstrong, W. (1939). The determinateness of the utility function. Econ. Journal, 49,
453467.
Aschenbrenner, K. (1981). Efficient sets, decision heuristics, and singlepeaked preferences.
Journal of Mathematical Psychology, 23, 227256.
Aschenbrenner, K. (1984). Moment versus dimensionoriented theories of risky choice: A
(fairly) general test involving singlepeaked preferences. Journal of Experimental Psy
chology: Learning, Memory, and Cognition, 10(3), 513535.
Barthélemy, J.P. (1990). Intransitivities of preferences, lexicographic shifts and the tran¬
sitive dimension of oriented graphs. British Journal of Mathematical and Statistical
Psychology, 43, 2937.
Beach, L., & Mitchell, T. (1978). A contingency model for the selection of decision strategies.
Academy of Management Review, 3, 439449.
Becker, G., DeGroot, M., & Marschak, J. (1963a). An experimental study of some stochastic
models for wagers. Behavioral Science, 8, 199202.
Becker, G., DeGroot, M., & Marschak, J. (1963b). Probabilities of choices among very
8, 306311.
similar objects. Behavioral Science,
187
188
BIBLIOGRAPHY
Becker, G., DeGroot, M., & Marschak, J. (1963c). Stochastic models of choice behavior.
Behavioral Science, 8, 4155.
Bell, D. (1982). Regret in decision making under uncertainty. Operations Research, 30,
961981.
Bergé, C. (1976). Graphs and Hypergraphs. NorthHolland, Amsterdam.
Bermond, J., & Thomassen, C. (1981). Cycles on digraphs  a survey. Journal of Graph
Theory, 5, 143.
Bernoulli, D. (1738). Specimen theoriae nova de mensura sortis. Commentarii Academiæe
Scientiarum Imperialis Petropolitanæ, 5, 175192. Translated by L. Sommer (1954).
Exposition of a new theory on the measurement of risk. Econometrica, 22:2336.
Bezembinder, T. (1981). Circularity and consistency in paired comparisons. British Journal
of Mathematical and Statistical Psychology, 34, 1637.
Biggs, N. (1993). Algebraic Graph Theory (2nd edition). Cambridge University Press,
Cambridge, UK.
Billings, R., & Marcus, S. (1983). Measures of compensatory and noncompensatory models
of decision behavior: Process tracing versus policy capturing. Organizational Behavior
and Human Decision Processes, 31, 331352.
Birkhoff, G. (1967). Lattice theory (3rd edition)., Vol. Colloquium Publication, XXV. Amer¬
ican Mathematical Society, Providence, R.E.
Birnbaum, M. (1992). Issues in utility measurement. Organizational Behavior and Human
Decision Processes, 52, 319330.
Birnbaum, M., Coffey, G., Mellers, B., & Weiss, R. (1992). Utility measurement: Configural¬
weight theory and the judge's point of view. Journal of Experimental Psychology:
Human Perception and Performance, 18, 331346.
Birnbaum, M., & Stegner, S. (1979). Source credibility in social judgement: Bias, expertise,
and the the judge's point of view. Journal of Personality and Social Psychology, 37,
4874.
Björner, A., Las Vergnas, M., Sturmfels, B., White, N., & Ziegler, G. (1993). Oriented
matroids. Encyclopedia of Mathematics and Its Applications. Cambridge University
Press, Cambridge, UK.
Block, H., & Marschak, J. (1960). Random orderings and stochastic theories of responses.
In Olkin, I., Ghurye, S., Hoeffding, W., Madow, W., & Mann, H. (Eds.), Contributions
to probability and statistics, chap. 10, pp. 97132. Stanford University Press, Stanford,
CA.
Bratko, I. (1990). Prolog programming for artificial intelligence (2nd edition). Addison¬
Wesley, Singapore.
189
BIBLIOGRAPHY
Brualdi, R., & Ryser, H. (1991). Combinatorial matrix theory, Vol. 39 of Encyclopedia of
Mathematics and its Applications. Cambridge University Press, Cambridge, NY.
Camerer, C. (1989). An experimental test of several generalized utility theories. Journal of
Risk and Uncertainty, 2, 61104.
Cameron, P. (1994). Combinatorics: Topics, techniques, algorithms. Cambridge University
Press, Cambridge, UK.
Cantor, G. (1895). Beiträge zur Begründung der transfiniten Mengenlehre. Mathematische
Annalen, 46, 481512.
Chubb, C. (1986). Collapsing binary data for algebraic multidimensional representation.
Journal of Mathematical Psychology, 30, 161187.
Condorcet, Marquis de (1785). Essai de l'application de l'analyse à la probilité des décisions
rendues à la pluralité des voix. Paris. Reprint by M.J.A. Caritat, (1974). Chelsea
Publishers, New York.
Coombs, C. (1964). A theory of data. Wiley, New York.
Coombs, C. (1969). Portfolio theory: A theory of risky decision making. La Decision. Centre
National de la Recherche Scientifique, Paris.
Coombs, C. (1975). Portfolio theory and the measurement of risk. In Kaplan, M., & Schwartz,
S. (Eds.), Human judgment and decision processes. Academic Press, New York.
Coombs, C., & Avrunin, G. (1977a). Singlepeaked functions and the theory of preference.
Psychological Review, 84, 216230.
Coombs, C., & Avrunin, G. (1977b). A theorem of singlepeaked functions in one dimension.
Journal of Mathematical Psychology, 16, 261266.
Costanza, M., & Afifi, A. (1979). Comparison of stopping rules in forward stepwise discrim¬
inant analysis. Journal of the American Statistical Association, 74, 777785.
Cramer, G. (1728). correspondence to N. Bernoulli and D. Bernoulli (1738). Commentarii
Academiæ Scientiarum Imperialis Petropolitanæ, 5, 175192. Translated by L. Sommer
(1954). Exposition of a new theory on the measurement of risk. Econometrica, 22:23
36.
Dahlstrand, V., & Montgomery, H. (1984). Information search and evaluation processes
in decisionmaking: A computer based process tracing study. Acta Psychologica, 56,
113123.
Davey, B., & Priestley, H. (1990). Introduction to lattices and order. Cambridge University
Press, Cambridge.
Davidson, D., & Marschak, J. (1959). Experimental tests of a stochastic decision theory.
In Churchman, C., & Ratoosh, P. (Eds.), Measurement: Definitions and theories, pp.
233269. Wiley, New York.
190
BIBLIOGRAPHY
Dawes, R. (1964). Social selection based on multidimensional criteria. Journal of Abnormal
and Social Psychology, 68, 104109.
Debreu, G. (1959). Topological methods in cardinal utility theory. In Arrow, K., Karlin, S.,
& Suppes, P. (Eds.), Mathematical methods in the social sciences. Stanford University
Press, Stanford, CA.
Doignon, J.P. (1995). A model for the emergence of preference relations (Wellgraded fam¬
ilies of semiorders). Paper presented at the 26th European Mathematical Psychology
Group Meeting in Regensburg, Germany.
Doignon, J.P., Ducamp, A., & Falmagne, J.C. (1984). On realizable biorders and the
biorder dimension of a relation. Journal of Mathematical Psychology, 28, 73109.
Doignon, J.P., & Falmagne, J.C. (1985). Spaces for the assessment of knowledge. Interna¬
tional Journal of ManMachine Studies, 23, 175196.
Doignon, J.P., & Falmagne, J.C. (1997). Wellgraded families of relations. Discrete Math¬
ematics, 173, 3544.
Edgell, S., Geisler, W., & Zinnes, J. (1973). A note on a paper by Rumelhart and Greeno.
Journal of Mathematical Psychology, 10, 8690.
Edwards, W. (1953). Probabilitypreferences in gambling. American Journal of Psychology,
66, 349364.
Edwards, W. (1954a). Probability preferences among bets with differing expected values.
American Journal of Psychology, 67, 5667.
Edwards, W. (1954b). The reliability of probability preferences. American Journal of Psy¬
chology, 67, 6895.
Edwards, W. (1954c). The theory of decision making. Psychological Bulletin, 51, 380417.
Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. Quarterly Journal of Eco¬
nomics, 75, 643669.
Ericsson, K., & Simon, H. (1984). Protocol analysis: Verbal reports as data. MIT Press,
Cambridge, Massachusetts.
Falmagne, J.C. (1985). Elements of psychophysical theory. Oxford Psychology Series No.6.
Oxford University Press, New York.
Falmagne, J.C., Koppen, M., Villano, M., Doignon, J.P., & Johannesen, L. (1990). In¬
troduction to knowledge spaces: How to build, test, and search them. Psychological
Review, 97(2), 201224.
Fechner, G. (1860). Elemente der Psychophysik. Breitkopf & Hartel, Leibzig.
Finetti, B. de (1980). Foresight: Its logical laws, its subjective sources. In Kyburg, Jr.,
H., & Smokler, H. (Eds.), Studies in subjective probability. Krieger, Huntington, NY.
Originally published 1937 in Ann. Inst. H. Poincaré.
BIBLIOGRAPHY
191
Fishburn, P. (1970). Intransitive indifference with unequal indifference intervals. Journal of
Mathematical Psychology, 7, 144149.
Fishburn, P. (1979). Utility theory for decision making (2nd edition). R.E. Krieger, Hunt¬
ington, New York.
Fishburn, P. (1982). Nontransitive measurable utility. Journal of Mathematical Psychology,
26, 3167.
Fishburn, P. (1992). Induced binary probabilities and the linear ordering polytope: A status
report. Mathematical Social Sciences, 23, 6780.
Ford, J., Schmitt, N., Schlechtman, S., Hults, B., & Doherty, M. (1989). Process tracing
methods: Contributions, problems, and neglected research questions. Organizational
Behavior and Human Decision Processes, 43, 75117.
Franklin, B. (1987). Letter to Joseph Priestly. In Writings, pp. 877878. The Library of
America, New York. Original letter written September 19, 1772.
Frobenius, G. (1912). Über Matrizen aus nicht negativen Elementen. Sitzungsberichte Preuss.
Akad. Wiss., Berlin, 457476.
Galluccio, A., & Loebl, M. (1996). (P,Q)odd digraphs. Journal of Graph Theory, 23(2),
175184.
Garey, M., & Johnson, D. (1979). Computers and intractability: A guide to the theory of
NPcompleteness. W.H. Freeman, San Francisco.
Gigerenzer, G., & Goldstein, D. (1996). Reasoning the fast and frugal way: models of
bounded rationality. Psychological Review, 103, 650669.
Gilboa, I. (1987). Expected utility with purely subjective nonadditive probabilities. Journal
of Mathematical Economics, 16, 6588.
Goldberg, M., & Moon, J. (1971). Arc mappings and tournament isomorphisms. Journal of
the London Mathematical Society, 3(2), 378384.
Grether, D., & Plott, C. (1979). Economic theory of choice and the preference reversal
phenomenon. American Economic Review, 69, 623638.
Guilford, J. (1954). Psychometric methods (2nd edition). Macmillan, New York.
Harary, F. (1969). Graph theory. AddisonWesley, London.
Heinemann, E. (1955). Simultaneous brightness induction as a function of inducing and
testfield luminances. Journal of Experimental Psychology, 50, 8996.
Herstein, I. (1975). Topics in algebra (2nd edition). John Wiley & Sons, New York.
Huber, O., Payne, J., & Puto, C. (1982). Adding asymmetrically dominated alternatives:
Violations of regularity and the similarity hypothesis. Journal of Consumer Research,
9, 9098.
BIBLIOGRAPHY
192
Huynh, H. (1978). Some approximate tests for repeated measurement designs. Psychome¬
trika, 43, 161175.
Huynh, H., & Feldt, L. (1976). Estimation of the Box correction for degrees of freedom
from sample data in randomized block and splitplot designs. Journal of Educational
Statistics, 1, 6982.
Jensen, N. (1967). An introduction to Bernoullian utility theory. I. Utility functions. Swedish
Journal of Economics, 69, 163183.
Jünger, M. (1985). Polyhedral combinatorics and the acyclic subgraph problem. Heldermann
Verlag, Berlin.
Kahneman, D., & Tversky, A. (1979). Prospect theory. Analysis of decision under risk.
Econometrica, 47, 263291.
Keeney, R., & Raiffa, H. (1976). Decisions with multiple objectives: Preferences and value
tradeoffs. Wiley, New York. republished 1993 by Cambridge University Press.
Kendall, M. (1970). Rank correlation methods (4th edition). Charles Griffin, London.
Kendall, M., & BabingtonSmith, B. (1940). On the method of paired comparisons.
Biometrika, 33, 239251.
Kernighan, B., & Ritchie, D. (1988). The C programming language (2nd edition). Prentice
Hall, Englewood Cliffs, NJ.
Klecka, W. (1980). Discriminant analysis. Sage University Paper Series on Quantitative
Applications in the Social Sciences, Series No. 07019. Sage Publications, beverly Hills.
Kolmogorow, A. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Ergebnisse der
Mathematik. Springer, Berlin.
König, D. (1936). Theorie der endlichen und unendlichen Graphen. Leipzig, Reprinted 1950,
Chelsea, New York.
Koppen, M. (1995). Random utility representation of binary choice probabilities: critical
graphs yielding critical necessary conditions. Journal of Mathematical Psychology, 39,
2139.
Koppen, M. (1987). On finding the bidimension of a relation. Journal of Mathematical
Psychology, 31, 155178.
Krantz, D., Luce, R., Suppes, P., & Tversky, A. (1971). Foundations of measurement, Vol. 1.
Academic Press, New York.
Kyburg Jr., H., & Smokler, H. (1980). Studies in subjective probabilities. Wiley, New York.
Lages, M. (1989). Choosing with binary operations: A theoretical note. Paper presented
on the 20th European Mathematical Psychology Group Meeting in Nijmegen, Nether
lands.
BIBLIOGRAPHY
193
Lages, M. (1991). Erhebung von Wissensstrukturen beim binären Wählen. Unpublished
thesis of diploma, Psychologisches Institut der Universität Heidelberg.
Lages, M. (1995). Algebraic decomposition of adjacency matrices. Paper presented on the
26th European Mathematical Psychology Group Meeting in Regensburg, Germany.
Loomes, G., & Sugden, R. (1982). Regret theory: An alternative theory of rational choice
under uncertainty. Economic Journal, 92, 805824.
Lopes, L. (1984). Risk and distributional inequality. Journal of Experimental Psychology:
Human Perception and Performance, 10, 465485.
Lopes, L. (1987). Between hope and fear: The psychology of risk. Advances in Experimental
Social Psychology, 20, 255295.
Lopes, L. (1981). Decision making in the short run. Journal of Human Experimental Psy¬
chology: Human Learning and Memory, 7, 377385.
Lopes, L. (1990). Remodeling risk aversion: A comparison of Bernoullian and rank de¬
pendent value approaches. In von Fuerstenberg, G. (Ed.), Acting under uncertainty:
Multidisciplinary conceptions, pp. 267299. Kluwer, Boston.
Lovász, L., & Plummer, M. (1986). Matching theory. Elsevier, NorthHolland.
D., & Winterfeldt, D. von (1994). What common ground exists for descriptive, pre¬
Luce,
scriptive and normative utility theories?. Management Science, 40, 263279.
Luce,
R. (1956). Semiorders and a theory of utility discrimination. Econometrica, 24, 178
191.
Luce,
R. (1959). Individual choice behavior: A theoretical analysis. Wiley, New York.
R. (1988). Rankdependent, subjective expectedutility representations. Journal of
Luce,
Risk and Uncertainty, 1, 305332.
Luce, R. (1990). Rational versus plausible accounting equivalences in preference judgments.
Psychological Science, 1(4), 225234.
R. (1991). Rank and signdependent linear utility models for binary gambles. Journal
Luce,
of Economic Theory, 53, 75100.
R. (1992). Where does subjective expected utility fail descriptively?. Journal of Risk
Luce,
and Uncertainty, 5, 527.
R. (1996). The ongoing dialog between empirical science and measurement theory.
Luce,
Journal of Mathematical Psychology, 40, 7898.
R., & Fishburn, P. (1991). Rank and signdependent linear utility models for finite
Luce,
firstorder gambles. Journal of Risk and Uncertainty, 4, 2959.
Luce, R., Krantz, D., Suppes, P., & Tversky, A. (1990). Foundations of measurement, Vol. 3.
Academic Press, San Diego, CA.
194
BIBLIOGRAPHY
Luce, R., & Narens, L. (1985). Classification of concatenation measurement structures
according to scale type. Journal of Mathematical Psychology, 29, 172.
Luce, R., & Suppes, P. (1965). Preference, utility, and subjective probability. In Luce,
R., Bush, R., & Galanter, E. (Eds.), Handbook of Mathematical Psychology, Vol. 3,
chap. 19, pp. 249410. Wiley, New York.
R., & Tukey, J. (1964). Simultaneous conjoint measurement: A new type of funda¬
Luce,
mental measurement. Journal of Mathematical Psychology, 1, 127.
Machina, M. (1983) In Stigum, B., & Wenstøp, F. (Eds.), Foundations of utility and risk
theory with applications, Vol. B, pp. 263293. Reidel, Dordrecht, Netherlands.
K. (1954). Intransitivity, utility, and the aggregation of preference patterns. Econ¬
May,
metrica, 22, 113.
Milgram, A. (1939). Partially ordered sets, separating systems and inductiveness. In Meyer,
K. (Ed.), Reports of a Mathematical Colloquium, Second Series, No.1, pp. University
of Notre Dame.
Moon, J. (1968). Topics on tournaments. Holt, Rinehart, and Winston, New York.
Mowshowitz, A. (1972). The characteristic polynomial of a graph. Journal of Combinatorial
Theory, 12(B), 177193.
Narens, L. (1985). Abstract measurement theory. MIT Press, Cambridge.
Narens, L. (1991). On the interpretation of strict utility choice models. In Doignon, J.P.,
& Falmagne, J.C. (Eds.), Mathematical Psychology: Current developments (Vol. 14).
SpringerVerlag, New York.
Neumann, J. von, & Morgenstern, O. (1944). Theory of games and economic behavior.
Princeton University Press, Princeton. 2nd edition, 1947; 3rd edition, 1953.
Ore, O. (1962). Theory of graphs, Vol. 28 of American Mathematical Society Colloquium
Publications. American Mathematical Society, Providence, Rhode Island.
Payne, J. (1976). Task complexity and contingent processing in decision making: An infor¬
mation search and protocol analysis. Organizational Behavior and Human Decision
Processes, 16, 366387.
Payne, J. (1982). Contingent decision behavior. Psychological Bulletin, 92, 382402.
Payne, J., Bettman, J., & Johnson, E. (1988). Adaptive strategy selection in decision making.
Journal of Experimental Psychology: Learning, Memory and Cognition, 14, 534552.
Payne, J., Bettman, J., & Johnson, E. (1990). The adaptive decision maker: effort and
accuracy in choice. In Hogarth, R. (Ed.), Insights in decision making: A tribute to
Hillel J. Einhorn, pp. 129153. University of Chicago Press, Chicago.
Pelli, D., & Zhang, L. (1991). Accurate control of contrast on microcomputer displays.
Vision Research, 31 (7/8), 13371350.
195
BIBLIOGRAPHY
Ramsey, F. (1931). Truth and probability. In Ramsey, F. (Ed.), The foundations of mathe¬
matics and other logical essays. Harcourt Brace, New York.
Reid, K., & Beineke, L. (1978). Tournaments. In Beineke, L., & Wilson, R. (Eds.), Selected
topics in graph theory, chap. 7, pp. 169204. Academic Press, New York.
Reinelt, G. (1985). The linear ordering problem: Algorithms and applications. Heldermann
Verlag, Berlin.
Restle, F. (1961). Psychology of judgment and choice: A theoretical essay. Wiley, New York.
Rice, J. (1988). Mathematical statistics and data analysis. Statistics/Probability Series.
Wadsworth & Brooks/Cole, Pacific Grove, CA.
Roberts, F. (1979). Measurement theory with applications to decision making, utility and the
social sciences. In Rota, G.C. (Ed.), Encyclopedia of mathematics and its applications:
Vol. 7, Mathematics and the social sciences. AddisonWesley, Reading, MA.
Roskam, E. (1987). Toward a psychometric theory of intelligence. In Roskam, E., & Suck, R.
(Eds.), Progress in mathematical psychology, Volume I, pp. 151174. Elsevier Science
Publishing Company, Amsterdam, NL.
Rumelhart, D., & Greeno, J. (1971). Similarity between stimuli: An experimental test of
the Luce and Restle choice models. Journal of Mathematical Psychology, 8, 370381.
Ryser, H. (1973). Indeterminates and incidence matrices. Linear Multilin. Algebra, 1, 149
157.
Sage,
A., & White, E. (1983). Decision and information structures in regret models of
judgment and choice. IEEE Transactions on Systems, Man and Cybernetics, SMC13,
136145.
Santen, J. (1978). A new axiomatization of portfolio theory. Journal of Mathematical
Psychology, 17, 1420.
Savage, L. (1954). The foundations of statistics. Wiley, New York.
Schmalhofer, F., Albert, D., Aschenbrenner, K., & Gertzen, H. (1986). Process traces of
binary choices: Evidence for selective and adaptive decision heuristics. The Quarterly
Journal of Experimental Psychology, 384, 5976.
Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econo¬
metrica, 57, 571587.
Schneider, H. (1977). The concepts of irreducibility and full indecomposability of a matix in
the works of Frobenius, König and Markov. Linear Algebra and its applications, 18,
139162.
Schneider, W. (1988). Micro Experimental Laboratory: An integrated system for IBM¬
PC compatibles. Behavior Research Methods, Instrumentation, and Computers, 20,
206217.
196
BIBLIOGRAPHY
Schneider, W. (1990). MEL user's guide: Computer techniques for real time experimentation.
Pittsburgh. Psychology Software Tools.
Schoemaker, P. (1982). The expected utility model: Its variants, purposes, evidence and
limitations. Journal of Economic Literature, 20, 529563.
Schrijver, A. (1986). Theory of Linear Programming and Integer Programming. John Wiley
& Sons, New York.
Scott, D. (1964). Measurement structures and linear inequalities. Journal of Mathematical
Psychology, 1, 233247.
Semmelroth, C. (1970). Predictions of lightness and brightness on different backgrounds.
Journal for the Optical Society of America, 60, pp.1685.
Simon, H. (1959). Theories of decision making in economics and behavioral science. American
Economic Review, 49, 253283.
Skiena, S. (1990). Implementing discrete mathematics: combinatorics and graph theory with
Mathematica. AddisonWesley, Redwood City, CA.
Slater, P. (1961). Inconsistencies in a schedule of paired comparisons. Biometrika, 48,
303312.
Slovic, P., & Lichtenstein, S. (1968). Relative importance of probabilities and payoffs in risk
taking. Journal of Experimental Psychology, 78, 118.
Slovic, P., & Tversky, A. (1974). Who accepts Savage's axiom?. Behavioral Science, 19,
368373.
Stevens, S. (1946). On the theory of scales of measurement. Science, 103, 677680.
Stevens, S. (1951). Mathematics, measurement and psychophysics. In Stevens, S. (Ed.),
Handbook of Experimental Psychology, pp. 149. Wiley, New York.
Street, A., & Street, D. (1987). Combinatorics of experimental design. Clarendon Press,
Oxford.
R. (1994). A theorem on order extensions: Embeddability of a system of weak orders
Suck,
to meet solvability constraints. Journal of Mathematical Psychology, 38, 128134.
Suck, R., & Getta, A. (1994). A reaxiomatization of portfolio theory. Journal of Mathematical
Psychology, 38, 115127.
Suppes, P., Krantz, D., Luce, R., & Tversky, A. (1989). Foundations of measurement, Vol. 2.
Academic Press, San Diego, CA.
Takasaki, H. (1966). Lightness change of grays induced by change in reflectance of gray
background. Journal for the Optical Society of America, 56, pp.504.
Tarjan, R. (1972). Depth first search and linear graph algorithms. SIAM Journal of Com¬
puting, 1, 146160.
BIBLIOGRAPHY
197
Thomassen, C. (1989). Whitney's 2switching theorem, cycle spaces, and arc mappings of
directed graphs. Journal of Combinatorial Theory, B 46, 257291.
Thurstone, L. (1927a). A law of comparative judgment. Psychological Review, 34, 273286.
Thurstone, L. (1927b). Psychological analysis. American Journal of Psychology, 38, 368389.
Treisman, M. (1983). A solution to the St.Petersburg Paradox. British Journal of Mathe¬
matical and Statistical Psychology, 36, 224227.
Tversky, A. (1969). Intransitivity of preferences. Psychological Review, 76, 3145.
Tversky, A. (1972a). Choice by elimination. Journal of Mathematical Psychology, 9, 341367.
Tversky, A. (1972b). Elimination by aspects: A theory of choice. Psychological Review, 79,
281299.
Tversky, A., & Kahneman, D. (1981). The framing of decisions and the psychology of choice.
Science, 211, 453458.
Tversky, A., & Kahneman, D. (1992). Advances in pospect theory: Cumulative representa¬
tion of uncertainty. Journal of Risk and Uncertainty, 5, 297323.
Vlek, C., & Wagenaar, W. (1979). Judgment and decision under uncertainty. In Michon, J.
(Ed.), Handbook of Psychonomics, Vol.2, pp. NorthHolland, Amsterdam.
Wakker, P. (1989). Additive representations of preferences. Kluwer, Dordrecht.
Weber, E. (1994). From subjective probabilities to decision weights: The effect of asymmet
ric loss functions on the evaluation of uncertain outcomes and events. Psychological
Bulletin, 115, 228242.
Wedell, D. (1991). Distinguishing among models of contextually induced preference reversals.
Journal of Experimental Psychology: Learning, Memory and Cognition, 17, 767778.
Whitney, H. (1932). Nonseparable and planar graphs. Transactions of the American Math¬
ematical Society, 34, 339362.
Wilkinson, J. (1988). The algebraic eigenvalue problem. Oxford University Press, New York.
Winer, B., Brown, D., & Michels, K. (1991). Statistical principles in experimental design
(3rd edition). McFrawHill, New York.
Wolfram, S. (1991). Mathematica: A system for doing mathematics by computer (2nd edi¬
tion). AddisonWesley, Reading, MA.
Yellott Jr., J. (1977). The relationship between Luce's choice axiom Thurstone's theory of
comparative judgment, and the double exponential distribution. Journal of Mathemat¬
ical Psychology, 15, 109144.
Zermelo, E. (1929). Die Berechnung der Turnierergebnisse als ein Maximumproblem der
Wahrscheinlichkeitsrechnung. Mathematische Zeitschrift, 29, 436460.
Index
NPcomplete, 43
acyclic subgraph problem, 43
Adams, E., 19
additive conjoint measurement, 23
adjacency matrix, 137
adjacent, 136
Afifi, A., 78
Albert, D., 25, 26, 130
Allais Paradox, 14
Allais, M., 14
Allerby, R., 137
antiisomorphism, 134
arc, 136
Arend, L., 109
Armstrong, W.E., 5
Aschenbrenner, K.M.
16, 25
Avrunin, G.H., 16
81
BabingtonSmith, B.
Barthélemy, J.P., 27
Beach, L.R., 24
Becker, G.M., 21
Beineke, L., 3537, 136
Bell, D., 17
Bergé, C., 136
Bermond, J., 35
Bernoulli, D., 2, 4, 11
Bettman, 26
Bettman, J.R., 26
bidimension, 26
Biggs, N., 137
Billings, R.S., 24
biorder, 10
Birkhoff, G., 9, 67
Birnbaum, M., 14.
15
Björner, A., 128
Block, H., 20
Bratko, I., 177
Brown, D., 76, 78
198
Brualdi, R., 38, 137
BTL model, 19, 24
Camerer, C.F., 13
Cameron, P., 137
Cantor, G., 9
chain, 56, 61
characteristic polynomial, 140
choice
inconsistent,
6
intransitive,
5
irrational, 4
rational, 4
transitive, 5
Chubb, C., 27
closure, 135
Coffey, G., 15
combinatorial matrix function, 42
complement, 134
completion by cuts, 63
Condorcet, Marquis de,
2
conjunctive decision rule,
22
constant utility model, 17
continuity axiom, 13
Coombs, C.H., 16, 22
Costanza, M., 78
Cramer, G., 11
cycle basis, 54
cycle space, 54
Dahlstrand, V., 24
Davey, B., 134, 135
Davidson, D., 20
Dawes, R.M., 22
de Finetti, B., 7
Debreu, G., 23
decomposition, 36
DedekindMacNeille completion, 63
DeGroot, M.H., 21
determinant, 138
INDEX
dicycle, 34
digraph, 136
directed cycle, 34, 136
directed graph, 136
directed walk, 136
disjunctive decision rule, 22
Doherty, M., 24
Doignon, J.P., 11, 26, 28, 128
domain, 134
dominance principle, 22
Ducamp, A., 11, 26
ear decomposition, 54
ear decomposition by sequence, 56
edge, 136
Edgell, S.E., 25
Edwards, W., 3, 4, 16
elimination by aspects.
25
Ellsberg Paradox, 14
Ellsberg, D., 14
equivalence
class, 135
relation, 135
Ericsson, K., 32, 129
expectancy, 8
expected utility model, 12
Falmagne, J.C., 11, 17, 26, 28, 128
Fechner, G.T., 18
Fechnerian utility model, 18
Fishburn, P.C., 10, 13, 15, 18, 23, 43
Ford, J.K., 24
Franklin, B., 2
Frobenius normal form, 38
Frobenius, G., 29, 38, 41
Gallucio, A., 54
Garey, M., 43
Geisler, W.S., 25
Gertzen, H., 26
Getta, A., 17
Gigerenzer, G., 24
Gilboa, I., 21
Goldberg, M., 46
Goldstein, D.G., 24
graph, 136
complete, 136
Greeno, J.G., 25
Grether, D.M., 15
Guilford, J.P., 21
Hadamard product, 44
Harary, F., 136, 138
Heinemann, E., 109
Herstein, I., 42, 137, 139
homomorphism, 134
Huber, O., 25
Hults, B., 24
image, 134
incidence matrix, 138
incidence vector, 138
eulerian, 138
inconsistency, 6
indegree, 138
independence axiom, 13
indifference, 5
infimum, 135
intransitive subchain, 62
family, 62
isomorphism, 134
Jünger, M., 43, 46
Jensen, N.E., 13
Johannesen, L., 128
Johnson, D., 43
Johnson, E.J., 26
König, D., 41
Kahneman, D., 16, 23, 25
Keeney, K., 3
80
Kendall's 7
81
Kendall's (,
Kendall, M., 80
Kernighan, B., 185
Klecka, W., 78
knowledge structures, 128
Kolmogorow, A., 7
Koppen, M., 21, 26, 128
Krantz, D.H., 9, 11, 22, 23
Kyburg Jr., H.E., 7
Lages, M., 28, 32, 130
Las Vergnas, M., 128
lattice, 136
lexicographic decomposition, 27
lexicographic rule, 24
199
200
lexicographic sum, 28
Lichtenstein, S., 23
linear order, 135
linear ordering problem, 43
link, 62
Loebl, M., 54
logistic model, 21
Loomes, G., 17
Lopes, L., 12, 14, 15
Lovász, L., 53
lower bound, 135
19
Luce'’s choice axiom
15, 17, 19, 2124, 131
Luce, R.D., 4, 811,
Machina, M., 16
majority rule, 23
mapping, 134
bijective, 134
injective, 134
surjective, 134
Marcus, S.A., 24
Marschak, J., 20, 21
matrix
irreducible, 138
reducible, 138
maximal element, 135
maximum, 135
May, K.O., 20, 24
Mellers, B., 15
Messick, S., 19
Michels, K., 76, 78
Milgram, A., 9
minimal element, 135
minimum, 135
Mitchell, T.R., 24
Montgomery, H., 24
Moon, J., 35, 36, 46, 136
Morgenstern, O., 12
Mowshowitz, A., 39
Narens, L., 8, 11, 15
Neumann, J. von, 12
nonexpected utility model, 15
Ore, O., 136
outdegree, 138
partial order, 135
INDEX
partial quasiorder, 135
partition of polynomial y, 42
Payne, J.W., 2426
permutation, 137
permutation cycle, 137
permutation matrix, 138
Plott, C.R., 15
Plummer, M., 53
polynomial
degree, 139
irreducible, 139
monic, 139
primitive, 139
reducible, 139
polynomial conjoint measurement, 23
polynomial ring, 139
portfolio theory, 16
Priestley, H., 134, 135
probability, 6
probability measure.
7
prospect theory, 16
Puto, C., 25
Raiffa, H., 3
Ramsey, F.P 7, 13
random block design, 74
20
random utility model
rank and signdependent utility model, 16
regret theory, 17
Reid, K., 3537, 136
Reinelt, G., 43
relation
nary, 134
antisymmetric, 134
asymmetric, 134
binary, 134
connected, 134
irreflexive, 134
negatively transitive, 134
reflexive, 134
symmetric, 134
transitive, 134
repetition block design,
73
resolution block design.
73
Restle, F., 25
Rice, J., 77
Ritchie, D., 185
INDEX
Roberts, F.S., 17
Roskam, E.E., 11
Rumelhart, D.L., 25
Ryser, H., 38, 41, 137
Sage, A.P., 17
Savage, L.J., 12, 13
Schlechtman, S., 24
Schmalhofer, F., 25, 26
Schmeidler, D., 21
Schmitt, N., 24
Schneider, H., 41, 42
Schoemaker, P.J., 14
Schrijver, A., 54
Scott, D., 13
semiorder, 9
shift, 62
Simon, H.A., 24, 32, 129
Skiena, S., 184
Slater's i, 32
Slater, P., 32
Slovic, P., 14, 23
Smokler, H.E., 7
Spehar, B., 109
St. Petersburg Paradox, 11
Stegner, S.E., 15
stepwise discriminant analysis, 78
Stevens, S., 4
73
Street, A.,
Street, D., 73
strict partial order, 10
strong, 34
strong component, 34, 136
strong tournament, 35
strong utility model, 18
strongly connected, 34, 136
Sturmfels, B., 128
subjectice expected utility model, 13
Suck, R., 17, 28
Sugden, R., 17
Suppes, P., 8, 9, 11, 17, 19, 22
supremum, 135
surething principle, 14
Tarjan, R., 38
Thomassen, C., 35, 46
Thurstone’s law of comparative judgment,
20
201
Thurstone, L.L., 21
tournament, 34
transitivity
moderate (stochastic), 18
strong (stochastic), 18
weak (stochastic), 18
Treisman, M., 12
Tukey, J.W., 23
Tversky, A., 6, 9, 11, 14, 16, 20, 22, 23, 25
unique factorization domain, 139
upper bound, 135
van Santen, J.P., 17
vertices, 136
Villano, M., 128
Vlek, C., 12
Wagenaar, W.A., 12
Wakker, P.P., 21
weak order, 9, 135
multiattribute
23
weak utility mode
18
Wedell, D.H., 25
Weiss, R., 15
White, E.B., 17
White, N., 128
Whitney, H., 53, 55
Wilkinson, J., 39, 140
Winer, B., 76, 78
Winterfeldt, von D., 15
Wolfram, S., 184
Yellott Jr, J.I., 21
Zermelo, E., 19
Ziegler, G.M., 128
62
61
60
65 Susanne A. Böhmig
Leistungspotentiale wertrelativierenden
Denkens.
Die Rolle einer wissensaktivierenden
Gedächtnisstrategie.
231 S. Erschienen 1998.
ISBN 3879850682
DM 27,
64 Jürgen Baumert, Wilfried Bos und
Rainer Watermann
TIMSS/III: Schülerleistungen in Mathematik
und den Naturwissenschaften am Ende der
Sekundarstufe II im internationalen Vergleich.
Zusammenfassung deskriptiver Ergebnisse.
40 S. Erschienen 1998.
ISBN 3879850674
DM 10,
63 Ursula Henz
Intergenerationale Mobilität.
Methodische und empirische Untersuchungen.
354 S. Erschienen 1996.
ISBN 3879850593
DM 32,
Andreas Maercker
Existentielle Konfrontation.
Eine Untersuchung im Rahmen eines
psychologischen Weisheitsparadigmas.
170 S. Erschienen 1995.
ISBN 3879850453
DM 19,
Alexandra M. Freund
Die Selbstdefinition alter Menschen.
Inhalt, Struktur und Funktion.
251 S. Erschienen 1995.
ISBN 3879850577
DM 17.
Klaus Schömann
The Dynamics of Labor Earnings over the Life
Course.
A Comparative and Longitudinal Analysis of
Germany and Poland.
188 S. Erschienen 1994.
ISBN 3879850569
DM 13,
59 Frieder R. Lang
Die Gestaltung informeller Hilfebeziehungen
im hohen Alter  Die Rolle von Elternschaft
und Kinderlosigkeit.
Eine empirische Studie zur sozialen Unterstützung
und deren Effekt auf die erlebte soziale Einbindung.
177 S. Erschienen 1994.
ISBN 3879850550
DM 13,
58
Ralf Th. Krampe
Maintaining Excellence.
CognitiveMotor Performance in Pianists
Differing in Age and Skill Level.
194 S. Erschienen 1994.
ISBN 3879850542
DM 14,
57 Ulrich Mayr
AgeBased Performance Limitations in Figural
Transformations.
The Effect of Task Complexity and Practice.
172 S. Erschienen 1993.
ISBN 3879850534
DM 13,
56
Marc Szydlik
Arbeitseinkommen und Arbeitsstrukturen.
Eine Analyse für die Bundesrepublik Deutschland
und die Deutsche Demokratische Republik.
255 S. Erschienen 1993.
ISBN 3879850526
DM 21.
55
Bernd Schellhas
Die Entwicklung der Ängstlichkeit in Kindheit
und Jugend.
Befunde einer Längsschnittstudie über die
Bedeutung der Ängstlichkeit für die Entwicklung
der Kognition und des Schulerfolgs.
205 S. Erschienen 1993
ISBN 3879850518
DM 13,
54 Falk Fabich
Forschungsfeld Schule: Wissenschaftsfreiheit,
Individualisierung und Persönlichkeitsrechte.
Ein Beitrag zur Geschichte
sozialwissenschaftlicher Forschung.
235 S. Erschienen 1993.
ISBN 387985050X
DM 22,
53 Helmut Köhler
Bildungsbeteiligung und Sozialstruktur in der
Bundesrepublik.
Zu Stabilität und Wandel der Ungleichheit von
Bildungschancen.
133 S. Erschienen 1992.
ISBN 3879850496
DM 10,
52 Ulman Lindenberger
Aging, Professional Expertise, and Cognitive
Plasticity.
The Sample Case of ImageryBased Memory
Functioning in Expert Graphic Designers.
130 S. Erschienen 1991.
ISBN 3608982574
DM 11,
51 Volker Hofmann
Die Entwicklung depressiver Reaktionen in
Kindheit und Jugend.
Eine entwicklungspsychopathologische Längs¬
schnittuntersuchung.
197 S. Erschienen 1991.
ISBN 3608982566
DM 14,
50 Georgios Papastefanou (vergriffen)
Familiengründung im Lebensverlauf.
Eine empirische Analyse sozialstruktureller Bedin¬
gungen der Familiengründung bei den Kohorten
192931, 1939—41 und 194951.
185 S. Erschienen 1990.
ISBN 3608982558
DM 15,
49 Jutta Allmendinger
Career Mobility Dynamics.
A Comparative Analysis of the United States,
Norway, and West Germany.
169 S. Erschienen 1989.
ISBN 360898254X
DM 13,
48 Doris Sowarka
Weisheit im Kontext von Person, Situation und
Handlung.
Eine empirische Untersuchung alltagspsycholo¬
gischer Konzepte alter Menschen.
—
275 S. Erschienen 1989.
ISBN 3608982531
DM 20,
47 Ursula M. Staudinger
The Study of Live Review.
An Approach to the Investigation of Intellectual
Development Across the Life Span.
211 S. Erschienen 1989.
ISBN 3608982523
DM 19,—
46 Detlef Oesterreich
Die Berufswahlentscheidung von jungen Lehrern.
115 S. Erschienen 1987.
ISBN 3608982515
DM 9,
45
HansPeter Füssel
Elternrecht und Schule.
Ein Beitrag zum Umfang des Elternrechts in der
Schule für Lerbehinderte.
501 S. Erschienen 1987.
ISBN 3608982493
DM 22,
44
Diether Hopf
Herkunft und Schulbesuch ausländischer Kinder.
Eine Untersuchung am Beispiel griechischer Schüler.
114 S. Erschienen 1987.
ISBN 3608982485
DM 8,
43 Eberhard Schröder
Entwicklungssequenzen konkreter Operationen.
Eine empirische Untersuchung individueller Ent¬
wicklungsverläufe der Kognition.
112 S. Erschienen 1986.
ISBN 3608982477
DM 13,
56
57
62 Jürgen Baumert, Wilfried Bos u. a. (Hrsg.)
Testaufgaben zu TIMSS/III
Mathematischnaturwissenschaftliche
Grundbildung und voruniversitäre
Mathematik und Physik der Abschluß¬
klassen der Sekundarstufe II
(Population 3). Im Druck
ISBN 3879850690
61
Jürgen Baumert, Rainer Lehmann u. a. (Hrsg.)
Testaufgaben Naturwissenschaften TIMSS
7./8. Klasse (Population 2).
111 S. Erschienen 1998.
DM 13.—
ISBN 3879850666
60
Jürgen Baumert, Rainer Lehmann u. a. (Hrsg.)
Testaufgaben Mathematik TIMSS
7./8. Klasse (Population 2).
131 S. Erschienen 1998.
ISBN 3879850658
DM 15.
59 Todd D. Little and Brigitte Wanner
The MultiCAM:
A Multidimensional Instrument to Assess
Children's ActionControl Motives, Beliefs, and
Behaviors.
194 S. Erschienen 1997.
ISBN 387985064X
DM 13.
58
Christine Schmid
Geschwister und die Entwicklung
soziomoralischen Verstehens.
Der Einfluß von Altersabstand und Geschlecht
jüngerer und älterer Geschwister im Entwicklungs
verlauf.
121 S. Erschienen 1997.
ISBN 3879850623
DM 10.
Kurt Kreppner und Manuela Ullrich
FamilienCodierSystem (FCS).
Beschreibung eines Codiersystems zur Beurteilung
von Kommunikationsverhalten in Familiendyaden.
94 S. Erschienen 1996.
ISBN 3879850615
DM 10.
Rosmarie Brendgen
Peer Rejection and Friendship Quality.
A View from Both Friends' Perspectives.
194 S. Erschienen 1996.
ISBN 3879850607
DM 21.
51
55 Siegfried Reuss und Günter Becker
Evaluation des Ansatzes von Lawrence
Kohlberg zur Entwicklung und Messung
moralischen Urteilens.
Immanente Kritik und Weiterentwicklung.
112 S. Erschienen 1996
ISBN 3879850488
DM 13,
54 Beate Krais und Luitgard Trommer
AkademikerBeschäftigung.
Sonderauswertung aus der Volkszählung 1987.
324 S. Erschienen 1995.
ISBN 387985047X
DM 33,
53
Marianne MüllerBrettel
Frieden und Krieg in der psychologischen
Forschung.
Historische Entwicklungen, Theorien und
Ergebnisse.
296 S. Erschienen 1995.
ISBN 3879850461
DM 32,
52 Harald Uhlendorff
Soziale Integration in den Freundeskreis.
Eltern und ihre Kinder.
130 S. Erschienen 1995.
ISBN 3879850445
DM 15,
Peter M. Roeder und Bernhard Schmitz
Der vorzeitige Abgang vom Gymnasium.
Teilstudie 1: Schulformwechsel vom Gymnasium
in den Klassen 5 bis 10.
Teilstudie II: Der Abgang von der Sekundarstufe I.
159 S. Erschienen 1995
ISBN 3879850437
DM 18,
50 Hannah Brückner
Surveys Don’t Lie, People Do?
An Analysis of Data Quality in a Retrospective
Life Course Study.
86 S. Erschienen 1995.
ISBN 3879850429
DM 7.
49 Todd D. Little, Gabriele Oettingen, and
Paul B. Baltes
The Revised Control, Agency, and Meansends
Interview (CAMI).
A MultiCultural Validity Assessment Using Mean
and Covariance Structures (MACS) Analyses.
97 S. Erschienen 1995.
ISBN 3879850410
DM 8,
48 Hannah Brückner und Karl Ulrich Mayer
Lebensverläufe und gesellschaftlicher Wandel.
Konzeption, Design und Methodik der Erhebung
von Lebensverläufen der Geburtsjahrgänge
19541956 und 19591961.
Teil I, Teil II, Teil III.
169 S., 224 S., 213 S.
Erschienen 1995.
ISBN 3879850399
DM 48.
46 Ursula M. Staudinger, Jacqui Smith und
Paul B. Baltes
Handbuch zur Erfassung von weisheits¬
bezogenem Wissen.
89 S. Deutsche Ausgabe
Manual for the Assessment of
WisdomRelated Knowledge.
83 S. Englische Ausgabe Erschienen 1994.
ISBN 3879850372
10,
DM
Jochen Fuchs
45
Internationale Kontakte im schulischen Sektor.
Zur Entwicklung und Situation des Schüleraustau¬
sches sowie von Schulpartnerschaften in der BRD.
74 S. Erschienen 1993.
ISBN 3879850356
DM 19,
44 Erika Brückner
Lebensverläufe und gesellschaftlicher Wandel.
Konzeption, Design und Methodik der Erhebung
von Lebensverläufen der Geburtsjahrgänge
19191921.
Teil I, Teil II, Teil III, Teil IV, Teil V.
235 S., 380 S., 200 S., 230 S., 141 S.
Erschienen 1993
ISBN 387985033X
DM 84,
43 ErnstH. Hoff und HansUwe Hohner
Methoden zur Erfassung von Kontroll¬
bewußtsein.
Textteil; Anhang.
99 S. und 178 S. Erschienen 1992.
ISBN 3879850321
DM 25,
38
37
41
39
36
35
42 Michael Corsten und Wolfgang Lempert
Moralische Dimensionen der Arbeitssphäre.
Literaturbericht, Fallstudien und Bedingungs¬
analysen zum betrieblichen und beruflichen
Handeln und Leren.
367 S. Erschienen 1992.
ISBN 3879850313
DM 20,
Armin Triebel
Zwei Klassen und die Vielfalt des Konsums.
Haushaltsbudgetierung bei abhängig Erwerbs¬
tätigen in Deutschland im ersten Drittel des
20. Jahrhunderts. Teil I, Teil II.
416 S., 383 S. Erschienen 1991.
ISBN 3879850305
DM 48.
Gundel Schümer
Medieneinsatz im Unterricht.
Bericht über Ziel, Anlage und Durchführung einer
Umfrage in allgemeinbildenden Schulen.
230 S. Erschienen 1991.
ISBN 3879850259
DM 24,
Clemens TeschRömer
Identitätsprojekte und Identitätstransforma
tionen im mittleren Erwachsenenalter.
312 S. Erschienen 1990.
ISBN 3879850267
DM 25,
Helmut Köhler
Neue Entwicklungen des relativen Schul und
Hochschulbesuchs.
Eine Analyse der Daten für 1975 bis 1978.
138 S. Erschienen 1990.
ISBN 3879850240
DM 10,
Wilfried Spang und Wolfgang Lempert
Analyse moralischer Argumentationen.
Beschreibung eines Auswertungsverfahrens.
Textteil: Grundlagen, Prozeduren, Evaluation.
Anhang: Interviewleitfaden, Tonbandtranskript und
Auswertungsbeispiele.
102 und 191 S. Erschienen 1989.
DM 29,
Karl Ulrich Mayer und Erika Brückner
Lebensverläufe und Wohlfahrtsentwicklung.
Konzeption, Design und Methodik der Erhebung
von Lebensverläufen der Geburtsjahrgänge
19291931, 19391941, 19491951.
Teil I, Teil II, Teil III.
261 S., unpaginiert, 175 S.
Erschienen 1989.
DM 39,
34 Christoph Droß und Wolfgang Lempert
Untersuchungen zur Sozialisation in der Arbeit
1977 bis 1988.
Ein Literaturbericht.
204 S. Erschienen 1988.
DM 12,
32 Friedrich Edding (Hrsg.)
Bildung durch Wissenschaft in neben und
nachberuflichen Studien.
Tagungsbericht.
157 S. Erschienen 1988.
DM 11,
29 Ulrich Trommer
Aufwendungen für Forschung und Entwicklung
in der Bundesrepublik Deutschland 1965 bis 1983.
Theoretische und empirischstatistische Probleme.
321 S. Erschienen 1987.
DM 32,
MaxPlanckInstitut für Bildungsforschung (Hrsg.
Reden zur Emeritierung von Wolfgang Edelstein
118 S. Berlin: MaxPlanckInstitut für Bildungs¬
forschung, 1997.
ISBN 3879850631
MaxPlanckInstitut für Bildungsforschung (Hrsg.
Pädagogik als empirische Wissenschaft.
Reden zur Emeritierung von Peter Martin Roeder
90 S. Berlin: MaxPlanckInstitut für Bildungs¬
forschung, 1996.
ISBN 3879850585
Ingo Richter, Peter M. Roeder, HansPeter Füssel (Eds.)
Pluralism and Education.
Current World Trends in Policy, Law, and
Administration.
345 S. Berkeley: University of California/USA, 1995.
DM 25.
MaxPlanckInstitut für Bildungsforschung (Hrsg.)
Bekenntnis und Dienst.
Reden zum 80. Geburtstag von Dietrich Goldschmidt
96 S. Berlin: MaxPlanckInstitut für Bildungs¬
forschung, 1995.
ISBN 3879850402
MaxPlanckInstitut für Bildungsforschung (Hrsg.)
Abschied von Hellmut Becker.
Reden auf der Trauerfeier am 18. Januar 1994.
47 S. Berlin: MaxPlanckInstitut für Bildungs¬
forschung, 1994.
ISBN 3879850364
MaxPlanckInstitut für Bildungsforschung (Hrsg.
Bildungsforschung und Bildungspolitik.
Reden zum 80. Geburtstag von Hellmut Becker
98 S. Berlin: MaxPlanckInstitut für Bildungs¬
forschung, 1993.
ISBN 3879850348
Wolfgang Schneider and Wolfgang Edelstein (Eds.)
Inventory of European Longitudinal Studies in the
Behavioral and Medical Sciences.
A Project Supported by the European Science Foun
dation.
557 S. Munich: Max Planck Institute for Psychological
Research, and Berlin: Max Planck Institute for Human
Development and Education, 1990.
ISBN 3879850283
DM 58,
MaxPlanckInstitut für Bildungsforschung (Hrsg.)
Entwicklung und Lernen.
Beiträge zum Symposium anläßlich des 60. Geburts¬
tages von Wolfgang Edelstein.
98 S. Berlin: MaxPlanckInstitut für Bildungs¬
forschung, 1990.
ISBN 3879850232
MaxPlanckInstitut für Bildungsforschung (Hrsg.)
Normative Voraussetzungen und ethische Implika¬
tionen sozialwissenschaftlicher Forschung.
Beiträge zum Symposium anläßlich des 75. Geburts¬
tages von Dietrich Goldschmidt.
108 S. Berlin: MaxPlanckInstitut für Bildungs¬
forschung, 1990.
ISBN 3879850275
MaxPlanckInstitut für Bildungsforschung (Hrsg.)
25 Jahre MaxPlanckInstitut für Bildungs
forschung.
Festvorträge.
48 S. Berlin: MaxPlanckInstitut für Bildungs
forschung, 1989.
Friedrich Edding
Mein Leben mit der Politik.
126 S. Berlin: MaxPlanckInstitut für Bildungs¬
forschung, 1989.
MaxPlanckInstitut für Bildungsforschung (Hrsg.)
Gewerbliche Unternehmen als Bildungsträger.
Beiträge zum Symposium anläßlich des 80. Geburts¬
tages von Friedrich Edding.
126 S. Berlin: MaxPlanckInstitut für Bildungs¬
forschung, 1989.
Hermann Avenarius, Jürgen Baumert, Hans Döbert
und HansPeter Füssel (Hrsg.)
Schule in erweiterter Verantwortung.
Positionsbestimmungen aus erziehungswissenschaft¬
licher, bildungspolitischer und verfassungsrechtliche
Sicht.
166 S. Neuwied: Luchterhand Verlag 1998.
Matthias Grundmann
Norm und Konstruktion.
Zur Dialektik von Bildungsvererbung und Bildungs¬
aneignung
231 S. Opladen: Leske + Budrich 1998.
Tobias Krettenauer
Gerechtigkeit als Solidarität.
Entwicklungsbedingungen sozialen Engagements im
Jugendalter
267 S. Weinheim: Deutscher Studien Verlag 1998.
Michael Wagner und Yvonne Schütze
Verwandtschaft.
Sozialwissenschaftliche Beiträge zu einem
vernachlässigten Thema.
281 S. Stuttgart: F. Enke Verlag 1998.
Kai Schnabel
Prüfungsangst und Lernen.
Empirische Analysen zum Einfluß fachspezifischer
Leistungsängstlichkeit auf schulischen Lerfortschritt.
201 S. New York, München, Berlin: Waxmann Verlag
1998.
Olaf Köller
Zielorientierungen und schulisches Lernen.
216 S. New York, München, Berlin: Waxmann Verlag
1998.
Michael Wagner
Scheidung in Ost und Westdeutschland.
Zum Verhältnis von Ehestabilität und
Sozialstruktur seit den 30er Jahren.
355 S. Frankfurt/M. Campus Verlag, 1997
Gero Lenhardt und Manfred Stock
Bildung, Bürger, Arbeitskraft.
Schulentwicklung und Sozialstruktur in
der BRD und der DDR
253 S. Frankfurt/M. Suhrkamp
Taschenbuch Verlag, 1997.
Michael Corsten und Wolfgang Lempert
Beruf und Moral
Exemplarische Analysen beruflicher Werdegänge,
betrieblicher Kontexte und sozialer Orientierungen
erwerbstätiger Lehrabsolventen.
200 S. Weinheim: BeltzDeutscher Studien Verlag,
1997.
Jürgen Baumert und Rainer Lehmann u.a
TIMSS  Mathematischnaturwissenschaftlicher
Unterricht im internationalen Vergleich.
Deskriptive Befunde
242 S. Leverkusen: LeskeBudrich, 1997.
Gabriele Oettingen
Psychologie des Zukunftsdenkens.
Erwartungen und Phantasien.
452 S. Göttingen/Bern/Toronto/ Seattle: Hogrefe, 1996.
Detlef Oesterreich
Flucht in die Sicherheit.
Zur Theorie des Autoritarismus und der autoritären
Reaktion.
250 S. Leverkusen: LeskeBudrich, 1996.
Karl Ulrich Mayer und Paul B. Baltes (Hrsg.)
Die Berliner Altersstudie.
(Ein Projekt der BerlinBrandenburgischen Akademie
der Wissenschaften)
672 S. Berlin: Akademie Verlag, 1996.
Paul B. Baltes and Ursula M. Staudinger (Eds.)
Interactive Minds.
LifeSpan Perspectives on the Social Foundation of
Cognition.
457 pp. New York: Cambridge University Press, 1996.
Monika Keller
Moralische Sensibilität: Entwicklung in
Freundschaft und Familie.
259 S. Weinheim: Psychologie Verlags Union, 1996.
Martin Diewald, Karl Ulrich Mayer (Hrsg.)
Zwischenbilanz der Wiedervereinigung.
Strukturwandel und Mobilität im
Transformationsprozeß.
352 S. Lerverkusen: LeskeBudrich, 1996.
Johannes Huinink, Karl Ulrich Mayer u.a.
Kollektiv und Eigensinn.
Lebensverläufe in der DDR und danach.
414 S. Berlin: Akademie Verlag, 1995.
Johannes Huinink
Warum noch Familie?
Zur Attraktivität von Partnerschaft und Elternschaft in
unserer Gesellschaft.
385 S. Frankfurt/M./New York: Campus, 1995.
Heike Trappe
Emanzipation oder Zwang?
Frauen in der DDR zwischen Beruf, Familie und
Sozialpolitik.
242 S. Berlin: Akademie Verlag, 1995.
Heike Solga
Auf dem Weg in eine klassenlose Gesellschaft?
Klassenlagen und Mobilität zwischen Generationen in
der DDR
265 S. Berlin: Akademie Verlag, 1995.
Lothar Krappmann und Hans Oswald
Alltag der Schulkinder.
Beobachtungen und Analysen von Interaktionen und
Sozialbeziehungen.
224 S. Weinheim/München: Juventa, 1995.
Freya DittmannKohli
Das persönliche Sinnsystem.
Ein Vergleich zwischen frühem und spätem
Erwachsenenalter.
402 S. Göttingen/Bern/Toronto/ Seattle: Hogrefe, 1995
Hartmut Zeiher und Helga Zeiher
Orte und Zeiten der Kinder.
Soziales Leben im Alltag von Großstadtkindern.
223 S. Weinheim/München: Juventa, 1994.
Christiane LangeKüttner
Gestalt und Konstruktion.
Die Entwicklung der grafischen Kompetenz beim
Kind.
242 S. Bern/Toronto: Huber, 1994.
Jutta Allmendinger
Lebensverlauf und Sozialpolitik.
Die Ungleichheit von Mann und Frau und ihr
öffentlicher Ertrag.
302 S. Frankfurt a. M./New York: Campus, 1994.
Wolfgang Lauterbach
Berufsverläufe von Frauen.
Erwerbstätigkeit, Unterbrechung und Wiedereintritt.
289 S. Frankfurt a. M./New York: Campus, 1994.
Arbeitsgruppe Bildungsbericht am
MaxPlanckInstitut für Bildungsforschung
Das Bildungswesen in der Bundesrepublik
Deutschland.
Strukturen und Entwicklungen im Überblick.
843 S. Reinbek: Rowohlt, 1994 (4., vollständig über¬
arbeitete und erweiterte Neuausgabe).
Hellmut Becker und Gerhard Kluchert
Die Bildung der Nation.
Schule, Gesellschaft und Politik vom Kaiserreich zur
Weimarer Republik.
538 S. Stuttgart: KlettCotta, 1993.
Rolf Becker
Staatsexpansion und Karrierechancen.
Berufsverläufe im öffentlichen Dienst und in der
Privatwirtschaft.
303 S. Frankfurt a.M./New York: Campus, 1993.
Wolfgang Edelstein und
Siegfried HoppeGraff (Hrsg.)
Die Konstruktion kognitiver Strukturen.
Perspektiven einer konstruktivistischen
Entwicklungspsychologie.
328 S. Bern/Stuttgart/Toronto: Huber, 1993.
Wolfgang Edelstein, Gertrud NunnerWinkler
und Gil Noam (Hrsg.)
Moral und Person.
418 S. Frankfurt a.M.: Suhrkamp, 1993.
Lothar Lappe
Berufsperspektiven junger Facharbeiter.
Eine qualitative Längsschnittanalyse zum Kernbereich
westdeutscher Industriearbeit.
394 S. Frankfurt a.M./New York: Campus, 1993.
Detlef Oesterreich
Autoritäre Persönlichkeit und Gesellschaftsordnung
Der Stellenwert psychischer Faktoren für politische
Einstellungen  eine empirische Untersuchung von
Jugendlichen in Ost und West.
243 S. Weinheim/München: Juventa, 1993.
Marianne MüllerBrettel
Bibliographie Friedensforschung und
Friedenspolitik:
Der Beitrag der Psychologie 19001991.
(Deutsch/Englisch)
383 S. München/London/New York/Paris: Saur, 1993.
Paul B. Baltes und Jürgen Mittelstraß (Hrsg.)
Zukunft des Alterns und gesellschaftliche
Entwicklung.
(= Forschungsberichte der Akademie der
Wissenschaften zu Berlin, 5.)
814 S. Berlin/New York: De Gruyter, 1992.
Matthias Grundmann
Familienstruktur und Lebensverlauf.
Historische und gesellschaftliche Bedingungen
individueller Entwicklung.
226 S. Frankfurt a.M./New York: Campus, 1992.
Karl Ulrich Mayer (Hrsg.
Generationsdynamik in der Forschung
1992.
245 S. Frankfurt a.M./New York: Campus,
Erika M. Hoerning
Zwischen den Fronten.
Berliner Grenzgänger und Grenzhändler 19481961
266 S. Köln/Weimar/Wien: Böhlau, 1992.
ErnstH. Hoff
Arbeit, Freizeit und Persönlichkeit.
Wissenschaftliche und alltägliche Vorstellungsmuster.
238 S. Heidelberg: Asanger Verlag, 1992 (2. über¬
arbeitete und aktualisierte Auflage).
Erika M. Hoerning
Biographieforschung und Erwachsenenbildung.
223 S. Bad Heilbrunn: Klinkhardt, 1991.
MaxPlanckInstitut für Bildungsforschung
Traditions et transformations.
Le système d'éducation en République fédérale
d'Allemagne.
341 S. Paris: Economica, 1991.
Dietrich Goldschmidt
Die gesellschaftliche Herausforderung der
Universität.
Historische Analysen, internationale Vergleiche,
globale Perspektiven.
297 S. Weinheim: Deutscher Studien Verlag, 1991
Uwe Henning und Achim Leschinsky (Hrsg.)
Enttäuschung und Widerspruch.
Die konservative Position Eduard Sprangers im
Nationalsozialismus. Analysen  Texte  Dokumente.
213 S. Weinheim: Deutscher Studien Verlag, 1991.
ErnstH. Hoff, Wolfgang Lempert und Lothar Lappe
Persönlichkeitsentwicklung in Facharbeiter
biographien.
282 S. Bern/Stuttgart/Toronto: Huber, 1991
Karl Ulrich Mayer, Jutta Allmendinger und
Johannes Huinink (Hrsg.)
Vom Regen in die Traufe: Frauen zwischen Beruf
und Familie.
483 S. Frankfurt a.M./New York: Campus, 1991
Maria von Salisch
Kinderfreundschaften.
Emotionale Kommunikation im Konflikt.
153 S. Göttingen/Toronto/Zürich: Hogrefe, 1991.
Paul B. Baltes and Margret M. Baltes (Eds.)
Successful Aging: Perspectives from the Behavioral
Sciences.
397 pp. Cambridge: Cambridge University Press, 1990.
Paul B. Baltes, David L. Featherman and
Richard M. Lerer (Eds.)
LifeSpan Development and Behavior.
368 pp. Vol. 10. Hillsdale, N.J.: Erlbaum, 1990.
Achim Leschinsky and Karl Ulrich Mayer (Eds.)
The Comprehensive School Experiment Revisited:
Evidence from Western Europe.
211 pp. Frankfurt a.M./Bern/New York/Paris: Lang
1990.
Karl Ulrich Mayer (Hrsg.)
Lebensverläufe und sozialer Wandel.
467 S. Opladen: Westdeutscher Verlag, 1990.
(= Sonderheft 31 der KZfSS).
Karl Ulrich Mayer and Nancy Brandon Tuma (Eds.)
Event History Analysis in Life Course Research.
320 pp. Madison, Wis.: The University of Wisconsin
Press, 1990.
Hans J. Nissen, Peter Damerow und Robert K. Englund
Frühe Schrift und Techniken der Wirtschafts¬
verwaltung im alten Vorderen Orient.
Informationsspeicherung und verarbeitung vor
5000 Jahren.
Katalog zur gleichnamigen Ausstellung Berlin¬
Charlottenburg, MaiJuli 1990.
222 S. Bad Salzdetfurth: Franzbecker, 1990.
(2. Aufl. 1991).
Peter Alheit und Erika M. Hoerning (Hrsg.)
Biographisches Wissen.
Beiträge zu einer Theorie lebensgeschichtlicher
Erfahrung.
284 S. Frankfurt a.M./New York: Campus, 1989.
Arbeitsgruppe am MaxPlanckInstitut für
Bildungsforschung
Das Bildungswesen in der Bundesrepublik
Deutschland.
Ein Überblick für Eltern, Lehrer und Schüler.
Japanische Ausgabe: 348 S. Tokyo: Toshindo
Publishing Co. Ltd., 1989.
HansPeter Blossfeld
Kohortendifferenzierung und Karriereprozeß.
Eine Längsschnittstudie über die Veränderung der
Bildungs und Berufschancen im Lebenslauf.
185 S. Frankfurt a.M./New York: Campus, 1989.
HansPeter Blossfeld, Alfred Hamerle and
Karl Ulrich Mayer
Event History Analysis.
Statistical Theory and Application in the Social
Sciences.
297 pp. Hillsdale, N.J.: Erlbaum, 1989.
Erika M. Hoerning und Hans Tietgens (Hrsg.)
Erwachsenenbildung: Interaktion mit der
Wirklichkeit.
200 S. Bad Heilbrunn: Klinkhardt, 1989.
Johannes Huinink
MehrebenensystemModelle in den Sozialwissen¬
schaften.
292 S. Wiesbaden: Deutscher UniversitätsVerlag, 1989.
Kurt Kreppner and Richard M. Lerner (Eds.)
Family Systems and LifeSpan Development.
416 pp. Hillsdale, N.J.: Erlbaum, 1989.
Bernhard Schmitz
Einführung in die Zeitreihenanalyse.
Modelle, Softwarebeschreibung, Anwendungen.
235 S. Bern/Stuttgart/ Toronto: Huber, 1989.
Eberhard Schröder
Vom konkreten zum formalen Denken.
Individuelle Entwicklungsverläufe von der Kindheit
zum Jugendalter.
328 S. Bern/Stuttgart/Toronto: Huber, 1989.
Michael Wagner
Räumliche Mobilität im Lebensverlauf.
Eine empirische Untersuchung sozialer Bedingungen
der Migration.
226 S. Stuttgart: Enke, 1989.
Paul B. Baltes, David L. Featherman and
Richard M. Lerner (Eds.)
LifeSpan Development and Behavior.
338 pp. Vol. 9. Hillsdale, N.J.: Erlbaum, 1988.
Paul B. Baltes, David L. Featherman and
Richard M. Lerner (Eds.)
LifeSpan Development and Behavior.
337 pp. Vol. 8. Hillsdale, N.J.: Erlbaum, 1988.
Lothar Krappmann
Soziologische Dimensionen der Identität.
Strukturelle Bedingungen für die Teilnahme an
Interaktionsprozessen.
231 S. Stuttgart: KlettCotta, 7: Aufl., 1988
(= Standardwerke der Psychologie).
Detlef Oesterreich
Lehrerkooperation und Lehrersozialisation.
159 S. Weinheim: Deutscher Studien Verlag, 1988.
Michael Bochow und Hans Joas
Wissenschaft und Karriere.
Der berufliche Verbleib des akademischen Mittelbaus.
172 und 37 S. Frankfurt a.M./New York: Campus, 1987.
HansUwe Hohner
Kontrollbewußtsein und berufliches Handeln.
Motivationale und identitätsbezogene Funktionen
subjektiver Kontrollkonzepte.
201 S. Bern/Stuttgart/Toronto: Huber, 1987.
Bernhard Schmitz
Zeitreihenanalyse in der Psychologie.
Verfahren zur Veränderungsmesung und Prozeß¬
diagnostik.
304 S. Weinheim/Basel: Deutscher Studien Verlag/
Beltz, 1987.
Margret M. Baltes and Paul B. Baltes (Eds.)
The Psychology of Control and Aging.
415 pp. Hillsdale, N.J.: Erlbaum, 1986.
Paul B. Baltes, David L. Featherman and
Richard M. Lerner (Eds.)
LifeSpan Development and Behavior.
334 pp. Vol. 7. Hillsdale, N.J.: Erlbaum, 1986
HansPeter Blossfeld, Alfred Hamerle und
Karl Ulrich Mayer
Ereignisanalyse.
Statistische Theorie und Anwendung in den
Wirtschafts und Sozialwissenschaften.
290 S. Frankfurt a.M./New York: Campus, 1986.
Axel Funke, Dirk Hartung, Beate Krais und
Reinhard Nuthmann
Karrieren außer der Reihe.
Bildungswege und Berufserfolge von Stipendiaten
der gewerkschaftlichen Studienförderung.
256 S. Köln: Bund, 1986.
ErnstH. Hoff, Lothar Lappe und
Wolfgang Lempert (Hrsg.)
Arbeitsbiographie und Persönlichkeitsentwicklung.
288 S. Bern/Stuttgart/ Toronto: Huber, 1986.
Klaus Hüfner, Jens Naumann, Helmut Köhler und
Gottfried Pfeffer
Hochkonjunktur und Flaute: Bildungspolitik in
der Bundesrepublik Deutschland 19671980.
361 S. Stuttgart: KlettCotta, 1986.
Jürgen Staupe
Parlamentsvorbehalt und Delegationsbefugnis.
Zur „Wesentlichkeitstheorie“ und zur Reichweite
legislativer Regelungskompetenz, insbesondere im
Schulrecht.
419 S. Berlin: Duncker & Humblot, 1986.
HansPeter Blossfeld
Bildungsexpansion und Berufschancen.
Empirische Analysen zur Lage der Berufsanfänger in
der Bundesrepublik.
191 S. Frankfurt a.M./New York: Campus, 1985.
Christel Hopf, Knut Nevermann und Ingrid Schmidt
Wie kamen die Nationalsozialisten an die Macht.
Eine empirische Analyse von Deutungen im Unterricht
344 S. Frankfurt a.M./New York: Campus, 1985.
John R. Nesselroade and Alexander von Eye (Eds.)
Individual Development and Social Change:
Explanatory Analysis.
380 pp. New York: Academic Press, 1985.
Michael Jenne
Music, Communication, Ideology.
185 pp. Princeton, N.J.: Birch Tree Group Ltd., 1984.
Gero Lenhardt
Schule und bürokratische Rationalität.
282 S. Frankfurt a.M.: Suhrkamp, 1984.
Achim Leschinsky und Peter Martin Roeder
Schule im historischen Prozeß.
Zum Wechselverhältnis von institutioneller Erziehung
und gesellschaftlicher Entwicklung.
545 S. Frankfurt a.M./Berlin/Wien: Ullstein, 1983.
Max Planck Institute for
Human Development and Education
Between Elite and Mass Education.
Education in the Federal Republic of Germany.
348 pp. Albany: State University of New York Press,
1983.
Margit Österloh
Handlungsspielräume und Informationsver¬
arbeitung.
369 S. Bern/Stuttgart/Toronto: Huber, 1983.
Knut Nevermann
Der Schulleiter.
Juristische und historische Aspekte zum Verhältnis
von Bürokratie und Pädagogik.
314 S. Stuttgart: KlettCotta, 1982.
Gerd Sattler
Englischunterricht im FEGAModell.
Eine empirische Untersuchung über inhaltliche und
methodische Differenzierung an Gesamtschulen.
355 S. Stuttgart: KlettCotta, 1981.
Christel Hopf, Knut Nevermann und Ingo Richter
Schulaufsicht und Schule.
Eine empirische Analyse der administrativen Bedin
gungen schulischer Erziehung.
428 S. Stuttgart: KlettCotta, 1980.
Diether Hopf
Mathematikunterricht.
Eine empirische Untersuchung zur Didaktik und
Unterrichtsmethode in der 7. Klasse des Gymnasiums.
251 S. Stuttgart: KlettCotta, 1980.
MaxPlanckInstitut für Bildungsforschung
Projektgruppe Bildungsbericht (Hrsg.)
Bildung in der Bundesrepublik Deutschland.
Daten und Analysen.
Bd. 1: Entwicklungen seit 1950.
Bd. 2: Gegenwärtige Probleme.
1404 S. Stuttgart: KlettCotta, 1980.
Dietrich Goldschmidt und Peter Martin Roeder (Hrsg.)
Alternative Schulen?
Gestalt und Funktion nichtstaatlicher Schulen im
Rahmen öffentlicher Bildungssysteme.
623 S. Stuttgart: KlettCotta, 1979.