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continued on ~ 85
Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
250
Marc Roubens
Philippe Vincke
(Preference Modelling)
Springer-Verlag
Berlin Heidelberg New York Tokyo
editorial Board
H. Albach M. Beckmann (Managing Editor) P. Dhrymes
G. Fandel J. Green W Hildenbrand W. Krelle(Managing Editor) H.P. Kunzi
G.L Nemhauser K. Ritter R. Sato U. Schittko P. Schonfeld R. Selten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W. Krelle
Institut fur Gesellschafts-und Wirtschaftswissenschaften
der Universitii.t Bonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
Authors
Prof. Dr. Marc Roubens
State University of Mons
Rue de Houdain 9, 7000 Mons, Belgium
Prof. Dr. Philippe Vincke
Free University of Brussels
CPo 210, Boulevard du Triomphe, 1050 Brussels, Belgium
ISBN-13: 978-3-540-15685-7 e-ISBN-13: 978-3-642-46550-5
001: 10.1007/978-3-642-46550-5
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IC by Springer-Verlag Berlin Heidelberg 1985
2142/3140-643210
To Annie and
~chele~
oU!" wi ve s .
INTROIlJCTI <J.I
Preference modelling is an inevitable step in a lot of fields, like
psychology, sociology, operational research, decision theory, •.••
econo~,
This step is sometimes implicit, like in operational research and economy,
where the preferences of the decision-maker are often represented by a func
tion to be optimized; sometimes it is studied in details and based on expe
riments or inquiries, like in multicriteria analysis and mathematical psycho
logy. In any case, more and more scientists have the feeling that this step
is fundamental and cannot be neglected.
It would be impossible to cover all the aspects of preference modelling
in about a hundred pages. The point of view adopted here is essentially moti
vated by recent researches in decision-aid and by the will to treat with well
known tools. Thi sis the reason why the so-ca 11 ed "preference structures" have
been studied according to three specific guidelines: gpaph pepPesentation,
numePioaZ (or functional) PepPesentation and opinion tableau oonfigUPation.
In particular, this text does not cover the following fields: preference data
collection, geometrical representation of preferences, measurement theory and
expected utility theory.
Chapters 1 and 2 form a ground work for the rest of these notes and
include most of the concepts needed later on.
Chapter 3 introduces the usual preference structures which are widely
used in the literature: tournaments, total orders, weak orders, interval
orders, semiorders, partial orders and quasi orders. They are studied according
to the three aspects defined before.
In chapter 4, we introduce two new preference structures, called partial
interval order and partial semi order. They coincide with the corresponding
complete structures when incomparability does not exist and they are compatible
with numerical representations which generalize the partial order and quasi
order cases.
Chapter 5 is devoted to the valued preference structures which appear
in probabilistic or fuzzy contexts. It presents the families of relations that
VI
can be built up on the basis of a valued (or probabilistic) relation. These
families contain, as particular cases, some of the concepts introduced by
ROBERTS and FISHBURN. They are also examined from the view point of numerical
representation.
In chapter 6, the particular case of valued preference structures with
only two distinct values (weak and strong preferences) is presented and some
specific results are established.
This text does not pre-suppose the possession of more than the basic
vocabulary of the relations and graphs theory. All the other concepts are
defined in the context. References have been provided at the end of each
chapter in order to simplify the task of the reader.
These notes have been used as a part of an interuniversity course on
"Preference Modelling and Decision Aid", sponsored by the F.N.R.S. (Fonds
National de 1a Recherche Scientifique). We want to thank the people who
actively participated to this semin r. Special thanks are due to D. BOUYSSOU
and B. MONJARDET for their comments on the manuscript. We wish also to express
our appreciation to Mrs MOBERS for excellent typing.
TABLE OF CONTENTS
CHAPTER 1 BINARY RELATIONS : DEFINITIONS, REPRESENTATIONS,
BASIC PROPERTIES.
1.1. Binary relations •••.••...•.
1.2. Gpaph rep~sentation of binary relations 2
1.3. Coding the binary relations . ..... . 2
1.4. Matrix ~p~sentation of binary relations 2
1.5. Basia properties of binary relations 3
1.6. Partiaular binary relations • .•.. 4
1.7. Graph interpretation of the properties 4
1.8. Algebraia interpretation of the properties 5
1 .9. Re ferenaes • • . • . . . • . . . . • . 5
CHAPTER 2 : THE CONCEPT OF PREFERENCE STRUCTURE.
2.1. Preferenae, indifferenae, inaomparability 6
2.2. Preferenae struat~ . ••....... 6
2.3. Important ag~ement ..•.•.•... 7
2.4. Charaateristia ~lation of a preferenae struature 7
2.5. Graph rep~sentation Of a preferenae struatu~. 8
2.6. Coding the pre ferenae struature 8
2.7. Example •. 9
2.8. Re ferenaes 10
CHAPTER 3 : USUAL PREFERENCE STRUCTURES
3.1. Tournament struatu~ 11
3.2. Total order stt'Uature 14
3.3. Weak order struat~ 18
3.4. Total interval order struatu~ 22
3.5. Total semiorder struature 35
3.6. Partial order struat~ 41
3.7. Quasi order struat~ 48
3.8. Referenaes 52
VIII
CHAPTER 4 : TWO NEW PREFERENCE STRUCTURES
4.1. PazotiaZ intervaZ Ol'del' stl'UCtuzte • 54
4.2. Paz>tiaZ semiOl'del' stl'UCtuzte 61
4.3. Refe~nces ••••••••• 64
CHAPTER 5 : COMPLETE VALUED PREFERENCE STRUCTURES
5.1. Dsfinition • • • 65
5.2. Impol'tant ~maztk • • 65
5.3. Paz>ticuZazo case 65
5.4. Graph l'ep~sentation 66
5.5. MatzticiaZ ~pl'esentation 66
5.6. Pal'ticuZazo compZete vaZued p~fel'ence stl'UCtuztes 67
5.7. Binazty l'eZations and vaztious pl'opel'ties ~Zated to a
compZete vaZued pl'efel'ence stl'Uctuz>e • • • • • • • • 68
5.8. Chazoactel'izations of the famiZies de~ned in section 5.6.. 74
5.9. FunctionaZ ~pl'8sentation of a vaZued pztefeztence stl'UCtUzte 77
5.10.Robel'm homogeneous famiZies of semiOl'del'S 82
5.11.FamiZies of~ak Ol'del'S 83
5. 12 •S U1TU1Iazty 84
5.13.EroampZes • 85
5. 14.Refeztences 86
CHAPTER 6 : COMPLETE TWO-VALUED PREFERENCE STRUCTURES
6.1. Intl'oduction • . ••••••••••••• 87
6.2. Two-vaZued pztefel'ence stl'Uctuz>es with constant ·thzteshoZds 88
6.3. EroampZe 91
6.4. Refeztences ••••••••••••••••••.••••• 92
CHAPTER 1
BINARY RELATIONS: DEFINITIONS. REPRESENTATIONS. BASIC PROPERTIES
1.1. BINARY RELATIONS
Let A denote a finite set of elements a. b. c. , ..
A binary relation S on the set A is a subset of the cartesian product
A x A. that is. a set of ordered pairs (a.b) such that a and b are in A :
SeA x A. If the ordered pair (a.b) belongs to S. we denote indifferently
(a.b) E S or a S b .
c - d
The complement S • the converse S and the dual S are respectively
defined as follows:
(a.b) E SC iff (a.b) ¢ S •
(a.b) E ~ iff (b.a) E S •
(a.b) E Sd iff (b.a) ¢ S
Example
A = {a.b.c.d}
S = {(a.a).(a.c).(b.c).(c.b).(c.d).(d.a).(d.d)}
SC = {(a.b).(a.d).(b.a).(b.b).(b.d).(c.a).(c.c).(d.b).(d.c)}
~ = {(a.a).(c.a).(c.b).(b.c).(d.c).(a.d).(d.d)}
Sd = {(a.b).(a.c).(b.a).(b.b).(b.d).(c.c).(c.d).(d.a).(d.b)}
Remark
Notations
Relations S. SC. ~. Sd being subsets of Ax A. we can use the set-theoric
notations as union. intersection •. ,.
Let Sand T be two relations on the same set A. We denote
2
inclusion SeT iff a S b- a T b, V a,b E A,
union a(S U T)b iff a S b or (inclusive) a T b
i ntersecti on a(S n T)b iff a S b and a T b ,
relative product a S.T b iff 3 c E A a S c and c T b ,
a S2 b iff 3 c E A a S c and c S b •
1.2. GRAPH REPRESENTATION OF BINARY RELATIONS
Every relation S on the set A may be represented using a digraph
(di rected graph) (A, S), where P. is the set of nodes and S the set of arcs.
There exists an arc from a to b iff a S b. When a S a, one undirected loop
is drawn on a (replacing obviously two directed loops).
Example: Graph representation for example of section 1.1.
a b
~
S
1.3. CODING THE BINARY RELATIONS
Coding the relation S comes to consider two different real numbers a
and a and to associate one of these numbers to the ordered pair (a,b) in
the following way:
CSab = a iff a S b ,
C!b = S iff a SC b
In general, the boolean rating is used: a = 1, a = O.
1.4. r~TRIX REPRESENTATION OF BINARY RELATIONS.
Starting with the code related to S, one can consider the tableau ~S
S
with entry a,b (line a, column b) taken to be Cab.
