Arrovian Aggregation Models

254 Pages·1999·16.032 MB·English

by  Fuad Aleskerov (auth.)

ARROVIAN AGGREGATION MODELS
THEORY AND DECISION LIBRARY
General Editors: W. Leinfellner (Vienna) and G. Eberlein (Munich)
Series A: Philosophy and Methodology of the Social Sciences
Series B: Mathematical and Statistical Methods
Series C: Game Theory, Mathematical Programming and Operations Research
SERIES B: MATHEMATICAL AND STATISTICAL METHODS
VOLUME39
Editor: H. J. Skala (Paderborn); Assistant Editor: M. Kraft (Paderborn); Editorial Board:
J. Aczel (Waterloo, Ont.), G. Bamberg (Augsburg), H. Drygas (Kassel), W. Eichhorn
(Karlsruhe), P. Fishburn (Murray Hill, N.J.), D. Fraser (Toronto), W. Janko (Vienna), P. de
Jong (Vancouver), T. Kariya (Tokyo), M. Machina (La Jolla, Calif.), A. Rapoport (Toronto),
M. Richter (Kaiserslautern), B. K. Sinha (Cattonsville, Md.), D. A. Sprott (Waterloo, Ont.),
P. Suppes (Stanford, Calif.), H. Theil (St. Augustine, Fla.), E. Trillas (Madrid), L.A. Zadeh
(Berkeley, Calif.).
Scope: The series focuses on the application of methods and ideas of logic, mathematics and
statistics to the social sciences. In particular, formal treatment of social phenomena, the
analysis of decision making, information theory and problems of inference will be central
themes of this part of the library. Besides theoretical results, empirical investigations and the
testing of theoretical models of real world problems will be subjects of interest. In addition
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dissemination of recent results.
The titles published in this series are listed at the end oft his volume.
FUAD ALESKEROV
Bogazi~i University, Bebek, Istanbul, Turkey
and
Russian Academy ofS ciences, Institute ofC ontrol Sciences, Moscow, Russia
ARROVIAN
AGGREGATION
MODELS
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-5079-6 ISBN 978-1-4757-4542-9 (eBook)
DOI 10.1007/978-1-4757-4542-9
Printed on acid-free paper
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© 1999 Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers, Boston in 1999
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TO MY PARENTS
TABLE OF CONTENTS
FOREWORD ix
ACKNOWLEDGEMENTS xi
1 AGGREGATION: A GENERAL DESCRIPTION 1
1.1 Introduction 1
1.2 Analysis of examples 2
1.3 Arrow's General Impossibility Theorem 3
1.4 Individual opinion: a formalization 8
1.5 Aggregation: the synthesis problem 11
1.6 Concluding remarks 16
2 RATIONALITY OF INDIVIDUAL OPINIONS AND SOCIAL
DECISIONS 17
2.1 Introduction 17
2.2 Binary relations 17
2.3 Criteria! model of choice 22
2.4 Expansion-Contraction Axioms 26
2.5 Relations between the classes of choice functions 35
2.6 Concluding remarks 41
3 SOCIAL DECISION FUNCTIONS 45
3.1 Introduction 45
3.2 Strong locality 46
3.3 Normative conditions 49
3.4 Rules from Central Class 53
3.5 Rationality constraints 56
3.6 Comparing classes in Ac 62
3.7 Arrow's General Impossibility Theorem 65
3.8 Rationality constraints: further results 67
3.9 Aggregation of equivalences 72
3.10 Non-monotonic strongly local SDFs 74
3.11 Locality 78
3.12 Normative conditions 80
3.13 Rules from Central Class 85
3.14 Rationality constraints 97
3.15 Comparing classes in Ac 110
vm TABLE OF CONTENTS
3.16 Concluding remarks 119
4 FUNCTIONAL AGGREGATION RULES 123
4.1 Introduction 123
4.2 Locality 124
4.3 Normative conditions 127
4.4 Rules from Central Class 133
4.5 Rationality constraints: non-emptiness 136
4.6 Rationality constraints: domains H, C, and 0 142
4.7 Comparing classes in A0 150
4.8 Rules from Basic Class 152
4.9 Non-monotonic rules 158
4.10 Non-monotonic rules: dual domains 165
4.11 Concluding remarks 174
5 SOCIAL CHOICE CORRESPONDENCES 177
5.1 Introduction 177
5.2 Locality 177
5.3 Normative conditions 183
5. 4 Boolean representation of Social Choice Correspondences 190
5.5 Rules from Central Class, I 192
5.6 Rules from Central Class, II 200
5. 7 Rules from Symmetrically Central Class 204
5. 8 Rationality constraints: single-valuedness 212
5. 9 Coalitional q-federation rules under rationality constraints 216
5.10 Rationality constraints: domains H, C, 0 219
5.11 Comparing classes in Ac 222
5.12 Concluding remarks 223
BIBLIOGRAPHY 227
INDEX 239
FOREWORD
Aggregation of individual opinions into a social decision is a problem widely observed
in everyday life. During centuries people tried to invent the 'best' aggregation rule. In
1951 young American scientist and future Nobel Prize winner Kenneth Arrow formu
lated the problem in an axiomatic way, i.e., he specified a set of axioms to which every
reasonable aggregation rule seems has to satisfy, and obtained that these axioms are
inconsistent. This result, often called Arrow's Paradox or General Impossibility The
orem, had become a corner stone of social choice theory, a vast and ever increasing
scientific field. In contrast to the earlier stage in the development of the theory, these
works focus on finding a solution to the problem within the framework ofthe axiomatic
approach with modified or weakened conditions than those proposed by K.Arrow.
The main condition used by K.Arrow was his famous Independence oflrrelevant Al
ternatives. This very condition pre-defmes the 'local' treatment of the alternatives (or
pairs of alternatives, or sets of alternatives, etc.) in aggregation procedures. The coun
terparts of that condition are called in the monograph as Locality condition.
Remaining within the framework of the axiomatic approach and based on the con
sideration oflocal rules, the book investigates three formulations of the aggregation
problem according to the form in which the individual opinions about the alternatives
are defmed, as well as to the form of desired social decision. In other words, we study
three aggregation models. What is common between them is that in all models some
analogue oflndependence oflrrelevant Alternatives condition is used. That is why we
call these models as Arrovian aggregation models.
Chapter 1 presents a general description of the problem of axiomatic synthesis oflocal
rules, and introduces problem formulations for various versions of formalization of
individual opinions and collective decision.
Chapter 2 formalizes precisely the notion of 'rationality' ofi ndividual opinions and so
cial decision. Various types of binary relations (preferences) are introduced and inves
tigated. The characteristic conditions (often called Expansion-Contraction Axioms)
identifying different classes of choice functions are defmed here, and the interrela
tions between them are established. Additionally, the choice functions rationalizable
by numerical (utility) functions are described, and interrelations between them and the
choice functions rationalizable via binary relations are determined.
Chapter 3 deals with the aggregation model for the case of individual opinions and so
cial decisions formalized as binary relations. Two types oflocal rules which are called
Social Decision Rules, or Social Decision Functions, are studied. The explicit forms
of those rules are completely investigated. Rules restricted by rationality constraints,
i.e., by constraints on domains and ranges of the rules, are studied as well.
Chapter 4 deals with Functional Aggregation Rules which transform into a social
X FOREWORD
choice function individual opinions defmed as choice functions. In doing so, ratio
nalizability of those choice functions is not assumed, that is, consideration is in gen
eral given to the non-classical choice functions. The explicit form of these rules is
obtained and rules which satisfy different rationality constraints - such as those gen
erating social choice functions from different classes in the set of choice functions -
are studied.
Chapter 5 considers another model- Social Choice Correspondences when the indi
vidual opinions are formalized as binary relations, and the collective decision is look
for as a choice function. The explicit form of rules is studied enabling a proper map
ping with and without additional rationality constraints on the social choice function.
The obtained classes comprise rules such as the generalized Pareto rules. Several new
classes of rules are introduced and analyzed.
Bibliography lists the major publications in the field ofa xiomatic synthesis ofA rrovian
aggregation models.