Table Of ContentPATTERNS AND CONFIGURATIONS IN ECONOMIC SCIENCE
PATTERNS
AND CONFIGURATIONS IN
ECONOMIC SCIENCE
by
J. M. BLIN
Graduate School of Management and The Center for Mathematical Studies
in Economics and Management Science. Northwestern University, Evanston, Ill., U.S.A.
D. REIDEL PUBLISHING COMPANY
DORDRECHT-HOLLAND / BOSTON-U.S.A.
Library of Congress Catalog Card Number 72-92525
ISBN- 13: 978-94-010-9591-4 e-ISBN-13: 978-94-010-9589-1
DOl: 10.1007/978-94-010-9589-1
Published by D. Reidel Publishing Company,
P.O. Box 17, Dordrecht, Holland
Sold and distributed in the U.S.A., Canada, and Mexico
by D. Reidel Publishing Company, Inc.
306 Dartmouth Street, Boston,
Mass. 02116, U.S.A.
All Rights Reserved
Copyright © 1973 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover 1s t edition 1973
No part of this book may be reproduced in any form, by print, photoprint, microfilm,
or any other means, without written permission from the publisher
D. Reidel, Dordrecht
To my parents
FOREWORD
This book was born from the belief that, in an era of narrowly specialized
experts, looking beyond the arbitrary borders on one's own field offered
the only chance for intellectual survival. The spectacular growth of
economics in the past forty years or so has been accompanied by the
rapid fragmentation process that seems to characterize the making of any
science. If we chose to view this as a symptom of scientific maturity,
we would be tempted to welcome this process. But, as economists, we
should think of its opportunity cost. In one sense this work attempts to
assess the price we have chosen to pay - consciously or not - to win
recognition of our scientific status from other disciplines. If, after going
through this book, the reader comes out with the feeling that this shadow
price is not all shadow after all, we will have achieved one of our goals!
Both the merits and defects of this book stem from the fact that we tried
to pursue a different approach in the study of economics. This line of
thought originates in the area of pattern recognition and machine learn
ing. For the past two decades, a number of applied mathematicans,
logicians, and engineers have systematically tried to mimic human intelli
gence in one of its most important functions viz. that of identifying and
recognizing patterns in an a priori unorganized and confusing environment.
Recognition of a sound, of a letter, of a person's handwriting, of a
composer's or a painter's style are but a few examples of this problem.
Clearly what is needed is a model allowing us to identify patterns for
future classification and recognition. Loosely speaking these 'model
patterns' would play the role of Plato's 'world of Ideas.' To successfully
carry out this process of abstraction, classification and generalization,
mathematics provides us with a wealth of models and structures. The
choice of a specific structure in a given problem is then guided by con
siderations of convenience, accuracy, and computational cost.
The concepts of pattern recognition that we just outlined enter in
economics in at least two different ways. On the one hand it is clear that
when we talk about a consumer preference pattern or a market form, for
VIII FOREWORD
instance, we have in mind - implicitly at least - some loosely defined
reference pattern(s) in reference to which we classify actual or hypothe
tical observations of consumer choice, or market phenomena. On the
other hand the concepts of pattern abstraction and classification are
represented in economics in an even more fundamental way under the
guise of the so-called aggregation problem. This problem occurs in all
areas of economics, whether the analysis be theoretical or empirical,
micro or macro. Here again we are faced with a classification decision that
can only be made on the basis of some prior study of pattern classes and
more generally abstract patterns chosen as viable representations of actual
observations. Because of this deeper connection between economic science
and pattern recognition we have chosen to concern ourselves, in this book,
only with the specific problem of consumer preference aggregation. For
one thing preference patterns are fairly easily abstracted and lend them
selves to many alternative mathematical structures, and thus provide more
flexible illustrations of the basic concepts involved. But, an even more
important consideration in guiding our choice of topics for this book
was the fact that the preference aggregation problem constitutes one of
the most celebrated forms of the general aggregation problem in economics.
Our current research has shown the usefulness of these tools in other
areas of economics and we hope to be able to present these results in an
integrated fashion in a separate book in the near future.
This book has greatly benefited from discussions with several indivi
duals who, directly or indirectly, showed interest in my work. Among all
those I wish to thank Professors King-Sun Fu and Andrew B. Whinston,
both at Purdue University. Professor K-S. Fu must be thanked for having
taught me all I know about pattern recognition and making many valuable
comments on my study. Learning to view the world as an intelligible
combination of patterns was, in itself, an unforgettable experience. His
interest in my work was most welcome, especially at times of doubt.
Professor A. B. Whinston must be thanked for arousing my interest in the
area of collective decision-making and choice theory through many stimu
lating discussions and for urging me to follow my natural inclination
towards pattern recognition.
My thanks also go to other friends and colleagues at Northwestern
University, especially Professors A. Camacho and M. Kamien who
provided challenging discussions that helped in the preparation of this
FOREWORD IX
work. To my wife, Bernadette, goes my appreciation of her encouragement.
To my daughter Valerie lowe a special debt of gratitude; her exuberant
intrusions into my work brought me more happiness than headaches!
And finally, Mrs. Tina Wieczorek should be thanked for her endless
patience and competence in typing various portions of this book: after
juggling with an undecipherable manuscript, she was able to turn it into
an intelligible typescript!
PREFACE
From time to time important advances in one discipline are stimulated by
work in an area thought to be quite apart. The work described in this
book represents an example of such a case. The area of pattern recogni
tion has a firm place in electrical engineering and computer science and
has contributed significantly to other allied physical science and engi
neering fields. Yet until now it has had scarcely any impact on economics
or other social sciences. Thus this book truly represents a pioneering
effort in its applications of pattern recognition theory to economic and in
particular, social choice theory. Since it is the first, we expect it to stimu
late a significant amount of new research in this area and are confident
that much of what is contained within this book will be further developed
in subsequent work.
Social choice theory is concerned with the problem of combining
individual preferences over a set of alternatives into a group or aggregate
preference ordering. It is an important area since it is related to almost all
of decision making, with the exception of choices made in a dictatorial
situation. What are 'good' or 'proper' or 'just' rules for making social
choices? How do these compare with the practice of using voting rules,
such as majority voting, or on the other hand, cost benefit analysis? With
important exceptions, most economists doing research in these problems
have studied refinements on essentially negative results showing the non
existence of 'just' rules of aggregation. These results have little impact on
the day to day practice. Blin's approach, on the other hand, is direct to
the problem of how to analyze actual practice and possibly improve it.
In our view, pattern recognition theory will have an increasing influence
on the development of economic theory. This book presents some aspects
of this new viewpoint and also suggestions that others may wish to pursue.
We feel sure that this book will have an important impetus in the devel
opment of economic theory and its application to social choice.
Lafayette, Indiana KING-SUN FU
July, 1972 ANDREW B. WHINSTON
TABLE OF CONTENTS
FOREWORD VII
PREFACE XI
LIST OF TABLES AND FIGURES XVI
INTRODUCTION 1
CHAPTER I / WELFARE ECONOMICS AND PUBLIC DECISION-
MAKING 4
1. Introduction 4
2. A Theoretical Approach: Arrow's General Impossibility
Theorem 7
3. A Practical Approach: Benefit-Cost Analysis 12
4. Some Heuristic Methods of Social Choice 18
A Statement of the Problem 18
The 'Electra' Method 19
Linear Scoring Models 20
CHAPTER II / CONSTITUTIONAL CHOICE AND MAJORITY
VOTING 22
1. Introduction 22
2. Statement of the Problem and Basic Assumptions 25
3. The Optimality Property of Majority Voting 31
4. Interpretation and Extension 38
CHAPTER III / OPTIMIZATION OF PUBLIC DECISIONS IN THE
LARGE: A PATTERN RECOGNITION APPROACH 40
1. Introduction 40
XIV TABLE OF CONTENTS
2. Consumer Preferences and the Concept of a Pattern 42
3. Discriminant Functions as Social Decision Rules 49
The Linear Classification Procedure 49
Piecewise Linear Decision Rule 55
Polynomial Decision Rules 56
4. A Characterization of Pareto-Optimal Patterns in the Feature
Space FN 57
The Basic Characterization Theorem for the Set of Pareto-
Optimal States in the Feature Space F:"i 58
The Interpretation of the Characterization Theorem 64
The Role of the Status Quo 64
Devising an Optimal Taxation Scheme for a Public
Goods Economy 64
CHAPTER IV / ALGEBRAIC FOUNDATIONS OF THE THEORY OF
AGGREGATION 69
1. Introduction 69
2. Some Properties of Tournament Matrices 76
3. Transitive Preference Patterns, Finite Groups and Lattices 83
Preliminary Notions on the Permutation Group of Order m 83
Transitive Patterns and Complete Lattices 88
Partially Ordered Sets and Lattices 89
The Set of Transitive Patterns as a Complete Lattice 90
Illustrationform=3 91
Definition of a Metric on the Lattice of All Transitive
Finite Patterns 96
CHAPTER V / OPTIMIZATION OF PUBLIC DECISIONS: NEW
RESULTS IN THE THEORY OF AGGREGATION
1. Introduction 99
2. Black-Coombs Orders and the Concept of Restricted Preference
Patterns 100
Generating Black Patterns: A Combinatorial Analysis 10 1
Aggregation of Black Patterns 109
