Table Of ContentSOCIETIES AND SOCIAL DECISION FUNCTIONS
THEOR Y AND DECISION LIBRARY
AN INTERNATIONAL SERIES
IN THE PHILOSOPHY AND METHODOLOGY OF THE
SOCIAL AND BEHAVIORAL SCIENCES
Editors
GERALD EBERLEIN, University of Technology, Munich
WERNER LEINFELLNER, University of Nebraska
Editorial Advisory Board:
K. BORCH, Norwegian School of Economics and Business Administration
M. BUNGE, McGill University
1. S. COLEMAN, University of Chicago
W. KROEBER-RIEL, University of Saarland
A. RAPOPORT, University of Toronto
F. SCHICK, Rutgers University
A. SEN, Oxford University
W. STEGMULLER, University of Munich
K. SZANIA WSKI, University of Warsaw
L. TONDL, Prague
A. TVERSKY, Stanford University
VOLUME 30
A. CAMACHO
Dept. of Economics, University of Illinois at Chicago Circle
SOCIETIES AND
SOCIAL DECISION FUNCTIONS
A Model with Fo eus on the Information Problem
D. REIDEL PUBLISHING COMPANY
DORDRECHT: HOLLAND/BOSTON: U.S.A.
LONDON: ENGLAND
Library of Congress Cataloging in Publication Data
Camacho, A., 1927-
Societies and social decision functions.
(Theory and decision library; v. 30)
Bibliography: p.
Includes indexes.
1. Social choice-Mathematical models. 2. Decision
making-Mathematical models. 3. Social prediction.
I. Title. II. Series.
HM73.C29 1982 302'.13 82-9866
ISBN-13: 978-94-009-7815-7 e-ISBN-13: 978-94-009-7813-3
001: 10.1007/978-94-009-7813-3 AACR2
Published by D. Reidel Publishing Company,
P.O. Box 17,3300 AA Dordrecht, Holland.
Sold and distributed in the U.S.A. and Canada
by Kluwer Boston Inc.,
190 Old Derby Street, Hingham, MA 02043, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, Holland.
D. Reidel Publishing Company is a member of the Kluwer Group.
All Rights Reserved
Copyright © 1982 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover 1s t edition 1982
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any informational storage and
retrieval system, without written permission from the copyright owner
To Manolita,
the Inspiration
of my Life and Work
TABLE OF CONTENTS
INTRODUCTION IX
ACKNOWLEDGEMENTS XV
CHAPTER II Comparison of Approaches to Social Choice
21
CHAPTER Intensity of Preferences and Cardinal Utility 11
31
CHAPTER Unsatisfactoriness of Ordinal Methods in
Dealing with Problems of Social Choice 24
CHAPTER 41 A System of Axioms for Cardinal Utility 36
51
CHAPTER A More General System of Axioms for
Cardinal Utility 52
CHAPTER 61 An Abstract Model of Society 62
CHAPTER 71 Social Decision Functions 76
81
CHAPTER A Theorem Proving the Unsatisfactoriness
of the Ordinal Approach to Social Choice 85
CHAPTER 91 Strengthening the Theorem Proved In
Chapter 8: Informal Discussion 92
1
CHAPTER 10 Unsatisfactoriness of the Ordinal Approach
to Social Choice: Further Results 10 I
Vll
TABLE OF CONTENTS
VI II
CHAPTER II/Justifying the Use of Ordinal Methods 115
CHAPTER 12/ Conclusion 123
ApPENDIX 1 /The Utility Differences Approach to Cardi-
nal Utility 127
ApPENDIX 2 /The Expected Utility Approach to Cardinal
Utility 135
BIBLIOGRAPHY 140
INDEX OF NAMES 141
INDEX OF SUBJECTS 142
INTRODUCTION
A model is an idealization. It is an abstract representation of a given
perceived reality. To construct a model one abstracts from the
unimportant features of that reality and replaces it by a formal
structure, whose properties, explicitly assumed or logically de
ducible from the stated assumptions, correspond to the interesting
relationships of the reality being studied.
The purpose of constructing a model is twofold: first, to help
better understand a complex reality; second, to help make pre
dictions with regard to still unobserved phenomena. The first
purpose will be satisfied if the constructor of the model is able to
identify and disregard the unimportant features of the reality being
studied and replaces this reality by an easier to understand formal
structure. By applying then the techniques of mathematics and logic
to this formal structure we might be able to reach conclusions with
regard to still unobserved phenomena, which will be of help in
making predictions.
The model to be developed in this monograph will satisfy, at least
to a certain degree, the two purposes stated above. It should be
stressed, however, that, contrary to the world of physics where it is
relatively easy to put conclusions to a test, this is not in general the
case within the social sciences. For this reason, emphasis on the
predictive power of a model, which is paramount in the physical
sciences, should be more subdued in the social sciences. Thus, in this
field, a model which illuminates and helps understand a complex
social reality is of great use even if its predictive powers are very
limited.
The model to be developed can be used to help clarify and
understand the complex and controversial issue of measurability or
cardinality of utility and the need to use such cardinal measure in
ix
x INTRODUCTION
dealing with problems of social choice. By clarifying this issue the
model can be of use in understanding why certain social decision
rules have failed, in some cases, to produce satisfactory results, and
in helping to select 'good' social decision rules to face future
situations as they unfold.
As stated before, a model is an abstract representation of a
perceived reality. To better understand an abstract model it is
helpful to provide first an intuitive informal description of the reality
envisioned by the constructor of the model.
The realities to be modeled are called societies and social decision
functions. A society is envisioned as a group of individuals facing
different sets of circumstances over which they have no control
whatsoever, and reacting to each of these sets of circumstances when
they happen by choosing ajoint action from among the set of actions
available when the set of circumstances does occur. A social decision
function, also to be called a social decision rule, or mechanism, is
seen as a rule that, taking into consideration the feelings of the
members of society with respect to the available alternative actions,
determines for each set of circumstances the joint action that society
will implement.
A social club periodically facing the problem of which appli
cations for membership to accept and which to reject, the United
States of America deciding every four years what person to elect as
its president, the board of directors of a business corporation
selecting the investment program that their company will adopt, are
real world counterpart examples of our conception of a society. The
ranking method or Borda count used in some sport events and social
clubs, the rules to elect a president every four years in the United
States of America, the unanimity rule adopted by certain legislative
bodies, etc. are real world counterpart examples of our idea of a
social decision function.
A society is envisioned not in a static state where it faces just one
set of circumstances, reacts to it by taking an action and that is the
INTRODUCTION XI
end of its social choice activity. Rather, a society is envisioned as
going through different sets of circumstances that keep occurring
and using its adopted social decision function to generate the
corresponding joint actions. The social decision function adopted by
a society is judged good or not by seeing whether, for any possible
sequence of sets of circumstances that may occur to this society, the
corresponding sequence of actions generated by the decision
function is, in some sense, satisfactory or not.
Thus, when judging whether the constitution of a country is good
or not we do not usually look at any isolated set of circumstances
and the corresponding action generated by the constitution. Instead
we look at how the circumstances of that country keep evolving and
the corresponding sequence of actions that the constitution
generates.
Similarly, when we consider the unanimity rule as an un
satisfactory social decision function, we do not have in mind an
isolated situation and the corresponding action that normally, under
this rule, would be the preservation of the status quo. We have in
mind a sequence of situations and the possibility that, each time,
some member of the group may block the motion to alter the status
quo, making any change almost impossible. 1
It is important to emphasize that this dynamic view of a society
and of its corresponding social decision function is mainly what
differentiates the approach to the social choice problem to be
proffered in this monograph from alternative approaches proposed
in the economic literature. Furthermore, this dynamic view allows
for the giving of a precise meaning to the notion of intensity of
preferences, the derivation of cardinal utility functions for each of the
members of society and the demonstration that these cardinal utility
functions must be used if we want to obtain satisfactory, in a sense to
be made clear later, social decision functions.
Chapter 1 will be devoted to summarizing and discussing
Bergson-Samuelson's social welfare function approach, Arrow's
