Additive and Polynomial Representations

594 Pages·1971·29.004 MB·English

by  David H. Krantz, R Duncan Luce and Patrick Suppes (Auth.)

Frontispiece by Ruth Weisberg
FOUNDATIONS OF
MEASUREMENT
DAVID H. KRANTZ R. DUNCAN LUCE
UNIVERSITY OF MICHIGAN THE INSTITUTE FOR ADVANCED STUDY
PATRICK SUPPES AMOS TVERSKY
STANFORD UNIVERSITY THE HEBREW UNIVERSITY OF JERUSALEM
VOLUME I
Additive and Polynomial Representations
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers
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COPYRIGHT © 1971, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
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The quotation on the opposite page is taken from Plato's Protagoras in The Dialogues
of Plato translated by Benjamin Jowett, 4th ed., 1953, Vol. 1, pp. 183-184. Copyright 1953
by the Clarendon Press, Oxford.
... What measure is there of the relations of pleasure to pain other than excess and
defect, which means that they become greater and smaller, and more and fewer, and
differ in degree? For if any one says: "Yes, Socrates, but immediate pleasure differs
widely from future pleasure and pain"—to that I should reply: And do they differ in any-
thing but pleasure and pain ? There can be no other measure of them. And do you,
like a skillful weigher, put into the balance the pleasures and the pains, and their
nearness and distance, and weigh them, and then say which outweighs the other. If
you weigh pleasures against pleasures, you of course take the more and greater; or
if you weigh pains against pains, you take the fewer and the less; or if pleasures against
pains, then you choose that course of action in which the painful is exceeded by the
pleasant, whether the distant by the near or the near by the distant; and you avoid that
course of action in which the pleasant is exceeded by the painful. Would you not admit,
my friends, that this is true? ...
Summary of Contents
Preface xvii
1. Introduction 1
2. Construction of Numerical Functions 38
3. Extensive Measurement 71
4. Difference Measurement 136
5. Probability Representations 199
6. Additive Conjoint Measurement 245
7. Polynomial Conjoint Measurement 316
8. Conditional Expected Utility 369
9. Measurement Inequalities 423
10. Dimensional Analysis and Numerical Laws 454
Answers and Hints to Selected Exercises 545
References 551
Author Index 565
Subject Index 570
vi
Preface
Scattered about the literatures of economics, mathematics, philosophy,
physics, psychology, and statistics are axiom systems and theorems that are
intended to explain why some attributes of objects, substances, and events
can reasonably be represented numerically. These results constitute the
mathematical foundations of measurement. Although such systems are of
some mathematical interest, they warrant our attention primarily as em-
pirical theories—as attempts to formulate properties that are observed to be
true about certain qualitative attributes. Some of the theories appropriate
to classical physics are so well accepted that they are usually considered
in the province of philosophy rather than physics, but this should not be
allowed to becloud the basic empirical character of any theory that purports,
for example, to justify treating mass as an additive numerical property.
From time to time, the empirical nature of basic measurement assumptions
is forcibly brought to everyone's attention—for example, when the theory
of relativity made clear that velocities do not combine additively; when
quantum mechanics made clear that the probability theory of elementary
particles is somewhat different from that appropriate to macroscopic
events; and when we recall that the attribute of hardness still lacks any
satisfactory measurement analysis.
In the nonphysical sciences, measurement has always been problematic,
and it has become increasingly evident to nearly everyone concerned that
we must devise theories somewhat different from those that have worked
XVll
XV111 PREFACE
in physics. Because of the active—and we believe crucial—concern with
measurement in these sciences, it is not terribly surprising that four behavioral
scientists might attempt to summarize the field. The methodology of measure-
ment has, of course, a long history in the physical sciences, and we have
also tried to cover the major foundational problems there, ranging from the
theory of extensive measurement to dimensional analysis.
We must emphasize that this book deals with Foundations and not with
the history or current practice of measurement in any field. To the extent
that we deal with empirical examples, it is to motivate theories and to illustrate
their testing. The reader who is interested in specific empirical material
will have to consult references in particular specialties. For example, an
excellent history of weight measurement, with many illustrations of apparatus,
is the volume by Kisch (1965).
As one explores the measurement literature it becomes clear that, in spite
of the fact that each proof makes some peculiar use of the structure in ques-
tion, many proofs are quite similar. Moreover, little is done to relate the
particular theorem being presented to any others except those having
exactly the same primitive concepts. Eventually, this becomes frustrating:
results are reproved ; it is extremely difficult to maintain a clear idea of the
structure of the field; it is uncertain how many really basic ideas there are;
and it is likely that the field gradually becomes inefficient in the sense that,
for about the same effort, stronger theorems could be proved if other results
were used instead of returning to first principles. We have attempted to
organize the central results in a cumulative fashion.
As a result of our attempt, we have concluded that at present there are
three distinct mathematical results (see Chapters 1 and 2) used to construct
numerical representations of qualitative structures. With some minor
exceptions, the remaining major theorems of this volume either reduce to
one of these results directly or indirectly via some other representation
theorem. By "reduce" we do not mean to suggest "reduce readily," for in
many cases the proofs are quite lengthy and it is not always immediately
obvious what to reduce to what. In organizing the material, we have un-
doubtedly made some arbitrary decisions about this, although as a matter of
fact we suspect that there maybe rather less flexibility than might first seem.
In the process, we have usually arrived at theorems somewhat better than
those previously published, and virtually all of our proofs of major results
are different from those in the literature. (Some of our new results have
been published independently of this book; any paper on measurement
by the authors dated 1967-1970 is a byproduct of work on the book.)
Since the book includes both new results and new proofs it is a research
monograph, but we hope that it is more than that. We have gone to pains
to make it a textbook—in fact, two textbooks. If one deletes Chapter 2 and
PREFACE XIX
all of the sections headed "Proofs," what remains is a self-contained book
that should serve as a comprehensive introduction to the mathematical
theory of measurement for nonspecialists who have enough mathematics
to understand the formulation of the problems (see below), but who have
no particular reason for studying the proofs. For those with the requisite
mathematical background, the proofs are there in separate sections. Rather
more detail is provided than a trained mathematician is likely to want,
but with his facility at skipping things that are routine to reconstruct, this
should not really bother him. For the novice, the detail is important because
he must learn to develop complete proofs, even though he may ultimately
present them only in abbreviated form. The reason is that, more than in
some areas of mathematics, it is all too easy to accept and use, as if it were
proved, a familiar property that has not been proved and may, given the
axioms, be exceedingly tricky or tedious to establish. The most "obvious"
properties can be major stumbling blocks or the source of erroneous proofs.
To further the usefulness of the book as a text, we have included a number
of problems which the student can use to exercise and test his developing
skills. Many of these exercises are quite easy for a person familiar with the
area, but a few are moderately difficult. In addition, a number of unsolved
problems are mentioned throughout the book. Some of these are difficult
and are suitable for dissertations; others, no doubt, will turn out to be
easier than we have thought.
The material that we have elected to cover and our mode of presentation
make it impossible to package it conveniently in a single volume. This
first volume covers all of the major representation theorems in which the
qualitative structure is reflected as some sort of a polynomial function of
one or more numerical functions defined on the basic entities. The simplest
and best-known examples are additive expressions of a single measure,
such as the probability of disjoint events being the sum of their probabilities,
and additive expressions of two measures, such as the logarithm of momentum
being the sum of log mass and log velocity terms. The second volume will
include representations in terms of distances in some sort of space, a treat-
ment of the exceedingly perplexing problems raised by errors of measurement,
and analyses of the philosophical issues that center about axiomatizability
and meaningfulness.
Mathematical Background
For a reader who plans to skip proofs, a modest background in mathe-
matics and some diligence should suffice. He certainly must know well the
elementary material on sets, relations, functions, and probability that is
provided in introductory courses at many colleges and universities and is
XX PREFACE
found in, among other books, Kershner and Wilcox (1950) and Suppes (1957).
With no more than that, there will be some difficult spots and a few sections
will be unintelligible because they depend upon calculus or topological
concepts. For the reader who plans to follow the proofs, a background in
calculus and in elementary abstract algebra is needed. In a few places,
some elementary topological material that can be found in, for example,
Kelley (1955) and some of the functional equations found in Aczél (1966)
are also required.
Selecting Among the Chapters
Within the present volume, various courses of study are possible by
selecting among the chapters. The main constraints are conceptual de-
pendencies and the logical development of the proofs. These two are dia-
grammed separately in Figure A. Obviously, many more options are avail-
Conceptual Logical
(a) (b)
FIGURE A. (a) Conceptual dependencies of chapters, (b) Logical development of
the proofs.
able to those who do not plan to read proofs, and so it may be useful for us
to state what we believe to be the minimum core of information for physical
and behavioral scientists. A physical scientist who plans to ignore proofs
should cover Chapters 1, 3, 6, and 10; he will find that the initial sections
of Chapters 4, 5, and parts of Chapter 9 are also valuable. To follow the
proofs, Chapter 2 must be added. For a behavioral scientist, the essential
chapters are 1, 4, 5, 6, 8, and 9, with Chapter 3 also being desirable. To
follow the proofs, Chapters 2 and 3 must be added. In addition, to guide
the reader further we have marked with a square (□) those sections of
Chapters 3-10 that we consider to be the basic core material.
Acknowledgments
Each of us has received substantial assistance while working on this book and we
gratefully acknowledge it.
Krantz: Grants GB-4947 and 8181 from the National Science Foundation to the Univer-
sity of Michigan; Public Health Service Special Postdoctoral Fellowship, 1968; and the
hospitality of the Center for Advanced Study in the Behavioral Sciences, Stanford, Cali-
fornia, during the summer of 1967 and all of 1970-1971.
Luce: Grant GB-6536 from the National Science Foundation to the University of
Pennsylvania; a National Science Foundation Senior Postdoctoral Fellowship, 1966-1967,
which with a sabbatical leave from the University of Pennsylvania permitted him to spend
that academic year at the Center for Advanced Study in the Behavioral Sciences, Stanford,
California; an Organization of American States Professorship for the academic year
1968-1969 at the Pontificia Universidade Catolica do Rio de Janeiro, Brazil; and a grant
from the Alfred P. Sloan Foundation to The Institute for Advanced Study, Princeton,
New Jersey.
Suppes: Grant GJ-443X from the National Science Foundation to Stanford University.
Tversky: Grant GB-6782 from the National Science Foundation to the University of
Michigan and the hospitality of the Center for Advanced Study in the Behavioral Sciences,
Stanford, California, during the summer of 1967 and all of 1970-1971.
Drafts of the chapters have been circulated among a small group of people whom we
know to be especially interested in these matters. We have benefited greatly from their
comments, some of which have been detailed and penetrating. We thank: J. Aczél,
Maya Bar-Hillel, R. L. Causey, R. M. Dawes, Z. Domotor, P. C. Fishburn, R. Giles,
E. W. Holman, M. V. Levine, K. R. MacCrimmon, A. A. J. Marley, F. S. Roberts,
R. J. Titiev, J. W. Tukey, and T. S. Wallsten.
XXI