the mit press preference belief and similarity selected writings dec 2003

Preference, Belief, and Similarity
Selected Writings
Amos Tversky
edited by Eldar Shafir
Preference, Belief, and Similarity
Preference, Belief, and Similarity
Selected Writings
Amos Tversky
edited by Eldar Shafir
Amos Tversky (1937–1996), a towering figure in cognitive and mathematical psychology, devoted his profes-
sional life to the study of similarity, judgment, and decision making. He had a unique ability to master the
technicalities of normative ideals and then to intuit and demonstrate experimentally how they are systemat-
ically violated by the vagaries and consequences of human information processing. He created new areas of
study and helped transform disciplines as varied as economics, law, medicine, political science, philosophy,
and statistics.
This book collects forty of Tversky’s articles, selected by him in collaboration with the editor during the
last months of Tversky’s life. Included are several articles written with his frequent collaborator, Nobel
Prize–winning economist Daniel Kahneman.
Eldar Shafir is Professor of Psychology and Public Affairs at Princeton University.
“Amos Tversky was one of the most important social scientists of the last century. This extraordinary collec-
tion demonstrates his range and brilliance, and in particular his genius for showing how and why human
intuitions go wrong. Is there a ‘hot hand’ in basketball? Is arthritis pain related to the weather? Why do we
exaggerate certain risks? Why are some conflicts so hard to resolve? Tversky’s answers will surprise you.
Indispensable reading, and full of implications, for everyone interested in social science.”
—Cass R. Sunstein, Law School and Department of Political Science, University of Chicago
“Amos Tversky’s research on preferences and beliefs has had a shattering and yet highly constructive influ-
ence on the development of economics. The vague complaints of psychologists and dissident economists
about the excessive rationality assumptions of standard economics, going back over a century, had little
impact. It required the careful accumulation of evidence, the clear sense that Tversky did not misunder-
stand what economists were assuming, and above all his formulation of useful alternative hypotheses to
change dissatisfaction into a revolutionary change in perspective.”

—Kenneth J. Arrow, Professor of Economics Emeritus, Stanford University
“Amos Tversky’s work has produced an ongoing revolution in our understanding of judgment and choice.
The articles in this book show why. They also show how: the articles are written with grace, wit, and a bril-
liance that frequently verges on the pyrotechnic.”
—Richard E. Nisbett, author of
The Geography of Thought: How Asians and Westerners Think Differently . . .
and Why
“Amos Tversky may have shown that basketball players do not have ‘hot hands,’ but he proved the opposite
for psychologists. Tversky always made his basket, and in the process changed psychology, and also eco-
nomics, forever.”
—George Akerlof, Koshland Professor of Economics, University of California, Berkeley, 2001 Nobel Laureate
in Economic Sciences
“It is deeply ironic that ‘similarity’ and ‘bounded rationality’ were two of the many topics that Amos Tversky
studied—ironic because he seemed to be unboundedly rational and similar to no one. No one shared his
combination of brilliance, precision, intuition, breadth, and enormous good humor. Few scholars change
their own disciplines before they reach 40, as Tversky did, and even fewer then transform other disciplines,
as he and Daniel Kahneman did for economics. Their influence on economics, recognized by the 2002 Nobel
Prize, is still growing, and the discipline will never be the same. Nor will anyone who reads these papers: it
is impossible to read Tversky without changing the way you think.”
—Richard H. Thaler, Robert P. Gwinn Professor of Economics and Behavioral Science, University of Chicago
“This collection offers the best of Tversky, the best of the best. It is amazing how many of these articles are
already classics, not only in the fields of choice and decision making, but in psychology in general.”
—Edward E. Smith, Arthur W. Melton Professor of Psychology, University of Michigan
A Bradford Book
The MIT Press
Massachusetts Institute of Technology
Cambridge, Massachusetts 02142

,!7IA2G2-haajdb!:t;K;k;K;k
0-262-70093-X

Tversky
Shafir, editor
tversky 12/16/03 12:58 PM Page 1
Preference, Belief, and Similarity

Preference, Belief, and Similarity
Selected Writings
by Amos Tversky
edited by Eldar Shafir
A Bradfor d Book
The MIT Press
Cambridge, Massachusetts
London, England
6 2004 Massachusetts Institute of Technology
All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical
means (including photocopying, recording, or information storage and retrieval) without permission in
writing from the publisher.
This book was set in Times New Roman on 3B2 by Asco Typesetters, Hong Kong and was printed and
bound in the United States of America.
Library of Congress Cataloging-in-Publication Data
Tversky, Amos.
Preference, belief, and similarity : selected writings / by Amos Tversky ; edited by Eldar Shafir.
p. cm.
‘‘A Bradford book.’’
Includes bibliographical references and index.
ISBN 0-262-20144-5 (alk. paper) — ISBN 0-262-70093-X (pbk. : alk. paper)
1. Cognitive psychology. 2. Decision making. 3. Judgment. 4. Tversky, Amos. I. Shafir, Eldar.
II. Title.
BF201 .T78 2003
153—dc21 2002032164

10987654321
Contents
Introduction and Biography ix
Sources xv
SIMILARITY 1
Editor’s Introductory Remarks 3
1 Features of Similarity 7
Amos Tversky
2 Additive Similarity Trees 47
Shmuel Sattath and Amos Tversky
3 Studies of Similarity 75
Amos Tversky and Itamar Gati
4 Weighting Common and Distinctive Features in Perceptual and
Conceptual Judgments
97
Itamar Gati and Amos Tversky
5 Nearest Neighbor Analysis of Psychological Spaces 129
Amos Tversky and J. Wesley Hutchinson
6 On the Relation between Common and Distinctive Feature Models 171
Shmuel Sattath and Amos Tversky
JUDGMENT 187
Editor’s Introductory Remarks 189
7 Belief in the Law of Small Numbers 193
Amos Tversky and Daniel Kahneman
8 Judgment under Uncertainty: Heuristics and Biases 203
Amos Tversky and Daniel Kahneman
9 Extensional vs. Intuitive Reasoning: The Conjunction Fallacy in
Probability Judgment
221
Amos Tversky and Daniel Kahneman

10 The Cold Facts about the ‘‘Hot Hand’’ in Bas ketball 257
Amos Tversky and Thomas Gilovich
Editor’s Introductory Remarks to Chapter 11 267
11 The ‘‘Hot Hand’’: Statistical Reality or Cognitive Illusion? 269
Amos Tversky and Thomas Gilovich
12 The Weighing of Evidence and the Determinants of Confidence 275
Dale Gri‰n and Amos Tversky
13 On the Evaluation of Probability Judgments: Calibration, Resolution,
and Monotonicity
301
Varda Liberman and Amos Tversky
14 Support The ory: A Nonextensional Representation of Subjective
Probability
329
Amos Tversky and Derek J. Koehler
15 On the Belief That Arthritis Pain Is Related to the Weather 377
Donald A. Redelmeier and Amos Tversky
16 Unpacking, Repacking, and Anchoring: Advances in Support Theory 383
Yuval Rottenstreich and Amos Tversky
PREFERENCE 403
Editor’s Introductory Remarks 405
Probabilistic Models of Choice 411
17 On the Optimal Number of Alternatives at a Choice Point 413
Amos Tversky
18 Substitutability and Similarity in Binary Choices 419
Amos Tversky and J. Edward Russo
19 The Intransitivity of Preferences 433
Amos Tversky
20 Elimination by Aspects: A Theory of Choice 463
Amos Tversky

21 Preference Trees 493
Amos Tversky and Shmuel Sattath
Choice under Risk and Uncertainty 547
22 Prospect Theory: An Analysis of Decision under Risk 549
Daniel Kahneman and Amos Tversky
vi Contents
23 On the Elicitation of Preferences for Alternative Therapies 583
Barbara J. McNeil, Stephen G. Pauker, Harold C. Sox, Jr., and
Amos Tversky
24 Rational Choice and the Framing of Decisions 593
Amos Tversky and Daniel Kahneman
25 Contrasting Rational and Psychological Analyses of Political Choice 621
George A. Quattrone and Amos Tversky
26 Preference and Belief: Ambiguity and Competence in Choice under
Uncertainty
645
Chip Heath and Amos Tversky
27 Advances in Prospect Theory: Cumulative Representation of
Uncertainty
673
Amos Tversky and Daniel Kahneman
28 Thinking through Uncertainty: Nonconsequential Reasoning and
Choice
703
Eldar Shafir and Amos Tversky
29 Conflict Resolution: A Cognitive Perspective 729
Daniel Kahneman and Amos Tversky
30 Weighing Risk and Uncertainty 747
Amos Tversky and Craig R. Fox
31 Ambiguity Aversion and Comparative Ignorance 777

Craig R. Fox and Amos Tversky
32 A Belief-Based Account of Decision under Uncertainty 795
Craig R. Fox and Amos Tversky
Contingent Preferences 823
33 Self-Deception and the Voter’s Illusion 825
George A. Quattrone and Amos Tversky
34 Contingent Weighting in Judgment and Choice 845
Amos Tversky, Shmuel Sattath, and Paul Slovic
35 Anomalies: Preference Reversals 875
Amos Tversky and Richard H. Thaler
Contents vii
36 Discrepancy between Medical Decisions for Individual Patients and for
Groups
887
Donald A. Redelmeier and Amos Tversky
37 Loss Aversion in Riskless Choice: A Reference-Dependent Model 895
Amos Tversky and Daniel Kahneman
38 Endowment and Contrast in Judgments of Well-Being 917
Amos Tversky and Dale Gri‰ n
39 Reason-Based Choice 937
Eldar Shafir, Itamar Simonson, and Amos Tversky
40 Context-Dependence in Legal Decision Making 963
Mark Kelman, Yuval Rottenstreich, and Amos Tversky
Amos Tversky’s Complete Bibliography 995
Index 1001
viii Contents
Introduction and Biography
Amos Tversky was a towering figure in the field of cognitive psychology and in the
decision sciences. His research had enormous influence; he created new areas of study
and helped transform related disciplines. His work was innovative, exciting, aes-

thetic, and ingenious. This book brings together forty of Tversky’s original articles,
which he and the editor chose together during the last months of Tversky’s life.
Because it includes only a fragment of Tversky’s published work, this book cannot
provide a full sense of his remarkable achievements. Instead, this collection of
favorites is intended to capture the essence of Tversky’s phenomenal mind for those
who did not have the fortune to know him, and will provide a cherished memento to
those whose lives he enriched.
Tversky was born on March 16, 1937, in Haifa, Israel. His father was a veterinar-
ian, and his mother was a social worker and later a member of the first Israeli
Parliament. He received his Bachelor of Arts degree, majoring in philosophy and
psychology, from Hebrew University in Jerusalem in 1961, and his Doctor of Phi-
losophy degree in psychology from the University of Michigan in 1965. Tversky
taught at Hebrew University (1966–1978) and at Stanford University (1978–1996),
where he was the inaugural Davis-Brack Professor of Behavioral Sciences and Prin-
cipal Investigator at the Stanford Center on Conflict and Negotiation. After 1992 he
also held an appointment as Senior Visiting Professor of Economics and Psychology
and Permanent Fellow of the Sackler Institute of Advanced Studies at Tel Aviv
University.
Tversky wrote his dissertation, which won the University of Michigan’s Marquis
Award, under the supervision of Clyde Coombs. His early work in mathematical
psychology focused on the study of individual choice behavior and the analysis of
psychological measurement. Almost from the beginning, Tversky’s work explored
the surprising implications of simple and intuitively compelling psychological
assumptions for theories that, until then, seemed self- evident. In one oft-cited early
work (chapter 19), Tversky showed how a series of pair-wise choices could yield
intransitive patterns of preference. To do this, he created a set of options such that
the di¤erence on an important dimension was negligible between adjacent alter-
natives, but proved to be consequential once it accumulated across a number of such
comparisons, yielding a reversal of preference between the first and the last. This was
of theoretical significance since the transitivity of preferences is one of the funda-

mental axioms of utility theory. At the same time, it provided a revealing glimpse
into the psychological processes involved in choices of that kind.
In his now-famous model of similarity (chapter 1), Tversky made a number of
simple psychol ogical assumptions: items are mentally represented as collect ions of
features, with the similarity between them an increasing function of the features th at
they have in common, and a decreasing functio n of their distinct features. In addi-
tion, feature weights are assumed to depend on the nature of the task so that, for
example, common features matter more in judgments of similarity, whereas distinc-
tive features receive more attention in judgments of dissimilarity. Among other
things, this simple theory was able to explain asymmetries in similarity judgments
(A may be more similar to B than B is to A), and the fact that item A may be per-
ceived as quite similar to item B and item B quite similar to item C, but items A and
C may nevertheless be perceived as highly dissimilar (chapter 3). In many ways, these
early papers foreshadowed the immensely elegant work to come. They were pre-
dicated on the technical mastery of relevant normative theories, and explored simple
and compe lling psychological principles until their unexpected theoretical implica-
tions bec ame apparent, and often striking.
Tversky’s long and extraordinarily influential collaboration with Daniel Kahne-
man began in 1969 and spanned the fields of judgment and decision making. (For a
sense of the impact, consider the fact that the two papers most representative of their
collaboration, chapters 8 and 22 in this book, register 3035 and 2810 citations,
respectively, in the Social Science Citation Index in the two decades spanning 1981–
2001.) Having recognized that intuitive predictions and judgments of probability do
not follow the principles of statistics or the laws of probability, Kahneman and
Tversky embarked on the study of biases as a method for investigating judgmental
heuristics. The beauty of the work was most apparent in the interplay of psycholog-
ical intuition with normative theory, accompanied by memorable demonstrations.
The research showed that people’s judgments often violate basic normative prin-
ciples. At the same time, it showed that they exhibit sensitivity to these principles’
normative appeal. The coexistence of fallible intuitions and an underlying apprecia-

tion for normative judgment yields a subtle picture of probab ilistic reasoning. An
important theme in Tversky’s work is a rejection of the claim that people are not
smart enough or sophisticated enough to grasp the relevant normative considera-
tions. Rather, Tversky attributes the recurrent and system atic errors that he finds to
people’s reliance on intuitive judgment and heuristic processes in situations where the
applicability of normative criteria is not immediately apparent. This approach runs
through much of Tversky’s work. The experimental demonstrations are noteworthy
not simply because they contradict a popular and highly influential normative
theory; rather, they are memorable precisely because people who exhibit these errors
typically find the demonstrations highly compelling, yet surprisingly inconsistent
with their own assumptions about how they make decisions.
Psychological common sense formed the basis for some of Tversky’s most
profound and original insights. A fundamental assumption underlying normative
theories is the extensionality principle: options that are extensionally equivalent are
x Introduction and Biography
assigned the same value, and extensionally equivalent events are assigned the same
probability. These theories, in other words, are about options and events in the
world: alternative descriptions of the same thing are still about th e same thing, and
hence similarly evaluated. Tversky’s analyses, on the other hand, focus on the mental
representations of the relevant constructs. The extensionality principle is deemed
descriptively invalid because alternative descriptions of the same options or events
often produce systematically di¤erent judgments and preferences. The way a decision
problem is described—for example, in terms of gains or losses—can trigg er conflict-
ing risk attitudes and thus lead to discrepant preferences with respect to the same
final outcomes (chapter 24); similarly, alternative descriptions of the same event
bring to mind di¤erent instances and thus can yield discrepant likelihood judgments
(chapter 14). Preferences as well as judgments appear to be constructed, not merely
revealed, in the elicitation process, and their construction depends on the framing of
the problem, the method of elicitation, and the valuations and attitudes that these
trigger.

Behavior, Tversky’s research made clear, is the outcome of normative ideals that
people endorse upon reflection, combined with psychological tendencies and pro-
cesses that intrude upon and shape behavior, independently of any deliberative
intent. Tversky had a unique ability to master the technicalities of the normative
requirements and to intuit, and then experimentally demonstrate, the vagaries and
consequences of the psychological processes that impinged on them. He was an
intellectual giant whose work has an exceptionally broad appeal; his research is
known to economists, philosophers, statisticians, political scientists, sociologists, and
legal theorists, among others. He published more than 120 papers and co-wrote or
co-edited 10 books. (A complete bibliography is printed at the back of this book.)
Tversky’s main research interests spa nned a large variety of topics, some of which
are better represented in this book than others, and can be roughly divided into three
general areas: similarity, judgment, and preference. The articles in this collected
volume are divided into corresponding sections and appear in chronological order
within each section.
Many of Tversky’s papers are both seminal and definitive. Reading a Tversky
paper o¤ers the pleasure of watching a craftsman at work: he provides a clear map of
a domain that had previously seemed confusing, and then o¤ers a new set of tools
and ideas for thinking about the problem. Tversky’s writings have had remarkable
longevity: the research he did early in his career has remained at the center of atten-
tion for several decades, and the work he was doing toward the end of his life will
a¤ect theory and research for a long time to come. Special issues of The Quarterly
Journal of Economics (1997), the Journal of Risk and Uncertainty (1998), and Cog-
nitive Psychology (1999) have been dedicated to Tversky’s memory, and various
Introduction and Biography xi
obituaries and articles about Tversky have appeared in places ranging from The Wall
Street Journal (1996) and The New York Times (1996), to the Journal of Medical
Decision Making (1996), American Psychologist (1998), Thinking & Reasoning (1997) ,
and The MIT Encyclopedia of Cognitive Science (1999), to name a few.
Tversky won many awards for his diverse accomplishments. As a young o‰cer in

a paratroops regiment, he earned Israel’s highest honor for bravery in 1956 for
rescuing a soldier. He won the Distinguished Scientific Contribution Aw ard of the
American Psychological Association in 1982, a MacArthur Prize in 1984, and the
Warren Medal of the Society of Experimental Psychologists in 1995. He was elected
to the American Academy of Arts and Sciences in 1980, to the Econometric Society
in 1993, and to the National Academy of Sciences as a foreign member in 1985. He
received honorary doctorates from the University of Goteborg, the State University
of New York at Bu¤alo, the University of Chicago, and Yale University.
Tversky managed to combine discipline and joy in the conduct of his life in a
manner that conveyed a great sense of freedom and autonomy. His habit of working
through the night helped protect him from interruptions and gave him the time to
engage at leisure in his research activities, as well as in other interests, including a
lifelong love of Hebrew literature, a fascination with modern physics, and an expert
interest in professional basketball. He was tactful but firm in rejecting commitments
that would distract him: ‘‘For those who like that sort of thing,’’ Amos would say
with his characteristic smile as he declined various engagements, ‘‘that’s the sort of
thing they like.’’ To his friends and collaborators, Amos was a delight. He found
great joy in sharing ideas and experiences with people close to him, and his joy was
contagious. Many friends became research collaborators, and many collaborators
became close friends. He would spend countless hours developing an idea, delighting
in it, refining it. ‘‘Let’s get this right,’’ he would say—and his ability to do so was
unequaled.
Amos Tversky continued his research and teaching until his illness made that
impossible, just a few weeks before his death. He died of metastatic melanoma on
June 2, 1996, at his home in Stanford, California. He was in the midst of an enor-
mously productive time, with over fifteen papers and several edited volumes in press.
Tversky is survived by his wife, Barbara, who was a fellow student at the University
of Michigan and then a fellow professor of psychology at Stanford University, and
by his three children, Oren, Tal, and Dona. This book is dedicated to them.
Eldar Shafir

xii Introduction and Biography
Postscript
In October 2002 The Royal Swedish Academy of Sciences awarded the Nobel Me-
morial Prize in Economic Sciences to Daniel Kahneman, ‘‘for having integrated
insights from psychological research into economic science, especially concerning
human judgment and decision-making under uncertainty.’’ The work Kahneman had
done together with Amos Tversky, the Nobel citation explained, formulated alter-
native theories that better account for observed behavior. The Royal Swedish Acad-
emy of Sciences does not award prizes posthumously, but took the unusual step of
acknowledging Tversky in the citation. ‘‘Certainly, we would have gotten this to-
gether,’’ Kahneman said on the day of the announcement. Less than two months
later, Amos Tversky posthumously won the prestigious 2003 Grawemeyer Award
together with Kahneman. The committee of the Grawemeyer Award, which recog-
nizes powerful ideas in the arts and sciences, noted, ‘‘It is di‰cult to identify a more
influential idea than that of Kahneman and Tversky in the human sciences.’’ React-
ing to the award, Kahneman said, ‘‘My joy is mixed with the sadness of not being
able to share the experience with Amos Tversky, with whom th e work was done.’’ It
is with a similar mixture of joy and sadness that we turn to Amos’s beautiful work.
Introduction and Biography xiii

Sources
1. Tversky, A. (1977). Features of similarity. Psychological Review, 84, 327–352. Copyright 6 1977 by the
American Psychological Association. Reprinted with permission.
2. Sattath, S., and Tversky, A. (1977). Additive similarity trees. Psychometrika, 42, 319–345.
3. Tversky, A., and Gati, I. (1978). Studies of similarity. In E. Rosch and B. Lloyd (Eds.), Cognition and
Categorization, (79–98), Hillsdale, N.J.: Erlbaum.
4. Gati, I., and Tversky, A. (1984). Weighting common and distinctive features in perceptual and concep-
tual judgments, Cognitive Psychology, 16, 341–370.
5. Tversky, A., and Hutchinson, J. W. (1986). Nearest neighbor analysis of psychological spaces. Psycho-
logical Review, 93, 3–22. Copyright 6 1986 by the American Psychological Association. Reprinted with

permission.
6. Sattath, S., and Tversky, A. (1987). On the relation between common and distinctive feature models.
Psychological Review, 94, 16–22. Copyright 6 1987 by the American Psychological Association. Reprinted
with permission.
7. Tversky, A., and Kahneman, D. (1971). Belief in the law of small numbers. Psychological Bulletin, 76,
105–110.
8. Tversky, A., and Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science,
185, 1124–1131. Reprinted with permission from Science. Copyright 1974 American Association for the
Advancement of Science.
9. Tversky, A., and Kahneman, D. (1983). Extensional vs. intuitive reasoning: The conjunction fallacy in
probability judgment. Psychological Review, 91, 293–315. Copyright 6 1983 by the American Psychologi-
cal Association. Reprinted with permission.
10. Tversky, A., and Gilovich, T. (1989). The cold facts about the ‘‘hot hand’’ in basketball. Chance, 2(1),
16–21. Reprinted with permission from CHANCE. Copyright 1989 by the American Statistical Associa-
tion. All rights reserved.
11. Tversky, A., and Gilovich, T. (1989). The hot hand: Statistical reality or cognitive illusion? Chance,
2(4), 31– 34. Reprinted with permission from CHANCE. Copyright 1989 by the American Statistical
Association. All rights reserved.
12. Gri‰n, D., and Tversky, A. (1992). The weighing of evidence and the determinants of confidence.
Cognitive Psychology, 24, 411–435.
13. Liberman, V., and Tversky, A. (1993). On the evaluation of probability judgments: Calibration, reso-
lution and monotonicity. Psychological Bulletin, 114, 162–173.
14. Tversky, A., and Koehler, D. J. (1994). Support theory: A nonextensional representation of subjective
probability. Psychological Review, 101, 547–567. Copyright 6 1994 by the American Psychological Asso-
ciation. Reprinted with permission.
15. Redelmeier, D. A., and Tversky, A. (1996). On the belief that arthritis pain is related to the weather.
Proc. Natl. Acad. Sci., 93, 2895–2896. Copyright 1996 National Academy of Sciences, U.S.A.
16. Rottenstreich, Y., and Tversky, A. (1997). Unpacking, repacking, and anchoring: Advances in support
theory. Psychological Review, 104(2), 406–415. Copyright 6 1997 by the American Psychological Associ-
ation. Reprinted with permission.

17. Tversky, A. (1964). On the optimal number of alternatives at a choice point. Journal of Mathematical
Psychology, 2, 386–391.
18. Tversky, A., and Russo, E. J. (1969). Substitutability and similarity in binary choices. Journal of
Mathematical Psychology, 6, 1–12.
19. Tversky, A. (1969). The intransitivity of preferences. Psychological Review, 76, 31–48. Copyright 6
1969 by the American Psychological Association. Reprinted with permission.
20. Tversky, A. (1972). Elimination by aspects: A theory of choice. Psychological Review, 79, 281–299.
Copyright 6 1972 by the American Psychological Association. Reprinted with permission.
21. Tversky, A., and Sattath, S. (1979). Preference trees. Psychological Review, 86, 542– 573. Copyright 6
1979 by the American Psychological Association. Reprinted with permission.
22. Kahneman, D., and Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econo-
metrica, 47, 263–291. Copyright The Econometric Society.
23. McNeil, B., Pauker, S., Sox, H. Jr., and Tversky, A. (1982). On the elicitation of preferences for
alternative therapies. New England Journal of Medicine, 306, 1259–1262. Copyright 6 1982 Massachusetts
Medical Society. All rights reserved.
24. Tversky, A., and Kahneman, D. (1986). Rational choice and the framing of decisions. The Journal of
Business, 59, Part 2, S251–S278.
25. Quattrone, G. A., and Tversky, A. (1988). Contrasting rational and psychological analyses of political
choice. American Political Science Review, 82(3), 719–736.
26. Heath, F., and Tversky, A. (1991). Preference and belief: Ambiguity and competence in choice under
uncertainty. Journal of Risk and Uncertainty, 4(1), 5–28. Reprinted with kind permission from Kluwer
Academic Publishers.
27. Tversky, A., and Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of
uncertainty. Journal of Risk and Uncertainty, 5, 297–323. Reprinted with kind permission from Kluwer
Academic Publishers.
28. Shafir, E., and Tversky, A. (1992). Thinking through uncertainty: Nonconsequential reasoning and
choice. Cognitive Psychology, 24(4), 449–474.
29. Kahneman, D., and Tversky, A. (1995). Conflict resolution: A cognitive perspective. In K. Arrow, R.
Mnookin, L. Ross, A. Tversky, and R. Wilson (Eds.), Barriers to the Negotiated Resolution of Conflict,
(49–67). New York: Norton.

30. Tversky, A., and Fox, C. R. (1995). Weighing risk and uncertainty. Psychological Review, 102(2),
269–283. Copyright 6 1995 by the American Psychological Association. Reprinted with permission.
31. Fox, C. R., and Tversky, A. (1995). Ambiguity aversion and comparative ignorance. Quarterly Jour-
nal of Economics, 110, 585–603. 6 1995 by the President and Fellows of Harvard College and the Massa-
chusetts Institute of Technology.
32. Fox, C. R., and Tversky, A. (1998). A belief-based account of decision under uncertainty. Manage-
ment Science, 44(7), 879–895.
33. Quattrone, G. A., and Tversky, A. (1986). Self-deception and the voter’s illusion. In Jon Elster (Ed.),
The Multiple Self, (35–58), New York: Cambridge University Press. Reprinted with permission of Cam-
bridge University Press.
34. Tversky, A., Sattath, S., and Slovic, P. (1988). Contingent weighting in judgment and choice. Psycho-
logical Review, 95(3), 371–384. Copyright 6 1988 by the American Psychological Association. Reprinted
with permission.
35. Tversky, A., and Thaler, R. (1990). Anomalies: Preference reversals. Journal of Economic Perspectives,
4(2), 201–211.
36. Redelmeier, D. A., and Tversky, A. (1990). Discrepancy between medical decisions for individual
patients and for groups. New England Journal of Medicine, 322, 1162–1164. Copyright 6 1990 Massa-
chusetts Medical Society. All rights reserved.
37. Tversky, A., and Kahneman, D. (1991). Loss aversion in riskless choice: A reference-dependent model.
Quarterly Journal of Economics, 107(4), 1039–1061. 6 1991 by the President and Fellows of Harvard
College and the Massachusetts Institute of Technology.
38. Tversky, A., and Gri‰n, D. (1991). Endowment and contrast in judgments of well-being. In F. Strack,
M. Argyle, and N. Schwartz (Eds.), Subjective Well-being: An Interdisciplinary Perspective (101–118).
Elmsford, NY: Pergamon Press.
39. Shafir, E., Simonson, I., and Tversky, A. (1993). Reason-based choice. Cognition, 49, 11–36.
Reprinted from Cognition with permission from Elsevier Science. Copyright 1993.
40. Kelman, M., Rottenstreich, Y., and Tversky, A. (1996). Context-dependence in legal decision making.
The Journal of Legal Studies, 25, 287–318. Reprinted with permission of the University of Chicago.
xvi Sources
SIMILARITY

Editor’s Introductory Remarks
Early in his career as a mathematical psychologist Tversky developed a deep interest
in the formalization and conceptualization of similarity. The notion of similarity is
ubiquitous in psychological theorizing, where it plays a fundamental role in theories
of learning, memory, knowledge, perception, and judgment, among others. When
Tversky began his work in this area, geometric models dominated the theoretical
analysis of similarity relations; in these models each object is represented by a point
is some multidimensional coordinate space, and the metric distances between points
reflect the similarities between the respective objects.
Tversky found it more intuitive to represent stimuli in terms of their many quali-
tative features rather than a few quantitative dimensions. In this contrast model of
similarity (chapter 1), Tversky challenges the assumptions that underlie the geometric
approach to similarity and dev elops an alternative approach based on feature
matching. He began with simple psychol ogical assumptions couched in an aesthetic
formal treatment, and was able to predict surprising facts about the perception of
similarity and to provide a compelling reinterpretation of previously known facts.
According to the contrast model, items are represented as collections of features.
The perceived similarity between items is an increasing function of the features that
they have in common, and a decreasing function of the features on which they di¤er.
In addition, each set of common and distinctive features is weighted di¤erentially,
depending on the context, the order of comparison, and the particular task at hand.
For example, common features are weighted relatively more in judgments of simi-
larity, whereas distinctive features receive more attention in judgments of dissimilar-
ity. Among other things, the theory is able to explain asymmetries in similarity
judgments (A can be more similar to B than B is to A), the non-complementary
nature of similarity and dissimilarity judgments (A and B may be both more similar
to one another and more di¤erent from one another than are C and D), and triangle
inequality (item A may be perceived as quite similar to item B and item B quite
similar to item C, but items A and C may nevertheless be perceived a s highly

dissimilar) (Tversky & Gati 1982). These patterns of similarity judgments, which
Tversky and his colleagues compellingly documented, are inconsistent with geo-
metric rep resentations (where, for example, the distance from A to B needs to be the
same as that between B and A, etc.). The logic and implications of the contrast
model are further summarized and given a less technical presentation by Tversky and
Itamar Gati in chapter 3.
In a further investigation of how best to represent similarity relations (chapter 2),
Shmuel Sattath and Tversky consider the representation of similarity data in the
form of additive trees, and compare it to alternative representational schemes, par-
ticularly spatial representations that are limited in some of the ways suggested above.
In the additive tree model, objects are represented by the external nodes of a tree,
and the dissimilarity between obj ects is the length of th e path joining them. As it
turns out, the e¤ect of common features can be better captured by trees than by
spatial representations. In fact, an additive tree can be interpreted as a feature tree,
with each object viewed as a set of features, and each arc representing the set of fea-
tures shared by all objects that follow from that arc. (An additive tree is a special
case of the contrast model in which symmetry and the triangle inequality hold, and
the feature space allows a tree structure.)
Further exploring the adequacy of the geometric models, chapter 5 applies a
‘‘nearest neighbor’’ analysis to similarity data. The technical analysis essentially
shows that geo metric models are severely restricted in the number of objects that
they can allow to share the same nearest (for example, mo st similar) neighbor. Using
one hundred di¤erent data sets, Tversky and Hutchinson show that while perceptual
data often satisfy the bounds imposed by geometric representations, the conceptual
data sets typically do not. In particular, in many semantic fields a focal element (such
as the superordinate category) is the nearest neighbor of most of the category
instances. Tversky shows that such a popular nearest neighbor, while inconsistent
with a geometric representation, can be captured by an additive tree in which the
category name (for example, fruit) is the nearest neighb or of all its instances.
Tversky and his coauthors conclude that some similarity data are better described

by a tree, whereas other data may be better captured by a spatial configuration.
Emotions or sound, for example, may be characterized by a few dimensions that
di¤er in intensity, and may thus be natural candidates for a dimensional representa-
tion. Other items, however, have a hierarchical classification involving various qual-
itative attributes, and may thus be better captured by tree representations.
A formal procedure based on the contrast model is developed in chapter 4 in order
to assess the relative weight of common to distinctive features. By adding the same
component (for example, cloud) to two stimuli (for example, landscapes) or to one of
the stimuli only, Gati and Tversky are able to assess the impact of that component as
a common or as a distinctive feature. Among other things, they find that in verbal
stimuli common features loom larger than distinctive features (as if the di¤erences
between stimuli are acknowledged and one focuses on the search for common fea-
tures), whereas in pictorial stimuli distinctive features loom larger than common
features (consistent with the notion that commonalities are treated as background
and the search is for distinctive features.)
The theoretical relationship between common- and distinctive-feature models
is explored in chapter 6, where Sattath and Tversky show that common-feature
models and distinctive-feature models can produce di¤erent orderings of dissimilarity
4 Shafir
between objects. They further show that the choice of a model and the specification
of a feature structure are not always determined by the dissimilarity data and, in
particular, that the relative weights of common and distinctive features observed in
chapter 4 can depend on the feature structure induced by the addition of dimensions.
Chapter 6 concludes with general commen tary regarding the observation that the
form of measurement models often is not dictated by the data. This touches on
the massive project on the foundations of measurement that Tversky co-authored
(Krantz, Luce, Suppes, and Tversky, 1971; Suppes, Krantz, Luce, and Tversky,
1989; Luce, Krantz, Suppes, and Tversky, 1990), but which is not otherwise repre-
sented in this collection.
Similarity: Editor’s Introductory Remarks 5

1Features of Similarity
Amos Tve rsky
Similarity plays a fundamental role in theories of knowledge and behavior. It serves
as an organizing principle by which individuals classify objects, form concepts, and
make generalizations. Indeed , the concept of similarity is ubiquitous in psychological
theory. It underlies the accounts of stimulus and response generalization in learning,
it is employed to explain errors in memory and pattern recognition, and it is central
to the analysis of connotative meaning.
Similarity or dissimilarity data appea r in di¤erent forms: ratings of pairs, sorting
of objects, communality between associations, errors of subst itution, and correlation
between occurrences. Analyses of these data attempt to explain the observed simi-
larity relations and to capture the underlying structure of the objects under study.
The theoretical ana lysis of similarity relations has been dominated by geometric
models. These models represent objects as points in some coordinate space such
that the observed dissimilarities between objects correspond to the metric distances
between the respective points. Practically all analyses of proximity data have been
metric in nature, although some (e.g., hierarchical clustering) yield tree-like struc-
tures rather than dimensionally organized spaces. However, most theoretical and
empirical analyses of similarity assume that objects can be adequately represented as
points in some coordinate space and that dissimilarity behaves like a metric distance
function. Both dimen sional and metric assumptions are open to question.
It has been argued by many authors that dimensional representations are appro-
priate for certain stimuli (e.g., colors, tones) but not for others. It seems more ap-
propriate to represent faces, countries, or personalities in terms of many qualitative
features than in terms of a few quantitative dimensions. The assessment of similarity
between such stimuli, therefore, may be better described as a comparison of features
rather than as the computation of metric distance between points.
A metric distance function, d, is a scale that assigns to every pair of points a non-
negative number, called their distance, in accord with the following three axioms:

Minimality: dða; bÞ b dða; aÞ¼0:
Symmetry: dða; bÞ¼dðb; aÞ:
The triangle inequality: dða; bÞþdðb; cÞ b dða; c Þ:
To evaluate the adequacy of the geometric approach, let us examine the validity of
the metric axioms when d is regarded as a measure of dissimilarity. The minimality
axiom implies that the similarity between an object and itself is the same for all
objects. This assumption, however, does not hold for some similarity measures. For
example, the probability of judging two identical stimuli as ‘‘same’’ rather that ‘‘dif-
ferent’’ is not constant for all stimuli. Moreover, in recognition experiments the o¤-
diagonal entries often exceed the diagonal entries; that is, an object is identified as
another object more frequently than it is identified as itself. If identification proba-
bility is interpreted as a measure of similarity, then these observations violate mini-
mality and are, therefore, incompatible with the distance model.
Similarity has been viewed by both philosophers and psychologists as a prime
example of a symmetric relation. Indeed, the assumption of symmetry underlies
essentially all theoretical treatments of similarity. Contrary to this tradition, the
present paper provides empirical evidence for asymmetric similarities and argues that
similarity should not be treated as a symmetric relation.
Similarity judgments can be regarded as extensions of similarity statements, that is,
statements of the form ‘‘a is like b.’’ Such a statement is directional; it has a subject,
a, and a referent, b, and it is not equivalent in general to the converse similarity
statement ‘‘b is like a.’’ In fact, the choice of subject and referent depends, at least in
part, on the relative salience of the objects. We tend to select the more salient stimu-
lus, or the prototype, as a referent, and the less salient stimulus, or the variant, as a
subject. We say ‘‘the portrait resembles the person’’ rather than ‘‘the person resem-
bles the portrait.’’ We say ‘‘the son resembles the father’’ rather than ‘‘the father
resembles the son.’’ We say ‘‘an ellipse is like a circle,’’ not ‘‘a circle is like an
ellipse,’’ and we say ‘‘North Korea is like Red China’ ’ rather than ‘‘Red China is like
North Korea.’’
As will be demonstrated later, this asymmetry in the choice of similarity statements

is associated with asymmetry in judgments of similarity. Thus, the judged similarity
of North Korea to Red China exceeds the judged similarity of Red China to North
Korea. Likewise, an ellipse is more similar to a circle than a circle is to an ellipse.
Apparently, the direction of asymmetry is determined by the relative salience of the
stimuli; the variant is more similar to the prototype than vice versa.
The directionality and asymmetry of similarity relations are particularly noticeable
in similies and metaphors. We say ‘‘Turks fight like tigers’’ and not ‘‘tigers fight like
Turks.’’ Since the tiger is renowned for its fighting spirit, it is used as the referent
rather than the subject of the simile. The poet writes ‘‘my love is as deep as the
ocean,’’ not ‘‘the ocean is as deep as my love,’’ because the ocean epitomizes depth.
Sometimes both directions are used but they carry di¤erent meanings. ‘‘A man is like
a tree’’ implies that man has roots; ‘‘a tree is like a man’’ implies that the tree has a
life history. ‘‘Life is like a play’’ says that people play roles. ‘‘A play is like life’’ says
that a play can capture the essential elements of human life. The relations between
8 Tversky