![By the way, when I conducted this experiment in class, there were
(good!) students who preferred C over D after they preferred A over
B. When asked to justify this “strange” choice, they pointed out that
C is equivalent to A and D is equivalent to B and referred to their
previous choice of A! These students demonstrated another com-
mon procedural element of decision making: The choice in one
problem is made in relation to decisions made previously in re-
sponse to other problems.
The Search for Reasons
In the next example (following Huber, Payne, and Puto [1982]),
(x, y) represents a holiday package that contains x days in Paris and
y days in London, all offered for the same price. All subjects agree
that a day in London and a day in Paris are desirable goods. Denote,
A = (7, 4), B = (4, 7), C = (6, 3) and D = (3, 6). Some of the subjects
were requested to choose between the three packages A, B, and C;
others had to choose between A, B, and D. The subjects exhibited
a clear tendency to choose A out of the set {A, B, C} and to choose
B out of the set {A, B, D}. Obviously, this behavior is not consistent
with the behavior of a “rational man.” Given the universal prefer-
ence of A over C and of B over D, the preferred element out of {A, B}
should be chosen from both {A, B, C} and {A, B, D}.
Once again, the beauty of this example is not its contradiction of
the rational man paradigm but its demonstration of a procedural
element that often appears in decision making. Decision makers
look for reasons to prefer A over B. Sometimes, those reasons relate
to the decision problem itself. In the current example, “dominating
another alternative” is a reason to prefer one alternative over the
other. Reasons that involve relationships to other alternatives may
therefore conºict with the rational man paradigm.
Another related, striking experiment was conducted by Tversky
and Shaªr (1992). A subject was shown a list of twelve cards. Each
card described one prize. Then the subject was given two cards and
asked whether he wanted to pay a certain fee for getting a third
Bounded Rationality in Choice 19](https://image.slidesharecdn.com/28193960-250517145457-abaa2391/85/Modeling-Bounded-Rationality-Ariel-Rubinstein-36-320.jpg)
![To illustrate, let us look at results of an experiment, reported in
Kahneman and Tversky (1982), that is similar to that of the Allais
paradox. The objects of choice in the experiment are simple lotter-
ies. A simple lottery (x,p) is a random variable that yields $x with
probability p and $0 with probability 1 − p. Thus, each object of
choice can be thought of as a vector of length 2.
In the experiment, some subjects were asked to choose between:
L3 = (4000, 0.2) and L4 = (3000, 0.25).
Most subjects chose L3. Another group of subjects was asked to
choose between:
L1 = (4000, 0.8) and L2 = (3000, 1.0).
The vast majority of subjects chose L2.
The choices L2 from {L1, L2} and L3 from {L3, L4} do not violate ra-
tionality. However, they do violate the von Neumann-Morgenstern
independence axiom. To see this, notice that the lotteries L3 and L4
can be presented as compound lotteries of L1, L2, and the degenerate
lottery [0], which yields the certain prize 0:
L3 = 0.25L1 + 0.75[0] and L4 = 0.25L2 + 0.75[0].
Therefore, the independence axiom requires that the choice be-
tween L3 and L4 be made according to the choice between L1 and
L2, in striking contrast to the experimental results.
The reasoning that probably guided many of the subjects was the
following: When comparing L3 to L4, a decision maker faces an
internal conºict due to the higher prize in L3 versus the higher
probability of getting a positive prize in L4. He tries to simplify the
choice problem so that one of the alternatives will be patently better.
With this aim, he checks the similarities of the probabilities and the
prizes that appear in the two lotteries. He considers the probability
numbers 0.25 and 0.2 to be similar, in contrast to the prizes $4000
and $3000, which are not. These similarity comparisons lead him to
26 Chapter 2](https://image.slidesharecdn.com/28193960-250517145457-abaa2391/85/Modeling-Bounded-Rationality-Ariel-Rubinstein-43-320.jpg)
![“simplify” the problem by “canceling” the probabilities and making
the choice between L3 and L4, based on the obvious choice between
$4000 and $3000. On the other hand, when comparing L1 and L2,
the decision maker cannot simplify the problem on the basis of
the cancellation of similar components since neither the prob-
abilities nor the prizes are perceived to be similar. He then invokes
another principle, presumably risk aversion, to arrive at the supe-
riority of L2.
Note that the attractiveness of the vNM independence axiom is
also related to its interpretation as an expression of a similar pro-
cedural element. When the lotteries L3 and L4 are represented ex-
plicitly in the form of the reduced lotteries 0.25L1 + 0.75[0] and
0.25L2 + 0.75[0], respectively, decision makers tend to simplify the
comparison between L3 and L4 by “canceling” the possibility that
the lotteries will yield the prize [0], then basing their choice on a
comparison between L1 and L2, thus choosing L4.
Hence, whether the lotteries are presented as simple or com-
pound, it seems that a major step in the deliberation is the “cancel-
lation of similar factors” and the consequent reduction of the
original complex choice to a simpler one. Activating this principle
when comparing L3 and L4 as simple lotteries leads to the choice of
L3; activating it when the lotteries are presented as compound
lotteries leads to the choice of L4. The way in which the principle
of reducing the complexity of a choice is applied, therefore, depends
on how the decision problem is framed. To avoid framing effects in
our analysis, we will retain the format of each alternative as ªxed.
Consequently, all objects of choice will be simple lotteries presented
as vectors of the type (x, p).
In the next three sections we will formulate and analyze a pro-
cedure for choosing between pairs of such lotteries that makes
explicit use of similarity relations. The presentation consists of the
following stages. First we will describe a scheme of choice proce-
dures between pairs of lotteries. Then we will ask two questions:
Modeling Procedural Decision Making 27](https://image.slidesharecdn.com/28193960-250517145457-abaa2391/85/Modeling-Bounded-Rationality-Ariel-Rubinstein-44-320.jpg)
![1. Does such a procedure necessarily conºict with the rational man
paradigm?
2. If not, what preference relations are consistent with the
procedure?
But ªrst we will detour to the world of similarity relations in order
to equip ourselves with the necessary tools.
2.2 Preparing the Tools: Similarity Relations
In this chapter, a similarity relation is taken to be a binary relation ∼
on the set I = [0, 1] that satisªes the following properties:
(S-1) Reºexivity For all a ∈ I, a ∼ a.
(S-2) Symmetry For all a, b ∈ I, if a ∼ b, then b ∼ a.
(S-3) Continuity The graph of the relation ∼ is closed in I × I.
(S-4) Betweenness If a ≤ b ≤ c ≤ d and a ∼ d, then b ∼ c.
(S-5) Nondegeneracy 0 ⬃ 1, and for all 0 < a < 1, there are b and c
so that b < a < c and a ∼ b and a ∼ c. For a = 1, there is b < a so that
a ∼ b. (For reasons which will soon become clear, no such require-
ment is made for a = 0.)
(S-6) Responsiveness Denote by a* and a
*
the largest and the small-
est elements in the set that are similar to a. Then a* and a
*
are strictly
increasing functions (in a) at any point at which they obtain a value
different from 0 or 1.
Although these axioms restrict the notion of similarity quite sig-
niªcantly, I ªnd them particularly suitable when the similarity
stands for a relation of the kind “approximately the same.” This
does not deny that there are contexts in which the notion of simi-
larity clearly does not satisfy the above axioms. For example, we
say that “Luxembourg is similar to Belgium,” but we do not say
28 Chapter 2](https://image.slidesharecdn.com/28193960-250517145457-abaa2391/85/Modeling-Bounded-Rationality-Ariel-Rubinstein-45-320.jpg)
![that “Belgium is similar to Luxembourg” (see Tversky [1977]). In
this example, we say that “a is similar to b” in the sense that b is a
“representative” of the class of elements to which both a and b
belong; this use of the term does not satisfy the symmetry
condition.
A leading example of a family of relations that satisªes all these
assumptions is the one consisting of the λ-ratio similarity relations
(with λ > 1) deªned by a ∼ b if 1/λ ≤ a/b ≤ λ. More generally, for
any number λ > 1 and for every strictly increasing continuous
function, H, on the unit interval, the relation a ∼ b if 1/λ ≤
H(a)/H(b) ≤ λ is a similarity relation. In fact, we can represent any
similarity relation in this way. We say that the pair (H, λ) represents
the similarity relation ∼ if, for all a,b ∈ I, a ∼ b if 1/λ ≤ H(a)/H(b) ≤
λ. One can show (see Project 6) that for every λ > 1 there is a strictly
increasing continuous function H with values in [0, 1], so that the
pair (H, λ) represents the similarity ∼. If 0 is not similar to any
positive number, we can ªnd a representation of the similarity
relation with a function H so that H(0) = 0. This proposition is
analogous to propositions in utility theory that show the existence
of a certain functional form of a utility representation.
Note that no equivalence relation is a similarity relation under
this deªnition. Consider, for example, the relation according to
which any two elements in I relate if, in their decimal presentation,
they have identical ªrst digits. This binary relation is an equivalence
relation that fails to comply with the continuity assumption, the
monotonicity assumption (because (.13)
*
= (.14)
*
, for example), and
the nondegeneracy condition (there is no x < 0.4 that relates to 0.4).
2.3 A Procedure of Choice between Vectors
In this section we analyze a family of decision procedures applied
to decision problems where the choice is made from a set of pairs
of lotteries in A = X × P = [0, 1] × [0, 1], where (x, p) ∈ A stands for
Modeling Procedural Decision Making 29](https://image.slidesharecdn.com/28193960-250517145457-abaa2391/85/Modeling-Bounded-Rationality-Ariel-Rubinstein-46-320.jpg)
![[The following extracts from the private correspondence of Sir
Charles Hanbury Williams will further illustrate the remark in the
text, and show the unfavourable view taken by him of the
Prussian Court and Frederick the Great.]
Extract of a Letter from Charles Hanbury Williams, from Berlin, 1750.
.... ’Tis incredible what care this Pater Patriæ takes of his people.
He is so good as to meddle in their family affairs, in their marriages,
in the education of their children, and in the disposition of their
estates. He hates that anybody should marry, especially an officer,
let him be of what degree soever, and from the moment they take a
wife, they are sure of never being preferred. All children are
registered as soon as born, and the parents are obliged to produce
either certificates of their deaths, or the male children themselves, at
the age of fourteen, in order to be enrolled, and to take the oath of
a soldier to the King; and if this is not done, or the children have
escaped, the parents are answerable for the escape, and are sent to
prison.
No man can sell land throughout all the Prussian dominions
without a special licence from the King: and as he does no more give
licences, nobody can now dispose of or alienate his possessions. If
they could, and were to find fools to purchase them, I believe he
would not have ten of his present subjects left in a year’s time. They
have really no liberty left but that of thinking. There is a general
constraint that runs through all sorts of people, and diffidence is
painted in every face. All their ambition and desire is to be permitted
to go to their Country Seats, where they need not be obliged to
converse with any but their own family. But this leave is not easily
obtained, because the father of his country insists upon their living
at Berlin, and making his Capital flourish. He is never here but from
the beginning of December to the end of January, and during that
time, Prussians, Silesians, and all his most distant subjects, are
obliged to come and make a figure here, and spend all they have](https://image.slidesharecdn.com/28193960-250517145457-abaa2391/85/Modeling-Bounded-Rationality-Ariel-Rubinstein-60-320.jpg)
![genteeler than Mons. Adrié, who you may remember to have seen
make so great a figure in England.
The other, Count Finkenstein, whom everybody calls Count Fink, is
very like the late Lord Hervey, and yet his face is the ugliest I ever
saw. But when he speaks, his affectation, the motion of his eyes and
shoulders, all his different gestures and grimaces, bring Lord Hervey
very strongly into my mind; and, like that Lord, he is the Queen’s
favourite (I mean the Queen Mother’s); and her Majesty, whether
seriously or otherwise I can’t tell, calls him “Mon beau Comte Fink.”
He has parts, and is what, at Berlin, is called sçavant, which is to
say, that he has read all the modern French story books, from Les
Egaremens down to the history of Prince Cocquetron.
The person who has certainly the greatest share of the King of
Prussia’s confidence is one Heichel. He is his Private Secretary, and
writes all that the King himself dictates. But this man I never saw,
and people that have lived here seven years have never seen him.
He is kept like a State Prisoner, is in constant waiting, and never has
half an hour to himself in the whole year.
[Then follows the account of Fredersdorff, to the same purpose,
and nearly in the same words as in the extracts printed above.]
He (Fredersdorff) is his Secretary for all small affairs for his
Prussian Majesty.—
Il fait tout par ses mains, et voit tout par ses yeux.
If a Courier is to be dispatched to Versailles, or a Minister to
Vienna, his Prussian Majesty draws, himself, the instructions for the
one, and writes the letters for the other. This, you’ll say, is great; but
if a Dancer at the Opera has disputes with a Singer, or if one of
those performers want a new pair of stockings, a plume for his
helmet, or a finer petticoat, ’tis the same King of Prussia that sits in
judgment on the cause, and that with his own hand answers the
Dancer’s or the Singer’s letter. His Prussian Majesty laid out 20,000l.](https://image.slidesharecdn.com/28193960-250517145457-abaa2391/85/Modeling-Bounded-Rationality-Ariel-Rubinstein-62-320.jpg)
![K.
(Vide page 225.)
Though poetry was certainly neither a point of their rivalship, nor
of their ambition, it may not be unwelcome to the curious to
compare these great men even in their poetic capacities. The
following sonnet was written by Sir R. Walpole when a very young
man; the elegy, by Lord Bolingbroke, rather past his middle age. Had
they climbed no mountain but Parnassus, it is obvious how far Lord
Bolingbroke would have ascended above his competitor, since, when
turned of fifty, he excelled in the province of youth.
TO THE HELIOTROPE.[257]
A SONG.](https://image.slidesharecdn.com/28193960-250517145457-abaa2391/85/Modeling-Bounded-Rationality-Ariel-Rubinstein-68-320.jpg)
![1.
Hail, pretty emblem of my fate!
Sweet flower, you still on Phœbus wait;
On him you look, and with him move,
By nature led, and constant love.
2.
Know, pretty flower, that I am he,
Who am in all so like to thee;
I, too, my fair one court, and where
She moves, my eyes I thither steer.
3.
But yet this difference still I find,
The sun to you is always kind;
Does always life and warmth bestow:
—Ah! would my fair one use me so!
4.
Ne’er would I wait till she arose
From her soft bed and sweet repose;
But leaving thee, dull plant, by night
I’d meet my Phillis with delight.
TO CLARA.[258]
BY HENRY, VISCOUNT BOLINGBROKE.](https://image.slidesharecdn.com/28193960-250517145457-abaa2391/85/Modeling-Bounded-Rationality-Ariel-Rubinstein-69-320.jpg)
![FOOTNOTES:
[252] The Princess.
[253] Allen, Lord Bathurst.
[254] Sir George Lyttelton, who was out of favour with the Prince,
made a parody on this copy of verses: two of the lines were,
No—’tis that all-consenting tongue,
That never puts me in the wrong.
[255] “This is a strange country, this England” (said his Royal
Highness once); “I am told Doddington is reckoned a clever man;
yet I got 5000l. out of him this morning, and he has no chance of
ever seeing it again.”
[256] His house is since called Brandenburgh House.
[257] I found this song in an old pocket-book belonging to my
father, who wrote it, as he told me himself, when he was a very
young man, on a sister of Sir William Carew.
[258] This was written on a common woman whom Lord
Bolingbroke took into keeping, and who, many years afterwards,
sold oranges in the Court of Requests.](https://image.slidesharecdn.com/28193960-250517145457-abaa2391/85/Modeling-Bounded-Rationality-Ariel-Rubinstein-72-320.jpg)
![TRANSCRIBER’S NOTE
Obvious typographical errors and punctuation errors have been corrected after
careful comparison with other occurrences within the text and consultation of
external sources.
Except for those changes noted below, all misspellings in the text, and
inconsistent or archaic usage, have been retained. For example, Nova-Scotia,
Nova Scotia; goodnature, good-nature; Lord-Lieutenant, Lord Lieutenant;
se’nnight; disculpate; unapt; deficience; altercate; preponderated.
Pg xix: ‘acknowleged to have’ replaced by ‘acknowledged to have’.
Pg 22: ‘he committed to’ replaced by ‘he was committed to’.
Pg 25: ‘John Burnard—Factions’ replaced by ‘John Barnard—Factions’.
Pg 37: ‘election at Weobly’ replaced by ‘election at Weobley’.
Pg 68: ‘19.—The’ replaced by ‘19th.—The’.
Pg 85: ‘3.—Palmer’ replaced by ‘3rd.—Palmer’.
Pg 95: ‘to wordly success’ replaced by ‘to worldly success’.
Pg 153: ‘worthy grammarians’ replaced by ‘worthy of grammarians’.
Pg 190: ‘Holdernesse—Murray’ replaced by ‘Holderness—Murray’.
Pg 204: ‘upon the recal’ replaced by ‘upon the recall’.
Pg 256: ‘consume the propriety’ replaced by ‘consume the property’.
Pg 280: ‘revenge offerred’ replaced by ‘revenge offered’.
Pg 345: ‘2.—A ’ replaced by ‘2d.—A ’.
Pg 383: ‘fidler, Nero’ replaced by ‘fiddler, Nero’.
Pg 394: ‘the Mississipi’ replaced by ‘the Mississippi’.
Pg 400: ‘as Aid-de-camp’ replaced by ‘as Aide-de-camp’.
Pg 414: ‘all the coolurs’ replaced by ‘all the colours’.
Pg 445: ‘being confidente of’ replaced by ‘being confidante of’.
Footnote [21]: ‘and and in 1745’ replaced by ‘and in 1745’.
Footnote [30]: ‘Stafford in 1706’ replaced by ‘Stafford in 1786’.
Footnote [48]: ‘been prefered to’ replaced by ‘been preferred to’.
Footnote [98]: ‘been Aid-de-camp’ replaced by ‘been Aide-de-camp’.](https://image.slidesharecdn.com/28193960-250517145457-abaa2391/85/Modeling-Bounded-Rationality-Ariel-Rubinstein-73-320.jpg)
Modeling Bounded Rationality Ariel Rubinstein
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- 7. Zeuthen Lecture Book Series Karl Gunnar Persson, editor Modeling Bounded Rationality, Ariel Rubinstein
- 8. Modeling Bounded Rationality Ariel Rubinstein The MIT Press Cambridge, Massachusetts London, England
- 9. © 1998 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 Palatino using Ventura Publisher under Windows 95 by Wellington Graphics. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Rubinstein, Ariel. Modeling bounded rationality / Ariel Rubinstein. p. cm. — (Zeuthen lecture book series) Includes bibliographical references (p. ) and index. ISBN 0-262-18187-8 (hardcover : alk. paper). — ISBN 0-262-68100-5 (pbk. : alk. paper) 1. Decision-making. 2. Economic man. 3. Game theory. 4. Rational expectations (Economic theory) I. Title. II. Series. HD30.23.R83 1998 97-40481 153.8′3—dc21 CIP
- 10. Contents Series Foreword ix Preface xi Introduction 1 1 “Modeling” and “Bounded Rationality” 1 2 The Aim of This Book 2 3 The State of the Art 3 4 A Personal Note 5 5 Bibliographic Notes 5 1 Bounded Rationality in Choice 7 1.1 The “Rational Man” 7 1.2 The Traditional Economist’s Position 10 1.3 The Attack on the Traditional Approach 13 1.4 Experimental Evidence 16 1.5 Comments 21 1.6 Bibliographic Notes 23 1.7 Projects 24 2 Modeling Procedural Decision Making 25 2.1 Motivation 25 2.2 Preparing the Tools: Similarity Relations 28 2.3 A Procedure of Choice Between Vectors 29 2.4 Analysis 31 2.5 Case-Based Theory 34 2.6 Bibliographic Notes 37 2.7 Projects 37
- 11. 3 Modeling Knowledge 41 3.1 Knowledge and Bounded Rationality 41 3.2 Information Structure 41 3.3 The Set-Theoretical Deªnition of Knowledge 46 3.4 Kripke’s Model 48 3.5 The Impact of the Timing of Decisions and Having More Information 52 3.6 On the Possibility of Speculative Trade 56 3.7 Bibliographic Notes 60 3.8 Projects 61 4 Modeling Limited Memory 63 4.1 Imperfect Recall 63 4.2 An Extensive Decision Making Model with Imperfect Information 64 4.3 Perfect and Imperfect Recall 68 4.4 Time Consistency 70 4.5 The Role of Randomization 75 4.6 The Multiselves Approach 78 4.7 On the Problematics of Using the Model 81 4.8 Bibliographic Notes 84 4.9 Projects 84 5 Choosing What to Know 87 5.1 Optimal Information Structures 87 5.2 What Is “High” and What Is “Low”? 89 5.3 Manipulating Informational Restrictions 93 5.4 Perceptrons 100 5.5 Bibliographic Notes 104 5.6 Projects 104 6 Modeling Complexity in Group Decisions 107 6.1 Introduction 107 6.2 The Model of a Team 108 6.3 Processing Information 113 6.4 Aggregating Preferences 117 6.5 Bibliographic Notes 119 6.6 Projects 120 vi Contents
- 12. 7 Modeling Bounded Rationality in Games 121 7.1 Introduction 121 7.2 Interaction Between Luce Players 122 7.3 A Game with Procedural Rational Players 124 7.4 Limited Foresight in Extensive Games 129 7.5 Bibliographic Notes 135 7.6 Projects 135 8 Complexity Considerations in Repeated Games 137 8.1 Introduction 137 8.2 The Model of Repeated Games: A Brief Review 138 8.3 Strategies as Machines in Inªnitely Repeated Games 143 8.4 Complexity Considerations in Repeated Games 149 8.5 The Structure of Machine Game Equilibria 152 8.6 Repeated Extensive Games 159 8.7 Concluding Remarks 161 8.8 Bibliographic Notes 163 8.9 Projects 163 9 Attempts to Resolve the Finite Horizon Paradoxes 165 9.1 Motivation 165 9.2 Implementation of Strategies by Machines 166 9.3 Counting Is Costly 168 9.4 Bounded Capability to Count 169 9.5 Machines Also Send Messages 170 9.6 The ε-Equilibrium Approach: A Deviation Is Costly 172 9.7 Conclusion 173 9.8 Bibliographic Notes 174 9.9 Projects 174 10 Computability Constraints in Games 175 10.1 Introduction 175 10.2 Informal Results on Computability 178 10.3 Is There a Rational Player? 181 10.4 Turing Machine Game 184 10.5 Bibliographic Notes 185 10.6 Projects 185 Contents vii
- 13. 11 Final Thoughts 187 11.1 Simon’s Critique 187 11.2 Response 190 References 195 Index 203 viii Contents
- 14. Series Foreword The Zeuthen Lectures offer a forum for leading scholars to develop and synthesize novel results in theoretical and applied economics. They aim to present advances in knowledge in a form accessible to a wide audience of economists and advanced students of econom- ics. The choice of topics will range from abstract theorizing to economic history. Regardless of the topic, the emphasis in the lec- ture series will be on originality and relevance. The Zeuthen Lec- tures are organized by the Institute of Economics, University of Copenhagen. The lecture series is named after Frederik Zeuthen, a former professor at the Institute of Economics, and it is only appropriate that the ªrst Zeuthen lecturer is Ariel Rubinstein, who has reªned and developed a research program to which Frederik Zeuthen made important initial contributions. Karl Gunnar Persson
- 16. Preface This book is a collection of notes I have developed over the last eight years and presented in courses and lectures at the London School of Economics (1989), Hebrew University (1989), University of Pennsylvania (1990), Columbia University (1991), Princeton Uni- versity (1992, 1995), University of Oslo (1994), Paris X (1995), Ober- wesel (1995), New York University (1996), and my home university, Tel Aviv (1990, 1994). I completed writing the book while I was a visiting scholar at the Russell Sage Foundation, New York. A pre- liminary version was presented as the Core Lectures at Louvain- La-Neuve in October 1995; this version served as the basis for my Zeuthen Lectures at the University of Copenhagen in December 1996. The book provides potential material for a one-term graduate course. The choice of material is highly subjective. Bibliographic notes appear at the end of each chapter. The projects that follow those notes contain speculative material and ideas that the reader should consider with caution. My thanks to my friends Bart Lipman and Martin Osborne for their detailed comments and encouragement. I am grateful to all those students, especially Dana Heller, Rani Spigeler, and Ehud Yampuler, who commented on drafts of several chapters, to Nina Reshef, who helped edit the English-language manuscript, to Dana Heller, who prepared the index, and to Gregory McNamee who copyedited the manuscript.
- 18. Introduction 1 “Modeling” and “Bounded Rationality” The series of lectures that constitute the chapters in this book con- cerns modeling bounded rationality. The choice of the title “modeling bounded rationality” rather than “models of bounded rationality” or “economic models of bounded rationality” emphasizes that the focus is not on substantive conclusions derived from the models but on the tools themselves. As to the term bounded rationality, putting fences around a ªeld is often viewed as a picky activity. Nonetheless, it is important in this case in that the term has been used in many ways, sometimes just to say that we deal with incom- plete (or bad) models. Lying within the domain of this investigation are models in which elements of the process of choice are embedded explicitly. Usually, economic models do not spell out the procedures by which decisions of the economic units are made; here, we are interested in models in which procedural aspects of decision mak- ing are explicitly included. I will not touch the growing literature on evolutionary economics for three reasons. First, the topic of evolutionary/learning models deserves a complete and separate series of lectures. Second, the mathematical methods involved in models of evolutionary econom- ics are quite different than those used here. Third, and most impor- tant, I want to place an admittedly vague dividing line between the
- 19. two bodies of research. Within the scope of our discussion, I wish to include models in which decision makers make deliberate deci- sions by applying procedures that guide their reasoning about “what” to do, and probably also about “how” to decide. In contrast, evolutionary models treat agents as automata, merely responding to changing environments, without deliberating about their decisions. 2 The Aim of This Book The basic motivation for studying models of bounded rationality springs from our dissatisfaction with the models that adhere to the “perfect rational man” paradigm. This dissatisfaction results from the strong tension arising from a comparison of the assumptions made by economic modelers about “perfect rationality” with obser- vations about human behavior. This situation would be much less disturbing if we were able to perceive microeconomic models as “miraculous machines” that produce empirical linkages between economic parameters. I doubt that this is the case. I adhere to the view that the main objective of economic theory is to deduce inter- esting relationships between concepts that appear in our reasoning on interactive situations. Adopting this approach makes it impor- tant to examine the plausibility of the assumptions, and not only the conclusions. The emphasis on the modeling process, rather than on the sub- stance, does not diminish the importance of the goal, which is to construct models that will be useful tools in providing explanations of economic (or other) phenomena that could not otherwise be explained (ideally comparable to results such as those achieved by Spence’s signaling model). The following are examples of basic in- tuitions that await proper explanation: • Advertising is an activity that is supposed to inºuence an eco- nomic agent’s decisions not only by supplying information and 2 Introduction
- 20. changing preferences, but also by inºuencing the way decisions are made. • Decision makers are not equally capable of analyzing a situation even when the information available to all of them is the same. The differences in their economic success can be attributed to these differences. • Many social institutions, like standard contracts and legal proce- dures, exist, or are structured as they are, in order to simplify decision making. 3 The State of the Art Dissatisfaction with classical theory and attempts to replace the basic model of rational man with alternative decision models are not new. Ideas of how to model bounded rationality have been lurking in the economics literature for many years. Papers written by Herbert Simon as early as the mid-1950s have inspired many proposals in this vein. Although Simon received worldwide recog- nition for his work, only recently has his call affected mainstream economic theory. Only a few of the modeling tools we will discuss here have been applied to economic settings. What is more, the usefulness of these models is still far from being established. In fact, I have the impression that many of us feel that the attempts to model bounded rationality have yet to ªnd the right track. It is difªcult to pinpoint any economic work not based on fully rational microeconomic behavior that yields results as rich, deep, and in- teresting as those achieved by standard models assuming full rationality. I consider these to be the three fundamental obstacles we have to overcome: • The construction of pertinent new theories of choice. We have clear, casual, and experimental observations that indicate Introduction 3
- 21. systematic deviations from the rational man paradigm. We look for models that will capture this evidence. • The reªnement of the notion of choice. Decision makers also make decisions about how and when to decide; we look for models that will relate to such decisions as well. • The transformation of the notion of equilibrium. Current solution concepts, especially those concerning strategic interactions and ra- tional expectations, are based on an implicit assumption that indi- viduals know the prevailing equilibrium. But decision makers also have to make inferences about the environment in which they operate, an activity dependent on their ability to analyze the situ- ation. We look for models in which the making of inferences will be the basic activity occupying the decision maker. The evaluation that very little has been achieved makes one wonder whether it is at all possible to construct interesting models without the assumption of substantive rationality. Is there some- thing fundamental that prevents us from constructing useful bounded rationality models, or have we been “brainwashed” by our conventional models? One intriguing idea is that substantive rationality is actually a constraint on the modeler rather than an assumption about the real world. The rationality of the decision maker can be seen as the minimal discipline to be imposed on the modeler. Our departure from the rational man paradigm represents a removal of those chains. However, there are an inªnite number of “plausible” models that can explain social phenomena; without such chains we are left with a strong sense of arbitrariness. Al- though I have nothing to contribute to the discussion of this issue, I think it is worth mentioning. In any case, even if one believes like Kenneth Arrow (1987), that “there is no general principle that prevents the creation of an eco- nomic theory based on other hypotheses than that of rationality,” 4 Introduction
- 22. the only way to prove the power of including the procedural as- pects of decision making in speciªc economic theories is by actually doing so. This is the challenge for scholars of “bounded rationality.” 4 A Personal Note This book is not intended to be a triumphal march of a ªeld of research but a journey into the dilemmas faced by economic theo- rists attempting to expand the scope of the theory in the direction of bounded rationality. Some of the ideas I discuss are only just evolving. By choosing such a topic for this series of lectures, I am taking the risk that my presentation will be less clear, less persuasive, and much more speculative than if I were discussing a more established topic. However, these attributes can also be advantageous, espe- cially to the students among the readers. Newcomers to economic theory are in the best position to pursue themes that require imagi- nation and invention. Students have a major advantage over us teachers in that, they are not (yet) indoctrinated by the body of literature so ªrmly rooted in the notion of rational man. Finally, within the wide borders I have tried to draw, the selection of material is strongly biased toward topics with which I have been personally involved, either as an author or as an interested ob- server. I have not tried to be objective in the choice of topics, nor have I tried to summarize views held by “the profession.” In this respect, the book is personal and aims at presenting my own views and knowledge of the subject. 5 Bibliographic Notes Some of the methodological issues regarding the construction of new models on hypotheses other than that of rationality are Introduction 5
- 23. discussed in Hogarth and Reder (1987). In particular, the reader is encouraged to review the four articles by Arrow, Lucas, Thaler, and Tversky and Kahneman. Selten (1989) proposes an alternative view of bounded rationality and provides an overview of some of the issues discussed up to the late 1980s. For other views on modeling rational and bounded- rational players, see Binmore (1987, 1988) and Aumann (1996). Lip- man (1995a) contains a short survey covering some of the topics discussed in this book. 6 Introduction
- 24. 1 Bounded Rationality in Choice 1.1 The “Rational Man” In economic theory, a rational decision maker is an agent who has to choose an alternative after a process of deliberation in which he answers three questions: • “What is feasible?” • “What is desirable?” • “What is the best alternative according to the notion of desirabil- ity, given the feasibility constraints?” This description lacks any predictive power regarding a single decision problem, inasmuch as one can always explain the choice of an alternative, from a given set, as an outcome of a process of deliberation in which that outcome is indeed considered the best. Herein lies a key assumption regarding the rational man: The op- eration of discovering the feasible alternatives and the operation of deªning the preferences are entirely independent. That is, if the decision maker ranks one alternative above another when facing a set of options that includes both, he will rank them identically when encountering any other decision problem in which these two alter- natives are available. Formally, the most abstract model of choice refers to a decision maker who faces choices from sets of alternatives that are subsets
- 25. of some “grand set” A. A choice problem, A, is a subset of A; the task of the decision maker is to single out one element of A. To conclude, the scheme of the choice procedure employed by the rational decision maker is as follows: (P-1) The rational man The primitive of the procedure is a prefer- ence relation i over a set A. Given a choice problem A ⊆ A, choose an element x* in A that is i-optimal (that is, x* i x for all x ∈ A). For simplicity, it will be assumed through the rest of this chapter that preferences are asymmetric (i.e., if a i b then not b i a). Thus, the decision maker has in mind a preference relation, i, over the set of alternatives A. Facing a problem A, the decision maker chooses an element in the set A, denoted by Ci(A), satisfying Ci(A) i x for all x ∈ A. Sometimes we replace the preference relation with a utility function, u: A → R, with the understanding that u(a) ≥ u(a′) is equivalent to a i a′. (Of course, some assumptions are needed for establishing the equivalence between the existence of preferences and the existence of a utility function). Let us uncover some of the assumptions buried in the rational man procedure: • Knowledge of the problem The decision maker has a clear picture of the choice problem he faces: he is fully aware of the set of alternatives from which he has to choose (facing the problem A, the decision maker can choose any x ∈ A, and the chosen x* cannot be less preferred than any other x ∈ A). He neither invents nor discov- ers new courses of actions (the chosen x* cannot be outside the set A). • Clear preferences The decision maker has a complete ordering over the entire set of alternatives. • Ability to optimize The decision maker has the skill necessary to make whatever complicated calculations are needed to discover his optimal course of action. His ability to calculate is unlimited, and 8 Chapter 1
- 26. he does not make mistakes. (The simplicity of the formula “maxa∈Au(a)” is misleading; the operation may, of course, be very complex.) • Indifference to logically equivalent descriptions of alternatives and choice sets The choice is invariant to logically equivalent changes of de- scriptions of alternatives. That is, replacing one “alternative” with another “alternative” that is “logically equivalent” does not affect the choice. If the sets A and B are equal, then the choice from A is the same as the choice from B. Comment Often the preferences on a set of alternatives are derived from a more detailed structure. For example, it is often the case that the decision maker bases his preferences, deªned on A, on the calculation of consequences yielded from A. That is, he perceives a set of possible consequences, C. He has a preference relation over C (probably represented by a numerical function, V: C → R). He perceives the causal dependence of a consequence on a chosen alternative, described by a consequence function, f: A → C. He then chooses, from any set A ⊆ A, the alternative in A that yields the best consequence—that is, he solves the optimization problem maxa∈AV(f(a)). In other words, the preference relation on A is in- duced from the composition of the consequence function and the preference relation on C. In order to deal with the situation in which the decision maker assumes that the connection between the action and the conse- quence has elements of uncertainty, we usually enrich the model. A space of states, ⍀, is added. One element of ⍀ represents the list of exogenous factors that are relevant to the decision maker’s inter- ests and are beyond his control. The consequence function is taken to depend on ⍀ as well; that is, f: A × ⍀ → C. Each action a ∈ A corresponds to an “act” (a function that speciªes an element in C for each state in ⍀) a(ω) = f(a, ω). The preference relation on A is induced from a preference on “acts.” A choice problem now is a Bounded Rationality in Choice 9
- 27. pair (A, Ω) where A ⊆ A is the set of alternatives, whereas Ω ⊆ ⍀ is the set of states not excluded by the information the decision maker receives. Usually, it is taken that the rational man’s choice is based on a belief on the set ⍀, a belief he updates by the Bayesian formula whenever he is informed that an event Ω ⊆ ⍀ happens. Note that underlying this structure, both with and without un- certainty, is the assumption that the decision maker clearly per- ceives the action–consequence relationship. 1.2 The Traditional Economist’s Position Economists have often been apologetic about the assumption that decision makers behave like the “rational man.” Introspection sug- gests that those assumptions are often unrealistic. This is probably the reason why economists argued long ago that the rational man paradigm has to be taken less literally. The “traditional” argument is roughly this: In economics, we are mainly interested in the behavior of the decision maker and not in the process leading to his decision. Even if the decision maker does not behave in the manner described by the rational man procedure, it still may be the case that his behavior can be described as if he follows such a procedure. This is sufªcient for the purpose of economics. A good demonstration of this “as if” idea is given in consumer theory. Imagine a consumer who operates in a world with two goods, 1 and 2, who has budget I, and who faces prices p1 and p2. Assume that the consumer allocates the fraction α of his income to good 1 and (1 − α) of the income to good 2 (for every I, p1 and p2). This behavior rule may be the result of activating a rule of thumb. Nonetheless, it may still be presented as if it is the outcome of the consumer’s maximization of the utility function x1 αx2 1−α. Let us return to the general framework. The following argument was designed to support the traditional point of view. Consider a 10 Chapter 1
- 28. decision maker whose behavior regarding choices from subsets of the set A is described by a function C whose domain is the set of all non-empty subsets of A and whose range is the set A. The ele- ment C(A) is interpreted as the decision maker’s choice whenever he confronts the decision problem A. For every A, C(A) ∈ A. (Note that for simplicity, and in contrast to some of the literature, it is required here that C(A) is a single element in A and not a subset of A). We now come to an important necessary and sufªcient condition for a choice function to be induced by a decision maker who be- haves like a rational man. It is said that the decision maker’s behav- ior function C satisªes the consistency condition (sometimes referred to as the “independence of irrelevant alternatives”) if for all A1 ⊆ A2 ⊆ A, if C(A2) ∈ A1 then C(A1) = C(A2). That is, if the element chosen from the large set (A2) is a member of the smaller set (A1), then the decision maker chooses this element from the smaller set as well. It is easy to see that C is consistent if and only if there exists a preference relation i over A such that for all A ⊆ A, C(A) is the i-maximal element in A. Proof Of course, if for every subset A the element C(A) is the i-maximal element in A, then the choice function C satisªes the consistency condition. Assume that C satisªes the consistency con- dition. Deªne a preference relation i by a i b if a = C({a, b}). We ªrst verify that i is transitive. If a i b and b i c, then a = C({a, b}) and b = C({b, c}). Then C({a, b, c}) = a; otherwise, the consistency condition is violated with respect to one of the sets, {a, b} or {b, c}. Therefore, by the consistency condition, C({a, c}) = a; that is, a i c. To verify that for every set A, C(A) is the i-maximal element in A, notice that for any element a ∈ A, {a, C(A)} ⊆ A and because C satisªes the consistency condition, C({a, C(A)}) = C(A), therefore by deªnition of i, C(A) i a. The conclusion from this simple analysis is that choice functions that satisfy the consistency condition, even if they are not derived Bounded Rationality in Choice 11
- 29. from a rational man procedure, can be described as if they are derived by some rational man. The signiªcance of this result de- pends on the existence of plausible procedures that satisfy the consistency condition even though they do not belong to the scheme (P-1) of choosing a maximal element. One such classic example is what Simon termed the satisªcing procedure: (P-2) The primitives of the procedure are O, an ordering of the set A, and a set S ⊆ A (as well as a tie-breaking rule; see below). For any decision problem A, sequentially examine the alternatives in A, according to the ordering O, until you confront an alternative that is a member of the set S, the set of “satisfactory” alternatives. Once you ªnd such an element, stop and choose it. For the case where no element of A belongs to S, use the tie-breaking rule that satisªes the consistency requirement (such as choosing the last element in A). Any procedure within the scheme (P-2) satisªes the consistency condition. To verify this, suppose that A1 ⊆ A2 and C(A2) ∈ A1, that is, C(A2) is the ªrst (according to the ordering O) satisfactory alter- native in A2, then it is also the ªrst satisfactory alternative in the subset A1. If C(A2) ∉ S, then A1 also does not include any element belonging to S, and because the tie-breaking rule satisªes the con- sistency condition, we have C(A2) = C(A1). A special case of (P-2) is one where the set S is derived from two parameters, a function V and a number v*, so that S = {a ∈ A | V(a) ≥ v*}. The function V assigns a number to each of the potential alter- natives, whereas v* is the aspiration level. The decision maker searches for an alternative that satisªes the condition that its value be above the aspiration level. For example, in the “ªnding a worker” problem, the set of alternatives is the set of candidates for a job, the ordering might be the alphabetical ordering of the candi- dates’ names or an enumeration of their social security numbers, V(a) may be the grade that candidate a gets in a test, and v* is the 12 Chapter 1
- 30. required minimal grade. Note that instead of having a maximiza- tion problem, “maxa∈AV(a),” the decision maker who follows (P-2) solves what seems to be a simpler problem: “Find an a ∈ A for which V(a) ≥ v*.” 1.3 The Attack on the Traditional Approach The fact that we have found a family of plausible procedures that are not similar to the rational man procedure yet consistent with rationality provides support for the traditional economic position. However, the problem with this position is that it is difªcult to pro- pose additional procedures for inducing consistent choice functions. To appreciate the difªculties in ªnding such examples, note that in (P-2) the ordering in which the alternatives are examined is ªxed independent of the particular choice set. However, if the ordering by which the alternatives are examined is dependent on the set, a clash with the consistency condition arises. Consider the following decision procedure scheme: (P-3) The primitives of the procedure are two different orderings of A, O1 and O2, a natural number n*, and a set S (plus a tie-breaking rule). For a choice problem A, employ (P-2) with the ordering O1 if the number of elements in A is below n* and with O2 if the number of alternatives in A is above n*. It is easy to see that a procedure within the scheme (P-3) will often not satisfy the consistency condition. The fact that an element is the ªrst element, by the ordering O2, belonging to S in a “large” set A2 does not guarantee that it is the ªrst, by the other ordering O1, belonging to S in a “smaller” subset A1. In the rest of this section, we will refer to three motives often underlying procedures of choice that may conºict with the rational man paradigm: “framing effects,” the “tendency to simplify prob- lems,” and the “search for reasons.” In the next section, we present Bounded Rationality in Choice 13
- 31. evidence from the psychological literature that conªrms that these motives systematically appear in human choice situations. Framing Effects By framing effects, we refer to phenomena rooted solely in the way that the decision problem is framed, not in the content of the choice problem. Recall that a choice problem is deªned as a choice of an element from a set. In practice, this set has to be described; the way that it is described may affect the choice. For example, the model does not allow distinct choices between the lists of alternatives (a, b, b) and (a, a, a, b, b) because the sets {a, b, b} and {a, a, a, b, b} are identical. If, however, the language in which the sets are speci- ªed is a language of “lists,” then the following procedural scheme is well deªned: (P-4) Choose the alternative that appears in the list most often (and apply some rule that satisªes the consistency condition for tie- breaking). Of course, such a procedure does not satisfy the consistency condi- tion. It does not even induce a well-deªned choice function. The Tendency to Simplify Decision Problems Decision makers tend to simplify choice problems, probably as a method of saving deliberation resources. An example of a proce- dure motivated by the simpliªcation effort is the following: (P-5) The primitives of the procedure are an ordering O and a preference relation i on the set A. Given a decision problem A, pick the ªrst and last elements (by the ordering O) among the set A and choose the better alternative (by the preference relation i) between the two. In this case, the decision maker does not consider all the elements in A but only those selected by a predetermined rule. From this 14 Chapter 1
- 32. sample, he then chooses the i-best alternative. If the alternatives are a, b, and c, the preference ranking is b Ɑ a Ɑ c, and the ordering O is alphabetical, then the alternative a will be chosen from among {a, b, c} and b from among {a, b}, a choice conºicting with the consistency condition. (Try to verify the plausibility of this proce- dural motive by examining the method by which you make a choice from a large catalog.) The Search for Reasons Choices are often made on the basis of reasons. If the reasons are independent of the choice problem, the fact that the decision maker is motivated by them does not cause any conºict with rationality. Sometimes, however, the reasons are “internal,” that is, dependent on the decision problem; in such a case, conºict with rationality is often unavoidable. For example, in the next scheme of decision procedures, the decision maker has in mind a partial ordering, D, deªned on A. The interpretation given to a D b is that a “clearly dominates” b. Given a decision problem, A, the decision maker selects an alternative that dominates over more alternatives than does any other alternative in the set A. (P-6) The primitive is a partial ordering D. Given a problem A, for each alternative a ∈ A, count the number N(a) of alternatives in A that are dominated (according to the partial ordering D). Select the alternative a* so that N(a*) ≥ N(a) for all a ∈ A (and use a rule that satisªes the consistency requirement for tie-breaking). By (P-6) a reason for choosing an alternative is the “large number of alternatives dominated by the chosen alternative.” This is an “internal reason” in the sense that the preference of one alternative over another is determined by the other elements in the set. Of course, (P-6) often does not satisfy the consistency condition. Bounded Rationality in Choice 15
- 33. 1.4 Experimental Evidence Economic theory relies heavily on intuitions and casual observa- tions of real life. However, despite being an economic theorist who rarely approaches data, I have to agree that an understanding of the procedural aspects of decision making should rest on an em- pirical or experimental exploration of the algorithms of decision. Too many routes diverge from the rational man paradigm, and the input of experimentation may offer some guides for moving onward. The refutation of the rational man paradigm by experimental evidence is not new. As early as 1955 Simon asserted, “Recent developments in economics . . . have raised great doubts as to whether this schematized model of economic man provides a suit- able foundation on which to erect a theory—whether it be a theory of how ªrms do behave or of how they ‘should’ rationally behave.” Since then, a great deal of additional experimental evidence has been accumulated, mainly by psychologists. Of particular interest is the enormous literature initiated by Daniel Kahneman, Amos Tversky, and their collaborators. We now have a fascinating com- pilation of experimental data demonstrating the circumstances un- der which rationality breaks down and other patterns of behavior emerge. I will brieºy dwell on a few examples that seem to me to be especially strong in the sense that they not only demonstrate a deviation from the rational man paradigm, but they also offer clues about where to look for systematic alternatives. The order of the examples parallels that of the discussion in the previous section. Framing Effects A rich body of literature has demonstrated circumstances under which the assumption that two logically equivalent alternatives are treated equally, does not hold. A beautiful demonstration of the 16 Chapter 1
- 34. framing effect is the following experiment taken from Tversky and Kahneman (1986): Subjects were told that an outbreak of a disease will cause six hundred people to die in the United States. Two mutually exclusive programs, yielding the following results, were considered: A. two hundred people will be saved. B. With a probability of 1/3, six hundred people will be saved; with a probability of 2/3, none will be saved. Another group of subjects were asked to choose between two pro- grams, yielding the results: C. four hundred people will die. D. With a probability of 1/3 no one will die; with a probability of 2/3 all six hundred will die. Although 72 percent of the subjects chose A from {A, B}, 78 percent chose D from {C, D}. This occurred in spite of the fact that any reasonable man would say that A and C are identical and B and D are identical! One explanation for this phenomenon is that the description of the choice between A and B in terms of gains prompted risk aversion, whereas the description in terms of losses prompted risk loving. Framing effects pose the most problematic challenges to the ra- tionality paradigm. Their existence leads to the conclusion that an alternative has to appear in the model with its verbal description. Doing so is a challenging task beyond our reach at the moment. The Tendency to Simplify a Problem The following experiment is taken from Tversky and Kahneman (1986). Consider the lotteries A and B. Both involve spinning a roulette wheel. The colors, the prizes, and their probabilities are speciªed below: Bounded Rationality in Choice 17
- 35. A Color white red green yellow Probability (%) 90 6 1 3 Prize ($) 0 45 30 −15 B Color white red green yellow Probability (%) 90 7 1 2 Prize ($) 0 45 −10 −15 Facing the choice between A and B, about 58 percent of the subjects preferred A. Now consider the two lotteries C and D: C Color white red green blue yellow Probability (%) 90 6 1 1 2 Prize ($) 0 45 30 −15 −15 D Color white red green blue yellow Probability (%) 90 6 1 1 2 Prize ($) 0 45 45 −10 −15 The lottery D dominates C, and all subjects indeed chose D. How- ever, notice that lottery B is, in all relevant respects, identical to lottery D (red and green in D are combined in B), and that A is the same as C (blue and yellow are combined in A). What happened? As stated, decision makers try to simplify prob- lems. “Similarity” relations are one of the basic tools they use for this purpose. When comparing A and B, many decision makers went through the following steps: 1. 6 and 7 percent, and likewise 2 and 3 percent, are similar; 2. The data about the probabilities and prizes for the colors white, red, and yellow is more or less the same for A and B, and 3. “Cancel” those components and you are left with comparing a gain of $30 with a loss of $10. This comparison, favoring A, is the decisive factor in determining that the lottery A is preferred to B. 18 Chapter 1
- 36. By the way, when I conducted this experiment in class, there were (good!) students who preferred C over D after they preferred A over B. When asked to justify this “strange” choice, they pointed out that C is equivalent to A and D is equivalent to B and referred to their previous choice of A! These students demonstrated another com- mon procedural element of decision making: The choice in one problem is made in relation to decisions made previously in re- sponse to other problems. The Search for Reasons In the next example (following Huber, Payne, and Puto [1982]), (x, y) represents a holiday package that contains x days in Paris and y days in London, all offered for the same price. All subjects agree that a day in London and a day in Paris are desirable goods. Denote, A = (7, 4), B = (4, 7), C = (6, 3) and D = (3, 6). Some of the subjects were requested to choose between the three packages A, B, and C; others had to choose between A, B, and D. The subjects exhibited a clear tendency to choose A out of the set {A, B, C} and to choose B out of the set {A, B, D}. Obviously, this behavior is not consistent with the behavior of a “rational man.” Given the universal prefer- ence of A over C and of B over D, the preferred element out of {A, B} should be chosen from both {A, B, C} and {A, B, D}. Once again, the beauty of this example is not its contradiction of the rational man paradigm but its demonstration of a procedural element that often appears in decision making. Decision makers look for reasons to prefer A over B. Sometimes, those reasons relate to the decision problem itself. In the current example, “dominating another alternative” is a reason to prefer one alternative over the other. Reasons that involve relationships to other alternatives may therefore conºict with the rational man paradigm. Another related, striking experiment was conducted by Tversky and Shaªr (1992). A subject was shown a list of twelve cards. Each card described one prize. Then the subject was given two cards and asked whether he wanted to pay a certain fee for getting a third Bounded Rationality in Choice 19
- 37. card from the deck. If he did not pay the fee, he had to choose one of the two prizes appearing on the cards in his hand. If he chose to pay the fee, he would have three cards, the two he had originally been dealt and the third he would now draw; he would then have to choose one among the three prizes. The different conªgurations of prizes which appeared on the two cards given to the subjects were as follows: 1. Prizes A and B, where A dominates B; 2. Prizes A and C, where A and C are such that neither dominates the other. A signiªcantly lower percentage of subjects chose to pay the fee in face of (1) than in face of (2). Thus, once the decision maker has an “internal” reason (the domination of one over another alternative) to choose one of the alternatives, he is no longer interested in enriching the set of options. Many subjects, when confronted with conºict while making a choice, were ready to pay a fee for receipt of a reason that would help them to make the choice. Remark One often hears criticism among economists of the experi- ments done by psychologists. Critics tend to focus blame on the fact that in the typical experimental design, subjects have no sufªcient incentive to make the conduct of the experiment or its results relevant for economics—the rewards given were too small and the subjects were not trained to deal with the problems they faced. I disagree with this criticism for the following reasons: • The experiments, I feel, simply conªrmed solid intuitions origi- nating from our own thought experiments. • Many of the real-life problems we face entail small rewards and many of our daily decisions are made in the context of nonrecurring situations. • When considering human behavior regarding “major” decisions, we observe severe conºicts with rationality as well. To illustrate, 20 Chapter 1
- 38. Benartzi and Thaler (1995) discuss a survey regarding the choices made by university professors on the allocation of their investments in stocks and bonds within the TIAA-CREF pension fund. Although this decision is among the more important annual ªnancial deci- sions made in the life of an American professor, the authors observe that the majority of investors divide their money between the two options in a manner almost impossible to rationalize as optimal in any form. To summarize this section, we have reviewed several experiments that demonstrate motives for choice that are inconsistent with the rational man paradigm. The number of experiments undertaken, the clarity of the motives elicited, and their conªrmation by our own “thought experiments” do not allow us to dismiss these ex- periments as curiosities. 1.5 Comments Procedural and Substantive Rationality The observation that behavior is not rational does not imply that it is chaotic. As already stated, the experiments discussed in the previous section hint at alternative elements of decision-making procedures that may establish the foundations for new economic models. Simon distinguishes between substantive rationality and pro- cedural rationality: on one hand, substantive rationality refers to behavior that “is appropriate to the achievement of given goals within the limits imposed by given conditions and constraints”; on the other hand, “behavior is procedurally rational when it is the outcome of appropriate deliberation.” That is, procedurally rational behavior is the outcome of some strategy of reasoning, whereas irrational behavior is an outcome of impulsive responses without adequate intervention of thought. In this book, we will drop the Bounded Rationality in Choice 21
- 39. assumption of substantive rationality but retain that of procedural rationality. Mistakes vs. Bounded Rationality Some have claimed that the phenomena demonstrated in the above experiments are uninteresting inasmuch as they express “mistakes” that disappear once the subjects learn of their existence. They con- tend that economists are not interested in traders who believe that 1 + 1 = 3; similarly, they should not be interested in agents who are subject to framing affects. I beg to differ. Labeling behavior as “mistakes” does make the behavior uninteresting. If there are many traders in a market who calculate 1 + 1 = 3, then their “mistake” may be economically relevant. The fact that behavior may be changed after the subjects have been informed of their “mistakes” is of interest, but so is behavior absent the revelation of mistakes because, in real life, explicit “mistake-identiªers” rarely exist. Rationalizing on a Higher Level As economists raised on the rational man paradigm, our natural response to the idea of describing a decision maker by starting from a decision procedure is akin to asking the question, “Where does the procedure come from?” One method of rationalizing the use of decision procedures inconsistent with the rational man paradigm is by expanding the context to that of a “richer” decision problem in which additional considerations (such as the cost of deliberation) are taken into account. Under such conditions, one may try to argue that what seems to be irrational is actually rational. In regard to the satisªcing procedure (P-2), for example, such a question was asked and answered by Simon himself. Simon proposed a search 22 Chapter 1
- 40. model with costs in order to derive the use of the procedure and to provide an explanation for the determination of the aspiration value. This is an interesting research program, but I do not see why we must follow it. Alternatively, we may treat the level of aspiration simply as one parameter of the decision maker’s problem (similar to the coefªcient in a Cobb-Douglas utility function in standard consumer theory), a parameter that is not selected by the decision maker but is given among his exogenous characteristics. We should probably view rationality as a property of behavior within the model. The fact that having an aspiration level is justiªable as rational behavior in one model does not mean that we can consider that behavior as rational within any other model. 1.6 Bibliographic Notes The pioneering works on bounded rationality are those of Herbert Simon. See, for example, Simon (1955, 1956, 1972, and 1976). (About the ªrst two papers Simon wrote: “If I were asked to select just two of my publications in economics for transmission to another galaxy where intelligent life had just been discovered, these are the two I would choose.”) All four papers are reprinted in Simon (1982). For the foundation of choice theory see, for example, Kreps (1988). The material on experimental decision theory has been surveyed recently in Camerer (1994) and Shaªr and Tversky (1995). For a more detailed discussion of the framing effect see Tversky and Kahneman (1986). The experiments on reason-based choice are summarized in Shaªr, Simonson, and Tversky (1993) and Tversky and Shaªr (1992). See also Simonson (1989) and Huber, Payne, and Puto (1982). The tendency to simplify complicated problems to more manageable ones is discussed in Payne, Battman, and Johnson (1988). Bounded Rationality in Choice 23
- 41. 1.7 Projects 1. Reading Shaªr, Diamond, and Tversky (1997) reports on a “framing effect” in the context of economic decisions in times of inºation. Read the paper and sug- gest another context in which a similar “framing effect” may inºuence economic behavior. 2. Reading Read Benartzi and Thaler (1995) on the decision of real investors to allocate their investments between stocks and bonds. Consider the following “ex- periment.” Subjects are split into two groups. At each of the two periods of the experiment, each subject gets a ªxed income that he must invest immediately in stocks and bonds. At the end of the ªrst period, an investor has access to informa- tion about that period’s yields. A subject cashes his investments at the end of the second period. At every period, a member of the ªrst group is asked to allocate only the income he receives that period, whereas a member of the second group is asked to reallocate his entire balance at that point. Guess the two typical responses. Can such an experiment establish that investors’ behaviors are not compatible with rationality? 3. Innovative Choose one of the axiomatizations of decision making under uncer- tainly (exclude the original expected utility axiomatization) and examine the axiom from a procedural point of view. 24 Chapter 1
- 42. 2 Modeling Procedural Decision Making 2.1 Motivation In the previous chapter, I argued that experimental evidence from the psychological literature demonstrates the existence of common procedural elements that are quite distinct from those involved in the rational man’s decision-making mechanism. In this chapter, we turn to a discussion of attempts to model formally some of these elements. Note that when we model procedural aspects of decision making, we are not necessarily aiming at the construction of models of choice that are incompatible with rationality. Our research program is to model formally procedures of choice that exhibit a certain procedural element, and then to investigate whether or not such procedures are compatible with rationality. If they are, we will try to identify restrictions on the space of preferences that are compat- ible with those procedures. We now return to a motive we mentioned in Chapter 1: Decision makers attempt to simplify decision problems. For simplicity, let us focus on choice problems that contain two alternatives, each de- scribed as a vector. One way to simplify such a problem is to apply similarity notions in order to “cancel” the components of the two alternatives that are alike, and thereby to reduce the number of elements involved in the descriptions of the two alternatives. This makes the comparison less cumbersome.
- 43. To illustrate, let us look at results of an experiment, reported in Kahneman and Tversky (1982), that is similar to that of the Allais paradox. The objects of choice in the experiment are simple lotter- ies. A simple lottery (x,p) is a random variable that yields $x with probability p and $0 with probability 1 − p. Thus, each object of choice can be thought of as a vector of length 2. In the experiment, some subjects were asked to choose between: L3 = (4000, 0.2) and L4 = (3000, 0.25). Most subjects chose L3. Another group of subjects was asked to choose between: L1 = (4000, 0.8) and L2 = (3000, 1.0). The vast majority of subjects chose L2. The choices L2 from {L1, L2} and L3 from {L3, L4} do not violate ra- tionality. However, they do violate the von Neumann-Morgenstern independence axiom. To see this, notice that the lotteries L3 and L4 can be presented as compound lotteries of L1, L2, and the degenerate lottery [0], which yields the certain prize 0: L3 = 0.25L1 + 0.75[0] and L4 = 0.25L2 + 0.75[0]. Therefore, the independence axiom requires that the choice be- tween L3 and L4 be made according to the choice between L1 and L2, in striking contrast to the experimental results. The reasoning that probably guided many of the subjects was the following: When comparing L3 to L4, a decision maker faces an internal conºict due to the higher prize in L3 versus the higher probability of getting a positive prize in L4. He tries to simplify the choice problem so that one of the alternatives will be patently better. With this aim, he checks the similarities of the probabilities and the prizes that appear in the two lotteries. He considers the probability numbers 0.25 and 0.2 to be similar, in contrast to the prizes $4000 and $3000, which are not. These similarity comparisons lead him to 26 Chapter 2
- 44. “simplify” the problem by “canceling” the probabilities and making the choice between L3 and L4, based on the obvious choice between $4000 and $3000. On the other hand, when comparing L1 and L2, the decision maker cannot simplify the problem on the basis of the cancellation of similar components since neither the prob- abilities nor the prizes are perceived to be similar. He then invokes another principle, presumably risk aversion, to arrive at the supe- riority of L2. Note that the attractiveness of the vNM independence axiom is also related to its interpretation as an expression of a similar pro- cedural element. When the lotteries L3 and L4 are represented ex- plicitly in the form of the reduced lotteries 0.25L1 + 0.75[0] and 0.25L2 + 0.75[0], respectively, decision makers tend to simplify the comparison between L3 and L4 by “canceling” the possibility that the lotteries will yield the prize [0], then basing their choice on a comparison between L1 and L2, thus choosing L4. Hence, whether the lotteries are presented as simple or com- pound, it seems that a major step in the deliberation is the “cancel- lation of similar factors” and the consequent reduction of the original complex choice to a simpler one. Activating this principle when comparing L3 and L4 as simple lotteries leads to the choice of L3; activating it when the lotteries are presented as compound lotteries leads to the choice of L4. The way in which the principle of reducing the complexity of a choice is applied, therefore, depends on how the decision problem is framed. To avoid framing effects in our analysis, we will retain the format of each alternative as ªxed. Consequently, all objects of choice will be simple lotteries presented as vectors of the type (x, p). In the next three sections we will formulate and analyze a pro- cedure for choosing between pairs of such lotteries that makes explicit use of similarity relations. The presentation consists of the following stages. First we will describe a scheme of choice proce- dures between pairs of lotteries. Then we will ask two questions: Modeling Procedural Decision Making 27
- 45. 1. Does such a procedure necessarily conºict with the rational man paradigm? 2. If not, what preference relations are consistent with the procedure? But ªrst we will detour to the world of similarity relations in order to equip ourselves with the necessary tools. 2.2 Preparing the Tools: Similarity Relations In this chapter, a similarity relation is taken to be a binary relation ∼ on the set I = [0, 1] that satisªes the following properties: (S-1) Reºexivity For all a ∈ I, a ∼ a. (S-2) Symmetry For all a, b ∈ I, if a ∼ b, then b ∼ a. (S-3) Continuity The graph of the relation ∼ is closed in I × I. (S-4) Betweenness If a ≤ b ≤ c ≤ d and a ∼ d, then b ∼ c. (S-5) Nondegeneracy 0 ⬃ 1, and for all 0 < a < 1, there are b and c so that b < a < c and a ∼ b and a ∼ c. For a = 1, there is b < a so that a ∼ b. (For reasons which will soon become clear, no such require- ment is made for a = 0.) (S-6) Responsiveness Denote by a* and a * the largest and the small- est elements in the set that are similar to a. Then a* and a * are strictly increasing functions (in a) at any point at which they obtain a value different from 0 or 1. Although these axioms restrict the notion of similarity quite sig- niªcantly, I ªnd them particularly suitable when the similarity stands for a relation of the kind “approximately the same.” This does not deny that there are contexts in which the notion of simi- larity clearly does not satisfy the above axioms. For example, we say that “Luxembourg is similar to Belgium,” but we do not say 28 Chapter 2
- 46. that “Belgium is similar to Luxembourg” (see Tversky [1977]). In this example, we say that “a is similar to b” in the sense that b is a “representative” of the class of elements to which both a and b belong; this use of the term does not satisfy the symmetry condition. A leading example of a family of relations that satisªes all these assumptions is the one consisting of the λ-ratio similarity relations (with λ > 1) deªned by a ∼ b if 1/λ ≤ a/b ≤ λ. More generally, for any number λ > 1 and for every strictly increasing continuous function, H, on the unit interval, the relation a ∼ b if 1/λ ≤ H(a)/H(b) ≤ λ is a similarity relation. In fact, we can represent any similarity relation in this way. We say that the pair (H, λ) represents the similarity relation ∼ if, for all a,b ∈ I, a ∼ b if 1/λ ≤ H(a)/H(b) ≤ λ. One can show (see Project 6) that for every λ > 1 there is a strictly increasing continuous function H with values in [0, 1], so that the pair (H, λ) represents the similarity ∼. If 0 is not similar to any positive number, we can ªnd a representation of the similarity relation with a function H so that H(0) = 0. This proposition is analogous to propositions in utility theory that show the existence of a certain functional form of a utility representation. Note that no equivalence relation is a similarity relation under this deªnition. Consider, for example, the relation according to which any two elements in I relate if, in their decimal presentation, they have identical ªrst digits. This binary relation is an equivalence relation that fails to comply with the continuity assumption, the monotonicity assumption (because (.13) * = (.14) * , for example), and the nondegeneracy condition (there is no x < 0.4 that relates to 0.4). 2.3 A Procedure of Choice between Vectors In this section we analyze a family of decision procedures applied to decision problems where the choice is made from a set of pairs of lotteries in A = X × P = [0, 1] × [0, 1], where (x, p) ∈ A stands for Modeling Procedural Decision Making 29
- 47. a simple lottery that awards the prizes $x with probability p and $0 with the residual probability 1 − p. (P-∗) The primitives of the procedure are two similarity relations, ∼x and ∼p, that relate to the objects in X and P, respectively. (Thus, we do not require that the same similarity relation be relevant to both dimensions.) When choosing between the two lotteries L1 = (x1, p1) and L2 = (x2, p2): Step 1 (Check domination): If both xi > xj and pi > pj, then choose Li; If Step 1 is not decisive, move to Step 2, in which the similarity relations are invoked. This step is the heart of our procedure in that it captures the intuitions gained from the psychological experiments. Step 2 (Check similarities): If pi ∼p pj and not xi ∼x xj, and xi > xj, then choose Li. If xi ∼x xj and not pi ∼p pj, and pi > pj, then choose Li. If Step 2 is also not decisive, then move to Step 3, which is not speciªed. We move on to study the compatibility of following (P-∗) with the rational man procedure. Note that all vectors of the type (x, 0) or (0, p) are identical lotteries that yield the prize 0 with certainty. Therefore, in the following, preferences on X × P are assumed to have an indifference curve that coincides with the axis. The follow- ing deªnition deªnes the compatibility of a preference relation with (P-∗). We say that a preference relation i is ∗(∼x, ∼p) consistent if for any pair of lotteries Li and Lj, if Li is chosen in one of the ªrst two steps of the procedure, then Li Lj. In other words, any of the following three conditions implies that Li Lj: 30 Chapter 2
- 48. 1. Both xi xj and pi pj 2. pi ∼p pj and not xi ∼x xj, and also xi xj 3. xi ∼x xj and not pi ∼p pj, and also pi pj. Example Let i be a preference represented by the utility function pxα. Then i is consistent with (P-∗) where ∼x and ∼p are the λ and λα ratio similarities. For example, condition (2) implies Li Lj because if pi ∼p pj, not xi ∼x xj, and xi xj, then pixi α pi(λxj)α = (piλα)xj α ≥ pjxj α. 2.4 Analysis We now turn to an analysis of the decision procedures deªned in the previous section. Our general program, applied to the current setting, includes the following questions: 1. Given a pair of similarity relations, are the decisions implied by Steps 1 and 2 of (P-∗) consistent with the optimization of any preference relation? 2. How does (P-∗) restrict the set of preferences that are consistent with the procedure? First, note the following simple observation. Unless we assume that there is no x so that 0 ∼x x and no p so that 0 ∼p p, there is no preference that is ∗(∼x, ∼p) consistent. Assume, for example, that x ∼x 0 and x ≠ 0. Then, if there is a preference that is ∗(∼x, ∼p) consistent, the degenerate lottery (0, 1) has to be preferred to (x, 1 * − ε) for some ε 0 (by Step 2) and (x, 1 * − ε) has to be preferred to (0, 0) (by Step 1). Thus (0, 0) cannot be indifferent to (0, 1), as we assumed. The next proposition provides an answer to the ªrst question. For any pair of similarity relations there are preferences that do not contradict the execution of the ªrst two steps of (P-∗) with those two similarity relations. Thus, (P-∗) does not necessarily conºict with the rational man paradigm. Modeling Procedural Decision Making 31
- 49. Proposition 2.1 Let ∼x and ∼p be similarity relations satisfying that there is no x ≠ 0 or p ≠ 0 with 0 ∼x x or 0 ∼p p. Then, there are functions u: X → R+ and g: P → R+, so that g(p)u(x) represents a preference on X × P that is ∗(∼x, ∼p) consistent. Proof Let λ 1. From the previous section, there exist non-negative strictly increasing continuous functions, u and g, with u(0) = g(0) = 0, so that (u, λ) and (g, λ) represent the similarities ∼x and ∼p, respectively. The function g(p)u(x) assigns the utility 0 to all lotteries on the axes. We will show that g(p)u(x) induces a preference that is ∗(∼x, ∼p) consistent. Assume that both xi xj and pi pj; then g(pi)u(xi) g(pj)u(xj), thus Li Lj. Assume that pi ∼p pj, not xi ∼x xj, and xi xj; then u(xi) λu(xj), g(pi) ≥ (1/λ)g(pj), and hence g(pi)u(xi) g(pj)u(xj); so that, Li Lj. ▫ Note that this proof implies that there are not only preferences consistent with the ªrst two steps of the procedure but also prefer- ences consistent with the ªrst two steps that have an additive utility representation. We now approach the second question. Proposition 2.3 shows that few preferences are consistent with (P-∗). For any pair of similarities ∼x and ∼p, the preference relation built in the last propo- sition is “the almost unique” preference that is ∗(∼x, ∼p) consistent. Thus the assumption that a decision maker uses a (P-∗) procedure with a pair of similarity relations narrows down the consistent preferences to almost a unique preference whose maximization explains the decision maker’s behavior. The following proposition provides the key argument: Proposition 2.2 Consider a preference i on X × P that is ∗(∼x, ∼p) consistent. For any (x, p) with x* 1 and p* 1, all lotteries that dominate (x*, p * ) (or (x * , p*)) are preferred to (x, p), and all lotteries that are dominated by (x*, p * ) (or (x * , p*)) are inferior to (x, p). (If 32 Chapter 2
- 50. the preference is continuous, then it follows that the preference assigns indifference between (x, p) and (x*, p * ).) Proof By Step 2 of the procedure, (x, p) (x∗ + ε, p * ) for all ε 0. Any lottery that dominates (x*, p * ) must also dominate some lottery (x∗ + ε, p * ) for ε small enough, thus is preferred to (x, p). Similarly, (x, p) (x*, p * − ε) for all ε 0; thus, we also obtain that (x, p) is preferred to any lottery that is dominated by (x*, p * ). ▫ We will now show that for any two preferences i and i′, which are ∗(∼x, ∼p) consistent, and for every pair of lotteries L1 and L2 so that L1 L2, there must be a lottery L2′ “close” to L2 so that L1 ′ L2′. Thus, although there may be many preference relations consis- tent with the ªrst two stages of (P-∗), they are all “close.” The two ªrst steps of the procedure “almost” determine a unique preference where closeness is evaluated in terms of the similarity relations. Proposition 2.3 If i and i′ are both consistent with the pair of similarities (∼x, ∼p), then for any (x1, p1) and (x2, p2) satisfying (x1, p1) (x2, p2), there are x2′ ∼x x2 and p2′ ∼p p2 such that (x1, p1) ′ (x2′, p2′). Proof Consider the ªgure 2.1. By Proposition 2.2, any preference that is ∗(∼x, ∼p) consistent must satisfy the condition that all points in area A are preferred to L1 = (x, p) and L1 is preferred to any point in area B. (Actually, if the preference is continuous, then its indifference curve, passing through L1, includes the lotteries indicated by dots.) Thus, if both i and i′ are ∗(∼x, ∼p) consistent, and L1 L2 and not L1 ′ L2, then L2 must be outside areas A and B. But then, there is a lottery L2′ “close to L2” in the sense that both the x and the p components of L2 and L2′ are ∼x and ∼p similar, so that L1 ′ L2′. ▫ Discussion This proposition shows, in my opinion, that Steps 1 and 2 “overdetermine” the preference. Even before specifying the con- tent of Step 3, we arrive at an almost unique preference that is Modeling Procedural Decision Making 33
- 51. consistent with the choices determined by applying only Steps 1 and 2. The overdetermination result casts doubts as to whether decision makers who use such a procedure can be described as optimizers of preferences. However, one can interpret the analysis of this section as a deri- vation of separable preference relations. Any preference consistent with (P-∗) must be close to a preference relation represented by a utility function of the type g(p)u(x). The key to this separability result is that the similarities are assumed to be “global” in the sense that when examining the two lotteries (x1, p1) and (x2, p2), the determination of whether x1 is similar to x2 is done independently of the values of the probabilities p1 and p2. 2.5 Case-Based Theory We now turn to a short discussion of a formalization of “case-based theory,” an interesting model of choice that captures procedural Figure 2.1 34 Chapter 2
- 52. elements of decision making that are quite different from the ingre- dients of the rational man procedure. Case-based theory is designed to describe a decision maker who bases decisions on the consequences derived from past actions taken in relevant, similar cases. Take, for instance, the American decision whether to send troops to Bosnia in late 1995. When considering this problem, decision makers had in mind several previous events during which American troops were sent on inter- ventionary missions on foreign soil (Vietnam, Lebanon and the Persian Gulf). Those instances were offered as relevant precedents for the proposed action. The decision whether to interfere in Bosnia was taken, to a large extent, on the basis of evaluations of the past events and the assessment of the similarity of those cases to the Bosnian case. In the model, a decision maker has to choose among members of a ªxed set A. Let P be a set whose elements are called problems. An element in P is a description of the circumstances under which an alternative from the set A has to be chosen. The problems in P are related in the sense that the experience of one problem is conceived by the decision maker as relevant for another problem. Let C be a set of consequences; for simplicity, we take C = R, the set of real numbers. Taking an action in a problem deterministically yields a consequence, but the connection between the action and the conse- quence is unknown to the decision maker unless he has already experienced it. An instance of experience, a case, is a triple (p, a, u) interpreted as an event in which, at the problem p, the action a was taken and yielded the consequence u. A memory, M, is a ªnite set of cases. Note that the notion of memory here abstracts from the temporal order of the experienced cases. An instance of decision is a pair (p*, M): the decision maker has to choose an element from the set A, at the problem p* ∈ P, given the memory M. We assume that for each memory, all problems are distinct, that is, for any (p, a, u) and Modeling Procedural Decision Making 35
- 53. (p′, a′, u′) in M, p ≠ p′ (compare with Project 8). Finally, a choice function assigns to each problem p* and memory M, an action in A. The procedure described by Gilboa and Schmeidler (1995) is as follows. The primitive of the procedure is a measure of closeness between problems, s(p, p′). Each s(p, p′) is a non-negative number with the interpretation that the higher the s(p, p′), more similar is p′ to p. Given a problem p* and a memory M, each action a ∈ A is evaluated by the number v(a, p*, M) = Σ(p,a,u)∈Ms(p*, p)u. In case action a was not examined in the memory M, v(a, p*, M) is taken to be 0. The decision maker chooses an action a ∈ A that maximizes v(a, p*, M) (given some tie-breaking rule). Recall that in this model, the set A is ªxed and a decision maker bases his decision regarding one problem on past experiences with other problems. The model allows phrasing of consistency condi- tions that link different memories rather than different choice sets as in the rational choice theory. Gilboa and Schmeidler offer several axiomatizations of the above procedure. The basic axiomatization is based on the following (strong) assumption: A decision maker facing the problem p* and having the memory M, “transforms” each action a into a vector z(a, M) ∈ Z = RP (the set of functions that assign a real number to each problem in P, the set of problems experienced in M). He does so as follows: If (p, a, u) ∈ M, (that is, if the action a was taken when confronting the problem p), then z(a, M)(p) = u; otherwise (that is, if the action a was not attempted at the problem p), we take z(a, M)(p) = 0. It is assumed that the decision maker has in mind a preference relation ip* deªned on the set Z so that at the problem p*, having the memory M, he chooses an action a* satisfying z(a*, M) ip* z(a, M) for all a ∈ A. Given this assumption, we are left with the need to axiomatize the preference relation on Z. We have to show that there are coefª- cients, {s(p, p*)}p,p∗∈P, so that this preference relation has a utility representation of the type Σp∈Ps(p, p*)zp. This requires additional 36 Chapter 2
- 54. assumptions that induce a linearity structure. This can be done in a variety of ways: for example, by requiring that i satisªes mono- tonicity, continuity, and, most important, a property called separability: for any x, y, w, z ∈ Z, if x i y and w i z, then x + w i y + z (with strict preference in case w z). This axiomatization is quite problematic. A preference is deªned on the large set Z. This implies that the decision maker is required to compare vectors that cannot be realized in any memory (the decision maker will never have two different cases, such as (p, a, u) and (p, a′, u′) in his memory; yet the preference on the set RP exhibits comparisons between vectors z and z′ with both zp ≠ 0 and z′p ≠ 0). The separability axiom is quite arbitrary. As to the interpretation of s(p, p*) as a “degree of similarity,” because the axiomatization treats the behavior at any two problems completely separately, there are no restrictions on the similarity function. It might be, for example, that s(p*, p∗∗) = 1, whereas s(p∗∗, p*) = 0 making the interpretation of the numbers {s(p, p′)} as a similarity measure questionable. 2.6 Bibliographic Notes Sections 1–4 are based on Rubinstein (1988). For previous related work, see Luce (1956) and Ng (1977). The role of similarities in human reasoning was emphasized by Amos Tversky in a series of papers. In particular, see Tversky (1969) and Tversky (1977). Section 5 is based on Gilboa and Schmeidler (1995, 1997). See also Matsui (1994). 2.7 Projects 1. Innovative Tversky (1977) shows that in some contexts similarity relations may be asymmetric relations. Suggest a context in which such asymmetry is relevant to choice. 2. Reading Why are two objects perceived to be similar? One response is that an object a is similar to an object b when the number of properties (unary relations) Modeling Procedural Decision Making 37
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- 56. his, at the beginning of his amour with Lady Yarmouth, he frequently said, “I know you will love the Walmoden, because she loves me.” Old Blackbourn, the Archbishop of York, told her one day, “That he had been talking to her Minister Walpole about the new mistress, and was glad to find that her Majesty was so sensible a woman as to like her husband should divert himself.” Yet with the affectation of content, it made her most miserable: she dreaded Lady Yarmouth’s arrival, and repented not having been able to resist the temptation of driving away Lady Suffolk the first instant she had an opportunity, though a rival so powerless, and so little formidable. The King was the most regular man in his hours: his time of going down to Lady Suffolk’s apartment was seven in the evening: he would frequently walk up and down the gallery, looking at his watch, for a quarter of an hour before seven, but would not go till the clock struck. The King had another passager amour (between the disgrace of Lady Suffolk and the arrival of Lady Yarmouth) with the Governess to the two youngest Princesses; a pretty idiot, with most of the vices of her own sex, and the additional one of ours, drinking. Yet this thing of convenience, on the arrival of Lady Yarmouth, put on all that dignity of passion, which even revolts real inclination.
- 57. F. G. H. (Vide page 204.) Extracts from Letters of Sir Charles Hanbury Williams, during his Ministry at Berlin. TO THE DUKE OF NEWCASTLE. Berlin, July 11-22nd, 1750. .... Count Podewils’s behaviour to me has been hitherto very cold, and when I meet him at third places, he contents himself with making me a bow, without speaking to me. I have made one visit to Monsieur Finkenstein, who is the second Minister of State for Foreign Affairs. He has very much the air of a French petit-maître manqué, and is extremely affected in everything he says and does: but from what I have been able hitherto to learn, his credit with the King of Prussia increases daily; and that of Count Podewils is not thought to be so good as it formerly was. The former has lately gained a point over the latter: Count Podewils’s kinsman, who is at Vienna, was named to be a Minister of State before Count Finkenstein; but Count Finkenstein has got into his employment, and when Count Podewils returns from Vienna, Count Finkenstein will take place of him. Not that his Prussian Majesty gives entire confidence either to Podewils or Finkenstein; he reserves that for two persons that constantly reside with him at Potsdam, and whose names are Heichel and Fredersdorff; the first of whom is his Prussian Majesty’s Private Secretary, and who is always kept under the same roof with his Prussian Majesty, and is so well watched, that a person may be at this Court seven years without once seeing him. The other, who is the great favourite, was once a common soldier, and the King took a fancy to him, while he was yet Prince Royal of Prussia, as he was standing sentinel at the door of his apartment. This person has two very odd titles joined together, for he is styled
- 58. valet de chambre, and grand tresorier du Roi. He keeps out of all people’s sight as much as Heichel. But there is lately arose another young man, who has undoubtedly a large share in the King of Prussia’s favours: his name is Sedoo: he was not long ago his page, then came to be a lieutenant, and is very lately made a major, and premier ecuyer de l’ecurie de Potsdam, and will undoubtedly soon rise much higher. Another Extract. .... On Thursday, by appointment, I went to Court at eleven o’clock; the King of Prussia arrived about twelve, and Count Podewils immediately introduced me into his closet, where I delivered his Majesty’s letters into the King of Prussia’s hands, and made the usual compliments to him in the best manner I was able. To which his Prussian Majesty replied, to the best of my remembrance, as follows: “I have the truest esteem for the King of Great Britain’s person, and I set the highest value upon his friendship. I have at different times received essential proofs of it; and I desire you would acquaint the King, your master, that I will never forget them.” His Prussian Majesty afterwards said something with respect to myself, and then asked me several questions about indifferent things and persons. He seemed to express a great deal of esteem for my Lord Chesterfield, and a great deal of kindness for Mr. Villiers, but did not once mention Lord Hyndford, or Mr. Legge. I was in the closet with his Majesty exactly five minutes and a half. After my audience was over, the King of Prussia came out into that room where the Foreign Ministers wait for his Prussian Majesty. He just said one word to Count de la Puebla (the Austrian Minister) as he came in, and afterwards addressed his discourse to the French, Swedish, and Danish Ministers; but did not say one word either to the Russian Minister or myself. Extract from another Letter, in Cipher. Berlin July, 28, 1750.
- 59. .... About four days ago, Mr. Voltaire, the French poet, arrived at Potsdam from Paris. The King of Prussia had wrote to him about three months ago to desire him to come to Berlin. Mr. Voltaire answered his Prussian Majesty, that he should always be glad of an opportunity of throwing himself at his Majesty’s feet, but at that time he was not in circumstances to take so long a journey; upon which the King of Prussia sent him back word, that he would bear his expenses; but Mr. Voltaire, not caring to trust the King of Prussia, would not leave Paris till his Prussian Majesty had sent him a bill of exchange upon a banker in that town for 4000 rix-dollars, and he did not begin the journey till he had actually received the money. All that I now write your grace was told me by the Princess Amalie.— (Author.)
- 60. [The following extracts from the private correspondence of Sir Charles Hanbury Williams will further illustrate the remark in the text, and show the unfavourable view taken by him of the Prussian Court and Frederick the Great.] Extract of a Letter from Charles Hanbury Williams, from Berlin, 1750. .... ’Tis incredible what care this Pater Patriæ takes of his people. He is so good as to meddle in their family affairs, in their marriages, in the education of their children, and in the disposition of their estates. He hates that anybody should marry, especially an officer, let him be of what degree soever, and from the moment they take a wife, they are sure of never being preferred. All children are registered as soon as born, and the parents are obliged to produce either certificates of their deaths, or the male children themselves, at the age of fourteen, in order to be enrolled, and to take the oath of a soldier to the King; and if this is not done, or the children have escaped, the parents are answerable for the escape, and are sent to prison. No man can sell land throughout all the Prussian dominions without a special licence from the King: and as he does no more give licences, nobody can now dispose of or alienate his possessions. If they could, and were to find fools to purchase them, I believe he would not have ten of his present subjects left in a year’s time. They have really no liberty left but that of thinking. There is a general constraint that runs through all sorts of people, and diffidence is painted in every face. All their ambition and desire is to be permitted to go to their Country Seats, where they need not be obliged to converse with any but their own family. But this leave is not easily obtained, because the father of his country insists upon their living at Berlin, and making his Capital flourish. He is never here but from the beginning of December to the end of January, and during that time, Prussians, Silesians, and all his most distant subjects, are obliged to come and make a figure here, and spend all they have
- 61. been saving for the other ten months. He hates that any subject of his should be rich or easy; and if he lives a few years longer, he will have accomplished his generous design. There are actually but four persons in this great town that live upon their own means, and they are people that can’t last long in their present condition. He (always meaning Pater Patriæ) gives very small salaries to all employments, and this is the cause that he can get no gentleman to serve him in a Foreign Legation. His Ministers at every Court are the scum of the earth, and have nothing but the insolence of their master to support them; and, indeed, the Prussian method of treating with every Court is such, as I wonder how Sovereign Princes can bear. Of this, if I had time, I could give you many provoking instances. His Prussian Majesty’s Ministers at Berlin—I mean those for Foreign Affairs—make the oddest figure of any in Europe. They seldom or never see any dispatches that are sent to the Prussian Ministers at Foreign Courts; and all letters that come to Berlin from Foreign Courts go directly to the King; so that Mr. Podewils and Count Finkenstein know no more of what passes in Europe than what they are informed of by the Gazettes. When any of us go to them on any business, the surprise they are in easily betrays their ignorance, and the only answer you ever get is, that they will lay what you say before their master, and give you an answer as soon as he shall have signified his pleasure to them. When you return to their houses for this answer, they tell you the exact words which the king has directed, and never one word more; nor are you permitted to argue any point. In short, they act the part of Ministers without being really so, as much as ever Cibber did that of Wolsey upon the stage, only not half so well. The first of them is reputed to be an honest man, but he is nothing less. He loses that appearance of credit he once had, daily; for verily I believe he never had real weight enough with his master to have made an Ensign in his Army, or a Postillion in one of his Posthouses. His face is the picture of Dullness when she smiles, and his figure is a mixture of a clown and a petit-maître. He is a little
- 62. genteeler than Mons. Adrié, who you may remember to have seen make so great a figure in England. The other, Count Finkenstein, whom everybody calls Count Fink, is very like the late Lord Hervey, and yet his face is the ugliest I ever saw. But when he speaks, his affectation, the motion of his eyes and shoulders, all his different gestures and grimaces, bring Lord Hervey very strongly into my mind; and, like that Lord, he is the Queen’s favourite (I mean the Queen Mother’s); and her Majesty, whether seriously or otherwise I can’t tell, calls him “Mon beau Comte Fink.” He has parts, and is what, at Berlin, is called sçavant, which is to say, that he has read all the modern French story books, from Les Egaremens down to the history of Prince Cocquetron. The person who has certainly the greatest share of the King of Prussia’s confidence is one Heichel. He is his Private Secretary, and writes all that the King himself dictates. But this man I never saw, and people that have lived here seven years have never seen him. He is kept like a State Prisoner, is in constant waiting, and never has half an hour to himself in the whole year. [Then follows the account of Fredersdorff, to the same purpose, and nearly in the same words as in the extracts printed above.] He (Fredersdorff) is his Secretary for all small affairs for his Prussian Majesty.— Il fait tout par ses mains, et voit tout par ses yeux. If a Courier is to be dispatched to Versailles, or a Minister to Vienna, his Prussian Majesty draws, himself, the instructions for the one, and writes the letters for the other. This, you’ll say, is great; but if a Dancer at the Opera has disputes with a Singer, or if one of those performers want a new pair of stockings, a plume for his helmet, or a finer petticoat, ’tis the same King of Prussia that sits in judgment on the cause, and that with his own hand answers the Dancer’s or the Singer’s letter. His Prussian Majesty laid out 20,000l.
- 63. to build a fine theatre, and his music and Singers cost him near the same sum every year; yet this same King, when an opera is performed, wont allow ten pounds per night to light up the theatre with wax candles; and the smoke that rises from the bad oil, and the horrid stink that flows from the tallow, make many of the audience sick, and actually spoil the whole entertainment. What I have thought about this Prince is very true; and I believe, after reading what I say about him, you will think so too. He is great in great things, and little in little ones. In the summer 1749, three Prussian Officers came, without previously asking leave, to see a Review of some Austrian troops in Moravia; upon which the Commanding Officer of those troops, suspecting they were not come so much out of curiosity to see the Review, as to debauch some of the soldiers into the King of Prussia’s service, sent them orders to retire. This being reported to his Prussian Majesty, he was much offended, and resolved to take some method to show his resentment, which he did as follows:—Last summer, an Austrian Captain, being in the Duchy of Mecklenburgh, met there with an old acquaintance, one Chapeau, who is in great favour with the King of Prussia. At that time, there was to be a great Review at Berlin, and as Berlin was in the Austrian’s road in his return to Vienna, Chapeau invited him to see the Review; but the Austrian replied, that he would willingly come, but was afraid of receiving some affront, in return for what had been done to the Prussian Officers the year before in Moravia; to which Chapeau replied, that if he would come to Berlin, he would undertake to get the King of Prussia’s special leave for him to be present at the Review. Encouraged by this, the Austrian came, and the night before the Review, Chapeau brought him word that the leave was granted, and he might come with all safety. He did accordingly come; but as soon as the King of Prussia had notice of his being there, he sent an Aide-de-camp to him to tell him to retire that moment, which he was forced to do, not without much indignation against Chapeau, who
- 64. had drawn him into the scrape. The next morning he went to Chapeau, with an intention to demand satisfaction for the affront which, through him, he had received. Chapeau said he would do as he pleased, but first desired him to give him leave to speak for himself; which he did. Chapeau then told him, that immediately upon hearing that he had been sent out of the field in that strange manner, he had rode up to the King, and asked his Majesty whether he had not given him orders to tell the Austrian Officer that he might come to the Review with all security? and that the King had replied, it was very true, he had given such orders; because, if he had not, the Austrian would hardly have ventured to come to the Review; and if he had not come there, he (the King) should not have had an opportunity of revenging the affront that had been offered to some Officers of his own the year before in Moravia. I must tell you a story of the King of Prussia’s regard for the law of nations. There was, some time ago, a Minister here from the Duke of Brunswick, whose name was Hoffman. He was a person of very good sense, and what we call well-intentioned, (which means being attached to the interests of the maritime powers and the House of Austria.) He was, besides, very active and dexterous in getting intelligence, which he constantly communicated to the Ministers of England and Austria. This the King of Prussia being well informed of, wrote a letter with his own hand to the Duke of Brunswick, to insist (and in case of refusal to threaten) that he should absolutely disavow Hoffman for his Minister. The Duke, who is the worthiest Prince upon earth, was so frightened with this letter, that he complied, though much against his will, with this haughty and cruel request. The moment the King of Prussia received this answer, he sent a party of Guards to Hoffman’s house, seized him, sent him prisoner to Madgeburgh, where he has now been for above four years chained to a wheel-barrow, and working at the fortifications of that town! He was very near doing the same by a Minister of the Margravine of Anspach’s, but that person got timely notice, and
- 65. escaped out of Berlin in the morning; and when the King of Prussia’s Guards came to seize him at night, the bird had luckily flown. There is at present here a Minister of the Duke of Brunswick, the successor of Hoffman, to whom, in his first audience, the King said, that he advised him to act very differently from his predecessor, and particularly to take care not to frequent those Foreign Ministers that he must know were disagreeable to him; for if he did, he might depend upon it he should deal with him in the same manner as he had done with Hoffman. I think Hamlet says in the play, “Denmark is a prison;” the whole Prussian territory is so in the literal sense of the word. No man can, or does pretend to go out of it without the knowledge of the King and his Ministers. Very hard is the fate of those who have estates in other dominions besides those of his Prussian Majesty; he will neither permit them to sell their estates in his countries, nor live upon those they have out of them. The distresses which are come on the Silesians (who had estates also in Bohemia) are prodigious. Many people have given them up, or sold them for a trifle, to get out of this land of Egypt—this house of bondage. Six hundred dollars make just one hundred guineas, and I know the King of Prussia thinks that just as much as any of his subjects ought to have, exclusive of what he may give them. In a very few years, I am convinced that no subject of his that has not estates elsewhere will have more left him. But from what he has already done, he begins to find that it is no longer possible to collect the heavy taxes which he imposes on his subjects. I know that the revenues of all his countries, except Silesia, have diminished every year, for these last five years. A Prussian will tell you, with a very grave face, that their present King is the most merciful Prince that ever reigned, and that he hates shedding blood. This is not true; there are often as cruel and tormenting executions in this country as ever were known under any Sicilian tyrant. ’Tis true, they are not done at Berlin, nor in the face of the world, but at Potsdam, in private. Since my arrival in this
- 66. cursed country, an old woman was quartered alive at Potsdam, for having assisted two soldiers to desert. But his Prussian Majesty generally punishes offenders with close imprisonment and very hard labour, keeping them naked in the coldest weather, and giving them nothing, for years together, but bread and water. Such mercy is cruelty. Many persons destroy themselves here out of mere despair; but all imaginable care is taken to conceal such suicides. I have heard of one of our Governors in the Indies, who was reproached by his friends, on his return to England, that he put a great number of persons to death; to which that humane Governor replied, “It is not true; I only used them so ill, that they hanged themselves.” * * *
- 67. I. (Vide page 217.) Deux Henris immolés par nos braves ayeux, L’un à la liberté, et Bourbon à nos Dieux, Te menacent, Louis, d’une pareille entreprise: Ils revivent en toi ces anciens tyrans: Crains notre désespoir: la noblesse a des Guises, Paris des Ravaillacs, le clergé des Cléments.
- 68. K. (Vide page 225.) Though poetry was certainly neither a point of their rivalship, nor of their ambition, it may not be unwelcome to the curious to compare these great men even in their poetic capacities. The following sonnet was written by Sir R. Walpole when a very young man; the elegy, by Lord Bolingbroke, rather past his middle age. Had they climbed no mountain but Parnassus, it is obvious how far Lord Bolingbroke would have ascended above his competitor, since, when turned of fifty, he excelled in the province of youth. TO THE HELIOTROPE.[257] A SONG.
- 69. 1. Hail, pretty emblem of my fate! Sweet flower, you still on Phœbus wait; On him you look, and with him move, By nature led, and constant love. 2. Know, pretty flower, that I am he, Who am in all so like to thee; I, too, my fair one court, and where She moves, my eyes I thither steer. 3. But yet this difference still I find, The sun to you is always kind; Does always life and warmth bestow: —Ah! would my fair one use me so! 4. Ne’er would I wait till she arose From her soft bed and sweet repose; But leaving thee, dull plant, by night I’d meet my Phillis with delight. TO CLARA.[258] BY HENRY, VISCOUNT BOLINGBROKE.
- 70. Dear thoughtless Clara, to my verse attend, Believe for once the lover and the friend; Heav’n to each sex has various gifts assign’d, And shown an equal care of human kind. Strength does to man’s imperial race belong; To yours, that beauty which subdues the strong. But as our strength, when misapplied, is lost, And what should save, urges our ruin most; Just so, when beauty prostituted lies, Of b***s the prey, of rakes the abandon’d prize, Women no more their empire can maintain, Nor hope, vile slaves of lust, by love to reign; Superior charms but make their case the worse, When what was meant their blessing, proves their curse. O nymph! that might, reclin’d on Cupid’s breast, Like Psyche, soothe the God of Love to rest; Or if ambition move thee, Jove enthral, Brandish his thunder, and direct its fall; Survey thyself, contemplate ev’ry grace Of that sweet form, of that angelic face; Then, Clara, say, were those delicious charms Meant for lewd brothels and rude ruffians’ arms? No, Clara, no; that person and that mind Were form’d by nature, and by Heav’n design’d For nobler ends; to these return, though late; Return to these, and so redress thy fate. Think, Clara, think (nor may that thought be vain!) Thy slave, thy Harry, doom’d to drag his chain, Of love ill treated and abus’d, that he From more inglorious chains might rescue thee. Thy drooping health restor’d by his fond cares, Once more thy beauty its full lustre wears. Mov’d by his love, by his example taught, Soon shall thy soul, once more with virtue fraught, With kind and generous truth thy bosom warm, And thy fair mind, like thy fair person, charm. To virtue thus and to thyself restor’d, By all admir’d, by one alone ador’d, Be to thy Harry ever kind and true, And live for him who more than died for you.
- 71. (Vide page 356.) The reader will find a very ludicrous anecdote relating to Mr. Nugent, during his election at Bristol, in a letter from our Author to Richard Bentley, Esq., dated July 9th, 1754. It is printed in the publication of his correspondence with that gentleman, but we do not venture to insert it here. END OF VOL. I. T. C. Savill, Printer, 4, Chandos-street, Covent-Garden.
- 72. FOOTNOTES: [252] The Princess. [253] Allen, Lord Bathurst. [254] Sir George Lyttelton, who was out of favour with the Prince, made a parody on this copy of verses: two of the lines were, No—’tis that all-consenting tongue, That never puts me in the wrong. [255] “This is a strange country, this England” (said his Royal Highness once); “I am told Doddington is reckoned a clever man; yet I got 5000l. out of him this morning, and he has no chance of ever seeing it again.” [256] His house is since called Brandenburgh House. [257] I found this song in an old pocket-book belonging to my father, who wrote it, as he told me himself, when he was a very young man, on a sister of Sir William Carew. [258] This was written on a common woman whom Lord Bolingbroke took into keeping, and who, many years afterwards, sold oranges in the Court of Requests.
- 73. TRANSCRIBER’S NOTE Obvious typographical errors and punctuation errors have been corrected after careful comparison with other occurrences within the text and consultation of external sources. Except for those changes noted below, all misspellings in the text, and inconsistent or archaic usage, have been retained. For example, Nova-Scotia, Nova Scotia; goodnature, good-nature; Lord-Lieutenant, Lord Lieutenant; se’nnight; disculpate; unapt; deficience; altercate; preponderated. Pg xix: ‘acknowleged to have’ replaced by ‘acknowledged to have’. Pg 22: ‘he committed to’ replaced by ‘he was committed to’. Pg 25: ‘John Burnard—Factions’ replaced by ‘John Barnard—Factions’. Pg 37: ‘election at Weobly’ replaced by ‘election at Weobley’. Pg 68: ‘19.—The’ replaced by ‘19th.—The’. Pg 85: ‘3.—Palmer’ replaced by ‘3rd.—Palmer’. Pg 95: ‘to wordly success’ replaced by ‘to worldly success’. Pg 153: ‘worthy grammarians’ replaced by ‘worthy of grammarians’. Pg 190: ‘Holdernesse—Murray’ replaced by ‘Holderness—Murray’. Pg 204: ‘upon the recal’ replaced by ‘upon the recall’. Pg 256: ‘consume the propriety’ replaced by ‘consume the property’. Pg 280: ‘revenge offerred’ replaced by ‘revenge offered’. Pg 345: ‘2.—A ’ replaced by ‘2d.—A ’. Pg 383: ‘fidler, Nero’ replaced by ‘fiddler, Nero’. Pg 394: ‘the Mississipi’ replaced by ‘the Mississippi’. Pg 400: ‘as Aid-de-camp’ replaced by ‘as Aide-de-camp’. Pg 414: ‘all the coolurs’ replaced by ‘all the colours’. Pg 445: ‘being confidente of’ replaced by ‘being confidante of’. Footnote [21]: ‘and and in 1745’ replaced by ‘and in 1745’. Footnote [30]: ‘Stafford in 1706’ replaced by ‘Stafford in 1786’. Footnote [48]: ‘been prefered to’ replaced by ‘been preferred to’. Footnote [98]: ‘been Aid-de-camp’ replaced by ‘been Aide-de-camp’.
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