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The dataset generation failed because of a cast error
Error code: DatasetGenerationCastError
Exception: DatasetGenerationCastError
Message: An error occurred while generating the dataset
All the data files must have the same columns, but at some point there are 1 new columns ({'text'}) and 4 missing columns ({'num_embeddings', 'embedding_offset', 'passage_offset', 'num_passages'}).
This happened while the json dataset builder was generating data using
hf://datasets/forcemultiplier/diverse_subjects_nov13/colbert/indexes/pdf_files_index/collection.json (at revision 71885d6a36bc82e9dc30ba65e58681d1384ac230)
Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1870, in _prepare_split_single
writer.write_table(table)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/arrow_writer.py", line 622, in write_table
pa_table = table_cast(pa_table, self._schema)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/table.py", line 2292, in table_cast
return cast_table_to_schema(table, schema)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/table.py", line 2240, in cast_table_to_schema
raise CastError(
datasets.table.CastError: Couldn't cast
text: string
-- schema metadata --
pandas: '{"index_columns": [], "column_indexes": [], "columns": [{"name":' + 192
to
{'passage_offset': Value(dtype='int64', id=None), 'num_passages': Value(dtype='int64', id=None), 'num_embeddings': Value(dtype='int64', id=None), 'embedding_offset': Value(dtype='int64', id=None)}
because column names don't match
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1417, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1049, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 924, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1000, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1741, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1872, in _prepare_split_single
raise DatasetGenerationCastError.from_cast_error(
datasets.exceptions.DatasetGenerationCastError: An error occurred while generating the dataset
All the data files must have the same columns, but at some point there are 1 new columns ({'text'}) and 4 missing columns ({'num_embeddings', 'embedding_offset', 'passage_offset', 'num_passages'}).
This happened while the json dataset builder was generating data using
hf://datasets/forcemultiplier/diverse_subjects_nov13/colbert/indexes/pdf_files_index/collection.json (at revision 71885d6a36bc82e9dc30ba65e58681d1384ac230)
Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)Need help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
passage_offset
int64 | num_passages
int64 | num_embeddings
int64 | embedding_offset
int64 | text
string |
|---|---|---|---|---|
0 | 22,909 | 6,115,434 | 0 | null |
null | null | null | null | AN INTRODUCTION TO
BOUNDARY LAYER METEOROLOGY |
null | null | null | null | ROLAND B. STULL
Atmospheric Science Programme, Department of Geography
The University of British Columbia, Vancouver, Canada.
An Introduction to
Boundary
Layer
Meteorology
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON |
null | null | null | null | Libfary 01 Congress Cataloging in Publication Data
SIUI I. RolinG S .. 1950-
An
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ISIIN,'3: 918-9().2n·2769-5
,. Bou nOlry I IY", ' MIII.ralcgv'
OCe90.4.B655194 ,999
551.!I--ocI9
ISBN-I): 978·90-2n-2769·5
001: IO.1007/978·94..(X)9·3027_8
c· ISBN·I): 978·94..((19..3027·8
aa- 15564
'"
Published by Kluwe. Academic Publishe.s.
P.O. 80x 17. 3300 AA Dordred1t. The NetheflandS.
Kluwer Academic Publishers incorporates
the publishing proglammes 01
D. Reidel. Martinus Nijhotl. Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic: Publishers.
tOt Philip Orive. Norwell, MA 02061. U.S.A.
In all othel countries. sold and distributed
by Kluwer Academic Publ ishers GrOl)P.
P.O. 80x 322, 3.300 AH Dordlecht. The Netherlands.
H~t publislled 1999
RCIlO"inted 1989. 199 1. 1993. 1994
Reprinted with cITata 1991
R""rintoo 1999. 2001. |
null | null | null | null | H~t publislled 1999
RCIlO"inted 1989. 199 1. 1993. 1994
Reprinted with cITata 1991
R""rintoo 1999. 2001. 2003
Printtd on acid-fru paptr
All Rights Reserved
© t 988 by Kh./We, ACademic Publishers
SoftOOvef reprint of the hardoover tst edition t988
No part 01 the material protected by this copyright nOlice may be reproduced or
uhlized In any lorm or by any means. electronic or mechanical
Includin g photocopying. recording or by any inlormation "Iorage arid
retrieval system , without written permissioo Irom the copyright owner |
null | null | null | null | C!oo[co[s
Preface
XI
Mean Boundary Layer Characteristics
Turbulent transport
Taylor's hypothesis
1.1 A bowlCjary-layer definition
1.2 Wind and flow
1.3
1.4
l.5 Virtual potential temperature
1.6 Boundary layer depth and slructure
1.7 Micrometeorology
1.8
1.9 General references
1.10 References for this chapter
1.11 Exercises
Significance of the boundary layer
2
3
4
5
7
9
19
21
23
25
26
Some Mathematical and Conceptual Tools: Part 1. Statistics
29
1
2
The significance of turbulence and its spectrum
Some basic statistical methods
Turbulence kinetic energy
2.1
2.2 The spectral gap
2.3 Mean and turbulent parts
2.4
2.5
2.6 Kinematic flux
2.7
Eddy flux
2.8
Summation notation
2.9
Stress
2.10 Friction velocity
2.11 References
2.12 Exercises
29
33
33
35
45
47
51
57
63
67
68
70 |
null | null | null | null | vi
3
4
5
BOUNDARYLAYER~OROLOGY
Application or the Governing Equations to Turbulent Flow
75
Simplifications, approximations, and scaling arguments
3.1 Methodology
3.2 Basic governing equations
3.3
3.4 Equations for mean variables in a turbulent flow
3.5 Summary of equations, with simplifications
3.6 Case studies
3.7 References
3.8 Exercises
76
76
80
87
93
97
110
111
Prognostic Equations for Turbulent Fluxes and Variances
115
Prognostic equations for the turbulent departures
Free convection scaling variables
Prognostic equations for variances
Prognostic equations for turbulent fluxes
4.1
4.2
4.3
4.4
4.5 References
4.6 Exercises
115
117
120
134
147
148
Turbulence Kinetic Energy, Stability, and Scaling
151
5.1 The TKE budget derivation
5.2 Contributions 10 the TKE budget
5.3 TKE budget contributions as a function of eddy size
5.4 Mean kinetic energy and its interaction with turbulence
Stability concepts
5.5
5.6 The Richardson number
5.7 The Obukhov length
5.8 Dimensionless gradients
5.9 Miscellaneous scaling parameters
5.10 Combined stability tables
5.11 References
5.12 Exercises
6
Turbulence Closure Techniques
6.1 The closure problem
6.2 Parameterization rules
6.3 Local closure -
6.4
Local closure -
6.5
Local closure -
6.6 Local closure -
6.7
Local closure -
6.8 Nonlocal closure -
6.9 Nonlocal closure -
6.10 References
6. |
null | null | null | null | 3 Local closure -
6.4
Local closure -
6.5
Local closure -
6.6 Local closure -
6.7
Local closure -
6.8 Nonlocal closure -
6.9 Nonlocal closure -
6.10 References
6.11 Exercises
zero and half order
first order
one-and·a·half order
second order
third order
transilient turbulence theory
spectral diffusivity theory
151
153
166
168
169
175
180
183
184
186
187
189
197
200
202
203
214
220
224
225
240
242
245
197 |
null | null | null | null | COmENTS
vii
7
8
Boundary Conditions and External Forcings
251
7.1 Effective surface turbulent flux
7.2 Heat budget at the surface
7.3 Radiation budget
7.4 Auxes at interfaces
7.5 Partitioning of flux into sensible and latent portions
7.6 Aux 10 and from the ground
7.7 References
7.8 Exercises
251
253
256
261
272
282
289
292
Some Mathematical and Conceptual Tools: Part 2. Time Series 295
Fast Fourier Transfonn
8.1 Time and space series
8.2 Autocorrelation
8.3
Structure function
8.4 Discrete Fourier transfonn
8.5
8.6 Energy spectrum
8.7 Spectral characteristics
8.8 Spectra of two variables
8.9
Periodogram
8.10 Nonlocal spectra
8.11 Spectral decomposition of the TKE equation
8.12 References
8.13 Exercises
9
Similarity Theory
9.1 An overview
9.2 Buckingham Pi dimensional analysis methods
9.3 Scaling variables
9.4
Stable boundary layer similarity relationship lislS
9.5 Neutral boundary layer similarity relationship lislS
9.6 Convective boundary layer similarity relationship IislS
9.7 The log wind profile
9.8 Rossby-number similarity and proftle matching
9.9 Spectral similarity
9.10 Similarity scaling domains
9.11 References
9.12 Exercises
295
296
300
303
310
312
318
329
335
336
340
342
344
347
350
354
360
364
368
376
386
389
394
395
399
347 |
null | null | null | null | viii
BOUNDARY LAYER MmTEOROLOGY
10 Measurement and Simulation Techniques
405
Instrument platforms
10.1 Sensor and measurement categories
10.2 Sensor lists
10.3 Active remote sensor observations of morphology
10.4
10.5 Field experiments
10.6 Simulation methods
10.7 Analysis methods
10.8 References
10.9 Exercises
11
Convective Mixed Layer
11.1 The unstable surface layer
11.2 The mixed layer
11.3 Enrrainment zone
11.4 Enrrainment velocity and its parameterization
11.5 Subsidence and advection
11.6 References
11.7 Exercises
12
Stable Boundary Layer
12.1 Mean Characteristics
12.2 Processc:;
12.3 Evolution
12.4 Other Depth Models
12 .5 Low-level (nocturnal) jet
12.6 Buoyancy (gravity) waves
12.7 Terrain slope and drainage winds
12.8 References
12.9 Exercises
13
Boundary Layer Clouds
13.1 Thermodynamics
13.2 Radiation
13.3 Cloud enrrainment mechanisms
13.4 Fair-weather cumulus
13.5 Stratocumulus
13 .6 Fog
13.7 References
13.8 Exercises
405
407
410
413
417
420
427
434
438
442
450
473
477
483
487
494
499
506
516
518
520
526
534
538
542
545
555
558
562
570
576
578
583
441
499
545 |
null | null | null | null | 14 Geographic Effects
14.1 Geographically generated local winds
14.2 Geographically modified flow
14.3 Urban heat island
14.4 References
14.5 Exercises
Appendices
Scaling variables and dimensionless groups
A.
B. Notation
C. Useful constants, parameters and conversion factors
D. Derivation of virtual potential temperature
Subject Index
Errata section
CONTENTS
ix
587
595
609
613
617
621
629
639
645
587
619
649
667 |
null | null | null | null | Preface
Part of the excitement of boundary-layer meteorology is the challenge in :;tudying and
one of the unsolved problems of classical physics.
understanding turbulent flow -
Additional excitement stems from the rich diversity of topics and research methods that we
collect under the umbrella of boundary-layer meteorology. That we live our lives within
the boundary layer makes it a subject that touches us, and allows us to touch it. I've tried
to capture some of the excitement, challenges and diversity within this book.
I wrote this book with a variety of goals in mind. First and foremost, this book is
designed as a textbook. Fundamental concepts and mathematics are presented prior to
their use, physical interpretations of equation tenns are given, sample data is shown,
examples are solved, and exercises are included. Second, the book is organized as a
reference, with tables of parameterizations, procedures, field experiments, useful
constants, and graphs of various phenomena in a variety of conditions. Third, the last
several chapters are presented as a literature review of the current ideas and methods
in boundary layer meteorology.
It is assumed that this book will be used at the beginning graduate level for students
with an undergraduate background in meteorology. However, a diversity in the
background of the readers is anticipated. Those with a strong mathematical background
can skip portions of Chapter 2 on statistics, and those with experience with time series can
skip Chapter 8. These two mathematics chapters were separated to offer the reader a
chance to apply the first dose of statistics to boundary layer applications before delving
back into more math. Some students might have had a course on geophysical turbulence
or statistical fluid mechanics, and .can skim through the fll'st 5 chapters to get to the
boundary-layer applications. By excluding a few chapters, instructors can easily fit the
remaining material into a one-semester course. |
null | null | null | null | By excluding a few chapters, instructors can easily fit the
remaining material into a one-semester course. With supplemental readings, the book can
serve as a two-semester sequence in atmospheric turbulence and boundary-layer
meteorology.
Notational diversity proved to be the greatest difficulty. Each subdiscipline appeared
to have its own set of notation, which often conflicted with the notation of other
subdisciplines. To use the published notation would have lead to confusion. I was
therefore forced to select a consistent set of notation to use throughout the book. In most
cases I've tried to retain the notation most frequently used in the literature, or to add
subscripts or make logical extensions to existing notation. In other cases, I had to depart
from previously published notation. Readers are referred to Appendix B for a
comprehensive list of notation.
xi |
null | null | null | null | ~ BOUNDARYLAYER~OROLOGY
I certainly cannot claim to be an expert in all the myriad subdisciplines of boundary
layer meteorology. yet I knew the book should be comprehensive to be useful. My
interest and enthusiasm in writing this book motivated many trips to the library. and
stimulated my analysis of many research papers to learn the underlying themes and
common tools used in the diverse areas of boundary layer meteorology. Unfortunately.
the limited space within this book necessitated some difficult decisions regarding the
amount and level of material to include. Hopefully I've presented sufficient background
to lay the building blocks upon which more advanced concepts can be built by other
instructors and researchers. Certain topics such as atmospheric dispersion and agricultural
micrometeorology are not covered here. because there are other excellent books on these
subjects.
Many colleagues and friends helped with this book and contributed significantly to its
final form and quality. Michelle Vandall deserves special recognition and thanks for
drawing most of the figures. designing the page and chapter headers. and for her overall
dedication to this project. Colleagues Steve Stage. Larry Mahrt. George Young. Jacq
Schols. Chandran Kaimal. Steve Silberberg. Beth Ebert. and Bruce Albrecht reviewed
various chapters and provided valuable suggestions.
Information regarding field
experiments was provided by Anton Beljaars. Ad Driedonks. Jean-Claude Andre. Jean
Paul Goutorbe. Anne Jochum. Steve Nelson. Bob Murphy. Ruwim Berkowicz. Peter
Hildebrand. Don Lenschow. and others. Eric Nelson proofread the manuscript. and
helped with the list of notation. Sam Zhang compiled the index. |
null | null | null | null | Bob Murphy. Ruwim Berkowicz. Peter
Hildebrand. Don Lenschow. and others. Eric Nelson proofread the manuscript. and
helped with the list of notation. Sam Zhang compiled the index. Some of the equations
were typeset by Camille Riner and Michelle Vlllldall (her name keeps reappearing). Four
years of students in my micrometeorology courses at Wisconsin graciously tolerated
various unfinished drafts of the book. and caught many mistakes. The patient editors of
Kluwer Academic Publishers (formerly D. Reidel) provided constant encouragement. and
are to be congratulated for their foresight and advice. The American Meteorological
Society is acknowledged for permission to reproduce figures 10.7. 12.10 and 12.19. To
all of these people and the many more to whom I apologize for not naming here. I thank
you.
To my wife. Linda. I give my gratitude for her devotion during this exciting and
exhausting episode of my life.
Roland B. Stull
Boundary Layer Research Team
Madison. Wisconsin |
null | null | null | null | Z
(m)
1000
500
0
(a)
z
(m)
500
250
o
(b)
1000
Range from
lidar (km)
Range from
lidar (km)
Frontispiece
lidar images of the aerosol-laden boundary layer, obtained during
the FIFE field experiment in Kansas. (a) Convective mixed layer
observed at 1030 local time on 1 July 1987, when winds were
generally less than 2 mls. (b) Slightly-stable boundary layer with
shear-generated turbulence, observed at 530 local time on 7 July
1987. Winds ranged from 5 mls near the surface to 15 mls near the
top of the boundary layer. Photographs from the Univ. of Wisconsin
lidar are courtesy of E. Eloranta, Boundary Layer Research Team. |
null | null | null | null | Claude V. Palisca
HUMANISM
IN
ITALIAN
RENAISSANCE
MUSICAL
THOUGHT
Yale University Press
New Haven and London |
null | null | null | null | Humanism in Italian Renaissance
Musical Thought |
null | null | null | null | To tile memory of my motller
Gisella Fleisch/lacker Palisca (1895-1944)
I'ublishcd wilh .1SsUtanCr from the Louis 51t'", Memori;al Fund.
Copyright 4) 1985 by Yale UniversilY. All righls reserved. This book
may 1101 br reproduced. in whole or in pUI. ill ;any form (beyolld
Ih;al copy illS permilled by Sections 1117 ;and \08 of Iht' U. S. Copyrighl
L.t w .and c"cepl by rc:viewers for II\!: public prcss). wilhout wrinc:n
permission Irom the publishers.
DcsigllC.'d by N.ancy Ovedovill lllli 5C:1 in Ilembo typc: by Rainsford
Type. I)rilllell in the United 51311:5 of America by Edwards Brothers.
Inc .• Ann Arhor. Michigan.
Library of Congrcss Cataloging ill I)ublicatioll Dala
l'alisC:I. Claude V.
Humanism in h.alian Rm.:aissance musical thOllghl
Bibliognphy: p.
Includc.'S index.
I. Music-haly-15th century-History ;and criticism. 2. Music
h.:aly-16th cmlury-History .and crilicism. 3. nen;aissance-haly. 4.
Humanism. I. Tide. |
null | null | null | null | I. Music-haly-15th century-History ;and criticism. 2. Music
h.:aly-16th cmlury-History .and crilicism. 3. nen;aissance-haly. 4.
Humanism. I. Tide.
ML2'JeI.2.1)34 1985781.745 as-8ICJO
ISDN 0-300-03302-8
The paprr in Ihis book meets thr guidelincs for permanence and du
rability of the Committee on Production Guidelines for Book Lon
gevily of Ihe Council on Libnry Resources.
10
I) 8 7 6 5 4 3 2
I |
null | null | null | null | Contents
Preface
ONE
Introduction: An Italian Renaissance in Music?
TWO
The Rediscovery of the Ancient Sources
THREE
The Earliest Musical Humanists: Pietro d'Abano
FOUR
The Earliest Musical Humanists: Giorgio Valla
FIVE
The Earliest Musical Humanists: Carlo Valgulio
The Proem to Plm:lrch's De tllllsica
A Reply to an Opponent of Music
The Translation of the De IHilsica of Plutarch
SIX
The Early Translators: Burana, Lconiceno, Augio
Giovanni Fr:mcesco Burana
Nicolo Leoniceno
Giovanni Battista Augio
SEVEN
Antonio Goga va
Ptolemy's Harmotlics
Pseudo-Aristotlc's De alldibiliblls
Aristoxcnus' Harmonic Elements
Bomigari's Corrections
vii
xi
23
51
67
88
111
133 |
null | null | null | null | Contents
TWELVE
A Natural New Alliance of the Arts
Grammar
Mei on tonic accent
Pietro Bembo
THIRTEEN
The Poetics of Music
Music as Poetry
Vincenzo Galilci
The Poetics of Imitation
The Case against Mimesis: Francesco Patrizi
Expressing the Affections
FOURTEEN
Theory of Dramatic Music
Francesco Patrizi
Girolamo Mei
Jacopo Peri
Works Cited
Index
ix
333
369
408
435
453
viii
Contents
EIGHT
Harmonics and Disharmonies of the Spheres
161
Ugolino of Orvieto
Giorgio Anselmi
Franchino Gaffurio
Gioseffo Zarlino
Johannes Tinctoris
Francisco de Salinas
Giovanni Battista Benedetti
Celestial Harmony as Myth and Metaphor
NINE
GatTurio as a Humanist
TEN
The Ancient Musica Speclliativa and
Renaissance Musical Science
Franchino Gaffurio
Hamos de Pareja
Giovanni Spataro
Lodovico Fogliano
Gioscffo Zarlino
Francisco de Salinas
Girolamo Fracastoro
Giov:mni Battista Benedetti
Girolamo Mei and Vincenzo Galilci
ELEVEN
Greek Tonality and Western Modality
Johannes Gallicus
Erasmus of Horitz
Giorgio Valla
Nicolo Leoniceno
Franchino Gaffurio
Gioseffo Zarlino
Francisco de Salinas
Girolamo Mei
Vincenzo Galilei
The Tonoi and the Waning of Modality
Giovanni Bardi
Giovanni Battista Doni
191
226
280 |
null | null | null | null | Preface
Music historians have long been aware of a link between the revival of
ancient learning and the changes in musical style and theory that occurred
during the Renaissance. But the ties to antiquity have been hard to pin
down. because ancient music could not be recreated as could ancient lit
erature and architecture. Instead. the objects of revival were ancient attitudes
and thoughts about music. The route by which these reached Renaissance
musicians and critics has not been studied with any precision or thorough
ness. Indeed. the men most rt.'sponsible for the transmission of Greek thought
about music have been practically ignored. Their names. some of which
head chapters or sections in this book-Pietro d·Abano. Giorgio Valla.
Carlo Valgulio. Antonio Gogava. Francesco Burana. Nicolo Leoniceno
are missing from even the most comprehensive accounts of the musical
culture of the Renaissance.
This book aims to document the debt that Renaissance musical thought
owes to ancient. particularly Greek. musical thought and to trace its path
of transmission in Italy. I have had to rely almost entirely on primary
sources. Because of this necessity. the previous literature on musical hu
manism and on music in the Renaissance has been given less attention than
it truly deserves. Therefore I want to express here my debt to those who
earlier explored musical humanism and lighted my way, particularly Ed
ward E. Lowinsky. PaulO. Kristeller. Nino Pirrotta, Leo Schrade, D. P.
Walker. |
null | null | null | null | PaulO. Kristeller. Nino Pirrotta, Leo Schrade, D. P.
Walker. and Edith Weber, for I have learned enormously from them.
In general the field has been dominated by the hunt for parallels between
musical manifestations and those in other ans and humanities that show a
strong reliance on ancient models. But even where parallels have been found.
there has been little direct evidence of relationships among the composers.
writers. philosophers. architects. and artists whose work is involved. I
cannot claim to have discovered many such associations either, so the search
must continue, for where no direct connections can be shown. the con-
r
xi |
null | null | null | null | Preface
xiii
University. enriched and enhanced this book in many ways. by lending me
microfilms of Greek manuscripts that once belonged to Giorgio Valla. by
letting me use some of the information in the catalog of Greek manuscripts
of music theory he is preparing for the RepertOire imernationale des sources
musicales, by reviewing my translations of the Latin versions of Greek trea
tises by Burana. Leoniceno. Gogava. and Augio and offering many prov
ident corrections and excellent suggestions. and. finally. by reading and
commenting on the entire manuscript. To these scholars I. and the reader
too. owe sincere thanks.
Among others who have stood behind this work. I give special thanks
to Edward Tripp. Editor-in-Chief of the Yale University Press. for his
encouragement and interest. to jean van Altena for her very attentive reading
and sympathetic editing. to Michael Pepper and jay Williams for their
resourceful recoding and production of the manuscript from its electronic
state. to my daughter Madeline for her punctilious drafting of the index.
and to my wife. Jane. for advice on many matters. big and small. and for
her confidence and unfailing support.
Branford October 1984
xii
Preface
current trends. like parallel lines. never meet. and we can learn little from
simply contemplating the striking analogil'S. I have avoided drawing such
parallels. limiting myself to those connections between music and ancient
thought that we know existed in the minds of Renaissance men because
they are recorded in writing. These considerations. too. |
null | null | null | null | limiting myself to those connections between music and ancient
thought that we know existed in the minds of Renaissance men because
they are recorded in writing. These considerations. too. explain why I have
not allocated much space to past literature on musical humanism. As a
consequence of this approach and the interdisciplinary scope of my study.
the secondary literature referred to in the footnotes is restricted to those
works that were specifically utilized for the material in the text. and the
bibliography lists only these.
Some chapters may strike the reader as almost anthologies of extracts
from Renaissance writings on music and related subjects. Since so many of
the works quoted are unpublished or extremely rare. this was the only way
I could let my authors speak for themselves. And since none of them wrote
in English. I wanted to let the reader experience the power of their own
words. with the aid of parallel translations. Whenever possible the material
in the two columns corresponds line for line. consequences of which are a
certain literalness and a ragged format. The translations are my own except
where I have indicated otherwise.
Many organizations and individuals have generously supported my re
search over the years. It was begun in Florence on a Guggenheim Fellowship
and completed on a second one twenty years later. In between. a Senior
Fellowship of the National Endowment for the Humanities permitted a year
in Paris at the remarkable collection of Renaissance books of the Biblio
theque Nationale. The Whitney Griswold Fund of Yale University aided
the preparation of the manuscript. And. of course. the Yale libraries. |
null | null | null | null | The Whitney Griswold Fund of Yale University aided
the preparation of the manuscript. And. of course. the Yale libraries. p~r
ticularly the Music Library and the Beinecke: Rare Book and Manuscnpt
Library. provided a solid home base for my investigations.
Several of my students at Yale have helped me during various stages.
joseph DiGiovanni. of the Renaissance Studies Program. transcribed parts
of Leoniceno's translation ofPtolemy's Harmonics. Deborah Narani. of the
Medieval Studies Program. checked my translations of Pietro d'Abano's
commentary on the pseudo-Aristotle Problems. Otto Stein mayer. of the
Classics Depanment. reviewed most of my translations from Latin and
made many essential improvements in them.
Of the many colleagues to whom I feel indebted. I should name seve~al.
jon Solomon. of the University of Arizona. kindly made available his trans
lation ofCleonides' Harmonic introduction. Frank d' Accone, of the University
of California, Los Angeles, and james Haar. of the University of North
Carolina, contributed to my thinking with their learned commentaries on
my first chapter when it was delivered as a lecture in honor of A. Tillman
Merritt's retirement from Harvard. Thomas J. Mathiesen, of Brigham Young |
null | null | null | null | It
ONE
Introduction: An Italian Renaissance in Music?
istorians generally view the Renaissance as a movement
that began in Italy and spread northward. Music histo
rians. however. have habitually begun the study of music
in the Renaissance with composers associated with France
Jnd the Low Countries. Gustave Reese organized his book
lHIIsi( i" th,' R"lldiSSI1I1((' on the premise that a central mus
ical language arose in the fifteenth and sixteenth centuries in France. the
Low Countries. and Italy. and spread to Spain. Portugal. Germany. Eng
land. and eastt:rn Europe. In the first part of the book he defines this language
in terms of the music of Dufay. Busnois. and Ockeghem. who were active
principally in the north. 1
Similarly Howard Mayer Brown takes the view that music in the: Ren
aissance "is a northern art. or at least an art by northerners. All of the great
composers of the fifteenth and early sixteenth centuries were born in what
is today northern France. Belgium. and Holland.":
Thus. while the impetus for the Renaissance in the visual arts. literature.
and philosophy is generally recognized to have come from Italy and moved
across the Alps, we are confronted in music history with the thesis that
music. of all artistic expressions. moved in the opposite direction. Now
contrary motion may be a praiseworthy polyphonic practice. but it is dis
concerting when applied to cultural historiography. If history in general
has a proverbial "problem of the Renaissance." how much more acute it is
in music history!
Heinrich Besseler. |
null | null | null | null | but it is dis
concerting when applied to cultural historiography. If history in general
has a proverbial "problem of the Renaissance." how much more acute it is
in music history!
Heinrich Besseler. reflecting on his own work in Renaissance studies since
I. Gustave: R.e:rse. MIlS;( ;11 IlIr RtI,a;ssalltt (New York. 1954). Pt. I.
2. Howard Mayer Brown. ''''lIs;( ;11 tI,r R.."a;ssallcr (Englewood Cliffs. 1976). p. 4. Leo
Schrade:. in "Renaissance: the Historical Conception of an Epoch." K"'~~Tfss·B('r;("t drT IIII(r.
IIa/;ollalt Gtstllscl,aft fiir Mllsikll·;swrstl",,,,fi. Utrtc/It 1951 (Amsterdam. 1953). pp. 19-32. took
a similar view: "In contrast to the bonae lillerae and to the visual arts as well. the rebirth of
music came to pass as an achievement of northern composers ... " (p. 3U) .•
r
1 |
null | null | null | null | Game Theory |
null | null | null | null | GAME THEORY
Analysis of Conflict
ROGER B. MYERSON
HARVARD UNIVERSITY PRESS
Cambridge, Massachusetts
London, England |
null | null | null | null | Copyright © 1991 by the President and Fellows of Harvard College
All rights reserved
Printed in the United States of America
First Harvard University Press paperback edition, 1997
Library of Congress Cataloging-in-Publication Data
Myerson, Roger B.
Game theory : analysis of conflict / Roger B. Myerson.
p. (cid:9)
cm.
Includes bibliographical references and index.
ISBN 0-674-34115-5 (cloth)
ISBN 0-674-34116-3 (pbk.)
1. Game Theory (cid:9)
I. Title
H61.25.M94 (cid:9)
519.3—dc20 (cid:9)
1991
90-42901
(cid:9)
(cid:9) |
null | null | null | null | For Gina, Daniel, and Rebecca
With the hope that a better understanding of conflict
may help create a safer and more peaceful world |
null | null | null | null | Contents
Preface (cid:9)
1 Decision-Theoretic Foundations (cid:9)
1.1 (cid:9) Game Theory, Rationality, and Intelligence
1.2 Basic Concepts of Decision Theory 5
1.3 Axioms 9
1.4 The Expected-Utility Maximization Theorem 12
1.5 (cid:9)
1.6 Bayesian Conditional-Probability Systems 21
1.7 Limitations of the Bayesian Model 22
1.8 Domination 26
1.9 Proofs of the Domination Theorems 31
Equivalent Representations 18
Exercises 33
2 Basic Models (cid:9)
2.1 Games in Extensive Form 37
2.2 Strategic Form and the Normal Representation 46
2.3 Equivalence of Strategic-Form Games 51
2.4 Reduced Normal Representations 54
2.5 Elimination of Dominated Strategies 57
2.6 Multiagent Representations 61
2.7 Common Knowledge 63
2.8 Bayesian Games 67
2.9 Modeling Games with Incomplete Information 74
Exercises 83
3 Equilibria of Strategic-Form Games (cid:9)
3.1 (cid:9) Domination and Rationalizability 88
3.2 Nash Equilibrium 91
xi
1
37
88
(cid:9)
(cid:9)
(cid:9) |
null | null | null | null | viii (cid:9)
Contents
3.3 Computing Nash Equilibria 99
3.4 Significance of Nash Equilibria 105
3.5 The Focal-Point Effect 108
3.6 The Decision-Analytic Approach to Games 114
3.7 Evolution, Resistance, and Risk Dominance 117
3.8 Two-Person Zero-Sum Games 122
3.9 (cid:9)
3.10 Purification of Randomized Strategies in Equilibria 129
3.11 Auctions 132
3.12 Proof of Existence of Equilibrium 136
3.13 Infinite Strategy Sets 140
Bayesian Equilibria 127
Exercises 148
4 Sequential Equilibria of Extensive - Form Games (cid:9)
154
4.1 (cid:9) Mixed Strategies and Behavioral Strategies 154
4.2 (cid:9)
4.3 (cid:9)
Equilibria in Behavioral Strategies 161
Sequential Rationality at Information States with Positive
Probability 163
4.4 Consistent Beliefs and Sequential Rationality at All Information
States 168
4.5 Computing Sequential Equilibria 177
4.6 Subgame-Perfect Equilibria 183
4.7 Games with Perfect Information 185
4.8 Adding Chance Events with Small Probability 187
4.9 Forward Induction 190
4.10 Voting and Binary Agendas 196
4.11 Technical Proofs 202
Exercises 208
5 Refinements of Equilibrium in Strategic Form (cid:9)
213
Introduction 213
Perfect Equilibria 216
Existence of Perfect and Sequential Equilibria 221
Proper Equilibria 222
Persistent Equilibria 230
Stable Sets of Equilibria 232
5.1 (cid:9)
5.2 (cid:9)
5.3 (cid:9)
5.4 (cid:9)
5.5 (cid:9)
5.6 (cid:9)
5.7 Generic Properties 239
5.8 Conclusions 240
Exercises 242
6 Games with Communication (cid:9)
244
Contracts and Correlated Strategies 244
6. |
null | null | null | null | 5 (cid:9)
5.6 (cid:9)
5.7 Generic Properties 239
5.8 Conclusions 240
Exercises 242
6 Games with Communication (cid:9)
244
Contracts and Correlated Strategies 244
6.1 (cid:9)
6.2 Correlated Equilibria 249
6.3 Bayesian Games with Communication 258
6.4 Bayesian Collective-Choice Problems and Bayesian Bargaining
Problems 263 |
null | null | null | null | Contents (cid:9)
ix
6.5 Trading Problems with Linear Utility 271
6.6 General Participation Constraints for Bayesian Games with
Contracts 281
6.7 Sender-Receiver Games 283
6.8 Acceptable and Predominant Correlated Equilibria 288
6.9 Communication in Extensive-Form and Multistage Games 294
Exercises 299
Bibliographic Note 307
7 Repeated Games (cid:9)
308
7.1 The Repeated Prisoners' Dilemma 308
7.2 A General Model of Repeated Games 310
7.3 Stationary Equilibria of Repeated Gaines with Complete State
Information and Discounting 317
7.4 Repeated Games with Standard Information: Examples 323
7.5 General Feasibility Theorems for Standard Repeated Games 331
7.6 Finitely Repeated Games and the Role of Initial Doubt 337
7.7 Imperfect Observability of Moves 342
7.8 Repeated Games in Large Decentralized Groups 349
7.9 Repeated Games with Incomplete Information 352
7.10 Continuous Time 361
7.11 Evolutionary Simulation of Repeated Games 364
Exercises 365
8 Bargaining and Cooperation in Two-Person Games (cid:9)
370
8.1 Noncooperative Foundations of Cooperative Game Theory 370
8.2 Two-Person Bargaining Problems and the Nash Bargaining
Solution 375
8.3 Interpersonal Comparisons of Weighted Utility 381
8.4 Transferable Utility 384
8.5 Rational Threats 385
8.6 Other Bargaining Solutions 390
8.7 An Alternating-Offer Bargaining Game 394
8.8 An Alternating-Offer Game with Incomplete Information 399
8.9 A Discrete Alternating-Offer Game 403
8.10 Renegotiation 408
Exercises 412
9 Coalitions in Cooperative Games (cid:9)
417
Introduction to Coalitional Analysis 417
9.1 (cid:9)
9.2 Characteristic Functions with Transferable Utility 422
9.3 The Core 427
9. |
null | null | null | null | 1 (cid:9)
9.2 Characteristic Functions with Transferable Utility 422
9.3 The Core 427
9.4 The Shapley Value 436
9.5 Values with Cooperation Structures 444
9.6 Other Solution Concepts 452
9.7 Coalitional Games with Nontransferable Utility 456 |
null | null | null | null | x (cid:9)
Contents
9.8 Cores without Transferable Utility 462
9.9 Values without Transferable Utility 468
Exercises 478
Bibliographic Note 481
10 Cooperation under Uncertainty (cid:9)
483
10.1 Introduction 483
10.2 Concepts of Efficiency 485
10.3 An Example 489
10.4 Ex Post Inefficiency and Subsequent Offers 493
10.5 Computing Incentive-Efficient Mechanisms 497
10.6 Inscrutability and Durability 502
10.7 Mechanism Selection by an Informed Principal 509
10.8 Neutral Bargaining Solutions 515
10.9 Dynamic Matching Processes with Incomplete Information 526
Exercises 534
Bibliography
Index
539
553 |
null | null | null | null | Preface
Game theory has a very general scope, encompassing questions that are
basic to all of the social sciences. It can offer insights into any economic,
political, or social situation that involves individuals who have different
goals or preferences. However, there is a fundamental unity and co-
herent methodology that underlies the large and growing literature on
game theory and its applications. My goal in this book is to convey both
the generality and the unity of game theory. I have tried to present
some of the most important models, solution concepts, and results of
game theory, as well as the methodological principles that have guided
game theorists to develop these models and solutions.
This book is written as a general introduction to game theory, in-
tended for both classroom use and self-study. It is based on courses that
I have taught at Northwestern University, the University of Chicago,
and the University of Paris—Dauphine. I have included here, however,
somewhat more cooperative game theory than I can actually cover in a
first course. I have tried to set an appropriate balance between non-
cooperative and cooperative game theory, recognizing the fundamental
primacy of noncooperative game theory but also the essential and com-
plementary role of the cooperative approach.
The mathematical prerequisite for this book is some prior exposure
to elementary calculus, linear algebra, and probability, at the basic un-
dergraduate level. It is not as important to know the theorems that may
be covered in such mathematics courses as it is to be familiar with the
basic ideas and notation of sets, vectors, functions, and limits. Where
more advanced mathematics is used, I have given a short, self-contained
explanation of the mathematical ideas. |
null | null | null | null | xii (cid:9)
Preface
In every chapter, there are some topics of a more advanced or spe-
cialized nature that may be omitted without loss of subsequent compre-
hension. I have not tried to "star" such sections or paragraphs. Instead,
I have provided cross-references to enable a reader to skim or pass over
sections that seem less interesting and to return to them if they are
needed later in other sections of interest. Page references for the im-
portant definitions are indicated in the index.
In this introductory text, I have not been able to cover every major
topic in the literature on game theory, and I have not attempted to
assemble a comprehensive bibliography. I have tried to exercise my best
judgment in deciding which topics to emphasize, which to mention
briefly, and which to omit; but any such judgment is necessarily subjec-
tive and controversial, especially in a field that has been growing and
changing as rapidly as game theory. For other perspectives and more
references to the vast literature on game theory, the reader may consult
some of the other excellent survey articles and books on game theory,
which include Aumann (1987b) and Shubik (1982).
A note of acknowledgment must begin with an expression of my debt
to Robert Aumann, John Harsanyi, John Nash, Reinhard Selten, and
Lloyd Shapley, whose writings and lectures taught and inspired all of
us who have followed them into the field of game theory. I have bene-
fited greatly from long conversations with Ehud Kalai and Robert Weber
about game theory and, specifically, about what should be covered in a
basic textbook on game theory. Discussions with Bengt Holmstrom, Paul
Milgrom, and Mark Satterthwaite have also substantially influenced the
development of this book. Myrna Wooders, Robert Marshall, Dov Mon-
derer, Gregory Pollock, Leo Simon, Michael Chwe, Gordon Green,
Akihiko Matsui, Scott Page, and Eun Soo Park read parts of the manu-
script and gave many valuable comments. |
null | null | null | null | Myrna Wooders, Robert Marshall, Dov Mon-
derer, Gregory Pollock, Leo Simon, Michael Chwe, Gordon Green,
Akihiko Matsui, Scott Page, and Eun Soo Park read parts of the manu-
script and gave many valuable comments. In writing the book, I have
also benefited from the advice and suggestions of Lawrence Ausubel,
Raymond Deneckere, Itzhak Gilboa, Ehud Lehrer, and other colleagues
in the Managerial Economics and Decision Sciences department at
Northwestern University. The final manuscript was ably edited by Jodi
Simpson, and was proofread by Scott Page, Joseph Riney, Ricard
Torres, Guangsug Hahn, Jose Luis Ferreira, loannis Tournas, Karl
Schlag, Keuk-Ryoul Yoo, Gordon Green, and Robert Lapson. This book
and related research have been supported by fellowships from the John
Simon Guggenheim Memorial Foundation and the Alfred P. Sloan |
null | null | null | null | Preface (cid:9)
xiii
Foundation, and by grants from the National Science Foundation and
the Dispute Resolution Research Center at Northwestern University.
Last but most, I must acknowledge the steady encouragement of my
wife, my children, and my parents, all of whom expressed a continual
faith in a writing project that seemed to take forever.
Evanston, Illinois
December 1990 |
null | null | null | null | Game Theory |
null | null | null | null | 1
Decision-Theoretic Foundations
1.1 Game Theory, Rationality, and Intelligence
Game theory can be defined as the study of mathematical models of
conflict and cooperation between intelligent rational decision-makers.
Game theory provides general mathematical techniques for analyzing
situations in which two or more individuals make decisions that will
influence one another's welfare. As such, game theory offers insights
of fundamental importance for scholars in all branches of the social
sciences, as well as for practical decision-makers. The situations that
game theorists study are not merely recreational activities, as the term
"game" might unfortunately suggest. "Conflict analysis" or "interactive
decision theory" might be more descriptively accurate names for the
subject, but the name "game theory" seems to be here to stay.
Modern game theory may be said to begin with the work of Zermelo
(1913), Borel (1921), von Neumann (1928), and the great seminal book
of von Neumann and Morgenstern (1944). Much of the early work on
game theory was done during World War II at Princeton, in the same
intellectual community where many leaders of theoretical physics were
also working (see Morgenstern, 1976). Viewed from a broader perspec-
tive of intellectual history, this propinquity does not seem coincidental.
Much of the appeal and promise of game theory is derived from its
position in the mathematical foundations of the social sciences. In this
century, great advances in the most fundamental and theoretical
branches of the physical sciences have created a nuclear dilemma that
threatens the survival of our civilization. People seem to have learned
more about how to design physical systems for exploiting radioactive
materials than about how to create social systems for moderating human |
null | null | null | null | 2 (cid:9)
1 • Decision-Theoretic Foundations
behavior in conflict. Thus, it may be natural to hope that advances in
the most fundamental and theoretical branches of the social sciences
might be able to provide the understanding that we need to match our
great advances in the physical sciences. This hope is one of the moti-
vations that has led many mathematicians and social scientists to work
in game theory during the past 50 years. Real proof of the power of
game theory has come in recent years from a prolific development of
important applications, especially in economics.
Game theorists try to understand conflict and cooperation by studying
quantitative models and hypothetical examples. These examples may
be unrealistically simple in many respects, but this simplicity may make
the fundamental issues of conflict and cooperation easier to see in these
examples than in the vastly more complicated situations of real life. Of
course, this is the method of analysis in any field of inquiry: to pose
one's questions in the context of a simplified model in which many of
the less important details of reality are ignored. Thus, even if one is
never involved in a situation in which people's positions are as clearly
defined as those studied by game theorists, one can still come to under-
stand real competitive situations better by studying these hypothetical
examples.
In the language of game theory, a game refers to any social situation
involving two or more individuals. The individuals involved in a game
may be called the players. As stated in the definition above, there are
two basic assumptions that game theorists generally make about players:
they are rational and they are intelligent. Each of these adjectives is
used here in a technical sense that requires some explanation.
A decision-maker is rational if he makes decisions consistently in pur-
suit of his own objectives. In game theory, building on the fundamental
results of decision theory, we assume that each player's objective is to
maximize the expected value of his own payoff, which is measured in
some utility scale. The idea that a rational decision-maker should make
decisions that will maximize his expected utility payoff goes back at least
to Bernoulli (1738), but the modern justification of this idea is due to
von Neumann and Morgenstern (1947). |
null | null | null | null | The idea that a rational decision-maker should make
decisions that will maximize his expected utility payoff goes back at least
to Bernoulli (1738), but the modern justification of this idea is due to
von Neumann and Morgenstern (1947). Using remarkably weak as-
sumptions about how a rational decision-maker should behave, they
showed that for any rational decision-maker there must exist some way
of assigning utility numbers to the various possible outcomes that he
cares about, such that he would always choose the option that maximizes |
null | null | null | null | 1.1 Rationality and Intelligence (cid:9)
3
his expected utility. We call this result the expected-utility maximization
theorem.
It should be emphasized here that the logical axioms that justify the
expected-utility maximization theorem are weak consistency assump-
tions. In derivations of this theorem, the key assumption is generally a
sure-thing or substitution axiom that may be informally paraphrased as
follows: "If a decision-maker would prefer option 1 over option 2 when
event A occurs, and he would prefer option 1 over option 2 when event
A does not occur, then he should prefer option 1 over option 2 even
before he learns whether event A will occur or not." Such an assump-
tion, together with a few technical regularity conditions, is sufficient to
guarantee that there exists some utility scale such that the decision-
maker always prefers the options that give the highest expected utility
value.
Consistent maximizing behavior can also be derived from models of
evolutionary selection. In a universe where increasing disorder is a
physical law, complex organisms (including human beings and, more
broadly speaking, social organizations) can persist only if they behave
in a way that tends to increase their probability of surviving and repro-
ducing themselves. Thus, an evolutionary-selection argument suggests
that individuals may tend to maximize the expected value of some
measure of general survival and reproductive fitness or success (see
Maynard Smith, 1982).
In general, maximizing expected utility payoff is not necessarily the
same as maximizing expected monetary payoff, because utility values
are not necessarily measured in dollars and cents. A risk-averse individ-
ual may get more incremental utility from an extra dollar when he is
poor than he would get from the same dollar were he rich. This obser-
vation suggests that, for many decision-makers, utility may be a nonlin-
ear function of monetary worth. For example, one model that is com-
monly used in decision analysis stipulates that a decision-maker's utility
payoff from getting x dollars would be u(x) = 1 — e', for some number
c that represents his index of risk aversion (see Pratt, 1964). |
null | null | null | null | More gener-
ally, the utility payoff of an individual may depend on many variables
besides his own monetary worth (including even the monetary worths
of other people for whom he feels some sympathy or antipathy).
When there is uncertainty, expected utilities can be defined and com-
puted only if all relevant uncertain events can be assigned probabilities, |
null | null | null | null | 4 (cid:9)
1 • Decision-Theoretic Foundations
which quantitatively measure the likelihood of each event. Ramsey
(1926) and Savage (1954) showed that, even where objective probabili-
ties cannot be assigned to some events, a rational decision-maker should
be able to assess all the subjective probability numbers that are needed
to compute these expected values.
In situations involving two or more decision-makers, however, a spe-
cial difficulty arises in the assessment of subjective probabilities. For
example, suppose that one of the factors that is unknown to some given
individual 1 is the action to be chosen by some other individual 2. To
assess the probability of each of individual 2's possible choices, individual
1 needs to understand 2's decision-making behavior, so 1 may try to
imagine himself in 2's position. In this thought experiment, 1 may
realize that 2 is trying to rationally solve a decision problem of her own
and that, to do so, she must assess the probabilities of each of l's possible
choices. Indeed, 1 may realize that 2 is probably trying to imagine
herself in l's position, to figure out what 1 will do. So the rational
solution to each individual's decision problem depends on the solution
to the other individual's problem. Neither problem can be solved with-
out understanding the solution to the other. Thus, when rational deci-
sion-makers interact, their decision problems must be analyzed together,
like a system of equations. Such analysis is the subject of game theory.
When we analyze a game, as game theorists or social scientists, we say
that a player in the game is intelligent if he knows everything that we
know about the game and he can make any inferences about the situ-
ation that we can make. In game theory, we generally assume that
players are intelligent in this sense. Thus, if we develop a theory that
describes the behavior of intelligent players in some game and we believe
that this theory is correct, then we must assume that each player in the
game will also understand this theory and its predictions.
For an example of a theory that assumes rationality but not intelli-
gence, consider price theory in economics. |
null | null | null | null | For an example of a theory that assumes rationality but not intelli-
gence, consider price theory in economics. In the general equilibrium
model of price theory, it is assumed that every individual is a rational
utility-maximizing decision-maker, but it is not assumed that individuals
understand the whole structure of the economic model that the price
theorist is studying. In price-theoretic models, individuals only perceive
and respond to some intermediating price signals, and each individual
is supposed to believe that he can trade arbitrary amounts at these
prices, even though there may not be anyone in the economy actually
willing to make such trades with him. |
null | null | null | null | 1.2 • Basic Concepts (cid:9)
5
Of course, the assumption that all individuals are perfectly rational
and intelligent may never be satisfied in any real-life situation. On the
other hand, we should be suspicious of theories and predictions that
are not consistent with this assumption. If a theory predicts that some
individuals will be systematically fooled or led into making costly mis-
takes, then this theory will tend to lose its validity when these individuals
learn (from experience or from a published version of the theory itself )
to better understand the situation. The importance of game theory in
the social sciences is largely derived from this fact.
1.2 Basic Concepts of Decision Theory
The logical roots of game theory are in Bayesian decision theory. In-
deed, game theory can be viewed as an extension of decision theory (to
the case of two or more decision-makers), or as its essential logical
fulfillment. Thus, to understand the fundamental ideas of game theory,
one should begin by studying decision theory. The rest of this chapter
is devoted to an introduction to the basic ideas of Bayesian decision
theory, beginning with a general derivation of the expected utility max-
imization theorem and related results.
At some point, anyone who is interested in the mathematical social
sciences should ask the question, Why should I expect that any simple
quantitative model can give a reasonable description of people's behav-
ior? The fundamental results of decision theory directly address this
question, by showing that any decision-maker who satisfies certain in-
tuitive axioms should always behave so as to maximize the mathematical
expected value of some utility function, with respect to some subjective
probability distribution. That is, any rational decision-maker's behavior
should be describable by a utility function, which gives a quantitative
characterization of his preferences for outcomes or prizes, and a subjec-
tive probability distribution, which characterizes his beliefs about all rele-
vant unknown factors. Furthermore, when new information becomes
available to such a decision-maker, his subjective probabilities should be
revised in accordance with Bayes's formula.
There is a vast literature on axiomatic derivations of the subjective
probability, expected-utility maximization, and Bayes's formula, begin-
ning with Ramsey (1926), von Neumann and Morgenstern (1947), and
Savage (1954). |
null | null | null | null | There is a vast literature on axiomatic derivations of the subjective
probability, expected-utility maximization, and Bayes's formula, begin-
ning with Ramsey (1926), von Neumann and Morgenstern (1947), and
Savage (1954). Other notable derivations of these results have been
offered by Herstein and Milnor (1953), Luce and Raiffa (1957), An- |
null | null | null | null | 6 (cid:9)
1 • Decision-Theoretic Foundations
scombe and Aumann (1963), and Pratt, Raiffa, and Schlaiffer (1964);
for a general overview, see Fishburn (1968). The axioms used here are
mainly borrowed from these earlier papers in the literature, and no
attempt is made to achieve a logically minimal set of axioms. (In fact, a
number of axioms presented in Section 1.3 are clearly redundant.)
Decisions under uncertainty are commonly described by one of two
models: a probability model or a state-variable model. In each case, we speak
of the decision-maker as choosing among lotteries, but the two models
differ in how a lottery is defined. In a probability model, lotteries are
probability distributions over a set of prizes. In a state-variable model,
lotteries are functions from a set of possible states into a set of prizes.
Each of these models is most appropriate for a specific class of appli-
cations.
A probability model is appropriate for describing gambles in which
the prizes will depend on events that have obvious objective probabili-
ties; we refer to such events as objective unknowns. These gambles are
the "roulette lotteries" of Anscombe and Aumann (1963) or the "risks"
of Knight (1921). For example, gambles that depend on the toss of a
fair coin, the spin of a roulette wheel, or the blind draw of a ball out
of an urn containing a known population of identically sized but dif-
ferently colored balls all could be adequately described in a probability
model. An important assumption being used here is that two objective
unknowns with the same probability are completely equivalent for de-
cision-making purposes. For example, if we describe a lottery by saying
that it "offers a prize of $100 or $0, each with probability 1/2," we are
assuming that it does not matter whether the prize is determined by
tossing a fair coin or by drawing a ball from an urn that contains 50
white and 50 black balls.
On the other hand, many events do not have obvious probabilities;
the result of a future sports event or the future course of the stock
market are good examples. |
null | null | null | null | On the other hand, many events do not have obvious probabilities;
the result of a future sports event or the future course of the stock
market are good examples. We refer to such events as subjective un-
knowns. Gambles that depend on subjective unknowns correspond to
the "horse lotteries" of Anscombe and Aumann (1963) or the "uncer-
tainties" of Knight (1921). They are more readily described in a state-
variable model, because these models allow us to describe how the prize
will be determined by the unpredictable events, without our having to
specify any probabilities for these events.
Here we define our lotteries to include both the probability and the
state-variable models as special cases. That is, we study lotteries in which |
null | null | null | null | 1.2 Basic Concepts (cid:9)
7
the prize may depend on both objective unknowns (which may be di-
rectly described by probabilities) and subjective unknowns (which must
be described by a state variable). (In the terminology of Fishburn, 1970,
we are allowing extraneous probabilities in our model.)
Let us now develop some basic notation. For any finite set Z, we let
A(Z) denote the set of probability distributions over the set Z. That is,
(1.1) (cid:9)
A(Z) = {q:Z (cid:9)
RI (cid:9)
yEZ
q(y) = 1 and q(z) (cid:9)
0, Vz E Z}.
(Following common set notation, "I" in set braces may be read as "such
that.")
Let X denote the set of possible prizes that the decision-maker could
ultimately get. Let 11 denote the set of possible states, one of which will
be the true state of the world. To simplify the mathematics, we assume
that X and 11 are both finite sets. We define a lottery to be any function
f that specifies a nonnegative real number f(xIt), for every prize x in X
and every state t in CI, such that IxEx f(x I t) = 1 for every t in 11. Let L
denote the set of all such lotteries. That is,
L = {f:11 ---> A(X)}.
For any state t in 11 and any lottery f in L, f(.1t) denotes the probability
distribution over X designated by f in state t. That is,
./-(' I (cid:9) = (f(x t)),,Ex E A(X).
Each number f(x I t) here is to be interpreted as the objective condi-
tional probability of getting prize x in lottery f if t is the true state of
the world. (Following common probability notation, "1" in parentheses
may be interpreted here to mean "given.") |
null | null | null | null | (Following common probability notation, "1" in parentheses
may be interpreted here to mean "given.") For this interpretation to
make sense, the state must be defined broadly enough to summarize all
subjective unknowns that might influence the prize to be received. Then,
once a state has been specified, only objective probabilities will remain,
and an objective probability distribution over the possible prizes can be
calculated for any well-defined gamble. So our formal definition of a
lottery allows us to represent any gamble in which the prize may depend
on both objective and subjective unknowns.
A prize in our sense could be any commodity bundle or resource
allocation. We are assuming that the prizes in X have been defined so
that they are mutually exclusive and exhaust the possible consequences
of the decision-maker's decisions. Furthermore, we assume that each |
null | null | null | null | •
•
8 (cid:9)
1 Decision-Theoretic Foundations
prize in X represents a complete specification of all aspects that the
decision-maker cares about in the situation resulting from his decisions.
Thus, the decision-maker should be able to assess a preference ordering
over the set of lotteries, given any information that he might have about
the state of the world.
The information that the decision-maker might have about the true
state of the world can be described by an event, which is a nonempty
subset of a We let El, denote the set of all such events, so that
= IS I S C SZ and S
01.
For any two lotteries f and g in L and any event S in (cid:9)
we write
f (cid:9)
g iff the lottery f would be at least as desirable as g, in the opinion
of the decision-maker, if he learned that the true state of the world was
in the set S. (Here iff means "if and only if.") That is, f g iff the
decision-maker would be willing to choose the lottery f when he has to
choose between f and g and he knows only that the event S has occurred.
Given this relation we define relations (>s) and (— s) so that
f — s g iff f
g and g f;
f >, g iff f
g and g f
s g means that the decision-maker would be indifferent
That is, f
between f and g, if he had to choose between them after learning S;
and f >, g means that he would strictly prefer f over g in this situation.
We may write >, and — for >n, and —n, respectively. That is,
when no conditioning event is mentioned, it should be assumed that we
are referring to prior preferences before any states in ft are ruled out
by observations. |
null | null | null | null | We may write >, and — for >n, and —n, respectively. That is,
when no conditioning event is mentioned, it should be assumed that we
are referring to prior preferences before any states in ft are ruled out
by observations.
Notice the assumption here that the decision-maker would have well-
defined preferences over lotteries conditionally on any possible event
in a In some expositions of decision theory, a decision-maker's condi-
tional preferences are derived (using Bayes's formula) from the prior
preferences that he would assess before making any observations; but
such derivations cannot generate rankings of lotteries conditionally on
events that have prior probability 0. In game-theoretic contexts, this
omission is not as innocuous as it may seem. Kreps and Wilson (1982)
have shown that the characterization of a rational decision-maker's be-
liefs and preferences after he observes a zero-probability event may be
crucial in the analysis of a game.
For any number a such that 0 5- a 5_ 1, and for any two lotteries f
and g in L, of + (1 — a)g denotes the lottery in L such that |
null | null | null | null | 1.3 • Axioms (cid:9)
9
(af + (1 — a)g)(x1t) = af(x1t) + (1 — a)g(x1t), Vx E X, Vt E
To interpret this definition, suppose that a ball is going to be drawn
from an urn in which a is the proportion of black balls and 1 — a is
the proportion of white balls. Suppose that if the ball is black then the
decision-maker will get to play lottery f and if the ball is white then the
decision-maker will get to play lottery g. Then the decision-maker's
ultimate probability of getting prize x if t is the true state is af(x I t) +
(1 — a)g(xjt). Thus, of + (1 — a)g represents the compound lottery that
is built up from f and g by this random lottery-selection process.
For any prize x, we let [x] denote the lottery that always gives prize x
for sure. That is, for every state t,
(1.2) (cid:9)
[x](ylt) = 1 if y = x, [x](ylt) = 0 if y (cid:9)
x.
Thus, a[x] + (1 — u)[y] denotes the lottery that gives either prize x or
prize y, with probabilities a and 1 — a, respectively.
1.3 Axioms
Basic properties that a rational decision-maker's preferences may be
expected to satisfy can be presented as a list of axioms. Unless otherwise
stated, these axioms are to hold for all lotteries e, f, g, and h in L, for
all events S and T in E, and for all numbers a and 13 between 0 and 1.
Axioms 1.1A and 1.1B assert that preferences should always form a
complete transitive order over the set of lotteries.
AXIOM 1. 1A (COMPLETENESS). f g or g s f.
AXIOM 1. 1B (TRANSITIVITY). |
null | null | null | null | AXIOM 1. 1A (COMPLETENESS). f g or g s f.
AXIOM 1. 1B (TRANSITIVITY). If f zs g and g (cid:9)
h then f -?-s h.
It is straightforward to check that Axiom 1.1B implies a number of
other transitivity results, such as if f — s g and g — s h then f— s h; and
if f >s g and g h then f>, h.
Axiom 1.2 asserts that only the possible states are relevant to the
decision-maker, so, given an event S, he would be indifferent between
two lotteries that differ only in states outside S.
AXIOM 1.2 (RELEVANCE).
If f(•1 t) = g(•It) Vt E S, then f — s g.
Axiom 1.3 asserts that a higher probability of getting a better lottery
is always better. |
null | null | null | null | 10 (cid:9)
1 • Decision-Theoretic Foundations
AXIOM 1.3 (MONOTONICITY). If f >s h and 0 5_ 13<a51, then
of + (1 — a)h >s I3f + ( 1 — (3)h.
Building on Axiom 1.3, Axiom 1.4 asserts that -yf + (1 — y)h gets
better in a continuous manner as y increases, so any lottery that is
ranked between f and h is just as good as some randomization between
f and h.
AXIOM 1.4 (CONTINUITY). If f (cid:9)
some number y such that 0 -y s 1 and g — s -yf + (1 — -y)h.
g and g (cid:9)
h, then there exists
The substitution axioms (also known as independence or sure-thing
axioms) are probably the most important in our system, in the sense
that they generate strong restrictions on what the decision-maker's pref-
erences must look like even without the other axioms. They should also
be very intuitive axioms. They express the idea that, if the decision-
maker must choose between two alternatives and if there are two mu-
tually exclusive events, one of which must occur, such that in each event
he would prefer the first alternative, then he must prefer the first
alternative before he learns which event occurs. (Otherwise, he would
be expressing a preference that he would be sure to want to reverse
after learning which of these events was true!) In Axioms 1.5A and
1.5B, these events are objective randomizations in a random lottery-
selection process, as discussed in the preceding section. In Axioms 1.6A
and 1.6B, these events are subjective unknowns, subsets of f/.
AXIOM 1.5A (OBJECTIVE SUBSTITUTION). |
null | null | null | null | In Axioms 1.6A
and 1.6B, these events are subjective unknowns, subsets of f/.
AXIOM 1.5A (OBJECTIVE SUBSTITUTION). If e f and g >s h
and 0 s a (cid:9)
1, then ae + (1 — a)g (cid:9)
of + (1 — a)h.
AXIOM 1.5B (STRICT OBJECTIVE SUBSTITUTION). If e >s f
h and 0 < a 5_ 1, then ae + (1 — a)g >s of + (1 — a)h.
and g (cid:9)
AXIOM 1.6A (SUBJECTIVE SUBSTITUTION). If f g and f
g and S fl T = 0, then f' suT g.
AXIOM 1.6B (STRICT SUBJECTIVE SUBSTITUTION). If f >s g
and f >T g and S fl T = 0, then f >suT g. |
null | null | null | null | 1.3 Axioms (cid:9)
11
To fully appreciate the importance of the substitution axioms, we may
find it helpful to consider the difficulties that arise in decision theory
when we try to drop them. For a simple example, suppose an individual
would prefer x over y, but he would also prefer .5[y]+ .5[z] over .5[x] +
.5[z], in violation of substitution. Suppose that w is some other prize
that he would consider better than .5[x] + .5[z] and worse than .5[y] +
.5[z]. That is,
x > y but .5[y] + .5[z] > [w] > .5[x] + .5[z].
Now consider the following situation. The decision-maker must first
decide whether to take prize w or not. If he does not take prize w, then
a coin will be tossed. If it comes up Heads, then he will get prize z; and
if it comes up Tails, then he will get a choice between prizes x and y.
What should this decision-maker do? He has three possible decision-
making strategies: (1) take w, (2) refuse w and take x if Tails, (3) refuse
w and take y if Tails. If he follows the first strategy, then he gets the
lottery [w]; if he follows the second, then he gets the lottery .5[x] +
.5[z]; and if he follows the third, then he gets the lottery .5[y] + .5[z].
Because he likes .5[y] + .5[z] best among these lotteries, the third
strategy would be best for him, so it may seem that he should refuse w.
However, if he refuses w and the coin comes up Tails, then his pref-
erences stipulate that he should choose x instead of y. So if he refuses
w, then he will actually end up with z if Heads or x if Tails. But this
lottery .5[x] + .5[z] is worse than w. So we get the contradictory conclu-
sion that he should have taken w in the first place. |
null | null | null | null | But this
lottery .5[x] + .5[z] is worse than w. So we get the contradictory conclu-
sion that he should have taken w in the first place.
Thus, if we are to talk about "rational" decision-making without sub-
stitution axioms, then we must specify whether rational decision-makers
are able to commit themselves to follow strategies that they would sub-
sequently want to change (in which case "rational" behavior would lead
to .5[y] + .5[z] in this example). If they cannot make such commitments,
then we must also specify whether they can foresee their future incon-
stancy (in which case the outcome of this example should be [w]) or not
(in which case the outcome of this example should be .5[x]+ .5[z]). If
none of these assumptions seem reasonable, then to avoid this dilemma
we must accept substitution axioms as a part of our definition of ration-
ality.
Axiom 1.7 asserts that the decision-maker is never indifferent between
all prizes. This axiom is just a regularity condition, to make sure that
there is something of interest that could happen in each state. |
null | null | null | null | 12 (cid:9)
1 Decision-Theoretic Foundations
AXIOM 1.7 (INTEREST). For every state t in S2, there exist prizes y and
z in X such that [y] >m [z].
Axiom 1.8 is optional in our analysis, in the sense that we can state a
version of our main result with or without this axiom. It asserts that the
decision-maker has the same preference ordering over objective gam-
bles in all states of' the world. If this axiom fails, it is because the same
prize might be valued differently in different states.
AXIOM 1.8 (STATE NEUTRALITY). For any two states r and t in ft
if f(lr) = f(.It) and g(.Ir) = g(.1t) and f Z'{r} g, then f (cid:9)
g.
1.4 The Expected-Utility Maximization Theorem
A conditional-probability function on 11 is any function p:!, ---> A(I),) that
specifies nonnegative conditional probabilities p(tIS) for every state t in
f2 and every event S, such that
p(tIS) = 0 if t 0 S, and (cid:9)
Nrls) = 1.
rES
Given any such conditional-probability function, we may write
p(RIS) = (cid:9)
p(rIs), VR C SZ, VS ( E.
>ER
A utility function can be any function from X x 11 into the real numbers
R. A utility function u:X x SZ —> R is state independent iff it does not
actually depend on the state, so there exists some function U:X —> R
such that u(x4) = U(x) for all x and t.
Given any such conditional-probability function p and any utility func-
tion u and given any lottery f in L and any event S in we let E p(u( f )IS)
denote the expected utility value of the prize determined by f, when
p(.IS) is the probability distribution for the true state of the world. |
null | null | null | null | That
is,
E p(u( f )1 S) =- (cid:9)
p(tIS) (cid:9)
u(x,t)f(x1t).
THEOREM 1.1. Axioms I.1AB, 1.2, 1.3, 1.4, 1.5AB, 1.6AB, and 1.7
are jointly satisfied if and only if there exists a utility function u:X X SI —) R
and a conditional-probability function
A(1-1) such that |
null | null | null | null | 1.4 • Expected-Utility Maximization Theorem (cid:9)
13
(1.3) (cid:9)
(1.4) (cid:9)
max u(x,t) = 1 and min u(x,t) = 0, bit E
sEX (cid:9)
xEX
p(R1T) = p(Rjs)p(siT), VR, VS, and VT such that
RCSCTCfland SOO;
(1.5) (cid:9)
f (cid:9)
g if and only if E p(u( f )IS) (cid:9)
E p(u(g)IS),
Vf,g E L, VS E
Furthermore, given these Axioms 1.1AB-1.7, Axiom 1.8 is also satisfied if and
only if conditions (1.3)—(1.5) here can be satisfied with a state-independent utility
function.
In this theorem, condition (1.3) is a normalization condition, asserting
that we can choose our utility functions to range between 0 and 1 in
every state. (Recall that X and Cl are assumed to be finite.) Condition
(1.4) is a version of Bayes's formula, which establishes how conditional
probabilities assessed in one event must be related to conditional prob-
abilities assessed in another. The most important part of the theorem
is condition (1.5), however, which asserts that the decision-maker always
prefers lotteries with higher expected utility. By condition (1.5), once
we have assessed u and p, we can predict the decision-maker's optimal
choice in any decision-making situation. He will choose the lottery with
the highest expected utility among those available to him, using his
subjective probabilities conditioned on whatever event in Cl he has ob-
served. Notice that, with X and Cl finite, there are only finitely many
utility and probability numbers to assess. Thus, the decision-maker's
preferences over all of the infinitely many lotteries in L can be com-
pletely characterized by finitely many numbers. |
null | null | null | null | Notice that, with X and Cl finite, there are only finitely many
utility and probability numbers to assess. Thus, the decision-maker's
preferences over all of the infinitely many lotteries in L can be com-
pletely characterized by finitely many numbers.
To apply this result in practice, we need a procedure for assessing
the utilities u(x,t) and the probabilities p(t1s), for all x, t, and S. As Raiffa
(1968) has emphasized, such procedures do exist, and they form the
basis of practical decision analysis. To define one such assessment pro-
cedure, and to prove Theorem 1.1, we begin by defining some special
lotteries, using the assumption that the decision-maker's preferences
satisfy Axioms 1.1AB-1.7.
Let a, be a lottery that gives the decision-maker one of the best prizes
in every state; and let ao be a lottery that gives him one of the worst
prizes in every state. That is, for every state t, a1(y1t) = 1 = ao(z1t) for
some prizes y and z such that, for every x in X, y x z. Such best |
null | null | null | null | 14 (cid:9)
1 • Decision-Theoretic Foundations
and worst prizes can be found in every state because the preference
relation (....{,}) forms a transitive ordering over the finite set X.
For any event S in E•„ let bs denote the lottery such that
bs(.1t) = a,(.1t) if t E S,
b5(.1t) = a„(•1t) if t 0 S.
That is, bs is a "bet on S" that gives the best possible prize if S occurs
and gives the worst possible prize otherwise.
For any prize x and any state t, let c„,, be the lottery such that
r) = [x](. I r) if r = t,
c,,,(• I r) = ao • I r) if r (cid:9)
t.
That is, co is the lottery that always gives the worst prize, except in state
t, when it gives prize x.
We can now define a procedure to assess the utilities and probabilities
that satisfy the theorem, given preferences that satisfy the axioms. For
each x and t, first ask the decision-maker, "For what number 13 would
you be indifferent between [x] and Ba, + (1 — 13)a, if you knew that t
was the true state of the world?" By the continuity axiom, such a number
must exist. Then let u(x,t) equal the number that he specifies, such that
[x] —{,} u(x,t)a, + (1 — u(x,t))ao.
For each t and 5, ask the decision-maker, "For what number y would
you be indifferent between b{,} and ya, + (1 — y)ao if you knew that
the true state was in S?" Again, such a number must exist, by the
continuity axiom. (The subjective substitution axiom guarantees that
b{1} ?s a0.) Then let p(tls) equal the number that he specifies, such
a,
that
b{,} (cid:9)
p(tIS)a, + (1 — p(t1S))ao. |
null | null | null | null | Then let p(tls) equal the number that he specifies, such
a,
that
b{,} (cid:9)
p(tIS)a, + (1 — p(t1S))ao.
In the proof of Theorem 1.1, we show that defining u and p in this way
does satisfy the conditions of the theorem. Thus, finitely many questions
suffice to assess the probabilities and utilities that completely character-
ize the decision-maker's preferences.
Proof of Theorem 1.1. Let p and u be as constructed above. First, we
derive condition ( 1.5) from the axioms. The relevance axiom and the
definition of u(x,t) implies that, for every state r, |
null | null | null | null | 1.4 .Expected-Utility Maximization Theorem (cid:9)
15
u(x,t)b{, } + (1 - u(x,t))ao.
Then subjective substitution implies that, for every event S,
s u(x,t)b{,} + (1 - u(x,t))ao.
Axioms 1.5A and 1.5B together imply that f (cid:9)
g if and only if
( 1 (cid:9)
i ) f + (1
1 (cid:9)
1,—(1-1 ) ao (cid:9)
1 (cid:9)
(4-1-1 ) g + (1 - 47) %.
1
(Here, 111,1 denotes the number of states in the set SZ.) Notice that
1 (cid:9)
(pi)f
+ (1 (cid:9)
1 (cid:9)
ao = (TO /51 xx f(xl
1
But, by repeated application of the objective substitution axiom,
( (cid:9)
(
I (cid:9)
s
, (cid:9)
%-s (cid:9)
'Ell xEX
f(X10kU(X,t)b{ t} + (1 — U(X,Oad
Efix10(24x,o(p(tis)a,
,E
xEX
+ (1 — p(tIS))aO ) + ( 1 - u(x,t))ao)
(1+1-i) (cid:9)
f(xl t)u(x,t)p(t I S)a,
1 - E E f(xjou(x,00ls)/Ifil)ao
+
(
tES1 xEX
= (Ep(U(PIS)/10ai + (1 (cid:9)
(Ep(U(f)1S)/11/1))a0•
Similarly,
(1/InOg + (1 - (1/111j))ao
-s (Ep(u(g)IS)/Ifil)a l + (1 - (Ep(u(g)15')/Ifil))ao•
Thus, by transitivity, f (cid:9)
g if and only if |
null | null | null | null | 16 (cid:9)
1 • Decision-Theoretic Foundations
(Ep(u(f )1 S)/1,111)a, + (1 — (Ep(u(f )1S)/IniDao
(Ei (it(g)1S)/ini)a l + (1 — (Ep(u(g)IS)/ini))ao•
But by monotonicity, this final relation holds if and only if
E p(u(f)1S) (cid:9)
E p(u(g)IS),
because interest and strict subjective substitution guarantee that
a, >5 ao. Thus, condition (1.5) is satisfied.
Next, we derive condition (1.4) from the axioms. For any events R
and S,
(IRS) bR (cid:9)
(1 (cid:9)
E
RI) 'ER
(I (cid:9)
b{,}
a° (cid:9)
(p(ds)a, + (1 — p(ris))a)
(IR' I)E
,ER
= (
)(P(R1s)a, + (1 — P(RiS))ao) (cid:9)
I R '
( 1
IR' I a'
by objective substitution. Ri is the number of states in the set R.) Then,
using Axioms 1.5A and 1.5B, we get
b,
s p(RIS)a, + (1 — p(RIS))ao.
By the relevance axiom, bs —, a, and, for any r not in S, (cid:9)
— 5 a,.
So the above formula implies (using monotonicity and interest) that
p(riS) = 0 if r S, and p(sls) = 1. Thus, p is a conditional-probability
function, as defined above.
Now, suppose that R CSC T. Using b, — s a, again, we get
b„ (cid:9)
p(RIS)b, + (1 — p(RIS))ao. |
null | null | null | null | Now, suppose that R CSC T. Using b, — s a, again, we get
b„ (cid:9)
p(RIS)b, + (1 — p(RIS))ao.
Furthermore, because b„, bs, and a, all give the same worst prize outside
5, relevance also implies
b„ --r\s p(Ris)b, + (1 — p(RIS))ao.
(Here T\S = {ti t E T, t 0 S}.) So, by subjective and objective substitution,
bR
T P(R I WS + ( 1 — P(R I S))aO
p(R1s)(p(sIT)a, + (1 — p(siT))ao) + (1 — p(RIS))a()
= p(RIS)p(SIT)a, + (1 — p(Ris)p(siT))ao.
(cid:9) |
null | null | null | null | 1.4 • Expected-Utility Maximization Theorem (cid:9)
17
p(RIT)a, + (1 — p(RiT))ao. Also, a, >7- a0, so monotonicity
But b, (cid:9)
implies that p(R1T) = p(Ris)p(sIT). Thus, Bayes's formula (1.4) follows
from the axioms.
If y is the best prize and z is the worst prize in state t, then [y]
{t} a,
and [z] a,), so that u(y,t) = 1 and u(z,t) = 0 by monotonicity. So the
range condition (1.3) is also satisfied by the utility function that we have
constructed.
If state neutrality is also given, then the decision-maker will give us
the same answer when we assess u(x,t) as when we assess u(x,r) for any
other state r (because [x] —{,} Pa, + (1 — (3)a, implies [x] pa, +
(1 — p)a(), and monotonicity and interest guarantee that his answer is
unique). So Axiom 1.8 implies that u is state-independent.
To complete the proof of the theorem, it remains to show that the
existence of functions u and p that satisfy conditions (1.3)—(1.5) in the
theorem is sufficient to imply all the axioms (using state independence
only for Axiom 1.8). If we use the basic mathematical properties of the
expected-utility formula, verification of the axioms is straightforward.
To illustrate, we show the proof of one axiom, subjective substitution,
and leave the rest as an exercise for the reader.
Suppose that f (cid:9)
g and f (cid:9)
E p(u(g)1S) and E p(u( f )IT) (cid:9)
g and S fl T = 0. By (1.5), E p(u(f)IS)
E p(u(g)IT). |
null | null | null | null | By (1.5), E p(u(f)IS)
E p(u(g)IT). But Bayes's formula (1.4)
implies that
Ep(u(f)1S U T) =
p(tis U T)f(x1t)u(x,t)
r
= E E p(ths)p(sis U T)f(x1t)u(x,t)
/ES vEX
+ (cid:9)
E POI nP(TIs U T)f(xlt)u(x,t)
1E7' vEX
= p(SIS U T)E p(u( f )1 S) + pals U T)E p(u( f )1S)
and
Ep(u(g)1S U T) = p(SIS U T)Ep(u(g)IS) + pals U T)E p(u(g)IS).
So Ep(u(f)IS U T) Ep(u(g)IS U T) and f z.sur g. • |
null | null | null | null | 18 (cid:9)
1 Decision-Theoretic Foundations
1.5 Equivalent Representations
When we drop the range condition (1.3), there can be more than one
pair of utility and conditional-probability functions that represent the
same decision-maker's preferences, in the sense of condition (1.5). Such
equivalent representations are completely indistinguishable in terms of
their decision-theoretic properties, so we should be suspicious of any
theory of economic behavior that requires distinguishing between such
equivalent representations. Thus, it may be theoretically important to
be able to recognize such equivalent representations.
Given any subjective event S, when we say that a utility function v
and a conditional-probability function q represent the preference order-
ing we mean that, for every pair of lotteries f and g, Eq(v(f)IS)
Eq(v(g)IS) if and only if f (cid:9)
g.
THEOREM 1.2. Let S in EI be any given subjective event. Suppose that the
decision-maker's preferences satisfy Axioms 1.1AB through 1.7, and let u and p
be utility and conditional-probability functions satisfying (1.3)—(1.5) in Theorem
1.1. Then v and q represent the preference ordering >s if and only if there
exists a positive number A and a function B:S —> R such that
q(tIS)v(x,t) = Ap(tIS)u(x,t) + B(t), Vt E S, Nix E X.
Proof. Suppose first that A and 13(•) exist as described in the theorem. |
null | null | null | null | Suppose first that A and 13(•) exist as described in the theorem.
Then, for any lottery f,
E q(v( f )1 S) (cid:9)
f(xlt)q(tIS)v(x,t)
IES xEX
= (cid:9)
E focloop(o)u(x,t) + B(0)
IES xEX
= AEIf(Xlop(tis)u(x,t) + (cid:9)
B(t) (cid:9)
foclo
tES xEX (cid:9)
tES (cid:9)
xEX
= AE p (u(f)IS) + (cid:9)
B(t),
because E„,„ f(x I = 1. So expected v-utility with respect to q is an
increasing linear function of expected u-utility with respect to p, because
A > 0. Thus, Ea(v(f)IS) (cid:9)
Ep(u(g)IS), and so v and q together represent the same preference
ordering over lotteries as u and p.
Eg(v(g)IS) if and only if Ep(u(f)IS) |
null | null | null | null | 1.5 • Equivalent Representations (cid:9)
19
Conversely, suppose now that v and q represent the same preference
ordering as u and p. Pick any prize x and state t, and let
X =
E q(v(cx
t)I S) — E q(v(ao)IS)
'
E q(v (a 1) I S) — E q(v(ao)I S)
Then, by the linearity of the expected -value operator,
Eq(v(ka, + (1 — X)a,) I S) = E q(v(a0)1 S) + X(E q(v(a 01 S) — E q(v(a0)1 S))
= E q(v(c ,t)I S) ,
so cx,, —, Xa, + (1 — X.)a,. In the proof of Theorem 1.1, we constructed
u and p so that
c„,, - - s u(x ,t)b{,} + (1 — u(x ,t))ao
—s u(x,t)(P(ti S)al + (1 — P(tIS))a,) + (1 — u(x ,t))ao
—s POI S)u(x ,t)c t 1 + (1 — p(t1s)u(x,t))ao.
The monotonicity axiom guarantees that only one randomization be-
tween a, and a, can be just as good as c„,„ so
X = p(t1s)u(x,t).
But c,, differs from a, only in state t, where it gives prize x instead of
the worst prize, so
Eq(v(c„,)1 5) — Eq(v(ao) I S) = q(t I S) (v(x a) — min v(z,t)) . |
null | null | null | null | zEX
Thus, going back to the definition of X, we get
p(tis)u(x,t) .---
q(t I S)(v(x a) — min v(z,t))
zEX
E q(v(a 1)1 S) — E q(v(a0)I S)
Now let
A = E q(v (a 01 S) — E q(v (601 S) ,
and let
B(t)' = q(tI S) min v(z,t).
zEX |
null | null | null | null | 20 (cid:9)
1 • Decision-Theoretic Foundations
Then
Ap(tIS)u(x,t) + B(t) = q(tIS)v(x,t).
Notice that A is independent of x and t and that B(t) is independent of
x. In addition, A > 0, because a, >s a, implies Eq(v(a,)IS) >
E ,(v (a 0)1S). n
It is easy to see from Theorem 1.2 that more than one probability
distribution can represent the decision-maker's beliefs given some event
S. In fact, we can make the probability distribution q(• IS) almost any-
thing and still satisfy the equation in Theorem 1.2, as long as we make
reciprocal changes in v, to keep the left-hand side of the equation the
same. The way to eliminate this indeterminacy is to assume Axiom 1.8
and require utility functions to be state independent.
THEOREM 1.3. Let S in 5, be any given subjective event. Suppose that the
decision-maker's preferences satisfy Axioms 1.IAB through 1.8, and let u and p
be the state-independent utility function and the conditional-probability function,
respectively, that satisfy conditions (1.3)—(1.5) in Theorem 1.1. Let v be a state-
independent utility function, let q be a conditional-probability function, and
suppose that v and q represent the preference ordering Then
q(11S) = (cid:9)
Vt E S,
and there exist numbers A and C such that A > 0 and
v(x) = Au(x) + C, Vx E X.
(For simplicity, we can write v(x) and u(x) here, instead of v(x,t) and u(x,t),
because both functions are state independent.)
Proof. Let A =
,(v(a S) —
from the proof of Theorem 1.2,
q(v(ao) I S), and let C = minzExv(z). |
null | null | null | null | Proof. Let A =
,(v(a S) —
from the proof of Theorem 1.2,
q(v(ao) I S), and let C = minzExv(z). Then,
Ap(t1S)u(x) + q(tIS)C = q(tIS)v(x), Vx E X, Vt E S.
Summing this equation over all t in S, we get Au(x) + C = v(x). Then,
substituting this equation back, and letting x be the best prize so u(x) =
1, we get
Ap(o) + q(tIS)C = Aq(t1S) + q(tIS)C.
Because A > 0, we get p(tis) = q(tIS). n |
null | null | null | null | 1.6 • Bayesian Conditional-Probability Systems (cid:9)
21
1.6 Bayesian Conditional-Probability Systems
We define a Bayesian conditional-probability system (or simply a conditional-
probability system) on the finite set f2 to be any conditional-probability
function p on SI that satisfies condition (1.4) (Bayes's formula). That is,
if p is a Bayesian conditional-probability system on 12, then, for every S
that is a nonempty subset of SI, p(.Is) is a probability distribution over
,i/ such that p(sIs) = 1 and
p(RIT) = p(Rls)p(sIT), VR C S, VT D S.
We let L1*(0) denote the set of all Bayesian conditional-probability sys-
tems on Cl.
For any finite set Z, we let A°(Z) denote the set of all probability
distributions on Z that assign positive probability to every element in Z,
so
(1.6) (cid:9)
AV) = {q E A(Z)lq(z) > 0, Vz E Z}.
Any probability distribution fi in A°(.11) generates a conditional-proba-
bility system p in A*(a) by the formula
p(tIs) —
p
if t E S,
P
(r)
rES
p(tIs) = 0 if t (cid:9)
S.
The conditional-probability systems that can be generated in this way
from distributions in A°(C2) do not include all of 0*(f2), but any other
Bayesian conditional-probability system in A*(f2) can be expressed as
the limit of conditional-probability systems generated in this way. This
fact is asserted by the following theorem. For the proof, see Myerson
(1986b).
THEOREM 1 . 4 . |
null | null | null | null | This
fact is asserted by the following theorem. For the proof, see Myerson
(1986b).
THEOREM 1 . 4 . The probability function p is a Bayesian conditional-prob-
ability system in
if and only if there exists a sequence of probability
distributions {fik },7_, in A°(S2) such that, for every nonempty subset S of Cl and
every t in Cl,
*(Cl)
A
fik(t)
PO's) = lim E fik(r)
if t E S,
rES
p(tIs) = 0 if tlf S. |
null | null | null | null | 22 (cid:9)
1 Decision-Theoretic Foundations
1.7 Limitations of the Bayesian Model
We have seen how expected-utility maximization can be derived from
axioms that seem intuitively plausible as a characterization of rational
preferences. Because of this result, mathematical social scientists have
felt confident that mathematical models of human behavior that are
based on expected-utility maximization should have a wide applicability
and relevance. This book is largely motivated by such confidence.
It is important to try to understand the range of applicability of
expected-utility maximization in real decision-making. In considering
this question, we must remember that any model of decision-making
can be used either descriptively or prescriptively. That is, we may use a
model to try to describe and predict what people will do, or we may use
a model as a guide to apply to our own (or our clients') decisions. The
predictive validity of a model can be tested by experimental or empirical
data. The prescriptive validity of a decision model is rather harder to
test; one can only ask whether a person who understands the model
would feel that he would be making a mistake if he did not make
decisions according to the model.
Theorem 1.1, which derives expected-utility maximization from in-
tuitive axioms, is a proof of the prescriptive validity of expected-utility
maximization, if any such proof is possible. Although other models of
decision-making have been proposed, few have been able to challenge
the logical appeal of expected-utility maximization for prescriptive pur-
poses.
There is, of course, a close relationship between the prescriptive and
predictive roles of any decision-making model. If a model is prescrip-
tively valid for a decision-maker, then he diverges from the model only
when he is making a mistake. People do make mistakes, but they try
not to. When a person has had sufficient time to learn about a situation
and think clearly about it, we can expect that he will make relatively
few mistakes. Thus, we can expect expected-utility maximization to be
predictively accurate in many situations. |
null | null | null | null | When a person has had sufficient time to learn about a situation
and think clearly about it, we can expect that he will make relatively
few mistakes. Thus, we can expect expected-utility maximization to be
predictively accurate in many situations.
However, experimental research on decision-making has revealed
some systematic violations of expected-utility maximization (see Allais
and Hagen, 1979; Kahneman and Tversky, 1979; and Kahneman,
Slovic, and Tversky, 1982). This research has led to suggestions of new
models of decision-making that may have greater descriptive accuracy
(see Kahneman and Tversky, 1979; and Machina, 1982). We discuss |
null | null | null | null | 1.7 • Limitations of the Bayesian Model (cid:9)
23
here three of the best-known examples in which people often seem to
violate expected-utility maximization: one in which utility functions
seem inapplicable, one in which subjective probability seems inapplica-
ble, and one in which any economic model seems inapplicable.
Consider first a famous paradox, due to M. Allais (see Allais and
Hagen, 1979). Let X = {$12 million, $1 million, $0}, and let
= .10[$12 million] + .90[$0],
f2 = .11[$1 million] + .89[$0],
f3 = [$1 million],
f, = .10[$12 million] + .89[$1 million] + .01[$0].
Many people will express the preferences f, > f2 and f3 > f4. (Recall
that no subscript on > means that we are conditioning on D..) Such
people may feel that $12 million is substantially better than $1 million,
so the slightly higher probability of winning in f2 compared with f, is
not worth the lower prize. On the other hand, they would prefer to
take the sure $1 million in f3, rather than accept a probability .01 of
getting nothing in exchange for a probability .10 of increasing the prize
to $12 million in f,.
Such preferences cannot be accounted for by any utility function. To
prove this, notice that
.5f, + .5f3 = .05[$12 million] + .5[$1 million] + .45[$0]
= .5f2
Thus, the common preferences f, > f2 and f3 > f4 must violate the strict
objective substitution axiom.
Other paradoxes have been generated that challenge the role of
subjective probability in decision theory, starting with a classic paper by
Ellsberg (1961). |
null | null | null | null | Other paradoxes have been generated that challenge the role of
subjective probability in decision theory, starting with a classic paper by
Ellsberg (1961). For a simple example of this kind, due to Raiffa (1968),
let X = {—$100,$100}, let SZ = {A,N}, and let
bA($1001A) = 1 = bA(—$1001N),
b,(—$1001A) = 1 = bN($100IN).
That is, bA is a $100 bet in which the decision-maker wins if A occurs,
and bN is a $100 bet in which the decision-maker wins if N occurs.
Suppose that A represents the state in which the American League will
win the next All-Star game (in American baseball) and that N represents |
null | null | null | null | 24 (cid:9)
1 • Decision-Theoretic Foundations
the state in which the National League will win the next All-Star game.
(One of these two leagues must win the All-Star game, because the rules
of baseball do not permit ties.)
Many people who feel that they know almost nothing about American
baseball express the preferences .5[$100] + .5[—$100] > 6, and .5[$100]
+ .5[ —$100] > b„,. That is, they would strictly prefer to bet $100 on
Heads in a fair coin toss than to bet $100 on either league in the All-
Star game. Such preferences cannot be accounted for by any subjective
probability distribution over II At least one state in I/ must have prob-
ability greater than or equal to .5, and the bet on the league that wins
in that state must give expected utility that is at least as great as the bet
on the fair coin toss. To see it another way, notice that
.506, + .506, = .5[$100] + .5[—$100]
= .50(.5[$100] + .5[—$100]) + .50(.5[$100] + .5[—$100]),
so the common preferences expressed above must violate the strict
objective substitution axiom.
To illustrate the difficulty of constructing a model of decision-making
that is both predictively accurate and prescriptively appealing, Kahne-
man and Tversky (1982) have suggested the following example. In
Situation A, you are arriving at a theatrical performance, for which you
have bought a pair of tickets that cost $40. You suddenly realize that
your tickets have fallen out of your pocket and are lost. You must decide
whether to buy a second pair of tickets for $40 (there are some similar
seats still available) or simply go home. In Situation B, you are arriving
at a theatrical performance for which a pair of tickets costs $40. You
did not buy tickets in advance, but you put $40 in your pocket when
you left home. You suddenly realize that the $40 has fallen out of your
pocket and is lost. |
null | null | null | null | You
did not buy tickets in advance, but you put $40 in your pocket when
you left home. You suddenly realize that the $40 has fallen out of your
pocket and is lost. You must decide whether to buy a pair of tickets for
$40 with your charge card (which you still have) or simply go home.
As Kahneman and Tversky (1982) report, most people say that they
would simply go home in Situation A but would buy the tickets in
Situation B. However, in each of these situations, the final outcomes
resulting from the two options are, on the one hand, seeing the perfor-
mance and being out $80 and, on the other hand, missing the perfor-
mance and being out $40. Thus, it is impossible to account for such
behavior in any economic model that assumes that the levels of monetary |
null | null | null | null | 1.7 • Limitations of the Bayesian Model (cid:9)
25
wealth and theatrical consumption are all that should matter to the
decision-maker in these situations.
Any analytical model must derive its power from simplifying assump-
tions that enable us to see different situations as analytically equivalent,
but such simplifying assumptions are always questionable. A model that
correctly predicts the common behavior in this example must draw
distinctions between situations on the basis of fine details in the order
of events that have no bearing on the final outcome. Such distinctions,
however, would probably decrease the normative appeal of the model
if it were applied for prescriptive purposes. (What would you think of
a consultant who told you that you should make a point of behaving
differently in Situations A and B?)
The explanatory power of expected-utility maximization can be ex-
tended to explain many of its apparent contradictions by the analysis
of salient perturbations. A perturbation of a given decision problem is any
other decision problem that is very similar to it (in some sense). For any
given decision problem, we say that a perturbation is salient if people
who actually face the given decision problem are likely to act as if they
think that they are in this perturbation. A particular perturbation of a
decision problem may be salient when people find the decision problem
to be hard to understand and the perturbation is more like the kind of
situations that they commonly experience. If we can predict the salient
perturbation of an individual's decision problem, then the decision that
maximizes his expected utility in this salient perturbation may be a more
accurate prediction of his behavior.
For example, let us reconsider the problem of betting on the All-Star
game. |
null | null | null | null | For example, let us reconsider the problem of betting on the All-Star
game. To get a decision-maker to express his preference ordering (.--11)
over fb,, 6,, .5[$100] + .5[—$100]}, we must ask him, for each pair in
this set, which bet would he choose if this pair of bet-options were
offered to him uninformatively, that is, in a manner that does not give
him any new information about the true state in I. That is, when we
ask him whether he would prefer to bet $100 on the American League
or on a fair coin toss, we are assuming that the mere fact of offering
this option to him does not change his information about the All-Star
game. However, people usually offer to make bets only when they have
some special information or beliefs. Thus, when someone who knows
little about baseball gets an offer from another person to bet on the
American League, it is usually because the other person has information
suggesting that the American League is likely to lose. In such situations, |
null | null | null | null | 26 (cid:9)
1 Decision-Theoretic Foundations
an opportunity to bet on one side of the All-Star game should (by Bayes's
formula) make someone who knows little about baseball decrease his
subjective probability of the event that this side will win, so he may well
prefer to bet on a fair coin toss. We can try to offer bets uninformatively
in controlled experiments, and we can even tell our experimental sub-
jects that the bets are being offered uninformatively, but this is so
unnatural that the experimental subjects may instead respond to the
salient perturbation in which we would only offer baseball bets that we
expected the subject to lose.
1.8 Domination
Sometimes decision-makers find subjective probabilities difficult to as-
sess. There are fundamental theoretical reasons why this should be
particularly true in games. In a game situation, the unknown environ-
ment or "state of the world" that confronts a decision-maker may in-
clude the outcome of decisions that are to be made by other people.
Thus, to assess his subjective probability distribution over this state
space, the decision-maker must think about everything that he knows
about other people's decision-making processes. To the extent that these
other people are concerned about his own decisions, his beliefs about
their behavior may be based at least in part on his beliefs about what
they believe that he will do himself. So assessing subjective probabilities
about others' behavior may require some understanding of the pre-
dicted outcome of his own decision-making process, part of which is his
probability assessment itself. The resolution of this seeming paradox is
the subject of game theory, to be developed in the subsequent chapters
of this book.
Sometimes, however, it is possible to say that some decision-options
could not possibly be optimal for a decision-maker, no matter what his
beliefs may be. In this section, before turning from decision theory to
game theory, we develop some basic results to show when such proba-
bility-independent statements can be made.
Consider a decision-maker who has a state-dependent utility function
u:X x SZ --> R and can choose any x in X. That is, let us reinterpret X
as the set of decision-options available to the decision-maker. |
null | null | null | null | Consider a decision-maker who has a state-dependent utility function
u:X x SZ --> R and can choose any x in X. That is, let us reinterpret X
as the set of decision-options available to the decision-maker. If his
subjective probability of each state t in Si were p(t) (that is, p(t) = 0111),
Vt E 14), then the decision-maker would choose some particular y in X
only if |
null | null | null | null | 1.8 Domination (cid:9)
27
(1.7) (cid:9)
E p(t)u(y,t) (cid:9) E p(t)u(x,t), vx E X.
(En (cid:9)
tEO
Convexity is an important property of many sets that arise in math-
ematical economics. A set of vectors is convex iff, for any two vectors p
and q and any number X between 0 and 1, if p is in the set and q is in
the set then the vector Xp + (1 — X)q must also be in the set. Geomet-
rically, convexity means that, for any two points in the set, the whole
line segment between them must also be contained in the set.
THEOREM 1.5. Given u:X X (cid:9)
in A(n) such that y is optimal is convex.
R and given y in X, the set of all p
Proof. Suppose that y would be optimal for the decision-maker with
beliefs p and q. Let X be any number between 0 and 1, and let r =
Xp + (1 — X)q. Then for any x in X
E r(t)u(y,t) = X E p(ou(y,t) + (1 — X) E q(t)u(y,t)
tEll (cid:9)
t(f2
X (cid:9)
p(t)u(x,t) + (1 — X) E q(t)u(x,t)
rE0 (cid:9)
(En
E r(t)u(x,t).
tES1
So y is optimal for beliefs r. n
For example, suppose X = {a,13,)/}, f = {01 ,02}, and the utility func-
tion u is as shown in Table 1.1. With only two states, p(01 ) = 1 — p(02). |
null | null | null | null | With only two states, p(01 ) = 1 — p(02).
The decision a is optimal for the decision-maker iff the following two
inequalities are both satisfied:
8p(01 ) + 1(1 — p(01 )) (cid:9)
5p(0,) + 3(1 — p(01 ))
8p(01 ) + 1(1 — p(01 )) > 4p(01 ) + 7(1 — p(01 )).
Table 1.1 Expected utility payoffs for states 0, and 02
Decision
a (cid:9)
e, (cid:9)
8 (cid:9)
5 (cid:9)
4 (cid:9)
02
1
3
7
(cid:9) |
null | null | null | null | 28 (cid:9)
1 Decision-Theoretic Foundations
The first of these inequalities asserts that the expected utility payoff
from a must be at least as much as from p, and the second asserts that
the expected utility payoff from a must be at least as much as from y.
By straightforward algebra, these inequalities imply that a is optimal
when p(01) 0.6.
Similarly, decision y would be optimal iff
4p(01) + 7(1 — p(01 )) > 8p(01 ) + 1(1 — p(01 ))
4p(01 ) + 7(1 — P(0I)) > 5p(01) + 3(1 — PA%
and these two inequalities are both satisfied when p(01) lc 0.6. Decision
13 would be optimal iff
8p(01 ) + 1(1 — p(01 ))
5p(01 ) + 3(1 — p(01 ))
5p(01) + 3(1 — p(01)) > 4p(0 1 ) + 7(1 — p(0,)),
but there is no value of p(0,) that satisfies both of these inequalities.
Thus, each decision is optimal over some convex interval of probabili-
ties, except that the interval where 13 is optimal is the empty set.
Thus, even without knowing p, we can conclude that 0 cannot possibly
be the optimal decision for the decision-maker. Such a decision-option
that could never be optimal, for any set of beliefs, is said to be strongly
dominated.
Recognizing such dominated options may be helpful in the analysis
of decision problems. Notice that a would be best if the decision-maker
were sure that the state was 0,, and y would be best if the decision-
maker were sure that the state was 02, so it is easy to check that neither
a nor -y is dominated in this sense. Option 13 is a kind of intermediate
decision, in that it is neither best nor worst in either column of the
payoff table. However, such intermediate decision-options are not nec-
essarily dominated. |
null | null | null | null | Option 13 is a kind of intermediate
decision, in that it is neither best nor worst in either column of the
payoff table. However, such intermediate decision-options are not nec-
essarily dominated. For example, if the utility payoffs from decision 13
were changed to 6 in both states, then 13 would be the optimal decision
whenever 5/7 p(01 ) 1/3. On the other hand, if the payoffs from
decision 13 were changed to 3 in both states, then it would be obvious
that 13 could never be optimal, because choosing y would always be
better than choosing 13.
There is another way to see that 13 is dominated, for the original
payoff table shown above. Suppose that the decision-maker considered
the following randomized strategy for determining his decision: toss a
coin, and choose a if it comes out Heads, and choose y if it comes out |
null | null | null | null | 1.8 Domination (cid:9)
29
Tails. We may denote this strategy by .5[a] + .5[y], because it gives a
probability of .5 to a and -y each. If the true state were 01, then this
randomized strategy would give the decision-maker an expected utility
payoff of .5 x 8 + .5 x 4 = 6, which is better than the payoff of 5 that
he would get from 13. (Recall that, because these payoffs are utilities,
higher expected values are always more preferred by the decision-
maker.) If the true state were 02, then this randomized strategy would
give an expected payoff of .5 x 1 + .5 x 7 = 4, which is better than
the payoff of 3 that he would get from 13. So no matter what the state
may be, the expected payoff from .5[a] + .5[y] is strictly higher than
the payoff from 13. Thus, we may argue that the decision-maker would
be irrational to choose 13 because, whatever his beliefs about the state
might be, he would get a higher expected payoff from the randomized
strategy .5[a] + .5[-A than from choosing 13. We may say that 13 is
strongly dominated by the randomized strategy .5[a] + .5[y].
In general, a randomized strategy is any probability distribution over
the set of decision options X. We may denote such a randomized strategy
in general by o- = (cr(x))xEx, where o-(x) represents the probability of
choosing x. Given the utility function u:X x SZ —> R, we may say that a
decision option y in X is strongly dominated by a randomized strategy o-
in A(X) such that
(1.8) (cid:9)
xEX
cr(x)u(x,t) > u(y,t), Vt E
That is, y is strongly dominated by o- if, no matter what the state might
be, o- would always be strictly better than y under the expected-utility
criterion.
We have now used the term "strongly dominated" in two different
senses. The following theorem asserts that they are equivalent. |
null | null | null | null | We have now used the term "strongly dominated" in two different
senses. The following theorem asserts that they are equivalent.
THEOREM 1.6. Given u:X x f/ —> R, where X and ft are nonempty finite
sets, and given any y in X, there exists a randomized strategy o- in A(X) such
that y is strongly dominated by cr, in the sense of condition (1.8), if and only if
there does not exist any probability distribution p in A(11) such that y is optimal
in the sense of condition (1.7).
The proof is deferred to Section 1.9.
Theorem 1.6 gives us our first application of the important concept
of a randomized strategy. Notice, however, that this result itself does |
End of preview.
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