Non-Cooperative Game Theory

263 Pages·2015·4.994 MB·English

by  Takako Fujiwara-Greve (auth.)

Monographs in Mathematical Economics 1
Takako Fujiwara-Greve
Non-Cooperative
Game Theory
Monographs in Mathematical Economics
Volume 1
Editor-in-chief
Toru Maruyama, Tokyo, Japan
Series editors
Shigeo Kusuoka, Tokyo, Japan
Jean-Michel Grandmont, Malakoff, France
R. Tyrrell Rockafellar, Seattle, USA
More information about this series at http://www.springer.com/series/13278
Takako Fujiwara-Greve
Non-Cooperative
Game Theory
123
Takako Fujiwara-Greve
Department ofEconomics
KeioUniversity
Minato-ku,Tokyo
Japan
ISSN 2364-8279 ISSN 2364-8287 (electronic)
Monographsin Mathematical Economics
ISBN978-4-431-55644-2 ISBN978-4-431-55645-9 (eBook)
DOI 10.1007/978-4-431-55645-9
LibraryofCongressControlNumber:2015939677
SpringerTokyoHeidelbergNewYorkDordrechtLondon
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To Henrich, Jan, and Ryo
Preface
This book is based on my Japanese book Hikyouryoku Game Riron (Non-
Cooperative Game Theory), published in 2011 by Chisen-Shokan, with some
modifications for international readers. Both books grew out of my lecture notes
used for teaching at Keio University, Waseda University, and the Norwegian
Business School (BI) over more than a decade. Like my lectures, this book covers
topics from basics to graduate-level ones. I also included many exercises that my
students did in the class.
I constructed the chapters according to solution concepts, which are ways to
predict outcomes, or, in other words, are theories on their own. Game theory is a
collectionofsuchtheories,andusersofgametheoryshouldchoose anappropriate
solutionconceptforeachsituationtomakeagoodprediction.However,thereisno
completeagreementofwhatsolutionconceptshouldbeusedforparticulargames,
even among game theorists. For example, I advocate using sequential equilibrium
for extensive form games with imperfect information, but some people might use
perfect Bayesian equilibrium. Therefore, in this book, I do not make a corre-
spondence between a class of games and a solution concept. Rather, I line up
solution concepts and let the readers decide which one to apply when they face a
game to analyze.
Although I made every effort to avoid incorrect or misleading expressions, no
bookisfreeoferrors.Evenmyownopinionmaychangeovertime.Therefore,Iset
up a website for corrections and clarifications at http://web.econ.keio.ac.jp/staff/
takakofg/gamebook.html.
(Pleasenotethat,inthelongrun,theURLmaychange.Thelifeofabookisusually
longer than the life of a web page.) I also regret that I could not cover some
important topics in non-cooperative game theory, such as epistemic game theory
and learning models.
For beginners, I recommend reading chapters and sections without a star and
doing some exercises. Juniors and seniors at universities can read chapters and
sections with a single star. For those who want to study game theory at a graduate
level,chaptersandsectionswithtwostarsareuseful,andafterthat,readersshould
go on to the research papers in the references.
vii
viii Preface
I thank Toru Maruyama for coordinating with Springer Japan to publish this
book.Twoanonymousrefereesprovidedvaluablecommentstoimprovethequality
ofthebook.ManyJapanesepeoplewhohelpedmeinwritingtheJapaneseversion
of this book are already acknowledged there, so here I list mainly non-Japanese
individuals who helped me in my study and career. But first of all, I must thank
Masahiro Okuno-Fujiwara, who wrote a recommendation letter that helped me get
into the Stanford Graduate School of Business (GSB) Ph.D. program. Without his
help, my whole career would not have developed like this. Now he is one of my
most important co-authors as well. My teachers at Stanford GSB, in particular,
David Kreps, John Roberts, Faruk Gul, and Robert Wilson, were helpful at many
points of my study there. After GSB days, I am grateful to Ehud Kalai, Geir
Asheim, Carsten Krabbe Nielsen, Christian Riis, Joel Watson, Nicholas Yannelis,
MarcusBerliant,MichihiroKandori,AkihikoMatsui,andMasakiAoyagifortheir
support.DaisukeOyama,whousedtheJapaneseversionofthebookinhiscourse,
gave me comments that led to improvements of this book. I also thank Masayuki
Yao, Hayato Shimura, and other students in my recent courses at Keio University
whocheckedthedraftcarefully.AllisonKoriyamaproofreadtheentiremanuscript
and corrected the language thoroughly. All remaining errors are mine. Finally, but
of course not the least, I thank my husband and a co-author, Henrich R. Greve for
his constant support from the first term of the Ph.D. course at GSB.
Tokyo Takako Fujiwara-Greve
March 2015
Contents
1 Games in Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Games in Game Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Non-cooperative and Cooperative Games. . . . . . . . . . . . . . . . 2
1.3 Components of a Game. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Strategic Dominance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Prisoner’s Dilemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Strict Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Common Knowledge of a Game. . . . . . . . . . . . . . . . . . . . . . 11
2.4 Iterative Elimination of Strictly Dominated Strategies . . . . . . . 11
2.5 Weak Dominance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Maximin Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.7 Matrix Representation of 3-Player Games . . . . . . . . . . . . . . . 18
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Nash Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Cournot Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Bertrand Game. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 When Products Are Differentiated. . . . . . . . . . . . . . . 30
3.3.2 When Products Are Perfect Substitutes . . . . . . . . . . . 31
3.4 Location Choice Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Strategic Dominance and Nash Equilibrium(cid:1) . . . . . . . . . . . . . 34
3.6 Existence of Nash Equilibrium and Mixed Strategies. . . . . . . . 36
3.7 Existence Theorem(cid:1)(cid:1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8 Rationalizability(cid:1)(cid:1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
ix
x Contents
4 Backward Induction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Extensive Form Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Strategies in an Extensive-Form Game . . . . . . . . . . . . . . . . . 63
4.3 Backward Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Chain Store Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Stackelberg Game. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6 Ultimatum Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.7 Alternating Offer Bargaining Game. . . . . . . . . . . . . . . . . . . . 76
4.8 Introduction of Nature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.9 Common Knowledge of Rationality
and Backward Induction(cid:1)(cid:1) . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 Subgame Perfect Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1 Subgame Perfect Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Capacity Choice Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 Prisoner’s Dilemma of Neuroeconomics . . . . . . . . . . . . . . . . 95
5.4 Finitely Repeated Games. . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5 Infinitely Repeated Games. . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.6 Equilibrium Collusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.7 Perfect Folk Theorem(cid:1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.8 Repeated Games with Non-simultaneous Moves(cid:1) . . . . . . . . . . 119
5.9 Repeated Games with Overlapping Generations(cid:1). . . . . . . . . . . 123
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6 Bayesian Nash Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.1 Formulation of Games with Incomplete Information . . . . . . . . 133
6.2 Bayesian Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2.1 Ex-Ante Optimization . . . . . . . . . . . . . . . . . . . . . . . 138
6.2.2 Optimization by ‘Type’ Players . . . . . . . . . . . . . . . . 140
6.3 Cournot Game with Incomplete Information. . . . . . . . . . . . . . 140
6.4 Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.5 Harsanyi’s Purification Theorem(cid:1) . . . . . . . . . . . . . . . . . . . . . 146
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7 Perfect Bayesian Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.1 Extensive-Form Games with Incomplete Information. . . . . . . . 153
7.2 Signaling Games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.3 Pooling and Separating Equilibrium . . . . . . . . . . . . . . . . . . . 160