Game Theory: An open access textbook with 165 solved exercises

2015·5.2 MB·English

by  Giacomo Bonanno

GAME THEORY
An open access textbook with 165 solved exercises

Giacomo Bonanno
University of California, Davis
http://www.econ.ucdavis.edu/faculty/bonanno/

© 2015 Giacomo Bonanno

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/4.0/. Other than as provided
by this license, no part of this book may be reproduced, transmitted, or displayed by any electronic or mechanical means
without permission from the author.
CONTENTS
Preface and acknowledgments .......................................................... vi
0. Introduction .................................................................................. 1
PART I: Games with ordinal payoffs ............................................... 5
1. Ordinal games in strategic form
1.1 Game frames and games ..................................................................... 6
1.2 Strict and weak dominance .............................................................. 14
1.3 Second-price auction ........................................................................ 21
1.4 The pivotal mechanism ................................................................... 24
1.5 Iterated deletion procedures ............................................................ 28
1.6 Nash equilibrium ............................................................................. 32
1.7 Games with infinite strategy sets ..................................................... 37
Appendix 1.A: Proofs of theorems ........................................................ 40
Appendix 1.E: Exercises [23 exercises] .................................................. 43
Appendix 1.S: Solutions to exercises ..................................................... 51
2. Dynamic games with perfect information
2.1 Trees, frames and games .................................................................. 65
2.2 Backward induction ......................................................................... 69
2.3 Strategies in perfect-information games ........................................... 73
2.4 Relationship between backward induction and
other solutions ................................................................................. 76
2.5 Perfect-information games with two players ................................... 81
Appendix 2.E: Exercises [13 exercises] .................................................. 84
Appendix 2.S: Solutions to exercises ..................................................... 92
3. General dynamic games
3.1 Imperfect information ................................................................... 107
3.2 Strategies ........................................................................................ 114
3.3 Subgames ....................................................................................... 116
3.4 Subgame-perfect equilibrium ......................................................... 119
3.5 Games with chance moves ............................................................. 127
Appendix 3.E: Exercises [14 exercises] ................................................ 132
Appendix 3.S: Solutions to exercises ................................................... 139
i
PART II: Games with cardinal payoffs ......................................... 157
4. Expected Utility
4.1 Money lotteries and attitudes to risk ............................................. 158
4.2 Expected utility: theorems ............................................................. 160
4.3 Expected utility: the axioms .......................................................... 169
Appendix 4.E: Exercises [14 exercises] ................................................ 177
Appendix 4.S: Solutions to exercises ................................................... 181

5. Mixed strategies in strategic-form games
5.1 Strategic-form games with cardinal payoffs ................................... 186
5.2 Mixed strategies ............................................................................. 190
5.3 Computing the mixed-strategy Nash equilibria ............................. 195
5.4 Strict dominance and rationalizability ........................................... 201
Appendix 5.E: Exercises [15 exercises] ................................................ 205
Appendix 5.S: Solutions to exercises ................................................... 213

6. Dynamic games with cardinal payoffs
6.1 Behavioral strategies in dynamic games ......................................... 224
6.2 Subgame-perfect equilibrium revisited ........................................... 231
6.3 Problems with subgame-perfect equilibrium ................................. 235
Appendix 6.E: Exercises [9 exercises] .................................................. 237
Appendix 6.S: Solutions to exercises ................................................... 243

ii
PART III: Advanced Topics I:
Knowledge, common knowledge, belief ....................................... 252
7. Knowledge and common knowledge
7.1 Individual knowledge .................................................................... 253
7.2 Interactive knowledge .................................................................... 257
7.3 Common Knowledge .................................................................... 265
Appendix 7.E: Exercises [14 exercises] ................................................ 272
Appendix 7.S: Solutions to exercises ................................................... 279

8. Adding beliefs to knowledge
8.1 Probabilistic beliefs ........................................................................ 289
8.2 Conditional probability, belief updating, Bayes’ rule .................... 292
8.3 Belief revision ................................................................................ 296
8.4 Harsanyi consistency of beliefs or like-mindedness ....................... 301
8.5 Agreeing to disagree ....................................................................... 306
Appendix 8.A: Proof of the Agreement Theorem .............................. 315
Appendix 8.E: Exercises [12 exercises] ................................................ 316
Appendix 8.S: Solutions to exercises ................................................... 323

9. Common knowledge of rationality
9.1 Models of strategic-form games ..................................................... 332
9.2 Common knowledge of rationality in strategic-form games ......... 336
9.3 Common knowledge of rationality in extensive-form games ........ 339
Appendix 9.A: Proofs .......................................................................... 342
Appendix 9.E: Exercises [7 exercises] .................................................. 344
Appendix 9.S: Solutions to exercises ................................................... 348
iii
PART IV: Advanced Topics II:
Refinements of subgame-perfect equilibrium ................................ 353
10. A First Attempt: Weak Sequential Equilibrium
10.1 Assessments and sequential rationality ........................................ 354
10.2 Bayesian updating at reached information sets ............................ 361
10.3 Weak sequential equilibrium ....................................................... 363
Appendix 10.E: Exercises [8 exercises] ................................................ 375
Appendix 10.S: Solutions to exercises ................................................. 380

11. Sequential Equilibrium
11.1 Consistent assessments ................................................................. 389
11.2 Sequential equilibrium ................................................................. 395
11.3 Is ‘consistency’ a good notion? .................................................... 397
Appendix 11.E: Exercises [6 exercises] ................................................ 400
Appendix 11.S: Solutions to exercises ................................................. 406

12. Perfect Bayesian Equilibrium
12.1 Belief revision and AGM consistency .......................................... 418
12.2 Bayesian consistency .................................................................... 423
12.3 Perfect Bayesian equilibrium ....................................................... 426
12.4 Adding independence................................................................... 431
12.5 Filling the gap between Perfect Bayesian equilibrium
and sequential equilibrium .......................................................... 434
Appendix 12.A: History-based definition of extensive-form game ..... 443
Appendix 12.B: Proof of Theorems 12.4 ............................................. 446
Appendix 12.E: Exercises [10 exercises] .............................................. 448
Appendix 12.S: Solutions to exercises ................................................. 456
iv
PART V: Advanced Topics III:
Incomplete Information ................................................................ 470
13. Incomplete Information: Static Games
13.1 Interactive situations with incomplete information .................... 471
13.2 One-sided incomplete information .............................................. 473
13.3 Two-sided incomplete information ............................................. 482
13.4 Multi-sided incomplete information ............................................ 486
Appendix 13.E: Exercises [8 exercises] ................................................ 489
Appendix 13.S: Solutions to exercises ................................................. 497

14. Incomplete Information: Dynamic Games
14.1 One-sided incomplete information .............................................. 510
14.2 Multi-sided incomplete information ............................................ 528
Appendix 14.E: Exercises [7 exercises] ................................................ 535
Appendix 14.S: Solutions to exercises ................................................. 541

15. Incomplete Information: the type-space approach
15.1 Types of players ........................................................................... 557
15.2 Types that know their own payoffs ............................................. 558
15.3 The general case ........................................................................... 563
Appendix 15.E: Exercises [4 exercises] ................................................ 566
Appendix 15.S: Solutions to exercises ................................................. 570

References ...................................................................................... 576

v
Preface and Acknowledgments
After teaching game theory (at both the undergraduate and graduate level) at
the University of California, Davis for 25 years, I decided to organize all my
teaching material in a textbook. There are many excellent textbooks in game
theory and there is hardly any need for a new one. However, there are two
1
distinguishing features of this textbook: (1) it is open access and thus free, and
(2) it contains an unusually large number of exercises (a total of 165) with
complete and detailed answers.
I tried to write the book in such a way that it would be accessible to anybody
with minimum knowledge of mathematics (high-school level algebra and some
elementary notions of probability) and no prior knowledge of game theory.
However, the book is intended to be rigorous and it includes several proofs. I
believe it is appropriate for an advanced undergraduate class in game theory and
also for a first-year graduate-level class.
I expect that there will be some typos and (hopefully minor) mistakes. If you
come across any typos or mistakes, I would be grateful if you could inform me:
I can be reached at [e-mail]. I will maintain an updated
version of the book on my web page at
http://www.econ.ucdavis.edu/faculty/bonanno/
I also intend to add, some time in the future, a further collection of exercises
and exam questions with a solution manual to be given only to instructors.
Details will appear on my web page.
I am very grateful to Elise Tidrick for meticulously going through each chapter
of the book and for suggesting numerous improvements. Her insightful and
constructive comments have considerably enhanced this book.
I would also like to thank Nicholas Bowden, Lester Lusher, Burkhard Schipper and Matthieu
Stigler for pointing out typos.
Davis, October 27, 2015

1 There may be several other free textbooks on game theory available. The only one I am aware
of is the excellent book by Ariel Rubinstein and Martin Osborne, MIT Press, 1994, which
can be downloaded for free from Ariel Rubinstein’s web page: http://arielrubinstein.tau.ac.il.
In my experience this book is too advanced for an undergraduate class in game theory.
vi
GAME THEORY – G. Bonanno
Chapter
0
Introduction
he discipline of game theory was pioneered in the early 20th century by
T
mathematicians Ernst Zermelo (1913) and John von Neumann (1928).
The breakthrough came with John von Neumann and Oscar
Morgenstern’s book, Theory of games and economic behavior, published
in 1944. This was followed by important work by John Nash (1950-51)
and Lloyd Shapley (1953). Game theory had a major influence on the
development of several branches of economics (industrial organization,
international trade, labor economics, macroeconomics, etc.). Over time the
impact of game theory extended to other branches of the social sciences
(political science, international relations, philosophy, sociology, anthropology,
etc.) as well as to fields outside the social sciences, such as biology, computer
science, logic, etc. In 1994 the Nobel prize in economics was given to three
game theorists, John Nash, John Harsanyi and Reinhardt Selten, for their
theoretical work in game theory which was very influential in economics. At
the same time, the US Federal Communications Commission was using game
theory to help it design a $7-billion auction of the radio spectrum for personal
communication services (naturally, the bidders used game theory too!). The
Nobel prize in economics was awarded to game theorists three more times: in
2006 to Robert Aumann and Thomas Schelling, in 2007 to Leonid Hurwicz,
Eric Maskin and Roger Myerson and in 2010 to Lloyd Shapley and Alvin Roth.
Game theory provides a formal language for the representation and analysis of
interactive situations, that is, situations where several “entities”, called players,
take actions that affect each other. The nature of the players varies depending
on the context in which the game theoretic language is invoked: in
evolutionary biology (see, for example, John Maynard Smith, 1982) players are
1
GAME THEORY – G. Bonanno

2
non-thinking living organisms; in computer science (see, for example, Shoham-
Leyton-Brown, 2008) players are artificial agents; in behavioral game theory
(see, for example, Camerer, 2003) players are “ordinary” human beings, etc.
Traditionally, however, game theory has focused on interaction among
intelligent, sophisticated and rational individuals. For example, Aumann
describes game theory as follows:
“Briefly put, game and economic theory are concerned with the
interactive behavior of Homo rationalis - rational man. Homo
rationalis is the species that always acts both purposefully and
logically, has well-defined goals, is motivated solely by the desire
to approach these goals as closely as possible, and has the
calculating ability required to do so.” (Aumann, 1985, p. 35.)
This book is concerned with the traditional interpretation of game theory.
Game theory is divided into two main branches. The first is cooperative game
theory, which assumes that the players can communicate, form coalitions and
sign binding agreements. Cooperative game theory has been used, for example,
to analyze voting behavior and other issues in political science and related
fields. We will deal exclusively with the other main branch, namely non-
cooperative game theory. Non-cooperative game theory models situations
where the players are either unable to communicate or are able to communicate
but cannot sign binding contracts. An example of the latter situation is the
interaction among firms in an industry in an environment where antitrust laws
make it illegal for firms to reach agreements concerning prices or production
quotas or other forms of collusive behavior.
The book is divided into three parts.
Part I deals with games with ordinal payoffs, that is, with games where the
players’ preferences over the possible outcomes are only specified in terms of
an ordinal ranking (outcome o is better than outcome o or o is just as good as
o). Chapter 1 covers strategic-form games, Chapter 2 deals with dynamic
games with perfect information and Chapter 3 with dynamic games with
(possibly) imperfect information.

2 Evolutionary game theory has been applied not only to the analysis of animal and insect
behavior but also to studying the “ most successful strategies” for tumor and cancer cells (see,
for example, Gerstung et al., 2011).
2
GAME THEORY – G. Bonanno

Part II is devoted to games with cardinal payoffs, that is, with games where the
players’ preferences extend to uncertain prospects or lotteries: players are
assumed to have a consistent ranking of the set of lotteries over basic outcomes.
Chapter 4 reviews the theory of expected utility, Chapter 5 discusses the notion
of mixed strategy in strategic-form games and of mixed-strategy Nash
equilibrium , while Chapter 6 deals with mixed strategies in dynamic games.
Parts III, IV and V cover a number of advanced topics.
Part III deals with the notions of knowledge, common knowledge and belief.
Chapter 7 explains how to model what an individual knows and what she is
uncertain about and how to extend the analysis to the interactive knowledge of
several individuals (e.g. what Individual 1 knows about what Individual 2
knows about some facts or about the state of knowledge of Individual 1). The
chapter ends with the notion of common knowledge. Chapter 8 adds
probabilistic beliefs to the knowledge structures of the previous chapter and
discusses the notions of Bayesian updating, belief revision, like-mindedness and
the possibility of “agreeing to disagree”. Chapter 9 uses the interactive
knowledge-belief structures of the previous two chapters to model the players’
state of mind in a possible play of a given game and studies the implications of
common knowledge of rationality in strategic-form games.
Part IV focuses on dynamic (or extensive-form) games and on the issue of how
to refine the notion of subgame-perfect equilibrium (which was introduced in
Chapters 3 and 6). Chapter 10 introduces a simple notion, called weak
sequential equilibrium, which achieves some desirable goals (such as the
elimination of strictly dominated choices) but fails to provide a refinement of
subgame-perfect equilibrium. Chapter 11 explains the more complex notion of
sequential equilibrium, which is extensively used in applications of game
theory. That notion, however, leaves to be desired from a practical point of
view (it is typically hard to show that an equilibrium is indeed a sequential
equilibrium) and also from a conceptual point of view (it appeals to a
topological condition, whose interpretation is not clear). Chapter 12
introduces an intermediate notion, called perfect Bayesian equilibrium, whose
conceptual justification is anchored in the so called AGM theory of belief
revision, extensively studied in philosophy and computer science, which was
pioneered by Carlos Alchourrón (a legal scholar), Peter Gärdenfors (a
philosopher) and David Makinson (a computer scientist) in 1985. In Chapter 12 we
also provide an alternative characterization of sequential equilibrium based on the
notion of perfect Bayesian equilibrium, which is free of topological conditions.
3