Games and information

              GAMES AND INFORMATION, FOURTH EDITION
An Introduction to Game Theory
Eric Rasmusen
Basil Blackwell
v
Contents1
(starred sections are less important)
List of Figures
List of Tables
Preface
Contents and Purpose
Changes in the Second Edition
Changes in the Third Edition
Using the Book
The Level of Mathematics
Other Books
Contact Information
Acknowledgements
Introduction
History
Game Theory’s Method
Exemplifying Theory
This Book’s Style
Notes
PART 1: GAME THEORY
1 The Rules of the Game
1.1 Definitions
1.2 Dominant Strategies: The Prisoner’s Dilemma
1.3 Iterated Dominance: The Battle of the Bismarck Sea
1.4 Nash Equilibrium: Boxed Pigs, The Battle of the Sexes, and Ranked Coordina-tion
1.5 Focal Points
Notes
Problems
1xxx February 2, 2000. December 12, 2003. 24 March 2005. Eric Rasmusen, Erasmuse@indiana.edu.
http://www.rasmusen.org/GI Footnotes starting with xxx are the author’s notes to himself. Comments
are welcomed.
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2 Information
2.1 The Strategic and Extensive Forms of a Game
2.2 Information Sets
2.3 Perfect, Certain, Symmetric, and Complete Information
2.4 The Harsanyi Transformation and Bayesian Games
2.5 Example: The Png Settlement Game
Notes
Problems
3 Mixed and Continuous Strategies
3.1 Mixed Strategies: The Welfare Game
3.2 Chicken, The War of Attrition, and Correlated Strategies
3.3 Mixed Strategies with General Parameters and N Players: The Civic Duty Game
3.4 Different Uses of Mixing and Randomizing: Minimax and the Auditing Game
3.5 Continuous Strategies: The Cournot Game
3.6 Continuous Strategies: The Bertrand Game, Strategic Complements, and Strate-gic
Subsitutes
3.7 Existence of Equilibrium
Notes
Problems
4 Dynamic Games with Symmetric Information
4.1 Subgame Perfectness
4.2 An Example of Perfectness: Entry Deterrence I
4.3 Credible Threats, Sunk Costs, and the Open-Set Problem in the Game of Nui-sance
Suits
*4.4 Recoordination to Pareto-dominant Equilibria in Subgames: Pareto Perfection
Notes
Problems
5 Reputation and Repeated Games with Symmetric Information
5.1 Finitely Repeated Games and the Chainstore Paradox
5.2 Infinitely Repeated Games, Minimax Punishments, and the Folk Theorem
5.3 Reputation: the One-sided Prisoner’s Dilemma
5.4 Product Quality in an Infinitely Repeated Game
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*5.5 Markov Equilibria and Overlapping Generations in the Game of Customer Switch-ing
Costs
*5.6 Evolutionary Equilibrium: The Hawk-Dove Game
Notes
Problems
6 Dynamic Games with Incomplete Information
6.1 Perfect Bayesian Equilibrium: Entry Deterrence II and III
6.2 Refining Perfect Bayesian Equilibrium: the PhD Admissions Game
6.3 The Importance of Common Knowledge: Entry Deterrence IV and V
6.4 Incomplete Information in the Repeated Prisoner’s Dilemma: The Gang of Four
Model
6.5 The Axelrod Tournament
*6.6 Credit and the Age of the Firm: The Diamond Model
Notes
Problems
PART 2: ASYMMETRIC INFORMATION
7 Moral Hazard: Hidden Actions
7.1 Categories of Asymmetric Information Models
7.2 A Principal-Agent Model: The Production Game
7.3 The Incentive Compatibility, Participation, and Competition Constraints
7.4 Optimal Contracts: The Broadway Game
Notes
Problems
8 Further Topics in Moral Hazard
8.1 Efficiency Wages
8.2 Tournaments
8.3 Institutions and Agency Problems
*8.4 Renegotiation: the Repossession Game
*8.5 State-space Diagrams: Insurance Games I and II
*8.6 Joint Production by Many Agents: the Holmstrom Teams Model
Notes
Problems
9 Adverse Selection
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9.1 Introduction: Production Game VI
9.2 Adverse Selection under Certainty: Lemons I and II
9.3 Heterogeneous Tastes: Lemons III and IV
9.4 Adverse Selection under Uncertainty: Insurance Game III
*9.5 Market Microstructure
*9.6 A Variety of Applications
Notes
Problems
10 Mechanism Design in Adverse Selection and in Moral Hazard with Hidden Informa-tion
10.1 The Revelation Principle and Moral Hazard with Hidden Knowledge
10.2 An Example of Moral Hazard with Hidden Knowledge: the Salesman Game
*10.3 Price Discrimination
*10.4 Rate-of-return Regulation and Government Procurement
*10.5 The Groves Mechanism
Notes
Problems
11 Signalling
11.1 The Informed Player Moves First: Signalling
11.2 Variants on the Signalling Model of Education
11.3 General Comments on Signalling in Education
11.4 The Informed Player Moves Second: Screening
*11.5 Two Signals: the Game of Underpricing New Stock Issues
*11.6 Signal Jamming and Limit Pricing
Notes
Problems
PART 3: APPLICATIONS
12 Bargaining
12.1 The Basic Bargaining Problem: Splitting a Pie
12.2 The Nash Bargaining Solution
12.3 Alternating Offers over Finite Time
12.4 Alternating Offers over Infinite Time
12.5 Incomplete Information
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*12.6 Setting up a Way to Bargain: the Myerson-Satterthwaite Mechanism
Notes
Problems
13 Auctions
13.1 Auction Classification and Private-Value Strategies
13.2 Comparing Auction Rules
13.3 Risk and Uncertainty over Values
13.4 Common-value Auctions and the Winner’s Curse
13.5 Information in Common-value Auctions
Notes
Problems
14 Pricing
14.1 Quantities as Strategies: Cournot Equilibrium Revisited
14.2 Prices as Strategies
14.3 Location Models
*14.4 Comparative Statics and Supermodular Games
*14.5 Durable Monopoly
Notes
Problems
*A Mathematical Appendix
*A.1 Notation
*A.2 The Greek Alphabet
*A.3 Glossary
*A.4 Formulas and Functions
*A.5 Probability Distributions
*A.6 Supermodularity
*A.7 Fixed Point Theorems
*A.8 Genericity
*A.9 Discounting
*A.10 Risk
References and Name Index
Subject Index
x
xxx September 6, 1999; February 2, 2000. February 9, 2000. May 24, 2002. Ariel Kem-per
August 6, 2003. 24 March 2005. Eric Rasmusen, Erasmuse@indiana.edu; Footnotes
starting with xxx are the author’s notes to himself. Comments are welcomed.
Preface
Contents and Purpose
This book is about noncooperative game theory and asymmetric information. In the In-troduction,
I will say why I think these subjects are important, but here in the Preface I
will try to help you decide whether this is the appropriate book to read if they do interest
you.
I write as an applied theoretical economist, not as a game theorist, and readers in
anthropology, law, physics, accounting, and management science have helped me to be
aware of the provincialisms of economics and game theory. My aim is to present the game
theory and information economics that currently exist in journal articles and oral tradition
in a way that shows how to build simple models using a standard format. Journal articles
are more complicated and less clear than seems necessary in retrospect; precisely because
it is original, even the discoverer rarely understands a truly novel idea. After a few dozen
successor articles have appeared, we all understand it and marvel at its simplicity. But
journal editors are unreceptive to new articles that admit to containing exactly the same
idea as old articles, just presented more clearly. At best, the clarification is hidden in
some new article’s introduction or condensed to a paragraph in a survey. Students, who
find every idea as complex as the originators of the ideas did when they were new, must
learn either from the confused original articles or the oral tradition of a top economics
department. This book tries to help.
Changes in the Second Edition, 1994
By now, just a few years later after the First Edition, those trying to learn game
theory have more to help them than just this book, and I will list a number of excellent
books below. I have also thoroughly revised Games and Information. George Stigler used
to say that it was a great pity Alfred Marshall spent so much time on the eight editions
of Principles of Economics that appeared between 1890 and 1920, given the opportunity
cost of the other books he might have written. I am no Marshall, so I have been willing to
sacrifice a Rasmusen article or two for this new edition, though I doubt I will keep it up
till 2019.
What I have done for the Second Edition is to add a number of new topics, increase
the number of exercises (and provide detailed answers), update the references, change
the terminology here and there, and rework the entire book for clarity. A book, like a
poem, is never finished, only abandoned (which is itself a good example of a fundamental
economic principle). The one section I have dropped is the somewhat obtrusive discussion
of existence theorems; I recommend Fudenberg & Tirole (1991a) on that subject. The new
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topics include auditing games, nuisance suits, recoordination in equilibria, renegotiation
in contracts, supermodularity, signal jamming, market microstructure, and government
procurement. The discussion of moral hazard has been reorganized. The total number of
chapters has increased by two, the topics of repeated games and entry having been given
their own chapters.
Changes in the Third Edition, 2001
Besides numerous minor changes in wording, I have added new material and reorga-nized
some sections of the book.
The new topics are 10.3 “Price Discrimination”; 12.6 “Setting up a Way to Bargain:
The Myerson-Satterthwaite Mechanism”; 13.3 “Risk and Uncertainty over Values” (for
private- value auctions) ; A.7 “Fixed-Point Theorems”; and A.8 “Genericity”.
To accommodate the additions, I have dropped 9.5 “Other Equilibrium Concepts:
Wilson Equilibrium and Reactive Equilibrium” (which is still available on the book’s web-site),
and Appendix A, “Answers to Odd-Numbered Problems”. These answers are very
important, but I have moved them to the website because most readers who care to look at
them will have web access and problem answers are peculiarly in need of updating. Ideally,
I would like to discuss all likely wrong answers as well as the right answers, but I learn the
wrong answers only slowly, with the help of new generations of students.
Chapter 10, “Mechanism Design in Adverse Selection and in Moral Hazard with Hid-den
Information”, is new. It includes two sections from chapter 8 (8.1 “Pooling versus
Separating Equilibrium and the Revelation Principle” is now section 10.1; 8.2 “An Exam-ple
of Moral Hazard with Hidden Knowledge: the Salesman Game” is now section 10.2)
and one from chapter 9 (9.6 “The Groves Mechanism” is now section 10.5).
Chapter 15 “The New Industrial Organization”, has been eliminated and its sections
reallocated. Section 15.1 “Why Established Firms Pay Less for Capital: The Diamond
Model” is now section 6.6; Section 15.2 “Takeovers and Greenmail” remains section 15.2;
section 15.3 “Market Microstructure and the Kyle Model” is now section 9.5; and section
15.4 “Rate-of-return Regulation and Government Procurement” is now section 10.4.
Topics that have been extensively reorganized or rewritten include 14.2 “Prices as
Strategies”; 14.3 “Location Models”; the Mathematical Appendix, and the Bibliography.
Section 4.5 “Discounting” is now in the Mathematical Appendix; 4.6 “Evolutionary Equi-librium:
The Hawk-Dove Game” is now section 5.6; 7.5 “State-space Diagrams: Insurance
Games I and II” is now section 8.5 and the sections in Chapter 8 are reordered; 14.2 “Signal
Jamming: Limit Pricing” is now section 11.6. I have recast 1.1 “Definitions”, taking out
the OPEC Game and using an entry deterrence game instead, to illustrate the difference
between game theory and decision theory. Every other chapter has also been revised in
minor ways.
Some readers preferred the First Edition to the Second because they thought the extra
topics in the Second Edition made it more difficult to cover. To help with this problem, I
have now starred the sections that I think are skippable. For reference, I continue to have
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those sections close to where the subjects are introduced.
The two most novel features of the book are not contained within its covers. One is
the website, at
Http://www.rasmusen.org/GI/index.html
The website includes answers to the odd-numbered problems, new questions and an-swers,
errata, files from my own teaching suitable for making overheads, and anything else
I think might be useful to readers of this book.
The second newfeature is a Reader— a prettified version of the course packet I use when
I teach this material. This is available from Blackwell Publishers, and contains scholarly
articles, news clippings, and cartoons arranged to correspond with the chapters of the book.
I have tried especially to include material that is somewhat obscure or hard to locate, rather
than just a collection of classic articles from leading journals.
If there is a fourth edition, three things I might add are (1) a long discussion of strategic
complements and substitutes in chapter 14, or perhaps even as a separate chapter; (2)
Holmstrom & Milgrom’s 1987 article on linear contracts; and (3) Holmstrom & Milgrom’s
1991 article on multi-task agency. Readers who agree, let me know and perhaps I’ll post
notes on these topics on the website.
Using the Book
The book is divided into three parts: Part I on game theory; Part II on information
economics; and Part III on applications to particular subjects. Parts I and II, but not Part
III, are ordered sets of chapters.
Part I by itself would be appropriate for a course on game theory, and sections from
Part III could be added for illustration. If students are already familiar with basic game
theory, Part II can be used for a course on information economics. The entire book would
be useful as a secondary text for a course on industrial organization. I teach material
from every chapter in a semester-long course for first- and second-year doctoral students
at Indiana University’s Kelley School of Business, including more or fewer chapter sections
depending on the progress of the class.
Exercises and notes follow the chapters. It is useful to supplement a book like this with
original articles, but I leave it to my readers or their instructors to follow up on the topics
that interest them rather than recommending particular readings. I also recommend that
readers try attending a seminar presentation of current research on some topic from the
book; while most of the seminar may be incomprehensible, there is a real thrill in hearing
someone attack the speaker with “Are you sure that equilibrium is perfect?” after just
learning the previous week what “perfect” means.
Some of the exercises at the end of each chapter put slight twists on concepts in the
text while others introduce new concepts. Answers to odd-numbered questions are given
at the website. I particularly recommend working through the problems for those trying
to learn this material without an instructor.
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The endnotes to each chapter include substantive material as well as recommendations
for further reading. Unlike the notes in many books, they are not meant to be skipped, since
many of them are important but tangential, and some qualify statements in the main text.
Less important notes supply additional examples or list technical results for reference. A
mathematical appendix at the end of the book supplies technical references, defines certain
mathematical terms, and lists some items for reference even though they are not used in
the main text.
The Level of Mathematics
In surveying the prefaces of previous books on game theory, I see that advising readers
how much mathematical background they need exposes an author to charges of being out
of touch with reality. The mathematical level here is about the same as in Luce & Raiffa
(1957), and I can do no better than to quote the advice on page 8 of their book:
Probably the most important prerequisite is that ill-defined quality: mathe-matical
sophistication. We hope that this is an ingredient not required in large
measure, but that it is needed to some degree there can be no doubt. The
reader must be able to accept conditional statements, even though he feels the
suppositions to be false; he must be willing to make concessions to mathemati-cal
simplicity; he must be patient enough to follow along with the peculiar kind
of construction that mathematics is; and, above all, he must have sympathy
with the method – a sympathy based upon his knowledge of its past sucesses
in various of the empirical sciences and upon his realization of the necessity for
rigorous deduction in science as we know it.
If you do not know the terms “risk averse,” “first order condition,” “utility function,”
“probability density,” and “discount rate,” you will not fully understand this book. Flipping
through it, however, you will see that the equation density is much lower than in first-year
graduate microeconomics texts. In a sense, game theory is less abstract than price
theory, because it deals with individual agents rather than aggregate markets and it is
oriented towards explaining stylized facts rather than supplying econometric specifications.
Mathematics is nonetheless essential. Professor Wei puts this well in his informal and
unpublished class notes:
My experience in learning and teaching convinces me that going through a
proof (which does not require much mathematics) is the most effective way in
learning, developing intuition, sharpening technical writing ability, and improv-ing
creativity. However it is an extremely painful experience for people with
simple mind and narrow interests.
Remember that a good proof should be smooth in the sense that any serious
reader can read through it like the way we read Miami Herald; should be precise
such that no one can add/delete/change a word–like the way we enjoy Robert
Frost’s poetry!
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I wouldn’t change a word of that.
Other Books
At the time of the first edition of this book, most of the topics covered were absent
from existing books on either game theory or information economics. Older books on
game theory included Davis (1970), Harris (1987), Harsanyi (1977), Luce & Raiffa (1957),
Moulin (1986a, 1986b), Ordeshook (1986), Rapoport (1960, 1970), Shubik (1982), Szep &
Forgo (1985), Thomas (1984), and Williams (1966). Books on information in economics
were mainly concerned with decision making under uncertainty rather than asymmetric
information. Since the First Edition, a spate of books on game theory has appeared. The
stream of new books has become a flood, and one of the pleasing features of this literature
is its variety. Each one is different, and both student and teacher can profit by owning an
assortment of them, something one cannot say of many other subject areas. We have not
converged, perhaps because teachers are still converting into books their own independent
materials from courses not taught with texts. I only wish I could say I had been able to
use all my competitors’ good ideas in the present edition.
Why, you might ask in the spirit of game theory, do I conveniently list all my com-petitor’s
books here, giving free publicity to books that could substitute for mine? For
an answer, you must buy this book and read chapter 11 on signalling. Then you will un-derstand
that only an author quite confident that his book compares well with possible
substitutes would do such a thing, and you will be even more certain that your decision to
buy the book was a good one. (But see problem 11.6 too.)
Some Books on Game Theory and its Applications
1988 Tirole, Jean, The Theory of Industrial Organization, Cambridge, Mass: MIT Press.
479 pages. Still the standard text for advanced industrial organization.
1989 Eatwell, John, Murray Milgate & Peter Newman, eds., The New Palgrave: Game
Theory. 264 pages. New York: Norton. A collection of brief articles on topics in game
theory by prominent scholars.
Schmalensee, Richard & Robert Willig, eds., The Handbook of Industrial Organiza-tion,
in two volumes, New York: North- Holland. A collection of not-so-brief articles
on topics in industrial organization by prominent scholars.
Spulber, Daniel Regulation and Markets, Cambridge, Mass: MIT Press. 690 pages.
Applications of game theory to rate of return regulation.
1990 Banks, Jeffrey, Signalling Games in Political Science. Chur, Switzerland: Harwood
Publishers. 90 pages. Out of date by now, but worth reading anyway.
Friedman, James, Game Theory with Applications to Economics, 2nd edition, Ox-ford:
Oxford University Press (First edition, 1986 ). 322 pages. By a leading expert
on repeated games.
Kreps, David, A Course in Microeconomic Theory. Princeton: Princeton University
Press. 850 pages. A competitor to Varian’s Ph.D. micro text, in a more conversational
style, albeit a conversation with a brilliant economist at a level of detail that scares
some students.
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Kreps, David, Game Theory and Economic Modeling, Oxford: Oxford University
Press. 195 pages. A discussion of Nash equilibrium and its problems.
Krouse, Clement, Theory of Industrial Economics, Oxford: Blackwell Publishers.
602 pages. A good book on the same topics as Tirole’s 1989 book, and largely over-shadowed
by it.
1991 Dixit, Avinash K. & Barry J. Nalebuff, Thinking Strategically: The Competitive Edge
in Business, Politics, and Everyday Life. New York: Norton. 393 pages. A book in
the tradition of popular science, full of fun examples but with serious ideas too. I
use this for my MBA students’ half-semester course, though newer books are offering
competition for that niche.
Fudenberg, Drew & Jean Tirole, Game Theory. Cambridge, Mass: MIT Press. 579
pages. This has become the standard text for second-year PhD courses in game theory.
(Though I hope the students are referring back to Games and Information for help in
getting through the hard parts.)
Milgrom, Paul and John Roberts, Economics of Organization and Management.
Englewood Cliffs, New Jersey: Prentice-Hall. 621 pages. A model for how to think
about organization and management. The authors taught an MBA course from this,
but I wonder whether that is feasible anywhere but Stanford Business School.
Myerson, Roger, Game Theory: Analysis of Conflict, Cambridge, Mass: Harvard
University Press. 568 pages. At an advanced level. In revising for the third edition, I
noticed how well Myerson’s articles are standing the test of time.
1992 Aumann, Robert & Sergiu Hart, eds., Handbook of Game Theory with Economic
Applications, Volume 1, Amsterdam: North- Holland. 733 pages. A collection of
articles by prominent scholars on topics in game theory.
Binmore, Ken, Fun and Games: A Text on Game Theory. Lexington, Mass: D.C.
Heath. 642 pages. No pain, no gain; but pain and pleasure can be mixed even in the
study of mathematics.
Gibbons, Robert, Game Theory for Applied Economists,. Princeton: Princeton Uni-versity
Press. 267 pages. Perhaps the main competitor to Games and Information.
Shorter and less idiosyncratic.
Hirshleifer, Jack & John Riley, The Economics of Uncertainty and Information,
Cambridge: Cambridge University Press. 465 pages. An underappreciated book that
emphasizes information rather than game theory.
McMillan, John, Games, Strategies, and Managers: How Managers Can Use Game
Theory to Make Better Business Decisions,. Oxford, Oxford University Press. 252
pages. Largely verbal, very well written, and an example of how clear thinking and
clear writing go together.
Varian, Hal, Microeconomic Analysis, Third edition. New York: Norton. (1st edition,
1978; 2nd edition, 1984.) 547 pages. Varian was the standard PhD micro text when
I took the course in 1980. The third edition is much bigger, with lots of game theory
and information economics concisely presented.
1993 Basu, Kaushik, Lectures in Industrial Organization Theory, . Oxford: Blackwell
Publishers. 236 pages. Lots of game theory as well as I.O.
Eichberger, Jurgen, Game Theory for Economists, San Diego: Academic Press. 315
pages. Focus on game theory, but with applications along the way for illustration.
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Laffont, Jean-Jacques & Jean Tirole, A Theory of Incentives in Procurement and
Regulation, Cambridge, Mass: MIT Press. 705 pages. If you like section 10.4 of
Games and Information, here is an entire book on the model.
Martin, Stephen, Advanced Industrial Economics, Oxford: Blackwell Publishers. 660
pages. Detailed and original analysis of particular models, and much more attention
to empirical articles than Krouse, Shy, and Tirole.
1994 Baird, Douglas, Robert Gertner & Randal Picker, Strategic Behavior and the Law:
The Role of Game Theory and Information Economics in Legal Analysis, Cambridge,
Mass: Harvard University Press. 330 pages. A mostly verbal but not easy exposition
of game theory using topics such as contracts, procedure, and tort.
Gardner, Roy, Games for Business and Economics, New York: JohnWiley and Sons.
480 pages. Indiana University has produced not one but two game theory texts.
Morris, Peter, Introduction to Game Theory, Berlin: Springer Verlag. 230 pages.
Not in my library yet.
Morrow, James, Game Theory for Political Scientists, Princeton, N.J. : Princeton
University Press. 376 pages. The usual topics, but with a political science slant, and
especially good on things such as utility theory.
Osborne, Martin and Ariel Rubinstein, A Course in Game Theory, Cambridge, Mass:
MIT Press. 352 pages. Similar in style to Eichberger’s 1993 book. See their excellent
“List of Results” on pages 313-19 which summarizes the mathematical propositions
without using specialized notation.
1995 Mas-Colell, Andreu Michael D. Whinston and Jerry R. Green, Microeconomic The-ory,
Oxford: Oxford University Press. 981 pages. This combines the topics of Varian’s
PhD micro text, those of Games and Information, and general equilibrium. Massive,
and a good reference.
Owen, Guillermo, Game Theory, New York: Academic Press, 3rd edition. (1st edi-tion,
1968; 2nd edition, 1982.) This book clearly lays out the older approach to game
theory, and holds the record for longevity in game theory books.
1996 Besanko, David, David Dranove and Mark Shanley, Economics of Strategy, New
York: John Wiley and Sons. This actually can be used with Indiana M.B.A. students,
and clearly explains some very tricky ideas such as strategic complements.
Shy, Oz, Industrial Organization, Theory and Applications, Cambridge, Mass: MIT
Press. 466 pages. A new competitor to Tirole’s 1988 book which is somewhat easier.
1997 Gates, Scott and Brian Humes, Games, Information, and Politics: Applying Game
Theoretic Models to Political Science, Ann Arbor: University of Michigan Press. 182
pages.
Ghemawat, Pankaj, Games Businesses Play: Cases and Models, Cambridge, Mass:
MIT Press. 255 pages. Analysis of six cases from business using game theory at the
MBA level. Good for the difficult task of combining theory with evidence.
Macho-Stadler, Ines and J. David Perez-Castillo, An Introduction to the Economics
of Information: Incentives and Contracts, Oxford: Oxford University Press. 277
pages. Entirely on moral hazard, adverse selection, and signalling.
Romp, Graham, Game Theory: Introduction and Applications, Oxford: Oxford Uni-versity
Press. 284 pages. With unusual applications (chapters on macroeconomics,
trade policy, and environmental economics) and lots of exercises with answers.
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Salanie, Bernard, The Economics of Contracts: A Primer, Cambridge, Mass: MIT
Press. 232 pages. Specialized to a subject of growing importance.
1998 Bierman, H. Scott & Luis Fernandez, Game Theory with Economic Applications.
Reading, Massachusetts: Addison Wesley, Second edition. (1st edition, 1993.) 452
pages. A text for undergraduate courses, full of good examples.
Dugatkin, Lee and Hudson Reeve, editors, Game Theory & Animal Behavior, Ox-ford:
Oxford University Press. 320 pages. Just on biology applications.
1999 Aliprantis, Charalambos & Subir Chakrabarti Games and Decisionmaking, Oxford:
Oxford University Press. 224 pages. An undergraduate text for game theory, decision
theory, auctions, and bargaining, the third game theory text to come out of Indiana.
Basar, Tamar & Geert Olsder Dynamic Noncooperative Game Theory, 2nd edition,
revised, Philadelphia: Society for Industrial and Applied Mathematics (1st edition
1982, 2nd edition 1995). This book is by and for mathematicians, with surprisingly
little overlap between its bibliography and that of the present book. Suitable for
people who like differential equations and linear algebra.
Dixit, Avinash & Susan Skeath, Games of Strategy, New York: Norton. 600 pages.
Nicely laid out with color and boldfacing. Game theory plus chapters on bargaining,
auctions, voting, etc. Detailed verbal explanations of many games.
Dutta, Prajit, Strategies and Games: Theory And Practice, Cambridge, Mass: MIT
Press. 450 pages.
Stahl, Saul, A Gentle Introduction to Game Theory, Providence, RI: American Math-ematical
Society. 176 pages. In the mathematics department tradition, with many
exercises and numerical answers.
Forthcoming Gintis, Herbert, Game Theory Evolving, Princeton: Princeton University Press.
(May 12, 1999 draft at www-unix.oit.umass.edu/∼gintis.) A wonderful book of prob-lems
and solutions, with much explanation and special attention to evolutionary biol-ogy.
Muthoo, Abhinay, Bargaining Theory With Applications, Cambridge: Cambridge
University Press.
Osborne, Martin, An Introduction to Game Theory, Oxford: Oxford University
Press. Up on the web via this book’s website if you’d like to check it out.
Rasmusen, Eric, editor, Readings in Games and Information, Oxford: Blackwell
Publishers. Journal and newspaper articles on game theory and information eco-nomics.
Rasmusen, Eric Games and Information. Oxford: Blackwell Publishers, Fourth
edition. (1st edition, 1989; 2nd edition, 1994, 3rd edition 2001.) Read on.
Contact Information
The website for the book is at
Http://www.rasmusen.org/GI/index.html
xxii
This site has the answers to the odd-numbered problems at the end of the chapters.
For answers to even-numbered questions, instructors or others needing them for good rea-sons
should email me at Erasmuse@Indiana.edu; send me snailmail at Eric Rasmusen,
Department of Business Economics and Public Policy, Kelley School of Business, Indi-ana
University, 1309 East 10th Street, Bloomington, Indiana 47405-1701; or fax me at
(812)855-3354.
If you wish to contact the publisher of this book, the addresses are 108 Cowley
Road, Oxford, England, OX4 1JF; or Blackwell Publishers, 350 Main Street, Malden,
Massachusetts 02148.
The text files on the website are two forms (a) *.te, LaTeX, which uses only ASCII
characters, but does not have the diagrams, and (b) *.pdf, Adobe Acrobat, which is format-ted
and can be read using a free reader program. I encourage readers to submit additional
homework problems as well as errors and frustrations. They can be sent to me by e-mail
at Erasmuse@Indiana.edu.
Acknowledgements
I would like to thank the many people who commented on clarity, suggested topics and
references, or found mistakes. I’ve put affiliations next to their names, but remember
that these change over time (A.B. was not a finance professor when he was my research
assistant!).
First Edition: Dean Amel (Board of Governors, Federal Reserve), Dan Asquith (S.E.C.),
Sushil Bikhchandani (UCLA business economics), Patricia Hughes Brennan (UCLA ac-counting),
Paul Cheng, Luis Fernandez (Oberlin economics), David Hirshleifer (Ohio State
finance), Jack Hirshleifer (UCLA economics), Steven Lippman (UCLA management sci-ence),
Ivan Png (Singapore), Benjamin Rasmusen (Roseland Farm), Marilyn Rasmusen
(Roseland Farm), Ray Renken (Central Florida physics), Richard Silver, Yoon Suh (UCLA
accounting), Brett Trueman (Berkeley accounting), Barry Weingast (Hoover) and students
in Management 200a made useful comments. D. Koh, Jeanne Lamotte, In-Ho Lee, Loi Lu,
Patricia Martin, Timothy Opler (Ohio State finance), Sang Tran, Jeff Vincent, Tao Yang,
Roy Zerner, and especially Emmanuel Petrakis (Crete economics) helped me with research
assistance at one stage or another. Robert Boyd (UCLA anthropology), Mark Ramseyer
(Harvard law), Ken Taymor, and John Wiley (UCLA law) made extensive comments in a
reading group as each chapter was written.
Second Edition: Jonathan Berk (U. British Columbia commerce), Mark Burkey (Ap-palachian
State economics), Craig Holden (Indiana finance), Peter Huang (Penn Law),
Michael Katz (Berkeley business), Thomas Lyon (Indiana business economics), Steve Postrel
(Northwestern business), Herman Quirmbach (Iowa State economics), H. Shifrin, George
Tsebelis (UCLA poli sci), Thomas Voss (Leipzig sociology), and Jong-ShinWei made useful
comments, and Alexander Butler (Louisiana State finance) and An- Sing Chen provided
research assistance. My students in Management 200 at UCLA and G601 at Indiana
University provided invaluable help, especially in suffering through the first drafts of the
homework problems.
xxiii
Third Edition: Kyung-Hwan Baik (Sung Kyun Kwan), Patrick Chen, Robert Dimand
(Brock economics), Mathias Erlei (Muenster), Francisco Galera, Peter-John Gordon (Uni-versity
of the West Indies), Erik Johannessen, Michael Mesterton-Gibbons (Pennsylvania),
David Rosenbaum (Nebraska economics), Richard Tucker, Hal Wasserman (Berkeley), and
Chad Zutter (Indiana finance) made comments that were helpful for the Third Edition.
Blackwell supplied anonymous reviewers of superlative quality. Scott Fluhr, Pankaj Jain
and John Spence provided research assistance and new generations of students in G601
were invaluable in helping to clarify my writing.
Eric Rasmusen
IU Foundation Professor of Business Economics and Public Policy
Kelley School of Business, Indiana University.
xxiv
Introduction
1
History
Not so long ago, the scoffer could say that econometrics and game theory were like Japan
and Argentina. In the late 1940s both disciplines and both economies were full of promise,
poised for rapid growth and ready to make a profound impact on the world. We all know
what happened to the economies of Japan and Argentina. Of the disciplines, econometrics
became an inseparable part of economics, while game theory languished as a subdiscipline,
interesting to its specialists but ignored by the profession as a whole. The specialists
in game theory were generally mathematicians, who cared about definitions and proofs
rather than applying the methods to economic problems. Game theorists took pride in the
diversity of disciplines to which their theory could be applied, but in none had it become
indispensable.
In the 1970s, the analogy with Argentina broke down. At the same time that Argentina
was inviting back Juan Peron, economists were beginning to discover what they could
achieve by combining game theory with the structure of complex economic situations.
Innovation in theory and application was especially useful for situations with asymmetric
information and a temporal sequence of actions, the two major themes of this book. During
the 1980s, game theory became dramatically more important to mainstream economics.
Indeed, it seemed to be swallowing up microeconomics just as econometrics had swallowed
up empirical economics.
Game theory is generally considered to have begun with the publication of von Neu-mann
& Morgenstern’s The Theory of Games and Economic Behaviour in 1944. Although
very little of the game theory in that thick volume is relevant to the present book, it
introduced the idea that conflict could be mathematically analyzed and provided the ter-minology
with which to do it. The development of the “Prisoner’s Dilemma” (Tucker
[unpub]) and Nash’s papers on the definition and existence of equilibrium (Nash [1950b,
1951]) laid the foundations for modern noncooperative game theory. At the same time,
cooperative game theory reached important results in papers by Nash (1950a) and Shapley
(1953b) on bargaining games and Gillies (1953) and Shapley (1953a) on the core.
By 1953 virtually all the game theory that was to be used by economists for the next
20 years had been developed. Until the mid 1970s, game theory remained an autonomous
field with little relevance to mainstream economics, important exceptions being Schelling’s
1960 book, The Strategy of Conflict, which introduced the focal point, and a series of papers
(of which Debreu & Scarf [1963] is typical) that showed the relationship of the core of a
game to the general equilibrium of an economy.
In the 1970s, information became the focus of many models as economists started
to put emphasis on individuals who act rationally but with limited information. When
1July 24, 1999. May 27, 2002. Ariel Kemper. August 6, 2003. 24 March 2005. Eric Rasmusen,
Erasmuse@indiana.edu. Http://www.rasmusen.org/GI. Footnotes starting with xxx are the author’s notes
to himself. Comments are welcomed. This section is zzz pages long.
1
attention was given to individual agents, the time ordering in which they carried out actions
began to be explicitly incorporated. With this addition, games had enough structure
to reach interesting and non-obvious results. Important “toolbox” references include the
earlier but long-unapplied articles of Selten (1965) (on perfectness) and Harsanyi (1967)
(on incomplete information), the papers by Selten (1975) and Kreps & Wilson (1982b)
extending perfectness, and the article by Kreps, Milgrom, Roberts & Wilson (1982) on
incomplete information in repeated games. Most of the applications in the present book
were developed after 1975, and the flow of research shows no sign of diminishing.
Game Theory’s Method
Game theory has been successful in recent years because it fits so well into the new method-ology
of economics. In the past, macroeconomists started with broad behavioral relation-ships
like the consumption function, and microeconomists often started with precise but
irrational behavioral assumptions such as sales maximization. Now all economists start
with primitive assumptions about the utility functions, production functions, and endow-ments
of the actors in the models (to which must often be added the available information).
The reason is that it is usually easier to judge whether primitive assumptions are sensible
than to evaluate high-level assumptions about behavior. Having accepted the primitive
assumptions, the modeller figures out what happens when the actors maximize their util-ity
subject to the constraints imposed by their information, endowments, and production
functions. This is exactly the paradigm of game theory: the modeller assigns payoff func-tions
and strategy sets to his players and sees what happens when they pick strategies to
maximize their payoffs. The approach is a combination of the “Maximization Subject to
Constraints” of MIT and the “No Free Lunch” of Chicago. We shall see, however, that
game theory relies only on the spirit of these two approaches: it has moved away from max-imization
by calculus, and inefficient allocations are common. The players act rationally,
but the consequences are often bizarre, which makes application to a world of intelligent
men and ludicrous outcomes appropriate.
Exemplifying Theory
Along with the trend towards primitive assumptions and maximizing behavior has been a
trend toward simplicity. I called this “no-fat modelling” in the First Edition, but the term
“exemplifying theory” from Fisher (1989) is more apt. This has also been called “modelling
by example” or “MIT-style theory.” A more smoothly flowing name, but immodest in its
double meaning, is “exemplary theory.” The heart of the approach is to discover the
simplest assumptions needed to generate an interesting conclusion– the starkest, barest
model that has the desired result. This desired result is the answer to some relatively
narrow question. Could education be just a signal of ability? Why might bid-ask spreads
exist? Is predatory pricing ever rational?
The modeller starts with a vague idea such as “People go to college to show they’re
smart.” He then models the idea formally in a simple way. The idea might survive intact;
it might be found formally meaningless; it might survive with qualifications; or its opposite
might turn out to be true. The modeller then uses the model to come up with precise
propositions, whose proofs may tell him still more about the idea. After the proofs, he
2
goes back to thinking in words, trying to understand more than whether the proofs are
mathematically correct.
Good theory of any kind uses Occam’s razor, which cuts out superfluous explanations,
and the ceteris paribus assumption, which restricts attention to one issue at a time. Ex-emplifying
theory goes a step further by providing, in the theory, only a narrow answer to
the question. As Fisher says, “Exemplifying theory does not tell us what must happen.
Rather it tells us what can happen.”
In the same vein, at Chicago I have heard the style called “Stories That Might be
True.” This is not destructive criticism if the modeller is modest, since there are also a
great many “Stories That Can’t Be True,” which are often used as the basis for decisions in
business and government. Just as the modeller should feel he has done a good day’s work
if he has eliminated most outcomes as equilibria in his model, even if multiple equilibria
remain, so he should feel useful if he has ruled out certain explanations for how the world
works, even if multiple plausible models remain. The aim should be to come up with one
or more stories that might apply to a particular situation and then try to sort out which
story gives the best explanation. In this, economics combines the deductive reasoning of
mathematics with the analogical reasoning of law.
A critic of the mathematical approach in biology has compared it to an hourglass
(Slatkin [1980]). First, a broad and important problem is introduced. Second, it is reduced
to a very special but tractable model that hopes to capture its essence. Finally, in the most
perilous part of the process, the results are expanded to apply to the original problem.
Exemplifying theory does the same thing.
The process is one of setting up “If-Then” statements, whether in words or symbols.
To apply such statements, their premises and conclusions need to be verified, either by
casual or careful empiricism. If the required assumptions seem contrived or the assump-tions
and implications contradict reality, the idea should be discarded. If “reality” is not
immediately obvious and data is available, econometric tests may help show whether the
model is valid. Predictions can be made about future events, but that is not usually the
primary motivation: most of us are more interested in explaining and understanding than
predicting.
The method just described is close to how, according to Lakatos (1976), mathematical
theorems are developed. It contrasts sharply with the common view that the researcher
starts with a hypothesis and proves or disproves it. Instead, the process of proof helps show
how the hypothesis should be formulated.
An important part of exemplifying theory is what Kreps & Spence (1984) have called
“blackboxing”: treating unimportant subcomponents of a model in a cursory way. The
game “Entry for Buyout” of section 15.4, for example, asks whether a new entrant would
be bought out by the industry’s incumbent producer, something that depends on duopoly
pricing and bargaining. Both pricing and bargaining are complicated games in themselves,
but if the modeller does not wish to deflect attention to those topics he can use the simple
Nash and Cournot solutions to those games and go on to analyze buyout. If the entire
focus of the model were duopoly pricing, then using the Cournot solution would be open
3
to attack, but as a simplifying assumption, rather than one that “drives” the model, it is
acceptable.
Despite the style’s drive towards simplicity, a certain amount of formalism and math-ematics
is required to pin down the modeller’s thoughts. Exemplifying theory treads a
middle path between mathematical generality and nonmathematical vagueness. Both al-ternatives
will complain that exemplifying theory is too narrow. But beware of calls for
more “rich,” “complex,” or “textured” descriptions; these often lead to theory which is
either too incoherent or too incomprehensible to be applied to real situations.
Some readers will think that exemplifying theory uses too little mathematical tech-nique,
but others, especially noneconomists, will think it uses too much. Intelligent laymen
have objected to the amount of mathematics in economics since at least the 1880s, when
George Bernard Shaw said that as a boy he (1) let someone assume that a = b, (2) per-mitted
several steps of algebra, and (3) found he had accepted a proof that 1 = 2. Forever
after, Shaw distrusted assumptions and algebra. Despite the effort to achieve simplicity (or
perhaps because of it), mathematics is essential to exemplifying theory. The conclusions
can be retranslated into words, but rarely can they be found by verbal reasoning. The
economist Wicksteed put this nicely in his reply to Shaw’s criticism:
Mr Shaw arrived at the sapient conclusion that there “was a screw loose somewhere”–
not in his own reasoning powers, but–“in the algebraic art”; and thenceforth
renounced mathematical reasoning in favour of the literary method which en-ables
a clever man to follow equally fallacious arguments to equally absurd
conclusions without seeing that they are absurd. This is the exact difference
between the mathematical and literary treatment of the pure theory of political
economy. (Wicksteed [1885] p. 732)
In exemplifying theory, one can still rig a model to achieve a wide range of results, but
it must be rigged by making strange primitive assumptions. Everyone familiar with the
style knows that the place to look for the source of suspicious results is the description at
the start of the model. If that description is not clear, the reader deduces that the model’s
counterintuitive results arise from bad assumptions concealed in poor writing. Clarity is
therefore important, and the somewhat inelegant Players-Actions-Payoffs presentation used
in this book is useful not only for helping the writer, but for persuading the reader.
This Book’s Style
Substance and style are closely related. The difference between a good model and a bad one
is not just whether the essence of the situation is captured, but also how much froth covers
the essence. In this book, I have tried to make the games as simple as possible. They often,
for example, allow each player a choice of only two actions. Our intuition works best with
such models, and continuous actions are technically more troublesome. Other assumptions,
such as zero production costs, rely on trained intuition. To the layman, the assumption
that output is costless seems very strong, but a little experience with these models teaches
that it is the constancy of the marginal cost that usually matters, not its level.
4
What matters more than what a model says is what we understand it to say. Just as
an article written in Sanskrit is useless to me, so is one that is excessively mathematical or
poorly written, no matter how rigorous it seems to the author. Such an article leaves me
with some new belief about its subject, but that belief is not sharp, or precisely correct.
Overprecision in sending a message creates imprecision when it is received, because precision
is not clarity. The result of an attempt to be mathematically precise is sometimes to
overwhelm the reader, in the same way that someone who requests the answer to a simple
question in the discovery process of a lawsuit is overwhelmed when the other side responds
with 70 boxes of tangentially related documents. The quality of the author’s input should
be judged not by some abstract standard but by the output in terms of reader processing
cost and understanding.
In this spirit, I have tried to simplify the structure and notation of models while
giving credit to their original authors, but I must ask pardon of anyone whose model has
been oversimplified or distorted, or whose model I have inadvertently replicated without
crediting them. In trying to be understandable, I have taken risks with respect to accuracy.
My hope is that the impression left in the readers’ minds will be more accurate than if a
style more cautious and obscure had left them to devise their own errors.
Readers may be surprised to find occasional references to newspaper and magazine
articles in this book. I hope these references will be reminders that models ought eventually
to be applied to specific facts, and that a great many interesting situations are waiting for
our analysis. The principal-agent problem is not found only in back issues of Econometrica:
it can be found on the front page of today’s Wall Street Journal if one knows what to look
for.
I make the occasional joke here and there, and game theory is a subject intrinsically
full of paradox and surprise. I want to emphasize, though, that I take game theory seriously,
in the same way that Chicago economists like to say that they take price theory seriously.
It is not just an academic artform: people do choose actions deliberately and trade off one
good against another, and game theory will help you understand how they do that. If it did
not, I would not advise you to study such a difficult subject; there are much more elegant
fields in mathematics, from an aesthetic point of view. As it is, I think it is important that
every educated person have some contact with the ideas in this book, just as they should
have some idea of the basic principles of price theory.
I have been forced to exercise more discretion over definitions than I had hoped. Many
concepts have been defined on an article-by-article basis in the literature, with no consis-tency
and little attention to euphony or usefulness. Other concepts, such as “asymmetric
information” and “incomplete information,” have been considered so basic as to not need
definition, and hence have been used in contradictory ways. I use existing terms whenever
possible, and synonyms are listed.
I have often named the players Smith and Jones so that the reader’s memory will be
less taxed in remembering which is a player and which is a time period. I hope also to
reinforce the idea that a model is a story made precise; we begin with Smith and Jones,
even if we quickly descend to s and j. Keeping this in mind, the modeller is less likely to
build mathematically correct models with absurd action sets, and his descriptions are more
5
pleasant to read. In the same vein, labelling a curve “U = 83” sacrifices no generality: the
phrase “U = 83 and U = 66” has virtually the same content as “U = α and U = β, where
α > β,” but uses less short-term memory.
A danger of this approach is that readers may not appreciate the complexity of some of
the material. While journal articles make the material seem harder than it is, this approach
makes it seem easier (a statement that can be true even if readers find this book difficult).
The better the author does his job, the worse this problem becomes. Keynes (1933) says
of Alfred Marshall’s Principles,
The lack of emphasis and of strong light and shade, the sedulous rubbing away
of rough edges and salients and projections, until what is most novel can appear
as trite, allows the reader to pass too easily through. Like a duck leaving water,
he can escape from this douche of ideas with scarce a wetting. The difficulties
are concealed; the most ticklish problems are solved in footnotes; a pregnant
and original judgement is dressed up as a platitude.
This book may well be subject to the same criticism, but I have tried to face up to difficult
points, and the problems at the end of each chapter will help to avoid making the reader’s
progress too easy. Only a certain amount of understanding can be expected from a book,
however. The efficient way to learn how to do research is to start doing it, not to read
about it, and after reading this book, if not before, many readers will want to build their
own models. My purpose here is to show them the big picture, to help them understand
the models intuitively, and give them a feel for the modelling process.
NOTES
• Perhaps the most important contribution of von Neumann & Morgenstern (1944) is the
theory of expected utility (see section 2.3). Although they developed the theory because
they needed it to find the equilibria of games, it is today heavily used in all branches of
economics. In game theory proper, they contributed the framework to describe games, and
the concept of mixed strategies (see section 3.1). A good historical discussion is Shubik
(1992) in the Weintraub volume mentioned in the next note.
• A number of good books on the history of game theory have appeared in recent years.
Norman Macrae’s John von Neumann and Sylvia Nasar’s A Beautiful Mind (on John Nash)
are extraordinarily good biographies of founding fathers, while Eminent Economists: Their
Life Philosophies and Passion and Craft: Economists at Work, edited by Michael Szenberg,
and Toward a History of Game Theory, edited by Roy Weintraub, contain autobiographical
essays by many scholars who use game theory, including Shubik, Riker, Dixit, Varian, and
Myerson. Dimand and Dimand’s A History of Game Theory, the first volume of which
appeared in 1996, is a more intensive look at the intellectual history of the field. See also
Myerson (1999).
• For articles from the history of mathematical economics, see the collection by Baumol &
Goldfeld (1968), Dimand and Dimand’s 1997 The Foundations of Game Theory in three
volumes, and Kuhn (1997).
6
• Collections of more recent articles include Rasmusen (2000a), Binmore & Dasgupta (1986),
Diamond & Rothschild (1978), and the immense Rubinstein (1990).
• On method, see the dialogue by Lakatos (1976), or Davis, Marchisotto & Hersh (1981),
chapter 6 of which is a shorter dialogue in the same style. Friedman (1953) is the classic
essay on a different methodology: evaluating a model by testing its predictions. Kreps &
Spence (1984) is a discussion of exemplifying theory.
• Because style and substance are so closely linked, how one writes is important. For advice
on writing, see McCloskey (1985, 1987) (on economics), Basil Blackwell (1985) (on books),
Bowersock (1985) (on footnotes), Fowler (1965), Fowler & Fowler (1949), Halmos (1970) (on
mathematical writing), Rasmusen (forthcoming), Strunk & White (1959), Weiner (1984),
and Wydick (1978).
• A fallacious proof that 1=2. Suppose that a = b. Then ab = b2 and ab − b2 = a2 − b2.
Factoring the last equation gives us b(a − b) = (a + b)(a − b), which can be simplified to
b = a+b. But then, using our initial assumption, b = 2b and 1 = 2. (The fallacy is division
by zero.)
7
xxx Footnotes starting with xxx are the author’s notes to himself. Comments are
welcomed. August 28, 1999. . September 21, 2004. 24 March 2005. Eric Rasmusen,
Erasmuse@indiana.edu. http://www.rasmusen.org/.
PART I GAME THEORY
9
1 The Rules of the Game
1.1: Definitions
Game theory is concerned with the actions of decision makers who are conscious that their
actions affect each other. When the only two publishers in a city choose prices for their
newspapers, aware that their sales are determined jointly, they are players in a game with
each other. They are not in a game with the readers who buy the newspapers, because each
reader ignores his effect on the publisher. Game theory is not useful when decisionmakers
ignore the reactions of others or treat them as impersonal market forces.
The best way to understand which situations can be modelled as games and which
cannot is to think about examples like the following:
1. OPEC members choosing their annual output;
2. General Motors purchasing steel from USX;
3. two manufacturers, one of nuts and one of bolts, deciding whether to use metric or
American standards;
4. a board of directors setting up a stock option plan for the chief executive officer;
5. the US Air Force hiring jet fighter pilots;
6. an electric company deciding whether to order a new power plant given its estimate
of demand for electricity in ten years.
The first four examples are games. In (1), OPEC members are playing a game because
Saudi Arabia knows that Kuwait’s oil output is based on Kuwait’s forecast of Saudi output,
and the output from both countries matters to the world price. In (2), a significant portion
of American trade in steel is between General Motors and USX, companies which realize
that the quantities traded by each of them affect the price. One wants the price low, the
other high, so this is a game with conflict between the two players. In (3), the nut and
bolt manufacturers are not in conflict, but the actions of one do affect the desired actions
of the other, so the situation is a game none the less. In (4), the board of directors chooses
a stock option plan anticipating the effect on the actions of the CEO.
Game theory is inappropriate for modelling the final two examples. In (5), each indi-vidual
pilot affects the US Air Force insignificantly, and each pilot makes his employment
decision without regard for the impact on the Air Force’s policies. In (6), the electric
company faces a complicated decision, but it does not face another rational agent. These
situations are more appropriate for the use of decision theory than game theory, decision
theory being the careful analysis of how one person makes a decision when he may be
10
faced with uncertainty, or an entire sequence of decisions that interact with each other,
but when he is not faced with having to interact strategically with other single decision
makers. Changes in the important economic variables could,however, turn examples (5)
and (6) into games. The appropriate model changes if the Air Force faces a pilots’ union
or if the public utility commission pressures the utility to change its generating capacity.
Game theory as it will be presented in this book is a modelling tool, not an axiomatic
system. The presentation in this chapter is unconventional. Rather than starting with
mathematical definitions or simple little games of the kind used later in the chapter, we
will start with a situation to be modelled, and build a game from it step by step.
Describing a Game
The essential elements of a game are players, actions, payoffs, and information— PAPI,
for short. These are collectively known as the rules of the game, and the modeller’s
objective is to describe a situation in terms of the rules of a game so as to explain what will
happen in that situation. Trying to maximize their payoffs, the players will devise plans
known as strategies that pick actions depending on the information that has arrived
at each moment. The combination of strategies chosen by each player is known as the
equilibrium. Given an equilibrium, the modeller can see what actions come out of the
conjunction of all the players’ plans, and this tells him the outcome of the game.
This kind of standard description helps both the modeller and his readers. For the
modeller, the names are useful because they help ensure that the important details of the
game have been fully specified. For his readers, they make the game easier to understand,
especially if, as with most technical papers, the paper is first skimmed quickly to see if it
is worth reading. The less clear a writer’s style, the more closely he should adhere to the
standard names, which means that most of us ought to adhere very closely indeed.
Think of writing a paper as a game between author and reader, rather than as a
single-player production process. The author, knowing that he has valuable information
but imperfect means of communication, is trying to convey the information to the reader.
The reader does not know whether the information is valuable, and he must choose whether
to read the paper closely enough to find out.1
To define the terms used above and to show the difference between game theory and
decision theory, let us use the example of an entrepreneur trying to decide whether to start
a dry cleaning store in a town already served by one dry cleaner. We will call the two firms
“NewCleaner” and “OldCleaner.” NewCleaner is uncertain about whether the economy
will be in a recession or not, which will affect how much consumers pay for dry cleaning,
and must also worry about whether OldCleaner will respond to entry with a price war or by
keeping its initial high prices. OldCleaner is a well-established firm, and it would survive
any price war, though its profits would fall. NewCleaner must itself decide whether to
1Once you have read to the end of this chapter: What are the possible equilibria of this game?
11
initiate a price war or to charge high prices, and must also decide what kind of equipment
to buy, how many workers to hire, and so forth.
Players are the individuals who make decisions. Each player’s goal is to maximize his
utility by choice of actions.
In the Dry Cleaners Game, let us specify the players to be NewCleaner and OldCleaner.
Passive individuals like the customers, who react predictably to price changes without
any thought of trying to change anyone’s behavior, are not players, but environmental
parameters. Simplicity is the goal in modelling, and the ideal is to keep the number of
players down to the minimum that captures the essence of the situation.
Sometimes it is useful to explicitly include individuals in the model called pseudo-players
whose actions are taken in a purely mechanical way.
Nature is a pseudo-player who takes random actions at specified points in the game with
specified probabilities.
In the Dry Cleaners Game, we will model the possibility of recession as a move by
Nature. With probability 0.3, Nature decides that there will be a recession, and with
probability 0.7 there will not. Even if the players always took the same actions, this
random move means that the model would yield more than just one prediction. We say
that there are different realizations of a game depending on the results of random moves.
An action or move by player i, denoted ai, is a choice he can make.
Player i’s action set, Ai = {ai}, is the entire set of actions available to him.
An action combination is an ordered set a = {ai}, (i = 1, . . . , n) of one action for each
of the n players in the game.
Again, simplicity is our goal. We are trying to determine whether Newcleaner will enter
or not, and for this it is not important for us to go into the technicalities of dry cleaning
equipment and labor practices. Also, it will not be in Newcleaner’s interest to start a price
war, since it cannot possibly drive out Oldcleaners, so we can exclude that decision from our
model. Newcleaner’s action set can be modelled very simply as {Enter, Stay Out}. Wewill
also specify Oldcleaner’s action set to be simple: it is to choose price from {Low,High}.
By player i’s payoff πi(s1, . . . , sn), we mean either:
(1) The utility player i receives after all players and Nature have picked their strategies and
the game has been played out; or
(2) The expected utility he receives as a function of the strategies chosen by himself and the
other players.
For the moment, think of “strategy” as a synonym for “action”. Definitions (1) and
(2) are distinct and different, but in the literature and this book the term “payoff” is used
12
for both the actual payoff and the expected payoff. The context will make clear which is
meant. If one is modelling a particular real-world situation, figuring out the payoffs is often
the hardest part of constructing a model. For this pair of dry cleaners, we will pretend
we have looked over all the data and figured out that the payoffs are as given by Table
1a if the economy is normal, and that if there is a recession the payoff of each player who
operates in the market is 60 thousand dollars lower, as shown in Table 1b.
Table 1a: The Dry Cleaners Game: Normal Economy
OldCleaner
Low price High price
Enter -100, -50 100, 100
NewCleaner
Stay Out 0,50 0,300
Payoffs to: (NewCleaner, OldCleaner) in thousands of dollars
Table 1b: The Dry Cleaners Game: Recession
OldCleaner
Low price High price
Enter -160, -110 40, 40
NewCleaner
Stay Out 0,-10 0,240
Payoffs to: (NewCleaner, OldCleaner) in thousands of dollars
Information is modelled using the concept of the information set, a concept which
will be defined more precisely in Section 2.2. For now, think of a player’s information set
as his knowledge at a particular time of the values of different variables. The elements
of the information set are the different values that the player thinks are possible. If the
information set has many elements, there are many values the player cannot rule out; if it
has one element, he knows the value precisely. A player’s information set includes not only
distinctions between the values of variables such as the strength of oil demand, but also
knowledge of what actions have previously been taken, so his information set changes over
the course of the game.
Here, at the time that it chooses its price, OldCleaner will know NewCleaner’s decision
about entry. But what do the firms know about the recession? If both firms know about the
recession we model that as Nature moving before NewCleaner; if only OldCleaner knows,
we put Nature’s move after NewCleaner; if neither firm knows whether there is a recession
at the time they must make their decisions, we put Nature’s move at the end of the game.
Let us do this last.
It is convenient to lay out information and actions together in an order of play. Here
is the order of play we have specified for the Dry Cleaners Game:
13
1 Newcleaner chooses its entry decision from {Enter, Stay Out}.
2 Oldcleaner chooses its price from {Low,High}.
3 Nature picks demand, D, to be Recession with probability 0.3 or Normal with proba-bility
0.7.
The purpose of modelling is to explain how a given set of circumstances leads to a
particular result. The result of interest is known as the outcome.
The outcome of the game is a set of interesting elements that the modeller picks from the
values of actions, payoffs, and other variables after the game is played out.
The definition of the outcome for any particular model depends on what variables
the modeller finds interesting. One way to define the outcome of the Dry Cleaners Game
would be as either Enter or Stay Out. Another way, appropriate if the model is being
constructed to help plan NewCleaner’s finances, is as the payoff that NewCleaner realizes,
which is, from Tables 1a and 1b, one element of the set {0, 100, -100, 40, -160}.
Having laid out the assumptions of the model, let us return to what is special about
the way game theory models a situation. Decision theory sets up the rules of the game
in much the same way as game theory, but its outlook is fundamentally different in one
important way: there is only one player. Return to NewCleaner’s decision about entry. In
decision theory, the standard method is to construct a decision tree from the rules of the
game, which is just a graphical way to depict the order of play.
Figure 1 shows a decision tree for the Dry Cleaners Game. It shows all the moves
available to NewCleaner, the probabilities of states of nature ( actions that NewCleaner
cannot control), and the payoffs to NewCleaner depending on its choices and what the
environment is like. Note that although we already specified the probabilities of Nature’s
move to be 0.7 for Normal, we also need to specify a probability for OldCleaner’s move,
which is set at probability 0.5 of Low price and probability 0.5 of High price.
14
Figure 1: The Dry Cleaners Game as a Decision Tree
Once a decision tree is set up, we can solve for the optimal decision which maximizes
the expected payoff. Suppose NewCleaner has entered. If OldCleaner chooses a high price,
then NewCleaner’s expected payoff is 82, which is 0.7(100) + 0.3(40). If OldCleaner chooses
a low price, then NewCleaner’s expected payoff is -118, which is 0.7(-100) + 0.3(-160). Since
there is a 50-50 chance of each move by OldCleaner, NewCleaner’s overall expected payoff
from Enter is -18. That is worse than the 0 which NewCleaner could get by choosing stay
out, so the prediction is that NewCleaner will stay out.
That, however, is wrong. This is a game, not just a decision problem. The flaw in the
reasoning I just went through is the assumption that OldCleaner will choose High price
with probability 0.5. If we use information about OldCleaner’ payoffs and figure out what
moves OldCleaner will take in solving its own profit maximization problem, we will come
to a different conclusion.
First, let us depict the order of play as a game tree instead of a decision tree. Figure
2 shows our model as a game tree, with all of OldCleaner’s moves and payoffs.
15
Figure 2: The Dry Cleaners Game as a Game Tree
Viewing the situation as a game, we must think about both players’ decision making.
Suppose NewCleaner has entered. If OldCleaner chooses High price, OldCleaner’s expected
profit is 82, which is 0.7(100) + 0.3(40). If OldCleaner chooses Low price, OldCleaner’s
expected profit is -68, which is 0.7(-50) + 0.3(-110). Thus, OldCleaner will choose High
price, and with probability 1.0, not 0.5. The arrow on the game tree for High price shows
this conclusion of our reasoning. This means, in turn, that NewCleaner can predict an
expected payoff of 82, which is 0.7(100) + 0.3(40), from Enter.
Suppose NewCleaner has not entered. If OldCleaner chooses High price, OldCleaner’
expected profit is 282, which is 0.7(300) + 0.3(240). If OldCleaner chooses Low price,
OldCleaner’s expected profit is 32, which is 0.7(50) + 0.3(-10). Thus, OldCleaner will
choose High price, as shown by the arrow on High price. If NewCleaner chooses Stay out,
NewCleaner will have a payoff of 0, and since that is worse than the 82 which NewCleaner
can predict from Enter, NewCleaner will in fact enter the market.
This switching back from the point of view of one player to the point of view of
another is characteristic of game theory. The game theorist must practice putting himself
in everybody else’s shoes. (Does that mean we become kinder, gentler people? — Or do we
just get trickier?)
Since so much depends on the interaction between the plans and predictions of different
players, it is useful to go a step beyond simply setting out actions in a game. Instead, the
modeller goes on to think about strategies, which are action plans.
Player i’s strategy si is a rule that tells him which action to choose at each instant of the
game, given his information set.
16
Player i’s strategy set or strategy space Si = {si} is the set of strategies available to
him.
A strategy profile s = (s1, . . . , sn) is an ordered set consisting of one strategy for each of
the n players in the game.2
Since the information set includes whatever the player knows about the previous ac-tions
of other players, the strategy tells him how to react to their actions. In the Dry
Cleaners Game, the strategy set for NewCleaner is just { Enter, Stay Out } , since New-
Cleaner moves first and is not reacting to any new information. The strategy set for
OldCleaner, though, is


High Price if NewCleaner Entered, Low Price if NewCleaner Stayed Out
Low Price if NewCleaner Entered, High Price if NewCleaner Stayed Out
High Price No Matter What
Low Price No Matter What


The concept of the strategy is useful because the action a player wishes to pick often
depends on the past actions of Nature and the other players. Only rarely can we predict
a player’s actions unconditionally, but often we can predict how he will respond to the
outside world.
Keep in mind that a player’s strategy is a complete set of instructions for him, which
tells him what actions to pick in every conceivable situation, even if he does not expect to
reach that situation. Strictly speaking, even if a player’s strategy instructs him to commit
suicide in 1989, it ought also to specify what actions he takes if he is still alive in 1990. This
kind of care will be crucial in Chapter 4’s discussion of “subgame perfect” equilibrium. The
completeness of the description also means that strategies, unlike actions, are unobservable.
An action is physical, but a strategy is only mental.
Equilibrium
To predict the outcome of a game, the modeller focusses on the possible strategy profiles,
since it is the interaction of the different players’ strategies that determines what happens.
The distinction between strategy profiles, which are sets of strategies, and outcomes, which
are sets of values of whichever variables are considered interesting, is a common source of
confusion. Often different strategy profiles lead to the same outcome. In the Dry Cleaners
Game, the single outcome of NewCleaner Enters would result from either of the following
two strategy profiles:
2I used “strategy combination” instead of “strategy profile” in the third edition, but “profile” seems
well enough established that I’m switching to it.
17
(
High Price if NewCleaner Enters, Low Price if NewCleaner Stays Out
Enter
)
(
Low Price if NewCleaner Enters, High Price if NewCleaner Stays Out
Enter
)
Predicting what happens consists of selecting one or more strategy profiles as being
the most rational behavior by the players acting to maximize their payoffs.
An equilibrium s∗ = (s∗1, . . . , s∗n) is a strategy profile consisting of a best strategy for each
of the n players in the game.
The equilibrium strategies are the strategies players pick in trying to maximize
their individual payoffs, as distinct from the many possible strategy profiles obtainable
by arbitrarily choosing one strategy per player. Equilibrium is used differently in game
theory than in other areas of economics. In a general equilibrium model, for example,
an equilibrium is a set of prices resulting from optimal behavior by the individuals in the
economy. In game theory, that set of prices would be the equilibrium outcome, but
the equilibrium itself would be the strategy profile– the individuals’ rules for buying and
selling– that generated the outcome.
People often carelessly say “equilibrium” when they mean “equilibrium outcome,” and
“strategy” when they mean “action.” The difference is not very important in most of the
games that will appear in this chapter, but it is absolutely fundamental to thinking like a
game theorist. Consider Germany’s decision on whether to remilitarize the Rhineland in
1936. France adopted the strategy: Do not fight, and Germany responded by remilitarizing,
leading to World War II a few years later. If France had adopted the strategy: Fight if
Germany remilitarizes; otherwise do not fight, the outcome would still have been that
France would not have fought. No war would have ensued,however, because Germany
would not remilitarized. Perhaps it was because he thought along these lines that John
von Neumann was such a hawk in the Cold War, as MacRae describes in his biography
(MacRae [1992]). This difference between actions and strategies, outcomes and equilibria,
is one of the hardest ideas to teach in a game theory class, even though it is trivial to state.
To find the equilibrium, it is not enough to specify the players, strategies, and payoffs,
because the modeller must also decide what “best strategy” means. He does this by defining
an equilibrium concept.
An equilibrium concept or solution concept F : {S1, . . . , Sn, π1, . . . , πn} → s∗ is a rule
that defines an equilibrium based on the possible strategy profiles and the payoff functions.
We have implicitly already used an equilibrium concept in the analysis above, which picked
one strategy for each of the two players as our prediction for the game (what we implicitly
18
used is the concept of subgame perfectness which will reappear in chapter 4). Only a few
equilibrium concepts are generally accepted, and the remaining sections of this chapter are
devoted to finding the equilibrium using the two best-known of them: dominant strategy
and Nash equilibrium.
Uniqueness
Accepted solution concepts do not guarantee uniqueness, and lack of a unique equilibrium
is a major problem in game theory. Often the solution concept employed leads us to believe
that the players will pick one of the two strategy profiles A or B, not C or D, but we cannot
say whether A or B is more likely. Sometimes we have the opposite problem and the game
has no equilibrium at all. By this is meant either that the modeller sees no good reason
why one strategy profile is more likely than another, or that some player wants to pick an
infinite value for one of his actions.
A model with no equilibrium or multiple equilibria is underspecified. The modeller
has failed to provide a full and precise prediction for what will happen. One option is to
admit that his theory is incomplete. This is not a shameful thing to do; an admission of
incompleteness like Section 5.2’s Folk Theorem is a valuable negative result. Or perhaps
the situation being modelled really is unpredictable. Another option is to renew the attack
by changing the game’s description or the solution concept. Preferably it is the description
that is changed, since economists look to the rules of the game for the differences between
models, and not to the solution concept. If an important part of the game is concealed
under the definition of equilibrium, in fact, the reader is likely to feel tricked and to charge
the modeller with intellectual dishonesty.
1.2 Dominated and Dominant Strategies: The Prisoner’s Dilemma
In discussing equilibrium concepts, it is useful to have shorthand for “all the other players’
strategies.”
For any vector y = (y1, . . . , yn), denote by y−i the vector (y1, . . . , yi−1, yi+1, . . . , yn), which
is the portion of y not associated with player i.
Using this notation, s−Smith, for instance, is the profile of strategies of every player except
player Smith. That profile is of great interest to Smith, because he uses it to help choose
his own strategy, and the new notation helps define his best response.
Player i’s best response or best reply to the strategies s−i chosen by the other players
is the strategy s∗i
that yields him the greatest payoff; that is,
πi(s∗i
, s−i) ≥ πi(s0i, s−i) ∀s0i 6= s∗i
. (1)
The best response is strongly best if no other strategies are equally good, and weakly best
otherwise.
19
The first important equilibrium concept is based on the idea of dominance.
The strategy sdi
is a dominated strategy if it is strictly inferior to some other strategy no
matter what strategies the other players choose, in the sense that whatever strategies they
pick, his payoff is lower with sdi
. Mathematically, sdi
is dominated if there exists a single s0i
such that
πi(sdi
, s−i) < πi(s0i, s−i) ∀s−i. (2)
Note that sdi
is not a dominated strategy if there is no s−i to which it is the best response,
but sometimes the better strategy is s0i and sometimes it is s00 i. In that case, sdi
could have
the redeeming feature of being a good compromise strategy for a player who cannot predict
what the other players are going to do. A dominated strategy is unambiguously inferior to
some single other strategy.
There is usually no special name for the superior strategy that beats a dominated
strategy. In unusual games, however, there is some strategy that beats every other strategy.
We call that a “dominant strategy”.
The strategy s∗i
is a dominant strategy if it is a player’s strictly best response to any
strategies the other players might pick, in the sense that whatever strategies they pick, his
payoff is highest with s∗i
. Mathematically,
πi(s∗i
, s−i) > πi(s0i, s−i) ∀s−i, ∀s0i 6= s∗i
. (3)
A dominant strategy equilibrium is a strategy profile consisting of each player’s dom-inant
strategy.
A player’s dominant strategy is his strictly best response even to wildly irrational
actions by the other players. Most games do not have dominant strategies, and the players
must try to figure out each others’ actions to choose their own.
The Dry Cleaners Game incorporated considerable complexity in the rules of the game
to illustrate such things as information sets and the time sequence of actions. To illustrate
equilibrium concepts, we will use simpler games, such as the Prisoner’s Dilemma. In the
Prisoner’s Dilemma, two prisoners, Messrs Row and Column, are being interrogated sep-arately.
If both confess, each is sentenced to eight years in prison; if both deny their
involvement, each is sentenced to one year.3 If just one confesses, he is released but the
other prisoner is sentenced to ten years. The Prisoner’s Dilemma is an example of a 2-by-2
game, because each of the two players– Row and Column– has two possible actions in his
action set: Confess and Deny. Table 2 gives the payoffs (The arrows represent a player’s
preference between actions, as will be explained in Section 1.4).
Table 2: The Prisoner’s Dilemma
3Another way to tell the story is to say that if both deny, then with probability 0.1 they are convicted
anyway and serve ten years, for an expected payoff of (−1, −1).
20
Column
Deny Confess
Deny -1,-1 → -10, 0
Row ↓ ↓ Confess 0,-10 → - 8,-8
Payoffs to: (Row,Column)
Each player has a dominant strategy. Consider Row. Row does not know which action
Column is choosing, but if Column chooses Deny, Row faces a Deny payoff of −1 and a
Confess payoff of 0, whereas if Column chooses Confess, Row faces a Deny payoff of −10
and a Confess payoff of −8. In either case Row does better with Confess. Since the
game is symmetric, Column’s incentives are the same. The dominant strategy equilibrium
is (Confess, Confess), and the equilibrium payoffs are (−8, −8), which is worse for both
players than (−1, −1). Sixteen, in fact, is the greatest possible combined total of years in
prison.
The result is even stronger than it seems, because it is robust to substantial changes
in the model. Because the equilibrium is a dominant strategy equilibrium, the information
structure of the game does not matter. If Column is allowed to know Row’s move before
taking his own, the equilibrium is unchanged. Row still chooses Confess, knowing that
Column will surely choose Confess afterwards.
The Prisoner’s Dilemma crops up in many different situations, including oligopoly
pricing, auction bidding, salesman effort, political bargaining, and arms races. Whenever
you observe individuals in a conflict that hurts them all, your first thought should be of
the Prisoner’s Dilemma.
The game seems perverse and unrealistic to many people who have never encountered
it before (although friends who are prosecutors assure me that it is a standard crime-fighting
tool). If the outcome does not seem right to you, you should realize that very
often the chief usefulness of a model is to induce discomfort. Discomfort is a sign that
your model is not what you think it is– that you left out something essential to the result
you expected and didn’t get. Either your original thought or your model is mistaken; and
finding such mistakes is a real if painful benefit of model building. To refuse to accept
surprising conclusions is to reject logic.
Cooperative and Noncooperative Games
What difference would it make if the two prisoners could talk to each other before making
their decisions? It depends on the strength of promises. If promises are not binding, then
although the two prisoners might agree to Deny, they would Confess anyway when the
time came to choose actions.
A cooperative game is a game in which the players can make binding commitments, as
opposed to a noncooperative game, in which they cannot.
21
This definition draws the usual distinction between the two theories of games, but
the real difference lies in the modelling approach. Both theories start off with the rules
of the game, but they differ in the kinds of solution concepts employed. Cooperative
game theory is axiomatic, frequently appealing to pareto-optimality,4 fairness, and equity.
Noncooperative game theory is economic in flavor, with solution concepts based on players
maximizing their own utility functions subject to stated constraints. Or, from a different
angle: cooperative game theory is a reduced-form theory, which focusses on properties of
the outcome rather than on the strategies that achieve the outcome, a method which is
appropriate if modelling the process is too complicated. Except for Section 12.2 in the
chapter on bargaining, this book is concerned exclusively with noncooperative games. For
a good defense of the importance of cooperative game theory, see the essay by Aumann
(1996).
In applied economics, the most commonly encountered use of cooperative games is
to model bargaining. The Prisoner’s Dilemma is a noncooperative game, but it could
be modelled as cooperative by allowing the two players not only to communicate but to
make binding commitments. Cooperative games often allow players to split the gains
from cooperation by making side-payments– transfers between themselves that change
the prescribed payoffs. Cooperative game theory generally incorporates commitments and
side-payments via the solution concept, which can become very elaborate, while noncoop-erative
game theory incorporates them by adding extra actions. The distinction between
cooperative and noncooperative games does not lie in conflict or absence of conflict, as is
shown by the following examples of situations commonly modelled one way or the other:
A cooperative game without conflict. Members of a workforce choose which of equally
arduous tasks to undertake to best coordinate with each other.
A cooperative game with conflict. Bargaining over price between a monopolist and a monop-sonist.
A noncooperative game with conflict. The Prisoner’s Dilemma.
A noncooperative game without conflict. Two companies set a product standard without
communication.
1.3 Iterated Dominance: The Battle of the Bismarck Sea
4If outcome X strongly pareto-dominates outcome Y , then all players have higher utility under
outcome X. If outcome X weakly pareto-dominates outcome Y , some player has higher utility under
X, and no player has lower utility. A zero-sum game does not have outcomes that even weakly pareto-dominate
other outcomes. All of its equilibria are pareto-efficient, because no player gains without another
player losing.
It is often said that strategy profile x “pareto dominates” or “dominates” strategy profile y. Taken
literally, this is meaningless, since strategies do not necessarily have any ordering at all– one could define
Deny as being bigger than Confess, but that would be arbitrary. The statement is really shorthand
for “The payoff profile resulting from strategy profile x pareto-dominates the payoff profile resulting from
strategy y.”
22
Very few games have a dominant strategy equilibrium, but sometimes dominance can still
be useful even when it does not resolve things quite so neatly as in the Prisoner’s Dilemma.
The Battle of the Bismarck Sea, a game I found in Haywood (1954), is set in the South
Pacific in 1943. General Imamura has been ordered to transport Japanese troops across the
Bismarck Sea to New Guinea, and General Kenney wants to bomb the troop transports.
Imamura must choose between a shorter northern route or a longer southern route to New
Guinea, and Kenney must decide where to send his planes to look for the Japanese. If
Kenney sends his planes to the wrong route he can recall them, but the number of days of
bombing is reduced.
The players are Kenney and Imamura, and they each have the same action set,
{North,South}, but their payoffs, given by Table 3, are never the same. Imamura loses ex-actly
what Kenney gains. Because of this special feature, the payoffs could be represented
using just four numbers instead of eight, but listing all eight payoffs in Table 3 saves the
reader a little thinking. The 2-by-2 form with just four entries is a matrix game, while
the equivalent table with eight entries is a bimatrix game. Games can be represented as
matrix or bimatrix games even if they have more than two moves, as long as the number
of moves is finite.
Table 3: The Battle of the Bismarck Sea
Imamura
North South
North 2,-2 ↔ 2, −2
Kenney ↑ ↓ South 1, −1 ← 3, −3
Payoffs to: (Kenney, Imamura)
Strictly speaking, neither player has a dominant strategy. Kenney would choose North
if he thought Imamura would choose North, but South if he thought Imamura would choose
South. Imamura would choose North if he thought Kenney would choose South, and he
would be indifferent between actions if he thought Kenney would choose North. This is
what the arrows are showing. But we can still find a plausible equilibrium, using the
concept of “weak dominance”.
Strategy s0i is weakly dominated if there exists some other strategy s00 i for player i which is
possibly better and never worse, yielding a higher payoff in some strategy profile and never
yielding a lower payoff. Mathematically, s0i is weakly dominated if there exists s00 i such that
πi(s00 i , s−i) ≥ πi(s0i, s−i) ∀s−i, and
πi(s00 i , s−i) > πi(s0i, s−i) forsomes−i.
(4)
One might define a weak dominance equilibrium as the strategy profile found by
deleting all the weakly dominated strategies of each player. Eliminating weakly dominated
23
strategies does not help much in the Battle of the Bismarck Sea, however. Imamura’s
strategy of South is weakly dominated by the strategy North because his payoff from North
is never smaller than his payoff from South, and it is greater if Kenney picks South. For
Kenney, however, neither strategy is even weakly dominated. The modeller must therefore
go a step further, to the idea of the iterated dominance equilibrium.
An iterated dominance equilibrium is a strategy profile found by deleting a weakly
dominated strategy from the strategy set of one of the players, recalculating to find which
remaining strategies are weakly dominated, deleting one of them, and continuing the process
until only one strategy remains for each player.
Applied to the Battle of the Bismarck Sea, this equilibrium concept implies that Ken-ney
decides that Imamura will pick North because it is weakly dominant, so Kenney elim-inates
“Imamura chooses South” from consideration. Having deleted one column of Table
3, Kenney has a strongly dominant strategy: he chooses North, which achieves payoffs
strictly greater than South. The strategy profile (North, North) is an iterated dominance
equilibrium, and indeed (North, North) was the outcome in 1943.
It is interesting to consider modifying the order of play or the information structure
in the Battle of the Bismarck Sea. If Kenney moved first, rather than simultaneously
with Imamura, (North, North) would remain an equilibrium, but (North, South) would also
become one. The payoffs would be the same for both equilibria, but the outcomes would
be different.
If Imamura moved first, (North, North) would be the only equilibrium. What is im-portant
about a player moving first is that it gives the other player more information before
he acts, not the literal timing of the moves. If Kenney has cracked the Japanese code and
knows Imamura’s plan, then it does not matter that the two players move literally simul-taneously;
it is better modelled as a sequential game. Whether Imamura literally moves
first or whether his code is cracked, Kenney’s information set becomes either {Imamura
moved North} or {Imamura moved South} after Imamura’s decision, so Kenney’s equilib-rium
strategy is specified as (North if Imamura moved North, South if Imamura moved
South).
Game theorists often differ in their terminology, and the terminology applied to the
idea of eliminating dominated strategies is particularly diverse. The equilibrium concept
used in the Battle of the Bismarck Sea might be called iterated dominance equilibrium
or iterated dominant strategy equilibrium, or one might say that the game is domi-nance
solvable, that it can be solved by iterated dominance, or that the equilibrium
strategy profile is serially undominated. Sometimes the terms are used to mean dele-tion
of strictly dominated strategies and sometimes to mean deletion of weakly dominated
strategies.
The significant difference is between strong and weak dominance. Everyone agrees
24
that no rational player would use a strictly dominated strategy, but it is harder to argue
against weakly dominated strategies. In economic models, firms and individuals are often
indifferent about their behavior in equilibrium. In standard models of perfect competition,
firms earn zero profits but it is crucial that some firms be active in the market and some
stay out and produce nothing. If a monopolist knows that customer Smith is willing to pay
up to ten dollars for a widget, the monopolist will charge exactly ten dollars to Smith in
equilibrium, which makes Smith indifferent about buying and not buying, yet there is no
equilibrium unless Smith buys. It is impractical, therefore, to rule out equilibria in which
a player is indifferent about his actions. This should be kept in mind later when we discuss
the “open-set problem” in Section 4.3.
Another difficulty is multiple equilibria. The dominant strategy equilibrium of any
game is unique if it exists. Each player has at most one strategy whose payoff in any
strategy profile is strictly higher than the payoff from any other strategy, so only one
strategy profile can be formed out of dominant strategies. A strong iterated dominance
equilibrium is unique if it exists. A weak iterated dominance equilibrium may not be,
because the order in which strategies are deleted can matter to the final solution. If all the
weakly dominated strategies are eliminated simultaneously at each round of elimination,
the resulting equilibrium is unique, if it exists, but possibly no strategy profile will remain.
Consider Table 4’s Iteration Path Game. The strategy profile (r1, c1) and (r1, c3) are
both iterated dominance equilibria, because each of those strategy profile can be found
by iterated deletion. The deletion can proceed in the order (r3, c3, c2, r2) or in the order
(r2, c2, c1, r3).
Table 4: The Iteration Path Game
Column
c1 c2 c3
r1 2,12 1,10 1,12
Row r2 0,12 0,10 0,11
r3 0,12 1,10 0,13
Payoffs to: (Row, Column)
Despite these problems, deletion of weakly dominated strategies is a useful tool, and
it is part of more complicated equilibrium concepts such as Section 4.1’s “subgame perfect-ness”.
If we may return to the Battle of the Bismarck Sea, that game is special because the
25
payoffs of the players always sum to zero. This feature is important enough to deserve a
name.
A zero-sum game is a game in which the sum of the payoffs of all the players is zero
whatever strategies they choose. A game which is not zero-sum is nonzero-sum game or
variable- sum.
In a zero-sum game, what one player gains, another player must lose. The Battle of
the Bismarck Sea is a zero-sum game, but the Prisoner’s Dilemma and the Dry Cleaners
Game are not, and there is no way that the payoffs in those games can be rescaled to make
them zero-sum without changing the essential character of the games.
If a game is zero-sum the utilities of the players can be represented so as to sum to
zero under any outcome. Since utility functions are to some extent arbitrary, the sum can
also be represented to be non-zero even if the game is zero-sum. Often modellers will refer
to a game as zero-sum even when the payoffs do not add up to zero, so long as the payoffs
add up to some constant amount. The difference is a trivial normalization.
Although zero-sum games have fascinated game theorists for many years, they are
uncommon in economics. One of the few examples is the bargaining game between two
players who divide a surplus, but even this is often modelled nowadays as a nonzero-sum
game in which the surplus shrinks as the players spend more time deciding how to divide
it. In reality, even simple division of property can result in loss– just think of how much
the lawyers take out when a divorcing couple bargain over dividing their possessions.
Although the 2-by-2 games in this chapter may seem facetious, they are simple enough
for use in modelling economic situations. The Battle of the Bismarck Sea, for example, can
be turned into a game of corporate strategy. Two firms, Kenney Company and Imamura
Incorporated, are trying to maximize their shares of a market of constant size by choosing
between the two product designs North and South. Kenney has a marketing advantage,
and would like to compete head-to-head, while Imamura would rather carve out its own
niche. The equilibrium is (North, North).
1.4 Nash Equilibrium: Boxed Pigs, the Battle of the Sexes, and Ranked Coordination
For the vast majority of games, which lack even iterated dominance equilibria, modellers use
Nash equilibrium, the most important and widespread equilibrium concept. To introduce
Nash equilibrium we will use the game Boxed Pigs from Baldwin & Meese (1979). Two
pigs are put in a box with a special control panel at one end and a food dispenser at the
other end. When a pig presses the panel, at a utility cost of 2 units, 10 units of food are
dispensed at the dispenser. One pig is “dominant” (let us assume he is bigger), and if he
gets to the dispenser first, the other pig will only get his leavings, worth 1 unit. If, instead,
the small pig is at the dispenser first, he eats 4 units, and even if they arrive at the same
time the small pig gets 3 units. Table 5 summarizes the payoffs for the strategies Press
26
the panel and Wait by the dispenser at the other end.
Table 5: Boxed Pigs
Small Pig
Press Wait
Press 5, 1 → 4 , 4
Big Pig ↓ ↑
Wait 9 , −1 → 0, 0
Payoffs to: (Big Pig, Small Pig)
Boxed Pigs has no dominant strategy equilibrium, because what the big pig chooses
depends on what he thinks the small pig will choose. If he believed that the small pig would
press the panel, the big pig would wait by the dispenser, but if he believed that the small
pig would wait, the big pig would press the panel. There does exist an iterated dominance
equilibrium, (Press,Wait), but we will use a different line of reasoning to justify that
outcome: Nash equilibrium.
Nash equilibrium is the standard equilibrium concept in economics. It is less obviously
correct than dominant strategy equilibrium but more often applicable. Nash equilibrium
is so widely accepted that the reader can assume that if a model does not specify which
equilibrium concept is being used it is Nash or some refinement of Nash.
The strategy profile s∗ is a Nash equilibrium if no player has incentive to deviate from
his strategy given that the other players do not deviate. Formally,
∀i, πi(s∗i
, s∗
−i) ≥ πi(s0i, s∗
−i), ∀s0i. (5)
The strategy profile (Press,Wait) is a Nash equilibrium. The way to approach Nash
equilibrium is to propose a strategy profile and test whether each player’s strategy is a best
response to the others’ strategies. If the big pig picks Press, the small pig, who faces a
choice between a payoff of 1 from pressing and 4 from waiting, is willing to wait. If the
small pig picks Wait, the big pig, who has a choice between a payoff of 4 from pressing and
0 from waiting, is willing to press. This confirms that (Press,Wait) is a Nash equilibrium,
and in fact it is the unique Nash equilibrium.5
It is useful to draw arrows in the tables when trying to solve for the equilibrium, since
the number of calculations is great enough to soak up quite a bit of mental RAM. Another
solution tip, illustrated in Boxed Pigs, is to circle payoffs that dominate other payoffs (or
5This game, too, has its economic analog. If Bigpig, Inc. introduces granola bars, at considerable
marketing expense in educating the public, then Smallpig Ltd. can imitate profitably without ruining
Bigpig’s sales completely. If Smallpig introduces them at the same expense, however, an imitating Bigpig
would hog the market.
27
box, them, as is especially suitable here). Double arrows or dotted circles indicate weakly
dominant payoffs. Any payoff profile in which every payoff is circled, or which has arrows
pointing towards it from every direction, is a Nash equilibrium. I like using arrows better
in 2-by-2 games, but circles are better for bigger games, since arrows become confusing
when payoffs are not lined up in order of magnitude in the table (see Chapter 2’s Table 2).
The pigs in this game have to be smarter than the players in the Prisoner’s Dilemma.
They have to realize that the only set of strategies supported by self-consistent beliefs is
(Press,Wait). The definition of Nash equilibrium lacks the “∀s−i” of dominant strategy
equilibrium, so a Nash strategy need only be a best response to the other Nash strategies,
not to all possible strategies. And although we talk of “best responses,” the moves are
actually simultaneous, so the players are predicting each others’ moves. If the game were
repeated or the players communicated, Nash equilibrium would be especially attractive,
because it is even more compelling that beliefs should be consistent.
Like a dominant strategy equilibrium, a Nash equilibrium can be either weak or strong.
The definition above is for a weak Nash equilibrium. To define strong Nash equilibrium,
make the inequality strict; that is, require that no player be indifferent between his equi-librium
strategy and some other strategy.
Every dominant strategy equilibrium is a Nash equilibrium, but not every Nash equi-librium
is a dominant strategy equilibrium. If a strategy is dominant it is a best response to
any strategies the other players pick, including their equilibrium strategies. If a strategy is
part of a Nash equilibrium, it need only be a best response to the other players’ equilibrium
strategies.
The Modeller’s Dilemma of Table 6 illustrates this feature of Nash equilibrium. The
situation it models is the same as the Prisoner’s Dilemma, with one major exception:
although the police have enough evidence to arrest the prisoner’s as the “probable cause”
of the crime, they will not have enough evidence to convict them of even a minor offense if
neither prisoner confesses. The northwest payoff profile becomes (0,0) instead of (−1, −1).
Table 6: The Modeller’s Dilemma
Column
Deny Confess
Deny 0 , 0 ↔ −10, 0
Row l ↓
Confess 0 ,-10 → -8 , -8
Payoffs to: (Row, Column)
The Modeller’s Dilemma does not have a dominant strategy equilibrium. It does
have what might be called a weak dominant strategy equilibrium, because Confess is still
a weakly dominant strategy for each player. Moreover, using this fact, it can be seen that
(Confess, Confess) is an iterated dominance equilibrium, and it is a strong Nash equilibrium
28
as well. So the case for (Confess, Confess) still being the equilibrium outcome seems very
strong.
There is, however, another Nash equilibrium in the Modeller’s Dilemma: (Deny,
Deny), which is a weak Nash equilibrium. This equilibrium is weak and the other Nash
equilibrium is strong, but (Deny, Deny) has the advantage that its outcome is pareto-superior:
(0, 0) is uniformly greater than (−8, −8). This makes it difficult to know which
behavior to predict.
The Modeller’s Dilemma illustrates a common difficulty for modellers: what to predict
when two Nash equilibria exist. The modeller could add more details to the rules of
the game, or he could use an equilibrium refinement, adding conditions to the basic
equilibrium concept until only one strategy profile satisfies the refined equilibrium concept.
There is no single way to refine Nash equilibrium. The modeller might insist on a strong
equilibrium, or rule out weakly dominated strategies, or use iterated dominance. All of
these lead to (Confess, Confess) in the Modeller’s Dilemma. Or he might rule out Nash
equilibria that are pareto-dominated by other Nash equilibria, and end up with (Deny,
Deny). Neither approach is completely satisfactory. In particular, do not be misled into
thinking that weak Nash equilibria are to be despised. Often, no Nash equilibrium at all will
exist unless the players have the expectation that player B chooses X when he is indifferent
between X and Y. It is not that we are picking the equilibrium in which it is assumed B
does X when he is indifferent. Rather, we are finding the only set of consistent expectations
about behavior. (You will read more about this in connection with the “open-set problem”
of Section 4.2.)
The Battle of the Sexes
The third game we will use to illustrate Nash equilibrium is the Battle of the Sexes, a
conflict between a man who wants to go to a prize fight and a woman who wants to go
to a ballet. While selfish, they are deeply in love, and would, if necessary, sacrifice their
preferences to be with each other. Less romantically, their payoffs are given by Table 7.
Table 7: The Battle of the Sexes 6
Woman
Prize Fight Ballet
Prize Fight 2,1 ← 0, 0
Man ↑ ↓ Ballet 0, 0 → 1,2
Payoffs to: (Man, Woman)
6Political correctness has led to bowdlerized versions of this game being presented in many game theory
books. This is the original, unexpurgated game.
29
The Battle of the Sexes does not have an iterated dominance equilibrium. It has two
Nash equilibria, one of which is the strategy profile (Prize Fight, Prize Fight). Given that
the man chooses Prize Fight, so does the woman; given that the woman chooses Prize
Fight, so does the man. The strategy profile (Ballet, Ballet) is another Nash equilibrium
by the same line of reasoning.
How do the players know which Nash equilibrium to choose? Going to the fight and
going to the ballet are both Nash strategies, but for different equilibria. Nash equilibrium
assumes correct and consistent beliefs. If they do not talk beforehand, the man might go
to the ballet and the woman to the fight, each mistaken about the other’s beliefs. But even
if the players do not communicate, Nash equilibrium is sometimes justified by repetition
of the game. If the couple do not talk, but repeat the game night after night, one may
suppose that eventually they settle on one of the Nash equilibria.
Each of the Nash equilibria in the Battle of the Sexes is pareto-efficient; no other
strategy profile increases the payoff of one player without decreasing that of the other. In
many games the Nash equilibrium is not pareto-efficient: (Confess, Confess), for example,
is the unique Nash equilibrium of the Prisoner’s Dilemma, although its payoffs of (−8, −8)
are pareto- inferior to the (−1, −1) generated by (Deny, Deny).
Who moves first is important in the Battle of the Sexes, unlike any of the three previous
games we have looked at. If the man could buy the fight ticket in advance, his commitment
would induce the woman to go to the fight. In many games, but not all, the player who
moves first (which is equivalent to commitment) has a first-mover advantage.
The Battle of the Sexes has many economic applications. One is the choice of an
industrywide standard when two firms have different preferences but both want a common
standard to encourage consumers to buy the product. A second is to the choice of language
used in a contract when two firms want to formalize a sales agreement but they prefer
different terms. Both sides might, for example, want to add a “liquidated damages” clause
which specifies damages for breach, rather than trust to the courts to estimate a number
later, but one firm wants the value to be $10,000 and the other firm wants $12,000.
Coordination Games
Sometimes one can use the size of the payoffs to choose between Nash equilibria. In
the following game, players Smith and Jones are trying to decide whether to design the
computers they sell to use large or small floppy disks. Both players will sell more computers
if their disk drives are compatible, as shown in Table 8.
Table 8: Ranked Coordination
30
Jones
Large Small
Large 2,2 ← −1, −1
Smith ↑ ↓ Small −1, −1 → 1,1
Payoffs to: (Smith, Jones)
The strategy profiles (Large, Large) and (Small, Small) are both Nash equilibria, but
(Large, Large) pareto-dominates (Small, Small). Both players prefer (Large, Large), and
most modellers would use the pareto- efficient equilibrium to predict the actual outcome.
We could imagine that it arises from pre-game communication between Smith and Jones
taking place outside of the specification of the model, but the interesting question is what
happens if communication is impossible. Is the pareto-efficient equilibrium still more plau-sible?
The question is really one of psychology rather than economics.
Ranked Coordination is one of a large class of games called coordination games,
which share the common feature that the players need to coordinate on one of multiple
Nash equilibria. Ranked Coordination has the additional feature that the equilibria
can be pareto ranked. Section 3.2 will return to problems of coordination to discuss the
concepts of “correlated strategies” and “cheap talk.” These games are of obvious relevance
to analyzing the setting of standards; see, e.g., Michael Katz & Carl Shapiro (1985) and
Joseph Farrell & Garth Saloner (1985). They can be of great importance to the wealth of
economies— just think of the advantages of standard weights and measures (or read Charles
Kindleberger (1983) on their history). Note, however, that not all apparent situations of
coordination on pareto-inferior equilibria turn out to be so. One oft-cited coordination
problem is that of the QWERTY typewriter keyboard, developed in the 1870s when typing
had to proceed slowly to avoid jamming. QWERTY became the standard, although it has
been claimed that the faster speed possible with the Dvorak keyboard would amortize the
cost of retraining full-time typists within ten days (David [1985]). Why large companies
would not retrain their typists is difficult to explain under this story, and Liebowitz &
Margolis (1990) show that economists have been too quick to accept claims that QWERTY
is inefficient. English language spelling is a better example.
Table 9 shows another coordination game, Dangerous Coordination, which has the
same equilibria as Ranked Coordination, but differs in the off-equilibrium payoffs. If
an experiment were conducted in which students played Dangerous Coordination against
each other, I would not be surprised if (Small,Small), the pareto-dominated equilibrium,
were the one that was played out. This is true even though (Large, Large) is still a Nash
equilibrium; if Smith thinks that Jones will pick Large, Smith is quite willing to pick
Large himself. The problem is that if the assumptions of the model are weakened, and
Smith cannot trust Jones to be rational, well-informed about the payoffs of the game, and
unconfused, then Smith will be reluctant to pick Large because his payoff if Jones picks
Small is then -1,000. He would play it safe instead, picking Small and ensuring a payoff
31
of at least −1. In reality, people do make mistakes, and with such an extreme difference
in payoffs, even a small probability of a mistake is important, so (Large, Large) would be
a bad prediction.
Table 9: Dangerous Coordination
Jones
Large Small
Large 2,2 ← −1000, −1
Smith ↑ ↓ Small −1, −1 → 1,1
Payoffs to: (Smith, Jones)
Games like Dangerous Coordination are a major concern in the 1988 book by Harsanyi
and Selten, two of the giants in the field of game theory. I will not try to describe their
approach here, except to say that it is different from my own. I do not consider the fact
that one of the Nash equilibria of Dangerous Coordination is a bad prediction as a heavy
blow against Nash equilibrium. The bad prediction is based on two things: using the
Nash equilibrium concept, and using the game Dangerous Coordination. If Jones might
be confused about the payoffs of the game, then the game actually being played out is not
Dangerous Coordination, so it is not surprising that it gives poor predictions. The rules of
the game ought to describe the probabilities that the players are confused, as well as the
payoffs if they take particular actions. If confusion is an important feature of the situation,
then the two-by-two game of Table 9 is the wrong model to use, and a more complicated
game of incomplete information of the kind described in Chapter 2 is more appropriate.
Again, as with the Prisoner’s Dilemma, the modeller’s first thought on finding that the
model predicts an odd result should not be “Game theory is bunk,” but the more modest
“Maybe I’m not describing the situation correctly” (or even “Maybe I should not trust my
‘common sense’ about what will happen”).
Nash equilibrium is more complicated but also more useful than it looks. Jumping
ahead a bit, consider a game slightly more complex than the ones we have seen so far.
Two firms are choosing outputs Q1 and Q2 simultaneously. The Nash equilibrium is a pair
of numbers (Q∗1,Q∗2) such that neither firm would deviate unilaterally. This troubles the
beginner, who says to himself,
“Sure, Firm 1 will pick Q∗1 if it thinks Firm 2 will pick Q∗2. But Firm 1 will realize
that if it makes Q1 bigger, then Firm 2 will react by making Q2 smaller. So the
situation is much more complicated, and (Q∗1,Q∗2) is not a Nash equilibrium. Or, if
it is, Nash equilibrium is a bad equilibrium concept.”
If there is a problem in this model, it is not Nash equilibrium but the model itself. Nash
equilibrium makes perfect sense as a stable outcome in this model. The beginner’s hy-pothetical
is false because if Firm 1 chooses something other than Q∗1, Firm 2 would not
32
observe the deviation till it was too late to change Q2— remember, this is a simultaneous
move game. The beginner’s worry is really about the rules of the game, not the equilib-rium
concept. He seems to prefer a game in which the firms move sequentially, or maybe
a repeated version of the game. If Firm 1 moved first, and then Firm 2, then Firm 1’s
strategy would still be a single number, Q1, but Firm 2’s strategy— its action rule— would
have to be a function, Q2(Q1). A Nash equilibrium would then consist of an equilibrium
number, Q∗∗ 1 , and an equilibrium function, Q∗∗ 2 (Q1). The two outputs actually chosen, Q∗∗ 1
and Q∗∗ 2 (Q∗∗ 1 ), will be different from the Q∗1 and Q∗2 in the original game. And they should
be different— the new model represents a very different real-world situation. Look ahead,
and you will see that these are the Cournot and Stackelberg models of Chapter 3.
One lesson to draw fromthis is that it is essential to figure out the mathematical form
the strategies take before trying to figure out the equilibrium. In the simultaneous move
game, the strategy profile is a pair of non-negative numbers. In the sequential game, the
strategy profile is one nonnegative number and one function defined over the nonnegative
numbers. Students invariably make the mistake of specifying Firm 2’s strategy as a number,
not a function. This is a far more important point than any beginner realizes. Trust me—
you’re going to make this mistake sooner or later, so it’s worth worrying about.
1.5 Focal Points
Schelling’s book, The Strategy of Conflict (1960) is a classic in game theory, even though it
contains no equations or Greek letters. Although it was published more than 40 years ago,
it is surprisingly modern in spirit. Schelling is not a mathematician but a strategist, and
he examines such things as threats, commitments, hostages, and delegation that we will
examine in a more formal way in the remainder of this book. He is perhaps best known for
his coordination games. Take a moment to decide on a strategy in each of the following
games, adapted from Schelling, which you win by matching your response to those of as
many of the other players as possible.
1 Circle one of the following numbers: 100, 14, 15, 16, 17, 18.
2 Circle one of the following numbers 7, 100, 13, 261, 99, 666.
3 Name Heads or Tails.
4 Name Tails or Heads.
5 You are to split a pie, and get nothing if your proportions add to more than 100 percent.
6 You are to meet somebody in New York City. When? Where?
Each of the games above has many Nash equilibria. In example (1), if each player
thinks every other player will pick 14, he will too, and this is self-confirming; but the same
is true if each player thinks every other player will pick 15. But to a greater or lesser extent
33
they also have Nash equilibria that seem more likely. Certain of the strategy profiles are
focal points: Nash equilibria which for psychological reasons are particularly compelling.
Formalizing what makes a strategy profile a focal point is hard and depends on the
context. In example (1), 100 is a focal point, because it is a number clearly different from
all the others, it is biggest, and it is first in the listing. In example (2), Schelling found
7 to be the most common strategy, but in a group of Satanists, 666 might be the focal
point. In repeated games, focal points are often provided by past history. Examples (3)
and (4) are identical except for the ordering of the choices, but that ordering might make a
difference. In (5), if we split a pie once, we are likely to agree on 50:50. But if last year we
split a pie in the ratio 60:40, that provides a focal point for this year. Example (6) is the
most interesting of all. Schelling found surprising agreement in independent choices, but
the place chosen depended on whether the players knew New York well or were unfamiliar
with the city.
The boundary is a particular kind of focal point. If player Russia chooses the action
of putting his troops anywhere from one inch to 100 miles away from the Chinese border,
player China does not react. If he chooses to put troops from one inch to 100 miles beyond
the border, China declares war. There is an arbitrary discontinuity in behavior at the
boundary. Another example, quite vivid in its arbitrariness, is the rallying cry, “Fifty-Four
Forty or Fight!,” which refers to the geographic parallel claimed as the boundary by jingoist
Americans in the Oregon dispute between Britain and the United States in the 1840s.7
Once the boundary is established it takes on additional significance because behavior
with respect to the boundary conveys information. When Russia crosses an established
boundary, that tells China that Russia intends to make a serious incursion further into
China. Boundaries must be sharp and well known if they are not to be violated, and
a large part of both law and diplomacy is devoted to clarifying them. Boundaries can
also arise in business: two companies producing an unhealthful product might agree not
to mention relative healthfulness in their advertising, but a boundary rule like “Mention
unhealthfulness if you like, but don’t stress it,” would not work.
Mediation and communication are both important in the absence of a clear focal
point. If players can communicate, they can tell each other what actions they will take,
and sometimes, as in Ranked Coordination, this works, because they have no motive to
lie. If the players cannot communicate, a mediator may be able to help by suggesting an
equilibrium to all of them. They have no reason not to take the suggestion, and they would
use the mediator even if his services were costly. Mediation in cases like this is as effective
as arbitration, in which an outside party imposes a solution.
One disadvantage of focal points is that they lead to inflexibility. Suppose the pareto-superior
equilibrium (Large, Large) were chosen as a focal point in Ranked Coordination,
but the game was repeated over a long interval of time. The numbers in the payoff matrix
7The threat was not credible: that parallel is now deep in British Columbia.
34
might slowly change until (Small, Small) and (Large, Large) both had payoffs of, say, 1.6,
and (Small, Small) started to dominate. When, if ever, would the equilibrium switch?
In Ranked Coordination, we would expect that after some time one firm would switch
and the other would follow. If there were communication, the switch point would be at the
payoff of 1.6. But what if the first firm to switch is penalized more? Such is the problem
in oligopoly pricing. If costs rise, so should the monopoly price, but whichever firm raises
its price first suffers a loss of market share.
35
NOTES
N1.2 Dominant Strategies: The Prisoner’s Dilemma
• Many economists are reluctant to use the concept of cardinal utility (see Starmer [2000]),
and even more reluctant to compare utility across individuals (see Cooter & Rappoport
[1984]). Noncooperative game theory never requires interpersonal utility comparisons, and
only ordinal utility is needed to find the equilibrium in the Prisoner’s Dilemma. So long
as each player’s rank ordering of payoffs in different outcomes is preserved, the payoffs can
be altered without changing the equilibrium. In general, the dominant strategy and pure
strategy Nash equilibria of games depend only on the ordinal ranking of the payoffs, but the
mixed strategy equilibria depend on the cardinal values. Compare Section 3.2’s Chicken
game with Sectio 5.6’s Hawk-Dove.
• If we consider only the ordinal ranking of the payoffs in 2-by-2 games, there are 78 distinct
games in which each player has strict preference ordering over the four outcomes and 726
distinct games if we allow ties in the payoffs. Rapoport, Guyer & Gordon’s 1976 book, The
2x2 Game, contains an exhaustive description of the possible games.
• The Prisoner’s Dilemma was so named by Albert Tucker in an unpublished paper, although
the particular 2-by-2 matrix, discovered by Dresher and Flood, was already well known.
Tucker was asked to give a talk on game theory to the psychology department at Stanford,
and invented a story to go with the matrix, as recounted in Straffin (1980), pp. 101-18 of
Poundstone (1992), and pp. 171-3 of Raiffa (1992).
• In the Prisoner’s Dilemma the notation cooperate and defect is often used for the moves.
This is bad notation, because it is easy to confuse with cooperative games and with devia-tions.
It is also often called the Prisoners’ Dilemma (rs’, not r’s) ; whether one looks at
from the point of the individual or the group, the prisoners have a problem.
• The Prisoner’s Dilemma is not always defined the same way. If we consider just ordinal
payoffs, then the game in Table 10 is a Prisoner’s Dilemma if T(temptation) > R(revolt) >
P(punishment) > S(Sucker), where the terms in parentheses are mnemonics. This is
standard notation; see, for example, Rapoport, Guyer & Gordon (1976), p. 400. If the
game is repeated, the cardinal values of the payoffs can be important. The requirement
2R > T +S > 2P should be added if the game is to be a standard Prisoner’s Dilemma,
in which (Deny,Deny) and (Confess,Confess) are the best and worst possible outcomes
in terms of the sum of payoffs. Section 5.3 will show that an asymmetric game called the
One-Sided Prisoner’s Dilemma has properties similar to the standard Prisoner’s Dilemma,
but does not fit this definition.
Sometimes the game in which 2R < T + S is also called a prisoner’s dilemma, but in it
the sumof the players’ payoffs is maximized when one confesses and the other denies. If the
game were repeated or the prisoners could use the correlated equilibria defined in Section
3.2, they would prefer taking turns being confessed against, which would make the game
a coordination game similar to the Battle of the Sexes. David Shimko has suggested the
name “Battle of the Prisoners” for this (or, perhaps, the “Sex Prisoners’ Dilemma”).
Table 10: A General Prisoner’s Dilemma
36
Column
Deny Confess
Deny R,R → S, T
Row ↓ ↓
Confess T,S → P,P
Payoffs to: (Row, Column)
• Herodotus (429 B.C., III-71) describes an early example of the reasoning in the Prisoner’s
Dilemma in a conspiracy against the Persian emperor. A group of nobles met and decided
to overthrow the emperor, and it was proposed to adjourn till another meeting. One of
them named Darius then spoke up and said that if they adjourned, he knew that one of
them would go straight to the emperor and reveal the conspiracy, because if nobody else
did, he would himself. Darius also suggested a solution– that they immediately go to the
palace and kill the emperor.
The conspiracy also illustrates a way out of coordination games. After killing the emperor,
the nobles wished to select one of themselves as the new emperor. Rather than fight, they
agreed to go to a certain hill at dawn, and whoever’s horse neighed first would become
emperor. Herodotus tells how Darius’s groom manipulated this randomization scheme to
make him the new emperor.
• Philosophers are intrigued by the Prisoner’s Dilemma: see Campbell & Sowden (1985),
a collection of articles on the Prisoner’s Dilemma and the related Newcombe’s paradox.
Game theory has even been applied to theology: if one player is omniscient or omnipotent,
what kind of equilibrium behavior can we expect? See Brams (1983).
N1.4 Nash Equilibrium: Boxed Pigs, the Battle of the Sexes, and Ranked Coordi-nation
• I invented the payoffs for Boxed Pigs from the description of one of the experiments in
Baldwin & Meese (1979). They do not think of this as an experiment in game theory, and
they describe the result in terms of “reinforcement.” The Battle of the Sexes is taken from
p. 90 of Luce & Raiffa (1957). I have changed their payoffs of (−1, −1) to (−5, −5) to fit
the story.
• Some people prefer the term “equilibrium point” to “Nash equilibrium,” but the latter is
more euphonious, since the discoverer’s name is “Nash” and not “Mazurkiewicz.”
• Bernheim (1984a) and Pearce (1984) use the idea of mutually consistent beliefs to arrive at
a different equilibrium concept than Nash. They define a rationalizable strategy to be a
strategy which is a best response for some set of rational beliefs in which a player believes
that the other players choose their best responses. The difference from Nash is that not
all players need have the same beliefs concerning which strategies will be chosen, nor need
their beliefs be consistent.
This idea is attractive in the context of Bertrand games (see Section 3.6). The Nash
equilibrium in the Bertrand game is weakly dominated– by picking any other price above
marginal cost, which yields the same profit of zero as does the equilibrium. Rationalizability
rules that out.
• Jack Hirshleifer (1982) uses the name “the Tender Trap” for a game essentially the same
as Ranked Coordination, and the name “the Assurance Game“ has also been used for it.
37
• O. Henry’s story,“The Gift of the Magi” is about a coordination game noteworthy for the
reason communication is ruled out. A husband sells his watch to buy his wife combs for
Christmas, wh