GAMES AND INFORMATION, FOURTH EDITION An Introduction to Game Theory Eric Rasmusen Basil Blackwell v 1 Contents (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 1 xxx February 2, 2000. December 12, 2003. 24 March 2005. Eric Rasmusen, 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 vii *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 viii 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 ix *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, 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 xvi 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 new feature 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. xvii 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! xviii